

<!DOCTYPE html>
<!-- Server: sfs-forge-5 -->


    

















<html lang="en" class="no-js">
<head>
    <meta content="text/html; charset=UTF-8" http-equiv="content-type"/>
    <title>
  XSB / Src  (SVN)-old /
  [r4265]
  /trunk/XSB/packages/altCDF/doc/semantics.tex
</title>
    

<meta name="viewport" content="width=1100">


<link rel="icon" sizes="180x180" href="https://a.fsdn.com/con/img/sandiego/logo-180x180.png" type="image/png">
<link rel="icon" sizes="any" href="https://a.fsdn.com/con/img/sandiego/svg/originals/sf-icon-orange-no_sf.svg" type="image/svg+xml">
<link rel="apple-touch-icon" sizes="180x180" href="https://a.fsdn.com/con/img/sandiego/logo-180x180.png">
<link rel="mask-icon" href="https://a.fsdn.com/con/img/sandiego/svg/originals/sf-icon-orange-no_sf.svg" color="#FF6600">


<script src="https://a.fsdn.com/allura/nf/1752257864/_ew_/theme/sftheme/js/sftheme/cmp2.js"></script>






    
    
    
    
    
    
    
    
    <link rel="stylesheet" href="https://a.fsdn.com/con/css/lato.css?">


<script>
if (!window.SF) { window.SF = {}; }
SF.sandiego = false;
SF.sandiego_chrome = true;
SF.cdn = "https://a.fsdn.com/con";
SF.fpid = '739ccba1-630a-4aff-90a1-2ab54b15236e';
SF.billboard_route = '/software/product/$slug/';
</script>
    
        <!-- ew:head_css -->

    
        <link rel="stylesheet"
                type="text/css"
                href="https://a.fsdn.com/allura/nf/1752257864/_ew_/_slim/css?href=allura%2Fcss%2Fforge%2Fhilite.css%3Ballura%2Fcss%2Fforge%2Ftooltipster.css"
                >
    
        <link rel="stylesheet"
                type="text/css"
                href="https://a.fsdn.com/allura/nf/1752257864/_ew_/allura/css/font-awesome.min.css"
                >
    
        <link rel="stylesheet"
                type="text/css"
                href="https://a.fsdn.com/allura/nf/1752257864/_ew_/theme/sftheme/css/forge.css"
                >
    
        
<!-- /ew:head_css -->

    
    
        <!-- ew:head_js -->

    
        <script type="text/javascript" src="https://a.fsdn.com/allura/nf/1752257864/_ew_/_slim/js?href=allura%2Fjs%2Fjquery-3.6.0.min.js%3Ballura%2Fjs%2Fjquery.cookie-1.4.1.js%3Btheme%2Fsftheme%2Fjs%2Fsftheme%2Fshared_head.js%3Btheme%2Fsftheme%2Fjs%2Fsftheme%2Ftypescript%2Fcompliance.js"></script>
    
        
<!-- /ew:head_js -->

    

    
        <style type="text/css">
            /* make URL &amp; Project custom fields in support tickets not get hidden as much */
.project-forge.mountpoint-site-support .view_holder .grid-4:nth-child(5),
.project-forge.mountpoint-site-support .view_holder .grid-4:nth-child(6) {
  overflow: visible;
  font-size: 22px;
  width: 340px;
  line-height: 1;
  margin-bottom: 5px;
}
.project-forge.mountpoint-site-support .view_holder .grid-4:nth-child(5) label, .project-forge.mountpoint-site-support .view_holder .grid-4:nth-child(6) label {
  font-size: 14px;
}
        </style>
    
    
    <link rel="alternate" type="application/rss+xml" title="RSS" href="/p/xsb/src/feed.rss"/>
    <link rel="alternate" type="application/atom+xml" title="Atom" href="/p/xsb/src/feed.atom"/>
    <style type="text/css">
        #access_urls .btn-set {
            
            min-width: 14em;
        }
    </style>

    <style>.XaKzwzDiQELFvDnsKoVcKwqvDrMKvOKKYwYd {
        display: none
    }</style>

    
    
    
        <link rel="canonical" href="https://sourceforge.net/p/xsb/src/4265/tree/trunk/XSB/packages/altCDF/doc/semantics.tex" />
    


    

<script>
    var data = {hem: null };
    bizx.uids.gatherAndClean(data);
</script>



<script>
    function initPiwik(){
      var _paq = window._paq = window._paq || [];
      _paq.push(['trackPageView', document.title, {
            dimension1: 'xsb',
            dimension2: 'svn'
      }]);
      _paq.push(['enableLinkTracking']);
      
          _paq.push(['enableHeartBeatTimer']);
      
      (function() {
        var u="//analytics.slashdotmedia.com/";
        _paq.push(['setTrackerUrl', u+'sf.php']);
        _paq.push(['setSiteId', 39]);
        
        var d=document, g=d.createElement('script'), s=d.getElementsByTagName('script')[0];
        g.type='text/javascript'; g.async=true; g.defer=true; g.src=u+'sf.js'; s.parentNode.insertBefore(g,s);
      })();
    }
    bizx.cmp.ifConsent({ purposes: ['storage', 'measurement'], vendors: 'sdm'}, initPiwik);
</script>






<!-- Hubspot tracking -->






 <script>
    $(function() {
        bizx.cmp.ifConsent({ purposes: 'all' , vendors: 'narrative'}, function() {
            var current_time = (new Date()).getTime();
            let imageUrl = "https://io.narrative.io/?companyId=2440&id=first_party%3A739ccba1-630a-4aff-90a1-2ab54b15236e&id=site_name%3Asourceforge.net&id=url%3A%2Fp%2Fxsb%2Fsrc%2F4265%2Ftree%2Ftrunk%2FXSB%2Fpackages%2FaltCDF%2Fdoc%2Fsemantics.tex&id=pagetitle%3A%24PAGE_TITLE&id=vertical%3AOpen+Source+Software&id=product%3AXSB";
            imageUrl = imageUrl.replace(encodeURIComponent("$PAGE_TITLE"), document.title);  

            imageUrl = URL.parse(imageUrl);
            let hem = bizx.uids.getHem();
            if (hem) {
                imageUrl.searchParams.append('id', 'hem:' + hem);
            }
            imageUrl.searchParams.append('rand', current_time);

            var image = new Image();
            image.src = imageUrl.toString();
            image.style.display = "none";
            image.style.height = 0;
            image.style.width = 0;
            document.body.appendChild(image);
        });
    });
</script>



</head>

<body class="
body_class sandiego_chrome  legacy_chrome  l-allow-natural-width 
" id="forge">

    
        <!-- ew:body_top_js -->

    
        
<!-- /ew:body_top_js -->

    





<div class="off-canvas position-right" id="offCanvas" data-off-canvas>
    <!-- Menu -->
    <ul class="header-nav-menulist">
        <li class="highlight search">
            
            
                
            
            
    <form method="get" action="/directory/" class="m-search-form">
    
        <input type="text" placeholder="Search for software or solutions" autocomplete="off" name="q" >
        
        <label >
            <input type="submit" class="bt" value="">
            


<svg  data-name="search" viewBox="0 0 1792 1792" xmlns="http://www.w3.org/2000/svg"><path d="M1216 832q0-185-131.5-316.5t-316.5-131.5-316.5 131.5-131.5 316.5 131.5 316.5 316.5 131.5 316.5-131.5 131.5-316.5zm512 832q0 52-38 90t-90 38q-54 0-90-38l-343-342q-179 124-399 124-143 0-273.5-55.5t-225-150-150-225-55.5-273.5 55.5-273.5 150-225 225-150 273.5-55.5 273.5 55.5 225 150 150 225 55.5 273.5q0 220-124 399l343 343q37 37 37 90z"/></svg>
        </label>
    </form>
        </li>
        
        <li class="highlight"><a href="/auth/">Join/Login</a></li>
        
        <li><a href="/software/">Business Software</a></li>
        <li><a href="/directory/">Open Source Software</a></li>
        <li><a  href="/software/vendors/" title="For Vendors">For Vendors</a></li>
        <li><a href="/blog/" title="Blog">Blog</a></li>
        <li><a href="/about">About</a></li>
        <li><a id="header-nav-more" data-toggle="header-nav-more header-nav-more-content" data-toggler=".toggled">More</a></li>
        <li>
            <ul id="header-nav-more-content" class="toggled" data-toggler=".toggled">
                
    

    
    
        <li><a href="/articles/">Articles</a></li>
    
    

    
 
                
                <li><a href="/create">Create</a></li>
                
                <li><a href="https://sourceforge.net/articles/category/sourceforge-podcast/">SourceForge Podcast</a></li>
                
                    <li><a href="https://sourceforge.net/p/forge/documentation/Docs%20Home/">Site Documentation</a></li>
                
                <li><a href="/user/newsletters">Subscribe to our Newsletter</a></li>
                <li><a href="/support">Support Request</a></li>
            </ul>
        </li>
    </ul>
</div>
<div class="off-canvas-content" data-off-canvas-content>

    




<script>
    SF.linkout_icon = '<svg  data-name="sf-linkout-icon" xmlns="http://www.w3.org/2000/svg" x="0px" y="0px" viewBox="0 0 300 300" style="enable-background:new 0 0 300 300;" xml:space="preserve"><polygon class="st0" points="243.2,243.2 56.8,243.2 56.8,56.8 123,56.8 123,9 9,9 9,291 291,291 291,179.4 243.2,179 "/><polygon class="st0" points="128.5,213 155,186.5 176,165.5 206.7,196.3 235.5,132.5 248.9,102.6 290.6,9.8 291,9 290.6,9.2 197.4,51.1 169.1,63.8 103.7,93.3 137,126.5 115.9,147.5 89.5,174 "/></svg>';
</script>


<div class="l-header-nav  sandiego">

    <section class="sandiego l-header-nav-top show-for-large">
        <div class="row">
            <a href="/" title="Home" class="sf-logo">
                
    
    <img src="https://a.fsdn.com/con/images/sandiego/sf-logo-full.svg"  alt="SourceForge logo" class="sf-logo-full"/>
            </a>
            <nav class="links">
                
                    <a href="/user/newsletters" title="Subscribe to our newsletter"><span class="newsletter-icon">


<svg  data-name="mmSF_11mail" xmlns="http://www.w3.org/2000/svg" x="0px" y="0px" viewBox="0 0 42 42" enable-background="new 0 0 42 42" xml:space="preserve"><path fill="#FFFFFF" d="M0,6v30h42V6H0z M24.2,21.2c-0.8,0.8-2.3,2-3.2,2c-0.9,0-2.4-1.2-3.2-2L5.8,9.7h30.3L24.2,21.2z M13.7,21l-9.9,9.4V11.6L13.7,21z M15.7,23L15.7,23c0.5,0.5,2.9,2.9,5.3,2.9c2.4,0,4.8-2.4,5.2-2.8l0.1-0.1l9.8,9.4H5.8L15.7,23z M28.3,21l9.9-9.5v18.9L28.3,21z"/></svg></span></a>
                
                <a href="/software/vendors/" title="For Vendors">For Vendors</a>
                
                
                    <a href="/support" title="Help">Help</a>
                    <a href="/create/" title="Create">Create</a>
                

                
                <a href="/user/registration" title="Join" >Join</a>
                <a href="/auth/" title="Login">Login</a>
                
            </nav>
        </div>
    </section>

    <section class="sandiego l-header-nav-top hide-for-large">
        <div class="row">
            <a href="/" title="Home" class="sf-logo">
                
    
    <img src="https://a.fsdn.com/con/images/sandiego/sf-logo-full.svg"  alt="SourceForge logo" class="sf-logo-full"/>
            </a>
            <div class="title-bar-right">
                <button type="button" aria-label="Toggle Main Menu" class="menu-icon" data-toggle="offCanvas"></button>
            </div>
        </div>
    </section>
    <section class="sandiego l-header-nav-bottom">
        <nav class="row">
            <a href="/" title="Home" class="sf-logo">
                
    
    <img src="https://a.fsdn.com/con/images/sandiego/sf-logo-full.svg"  alt="SourceForge logo" class="sf-logo-full"/>
            </a>
            <div class="links">
                
    
        <div class="nav-dropdown">
            <a href="/software/">Business Software</a>
            
        </div>
        <div class="nav-dropdown">
            <a href="/directory/" title="Browse">Open Source Software</a>
            
        </div>
        <div class="nav-dropdown">
            <a href="https://sourceforge.net/articles/category/sourceforge-podcast/">SourceForge Podcast</a>
        </div>
        <div class="nav-dropdown">
            <a>Resources</a>
            <ul class="nav-dropdown-menu">
                  <li><a href="/articles/">Articles</a></li>
                  
                  <li><a href="/software/case-studies/">Case Studies</a></li>
                  
                  <li><a href="/blog/">Blog</a></li>
            </ul>
        </div>
    

                <div class="dev-menu-when-stuck">
                    Menu
                    <ul class="dev-menu-dropdown header-nav-menulist">
                        <li><a href="/support">Help</a></li>
                        <li><a href="/create">Create</a></li>
                        <li><a href="/user/registration/" title="Join" >Join</a></li>
                        <li><a href="/auth/" title="Login">Login</a></li>
                    </ul>
                </div>
                <div class="search-toggle-when-stuck">
                    <a class="search-toggle">
                        


<svg  data-name="search" viewBox="0 0 1792 1792" xmlns="http://www.w3.org/2000/svg"><path d="M1216 832q0-185-131.5-316.5t-316.5-131.5-316.5 131.5-131.5 316.5 131.5 316.5 316.5 131.5 316.5-131.5 131.5-316.5zm512 832q0 52-38 90t-90 38q-54 0-90-38l-343-342q-179 124-399 124-143 0-273.5-55.5t-225-150-150-225-55.5-273.5 55.5-273.5 150-225 225-150 273.5-55.5 273.5 55.5 225 150 150 225 55.5 273.5q0 220-124 399l343 343q37 37 37 90z"/></svg>
                    </a>
                </div>
            </div>

            <div class="search">
                
                
    
    
        <div class="main-nav-link">
            
            
            <a href="https://www.avast.com/lp-free-av?full_trSrc=mmm_srf_dlp_008_406_a&amp;utm_medium=referral&amp;utm_source=sourceforge&amp;utm_campaign=srf_dlp_008_406_a:site_banner" rel="nofollow" target="_blank" id="main-nav-badge-link" data-label="avastjuly">
                <img src="https://a.fsdn.com/con/assets/maxnav/sourceforge/ai-protection-32-75dd98b2.png"  srcset="https://a.fsdn.com/con/assets/maxnav/sourceforge/ai-protection-75de80f6.png 2x"  alt="" id="main-nav-image"/>
            </a>
        </div>
    


                
                    
                
                
                
                
    <form method="get" action="/directory/" class="m-search-form">
    
    <div class="typeahead__container">
      <div class="typeahead__field">
        <div class="typeahead__query">
        
        <input type="text" placeholder="Search for software or solutions" autocomplete="off" name="q" >
        
        </div>
        
        <label >
            <input type="submit" class="bt" value="">
            


<svg  data-name="search" viewBox="0 0 1792 1792" xmlns="http://www.w3.org/2000/svg"><path d="M1216 832q0-185-131.5-316.5t-316.5-131.5-316.5 131.5-131.5 316.5 131.5 316.5 316.5 131.5 316.5-131.5 131.5-316.5zm512 832q0 52-38 90t-90 38q-54 0-90-38l-343-342q-179 124-399 124-143 0-273.5-55.5t-225-150-150-225-55.5-273.5 55.5-273.5 150-225 225-150 273.5-55.5 273.5 55.5 225 150 150 225 55.5 273.5q0 220-124 399l343 343q37 37 37 90z"/></svg>
        </label>
      </div>
    </div>
    
    </form>
                
            </div>
        </nav>
        
    </section>
    
</div>
    
    
    <div class="notification-on-project-page">
    
        
    
    
    
    </div>
    
    


    <section id="page-body" class=" neighborhood-Projects project-xsb mountpoint-src 

">
        <div id="nav_menu_holder">
            
                
                    
    <div class="sandiego"> 
            
            
            
            
<section class="project-masthead"> 
    
    

    
<div class="backdrop" style="box-sizing: content-box; padding-bottom: 24px"></div>

    <div class="content">
    
        
    <nav id="breadcrumbs" class="breadcrumbs rtl">
        <ul itemscope itemtype="http://schema.org/BreadcrumbList">
            <li itemprop="itemListElement" itemscope itemtype="http://schema.org/ListItem">
                <a itemprop="item" href="/"><span itemprop="name">Home</span></a>
                <meta itemprop="position" content="1"/>
            </li>
            <li itemprop="itemListElement" itemscope itemtype="http://schema.org/ListItem">
                <a itemprop="item" href="/directory/"><span itemprop="name">Browse</span></a>
                <meta itemprop="position" content="2"/>
            </li>
            
            
                
                
            
            
                
            
            
                <li itemprop="itemListElement" itemscope itemtype="http://schema.org/ListItem">
                    <a itemprop="item" href="/p/xsb/"><span itemprop="name">XSB</span></a>
                <meta itemprop="position" content="3"/>
                </li>
            
                <li itemprop="itemListElement" itemscope itemtype="http://schema.org/ListItem">
                    <span itemprop="name">Src  (SVN)-old</span>
                <meta itemprop="position" content="4"/>
                </li>
            
        </ul>
    </nav>
    
    
    
        
                
                
<div class="overview">
    


<div class="project-icon   " >
    
    
    <img itemprop="image" alt="XSB" title="XSB" 
    
    
    src="https://a.fsdn.com/allura/p/xsb/icon?1368563712"
    /></div>


    
    <div class="title "> 

        <meta itemprop="name" content="XSB"/>
        <h1  >XSB Src  (SVN)-old
        </h1>
        
            
            
          
        
            
            
            <div class="as-h3 brought-by">
                
                Brought to you by:
                
                    <a href="/u/carlandersen/profile/">carlandersen</a>,
                    
                
                    <a href="/u/dwarren/profile/">dwarren</a>,
                    
                
                    <a href="/u/evansbj/profile/">evansbj</a>,
                    
                
                    <a href="/u/kifer/profile/">kifer</a>,
                    
                
                    
                        and <a href="/p/xsb/_members/">3 others</a>
                    
                
            </div>
            
        

        

        
    </div>


    

</div>

                
                
    
  
            
            
        
    </div>
</section>

        </div>
    

                
            
        </div>
        <div id="top_nav" class="">
            
                
<div id="top_nav_admin">
<ul class="dropdown">
  
  
    <li class="">
        <a href="/projects/xsb/" class="tool-summary-32" >
            Summary
        </a>
        
        
    </li>
  
    <li class="">
        <a href="/projects/xsb/files/" class="tool-files-sf-32" >
            Files
        </a>
        
        
    </li>
  
    <li class="">
        <a href="/projects/xsb/reviews/" class="tool-reviews-32" >
            Reviews
        </a>
        
        
    </li>
  
    <li class="">
        <a href="/projects/xsb/support" class="tool-support-32" >
            Support
        </a>
        
        
    </li>
  
    <li class="">
        <a href="/p/xsb/mailman/" class="tool-mailman-32" >
            Mail Lists
        </a>
        
        
    </li>
  
    <li class="">
        <a href="/p/xsb/_list/tickets" class="tool-tickets-32" >
            Tickets ▾
        </a>
        
        
            <ul>
                
                    <li class=""><a href="/p/xsb/bugs/" >Bugs</a></li>
                
                    <li class=""><a href="/p/xsb/feature-requests/" >Feature Requests</a></li>
                
                    <li class=""><a href="/p/xsb/support-requests/" >Support Requests</a></li>
                
                    <li class=""><a href="/p/xsb/patches/" >Patches</a></li>
                
            </ul>
        
    </li>
  
    <li class="">
        <a href="/p/xsb/news/" class="tool-blog-32" >
            News
        </a>
        
        
    </li>
  
    <li class="">
        <a href="/p/xsb/code/" class="tool-git-32" >
            Sources (GIT)
        </a>
        
        
    </li>
  
    <li class="selected">
        <a href="/p/xsb/src/" class="tool-svn-32" >
            Src  (SVN)-old
        </a>
        
        
    </li>
  
  
</ul>
</div>


            
        </div>
        <div id="content_base">
            

    
                
                    
<a id="sidebar-activate" href="#">
    <span class="btn-label">Menu</span>
    <span class="btn-arrow-down">▾</span>
    <span class="btn-arrow-up">▴</span>
</a>

                        


<div id="sidebar">
  
    <div class="placeholder-no-searchbox">&nbsp;</div>
  
    
    
      
      
        
    
      <ul class="sidebarmenu">
      
    
  <li>
      
        <a class="icon" href="/p/xsb/src/commit_browser" rel="nofollow" title="Browse Commits"><i class="fa fa-list"></i>
      
      <span>Browse Commits</span>
      </a>
  </li>
  
      
        
    
  <li>
      
        <a href="/p/xsb/src/"  rel="nofollow" >
      
      <span>Browse Files</span>
      </a>
  </li>
  
      
    
    
      </ul>
      
    
    
</div>



<script>
$('#sidebar-admin-header').on('click', function(e){
    $("#sidebar-admin-menu").toggleClass("hidden");
    $(this).toggleClass("expanded");
    e.preventDefault();
});
</script>

                    

                    
                
                
                    
                
                <div class="grid-20 pad">
                    <h2 class="dark title">
                        <span>
<a href="/p/xsb/src/4265/" rel="nofollow">[r4265]</a>:

  
      <a href="./../../../../../" rel="nofollow"><b class="fa fa-folder-open-o" title="Root directory"></b></a> /
  
  
    <a href="./../../../../" rel="nofollow">trunk</a> /
  
    <a href="./../../../" rel="nofollow">XSB</a> /
  
    <a href="./../../" rel="nofollow">packages</a> /
  
    <a href="./../" rel="nofollow">altCDF</a> /
  
    <a href="./" rel="nofollow">doc</a> /
  
 semantics.tex
</span>
                        <!-- actions -->
                        <small>
                            

    
    <a class="icon" href="#" id="maximize-content" title="Maximize"><i class="fa fa-expand"></i>&nbsp;Maximize</a>
    <a class="icon" href="#" id="restore-content" title="Restore"><i class="fa fa-compress"></i>&nbsp;Restore</a>
<a class="icon" href="/p/xsb/src/4265/log/?path=/trunk/XSB/packages/altCDF/doc/semantics.tex" rel="nofollow" title="History"><i class="fa fa-calendar"></i>&nbsp;History</a>

                        </small>
                        <!-- /actions -->
                    </h2>
                    
                    <div>
                        
  

                        
  
    <p><a rel="nofollow" href="?format=raw">Download this file</a></p>
    <div class="clip grid-19 codebrowser">
      <h3>
        591 lines (543 with data), 25.1 kB
      </h3>
      
        <div class="codehilite"><table class="codehilitetable"><tr><td class="linenos"><div class="linenodiv"><pre><span class="normal">  1</span>
<span class="normal">  2</span>
<span class="normal">  3</span>
<span class="normal">  4</span>
<span class="normal">  5</span>
<span class="normal">  6</span>
<span class="normal">  7</span>
<span class="normal">  8</span>
<span class="normal">  9</span>
<span class="normal"> 10</span>
<span class="normal"> 11</span>
<span class="normal"> 12</span>
<span class="normal"> 13</span>
<span class="normal"> 14</span>
<span class="normal"> 15</span>
<span class="normal"> 16</span>
<span class="normal"> 17</span>
<span class="normal"> 18</span>
<span class="normal"> 19</span>
<span class="normal"> 20</span>
<span class="normal"> 21</span>
<span class="normal"> 22</span>
<span class="normal"> 23</span>
<span class="normal"> 24</span>
<span class="normal"> 25</span>
<span class="normal"> 26</span>
<span class="normal"> 27</span>
<span class="normal"> 28</span>
<span class="normal"> 29</span>
<span class="normal"> 30</span>
<span class="normal"> 31</span>
<span class="normal"> 32</span>
<span class="normal"> 33</span>
<span class="normal"> 34</span>
<span class="normal"> 35</span>
<span class="normal"> 36</span>
<span class="normal"> 37</span>
<span class="normal"> 38</span>
<span class="normal"> 39</span>
<span class="normal"> 40</span>
<span class="normal"> 41</span>
<span class="normal"> 42</span>
<span class="normal"> 43</span>
<span class="normal"> 44</span>
<span class="normal"> 45</span>
<span class="normal"> 46</span>
<span class="normal"> 47</span>
<span class="normal"> 48</span>
<span class="normal"> 49</span>
<span class="normal"> 50</span>
<span class="normal"> 51</span>
<span class="normal"> 52</span>
<span class="normal"> 53</span>
<span class="normal"> 54</span>
<span class="normal"> 55</span>
<span class="normal"> 56</span>
<span class="normal"> 57</span>
<span class="normal"> 58</span>
<span class="normal"> 59</span>
<span class="normal"> 60</span>
<span class="normal"> 61</span>
<span class="normal"> 62</span>
<span class="normal"> 63</span>
<span class="normal"> 64</span>
<span class="normal"> 65</span>
<span class="normal"> 66</span>
<span class="normal"> 67</span>
<span class="normal"> 68</span>
<span class="normal"> 69</span>
<span class="normal"> 70</span>
<span class="normal"> 71</span>
<span class="normal"> 72</span>
<span class="normal"> 73</span>
<span class="normal"> 74</span>
<span class="normal"> 75</span>
<span class="normal"> 76</span>
<span class="normal"> 77</span>
<span class="normal"> 78</span>
<span class="normal"> 79</span>
<span class="normal"> 80</span>
<span class="normal"> 81</span>
<span class="normal"> 82</span>
<span class="normal"> 83</span>
<span class="normal"> 84</span>
<span class="normal"> 85</span>
<span class="normal"> 86</span>
<span class="normal"> 87</span>
<span class="normal"> 88</span>
<span class="normal"> 89</span>
<span class="normal"> 90</span>
<span class="normal"> 91</span>
<span class="normal"> 92</span>
<span class="normal"> 93</span>
<span class="normal"> 94</span>
<span class="normal"> 95</span>
<span class="normal"> 96</span>
<span class="normal"> 97</span>
<span class="normal"> 98</span>
<span class="normal"> 99</span>
<span class="normal">100</span>
<span class="normal">101</span>
<span class="normal">102</span>
<span class="normal">103</span>
<span class="normal">104</span>
<span class="normal">105</span>
<span class="normal">106</span>
<span class="normal">107</span>
<span class="normal">108</span>
<span class="normal">109</span>
<span class="normal">110</span>
<span class="normal">111</span>
<span class="normal">112</span>
<span class="normal">113</span>
<span class="normal">114</span>
<span class="normal">115</span>
<span class="normal">116</span>
<span class="normal">117</span>
<span class="normal">118</span>
<span class="normal">119</span>
<span class="normal">120</span>
<span class="normal">121</span>
<span class="normal">122</span>
<span class="normal">123</span>
<span class="normal">124</span>
<span class="normal">125</span>
<span class="normal">126</span>
<span class="normal">127</span>
<span class="normal">128</span>
<span class="normal">129</span>
<span class="normal">130</span>
<span class="normal">131</span>
<span class="normal">132</span>
<span class="normal">133</span>
<span class="normal">134</span>
<span class="normal">135</span>
<span class="normal">136</span>
<span class="normal">137</span>
<span class="normal">138</span>
<span class="normal">139</span>
<span class="normal">140</span>
<span class="normal">141</span>
<span class="normal">142</span>
<span class="normal">143</span>
<span class="normal">144</span>
<span class="normal">145</span>
<span class="normal">146</span>
<span class="normal">147</span>
<span class="normal">148</span>
<span class="normal">149</span>
<span class="normal">150</span>
<span class="normal">151</span>
<span class="normal">152</span>
<span class="normal">153</span>
<span class="normal">154</span>
<span class="normal">155</span>
<span class="normal">156</span>
<span class="normal">157</span>
<span class="normal">158</span>
<span class="normal">159</span>
<span class="normal">160</span>
<span class="normal">161</span>
<span class="normal">162</span>
<span class="normal">163</span>
<span class="normal">164</span>
<span class="normal">165</span>
<span class="normal">166</span>
<span class="normal">167</span>
<span class="normal">168</span>
<span class="normal">169</span>
<span class="normal">170</span>
<span class="normal">171</span>
<span class="normal">172</span>
<span class="normal">173</span>
<span class="normal">174</span>
<span class="normal">175</span>
<span class="normal">176</span>
<span class="normal">177</span>
<span class="normal">178</span>
<span class="normal">179</span>
<span class="normal">180</span>
<span class="normal">181</span>
<span class="normal">182</span>
<span class="normal">183</span>
<span class="normal">184</span>
<span class="normal">185</span>
<span class="normal">186</span>
<span class="normal">187</span>
<span class="normal">188</span>
<span class="normal">189</span>
<span class="normal">190</span>
<span class="normal">191</span>
<span class="normal">192</span>
<span class="normal">193</span>
<span class="normal">194</span>
<span class="normal">195</span>
<span class="normal">196</span>
<span class="normal">197</span>
<span class="normal">198</span>
<span class="normal">199</span>
<span class="normal">200</span>
<span class="normal">201</span>
<span class="normal">202</span>
<span class="normal">203</span>
<span class="normal">204</span>
<span class="normal">205</span>
<span class="normal">206</span>
<span class="normal">207</span>
<span class="normal">208</span>
<span class="normal">209</span>
<span class="normal">210</span>
<span class="normal">211</span>
<span class="normal">212</span>
<span class="normal">213</span>
<span class="normal">214</span>
<span class="normal">215</span>
<span class="normal">216</span>
<span class="normal">217</span>
<span class="normal">218</span>
<span class="normal">219</span>
<span class="normal">220</span>
<span class="normal">221</span>
<span class="normal">222</span>
<span class="normal">223</span>
<span class="normal">224</span>
<span class="normal">225</span>
<span class="normal">226</span>
<span class="normal">227</span>
<span class="normal">228</span>
<span class="normal">229</span>
<span class="normal">230</span>
<span class="normal">231</span>
<span class="normal">232</span>
<span class="normal">233</span>
<span class="normal">234</span>
<span class="normal">235</span>
<span class="normal">236</span>
<span class="normal">237</span>
<span class="normal">238</span>
<span class="normal">239</span>
<span class="normal">240</span>
<span class="normal">241</span>
<span class="normal">242</span>
<span class="normal">243</span>
<span class="normal">244</span>
<span class="normal">245</span>
<span class="normal">246</span>
<span class="normal">247</span>
<span class="normal">248</span>
<span class="normal">249</span>
<span class="normal">250</span>
<span class="normal">251</span>
<span class="normal">252</span>
<span class="normal">253</span>
<span class="normal">254</span>
<span class="normal">255</span>
<span class="normal">256</span>
<span class="normal">257</span>
<span class="normal">258</span>
<span class="normal">259</span>
<span class="normal">260</span>
<span class="normal">261</span>
<span class="normal">262</span>
<span class="normal">263</span>
<span class="normal">264</span>
<span class="normal">265</span>
<span class="normal">266</span>
<span class="normal">267</span>
<span class="normal">268</span>
<span class="normal">269</span>
<span class="normal">270</span>
<span class="normal">271</span>
<span class="normal">272</span>
<span class="normal">273</span>
<span class="normal">274</span>
<span class="normal">275</span>
<span class="normal">276</span>
<span class="normal">277</span>
<span class="normal">278</span>
<span class="normal">279</span>
<span class="normal">280</span>
<span class="normal">281</span>
<span class="normal">282</span>
<span class="normal">283</span>
<span class="normal">284</span>
<span class="normal">285</span>
<span class="normal">286</span>
<span class="normal">287</span>
<span class="normal">288</span>
<span class="normal">289</span>
<span class="normal">290</span>
<span class="normal">291</span>
<span class="normal">292</span>
<span class="normal">293</span>
<span class="normal">294</span>
<span class="normal">295</span>
<span class="normal">296</span>
<span class="normal">297</span>
<span class="normal">298</span>
<span class="normal">299</span>
<span class="normal">300</span>
<span class="normal">301</span>
<span class="normal">302</span>
<span class="normal">303</span>
<span class="normal">304</span>
<span class="normal">305</span>
<span class="normal">306</span>
<span class="normal">307</span>
<span class="normal">308</span>
<span class="normal">309</span>
<span class="normal">310</span>
<span class="normal">311</span>
<span class="normal">312</span>
<span class="normal">313</span>
<span class="normal">314</span>
<span class="normal">315</span>
<span class="normal">316</span>
<span class="normal">317</span>
<span class="normal">318</span>
<span class="normal">319</span>
<span class="normal">320</span>
<span class="normal">321</span>
<span class="normal">322</span>
<span class="normal">323</span>
<span class="normal">324</span>
<span class="normal">325</span>
<span class="normal">326</span>
<span class="normal">327</span>
<span class="normal">328</span>
<span class="normal">329</span>
<span class="normal">330</span>
<span class="normal">331</span>
<span class="normal">332</span>
<span class="normal">333</span>
<span class="normal">334</span>
<span class="normal">335</span>
<span class="normal">336</span>
<span class="normal">337</span>
<span class="normal">338</span>
<span class="normal">339</span>
<span class="normal">340</span>
<span class="normal">341</span>
<span class="normal">342</span>
<span class="normal">343</span>
<span class="normal">344</span>
<span class="normal">345</span>
<span class="normal">346</span>
<span class="normal">347</span>
<span class="normal">348</span>
<span class="normal">349</span>
<span class="normal">350</span>
<span class="normal">351</span>
<span class="normal">352</span>
<span class="normal">353</span>
<span class="normal">354</span>
<span class="normal">355</span>
<span class="normal">356</span>
<span class="normal">357</span>
<span class="normal">358</span>
<span class="normal">359</span>
<span class="normal">360</span>
<span class="normal">361</span>
<span class="normal">362</span>
<span class="normal">363</span>
<span class="normal">364</span>
<span class="normal">365</span>
<span class="normal">366</span>
<span class="normal">367</span>
<span class="normal">368</span>
<span class="normal">369</span>
<span class="normal">370</span>
<span class="normal">371</span>
<span class="normal">372</span>
<span class="normal">373</span>
<span class="normal">374</span>
<span class="normal">375</span>
<span class="normal">376</span>
<span class="normal">377</span>
<span class="normal">378</span>
<span class="normal">379</span>
<span class="normal">380</span>
<span class="normal">381</span>
<span class="normal">382</span>
<span class="normal">383</span>
<span class="normal">384</span>
<span class="normal">385</span>
<span class="normal">386</span>
<span class="normal">387</span>
<span class="normal">388</span>
<span class="normal">389</span>
<span class="normal">390</span>
<span class="normal">391</span>
<span class="normal">392</span>
<span class="normal">393</span>
<span class="normal">394</span>
<span class="normal">395</span>
<span class="normal">396</span>
<span class="normal">397</span>
<span class="normal">398</span>
<span class="normal">399</span>
<span class="normal">400</span>
<span class="normal">401</span>
<span class="normal">402</span>
<span class="normal">403</span>
<span class="normal">404</span>
<span class="normal">405</span>
<span class="normal">406</span>
<span class="normal">407</span>
<span class="normal">408</span>
<span class="normal">409</span>
<span class="normal">410</span>
<span class="normal">411</span>
<span class="normal">412</span>
<span class="normal">413</span>
<span class="normal">414</span>
<span class="normal">415</span>
<span class="normal">416</span>
<span class="normal">417</span>
<span class="normal">418</span>
<span class="normal">419</span>
<span class="normal">420</span>
<span class="normal">421</span>
<span class="normal">422</span>
<span class="normal">423</span>
<span class="normal">424</span>
<span class="normal">425</span>
<span class="normal">426</span>
<span class="normal">427</span>
<span class="normal">428</span>
<span class="normal">429</span>
<span class="normal">430</span>
<span class="normal">431</span>
<span class="normal">432</span>
<span class="normal">433</span>
<span class="normal">434</span>
<span class="normal">435</span>
<span class="normal">436</span>
<span class="normal">437</span>
<span class="normal">438</span>
<span class="normal">439</span>
<span class="normal">440</span>
<span class="normal">441</span>
<span class="normal">442</span>
<span class="normal">443</span>
<span class="normal">444</span>
<span class="normal">445</span>
<span class="normal">446</span>
<span class="normal">447</span>
<span class="normal">448</span>
<span class="normal">449</span>
<span class="normal">450</span>
<span class="normal">451</span>
<span class="normal">452</span>
<span class="normal">453</span>
<span class="normal">454</span>
<span class="normal">455</span>
<span class="normal">456</span>
<span class="normal">457</span>
<span class="normal">458</span>
<span class="normal">459</span>
<span class="normal">460</span>
<span class="normal">461</span>
<span class="normal">462</span>
<span class="normal">463</span>
<span class="normal">464</span>
<span class="normal">465</span>
<span class="normal">466</span>
<span class="normal">467</span>
<span class="normal">468</span>
<span class="normal">469</span>
<span class="normal">470</span>
<span class="normal">471</span>
<span class="normal">472</span>
<span class="normal">473</span>
<span class="normal">474</span>
<span class="normal">475</span>
<span class="normal">476</span>
<span class="normal">477</span>
<span class="normal">478</span>
<span class="normal">479</span>
<span class="normal">480</span>
<span class="normal">481</span>
<span class="normal">482</span>
<span class="normal">483</span>
<span class="normal">484</span>
<span class="normal">485</span>
<span class="normal">486</span>
<span class="normal">487</span>
<span class="normal">488</span>
<span class="normal">489</span>
<span class="normal">490</span>
<span class="normal">491</span>
<span class="normal">492</span>
<span class="normal">493</span>
<span class="normal">494</span>
<span class="normal">495</span>
<span class="normal">496</span>
<span class="normal">497</span>
<span class="normal">498</span>
<span class="normal">499</span>
<span class="normal">500</span>
<span class="normal">501</span>
<span class="normal">502</span>
<span class="normal">503</span>
<span class="normal">504</span>
<span class="normal">505</span>
<span class="normal">506</span>
<span class="normal">507</span>
<span class="normal">508</span>
<span class="normal">509</span>
<span class="normal">510</span>
<span class="normal">511</span>
<span class="normal">512</span>
<span class="normal">513</span>
<span class="normal">514</span>
<span class="normal">515</span>
<span class="normal">516</span>
<span class="normal">517</span>
<span class="normal">518</span>
<span class="normal">519</span>
<span class="normal">520</span>
<span class="normal">521</span>
<span class="normal">522</span>
<span class="normal">523</span>
<span class="normal">524</span>
<span class="normal">525</span>
<span class="normal">526</span>
<span class="normal">527</span>
<span class="normal">528</span>
<span class="normal">529</span>
<span class="normal">530</span>
<span class="normal">531</span>
<span class="normal">532</span>
<span class="normal">533</span>
<span class="normal">534</span>
<span class="normal">535</span>
<span class="normal">536</span>
<span class="normal">537</span>
<span class="normal">538</span>
<span class="normal">539</span>
<span class="normal">540</span>
<span class="normal">541</span>
<span class="normal">542</span>
<span class="normal">543</span>
<span class="normal">544</span>
<span class="normal">545</span>
<span class="normal">546</span>
<span class="normal">547</span>
<span class="normal">548</span>
<span class="normal">549</span>
<span class="normal">550</span>
<span class="normal">551</span>
<span class="normal">552</span>
<span class="normal">553</span>
<span class="normal">554</span>
<span class="normal">555</span>
<span class="normal">556</span>
<span class="normal">557</span>
<span class="normal">558</span>
<span class="normal">559</span>
<span class="normal">560</span>
<span class="normal">561</span>
<span class="normal">562</span>
<span class="normal">563</span>
<span class="normal">564</span>
<span class="normal">565</span>
<span class="normal">566</span>
<span class="normal">567</span>
<span class="normal">568</span>
<span class="normal">569</span>
<span class="normal">570</span>
<span class="normal">571</span>
<span class="normal">572</span>
<span class="normal">573</span>
<span class="normal">574</span>
<span class="normal">575</span>
<span class="normal">576</span>
<span class="normal">577</span>
<span class="normal">578</span>
<span class="normal">579</span>
<span class="normal">580</span>
<span class="normal">581</span>
<span class="normal">582</span>
<span class="normal">583</span>
<span class="normal">584</span>
<span class="normal">585</span>
<span class="normal">586</span>
<span class="normal">587</span>
<span class="normal">588</span>
<span class="normal">589</span></pre></div></td><td class="code"><div><pre><div id="l1" class="code_block"><span class="k">\chapter</span><span class="nb">{</span>The Meaning of Type-0 CDF Instances<span class="nb">}</span>
</div><div id="l2" class="code_block"><span class="k">\label</span><span class="nb">{</span>chap:type0<span class="nb">}</span> 
</div><div id="l3" class="code_block">
</div><div id="l4" class="code_block"><span class="c">%-------------------------------------------------------------------------------</span>
</div><div id="l5" class="code_block"><span class="k">\mycomment</span><span class="nb">{</span>
</div><div id="l6" class="code_block"><span class="k">\subsection</span><span class="nb">{</span>Terminology<span class="nb">}</span>
</div><div id="l7" class="code_block">
</div><div id="l8" class="code_block">In this paper we use standard logic programming terminology and
</div><div id="l9" class="code_block">notation (e.g. <span class="k">\cite</span><span class="nb">{</span>Lloy84<span class="nb">}</span>), extended with the well-founded
</div><div id="l10" class="code_block">semantics~<span class="k">\cite</span><span class="nb">{</span>VGR99<span class="nb">}</span>. In addition, we commonly make use of notation
</div><div id="l11" class="code_block">for sentences and formulas in first-order logic.  While most of this
</div><div id="l12" class="code_block">notation is readily apparent, we sometimes write a first-order formula
</div><div id="l13" class="code_block"><span class="s">$</span><span class="nv">\phi</span><span class="s">$</span> as <span class="s">$</span><span class="nv">\phi</span><span class="o">[</span><span class="nv">\overline</span><span class="nb">{X}</span><span class="o">]</span><span class="s">$</span>, where <span class="s">$</span><span class="nv">\overline</span><span class="nb">{X}</span><span class="s">$</span> is a <span class="nb">{</span><span class="k">\em</span>
</div><div id="l14" class="code_block">superset<span class="nb">}</span> of the free variables in <span class="s">$</span><span class="nv">\phi</span><span class="s">$</span>.
</div><div id="l15" class="code_block"><span class="nb">}</span>
</div><div id="l16" class="code_block"><span class="c">%-------------------------------------------------------------------------------</span>
</div><div id="l17" class="code_block">
</div><div id="l18" class="code_block">Facts in both Type-0 and Type-1 CDF instances are closely related to
</div><div id="l19" class="code_block">class expressions in description logics.  However, because CDF often
</div><div id="l20" class="code_block">stores class expressions as Prolog facts in an unconventional way, and
</div><div id="l21" class="code_block">because description logics may not be familiar to a logic programming
</div><div id="l22" class="code_block">audience, we present here a somewhat formal introduction to how CDF
</div><div id="l23" class="code_block">represents knowledge.  Users without a mathematical background can
</div><div id="l24" class="code_block">ignore the various axioms and formal definitions that are presented in
</div><div id="l25" class="code_block">this chapter.  Our approach is to introduce a semantics of CDF based
</div><div id="l26" class="code_block">on a translation of a <span class="nb">{</span><span class="k">\em</span> CDF instance<span class="nb">}</span> into a set of first-order
</div><div id="l27" class="code_block">logic sentences that constitute an <span class="nb">{</span><span class="k">\em</span> Ontology Theory<span class="nb">}</span> whose models
</div><div id="l28" class="code_block">are the models of a CDF instance.  For simplicity of presentation the
</div><div id="l29" class="code_block">description of CDF instances in this section omits certain details
</div><div id="l30" class="code_block">about components, extensional facts and intenstional rules, and other
</div><div id="l31" class="code_block">topics that will be introduced in later chapters.
</div><div id="l32" class="code_block">
</div><div id="l33" class="code_block">We illustrate aspects of CDF by means of an example drawn from
</div><div id="l34" class="code_block">electronic commerce.  Health care organizations, such as hospitals,
</div><div id="l35" class="code_block">clinics, etc., have difficulties in buying disposable medical devices
</div><div id="l36" class="code_block">such as sutures, bandages, gloves, and so on.  The difficulty arises
</div><div id="l37" class="code_block">from the fact that these devices may be quite specialized, for intance
</div><div id="l38" class="code_block">when they are used in surgery.  At the same time, since these devices
</div><div id="l39" class="code_block">are disposable, they may need to be purchased frequently.  We consider
</div><div id="l40" class="code_block">concretly the class of <span class="nb">{</span><span class="k">\em</span> absorbable sutures<span class="nb">}</span>, which are used for
</div><div id="l41" class="code_block">stitching and securing tissues, and which can be absorbed by the human
</div><div id="l42" class="code_block">body.  Information below is adapted from he U.S. Defence Logistics
</div><div id="l43" class="code_block">Information Service <span class="k">\url</span><span class="nb">{</span>http://www.dlis.mil<span class="nb">}</span>, from the Universal
</div><div id="l44" class="code_block">Standard Products and Services Classification~<span class="k">\cite</span><span class="nb">{</span>UNSPSC<span class="nb">}</span>, as well
</div><div id="l45" class="code_block">as from websites of various commercial medical supply companies.
</div><div id="l46" class="code_block">
</div><div id="l47" class="code_block"><span class="k">\section</span><span class="nb">{</span>Type-0 CDF Instances<span class="nb">}</span>
</div><div id="l48" class="code_block">
</div><div id="l49" class="code_block">We begin with the syntax of Type-0 instances~<span class="k">\footnote</span><span class="nb">{</span>The syntax for
</div><div id="l50" class="code_block">identfiers differs in the actual CDF implementation. See
</div><div id="l51" class="code_block">Section~<span class="k">\ref</span><span class="nb">{</span>sec:instance<span class="nb">}</span>.<span class="nb">}</span>: 
</div><div id="l52" class="code_block">
</div><div id="l53" class="code_block"><span class="k">\begin</span><span class="nb">{</span>definition<span class="nb">}</span>[Type-0 Instances: Semantic Level] <span class="k">\label</span><span class="nb">{</span>def:ids<span class="nb">}</span>
</div><div id="l54" class="code_block">A <span class="nb">{</span><span class="k">\em</span> Type-0 CDF instance<span class="nb">}</span> is a finite set of ground facts for the
</div><div id="l55" class="code_block">predicates <span class="k">\pred</span><span class="nb">{</span>isa/2<span class="nb">}</span>, <span class="k">\pred</span><span class="nb">{</span>hasAttr/3<span class="nb">}</span>, <span class="k">\pred</span><span class="nb">{</span>allAttr/3<span class="nb">}</span>,
</div><div id="l56" class="code_block"><span class="k">\pred</span><span class="nb">{</span>classHasAttr/3<span class="nb">}</span>, <span class="k">\pred</span><span class="nb">{</span>minAttr/4<span class="nb">}</span>, and <span class="k">\pred</span><span class="nb">{</span>maxAttr/4<span class="nb">}</span>.  An <span class="nb">{</span><span class="k">\em</span>
</div><div id="l57" class="code_block">identifier<span class="nb">}</span> is either a constant or a term.  The arguments of these
</div><div id="l58" class="code_block">predicates are <span class="nb">{</span><span class="k">\em</span> concrete identifiers<span class="nb">}</span>, where a term <span class="s">$</span><span class="nb">T</span><span class="s">$</span> is an
</div><div id="l59" class="code_block">identifier iff <span class="s">$</span><span class="nb">T</span><span class="s">$</span> has the functor symbol <span class="nb">{</span><span class="k">\tt</span> cid/1<span class="nb">}</span>, <span class="nb">{</span><span class="k">\tt</span> oid/1<span class="nb">}</span>,
</div><div id="l60" class="code_block"><span class="nb">{</span><span class="k">\tt</span> rid/1<span class="nb">}</span>, or <span class="nb">{</span><span class="k">\tt</span> crid/1<span class="nb">}</span> whose argument is either
</div><div id="l61" class="code_block">
</div><div id="l62" class="code_block"><span class="k">\begin</span><span class="nb">{</span>enumerate<span class="nb">}</span>
</div><div id="l63" class="code_block"><span class="k">\item</span> a constant; or 
</div><div id="l64" class="code_block"><span class="k">\item</span> a term <span class="s">$</span><span class="nb">f</span><span class="o">(</span><span class="nb">I_</span><span class="m">1</span><span class="nb">,</span><span class="nv">\ldots</span><span class="nb">,I_n</span><span class="o">)</span><span class="s">$</span> where <span class="s">$</span><span class="nb">I_</span><span class="m">1</span><span class="nb">,</span><span class="nv">\ldots</span><span class="nb">,I_n</span><span class="s">$</span> are identifiers.
</div><div id="l65" class="code_block"><span class="k">\end</span><span class="nb">{</span>enumerate<span class="nb">}</span>
</div><div id="l66" class="code_block">In the first case, an identifier is called <span class="nb">{</span><span class="k">\em</span> atomic<span class="nb">}</span>; in the second
</div><div id="l67" class="code_block">it is called a <span class="nb">{</span><span class="k">\em</span> product identifier<span class="nb">}</span>.
</div><div id="l68" class="code_block"><span class="k">\end</span><span class="nb">{</span>definition<span class="nb">}</span>
</div><div id="l69" class="code_block">
</div><div id="l70" class="code_block">Despite the simple syntax of Type-0 CDF instances, their semantics
</div><div id="l71" class="code_block">differs from the usual semantics assigned to facts in Prolog.
</div><div id="l72" class="code_block">Identifiers identify sets of objects, or binary relations between
</div><div id="l73" class="code_block">objects.  Furthermore, the facts of a Type-0 CDF instance can
</div><div id="l74" class="code_block">implicitly denote inheritance of various relationships among classes
</div><div id="l75" class="code_block">and objects, as well as inheritance constraints about what
</div><div id="l76" class="code_block">relationships are allowed.
</div><div id="l77" class="code_block">
</div><div id="l78" class="code_block"><span class="k">\subsection</span><span class="nb">{</span>Simple Taxonomies in CDF<span class="nb">}</span>
</div><div id="l79" class="code_block">
</div><div id="l80" class="code_block"><span class="k">\begin</span><span class="nb">{</span>example<span class="nb">}</span> <span class="k">\rm</span> <span class="k">\label</span><span class="nb">{</span>ex:suture1<span class="nb">}</span>
</div><div id="l81" class="code_block">The following CDF instance illustrates a fragment of a taxonomy for
</div><div id="l82" class="code_block">medical equipment.
</div><div id="l83" class="code_block"><span class="c">%-------------------------------------------</span>
</div><div id="l84" class="code_block"><span class="nb">{</span><span class="k">\tt</span>  <span class="nb">{</span><span class="k">\small</span> 
</div><div id="l85" class="code_block"><span class="k">\begin</span><span class="nb">{</span>tabbing<span class="nb">}</span>
</div><div id="l86" class="code_block">foo<span class="k">\=</span>foo<span class="k">\=</span>foo<span class="k">\=</span>foo<span class="k">\=</span>foo<span class="k">\=</span>foo<span class="k">\=</span>foooo<span class="k">\=</span>foooooooooooooooo<span class="k">\=\kill</span>
</div><div id="l87" class="code_block">isa(cid(medicalEquipment),cid(&#39;CDF Classes&#39;))  <span class="k">\\</span>
</div><div id="l88" class="code_block"><span class="k">\&gt;</span> isa(cid(woundCareProducts),cid(medicalEquipment)) <span class="k">\\</span>
</div><div id="l89" class="code_block"><span class="k">\&gt;</span> <span class="k">\&gt;</span> <span class="k">\&gt;</span>  isa(cid(suturesAndRelatedProducts),cid(woundCareProducts)) <span class="k">\\</span>
</div><div id="l90" class="code_block"><span class="k">\&gt;</span> <span class="k">\&gt;</span> <span class="k">\&gt;</span> isa(cid(sutures),cid(suturesAndRelatedProducts))  <span class="k">\\</span>
</div><div id="l91" class="code_block"><span class="k">\&gt;</span> <span class="k">\&gt;</span> <span class="k">\&gt;</span> <span class="k">\&gt;</span> isa(cid(absorbableSutures),cid(sutures))  <span class="k">\\</span>
</div><div id="l92" class="code_block"><span class="k">\&gt;</span> <span class="k">\&gt;</span> <span class="k">\&gt;</span> <span class="k">\&gt;</span> isa(cid(nonAbsorbableSutures),cid(sutures)) <span class="k">\\</span>
</div><div id="l93" class="code_block"><span class="k">\&gt;</span> <span class="k">\\</span>
</div><div id="l94" class="code_block"><span class="k">\&gt;</span> <span class="k">\&gt;</span> <span class="k">\&gt;</span>   isa(oid(sutureU245H),cid(absorbableSutures))  <span class="k">\\</span>
</div><div id="l95" class="code_block"><span class="k">\&gt;</span> <span class="k">\&gt;</span> <span class="k">\&gt;</span> <span class="k">\&gt;</span> <span class="k">\&gt;</span>   isa(oid(suture547466),cid(sutures)) 
</div><div id="l96" class="code_block"><span class="k">\end</span><span class="nb">{</span>tabbing<span class="nb">}</span> <span class="nb">}}</span> 
</div><div id="l97" class="code_block"><span class="c">%-------------------------------------------</span>
</div><div id="l98" class="code_block"><span class="k">\noindent</span>
</div><div id="l99" class="code_block">In CDF, sets of objects are termed <span class="nb">{</span><span class="k">\em</span> classes<span class="nb">}</span> to stress the
</div><div id="l100" class="code_block">informality of its sets from the perspective of set theory, and class
</div><div id="l101" class="code_block">identifiers have the functor <span class="nb">{</span><span class="k">\tt</span> cid/1<span class="nb">}</span>.  One can read the fact
</div><div id="l102" class="code_block"><span class="k">\begin</span><span class="nb">{</span>verbatim<span class="nb">}</span>
</div><div id="l103" class="code_block">      isa(cid(nonAbsorbableSutures),cid(sutures))
</div><div id="l104" class="code_block"><span class="k">\end</span><span class="nb">{</span>verbatim<span class="nb">}</span>
</div><div id="l105" class="code_block">as ``all elements in the class <span class="nb">{</span><span class="k">\tt</span> cid(nonAbsorbableSutures)<span class="nb">}</span> are
</div><div id="l106" class="code_block">also in the class <span class="nb">{</span><span class="k">\tt</span> cid(sutures)<span class="nb">}</span>&#39;&#39; --- in other words, that <span class="nb">{</span><span class="k">\tt</span>
</div><div id="l107" class="code_block">cid(nonAbsorbableSutures)<span class="nb">}</span> is a subclass of <span class="nb">{</span><span class="k">\tt</span> cid(sutures)<span class="nb">}</span>.
</div><div id="l108" class="code_block">Object identifiers have the functor <span class="nb">{</span><span class="k">\tt</span> oid/1<span class="nb">}</span>, and denote classes
</div><div id="l109" class="code_block">with cardinality 1, or <span class="nb">{</span><span class="k">\em</span> singleton classes<span class="nb">}</span>.  The fact
</div><div id="l110" class="code_block"><span class="k">\begin</span><span class="nb">{</span>verbatim<span class="nb">}</span>
</div><div id="l111" class="code_block">      isa(oid(sutureU245H),cid(absorbableSutures))
</div><div id="l112" class="code_block"><span class="k">\end</span><span class="nb">{</span>verbatim<span class="nb">}</span>
</div><div id="l113" class="code_block">can be read as ``the element of the singleton class <span class="nb">{</span><span class="k">\tt</span>
</div><div id="l114" class="code_block">oid(sutureU245H)<span class="nb">}</span> is in the class <span class="nb">{</span><span class="k">\tt</span> cid(absorbableSutures)<span class="nb">}</span>&#39;&#39;.
</div><div id="l115" class="code_block">Note that <span class="nb">{</span><span class="k">\tt</span> cid(absorbableSutures)<span class="nb">}</span> is (potentially) more specific
</div><div id="l116" class="code_block">than the class <span class="nb">{</span><span class="k">\tt</span> cid(sutures)<span class="nb">}</span>, to which <span class="nb">{</span><span class="k">\tt</span> oid(suture547466)<span class="nb">}</span>
</div><div id="l117" class="code_block">belongs.  The class <span class="nb">{</span><span class="k">\tt</span> cid(&#39;CDF Classes&#39;)<span class="nb">}</span> is taken to contain all
</div><div id="l118" class="code_block">objects in the domain of discourse.  All identifiers in this simple
</div><div id="l119" class="code_block">taxonomy are atomic.
</div><div id="l120" class="code_block">
</div><div id="l121" class="code_block">The decision of whether to denote an entity as an object or as a class
</div><div id="l122" class="code_block">depends on the use of a given CDF instance.  Here, a given part number
</div><div id="l123" class="code_block">can specify a number of physical parts, but the physical parts are
</div><div id="l124" class="code_block">taken to be identical for the purposes of this instance.  However, if
</div><div id="l125" class="code_block">we were constructing a CDF instance for warehouse management, the
</div><div id="l126" class="code_block">above objects might be better represented as classes, and the physical
</div><div id="l127" class="code_block">objects represented as CDF objects.
</div><div id="l128" class="code_block"><span class="k">\end</span><span class="nb">{</span>example<span class="nb">}</span> 
</div><div id="l129" class="code_block">
</div><div id="l130" class="code_block">Implicit in the above example is the fact that we use the term <span class="nb">{</span><span class="k">\em</span>
</div><div id="l131" class="code_block">object identifiers<span class="nb">}</span> or <span class="nb">{</span><span class="k">\em</span> objects<span class="nb">}</span> to denote singleton classes,
</div><div id="l132" class="code_block">class identifiers to denote all classes (including singleton classes).
</div><div id="l133" class="code_block">Elements cannot be denoted directly by CDF facts, but only through
</div><div id="l134" class="code_block">singleton classes that contain them (and are isomorphic to them).  At
</div><div id="l135" class="code_block">this point, we can begin defining the semantics of Type-0 CDF
</div><div id="l136" class="code_block">instances.
</div><div id="l137" class="code_block">
</div><div id="l138" class="code_block"><span class="c">%-------------------</span>
</div><div id="l139" class="code_block"><span class="k">\begin</span><span class="nb">{</span>definition<span class="nb">}</span> <span class="k">\label</span><span class="nb">{</span>def:ontolang<span class="nb">}</span>
</div><div id="l140" class="code_block"><span class="k">\index</span><span class="nb">{</span>ontology language<span class="nb">}</span> <span class="k">\index</span><span class="nb">{</span>ontology structure<span class="nb">}</span> <span class="k">\index</span><span class="nb">{</span>ontology
</div><div id="l141" class="code_block">theory<span class="nb">}</span>
</div><div id="l142" class="code_block"><span class="c">%</span>
</div><div id="l143" class="code_block">An <span class="nb">{</span><span class="k">\em</span> ontology language<span class="nb">}</span> is a first-order language with equality
</div><div id="l144" class="code_block">containing the predicates: <span class="nb">{</span><span class="k">\em</span> isClass/1, isElt/1, isRel/1, isCrel/1,
</div><div id="l145" class="code_block">elt/2, rel/3, and crel/3<span class="nb">}</span>, and a countable set of constants.  An <span class="nb">{</span><span class="k">\em</span>
</div><div id="l146" class="code_block">ontology structure<span class="nb">}</span> is a structure defined over an ontology language.
</div><div id="l147" class="code_block">An <span class="nb">{</span><span class="k">\em</span> ontology theory<span class="nb">}</span> is a set of first-order sentences formed over
</div><div id="l148" class="code_block">an ontology language that includes a set of <span class="nb">{</span><span class="k">\em</span> core axioms<span class="nb">}</span>, defined
</div><div id="l149" class="code_block">below, along with the defined predicate:
</div><div id="l150" class="code_block"><span class="sb">\[</span>
</div><div id="l151" class="code_block"><span class="nb">isObj</span><span class="o">(</span><span class="nb">X</span><span class="o">)</span><span class="nb"> </span><span class="o">=</span><span class="nb">_{def} isClass</span><span class="o">(</span><span class="nb">X</span><span class="o">)</span><span class="nb"> </span><span class="nv">\wedge</span><span class="nb"> </span>
</div><div id="l152" class="code_block"><span class="nb">	</span><span class="o">((</span><span class="nb">elt</span><span class="o">(</span><span class="nb">E_</span><span class="m">1</span><span class="nb">,X</span><span class="o">)</span><span class="nb"> </span><span class="nv">\wedge</span><span class="nb"> elt</span><span class="o">(</span><span class="nb">E_</span><span class="m">2</span><span class="nb">,X</span><span class="o">))</span><span class="nb"> </span><span class="nv">\Rightarrow</span><span class="nb"> E_</span><span class="m">1</span><span class="nb"> </span><span class="o">=</span><span class="nb"> E_</span><span class="m">2</span><span class="o">)</span>
</div><div id="l153" class="code_block"><span class="s">\]</span>
</div><div id="l154" class="code_block">If <span class="s">$</span><span class="nv">\cT</span><span class="s">$</span> is an ontology theory formed over an ontology
</div><div id="l155" class="code_block">language <span class="s">$</span><span class="nv">\cL</span><span class="s">$</span>, an ontology structure <span class="s">$</span><span class="nb">S</span><span class="s">$</span> over <span class="s">$</span><span class="nv">\cL</span><span class="s">$</span> is a model of
</div><div id="l156" class="code_block"><span class="s">$</span><span class="nv">\cT</span><span class="s">$</span> if every sentence of <span class="s">$</span><span class="nv">\cT</span><span class="s">$</span> is satisfied in <span class="s">$</span><span class="nb">S</span><span class="s">$</span>.
</div><div id="l157" class="code_block"><span class="k">\end</span><span class="nb">{</span>definition<span class="nb">}</span>
</div><div id="l158" class="code_block"><span class="c">%-------------------</span>
</div><div id="l159" class="code_block">
</div><div id="l160" class="code_block"><span class="k">\index</span><span class="nb">{</span>sorting predicates<span class="nb">}</span>
</div><div id="l161" class="code_block">By convention we assume that the variables in an ontology language are
</div><div id="l162" class="code_block">indexed by the set of natural numbers.  In this paper we will restrict
</div><div id="l163" class="code_block">our attention to ontology languages whose constant and function
</div><div id="l164" class="code_block">symbols are identifiers as described in Definition~<span class="k">\ref</span><span class="nb">{</span>def:ids<span class="nb">}</span>.
</div><div id="l165" class="code_block">Informally <span class="s">$</span><span class="nb">isClass</span><span class="o">/</span><span class="m">1</span><span class="s">$</span> indicates that an identifier <span class="s">$</span><span class="nb">I</span><span class="s">$</span> is a class
</div><div id="l166" class="code_block">name or <span class="nb">{</span><span class="k">\em</span> class identifier<span class="nb">}</span>; <span class="s">$</span><span class="nb">isElt</span><span class="o">/</span><span class="m">1</span><span class="s">$</span> indicates that an identifier
</div><div id="l167" class="code_block"><span class="s">$</span><span class="nb">I</span><span class="s">$</span> is an element of a class; <span class="s">$</span><span class="nb">isRel</span><span class="o">/</span><span class="m">1</span><span class="s">$</span> that <span class="s">$</span><span class="nb">I</span><span class="s">$</span> is a <span class="nb">{</span><span class="k">\em</span> relation
</div><div id="l168" class="code_block">identifier<span class="nb">}</span>; and <span class="s">$</span><span class="nb">isCrel</span><span class="o">/</span><span class="m">1</span><span class="s">$</span> that <span class="s">$</span><span class="nb">I</span><span class="s">$</span> is a <span class="nb">{</span><span class="k">\em</span> class-relation
</div><div id="l169" class="code_block">identifier<span class="nb">}</span>; and <span class="s">$</span><span class="nb">isObj</span><span class="o">/</span><span class="m">1</span><span class="s">$</span> that <span class="s">$</span><span class="nb">I</span><span class="s">$</span> is an <span class="nb">{</span><span class="k">\em</span> object identifier<span class="nb">}</span>.  We
</div><div id="l170" class="code_block">sometimes call these 5 predicates <span class="nb">{</span><span class="k">\em</span> sorting predicates<span class="nb">}</span>.
</div><div id="l171" class="code_block"><span class="s">$</span><span class="nb">elt</span><span class="o">(</span><span class="nb">E,C</span><span class="o">)</span><span class="s">$</span> indicates that an element <span class="s">$</span><span class="nb">O</span><span class="s">$</span> is a member of class
</div><div id="l172" class="code_block">identifier <span class="s">$</span><span class="nb">C</span><span class="s">$</span>; <span class="s">$</span><span class="nb">rel</span><span class="o">(</span><span class="nb">O_</span><span class="m">1</span><span class="nb">,R,O_</span><span class="m">2</span><span class="o">)</span><span class="s">$</span> indicates that an element <span class="s">$</span><span class="nb">E_</span><span class="m">1</span><span class="s">$</span> has a
</div><div id="l173" class="code_block"><span class="s">$</span><span class="nb">R</span><span class="s">$</span> relation to an element <span class="s">$</span><span class="nb">E_</span><span class="m">2</span><span class="s">$</span>; and <span class="s">$</span><span class="nb">crel</span><span class="o">(</span><span class="nb">C_</span><span class="m">1</span><span class="nb">,R,E</span><span class="o">)</span><span class="s">$</span> indicates that
</div><div id="l174" class="code_block">the class identifier <span class="s">$</span><span class="nb">C_</span><span class="m">1</span><span class="s">$</span> has a <span class="s">$</span><span class="nb">R</span><span class="s">$</span> relation to an object identifier
</div><div id="l175" class="code_block"><span class="s">$</span><span class="nb">E</span><span class="s">$</span>.
</div><div id="l176" class="code_block">
</div><div id="l177" class="code_block">The first core axiom ensures that objects, classes, relations and
</div><div id="l178" class="code_block">class-relations all have distinct identifiers within an ontology
</div><div id="l179" class="code_block">language.
</div><div id="l180" class="code_block"><span class="c">%-----------</span>
</div><div id="l181" class="code_block"><span class="k">\begin</span><span class="nb">{</span>axiom<span class="nb">}</span>[Distinct Identifiers] <span class="k">\label</span><span class="nb">{</span>ax:distinct<span class="nb">}</span>
</div><div id="l182" class="code_block"><span class="k">\index</span><span class="nb">{</span>axioms!distinct identifiers<span class="nb">}</span> 
</div><div id="l183" class="code_block"><span class="k">\</span> <span class="k">\\</span>
</div><div id="l184" class="code_block"><span class="k">\begin</span><span class="nb">{</span>tabbing<span class="nb">}</span>
</div><div id="l185" class="code_block">foo<span class="k">\=</span>foo<span class="k">\=</span>foo<span class="k">\=</span>foo<span class="k">\=</span>foo<span class="k">\=</span>foo<span class="k">\=</span>foooo<span class="k">\=</span>foooooooooooooooo<span class="k">\=\kill</span>
</div><div id="l186" class="code_block"><span class="k">\&gt;</span> <span class="s">$</span><span class="nv">\neg</span><span class="nb"> </span><span class="nv">\exists</span><span class="nb"> Id </span><span class="o">[</span><span class="nb">isClass</span><span class="o">(</span><span class="nb">Id</span><span class="o">)</span><span class="nb"> </span><span class="nv">\wedge</span><span class="nb"> </span><span class="o">(</span><span class="nb">isElt</span><span class="o">(</span><span class="nb">Id</span><span class="o">)</span><span class="nb"> </span><span class="nv">\vee</span><span class="nb"> isRel</span><span class="o">(</span><span class="nb">Id</span><span class="o">)</span><span class="nb"> </span>
</div><div id="l187" class="code_block"><span class="nb">	                                 </span><span class="nv">\vee</span><span class="nb"> isCrel</span><span class="o">(</span><span class="nb">Id</span><span class="o">))]</span><span class="nb"> </span><span class="nv">\wedge</span><span class="nb"> </span><span class="s">$</span> <span class="k">\\</span>
</div><div id="l188" class="code_block"><span class="k">\&gt;</span> <span class="s">$</span><span class="nv">\neg</span><span class="nb"> </span><span class="nv">\exists</span><span class="nb"> Id </span><span class="o">[</span><span class="nb">isObj</span><span class="o">(</span><span class="nb">Id</span><span class="o">)</span><span class="nb"> </span><span class="nv">\wedge</span><span class="nb"> </span><span class="o">(</span><span class="nb">isElt</span><span class="o">(</span><span class="nb">Id</span><span class="o">)</span><span class="nb"> </span><span class="nv">\vee</span><span class="nb"> isCrel</span><span class="o">(</span><span class="nb">Id</span><span class="o">))]</span><span class="nb"> </span><span class="nv">\wedge</span><span class="nb"> </span><span class="s">$</span> <span class="k">\\</span>
</div><div id="l189" class="code_block"><span class="k">\&gt;</span> <span class="s">$</span><span class="nv">\neg</span><span class="nb"> </span><span class="nv">\exists</span><span class="nb"> Id </span><span class="o">[</span><span class="nb">isElt</span><span class="o">(</span><span class="nb">Id</span><span class="o">)</span><span class="nb"> </span><span class="nv">\wedge</span><span class="nb"> isCrel</span><span class="o">(</span><span class="nb">Id</span><span class="o">)]</span><span class="nb"> </span><span class="s">$</span> 
</div><div id="l190" class="code_block"><span class="k">\end</span><span class="nb">{</span>tabbing<span class="nb">}</span>
</div><div id="l191" class="code_block"><span class="k">\end</span><span class="nb">{</span>axiom<span class="nb">}</span> 
</div><div id="l192" class="code_block"><span class="c">%-----------</span>
</div><div id="l193" class="code_block">
</div><div id="l194" class="code_block"><span class="s">$</span><span class="nb">isClass</span><span class="o">/</span><span class="m">1</span><span class="s">$</span>, <span class="s">$</span><span class="nb">isElt</span><span class="o">/</span><span class="m">1</span><span class="s">$</span>, <span class="s">$</span><span class="nb">isRel</span><span class="o">/</span><span class="m">1</span><span class="s">$</span>, and <span class="s">$</span><span class="nb">isCrel</span><span class="o">/</span><span class="m">1</span><span class="s">$</span> provide an effective
</div><div id="l195" class="code_block">sorting that extends to all predicates, as the next axiom indicates.
</div><div id="l196" class="code_block">
</div><div id="l197" class="code_block"><span class="k">\begin</span><span class="nb">{</span>axiom<span class="nb">}</span>[Predicate Sorts] <span class="k">\rm</span> <span class="k">\label</span><span class="nb">{</span>ax:sorts<span class="nb">}</span>
</div><div id="l198" class="code_block"><span class="k">\index</span><span class="nb">{</span>axioms!predicate sorts<span class="nb">}</span> 
</div><div id="l199" class="code_block"><span class="k">\</span> <span class="k">\\</span>
</div><div id="l200" class="code_block"><span class="k">\begin</span><span class="nb">{</span>tabbing<span class="nb">}</span>
</div><div id="l201" class="code_block">foo<span class="k">\=</span>foo<span class="k">\=</span>foo<span class="k">\=</span>foo<span class="k">\=</span>foo<span class="k">\=</span>foo<span class="k">\=</span>foooo<span class="k">\=</span>foooooooooooooooo<span class="k">\=\kill</span>
</div><div id="l202" class="code_block"><span class="k">\&gt;</span> <span class="k">\&gt;</span> <span class="s">$</span><span class="o">(</span><span class="nv">\forall</span><span class="nb"> X,Y</span><span class="o">)</span><span class="nb"> </span><span class="o">[</span><span class="nb">elt</span><span class="o">(</span><span class="nb">X,Y</span><span class="o">)</span><span class="nb"> </span><span class="nv">\Rightarrow</span><span class="nb"> </span><span class="o">(</span><span class="nb">isElt</span><span class="o">(</span><span class="nb">X</span><span class="o">)</span><span class="nb"> </span><span class="nv">\wedge</span><span class="nb"> isClass</span><span class="o">(</span><span class="nb">Y</span><span class="o">))]</span>
</div><div id="l203" class="code_block"><span class="nb">								</span><span class="nv">\wedge</span><span class="nb"> </span><span class="s">$</span> <span class="k">\\</span>
</div><div id="l204" class="code_block"><span class="k">\&gt;</span> <span class="k">\&gt;</span> <span class="s">$</span><span class="o">(</span><span class="nv">\forall</span><span class="nb"> X,Y,Z</span><span class="o">)</span><span class="nb"> </span><span class="o">[</span><span class="nb">rel</span><span class="o">(</span><span class="nb">X,Y,Z</span><span class="o">)</span><span class="nb"> </span><span class="nv">\Rightarrow</span><span class="nb"> </span><span class="o">(</span><span class="nb">isElt</span><span class="o">(</span><span class="nb">X</span><span class="o">)</span><span class="nb"> </span><span class="nv">\wedge</span><span class="nb"> isRel</span><span class="o">(</span><span class="nb">Y</span><span class="o">)</span>
</div><div id="l205" class="code_block"><span class="nb">					   </span><span class="nv">\wedge</span><span class="nb"> isElt</span><span class="o">(</span><span class="nb">Z</span><span class="o">))]</span><span class="nb"> </span><span class="nv">\wedge</span><span class="nb">  </span><span class="s">$</span> <span class="k">\\</span>
</div><div id="l206" class="code_block"><span class="k">\&gt;</span> <span class="k">\&gt;</span> <span class="s">$</span><span class="o">(</span><span class="nv">\forall</span><span class="nb"> X,Y,Z</span><span class="o">)</span><span class="nb"> </span><span class="o">[</span><span class="nb">crel</span><span class="o">(</span><span class="nb">X,Y,Z</span><span class="o">)</span><span class="nb"> </span><span class="nv">\Rightarrow</span><span class="nb"> </span><span class="o">(</span><span class="nb">isClass</span><span class="o">(</span><span class="nb">X</span><span class="o">)</span><span class="nb"> </span><span class="nv">\wedge</span><span class="nb"> isRel</span><span class="o">(</span><span class="nb">Y</span><span class="o">)</span>
</div><div id="l207" class="code_block"><span class="nb">					   </span><span class="nv">\wedge</span><span class="nb"> isElt</span><span class="o">(</span><span class="nb">Z</span><span class="o">))]</span><span class="nb"> </span><span class="s">$</span> <span class="k">\\</span>
</div><div id="l208" class="code_block"><span class="k">\end</span><span class="nb">{</span>tabbing<span class="nb">}</span>
</div><div id="l209" class="code_block"><span class="k">\end</span><span class="nb">{</span>axiom<span class="nb">}</span> 
</div><div id="l210" class="code_block">
</div><div id="l211" class="code_block"><span class="c">%-----------</span>
</div><div id="l212" class="code_block">
</div><div id="l213" class="code_block">The following definition of <span class="s">$</span><span class="nb">IdSort</span><span class="s">$</span> relates the functors of
</div><div id="l214" class="code_block">identifiers in a Type-0 CDF instance to their sort in an ontology
</div><div id="l215" class="code_block">theory.  It will be used in the various instance axioms to enforce
</div><div id="l216" class="code_block">proper sorting of product identifiers.
</div><div id="l217" class="code_block">
</div><div id="l218" class="code_block"><span class="k">\begin</span><span class="nb">{</span>definition<span class="nb">}{</span><span class="k">\bf</span> [IdSort]<span class="nb">}</span> <span class="k">\label</span><span class="nb">{</span>def:IdSort<span class="nb">}</span>
</div><div id="l219" class="code_block">Let <span class="nb">{</span><span class="k">\em</span> I<span class="nb">}</span> is be an identifier. Then <span class="nb">{</span><span class="k">\em</span> IdSort(I)<span class="nb">}</span> is equal to <span class="nb">{</span><span class="k">\em</span>
</div><div id="l220" class="code_block">isClass(I)<span class="nb">}</span> if the main functor symbol of <span class="nb">{</span><span class="k">\em</span> I<span class="nb">}</span> is <span class="nb">{</span><span class="k">\tt</span> cid/1<span class="nb">}</span>; <span class="nb">{</span><span class="k">\em</span>
</div><div id="l221" class="code_block">isObj(I)<span class="nb">}</span> if the main functor symbol of <span class="nb">{</span><span class="k">\em</span> I<span class="nb">}</span> is <span class="nb">{</span><span class="k">\tt</span> oid/1<span class="nb">}</span>; <span class="nb">{</span><span class="k">\em</span>
</div><div id="l222" class="code_block">isRel(I)<span class="nb">}</span> if the main functor symbol of <span class="nb">{</span><span class="k">\em</span> I<span class="nb">}</span> is <span class="nb">{</span><span class="k">\tt</span> rid/1<span class="nb">}</span>; and
</div><div id="l223" class="code_block"><span class="nb">{</span><span class="k">\em</span> isCrel(I)<span class="nb">}</span> if the main functor symbol of <span class="nb">{</span><span class="k">\em</span> I<span class="nb">}</span> is <span class="nb">{</span><span class="k">\tt</span> crid/1<span class="nb">}</span>.
</div><div id="l224" class="code_block"><span class="k">\end</span><span class="nb">{</span>definition<span class="nb">}</span>
</div><div id="l225" class="code_block">
</div><div id="l226" class="code_block"><span class="c">%-----------------------------------------------------------------------------------------</span>
</div><div id="l227" class="code_block"><span class="k">\mycomment</span><span class="nb">{</span>
</div><div id="l228" class="code_block"><span class="k">\begin</span><span class="nb">{</span>definition<span class="nb">}{</span><span class="k">\bf</span> [IdSort]<span class="nb">}</span>  Let <span class="nb">{</span><span class="k">\em</span> I<span class="nb">}</span> is be an identifier, and
</div><div id="l229" class="code_block"><span class="s">$</span><span class="nv">\cI</span><span class="s">$</span> be the set of identifiers occurring in <span class="nb">{</span><span class="k">\em</span> I<span class="nb">}</span>. Then
</div><div id="l230" class="code_block"><span class="c">%</span>
</div><div id="l231" class="code_block"><span class="sb">\[</span><span class="nb"> IdSort</span><span class="o">(</span><span class="nb">I</span><span class="o">)</span><span class="nb"> </span><span class="o">=</span><span class="nb">_{def} </span><span class="nv">\bigwedge</span><span class="nb">_{I&#39; </span><span class="nv">\in</span><span class="nb"> </span><span class="nv">\cI</span><span class="nb">} Sort</span><span class="o">(</span><span class="nb">I&#39;</span><span class="o">)</span><span class="nb"> </span><span class="s">\]</span>
</div><div id="l232" class="code_block"><span class="c">%</span>
</div><div id="l233" class="code_block">where <span class="nb">{</span><span class="k">\em</span> Sort(I)<span class="nb">}</span> is equal to <span class="nb">{</span><span class="k">\em</span> isClass(I)<span class="nb">}</span> if the main functor
</div><div id="l234" class="code_block">symbol of <span class="nb">{</span><span class="k">\em</span> I<span class="nb">}</span> is <span class="nb">{</span><span class="k">\tt</span> cid/1<span class="nb">}</span>; <span class="nb">{</span><span class="k">\em</span> isObj(I)<span class="nb">}</span> if the main functor
</div><div id="l235" class="code_block">symbol of <span class="nb">{</span><span class="k">\em</span> I<span class="nb">}</span> is <span class="nb">{</span><span class="k">\tt</span> oid/1<span class="nb">}</span>; <span class="nb">{</span><span class="k">\em</span> isRel(I)<span class="nb">}</span> if the main functor
</div><div id="l236" class="code_block">symbol of <span class="nb">{</span><span class="k">\em</span> I<span class="nb">}</span> is <span class="nb">{</span><span class="k">\tt</span> rid/1<span class="nb">}</span>; and <span class="nb">{</span><span class="k">\em</span> isCrel(I)<span class="nb">}</span> if the main
</div><div id="l237" class="code_block">functor symbol of <span class="nb">{</span><span class="k">\em</span> I<span class="nb">}</span> is <span class="nb">{</span><span class="k">\tt</span> crid/1<span class="nb">}</span>.
</div><div id="l238" class="code_block"><span class="k">\end</span><span class="nb">{</span>definition<span class="nb">}</span>
</div><div id="l239" class="code_block"><span class="nb">}</span>
</div><div id="l240" class="code_block"><span class="c">%-----------------------------------------------------------------------------------------</span>
</div><div id="l241" class="code_block">
</div><div id="l242" class="code_block">From Definitions~<span class="k">\ref</span><span class="nb">{</span>def:IdSort<span class="nb">}</span> and~<span class="k">\ref</span><span class="nb">{</span>def:ontolang<span class="nb">}</span>, it is easy
</div><div id="l243" class="code_block">to see that for any object identifier <span class="s">$</span><span class="nb">O</span><span class="s">$</span>, <span class="s">$</span><span class="nb">IdSort</span><span class="o">(</span><span class="nb">O</span><span class="o">)</span><span class="nb"> </span><span class="o">=</span><span class="nb"> isObj</span><span class="o">(</span><span class="nb">O</span><span class="o">)</span><span class="s">$</span>, and
</div><div id="l244" class="code_block"><span class="s">$</span><span class="nb">isObj</span><span class="o">(</span><span class="nb">O</span><span class="o">)</span><span class="nb"> </span><span class="nv">\Rightarrow</span><span class="nb"> isClass</span><span class="o">(</span><span class="nb">O</span><span class="o">)</span><span class="s">$</span>.
</div><div id="l245" class="code_block">
</div><div id="l246" class="code_block"><span class="c">%-----------</span>
</div><div id="l247" class="code_block"><span class="k">\begin</span><span class="nb">{</span>instance<span class="nb">}</span> [Translation of <span class="nb">{</span><span class="k">\tt</span> isa/2<span class="nb">}</span>] <span class="k">\rm</span> 
</div><div id="l248" class="code_block">For each fact of the form <span class="nb">{</span><span class="k">\tt</span> isa(Id<span class="s">$</span><span class="nb">_</span><span class="m">1</span><span class="s">$</span>,Id<span class="s">$</span><span class="nb">_</span><span class="m">2</span><span class="s">$</span>)<span class="nb">}</span> add the axiom
</div><div id="l249" class="code_block"><span class="k">\</span> <span class="k">\\</span>
</div><div id="l250" class="code_block"><span class="k">\begin</span><span class="nb">{</span>tabbing<span class="nb">}</span>
</div><div id="l251" class="code_block">foo<span class="k">\=</span>foo<span class="k">\=</span>foo<span class="k">\=</span>foo<span class="k">\=</span>foo<span class="k">\=</span>foo<span class="k">\=</span>foooo<span class="k">\=</span>foooooooooooooooo<span class="k">\=\kill</span>
</div><div id="l252" class="code_block"><span class="k">\&gt;</span> <span class="s">$</span><span class="nb"> IdSort</span><span class="o">(</span><span class="nb">Id_</span><span class="m">1</span><span class="o">)</span><span class="nb"> </span><span class="nv">\wedge</span><span class="nb"> IdSort</span><span class="o">(</span><span class="nb">Id_</span><span class="m">2</span><span class="o">)</span><span class="nb"> </span><span class="nv">\wedge</span><span class="nb"> </span><span class="s">$</span> <span class="k">\\</span>
</div><div id="l253" class="code_block">
</div><div id="l254" class="code_block"><span class="k">\&gt;</span> <span class="k">\&gt;</span> <span class="s">$</span><span class="nb"> </span><span class="o">(((</span><span class="nb">isClass</span><span class="o">(</span><span class="nb">Id_</span><span class="m">1</span><span class="o">)</span><span class="nb"> </span><span class="nv">\wedge</span><span class="nb"> isClass</span><span class="o">(</span><span class="nb">Id_</span><span class="m">2</span><span class="o">))</span><span class="nb"> </span><span class="nv">\wedge</span>
</div><div id="l255" class="code_block"><span class="nb">	</span><span class="o">(</span><span class="nv">\forall</span><span class="nb"> X</span><span class="o">)</span><span class="nb"> </span><span class="o">[</span><span class="nb">elt</span><span class="o">(</span><span class="nb">X,Id_</span><span class="m">1</span><span class="o">)</span><span class="nb"> </span><span class="nv">\Rightarrow</span><span class="nb"> elt</span><span class="o">(</span><span class="nb">X,Id_</span><span class="m">2</span><span class="o">)])</span><span class="nb"> </span><span class="nv">\vee</span><span class="nb"> </span><span class="s">$</span> <span class="k">\\</span>
</div><div id="l256" class="code_block">
</div><div id="l257" class="code_block"><span class="c">%\&gt; \&gt; $ ((isObj(Id_1) \wedge isClass(Id_2)) \wedge elt(Id_1,Id_2)) \vee $ \\</span>
</div><div id="l258" class="code_block">
</div><div id="l259" class="code_block"><span class="k">\&gt;</span> <span class="k">\&gt;</span> <span class="s">$</span><span class="nb"> </span><span class="o">((</span><span class="nb">isRel</span><span class="o">(</span><span class="nb">Id_</span><span class="m">1</span><span class="o">)</span><span class="nb"> </span><span class="nv">\wedge</span><span class="nb"> isRel</span><span class="o">(</span><span class="nb">Id_</span><span class="m">2</span><span class="o">))</span><span class="nb"> </span><span class="nv">\wedge</span>
</div><div id="l260" class="code_block"><span class="nb">	</span><span class="o">(</span><span class="nv">\forall</span><span class="nb"> X,Y</span><span class="o">)[</span><span class="nb">rel</span><span class="o">(</span><span class="nb">X,Id_</span><span class="m">1</span><span class="nb">,Y</span><span class="o">)</span><span class="nb"> </span><span class="nv">\Rightarrow</span><span class="nb"> rel</span><span class="o">(</span><span class="nb">X,Id_</span><span class="m">2</span><span class="nb">,Y</span><span class="o">)])</span><span class="nb"> </span><span class="nv">\vee</span><span class="nb"> </span><span class="s">$</span> <span class="k">\\</span>
</div><div id="l261" class="code_block">
</div><div id="l262" class="code_block"><span class="k">\&gt;</span> <span class="k">\&gt;</span> <span class="s">$</span><span class="nb"> </span><span class="o">((</span><span class="nb">isCrel</span><span class="o">(</span><span class="nb">Id_</span><span class="m">1</span><span class="o">)</span><span class="nb"> </span><span class="nv">\wedge</span><span class="nb"> isCrel</span><span class="o">(</span><span class="nb">Id_</span><span class="m">2</span><span class="o">))</span><span class="nb"> </span><span class="nv">\wedge</span>
</div><div id="l263" class="code_block"><span class="nb">	</span><span class="o">(</span><span class="nv">\forall</span><span class="nb"> X,Y</span><span class="o">)[</span><span class="nb">crel</span><span class="o">(</span><span class="nb">X,Id_</span><span class="m">1</span><span class="nb">,Y</span><span class="o">)</span><span class="nb"> </span><span class="nv">\Rightarrow</span><span class="nb"> crel</span><span class="o">(</span><span class="nb">X,Id_</span><span class="m">2</span><span class="nb">,Y</span><span class="o">)]))</span><span class="nb"> </span><span class="s">$</span> 
</div><div id="l264" class="code_block"><span class="k">\end</span><span class="nb">{</span>tabbing<span class="nb">}</span>
</div><div id="l265" class="code_block">denoted as <span class="nb">{</span><span class="k">\tt</span> isa(Id<span class="s">$</span><span class="nb">_</span><span class="m">1</span><span class="s">$</span>,Id<span class="s">$</span><span class="nb">_</span><span class="m">2</span><span class="s">$</span>)<span class="nb">}</span><span class="s">$</span><span class="nb">^{</span><span class="nv">\cI</span><span class="nb">}</span><span class="s">$</span>.
</div><div id="l266" class="code_block"><span class="k">\end</span><span class="nb">{</span>instance<span class="nb">}</span>
</div><div id="l267" class="code_block"><span class="c">%-----------</span>
</div><div id="l268" class="code_block">
</div><div id="l269" class="code_block">Note that the reflexive and transitive closure of <span class="nb">{</span><span class="k">\tt</span> isa/2<span class="nb">}</span> is an
</div><div id="l270" class="code_block">immediate consequence of its translation rule.  The next axiom is
</div><div id="l271" class="code_block">technical.  It is important for the semantics of relations that each
</div><div id="l272" class="code_block">class have at least one member.
</div><div id="l273" class="code_block"><span class="c">%-----------</span>
</div><div id="l274" class="code_block"><span class="k">\begin</span><span class="nb">{</span>axiom<span class="nb">}</span>[Non-Empty Classes] <span class="k">\label</span><span class="nb">{</span>ax:nonnull<span class="nb">}</span>
</div><div id="l275" class="code_block"><span class="k">\index</span><span class="nb">{</span>axioms!non-empty classes<span class="nb">}</span> 
</div><div id="l276" class="code_block"><span class="sb">\[</span><span class="nb"> </span><span class="o">(</span><span class="nv">\forall</span><span class="nb"> X</span><span class="o">)</span><span class="nb"> </span><span class="o">[</span><span class="nb">isClass</span><span class="o">(</span><span class="nb">X</span><span class="o">)</span><span class="nb"> </span><span class="nv">\Rightarrow</span><span class="nb"> </span><span class="o">(</span><span class="nv">\exists</span><span class="nb"> Y</span><span class="o">)</span><span class="nb"> </span><span class="o">[</span><span class="nb">elt</span><span class="o">(</span><span class="nb">Y,X</span><span class="o">)]]</span><span class="nb"> </span><span class="s">\]</span>
</div><div id="l277" class="code_block"><span class="k">\end</span><span class="nb">{</span>axiom<span class="nb">}</span> 
</div><div id="l278" class="code_block"><span class="c">%-----------</span>
</div><div id="l279" class="code_block">
</div><div id="l280" class="code_block">The last core axiom for these predicates ensures is that each class is
</div><div id="l281" class="code_block">a subclass of <span class="nb">{</span><span class="k">\tt</span> cid(&#39;CDF Classes&#39;)<span class="nb">}</span>.
</div><div id="l282" class="code_block">
</div><div id="l283" class="code_block"><span class="c">%-----------</span>
</div><div id="l284" class="code_block"><span class="k">\begin</span><span class="nb">{</span>axiom<span class="nb">}</span>[Domain Containment] <span class="k">\label</span><span class="nb">{</span>ax:contained<span class="nb">}</span>
</div><div id="l285" class="code_block"><span class="k">\index</span><span class="nb">{</span>axioms!domain containment<span class="nb">}</span> 
</div><div id="l286" class="code_block"><span class="sb">\[</span><span class="nb"> </span><span class="o">(</span><span class="nv">\forall</span><span class="nb"> X</span><span class="o">)</span><span class="nb"> </span><span class="o">[</span><span class="nb">isElt</span><span class="o">(</span><span class="nb">X</span><span class="o">)</span><span class="nb"> </span><span class="nv">\Rightarrow</span><span class="nb"> elt</span><span class="o">(</span><span class="nb">X,cid</span><span class="o">(</span><span class="nb">&#39;CDF</span><span class="nv">\</span><span class="nb"> Classes&#39;</span><span class="o">))]</span><span class="nb"> </span><span class="s">\]</span>
</div><div id="l287" class="code_block"><span class="k">\end</span><span class="nb">{</span>axiom<span class="nb">}</span> 
</div><div id="l288" class="code_block"><span class="c">%-----------</span>
</div><div id="l289" class="code_block">
</div><div id="l290" class="code_block"><span class="k">\subsection</span><span class="nb">{</span>General Relations between Objects in  Classes<span class="nb">}</span>
</div><div id="l291" class="code_block">
</div><div id="l292" class="code_block"><span class="k">\begin</span><span class="nb">{</span>example<span class="nb">}</span> <span class="k">\rm</span> <span class="k">\label</span><span class="nb">{</span>ex:suture2<span class="nb">}</span>
</div><div id="l293" class="code_block">The class <span class="nb">{</span><span class="k">\tt</span> cid(sutures)<span class="nb">}</span> can be further defined by its relations
</div><div id="l294" class="code_block">to other classes.  For instance, an object in <span class="nb">{</span><span class="k">\tt</span> cid(sutures)<span class="nb">}</span> may
</div><div id="l295" class="code_block">have a designation of its needle design indicating whether it is to be
</div><div id="l296" class="code_block">used for abdominal surgeries, thoracic surgeries, or other purposes.
</div><div id="l297" class="code_block">This information is indicated by the following CDF facts:
</div><div id="l298" class="code_block"><span class="c">%</span>
</div><div id="l299" class="code_block"><span class="nb">{</span><span class="k">\small</span>
</div><div id="l300" class="code_block"><span class="k">\begin</span><span class="nb">{</span>tabbing<span class="nb">}</span>
</div><div id="l301" class="code_block">fooo<span class="k">\=</span>foo<span class="k">\=</span>foo<span class="k">\=</span>foo<span class="k">\=</span>fooooooooooooooooooooooooooooooo<span class="k">\=</span>ooooooooooooo<span class="k">\=\kill</span>
</div><div id="l302" class="code_block"><span class="k">\&gt;</span> <span class="nb">{</span><span class="k">\tt</span> isa(rid(hasNeedleDesign),rid(&#39;CDF Relations&#39;))<span class="nb">}</span> <span class="k">\\</span>
</div><div id="l303" class="code_block"><span class="k">\\</span>
</div><div id="l304" class="code_block"><span class="k">\&gt;</span> <span class="nb">{</span><span class="k">\tt</span> isa(cid(domainTypes),cid(&#39;CDF Classes&#39;))<span class="nb">}</span> <span class="k">\\</span>
</div><div id="l305" class="code_block"><span class="k">\&gt;</span> <span class="k">\&gt;</span> <span class="nb">{</span><span class="k">\tt</span> isa(cid(needleDesignTypes),cid(domainTypes))<span class="nb">}</span> <span class="k">\\</span>
</div><div id="l306" class="code_block"><span class="k">\&gt;</span> <span class="k">\&gt;</span> <span class="k">\&gt;</span> <span class="nb">{</span><span class="k">\tt</span> isa(cid(Abdominal),cid(needleDesign))<span class="nb">}</span> <span class="k">\\</span>
</div><div id="l307" class="code_block"><span class="k">\&gt;</span> <span class="k">\&gt;</span> <span class="k">\&gt;</span> <span class="nb">{</span><span class="k">\tt</span> isa(cid(Abscisson),cid(needleDesign))<span class="nb">}</span> <span class="k">\\</span>
</div><div id="l308" class="code_block"><span class="k">\&gt;</span> <span class="k">\&gt;</span> <span class="k">\&gt;</span> <span class="nb">{</span><span class="k">\tt</span> isa(cid(&#39;Adson Dura&#39;),cid(needleDesign))<span class="nb">}</span> <span class="k">\\</span>
</div><div id="l309" class="code_block"><span class="k">\&gt;</span> <span class="nb">{</span><span class="k">\it</span> <span class="k">\%</span> 126 other values..<span class="nb">}</span>  <span class="k">\\</span>
</div><div id="l310" class="code_block"><span class="k">\\</span>
</div><div id="l311" class="code_block"><span class="k">\&gt;</span> <span class="nb">{</span><span class="k">\tt</span> allAttr(cid(sutures),rid(hasNeedleDesign),cid(needleDesign))<span class="nb">}</span> 
</div><div id="l312" class="code_block"><span class="k">\end</span><span class="nb">{</span>tabbing<span class="nb">}</span>
</div><div id="l313" class="code_block"><span class="nb">}</span>
</div><div id="l314" class="code_block"><span class="c">%</span>
</div><div id="l315" class="code_block"><span class="k">\noindent</span>
</div><div id="l316" class="code_block">The <span class="nb">{</span><span class="k">\tt</span> allAttr/3<span class="nb">}</span> fact above can be read as ``if any object in <span class="nb">{</span><span class="k">\tt</span>
</div><div id="l317" class="code_block">cid(absorbableSutures)<span class="nb">}</span> has a <span class="nb">{</span><span class="k">\tt</span> rid(hasNeedleDesign)<span class="nb">}</span> relation, it
</div><div id="l318" class="code_block">must be to an object in the class <span class="nb">{</span><span class="k">\tt</span> cid(needleDesign)<span class="nb">}</span>&#39;&#39;.  That
</div><div id="l319" class="code_block"><span class="nb">{</span><span class="k">\tt</span> hasNeedleDesign<span class="nb">}</span> is a relation is indicated by clothing it in the
</div><div id="l320" class="code_block">functor <span class="nb">{</span><span class="k">\tt</span> rid/1<span class="nb">}</span>.  This relation is an immediate subclass of all
</div><div id="l321" class="code_block"><span class="nb">{</span><span class="k">\tt</span> CDF Relations<span class="nb">}</span> which in turn is taken to represent the universal
</div><div id="l322" class="code_block">binary relation over the domain of discourse.  Thus the <span class="nb">{</span><span class="k">\tt</span>
</div><div id="l323" class="code_block">allAttr/3<span class="nb">}</span> fact effectively types the range of <span class="nb">{</span><span class="k">\tt</span>
</div><div id="l324" class="code_block">rid(hasNeedleDesign)<span class="nb">}</span> relations, stemming from objects in the class
</div><div id="l325" class="code_block"><span class="nb">{</span><span class="k">\tt</span> cid(absorbableSutures)<span class="nb">}</span>, but it does not indicate the existence
</div><div id="l326" class="code_block">of such a relationship.  From a user&#39;s point of view, the <span class="nb">{</span><span class="k">\tt</span>
</div><div id="l327" class="code_block">rid(hasNeedleDesign)<span class="nb">}</span> relation can be thought of as an optional
</div><div id="l328" class="code_block">attribute for a given <span class="nb">{</span><span class="k">\tt</span> cid(absorbableSutures)<span class="nb">}</span> object.  Sample
</div><div id="l329" class="code_block">values for <span class="nb">{</span><span class="k">\tt</span> cid(hasNeedleDesignTypes)<span class="nb">}</span> are also given.
</div><div id="l330" class="code_block"><span class="k">\end</span><span class="nb">{</span>example<span class="nb">}</span> 
</div><div id="l331" class="code_block">
</div><div id="l332" class="code_block"><span class="c">%-------------------</span>
</div><div id="l333" class="code_block">
</div><div id="l334" class="code_block"><span class="nb">{</span><span class="k">\tt</span> allAttr/3<span class="nb">}</span> provides a simple but powerful mechanism for
</div><div id="l335" class="code_block">inheritance of typing among CDF classes:
</div><div id="l336" class="code_block"><span class="c">%---------------------------------------------------------------------------</span>
</div><div id="l337" class="code_block"><span class="k">\mycomment</span><span class="nb">{</span>
</div><div id="l338" class="code_block">, as can be seen from the
</div><div id="l339" class="code_block">following translation rule, which uses the defined formula
</div><div id="l340" class="code_block">
</div><div id="l341" class="code_block"><span class="sb">\[</span><span class="nb"> </span>
</div><div id="l342" class="code_block"><span class="nb">elt</span><span class="o">(</span><span class="nb">X,Y</span><span class="o">)</span><span class="nb"> </span><span class="o">=</span><span class="nb">_{def} </span><span class="o">((</span><span class="nb">isClass</span><span class="o">(</span><span class="nb">Y</span><span class="o">)</span><span class="nb"> </span><span class="nv">\wedge</span><span class="nb"> elt</span><span class="o">(</span><span class="nb">X,Y</span><span class="o">))</span><span class="nb"> </span><span class="nv">\vee</span><span class="nb"> </span><span class="o">(</span><span class="nb">isObj</span><span class="o">(</span><span class="nb">Y</span><span class="o">)</span><span class="nb"> </span>
</div><div id="l343" class="code_block"><span class="nb">			</span><span class="nv">\wedge</span><span class="nb"> X </span><span class="o">=</span><span class="nb"> Y</span><span class="o">))</span>
</div><div id="l344" class="code_block"><span class="s">\]</span> <span class="nb">}</span>
</div><div id="l345" class="code_block"><span class="c">%---------------------------------------------------------------------------</span>
</div><div id="l346" class="code_block">
</div><div id="l347" class="code_block"><span class="k">\begin</span><span class="nb">{</span>instance<span class="nb">}</span> [Translation of <span class="nb">{</span><span class="k">\tt</span> allAttr/3<span class="nb">}</span>] <span class="k">\rm</span> 
</div><div id="l348" class="code_block">For each fact of the form <span class="nb">{</span><span class="k">\tt</span> allAttr(Id<span class="s">$</span><span class="nb">_</span><span class="m">1</span><span class="s">$</span>,Rid,Id<span class="s">$</span><span class="nb">_</span><span class="m">2</span><span class="s">$</span>)<span class="nb">}</span> add the instance
</div><div id="l349" class="code_block">axiom: 
</div><div id="l350" class="code_block"><span class="k">\begin</span><span class="nb">{</span>tabbing<span class="nb">}</span>
</div><div id="l351" class="code_block">foo<span class="k">\=</span>foo<span class="k">\=</span>foo<span class="k">\=</span>foo<span class="k">\=</span>foo<span class="k">\=</span>foo<span class="k">\=</span>foooo<span class="k">\=</span>foooooooooooooooo<span class="k">\=\kill</span>
</div><div id="l352" class="code_block"><span class="k">\&gt;</span> <span class="s">$</span><span class="nb"> IdSort</span><span class="o">(</span><span class="nb">Id_</span><span class="m">1</span><span class="o">)</span><span class="nb"> </span><span class="nv">\wedge</span><span class="nb"> IdSort</span><span class="o">(</span><span class="nb">Rid</span><span class="o">)</span><span class="nb"> </span><span class="nv">\wedge</span><span class="nb"> IdSort</span><span class="o">(</span><span class="nb">Id_</span><span class="m">2</span><span class="o">)</span><span class="nb"> </span><span class="nv">\wedge</span><span class="nb"> </span><span class="s">$</span> <span class="k">\\</span>
</div><div id="l353" class="code_block"><span class="k">\&gt;</span> <span class="k">\&gt;</span> <span class="s">$</span><span class="nb"> IsClass</span><span class="o">(</span><span class="nb">Id_</span><span class="m">1</span><span class="o">)</span><span class="nb"> </span><span class="nv">\wedge</span><span class="nb"> IsRel</span><span class="o">(</span><span class="nb">Rid</span><span class="o">)</span><span class="nb"> </span><span class="nv">\wedge</span><span class="nb"> IsClass</span><span class="o">(</span><span class="nb">Id_</span><span class="m">2</span><span class="o">)</span><span class="nb"> </span><span class="nv">\wedge</span><span class="nb"> </span><span class="s">$</span> <span class="k">\\</span>
</div><div id="l354" class="code_block"><span class="k">\&gt;</span> <span class="k">\&gt;</span> <span class="s">$</span><span class="o">(</span><span class="nv">\forall</span><span class="nb"> X, Y</span><span class="o">)</span><span class="nb"> </span><span class="o">[(</span><span class="nb">elt</span><span class="o">(</span><span class="nb">X,Id_</span><span class="m">1</span><span class="o">)</span><span class="nb"> </span><span class="nv">\wedge</span><span class="nb"> rel</span><span class="o">(</span><span class="nb">X,Rid,Y</span><span class="o">))</span>
</div><div id="l355" class="code_block"><span class="nb">					</span><span class="nv">\Rightarrow</span><span class="nb"> elt</span><span class="o">(</span><span class="nb">Y,Id_</span><span class="m">2</span><span class="o">)]</span><span class="nb"> </span><span class="s">$</span>
</div><div id="l356" class="code_block"><span class="k">\end</span><span class="nb">{</span>tabbing<span class="nb">}</span>
</div><div id="l357" class="code_block">denoted as <span class="nb">{</span><span class="k">\tt</span> allAttr(Id<span class="s">$</span><span class="nb">_</span><span class="m">1</span><span class="s">$</span>,Rid,Id<span class="s">$</span><span class="nb">_</span><span class="m">2</span><span class="s">$</span>)<span class="s">$</span><span class="nb">^{</span><span class="nv">\cI</span><span class="nb">}</span><span class="s">$</span><span class="nb">}</span>.
</div><div id="l358" class="code_block"><span class="k">\end</span><span class="nb">{</span>instance<span class="nb">}</span>
</div><div id="l359" class="code_block">
</div><div id="l360" class="code_block"><span class="c">%----------------------</span>
</div><div id="l361" class="code_block"><span class="k">\mycomment</span><span class="nb">{</span>
</div><div id="l362" class="code_block"><span class="k">\begin</span><span class="nb">{</span>tabbing<span class="nb">}</span>
</div><div id="l363" class="code_block">foo<span class="k">\=</span>foo<span class="k">\=</span>foo<span class="k">\=</span>foo<span class="k">\=</span>foo<span class="k">\=</span>foo<span class="k">\=</span>foooo<span class="k">\=</span>foooooooooooooooo<span class="k">\=\kill</span>
</div><div id="l364" class="code_block"><span class="k">\&gt;</span> <span class="s">$</span><span class="nb"> IdSort</span><span class="o">(</span><span class="nb">Id_</span><span class="m">1</span><span class="o">)</span><span class="nb"> </span><span class="nv">\wedge</span><span class="nb"> IdSort</span><span class="o">(</span><span class="nb">Rid</span><span class="o">)</span><span class="nb"> </span><span class="nv">\wedge</span><span class="nb"> IdSort</span><span class="o">(</span><span class="nb">Id_</span><span class="m">2</span><span class="o">)</span><span class="nb"> </span><span class="nv">\wedge</span><span class="nb"> </span><span class="s">$</span> <span class="k">\\</span>
</div><div id="l365" class="code_block"><span class="k">\&gt;</span> <span class="k">\&gt;</span> <span class="s">$</span><span class="nb"> </span><span class="o">(</span><span class="nb">IsClass</span><span class="o">(</span><span class="nb">Id_</span><span class="m">1</span><span class="o">)</span><span class="nb"> </span><span class="nv">\vee</span><span class="nb"> isObj</span><span class="o">(</span><span class="nb">Id_</span><span class="m">1</span><span class="o">))</span><span class="nb"> </span><span class="nv">\wedge</span><span class="nb"> IsRel</span><span class="o">(</span><span class="nb">Rid</span><span class="o">)</span><span class="nb"> </span><span class="nv">\wedge</span><span class="nb"> </span>
</div><div id="l366" class="code_block"><span class="nb">	 </span><span class="o">(</span><span class="nb">IsClass</span><span class="o">(</span><span class="nb">Id_</span><span class="m">2</span><span class="o">)</span><span class="nb"> </span><span class="nv">\vee</span><span class="nb"> isObj</span><span class="o">(</span><span class="nb">Id_</span><span class="m">2</span><span class="o">))</span><span class="nb"> </span><span class="nv">\wedge</span><span class="nb"> </span><span class="s">$</span> <span class="k">\\</span>
</div><div id="l367" class="code_block"><span class="k">\&gt;</span> <span class="k">\&gt;</span> <span class="s">$</span><span class="o">(</span><span class="nv">\forall</span><span class="nb"> X, Y</span><span class="o">)</span><span class="nb"> </span><span class="o">[(</span><span class="nb">elt</span><span class="o">(</span><span class="nb">X,Id_</span><span class="m">1</span><span class="o">)</span><span class="nb"> </span><span class="nv">\wedge</span><span class="nb"> rel</span><span class="o">(</span><span class="nb">X,Rid,Y</span><span class="o">))</span>
</div><div id="l368" class="code_block"><span class="nb">					</span><span class="nv">\Rightarrow</span><span class="nb"> elt</span><span class="o">(</span><span class="nb">Y,Id_</span><span class="m">2</span><span class="o">)]</span><span class="nb"> </span><span class="s">$</span>
</div><div id="l369" class="code_block"><span class="k">\end</span><span class="nb">{</span>tabbing<span class="nb">}</span>
</div><div id="l370" class="code_block"><span class="nb">}</span>
</div><div id="l371" class="code_block"><span class="c">%----------------------</span>
</div><div id="l372" class="code_block"><span class="k">\begin</span><span class="nb">{</span>example<span class="nb">}</span> <span class="k">\rm</span> <span class="k">\label</span><span class="nb">{</span>ex:hasAttr<span class="nb">}</span>
</div><div id="l373" class="code_block">While <span class="nb">{</span><span class="k">\tt</span> allAttr/3<span class="nb">}</span> indicates a typing for relations, it does not
</div><div id="l374" class="code_block">indicate that a relation must exist for elements of a class.  This
</div><div id="l375" class="code_block">statement is made by <span class="k">\pred</span><span class="nb">{</span>hasAttr/3<span class="nb">}</span>.  The relation <span class="nb">{</span><span class="k">\tt</span>
</div><div id="l376" class="code_block">rid(hasPointStyle)<span class="nb">}</span> for the class <span class="nb">{</span><span class="k">\tt</span> cid(absorbableSutures)<span class="nb">}</span> is
</div><div id="l377" class="code_block">taken to be required in this schema.  The facts
</div><div id="l378" class="code_block"><span class="c">%</span>
</div><div id="l379" class="code_block"><span class="nb">{</span><span class="k">\small</span>
</div><div id="l380" class="code_block"><span class="k">\begin</span><span class="nb">{</span>tabbing<span class="nb">}</span>
</div><div id="l381" class="code_block">fooo<span class="k">\=</span>foo<span class="k">\=</span>foo<span class="k">\=</span>foo<span class="k">\=</span>foooooooooooooooooooooooooooo<span class="k">\=</span>ooooooooooooo<span class="k">\=\kill</span>
</div><div id="l382" class="code_block"><span class="k">\&gt;</span> <span class="nb">{</span><span class="k">\tt</span> allAttr(cid(absorbableSutures),rid(hasPointStyle),cid(pointStyle)) <span class="nb">}</span> <span class="k">\\</span>
</div><div id="l383" class="code_block"><span class="k">\&gt;</span> <span class="nb">{</span><span class="k">\tt</span> hasAttr(cid(absorbableSutures),rid(hasPointStyle),cid(pointStyle)) <span class="nb">}</span>
</div><div id="l384" class="code_block"><span class="k">\end</span><span class="nb">{</span>tabbing<span class="nb">}</span>
</div><div id="l385" class="code_block"><span class="nb">}</span>
</div><div id="l386" class="code_block"><span class="c">%</span>
</div><div id="l387" class="code_block"><span class="k">\noindent</span>
</div><div id="l388" class="code_block">indicate not only the range of such relationships, but that such a
</div><div id="l389" class="code_block">relationship must exist.  Indeed, the <span class="nb">{</span><span class="k">\tt</span> hasAttr/3<span class="nb">}</span> fact can be read
</div><div id="l390" class="code_block">as ``all objects in the class <span class="nb">{</span><span class="k">\tt</span> cid(absorbableSutures)<span class="nb">}</span> have a
</div><div id="l391" class="code_block">relation <span class="nb">{</span><span class="k">\tt</span> rid(hasPointStyle)<span class="nb">}</span> to an object in the class <span class="nb">{</span><span class="k">\tt</span>
</div><div id="l392" class="code_block">cid(pointStyle)<span class="nb">}</span>&#39;&#39;.  The facts below also give information about the
</div><div id="l393" class="code_block"><span class="nb">{</span><span class="k">\tt</span> rid(hasPointStyle)<span class="nb">}</span> relation.
</div><div id="l394" class="code_block"><span class="c">%</span>
</div><div id="l395" class="code_block"><span class="nb">{</span><span class="k">\small</span>
</div><div id="l396" class="code_block"><span class="k">\begin</span><span class="nb">{</span>tabbing<span class="nb">}</span>
</div><div id="l397" class="code_block">fooo<span class="k">\=</span>foo<span class="k">\=</span>foo<span class="k">\=</span>foo<span class="k">\=</span>foooooooooooboooooooooooooooo<span class="k">\=</span>ooooooooooooo<span class="k">\=\kill</span>
</div><div id="l398" class="code_block"><span class="k">\&gt;</span> <span class="nb">{</span><span class="k">\tt</span> isa(cid(pointStyle),cid(domainTypes))<span class="nb">}</span> <span class="k">\\</span>
</div><div id="l399" class="code_block"><span class="k">\&gt;</span> <span class="nb">{</span><span class="k">\tt</span> isa(cid(regularCuttingEdge),cid(pointStyle))<span class="nb">}</span> <span class="k">\\</span>
</div><div id="l400" class="code_block"><span class="k">\&gt;</span> <span class="nb">{</span><span class="k">\tt</span> isa(cid(reverseCuttingEdge),cid(pointStyle))<span class="nb">}</span> <span class="k">\\</span>
</div><div id="l401" class="code_block"><span class="k">\&gt;</span> <span class="nb">{</span><span class="k">\tt</span> isa(cid(scalpelPoint),cid(pointStyle))<span class="nb">}</span> <span class="k">\\</span>
</div><div id="l402" class="code_block"><span class="k">\&gt;</span> <span class="nb">{</span><span class="k">\it</span> <span class="k">\%</span> 10 other values.<span class="nb">}</span> <span class="k">\\</span>
</div><div id="l403" class="code_block"><span class="k">\\</span>
</div><div id="l404" class="code_block"><span class="k">\&gt;</span> <span class="nb">{</span><span class="k">\tt</span> hasAttr(oid(sutureU245H),rid(hasPointStyle),cid(regularCuttingEdge))<span class="nb">}</span>
</div><div id="l405" class="code_block"><span class="k">\end</span><span class="nb">{</span>tabbing<span class="nb">}</span>
</div><div id="l406" class="code_block"><span class="nb">}</span> 
</div><div id="l407" class="code_block"><span class="k">\noindent</span>
</div><div id="l408" class="code_block">The last of the above facts can be read as ``the object <span class="nb">{</span><span class="k">\tt</span>
</div><div id="l409" class="code_block">oid(sutureU245H)<span class="nb">}</span> has a <span class="nb">{</span><span class="k">\tt</span> rid(hasPointStyle)<span class="nb">}</span> relation to an object
</div><div id="l410" class="code_block">in the class <span class="nb">{</span><span class="k">\tt</span> cid(pointStyle)<span class="nb">}</span>&#39;&#39;.  
</div><div id="l411" class="code_block"><span class="k">\end</span><span class="nb">{</span>example<span class="nb">}</span>
</div><div id="l412" class="code_block">
</div><div id="l413" class="code_block">Not surprisingly, the definition of <span class="nb">{</span><span class="k">\tt</span> hasAttr/3<span class="nb">}</span> bears some
</div><div id="l414" class="code_block">similarity to that of <span class="nb">{</span><span class="k">\tt</span> allAttr/3<span class="nb">}</span>.
</div><div id="l415" class="code_block">
</div><div id="l416" class="code_block"><span class="k">\begin</span><span class="nb">{</span>instance<span class="nb">}</span> [Translation of <span class="k">\pred</span><span class="nb">{</span>hasAttr/3<span class="nb">}</span>] <span class="k">\rm</span> 
</div><div id="l417" class="code_block">For each fact of the form <span class="nb">{</span><span class="k">\tt</span> hasAttr(Id<span class="s">$</span><span class="nb">_</span><span class="m">1</span><span class="s">$</span>,Rid,Id<span class="s">$</span><span class="nb">_</span><span class="m">2</span><span class="s">$</span>)<span class="nb">}</span> add the instance
</div><div id="l418" class="code_block">axiom: 
</div><div id="l419" class="code_block"><span class="k">\begin</span><span class="nb">{</span>tabbing<span class="nb">}</span>
</div><div id="l420" class="code_block">foo<span class="k">\=</span>foo<span class="k">\=</span>foo<span class="k">\=</span>foo<span class="k">\=</span>foo<span class="k">\=</span>foo<span class="k">\=</span>foooo<span class="k">\=</span>foooooooooooooooo<span class="k">\=\kill</span>
</div><div id="l421" class="code_block"><span class="k">\&gt;</span> <span class="s">$</span><span class="nb"> IdSort</span><span class="o">(</span><span class="nb">Id_</span><span class="m">1</span><span class="o">)</span><span class="nb"> </span><span class="nv">\wedge</span><span class="nb"> IdSort</span><span class="o">(</span><span class="nb">Id_</span><span class="m">2</span><span class="o">)</span><span class="nb"> </span><span class="nv">\wedge</span><span class="nb"> IdSort</span><span class="o">(</span><span class="nb">Id_</span><span class="m">2</span><span class="o">)</span><span class="nb"> </span><span class="nv">\wedge</span><span class="nb"> </span><span class="s">$</span> <span class="k">\\</span>
</div><div id="l422" class="code_block"><span class="k">\&gt;</span> <span class="k">\&gt;</span> <span class="s">$</span><span class="nb"> IsClass</span><span class="o">(</span><span class="nb">Id_</span><span class="m">1</span><span class="o">)</span><span class="nb"> </span><span class="nv">\wedge</span><span class="nb"> IsRel</span><span class="o">(</span><span class="nb">Rid</span><span class="o">)</span><span class="nb"> </span><span class="nv">\wedge</span>
</div><div id="l423" class="code_block"><span class="nb">	 IsClass</span><span class="o">(</span><span class="nb">Id_</span><span class="m">2</span><span class="o">)</span><span class="nb"> </span><span class="nv">\wedge</span><span class="nb"> </span><span class="s">$</span> <span class="k">\\</span>
</div><div id="l424" class="code_block"><span class="k">\&gt;</span> <span class="k">\&gt;</span> <span class="k">\&gt;</span> <span class="s">$</span><span class="nb"> </span><span class="o">(</span><span class="nv">\forall</span><span class="nb"> X</span><span class="o">)</span><span class="nb"> </span><span class="o">[</span><span class="nb">elt</span><span class="o">(</span><span class="nb">X,Id_</span><span class="m">1</span><span class="o">)</span><span class="nb"> </span><span class="nv">\Rightarrow</span><span class="nb"> </span><span class="nv">\exists</span><span class="nb"> Y </span><span class="o">[</span><span class="nb">rel</span><span class="o">(</span><span class="nb">X,Rid,Y</span><span class="o">)</span><span class="nb"> </span>
</div><div id="l425" class="code_block"><span class="nb">					</span><span class="nv">\wedge</span><span class="nb"> elt</span><span class="o">(</span><span class="nb">Y,Id_</span><span class="m">2</span><span class="o">)]]</span><span class="s">$</span>
</div><div id="l426" class="code_block"><span class="k">\end</span><span class="nb">{</span>tabbing<span class="nb">}</span>
</div><div id="l427" class="code_block">denoted as <span class="nb">{</span><span class="k">\tt</span> hasAttr(Id<span class="s">$</span><span class="nb">_</span><span class="m">1</span><span class="s">$</span>,Rid,Id<span class="s">$</span><span class="nb">_</span><span class="m">2</span><span class="s">$</span>)<span class="s">$</span><span class="nb">^{</span><span class="nv">\cI</span><span class="nb">}</span><span class="s">$</span><span class="nb">}</span>.
</div><div id="l428" class="code_block"><span class="k">\end</span><span class="nb">{</span>instance<span class="nb">}</span>
</div><div id="l429" class="code_block">
</div><div id="l430" class="code_block">We next turn to relational axioms that indicate the cardinality of
</div><div id="l431" class="code_block">various relations.
</div><div id="l432" class="code_block">
</div><div id="l433" class="code_block"><span class="k">\begin</span><span class="nb">{</span>example<span class="nb">}</span> <span class="k">\rm</span>  <span class="k">\label</span><span class="nb">{</span>ex:maxAttr<span class="nb">}</span>
</div><div id="l434" class="code_block">For our purposes, an object in <span class="nb">{</span><span class="k">\tt</span> cid(absorbableSutures)<span class="nb">}</span> can be
</div><div id="l435" class="code_block">thought of as consisting of a needle and a thread~<span class="k">\footnote</span><span class="nb">{</span>The thread
</div><div id="l436" class="code_block">is often called a suture.  We are assuming for purposes of
</div><div id="l437" class="code_block">illustration that all sutures are --- in suture-speak --- ``armed&#39;&#39;.<span class="nb">}</span>.  This is
</div><div id="l438" class="code_block">represented by the facts:
</div><div id="l439" class="code_block"><span class="c">%</span>
</div><div id="l440" class="code_block"><span class="nb">{</span><span class="k">\small</span>
</div><div id="l441" class="code_block"><span class="k">\begin</span><span class="nb">{</span>tabbing<span class="nb">}</span>
</div><div id="l442" class="code_block">foo<span class="k">\=</span>foo<span class="k">\=</span>foo<span class="k">\=</span>foo<span class="k">\=</span>foo<span class="k">\=</span>foo<span class="k">\=</span>foooo<span class="k">\=</span>foooooooooooooooo<span class="k">\=\kill</span>
</div><div id="l443" class="code_block"><span class="k">\&gt;</span> <span class="nb">{</span><span class="k">\tt</span> allAttr(cid(absorbableSutures),rid(hasImmedPart),cid(absSutPart)) <span class="nb">}</span> <span class="k">\\</span>
</div><div id="l444" class="code_block"><span class="k">\&gt;</span> <span class="nb">{</span><span class="k">\tt</span> hasAttr(cid(absorbableSutures),rid(hasImmedPart),cid(absSutNeedle)) <span class="nb">}</span> <span class="k">\\</span>
</div><div id="l445" class="code_block"><span class="k">\&gt;</span> <span class="nb">{</span><span class="k">\tt</span> hasAttr(cid(absorbableSutures),rid(hasImmedPart),cid(absSutThread)) <span class="nb">}</span> <span class="k">\\</span>
</div><div id="l446" class="code_block"><span class="k">\\</span>
</div><div id="l447" class="code_block"><span class="k">\&gt;</span> <span class="nb">{</span><span class="k">\tt</span> isa(cid(absSutPart),cid(suturesAndRelatedProducts))<span class="nb">}</span>  <span class="k">\\</span>
</div><div id="l448" class="code_block"><span class="k">\&gt;</span> <span class="k">\&gt;</span> <span class="nb">{</span><span class="k">\tt</span> isa(cid(absSutNeedle),cid(absSutPart)) <span class="nb">}</span> <span class="k">\\</span>
</div><div id="l449" class="code_block"><span class="k">\&gt;</span> <span class="k">\&gt;</span> <span class="nb">{</span><span class="k">\tt</span> isa(cid(absSutSuture),cid(absSutPart)) <span class="nb">}</span> 
</div><div id="l450" class="code_block"><span class="k">\end</span><span class="nb">{</span>tabbing<span class="nb">}</span>
</div><div id="l451" class="code_block"><span class="nb">}</span>
</div><div id="l452" class="code_block"><span class="c">%</span>
</div><div id="l453" class="code_block"><span class="k">\noindent</span>
</div><div id="l454" class="code_block">A needle for an absorbable suture is typically made of a different
</div><div id="l455" class="code_block">material than the thread to which the needle is attached.  Each of
</div><div id="l456" class="code_block">these materials may be important in choosing an absorbable suture, and
</div><div id="l457" class="code_block">each of these materials are unique.  The facts
</div><div id="l458" class="code_block"><span class="c">%</span>
</div><div id="l459" class="code_block"><span class="nb">{</span><span class="k">\small</span>
</div><div id="l460" class="code_block"><span class="k">\begin</span><span class="nb">{</span>tabbing<span class="nb">}</span>
</div><div id="l461" class="code_block">foo<span class="k">\=</span>foo<span class="k">\=</span>foo<span class="k">\=</span>foo<span class="k">\=</span>foo<span class="k">\=</span>foo<span class="k">\=</span>foooo<span class="k">\=</span>foooooooooooooooo<span class="k">\=\kill</span>
</div><div id="l462" class="code_block"><span class="k">\&gt;</span> <span class="nb">{</span><span class="k">\tt</span> hasAttr(cid(absSutPart),rid(hasMaterial),cid(absSutMaterial)) <span class="nb">}</span> <span class="k">\\</span>
</div><div id="l463" class="code_block"><span class="k">\&gt;</span> <span class="nb">{</span><span class="k">\tt</span> maxAttr(cid(absSutPart),rid(hasMaterial),cid(absSutMaterial),1) <span class="nb">}</span> <span class="k">\\</span>
</div><div id="l464" class="code_block"><span class="k">\\</span>
</div><div id="l465" class="code_block"><span class="k">\&gt;</span> <span class="nb">{</span><span class="k">\tt</span> isa(cid(material),cid(domainTypes))<span class="nb">}</span> <span class="k">\\</span>
</div><div id="l466" class="code_block"><span class="k">\&gt;</span> <span class="nb">{</span><span class="k">\tt</span> isa(cid(absSutMaterial),cid(material)<span class="nb">}</span> <span class="k">\\</span>
</div><div id="l467" class="code_block"><span class="k">\&gt;</span> <span class="nb">{</span><span class="k">\tt</span> isa(cid(gut),cid(absSutMaterial))<span class="nb">}</span> <span class="k">\\</span>
</div><div id="l468" class="code_block"><span class="k">\&gt;</span> <span class="nb">{</span><span class="k">\tt</span> isa(cid(polyglyconate),cid(absSutMaterial))<span class="nb">}</span> <span class="k">\\</span>
</div><div id="l469" class="code_block"><span class="k">\&gt;</span> <span class="nb">{</span><span class="k">\tt</span> isa(cid(polyglyconicAcid),cid(absSutMaterial)) <span class="nb">}</span> 
</div><div id="l470" class="code_block"><span class="k">\end</span><span class="nb">{</span>tabbing<span class="nb">}</span>
</div><div id="l471" class="code_block"><span class="nb">}</span>
</div><div id="l472" class="code_block"><span class="c">%</span>
</div><div id="l473" class="code_block"><span class="k">\noindent</span>
</div><div id="l474" class="code_block">indicate that each <span class="nb">{</span><span class="k">\tt</span> cid(absSutPart)<span class="nb">}</span> has a unique material.  The
</div><div id="l475" class="code_block"><span class="nb">{</span><span class="k">\tt</span> maxAttr/4<span class="nb">}</span> fact can be read as ``Each object in the class <span class="nb">{</span><span class="k">\tt</span>
</div><div id="l476" class="code_block">cid(absSutPart)<span class="nb">}</span> has at most 1 <span class="nb">{</span><span class="k">\tt</span> rid(hasMaterial)<span class="nb">}</span> relation to an
</div><div id="l477" class="code_block">object in the class <span class="nb">{</span><span class="k">\tt</span> cid(absSutMaterial)<span class="nb">}</span>&#39;&#39;.
</div><div id="l478" class="code_block"><span class="k">\end</span><span class="nb">{</span>example<span class="nb">}</span>
</div><div id="l479" class="code_block">
</div><div id="l480" class="code_block">In order to define the semantics of <span class="nb">{</span><span class="k">\tt</span> maxAttr/4<span class="nb">}</span>, let
</div><div id="l481" class="code_block"><span class="s">$</span><span class="nv">\exists</span><span class="nb">^{</span><span class="nv">\leq</span><span class="nb"> n}X_m</span><span class="o">[</span><span class="nb"> </span><span class="nv">\phi</span><span class="o">(</span><span class="nb">X,Z</span><span class="o">)]</span><span class="s">$</span> be defined as an abbreviation for
</div><div id="l482" class="code_block">the formula
</div><div id="l483" class="code_block"><span class="sb">\[</span>
</div><div id="l484" class="code_block"><span class="nb">  </span><span class="nv">\exists</span><span class="nb"> x_m,...,</span><span class="nv">\exists</span><span class="nb"> x_{m</span><span class="o">+</span><span class="nb">n} </span><span class="o">[</span><span class="nv">\bigwedge</span><span class="nb">_{m </span><span class="nv">\leq</span><span class="nb"> i </span><span class="nv">\leq</span><span class="nb"> m</span><span class="o">+</span><span class="nb">n} </span>
</div><div id="l485" class="code_block"><span class="nb">	</span><span class="nv">\phi</span><span class="o">(</span><span class="nb">x_i,</span><span class="nv">\overline</span><span class="nb">{z}</span><span class="o">)</span><span class="nb"> </span>
</div><div id="l486" class="code_block"><span class="nb">	</span><span class="nv">\Rightarrow</span><span class="nb"> </span><span class="nv">\bigvee</span><span class="nb">_{m </span><span class="nv">\leq</span><span class="nb"> i &lt; j </span><span class="nv">\leq</span><span class="nb"> m</span><span class="o">+</span><span class="nb">n} x_i  </span><span class="o">=</span><span class="nb"> x_j</span><span class="o">]</span>
</div><div id="l487" class="code_block"><span class="s">\]</span>
</div><div id="l488" class="code_block">i.e., for the formula indicating that there are at most <span class="s">$</span><span class="nb">N</span><span class="s">$</span> non-equal
</div><div id="l489" class="code_block">elements satisfying <span class="s">$</span><span class="nv">\phi</span><span class="o">(</span><span class="nb">x,z</span><span class="o">)</span><span class="s">$</span>.  The abbreviation <span class="s">$</span><span class="nv">\exists</span><span class="nb">^{</span><span class="nv">\geq</span><span class="nb"> N}</span><span class="s">$</span>
</div><div id="l490" class="code_block">is defined similarly to indicate that a formula is satisfied by at
</div><div id="l491" class="code_block">least <span class="s">$</span><span class="nb">N</span><span class="s">$</span> non-equal elements.
</div><div id="l492" class="code_block">
</div><div id="l493" class="code_block"><span class="k">\begin</span><span class="nb">{</span>instance<span class="nb">}</span> [Translation of <span class="nb">{</span><span class="k">\tt</span> maxAttr/4<span class="nb">}</span>] <span class="k">\rm</span> 
</div><div id="l494" class="code_block">For each fact of the form <span class="nb">{</span><span class="k">\tt</span> maxAttr(Id<span class="s">$</span><span class="nb">_</span><span class="m">1</span><span class="s">$</span>,Rid,Id<span class="s">$</span><span class="nb">_</span><span class="m">2</span><span class="s">$</span>,N)<span class="nb">}</span> add the instance
</div><div id="l495" class="code_block">axiom: 
</div><div id="l496" class="code_block"><span class="k">\begin</span><span class="nb">{</span>tabbing<span class="nb">}</span>
</div><div id="l497" class="code_block">foo<span class="k">\=</span>foo<span class="k">\=</span>foo<span class="k">\=</span>foo<span class="k">\=</span>foo<span class="k">\=</span>foo<span class="k">\=</span>foooo<span class="k">\=</span>foooooooooooooooo<span class="k">\=\kill</span>
</div><div id="l498" class="code_block"><span class="k">\&gt;</span> <span class="s">$</span><span class="nb"> IdSort</span><span class="o">(</span><span class="nb">Id_</span><span class="m">1</span><span class="o">)</span><span class="nb"> </span><span class="nv">\wedge</span><span class="nb"> IdSort</span><span class="o">(</span><span class="nb">Id_</span><span class="m">2</span><span class="o">)</span><span class="nb"> </span><span class="nv">\wedge</span><span class="nb"> IdSort</span><span class="o">(</span><span class="nb">Id_</span><span class="m">2</span><span class="o">)</span><span class="nb"> </span><span class="nv">\wedge</span><span class="nb"> </span><span class="s">$</span> <span class="k">\\</span>
</div><div id="l499" class="code_block"><span class="k">\&gt;</span> <span class="k">\&gt;</span> <span class="s">$</span><span class="nb"> IsClass</span><span class="o">(</span><span class="nb">Id_</span><span class="m">1</span><span class="o">)</span><span class="nb"> </span><span class="nv">\wedge</span><span class="nb"> IsRel</span><span class="o">(</span><span class="nb">Rid</span><span class="o">)</span><span class="nb"> </span><span class="nv">\wedge</span>
</div><div id="l500" class="code_block"><span class="nb">	 </span><span class="o">(</span><span class="nb">IsClass</span><span class="o">(</span><span class="nb">Id_</span><span class="m">2</span><span class="o">)</span><span class="nb"> </span><span class="nv">\wedge</span><span class="nb"> </span><span class="s">$</span> <span class="k">\\</span>
</div><div id="l501" class="code_block"><span class="k">\&gt;</span> <span class="k">\&gt;</span> <span class="k">\&gt;</span> <span class="s">$</span><span class="nb"> </span><span class="o">(</span><span class="nv">\forall</span><span class="nb"> X</span><span class="o">)</span><span class="nb"> </span><span class="o">[</span><span class="nb">elt</span><span class="o">(</span><span class="nb">X,Id_</span><span class="m">1</span><span class="o">)</span><span class="nb"> </span><span class="nv">\Rightarrow</span><span class="nb"> </span><span class="nv">\exists</span><span class="nb">^{</span><span class="nv">\leq</span><span class="nb"> N} Y </span><span class="o">[</span><span class="nb">rel</span><span class="o">(</span><span class="nb">X,Rid,Y</span><span class="o">)</span><span class="nb"> </span>
</div><div id="l502" class="code_block"><span class="nb">					</span><span class="nv">\wedge</span><span class="nb"> elt</span><span class="o">(</span><span class="nb">Y,Id_</span><span class="m">2</span><span class="o">)]]</span><span class="s">$</span>
</div><div id="l503" class="code_block"><span class="k">\end</span><span class="nb">{</span>tabbing<span class="nb">}</span>
</div><div id="l504" class="code_block">denoted as <span class="nb">{</span><span class="k">\tt</span> maxAttr(Id<span class="s">$</span><span class="nb">_</span><span class="m">1</span><span class="s">$</span>,Rid,Id<span class="s">$</span><span class="nb">_</span><span class="m">2</span><span class="s">$</span>,N)<span class="s">$</span><span class="nb">^{</span><span class="nv">\cI</span><span class="nb">}</span><span class="s">$</span><span class="nb">}</span>.
</div><div id="l505" class="code_block"><span class="k">\end</span><span class="nb">{</span>instance<span class="nb">}</span>
</div><div id="l506" class="code_block">
</div><div id="l507" class="code_block">A corresponding predicate, <span class="nb">{</span><span class="k">\tt</span> minAttr/4<span class="nb">}</span> is used to indicate a
</div><div id="l508" class="code_block">minimality restriction on a relation.  <span class="nb">{</span><span class="k">\tt</span> minAttr/4<span class="nb">}</span> is defined
</div><div id="l509" class="code_block">similarly to <span class="nb">{</span><span class="k">\tt</span> maxAttr/4<span class="nb">}</span>, but using <span class="s">$</span><span class="nv">\exists</span><span class="nb">^{</span><span class="nv">\geq</span><span class="nb"> N}</span><span class="s">$</span> rather than
</div><div id="l510" class="code_block"><span class="s">$</span><span class="nv">\exists</span><span class="nb">^{</span><span class="nv">\leq</span><span class="nb"> N}</span><span class="s">$</span>.  Indeed, the predicate
</div><div id="l511" class="code_block"><span class="c">%</span>
</div><div id="l512" class="code_block"><span class="nb">{</span><span class="k">\small</span> <span class="nb">{</span><span class="k">\tt</span> 
</div><div id="l513" class="code_block"><span class="k">\begin</span><span class="nb">{</span>tabbing<span class="nb">}</span>
</div><div id="l514" class="code_block">foo<span class="k">\=</span>foo<span class="k">\=</span>foo<span class="k">\=</span>foo<span class="k">\=</span>foo<span class="k">\=</span>foo<span class="k">\=</span>foooo<span class="k">\=</span>foooooooooooooooo<span class="k">\=\kill</span>
</div><div id="l515" class="code_block"><span class="k">\&gt;</span> hasAttr(cid(absSutPart),rid(hasMaterial),cid(absSutMaterial)).
</div><div id="l516" class="code_block"><span class="k">\end</span><span class="nb">{</span>tabbing<span class="nb">}</span>
</div><div id="l517" class="code_block"><span class="nb">}</span> <span class="nb">}</span>
</div><div id="l518" class="code_block"><span class="c">%</span>
</div><div id="l519" class="code_block"><span class="k">\noindent</span>
</div><div id="l520" class="code_block">could be replaced by the predicate
</div><div id="l521" class="code_block"><span class="c">%</span>
</div><div id="l522" class="code_block"><span class="nb">{</span><span class="k">\small</span> <span class="nb">{</span><span class="k">\tt</span> 
</div><div id="l523" class="code_block"><span class="k">\begin</span><span class="nb">{</span>tabbing<span class="nb">}</span>
</div><div id="l524" class="code_block">foo<span class="k">\=</span>foo<span class="k">\=</span>foo<span class="k">\=</span>foo<span class="k">\=</span>foo<span class="k">\=</span>foo<span class="k">\=</span>foooo<span class="k">\=</span>foooooooooooooooo<span class="k">\=\kill</span>
</div><div id="l525" class="code_block"><span class="k">\&gt;</span> minAttr(cid(absSutPart),rid(hasMaterial),cid(absSutMaterial),1).
</div><div id="l526" class="code_block"><span class="k">\end</span><span class="nb">{</span>tabbing<span class="nb">}</span>
</div><div id="l527" class="code_block"><span class="nb">}</span> <span class="nb">}</span>
</div><div id="l528" class="code_block"><span class="c">%</span>
</div><div id="l529" class="code_block"><span class="k">\subsection</span><span class="nb">{</span>Class Relations<span class="nb">}</span>
</div><div id="l530" class="code_block">Each of the predicates discussed so far are inheritable in their first
</div><div id="l531" class="code_block">argument.  For instance, the fact
</div><div id="l532" class="code_block"><span class="nb">{</span><span class="k">\small</span> <span class="nb">{</span><span class="k">\tt</span> 
</div><div id="l533" class="code_block"><span class="k">\begin</span><span class="nb">{</span>tabbing<span class="nb">}</span>
</div><div id="l534" class="code_block">foo<span class="k">\=</span>foo<span class="k">\=</span>foo<span class="k">\=</span>foo<span class="k">\=</span>foo<span class="k">\=</span>foo<span class="k">\=</span>foooo<span class="k">\=</span>foooooooooooooooo<span class="k">\=\kill</span>
</div><div id="l535" class="code_block"><span class="k">\&gt;</span> hasAttr(cid(absSutPart),rid(hasMaterial),cid(absSutMaterial)).
</div><div id="l536" class="code_block"><span class="k">\end</span><span class="nb">{</span>tabbing<span class="nb">}</span>
</div><div id="l537" class="code_block"><span class="nb">}</span> <span class="nb">}</span> 
</div><div id="l538" class="code_block"><span class="c">%</span>
</div><div id="l539" class="code_block"><span class="k">\noindent</span>
</div><div id="l540" class="code_block">implies that every subclass of <span class="nb">{</span><span class="k">\tt</span> cid(absSutures)<span class="nb">}</span> will have a
</div><div id="l541" class="code_block">material in the class <span class="nb">{</span><span class="k">\tt</span> cid(absSutMaterial)<span class="nb">}</span>.  However, classes may
</div><div id="l542" class="code_block">have relations that do <span class="nb">{</span><span class="k">\em</span> not<span class="nb">}</span> hold for their subclasses or members.
</div><div id="l543" class="code_block">For instance, a finite set may have a given cardinality, but its
</div><div id="l544" class="code_block">proper subsets will have a different cardinality.  Such relations are
</div><div id="l545" class="code_block">termed <span class="nb">{</span><span class="k">\em</span> class relations<span class="nb">}</span>.
</div><div id="l546" class="code_block">
</div><div id="l547" class="code_block"><span class="c">%-------------------</span>
</div><div id="l548" class="code_block"><span class="k">\begin</span><span class="nb">{</span>example<span class="nb">}</span> <span class="k">\label</span><span class="nb">{</span>ex:strel<span class="nb">}</span> <span class="k">\rm</span> 
</div><div id="l549" class="code_block">A practical example of a class relation comes from an application that
</div><div id="l550" class="code_block">may be called part equivalency matching.  In this application, the
</div><div id="l551" class="code_block">possible attributes for a class of parts are given various weights.
</div><div id="l552" class="code_block">Two parts match if the sum of the weights of their attributes that
</div><div id="l553" class="code_block">match are above a given threshold.  The weighting for the <span class="nb">{</span><span class="k">\tt</span>
</div><div id="l554" class="code_block">cid(pointStyle)<span class="nb">}</span> of sutures might be given as:
</div><div id="l555" class="code_block"><span class="c">%----------------------------------</span>
</div><div id="l556" class="code_block"><span class="nb">{</span><span class="k">\small</span> 
</div><div id="l557" class="code_block"><span class="nb">{</span><span class="k">\tt</span> 
</div><div id="l558" class="code_block"><span class="k">\begin</span><span class="nb">{</span>tabbing<span class="nb">}</span>
</div><div id="l559" class="code_block">foo<span class="k">\=</span>foo<span class="k">\=</span>foo<span class="k">\=</span>foo<span class="k">\=</span>foo<span class="k">\=</span>foo<span class="k">\=</span>foooo<span class="k">\=</span>foooooooooooooooo<span class="k">\=\kill</span>
</div><div id="l560" class="code_block"><span class="k">\&gt;</span> isa(cid(pointStyleWeight),cid(&#39;CDF Classes&#39;)) <span class="k">\\</span>
</div><div id="l561" class="code_block"><span class="k">\&gt;</span> <span class="k">\&gt;</span> isa(cid(highWeight),cid(pointStyleWeight)) <span class="k">\\</span>
</div><div id="l562" class="code_block"><span class="k">\&gt;</span> <span class="k">\&gt;</span> isa(cid(lowWeight),cid(pointStyleWeight)) <span class="k">\\</span>
</div><div id="l563" class="code_block"><span class="k">\\</span>
</div><div id="l564" class="code_block"><span class="k">\&gt;</span> classHasAttr(cid(sutures),crid(pointStyleWeight),cid(highWeight))
</div><div id="l565" class="code_block"><span class="k">\end</span><span class="nb">{</span>tabbing<span class="nb">}</span>
</div><div id="l566" class="code_block"><span class="nb">}</span> <span class="nb">}</span>
</div><div id="l567" class="code_block"><span class="c">%----------------------------------</span>
</div><div id="l568" class="code_block"><span class="k">\noindent</span>
</div><div id="l569" class="code_block">The <span class="nb">{</span><span class="k">\tt</span> classHasAttr/3<span class="nb">}</span> fact can be read as ``the class <span class="nb">{</span><span class="k">\tt</span>
</div><div id="l570" class="code_block">cid(sutures)<span class="nb">}</span> has a <span class="nb">{</span><span class="k">\tt</span> crid(pointStyleWeight)<span class="nb">}</span> relation to an object
</div><div id="l571" class="code_block">in <span class="nb">{</span><span class="k">\tt</span> cid(highWeight)<span class="nb">}</span>.  Matching weights are made non-inheritable
</div><div id="l572" class="code_block">via <span class="nb">{</span><span class="k">\tt</span> classHasAttr/3<span class="nb">}</span> because a weight may depend on a given
</div><div id="l573" class="code_block">classification of a part.  For instance if a part were classified as a
</div><div id="l574" class="code_block"><span class="nb">{</span><span class="k">\tt</span> cid(nonAbsorbableSuture)<span class="nb">}</span>, its <span class="nb">{</span><span class="k">\tt</span> cid(pointStyle)<span class="nb">}</span> might weigh
</div><div id="l575" class="code_block">less (or more) for determining whether two sutures are equivalent.
</div><div id="l576" class="code_block"><span class="k">\end</span><span class="nb">{</span>example<span class="nb">}</span>
</div><div id="l577" class="code_block"><span class="c">%------------------</span>
</div><div id="l578" class="code_block"><span class="k">\begin</span><span class="nb">{</span>instance<span class="nb">}</span> [Translation of <span class="nb">{</span><span class="k">\tt</span> classHasAttr/3<span class="nb">}</span>] <span class="k">\rm</span> 
</div><div id="l579" class="code_block">For each fact of the form <span class="nb">{</span><span class="k">\tt</span> classHasAttr(Id<span class="s">$</span><span class="nb">_</span><span class="m">1</span><span class="s">$</span>,CRid,Id<span class="s">$</span><span class="nb">_</span><span class="m">2</span><span class="s">$</span>)<span class="nb">}</span> add the
</div><div id="l580" class="code_block">instance axiom: 
</div><div id="l581" class="code_block"><span class="k">\begin</span><span class="nb">{</span>tabbing<span class="nb">}</span>
</div><div id="l582" class="code_block">foo<span class="k">\=</span>foo<span class="k">\=</span>foo<span class="k">\=</span>foo<span class="k">\=</span>foo<span class="k">\=</span>foo<span class="k">\=</span>foooo<span class="k">\=</span>foooooooooooooooo<span class="k">\=\kill</span>
</div><div id="l583" class="code_block"><span class="k">\&gt;</span> <span class="s">$</span><span class="nb"> IdSort</span><span class="o">(</span><span class="nb">Id_</span><span class="m">1</span><span class="o">)</span><span class="nb"> </span><span class="nv">\wedge</span><span class="nb"> IdSort</span><span class="o">(</span><span class="nb">CRid</span><span class="o">)</span><span class="nb"> </span><span class="nv">\wedge</span><span class="nb"> IdSort</span><span class="o">(</span><span class="nb">Id_</span><span class="m">2</span><span class="o">)</span><span class="nb"> </span><span class="nv">\wedge</span><span class="nb"> </span><span class="s">$</span> <span class="k">\\</span>
</div><div id="l584" class="code_block"><span class="k">\&gt;</span> <span class="k">\&gt;</span> <span class="s">$</span><span class="nb"> IsClass</span><span class="o">(</span><span class="nb">Id_</span><span class="m">1</span><span class="o">)</span><span class="nb"> </span><span class="nv">\wedge</span><span class="nb"> IsCrel</span><span class="o">(</span><span class="nb">CRid</span><span class="o">)</span><span class="nb"> </span><span class="nv">\wedge</span><span class="nb"> isClass</span><span class="o">(</span><span class="nb">Id_</span><span class="m">2</span><span class="o">)</span><span class="nb"> </span><span class="nv">\wedge</span><span class="nb"> </span><span class="s">$</span> <span class="k">\\</span>
</div><div id="l585" class="code_block"><span class="k">\&gt;</span> <span class="k">\&gt;</span> <span class="k">\&gt;</span> <span class="s">$</span><span class="nb"> </span><span class="o">(</span><span class="nv">\exists</span><span class="nb"> X</span><span class="o">)</span><span class="nb"> </span><span class="o">[</span><span class="nb">elt</span><span class="o">(</span><span class="nb">X,Id_</span><span class="m">2</span><span class="o">)</span><span class="nb"> </span><span class="nv">\wedge</span><span class="nb"> crel</span><span class="o">(</span><span class="nb">Id_</span><span class="m">1</span><span class="nb">,CRid,X</span><span class="o">)]</span><span class="nb"> </span><span class="s">$</span>
</div><div id="l586" class="code_block"><span class="k">\end</span><span class="nb">{</span>tabbing<span class="nb">}</span>
</div><div id="l587" class="code_block">denoted as <span class="nb">{</span><span class="k">\tt</span> classHasAttr(Id<span class="s">$</span><span class="nb">_</span><span class="m">1</span><span class="s">$</span>,Rid,Id<span class="s">$</span><span class="nb">_</span><span class="m">2</span><span class="s">$</span>)<span class="s">$</span><span class="nb">^{</span><span class="nv">\cI</span><span class="nb">}</span><span class="s">$</span><span class="nb">}</span>.
</div><div id="l588" class="code_block"><span class="k">\end</span><span class="nb">{</span>instance<span class="nb">}</span>
</div><div id="l589" class="code_block"><span class="c">%----------</span>
</div></pre></div></td></tr></table></div>

      
    </div>
  

                    </div>
                    
                    
                </div>
            



        </div>
    </section>
      

    <footer class="sandiego">
    <div class="as-row">
        <div class="footer-wrapper">
            <nav aria-label="Site Links" role="navigation">
                <section>
                    <div class="as-h2">SourceForge</div>
                    <ul>
                    
                        <li><a href="/create/" title="Create a Project">Create a Project</a></li>
                    
                        <li><a href="/directory/" title="Open Source Software Directory">Open Source Software</a></li>
                        <li><a href="/software/" title="Business Software Directory">Business Software</a></li>
                        
                            
                            <li><a href="/top" title="Top Open Source Projects">Top Downloaded Projects</a></li>
                        

                    </ul>
                </section>
            </nav>
            <nav aria-label="Company Links" role="navigation">
                <section>
                    <div class="as-h2">Company</div>
                    <ul>
                        <li><a href="/about">About</a></li>
                        <li><a href="/about/leadership" title="Open Source Software Directory">Team</a></li>
                        <li class="h-card">
                            <address>
                                <span class="p-name p-org">SourceForge Headquarters</span><br>
                                <span class="p-street-address">225 Broadway Suite 1600</span><br>
                                <span class="p-locality">San Diego, CA <span class="p-postal-code">92101</span></span><br>
                                <span class="p-tel">+1 (858) 422-6466</span><br>
                            </address>
                        </li>
                        <li id="social">
                            
<span></span>
<a href="https://twitter.com/sourceforge" class="twitter" rel="nofollow" target="_blank" title="SourceForge on X">


<svg  class="vertical-icon-fix" data-name="twitter" viewBox="0 0 1792 1792" xmlns="http://www.w3.org/2000/svg"><path d="M1684 408q-67 98-162 167 1 14 1 42 0 130-38 259.5t-115.5 248.5-184.5 210.5-258 146-323 54.5q-271 0-496-145 35 4 78 4 225 0 401-138-105-2-188-64.5t-114-159.5q33 5 61 5 43 0 85-11-112-23-185.5-111.5t-73.5-205.5v-4q68 38 146 41-66-44-105-115t-39-154q0-88 44-163 121 149 294.5 238.5t371.5 99.5q-8-38-8-74 0-134 94.5-228.5t228.5-94.5q140 0 236 102 109-21 205-78-37 115-142 178 93-10 186-50z"/></svg></a>
<a href="https://fosstodon.org/@sourceforge" rel="me nofollow" target="_blank" title="SourceForge on Mastodon">


<svg  class="vertical-icon-fix" data-name="mastodon" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 448 512"><path d="M433 179.11c0-97.2-63.71-125.7-63.71-125.7-62.52-28.7-228.56-28.4-290.48 0 0 0-63.72 28.5-63.72 125.7 0 115.7-6.6 259.4 105.63 289.1 40.51 10.7 75.32 13 103.33 11.4 50.81-2.8 79.32-18.1 79.32-18.1l-1.7-36.9s-36.31 11.4-77.12 10.1c-40.41-1.4-83-4.4-89.63-54a102.54 102.54 0 0 1-.9-13.9c85.63 20.9 158.65 9.1 178.75 6.7 56.12-6.7 105-41.3 111.23-72.9 9.8-49.8 9-121.5 9-121.5zm-75.12 125.2h-46.63v-114.2c0-49.7-64-51.6-64 6.9v62.5h-46.33V197c0-58.5-64-56.6-64-6.9v114.2H90.19c0-122.1-5.2-147.9 18.41-175 25.9-28.9 79.82-30.8 103.83 6.1l11.6 19.5 11.6-19.5c24.11-37.1 78.12-34.8 103.83-6.1 23.71 27.3 18.4 53 18.4 175z"/></svg></a>
<a href="https://www.facebook.com/sourceforgenet/" class="facebook" rel="nofollow" target="_blank" title="SourceForge on Facebook">


<svg  data-name="facebook" viewBox="0 0 1792 1792" xmlns="http://www.w3.org/2000/svg"><path d="M1343 12v264h-157q-86 0-116 36t-30 108v189h293l-39 296h-254v759h-306v-759h-255v-296h255v-218q0-186 104-288.5t277-102.5q147 0 228 12z"/></svg></a>
<a href="https://www.linkedin.com/company/sourceforge.net" class="linkedin" rel="nofollow" target="_blank" title="SourceForge on LinkedIn">


<svg  data-name="linkedin" viewBox="0 0 1792 1792" xmlns="http://www.w3.org/2000/svg"><path d="M477 625v991h-330v-991h330zm21-306q1 73-50.5 122t-135.5 49h-2q-82 0-132-49t-50-122q0-74 51.5-122.5t134.5-48.5 133 48.5 51 122.5zm1166 729v568h-329v-530q0-105-40.5-164.5t-126.5-59.5q-63 0-105.5 34.5t-63.5 85.5q-11 30-11 81v553h-329q2-399 2-647t-1-296l-1-48h329v144h-2q20-32 41-56t56.5-52 87-43.5 114.5-15.5q171 0 275 113.5t104 332.5z"/></svg></a>

<a href="/user/newsletters" rel=nofollow class="newsletter" title="Subscribe to our newsletter">


<svg  class="vertical-icon-fix" data-name="mmSF_11mail" xmlns="http://www.w3.org/2000/svg" x="0px" y="0px" viewBox="0 0 42 42" enable-background="new 0 0 42 42" xml:space="preserve"><path fill="#FFFFFF" d="M0,6v30h42V6H0z M24.2,21.2c-0.8,0.8-2.3,2-3.2,2c-0.9,0-2.4-1.2-3.2-2L5.8,9.7h30.3L24.2,21.2z M13.7,21l-9.9,9.4V11.6L13.7,21z M15.7,23L15.7,23c0.5,0.5,2.9,2.9,5.3,2.9c2.4,0,4.8-2.4,5.2-2.8l0.1-0.1l9.8,9.4H5.8L15.7,23z M28.3,21l9.9-9.5v18.9L28.3,21z"/></svg></a>

<span></span>
                        </li>
                    </ul>
                </section>
            </nav>
             <nav aria-label="Resources Links" role="navigation">
                <section>
                    <div class="as-h2">Resources</div>
                    <ul>
                        
                            <li><a href="/support" title="Support Section">Support</a></li>
                            <li><a href="/p/forge/documentation/Docs%20Home/" title="Site Documentation">Site Documentation</a></li>
                        
                        <li><a href="https://fosstodon.org/@sourceforgestatus" title="Site Status" rel="me nofollow">Site Status</a></li>
                        <li><a href="/reviews" title="Reviews" rel="me nofollow">SourceForge Reviews</a></li>

                    </ul>
                </section>
            </nav>
            <section class="footer-logo">
                <a href="/" title="Home" class="sf-logo">
                    
    
    <img src="https://a.fsdn.com/con/images/sandiego/sf-logo-full.svg"  alt="SourceForge logo" class="sf-logo-full"/>
                </a>
            </section>
        </div>
    </div>
    <section class="l-nav-bottom">
        <nav class="row">
            
                
            
        
    <div class="columns small-12 large-6 copyright-notice">
        &copy; 2025 Slashdot Media. All Rights Reserved.
    </div>
    <div class="columns large-6 links links-right">
        

    
    

    <a href="https://slashdotmedia.com/terms-of-use" target="_blank" title="Terms" rel="nofollow">Terms</a>
    <a href="https://slashdotmedia.com/privacy-statement/" target="_blank" title="Privacy" rel="nofollow">Privacy</a>

    
    
    

    
        <a href="https://slashdotmedia.com/opt-out-choices/" target="_blank" title="Opt Out" rel="nofollow">Opt Out</a>
    

    
        <a href="https://slashdotmedia.com/contact/" target="_blank" title="Advertise" rel="nofollow">Advertise</a>
    
    </div>
        </nav>
    </section>

    </footer>
    





    
    
    <div id="newsletter-floating" class="sandiego newsletter-floating-new">
        <a class="close-button btn-closer" data-close aria-label="Close modal">
            <span aria-hidden="true">&times;</span>
        </a>
        <div class="as-h2">Want the latest updates on software, tech news, and AI?</div>
        <div class="row">
            <div class="column small-2 medium-2 large-2" style="padding-left: 0px; padding-right: 0px;"><img src="/images/sandiego/sf_email_icon.svg"  alt="" /></div>
            <div class="as-h3 column small-10 medium-10 large-10">Get latest updates about software, tech news, and AI from SourceForge directly in your inbox once a month.</div>
        </div>
        <form action="/user/newsletters/" method="post">
            <div class="row">
              <input class="newsletter-input" type="email" placeholder="Enter your email address" autocomplete="off" name="newsletter_email"
                     value="">
              <input type="hidden" name="source" value="floating">
              <button type="submit" class="button blue newsletter-button">Submit</button>
            </div>
        </form>
    </div>
    






</div>


<div id="messages">
    
</div>


    <!-- ew:body_js -->


    <script type="text/javascript" src="https://a.fsdn.com/allura/nf/1752257864/_ew_/_slim/js?href=allura%2Fjs%2Fjquery.notify.js%3Ballura%2Fjs%2Fjquery.tooltipster.js%3Ballura%2Fjs%2Fallura-base.js%3Ballura%2Fjs%2Fchecklist.js%3Ballura%2Fjs%2Fadmin_modal.js%3Bjs%2Fjquery.lightbox_me.js%3Ballura%2Fjs%2Fmemorable.js%3Btheme%2Fsftheme%2Fjs%2Fsftheme%2Fshared.js%3Btheme%2Fsftheme%2Fjs%2Fsftheme%2Fsticky.js%3Btheme%2Fsftheme%2Fjs%2Fsftheme%2Fsandiego%2Fchrome.js%3Btheme%2Fsftheme%2Fjs%2Fsftheme%2Fsandiego%2Futilities.js%3Btheme%2Fsftheme%2Fjs%2Fsftheme%2Fvendor%2Ffoundation6%2Fplugins%2Ffoundation.core.min.js%3Btheme%2Fsftheme%2Fjs%2Fsftheme%2Fvendor%2Ffoundation6%2Fplugins%2Ffoundation.util.mediaQuery.min.js%3Btheme%2Fsftheme%2Fjs%2Fsftheme%2Fvendor%2Ffoundation6%2Fplugins%2Ffoundation.util.keyboard.min.js%3Btheme%2Fsftheme%2Fjs%2Fsftheme%2Fvendor%2Ffoundation6%2Fplugins%2Ffoundation.util.box.min.js%3Btheme%2Fsftheme%2Fjs%2Fsftheme%2Fvendor%2Ffoundation6%2Fplugins%2Ffoundation.util.nest.min.js%3Btheme%2Fsftheme%2Fjs%2Fsftheme%2Fvendor%2Ffoundation6%2Fplugins%2Ffoundation.dropdownMenu.min.js%3Btheme%2Fsftheme%2Fjs%2Fsftheme%2Fvendor%2Ffoundation6%2Fplugins%2Ffoundation.offcanvas.min.js%3Btheme%2Fsftheme%2Fjs%2Fsftheme%2Fvendor%2Ffoundation6%2Fplugins%2Ffoundation.toggler.min.js%3Btheme%2Fsftheme%2Fjs%2Fsftheme%2Fsandiego%2Fproject.js%3Btheme%2Fsftheme%2Fjs%2Fsftheme%2Fvendor%2Fjquery.typeahead.min.js%3Ballura%2Fjs%2Fmaximize-content.js"></script>

    
<!-- /ew:body_js -->



    <!-- ew:body_js_tail -->


    <script type="text/javascript" src="https://a.fsdn.com/allura/nf/1752257864/_ew_/_slim/js?href=theme%2Fsftheme%2Fjs%2Fsftheme%2Fvendor%2Ffoundation6%2Fplugins%2Ffoundation.util.triggers.min.js%3Btheme%2Fsftheme%2Fjs%2Fsftheme%2Fvendor%2Ffoundation6%2Fplugins%2Ffoundation.reveal.min.js"></script>

    
<!-- /ew:body_js_tail -->




<script type="text/javascript">(function() {
  $('#access_urls .btn').click(function(evt){
    evt.preventDefault();
    var parent = $(this).parents('.btn-bar');
    var checkout_cmd = $(this).attr('data-url');
    $(parent).find('input').val(checkout_cmd);
    $(parent).find('span').text($(this).attr('title')+' access');
    $(this).parent().children('.btn').removeClass('active');
    $(this).addClass('active');
    if (checkout_cmd.indexOf(' http://') !== -1 || checkout_cmd.indexOf(' https://') !== -1 ) {
      $('#http-2fa-msg').show();
    } else {
      $('#http-2fa-msg').hide();
    }
  });
  $('#access_urls .btn').first().click();

  
  var repo_status = document.getElementById('repo_status');
  // The repo_status div will only be present if repo.status != 'ready'
  if (repo_status) {
    $('.spinner').show()
    var delay = 500;
    function check_status() {
        $.get('/p/xsb/src/status', function(data) {
            if (data.status === 'ready') {
                $('.spinner').hide()
                $('#repo_status h2').html('Repo status: ready. <a href="." rel="nofollow">Click here to refresh this page.</a>');
            }
            else {
                $('#repo_status h2 span').html(data.status);
                if (delay < 60000){
                    delay = delay * 2;
                }
                window.setTimeout(check_status, delay);
            }
        });
    }
    var status_checker = window.setTimeout(check_status, delay);
    
  }
}());
</script>

<script type="text/javascript">(function() {
  $(window).on('hashchange', function(e) {
    var hash = window.location.hash.substring(1);
    if ('originalEvent' in e && 'oldURL' in e.originalEvent) {
      $('#' + e.originalEvent.oldURL.split('#')[1]).css('background-color', 'transparent');
    }
    if (hash !== '' && hash.substring(0, 1) === 'l' && !isNaN(hash.substring(1))) {
      $('#' + hash).css('background-color', '#ffff99');
    }
  }).trigger('hashchange');

  var clicks = 0;
  $('.code_block').each(function(index, element) {
    $(element).on('click', function() {
      // Trick to ignore double and triple clicks
      clicks++;
      if (clicks === 1) {
        setTimeout(function() {
          if (clicks === 1) {
            var hash = window.location.hash.substring(1);
            if (hash !== '' && hash.substring(0, 1) === 'l' && !isNaN(hash.substring(1))) {
              $('#' + hash).css('background-color', 'transparent');
            }
            $(element).css('background-color', '#ffff99');
            window.history.pushState({}, document.title, window.location.pathname + window.location.search + '#' + $(element).attr('id'));
          }
          clicks = 0;
        }, 500);
      }
    });
  });
}());
</script>


    


    
    
    <noscript><p><img src="https://analytics.slashdotmedia.com/index.php?idsite=39" style="border:0;position:absolute;top:0;" alt="" /></p></noscript>
    
    <script>
        $(document).foundation();
    </script>
    








    

<script>
    (function () {
        function load_bombora() {
            /*global _ml:true, window */
            _ml = window._ml || {};
            
            _ml.eid = '771';
            _ml.fp = '739ccba1-630a-4aff-90a1-2ab54b15236e';  
            var s = document.getElementsByTagName('script')[0], cd = new Date(), mltag = document.createElement('script');
            mltag.type = 'text/javascript';
            mltag.async = true;
            mltag.defer = true;
            mltag.src = '//ml314.com/tag.aspx?' + cd.getDate() + cd.getMonth() + cd.getFullYear();
            s.parentNode.insertBefore(mltag, s);
        }
        bizx.cmp.ifConsent({ purposes: 'all', vendors: 'bombora'},function() {
                    if (document.hidden) {
                        document.addEventListener('visibilitychange', function () {
                            if (!document.hidden) {
                                load_bombora();
                            }
                        }, {once: true});
                    } else {
                        load_bombora();
                    }
                });
    })();
</script>




<script>
    $(document).ready(function () {
        $(".tooltip, .m-tooltip").each(function(){
            let $el = $(this);
            $el.tooltipster({
                animation: 'fade',
                delay: 200,
                theme: 'tooltipster-light',
                trigger: 'hover',
                position: $el.data('position') || 'right',
                iconCloning: false,
                maxWidth: 300,
                contentAsHTML: true,
                interactive: true
            }).focus(function () {
                $(this).tooltipster('show');
            }).blur(function () {
                $(this).tooltipster('hide');
            });
        });
    });
</script>

    



<script type="text/javascript">
    bizx.cmp.ifConsent({ purposes: ['storage', 'measurement'], vendors: 'google-analytics'}, function () {
        /* jshint ignore:start */
        (function(i,s,o,g,r,a,m){i['GoogleAnalyticsObject']=r;i[r]=i[r]||function(){
                (i[r].q=i[r].q||[]).push(arguments)},i[r].l=1*new Date();a=s.createElement(o),
            m=s.getElementsByTagName(o)[0];a.async=1;a.src=g;m.parentNode.insertBefore(a,m)
        })(window,document,'script','//www.google-analytics.com/analytics.js','ga');
        /* jshint ignore:end */

        window.dataLayer = window.dataLayer || [];
        function gtag(){ window.dataLayer.push(arguments); }
        window.gtag = window.gtag || gtag;
        bizx.cmp.embedScript("https://www.googletagmanager.com/gtag/js");
        gtag('js', new Date());
        gtag('set', {
            'page_location': 'https://sourceforge.net/p/xsb/src/4265/tree/trunk/XSB/packages/altCDF/doc/semantics.tex',
        });
    });
</script>










<script>
        function gam(id){
            bizx.cmp.ifConsent({ purposes: 'all' }, function () {
                bizx.cmp.embedScript(`https://pagead2.googlesyndication.com/pagead/js/pcd.js?${id}`,
                    true,  // async
                    'head', // location
                    null, // callback
                    false, // defer
                    {id: `google-pcd-tag-${id}`, 'data-audience-pixel': 'dc_iu=/41014381/DFPAudiencePixel;dc_seg=' + id});
            });
        }

        
            gam("8901705213");gam("8901712041");</script>

<script type="text/javascript">
    /*global ga, gtag */

    
        
        bizx.cmp.ifConsent({ purposes: ['storage', 'measurement'], vendors: 'google-analytics'}, function () {
            gtag('config', 'G-1H226E4E4L', {
                
                send_page_view: false,
                'SF_Page_Type': '', 
                user_properties: {
                    'SF_Logged_in': 'Logged Out', 
                },
                'devicePixelRatio': SF.devicePixelRatio, 
            });
            gtag('event', 'page_view', { send_to: 'G-1H226E4E4L' });
        });
    
    
    
        
</script>

</body>
</html>