<!DOCTYPE html>
<html lang="en">
<head>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<meta name="generator" content="AsciiDoc 8.6.9">
<title>Fixpoints</title>
<link rel="stylesheet" href="./asciidoc.css" type="text/css">
<link rel="stylesheet" href="./pygments.css" type="text/css">


<script type="text/javascript" src="./asciidoc.js"></script>
<script type="text/javascript">
/*<![CDATA[*/
asciidoc.install();
/*]]>*/
</script>
<link rel="stylesheet" href="./mlton.css" type="text/css"/>
</head>
<body class="article">
<div id="banner">
<div id="banner-home">
<a href="./Home">MLton 20130715</a>
</div>
</div>
<div id="header">
<h1>Fixpoints</h1>
</div>
<div id="content">
<div id="preamble">
<div class="sectionbody">
<div class="paragraph"><p>This page discusses a framework that makes it possible to compute
fixpoints over arbitrary products of abstract types.  The code is from
an Extended Basis library
(<a href="https://github.com/MLton/mltonlib/blob/master/com/ssh/extended-basis/unstable/README"><span class="monospaced">README</span></a>).</p></div>
<div class="paragraph"><p>First the signature of the framework
(<a href="https://github.com/MLton/mltonlib/blob/master/com/ssh/extended-basis/unstable/public/generic/tie.sig"><span class="monospaced">tie.sig</span></a>):</p></div>
<div class="listingblock">
<div class="content"><div class="highlight"><pre><span></span>
</pre></div></div></div>
<div class="paragraph"><p><span class="monospaced">fix</span> is a <a href="TypeIndexedValues">type-indexed</a> function.  The type-index
parameter to <span class="monospaced">fix</span> is called a "witness".  To compute fixpoints over
products, one uses the <span class="monospaced">*&grave;</span> operator to combine witnesses.  To provide
a fixpoint combinator for an abstract type, one implements a witness
providing a thunk whose instantiation allocates a fresh, mutable proxy
and a procedure for updating the proxy with the solution.  Naturally
this means that not all possible ways of computing a fixpoint of a
particular type are possible under the framework.  The <span class="monospaced">pure</span>
combinator is a generalization of <span class="monospaced">tier</span>.  The <span class="monospaced">iso</span> combinator is
provided for reusing existing witnesses.</p></div>
<div class="paragraph"><p>Note that instead of using an infix operator, we could alternatively
employ an interface using <a href="Fold">Fold</a>.  Also, witnesses are eta-expanded
to work around the <a href="ValueRestriction">value restriction</a>, while
maintaining abstraction.</p></div>
<div class="paragraph"><p>Here is the implementation
(<a href="https://github.com/MLton/mltonlib/blob/master/com/ssh/extended-basis/unstable/detail/generic/tie.sml"><span class="monospaced">tie.sml</span></a>):</p></div>
<div class="listingblock">
<div class="content"><div class="highlight"><pre><span></span>
</pre></div></div></div>
<div class="paragraph"><p>Let&#8217;s then take a look at a couple of additional examples.</p></div>
<div class="paragraph"><p>Here is a naive implementation of lazy promises:</p></div>
<div class="listingblock">
<div class="content"><div class="highlight"><pre><span></span><span class="kr">structure</span> <span class="nn">Promise</span> <span class="p">:&gt;</span> <span class="kr">sig</span>
   <span class="kr">type</span> <span class="nd">&#39;a</span> <span class="kt">t</span>
   <span class="kr">val</span> <span class="nv">lazy</span> <span class="p">:</span> <span class="nd">&#39;a</span> <span class="nn">Thunk</span><span class="p">.</span><span class="n">t</span> <span class="p">-&gt;</span> <span class="nd">&#39;a</span> <span class="n">t</span>
   <span class="kr">val</span> <span class="nv">force</span> <span class="p">:</span> <span class="nd">&#39;a</span> <span class="n">t</span> <span class="p">-&gt;</span> <span class="nd">&#39;a</span>
   <span class="kr">val</span> <span class="nv">Y</span> <span class="p">:</span> <span class="nd">&#39;a</span> <span class="n">t</span> <span class="nn">Tie</span><span class="p">.</span><span class="n">t</span>
<span class="kr">end</span> <span class="p">=</span> <span class="kr">struct</span>
   <span class="kr">datatype</span> <span class="nd">&#39;a</span> <span class="kt">t&#39;</span> <span class="p">=</span>
      <span class="nc">EXN</span> <span class="kr">of</span> <span class="n">exn</span>
    <span class="p">|</span> <span class="nc">THUNK</span> <span class="kr">of</span> <span class="nd">&#39;a</span> <span class="nn">Thunk</span><span class="p">.</span><span class="n">t</span>
    <span class="p">|</span> <span class="nc">VALUE</span> <span class="kr">of</span> <span class="nd">&#39;a</span>
   <span class="kr">type</span> <span class="nd">&#39;a</span> <span class="kt">t</span> <span class="p">=</span> <span class="nd">&#39;a</span> <span class="n">t&#39;</span> <span class="nn">Ref</span><span class="p">.</span><span class="n">t</span>
   <span class="kr">fun</span> <span class="nf">lazy</span> <span class="n">f</span> <span class="p">=</span> <span class="n">ref</span> <span class="p">(</span><span class="n">THUNK</span> <span class="n">f</span><span class="p">)</span>
   <span class="kr">fun</span> <span class="nf">force</span> <span class="n">t</span> <span class="p">=</span>
      <span class="kr">case</span> <span class="n">!t</span>
       <span class="kr">of</span> <span class="n">EXN</span> <span class="n">e</span>   <span class="p">=&gt;</span> <span class="kr">raise</span> <span class="n">e</span>
        <span class="n">|</span> <span class="n">THUNK</span> <span class="n">f</span> <span class="p">=&gt;</span> <span class="p">(</span><span class="n">t</span> <span class="n">:=</span> <span class="n">VALUE</span> <span class="p">(</span><span class="n">f</span> <span class="p">())</span> <span class="kr">handle</span> <span class="n">e</span> <span class="p">=&gt;</span> <span class="n">t</span> <span class="n">:=</span> <span class="n">EXN</span> <span class="n">e</span> <span class="p">;</span> <span class="n">force</span> <span class="n">t</span><span class="p">)</span>
        <span class="n">|</span> <span class="n">VALUE</span> <span class="n">v</span> <span class="p">=&gt;</span> <span class="n">v</span>
   <span class="kr">fun</span> <span class="nf">Y</span> <span class="n">?</span> <span class="p">=</span> <span class="nn">Tie</span><span class="p">.</span><span class="n">tier</span> <span class="p">(</span><span class="kr">fn</span> <span class="p">()</span> <span class="p">=&gt;</span> <span class="kr">let</span>
                             <span class="kr">val</span> <span class="nv">r</span> <span class="p">=</span> <span class="n">lazy</span> <span class="p">(</span><span class="n">raising</span> <span class="nn">Fix</span><span class="p">.</span><span class="n">Fix</span><span class="p">)</span>
                          <span class="kr">in</span>
                             <span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">r</span> <span class="n">&lt;\</span> <span class="kr">op</span> <span class="n">:=</span> <span class="n">o</span> <span class="n">!</span><span class="p">)</span>
                          <span class="kr">end</span><span class="p">)</span> <span class="n">?</span>
<span class="kr">end</span>
</pre></div></div></div>
<div class="paragraph"><p>An example use of our naive lazy promises is to implement equally naive
lazy streams:</p></div>
<div class="listingblock">
<div class="content"><div class="highlight"><pre><span></span><span class="kr">structure</span> <span class="nn">Stream</span> <span class="p">:&gt;</span> <span class="kr">sig</span>
   <span class="kr">type</span> <span class="nd">&#39;a</span> <span class="kt">t</span>
   <span class="kr">val</span> <span class="nv">cons</span> <span class="p">:</span> <span class="nd">&#39;a</span> <span class="n">*</span> <span class="nd">&#39;a</span> <span class="n">t</span> <span class="p">-&gt;</span> <span class="nd">&#39;a</span> <span class="n">t</span>
   <span class="kr">val</span> <span class="nv">get</span> <span class="p">:</span> <span class="nd">&#39;a</span> <span class="n">t</span> <span class="p">-&gt;</span> <span class="p">(</span><span class="nd">&#39;a</span> <span class="n">*</span> <span class="nd">&#39;a</span> <span class="n">t</span><span class="p">)</span> <span class="nn">Option</span><span class="p">.</span><span class="n">t</span>
   <span class="kr">val</span> <span class="nv">Y</span> <span class="p">:</span> <span class="nd">&#39;a</span> <span class="n">t</span> <span class="nn">Tie</span><span class="p">.</span><span class="n">t</span>
<span class="kr">end</span> <span class="p">=</span> <span class="kr">struct</span>
   <span class="kr">datatype</span> <span class="nd">&#39;a</span> <span class="kt">t</span> <span class="p">=</span> <span class="nc">IN</span> <span class="kr">of</span> <span class="p">(</span><span class="nd">&#39;a</span> <span class="n">*</span> <span class="nd">&#39;a</span> <span class="n">t</span><span class="p">)</span> <span class="nn">Option</span><span class="p">.</span><span class="n">t</span> <span class="nn">Promise</span><span class="p">.</span><span class="n">t</span>
   <span class="kr">fun</span> <span class="nf">cons</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">xs</span><span class="p">)</span> <span class="p">=</span> <span class="n">IN</span> <span class="p">(</span><span class="nn">Promise</span><span class="p">.</span><span class="n">lazy</span> <span class="p">(</span><span class="kr">fn</span> <span class="p">()</span> <span class="p">=&gt;</span> <span class="n">SOME</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">xs</span><span class="p">)))</span>
   <span class="kr">fun</span> <span class="nf">get</span> <span class="p">(</span><span class="n">IN</span> <span class="n">p</span><span class="p">)</span> <span class="p">=</span> <span class="nn">Promise</span><span class="p">.</span><span class="n">force</span> <span class="n">p</span>
   <span class="kr">fun</span> <span class="nf">Y</span> <span class="n">?</span> <span class="p">=</span> <span class="nn">Tie</span><span class="p">.</span><span class="n">iso</span> <span class="nn">Promise</span><span class="p">.</span><span class="n">Y</span> <span class="p">(</span><span class="kr">fn</span> <span class="n">IN</span> <span class="n">p</span> <span class="p">=&gt;</span> <span class="n">p</span><span class="p">,</span> <span class="n">IN</span><span class="p">)</span> <span class="n">?</span>
<span class="kr">end</span>
</pre></div></div></div>
<div class="paragraph"><p>Note that above we make use of the <span class="monospaced">iso</span> combinator.  Here is a finite
representation of an infinite stream of ones:</p></div>
<div class="listingblock">
<div class="content"><div class="highlight"><pre><span></span><span class="kr">val</span> <span class="nv">ones</span> <span class="p">=</span> <span class="kr">let</span>
   <span class="kr">open</span> <span class="nn">Tie</span> <span class="nn">Stream</span>
<span class="kr">in</span>
   <span class="n">fix</span> <span class="n">Y</span> <span class="p">(</span><span class="kr">fn</span> <span class="n">ones</span> <span class="p">=&gt;</span> <span class="n">cons</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">ones</span><span class="p">))</span>
<span class="kr">end</span>
</pre></div></div></div>
</div>
</div>
</div>
<div id="footnotes"><hr></div>
<div id="footer">
<div id="footer-text">
</div>
<div id="footer-badges">
</div>
</div>
</body>
</html>
