doc/guide/localhost/Fixpoints

   1 <!DOCTYPE html>
   2 <html lang="en">
   3 <head>
   4 <meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
   5 <meta name="generator" content="AsciiDoc 8.6.9">
   6 <title>Fixpoints</title>
   7 <link rel="stylesheet" href="./asciidoc.css" type="text/css">
   8 <link rel="stylesheet" href="./pygments.css" type="text/css">
   9
  10
  11 <script type="text/javascript" src="./asciidoc.js"></script>
  12 <script type="text/javascript">
  13 /*<![CDATA[*/
  14 asciidoc.install();
  15 /*]]>*/
  16 </script>
  17 <link rel="stylesheet" href="./mlton.css" type="text/css">
  18 </head>
  19 <body class="article">
  20 <div id="banner">
  21 <div id="banner-home">
  22 <a href="./Home">MLton 20180207</a>
  23 </div>
  24 </div>
  25 <div id="header">
  26 <h1>Fixpoints</h1>
  27 </div>
  28 <div id="content">
  29 <div id="preamble">
  30 <div class="sectionbody">
  31 <div class="paragraph"><p>This page discusses a framework that makes it possible to compute
  32 fixpoints over arbitrary products of abstract types.  The code is from
  33 an Extended Basis library
  34 (<a href="https://github.com/MLton/mltonlib/blob/master/com/ssh/extended-basis/unstable/README"><span class="monospaced">README</span></a>).</p></div>
  35 <div class="paragraph"><p>First the signature of the framework
  36 (<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>
  37 <div class="listingblock">
  38 <div class="content"><div class="highlight"><pre><span class="cm">(**</span>
  39 <span class="cm"> * A framework for computing fixpoints.</span>
  40 <span class="cm"> *</span>
  41 <span class="cm"> * In a strict language you sometimes want to provide a fixpoint</span>
  42 <span class="cm"> * combinator for an abstract type {t} to make it possible to write</span>
  43 <span class="cm"> * recursive definitions.  Unfortunately, a single combinator {fix} of the</span>
  44 <span class="cm"> * type {(t -&gt; t) -&gt; t} does not support mutual recursion.  To support</span>
  45 <span class="cm"> * mutual recursion, you would need to provide a family of fixpoint</span>
  46 <span class="cm"> * combinators having types of the form {(u -&gt; u) -&gt; u} where {u} is a</span>
  47 <span class="cm"> * type of the form {t * ... * t}.  Unfortunately, even such a family of</span>
  48 <span class="cm"> * fixpoint combinators does not support mutual recursion over different</span>
  49 <span class="cm"> * abstract types.</span>
  50 <span class="cm"> *)</span><span class="w"></span>
  51 <span class="k">signature</span><span class="w"> </span><span class="n">TIE</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="k">sig</span><span class="w"></span>
  52 <span class="w">   </span><span class="k">include</span><span class="w"> </span><span class="n">ETAEXP&#39;</span><span class="w"></span>
  53 <span class="w">   </span><span class="k">type</span><span class="w"> </span><span class="n">&#39;a</span><span class="w"> </span><span class="n">t</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">&#39;a</span><span class="w"> </span><span class="n">etaexp</span><span class="w"></span>
  54 <span class="w">   </span><span class="cm">(** The type of fixpoint witnesses. *)</span><span class="w"></span>
  55
  56 <span class="w">   </span><span class="k">val</span><span class="w"> </span><span class="n">fix</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="n">&#39;a</span><span class="w"> </span><span class="n">t</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="n">&#39;a</span><span class="w"> </span><span class="n">Fix</span><span class="p">.</span><span class="n">t</span><span class="w"></span>
  57 <span class="w">   </span><span class="cm">(**</span>
  58 <span class="cm">    * Produces a fixpoint combinator from the given witness.  For example,</span>
  59 <span class="cm">    * one can make a mutually recursive definition of functions:</span>
  60 <span class="cm">    *</span>
  61 <span class="cm">    *&gt; val isEven &amp; isOdd =</span>
  62 <span class="cm">    *&gt;     let open Tie in fix (function *` function) end</span>
  63 <span class="cm">    *&gt;        (fn isEven &amp; isOdd =&gt;</span>
  64 <span class="cm">    *&gt;            (fn 0 =&gt; true</span>
  65 <span class="cm">    *&gt;              | 1 =&gt; false</span>
  66 <span class="cm">    *&gt;              | n =&gt; isOdd (n-1)) &amp;</span>
  67 <span class="cm">    *&gt;            (fn 0 =&gt; false</span>
  68 <span class="cm">    *&gt;              | 1 =&gt; true</span>
  69 <span class="cm">    *&gt;              | n =&gt; isEven (n-1)))</span>
  70 <span class="cm">    *)</span><span class="w"></span>
  71
  72 <span class="w">   </span><span class="cm">(** == Making New Witnesses == *)</span><span class="w"></span>
  73
  74 <span class="w">   </span><span class="k">val</span><span class="w"> </span><span class="n">pure</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="p">(</span><span class="n">&#39;a</span><span class="w"> </span><span class="n">*</span><span class="w"> </span><span class="n">&#39;a</span><span class="w"> </span><span class="n">UnOp</span><span class="p">.</span><span class="n">t</span><span class="p">)</span><span class="w"> </span><span class="n">Thunk</span><span class="p">.</span><span class="n">t</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="n">&#39;a</span><span class="w"> </span><span class="n">t</span><span class="w"></span>
  75 <span class="w">   </span><span class="cm">(**</span>
  76 <span class="cm">    * {pure} is a more general version of {tier}.  It is mostly useful for</span>
  77 <span class="cm">    * computing fixpoints in a non-imperative manner.</span>
  78 <span class="cm">    *)</span><span class="w"></span>
  79
  80 <span class="w">   </span><span class="k">val</span><span class="w"> </span><span class="n">tier</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="p">(</span><span class="n">&#39;a</span><span class="w"> </span><span class="n">*</span><span class="w"> </span><span class="n">&#39;a</span><span class="w"> </span><span class="n">Effect</span><span class="p">.</span><span class="n">t</span><span class="p">)</span><span class="w"> </span><span class="n">Thunk</span><span class="p">.</span><span class="n">t</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="n">&#39;a</span><span class="w"> </span><span class="n">t</span><span class="w"></span>
  81 <span class="w">   </span><span class="cm">(**</span>
  82 <span class="cm">    * {tier} is used to define fixpoint witnesses for new abstract types</span>
  83 <span class="cm">    * by providing a thunk whose instantiation allocates a mutable proxy</span>
  84 <span class="cm">    * and a procedure for updating it with the result.</span>
  85 <span class="cm">    *)</span><span class="w"></span>
  86
  87 <span class="w">   </span><span class="k">val</span><span class="w"> </span><span class="n">id</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="n">&#39;a</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="n">&#39;a</span><span class="w"> </span><span class="n">t</span><span class="w"></span>
  88 <span class="w">   </span><span class="cm">(** {id x} is equivalent to {pure (const (x, id))}. *)</span><span class="w"></span>
  89
  90 <span class="w">   </span><span class="cm">(** == Combining Existing Witnesses == *)</span><span class="w"></span>
  91
  92 <span class="w">   </span><span class="k">val</span><span class="w"> </span><span class="n">iso</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="n">&#39;b</span><span class="w"> </span><span class="n">t</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="p">(</span><span class="n">&#39;a</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;b</span><span class="p">)</span><span class="w"> </span><span class="n">Iso</span><span class="p">.</span><span class="n">t</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="n">&#39;a</span><span class="w"> </span><span class="n">t</span><span class="w"></span>
  93 <span class="w">   </span><span class="cm">(**</span>
  94 <span class="cm">    * Given an isomorphism between {&#39;a} and {&#39;b} and a witness for {&#39;b},</span>
  95 <span class="cm">    * produces a witness for {&#39;a}.  This is useful when you have a new</span>
  96 <span class="cm">    * type that is isomorphic to some old type for which you already have</span>
  97 <span class="cm">    * a witness.</span>
  98 <span class="cm">    *)</span><span class="w"></span>
  99
 100 <span class="w">   </span><span class="k">val</span><span class="w"> </span><span class="n">product</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="n">&#39;a</span><span class="w"> </span><span class="n">t</span><span class="w"> </span><span class="n">*</span><span class="w"> </span><span class="p">(</span><span class="n">&#39;a</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="n">&#39;b</span><span class="w"> </span><span class="n">t</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="p">(</span><span class="n">&#39;a</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;b</span><span class="p">)</span><span class="w"> </span><span class="n">Product</span><span class="p">.</span><span class="n">t</span><span class="w"> </span><span class="n">t</span><span class="w"></span>
 101 <span class="w">   </span><span class="cm">(**</span>
 102 <span class="cm">    * Dependent product combinator.  Given a witness for {&#39;a} and a</span>
 103 <span class="cm">    * constructor from a {&#39;a} to witness for {&#39;b}, produces a witness for</span>
 104 <span class="cm">    * the product {(&#39;a, &#39;b) Product.t}.  The constructor for {&#39;b} should</span>
 105 <span class="cm">    * not access the (proxy) value {&#39;a} before it has been fixed.</span>
 106 <span class="cm">    *)</span><span class="w"></span>
 107
 108 <span class="w">   </span><span class="k">val</span><span class="w"> </span><span class="n">*`</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="n">&#39;a</span><span class="w"> </span><span class="n">t</span><span class="w"> </span><span class="n">*</span><span class="w"> </span><span class="n">&#39;b</span><span class="w"> </span><span class="n">t</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="p">(</span><span class="n">&#39;a</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;b</span><span class="p">)</span><span class="w"> </span><span class="n">Product</span><span class="p">.</span><span class="n">t</span><span class="w"> </span><span class="n">t</span><span class="w"></span>
 109 <span class="w">   </span><span class="cm">(** {a *` b} is equivalent to {product (a, const b)}. *)</span><span class="w"></span>
 110
 111 <span class="w">   </span><span class="k">val</span><span class="w"> </span><span class="n">tuple2</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="n">&#39;a</span><span class="w"> </span><span class="n">t</span><span class="w"> </span><span class="n">*</span><span class="w"> </span><span class="n">&#39;b</span><span class="w"> </span><span class="n">t</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="p">(</span><span class="n">&#39;a</span><span class="w"> </span><span class="n">*</span><span class="w"> </span><span class="n">&#39;b</span><span class="p">)</span><span class="w"> </span><span class="n">t</span><span class="w"></span>
 112 <span class="w">   </span><span class="cm">(**</span>
 113 <span class="cm">    * Given witnesses for {&#39;a} and {&#39;b} produces a witness for the product</span>
 114 <span class="cm">    * {&#39;a * &#39;b}.</span>
 115 <span class="cm">    *)</span><span class="w"></span>
 116
 117 <span class="w">   </span><span class="cm">(** == Particular Witnesses == *)</span><span class="w"></span>
 118
 119 <span class="w">   </span><span class="k">val</span><span class="w"> </span><span class="n">function</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="p">(</span><span class="n">&#39;a</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="n">&#39;b</span><span class="p">)</span><span class="w"> </span><span class="n">t</span><span class="w"></span>
 120 <span class="w">   </span><span class="cm">(** Witness for functions. *)</span><span class="w"></span>
 121 <span class="k">end</span><span class="w"></span>
 122 </pre></div></div></div>
 123 <div class="paragraph"><p><span class="monospaced">fix</span> is a <a href="TypeIndexedValues">type-indexed</a> function.  The type-index
 124 parameter to <span class="monospaced">fix</span> is called a "witness".  To compute fixpoints over
 125 products, one uses the <span class="monospaced">*&grave;</span> operator to combine witnesses.  To provide
 126 a fixpoint combinator for an abstract type, one implements a witness
 127 providing a thunk whose instantiation allocates a fresh, mutable proxy
 128 and a procedure for updating the proxy with the solution.  Naturally
 129 this means that not all possible ways of computing a fixpoint of a
 130 particular type are possible under the framework.  The <span class="monospaced">pure</span>
 131 combinator is a generalization of <span class="monospaced">tier</span>.  The <span class="monospaced">iso</span> combinator is
 132 provided for reusing existing witnesses.</p></div>
 133 <div class="paragraph"><p>Note that instead of using an infix operator, we could alternatively
 134 employ an interface using <a href="Fold">Fold</a>.  Also, witnesses are eta-expanded
 135 to work around the <a href="ValueRestriction">value restriction</a>, while
 136 maintaining abstraction.</p></div>
 137 <div class="paragraph"><p>Here is the implementation
 138 (<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>
 139 <div class="listingblock">
 140 <div class="content"><div class="highlight"><pre><span class="k">structure</span><span class="w"> </span><span class="n">Tie</span><span class="w"> </span><span class="p">:&gt;</span><span class="w"> </span><span class="n">TIE</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="k">struct</span><span class="w"></span>
 141 <span class="w">   </span><span class="k">open</span><span class="w"> </span><span class="n">Product</span><span class="w"></span>
 142 <span class="w">   </span><span class="k">infix</span><span class="w"> </span><span class="n">&amp;</span><span class="w"></span>
 143 <span class="w">   </span><span class="k">type</span><span class="w"> </span><span class="n">&#39;a</span><span class="w"> </span><span class="n">etaexp_dom</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">Unit</span><span class="p">.</span><span class="n">t</span><span class="w"></span>
 144 <span class="w">   </span><span class="k">type</span><span class="w"> </span><span class="n">&#39;a</span><span class="w"> </span><span class="n">etaexp_cod</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="p">(</span><span class="n">&#39;a</span><span class="w"> </span><span class="n">*</span><span class="w"> </span><span class="n">&#39;a</span><span class="w"> </span><span class="n">UnOp</span><span class="p">.</span><span class="n">t</span><span class="p">)</span><span class="w"> </span><span class="n">Thunk</span><span class="p">.</span><span class="n">t</span><span class="w"></span>
 145 <span class="w">   </span><span class="k">type</span><span class="w"> </span><span class="n">&#39;a</span><span class="w"> </span><span class="n">etaexp</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">&#39;a</span><span class="w"> </span><span class="n">etaexp_dom</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="n">&#39;a</span><span class="w"> </span><span class="n">etaexp_cod</span><span class="w"></span>
 146 <span class="w">   </span><span class="k">type</span><span class="w"> </span><span class="n">&#39;a</span><span class="w"> </span><span class="n">t</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">&#39;a</span><span class="w"> </span><span class="n">etaexp</span><span class="w"></span>
 147 <span class="w">   </span><span class="k">fun</span><span class="w"> </span><span class="n">fix</span><span class="w"> </span><span class="n">aT</span><span class="w"> </span><span class="n">f</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="k">let</span><span class="w"> </span><span class="k">val</span><span class="w"> </span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="w"> </span><span class="n">ta</span><span class="p">)</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">aT</span><span class="w"> </span><span class="p">()</span><span class="w"> </span><span class="p">()</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">ta</span><span class="w"> </span><span class="p">(</span><span class="n">f</span><span class="w"> </span><span class="n">a</span><span class="p">)</span><span class="w"> </span><span class="k">end</span><span class="w"></span>
 148 <span class="w">   </span><span class="k">val</span><span class="w"> </span><span class="n">pure</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">Thunk</span><span class="p">.</span><span class="n">mk</span><span class="w"></span>
 149 <span class="w">   </span><span class="k">fun</span><span class="w"> </span><span class="n">iso</span><span class="w"> </span><span class="n">bT</span><span class="w"> </span><span class="p">(</span><span class="n">iso</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="p">(_,</span><span class="w"> </span><span class="n">b2a</span><span class="p">))</span><span class="w"> </span><span class="p">()</span><span class="w"> </span><span class="p">()</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="k">let</span><span class="w"></span>
 150 <span class="w">      </span><span class="k">val</span><span class="w"> </span><span class="p">(</span><span class="n">b</span><span class="p">,</span><span class="w"> </span><span class="n">fB</span><span class="p">)</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">bT</span><span class="w"> </span><span class="p">()</span><span class="w"> </span><span class="p">()</span><span class="w"></span>
 151 <span class="w">   </span><span class="k">in</span><span class="w"></span>
 152 <span class="w">      </span><span class="p">(</span><span class="n">b2a</span><span class="w"> </span><span class="n">b</span><span class="p">,</span><span class="w"> </span><span class="n">Fn</span><span class="p">.</span><span class="n">map</span><span class="w"> </span><span class="n">iso</span><span class="w"> </span><span class="n">fB</span><span class="p">)</span><span class="w"></span>
 153 <span class="w">   </span><span class="k">end</span><span class="w"></span>
 154 <span class="w">   </span><span class="k">fun</span><span class="w"> </span><span class="n">product</span><span class="w"> </span><span class="p">(</span><span class="n">aT</span><span class="p">,</span><span class="w"> </span><span class="n">a2bT</span><span class="p">)</span><span class="w"> </span><span class="p">()</span><span class="w"> </span><span class="p">()</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="k">let</span><span class="w"></span>
 155 <span class="w">      </span><span class="k">val</span><span class="w"> </span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="w"> </span><span class="n">fA</span><span class="p">)</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">aT</span><span class="w"> </span><span class="p">()</span><span class="w"> </span><span class="p">()</span><span class="w"></span>
 156 <span class="w">      </span><span class="k">val</span><span class="w"> </span><span class="p">(</span><span class="n">b</span><span class="p">,</span><span class="w"> </span><span class="n">fB</span><span class="p">)</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">a2bT</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="p">()</span><span class="w"> </span><span class="p">()</span><span class="w"></span>
 157 <span class="w">   </span><span class="k">in</span><span class="w"></span>
 158 <span class="w">      </span><span class="p">(</span><span class="n">a</span><span class="w"> </span><span class="n">&amp;</span><span class="w"> </span><span class="n">b</span><span class="p">,</span><span class="w"> </span><span class="n">Product</span><span class="p">.</span><span class="n">map</span><span class="w"> </span><span class="p">(</span><span class="n">fA</span><span class="p">,</span><span class="w"> </span><span class="n">fB</span><span class="p">))</span><span class="w"></span>
 159 <span class="w">   </span><span class="k">end</span><span class="w"></span>
 160 <span class="w">   </span><span class="cm">(* The rest are not primitive operations. *)</span><span class="w"></span>
 161 <span class="w">   </span><span class="k">fun</span><span class="w"> </span><span class="k">op</span><span class="w"> </span><span class="n">*`</span><span class="w"> </span><span class="p">(</span><span class="n">aT</span><span class="p">,</span><span class="w"> </span><span class="n">bT</span><span class="p">)</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">product</span><span class="w"> </span><span class="p">(</span><span class="n">aT</span><span class="p">,</span><span class="w"> </span><span class="n">Fn</span><span class="p">.</span><span class="n">const</span><span class="w"> </span><span class="n">bT</span><span class="p">)</span><span class="w"></span>
 162 <span class="w">   </span><span class="k">fun</span><span class="w"> </span><span class="n">tuple2</span><span class="w"> </span><span class="n">ab</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">iso</span><span class="w"> </span><span class="p">(</span><span class="k">op</span><span class="w"> </span><span class="n">*`</span><span class="w"> </span><span class="n">ab</span><span class="p">)</span><span class="w"> </span><span class="n">Product</span><span class="p">.</span><span class="n">isoTuple2</span><span class="w"></span>
 163 <span class="w">   </span><span class="k">fun</span><span class="w"> </span><span class="n">tier</span><span class="w"> </span><span class="n">th</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">pure</span><span class="w"> </span><span class="p">((</span><span class="k">fn</span><span class="w"> </span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="w"> </span><span class="n">ua</span><span class="p">)</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="w"> </span><span class="n">Fn</span><span class="p">.</span><span class="n">const</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="n">o</span><span class="w"> </span><span class="n">ua</span><span class="p">))</span><span class="w"> </span><span class="n">o</span><span class="w"> </span><span class="n">th</span><span class="p">)</span><span class="w"></span>
 164 <span class="w">   </span><span class="k">fun</span><span class="w"> </span><span class="n">id</span><span class="w"> </span><span class="n">x</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">pure</span><span class="w"> </span><span class="p">(</span><span class="n">Fn</span><span class="p">.</span><span class="n">const</span><span class="w"> </span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="w"> </span><span class="n">Fn</span><span class="p">.</span><span class="n">id</span><span class="p">))</span><span class="w"></span>
 165 <span class="w">   </span><span class="k">fun</span><span class="w"> </span><span class="n">function</span><span class="w"> </span><span class="n">?</span><span class="w"> </span><span class="p">=</span><span class="w"></span>
 166 <span class="w">       </span><span class="n">pure</span><span class="w"> </span><span class="p">(</span><span class="k">fn</span><span class="w"> </span><span class="p">()</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="k">let</span><span class="w"></span>
 167 <span class="w">                   </span><span class="k">val</span><span class="w"> </span><span class="n">r</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">ref</span><span class="w"> </span><span class="p">(</span><span class="n">Basic</span><span class="p">.</span><span class="n">raising</span><span class="w"> </span><span class="n">Fix</span><span class="p">.</span><span class="n">Fix</span><span class="p">)</span><span class="w"></span>
 168 <span class="w">                </span><span class="k">in</span><span class="w"></span>
 169 <span class="w">                   </span><span class="p">(</span><span class="k">fn</span><span class="w"> </span><span class="n">x</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="n">!r</span><span class="w"> </span><span class="n">x</span><span class="p">,</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="n">f</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="p">(</span><span class="n">r</span><span class="w"> </span><span class="n">:=</span><span class="w"> </span><span class="n">f</span><span class="w"> </span><span class="p">;</span><span class="w"> </span><span class="n">f</span><span class="p">))</span><span class="w"></span>
 170 <span class="w">                </span><span class="k">end</span><span class="p">)</span><span class="w"> </span><span class="n">?</span><span class="w"></span>
 171 <span class="k">end</span><span class="w"></span>
 172 </pre></div></div></div>
 173 <div class="paragraph"><p>Let&#8217;s then take a look at a couple of additional examples.</p></div>
 174 <div class="paragraph"><p>Here is a naive implementation of lazy promises:</p></div>
 175 <div class="listingblock">
 176 <div class="content"><div class="highlight"><pre><span class="k">structure</span><span class="w"> </span><span class="n">Promise</span><span class="w"> </span><span class="p">:&gt;</span><span class="w"> </span><span class="k">sig</span><span class="w"></span>
 177 <span class="w">   </span><span class="k">type</span><span class="w"> </span><span class="n">&#39;a</span><span class="w"> </span><span class="n">t</span><span class="w"></span>
 178 <span class="w">   </span><span class="k">val</span><span class="w"> </span><span class="n">lazy</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="n">&#39;a</span><span class="w"> </span><span class="n">Thunk</span><span class="p">.</span><span class="n">t</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="n">&#39;a</span><span class="w"> </span><span class="n">t</span><span class="w"></span>
 179 <span class="w">   </span><span class="k">val</span><span class="w"> </span><span class="n">force</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="n">&#39;a</span><span class="w"> </span><span class="n">t</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="n">&#39;a</span><span class="w"></span>
 180 <span class="w">   </span><span class="k">val</span><span class="w"> </span><span class="n">Y</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="n">&#39;a</span><span class="w"> </span><span class="n">t</span><span class="w"> </span><span class="n">Tie</span><span class="p">.</span><span class="n">t</span><span class="w"></span>
 181 <span class="k">end</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="k">struct</span><span class="w"></span>
 182 <span class="w">   </span><span class="k">datatype</span><span class="w"> </span><span class="n">&#39;a</span><span class="w"> </span><span class="n">t&#39;</span><span class="w"> </span><span class="p">=</span><span class="w"></span>
 183 <span class="w">      </span><span class="n">EXN</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="n">exn</span><span class="w"></span>
 184 <span class="w">    </span><span class="p">|</span><span class="w"> </span><span class="n">THUNK</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="n">&#39;a</span><span class="w"> </span><span class="n">Thunk</span><span class="p">.</span><span class="n">t</span><span class="w"></span>
 185 <span class="w">    </span><span class="p">|</span><span class="w"> </span><span class="n">VALUE</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="n">&#39;a</span><span class="w"></span>
 186 <span class="w">   </span><span class="k">type</span><span class="w"> </span><span class="n">&#39;a</span><span class="w"> </span><span class="n">t</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">&#39;a</span><span class="w"> </span><span class="n">t&#39;</span><span class="w"> </span><span class="n">Ref</span><span class="p">.</span><span class="n">t</span><span class="w"></span>
 187 <span class="w">   </span><span class="k">fun</span><span class="w"> </span><span class="n">lazy</span><span class="w"> </span><span class="n">f</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">ref</span><span class="w"> </span><span class="p">(</span><span class="n">THUNK</span><span class="w"> </span><span class="n">f</span><span class="p">)</span><span class="w"></span>
 188 <span class="w">   </span><span class="k">fun</span><span class="w"> </span><span class="n">force</span><span class="w"> </span><span class="n">t</span><span class="w"> </span><span class="p">=</span><span class="w"></span>
 189 <span class="w">      </span><span class="k">case</span><span class="w"> </span><span class="n">!t</span><span class="w"></span>
 190 <span class="w">       </span><span class="k">of</span><span class="w"> </span><span class="n">EXN</span><span class="w"> </span><span class="n">e</span><span class="w">   </span><span class="p">=&gt;</span><span class="w"> </span><span class="k">raise</span><span class="w"> </span><span class="n">e</span><span class="w"></span>
 191 <span class="w">        </span><span class="p">|</span><span class="w"> </span><span class="n">THUNK</span><span class="w"> </span><span class="n">f</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="p">(</span><span class="n">t</span><span class="w"> </span><span class="n">:=</span><span class="w"> </span><span class="n">VALUE</span><span class="w"> </span><span class="p">(</span><span class="n">f</span><span class="w"> </span><span class="p">())</span><span class="w"> </span><span class="k">handle</span><span class="w"> </span><span class="n">e</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="n">t</span><span class="w"> </span><span class="n">:=</span><span class="w"> </span><span class="n">EXN</span><span class="w"> </span><span class="n">e</span><span class="w"> </span><span class="p">;</span><span class="w"> </span><span class="n">force</span><span class="w"> </span><span class="n">t</span><span class="p">)</span><span class="w"></span>
 192 <span class="w">        </span><span class="p">|</span><span class="w"> </span><span class="n">VALUE</span><span class="w"> </span><span class="n">v</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="n">v</span><span class="w"></span>
 193 <span class="w">   </span><span class="k">fun</span><span class="w"> </span><span class="n">Y</span><span class="w"> </span><span class="n">?</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">Tie</span><span class="p">.</span><span class="n">tier</span><span class="w"> </span><span class="p">(</span><span class="k">fn</span><span class="w"> </span><span class="p">()</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="k">let</span><span class="w"></span>
 194 <span class="w">                             </span><span class="k">val</span><span class="w"> </span><span class="n">r</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">lazy</span><span class="w"> </span><span class="p">(</span><span class="n">raising</span><span class="w"> </span><span class="n">Fix</span><span class="p">.</span><span class="n">Fix</span><span class="p">)</span><span class="w"></span>
 195 <span class="w">                          </span><span class="k">in</span><span class="w"></span>
 196 <span class="w">                             </span><span class="p">(</span><span class="n">r</span><span class="p">,</span><span class="w"> </span><span class="n">r</span><span class="w"> </span><span class="n">&lt;\</span><span class="w"> </span><span class="k">op</span><span class="w"> </span><span class="n">:=</span><span class="w"> </span><span class="n">o</span><span class="w"> </span><span class="n">!</span><span class="p">)</span><span class="w"></span>
 197 <span class="w">                          </span><span class="k">end</span><span class="p">)</span><span class="w"> </span><span class="n">?</span><span class="w"></span>
 198 <span class="k">end</span><span class="w"></span>
 199 </pre></div></div></div>
 200 <div class="paragraph"><p>An example use of our naive lazy promises is to implement equally naive
 201 lazy streams:</p></div>
 202 <div class="listingblock">
 203 <div class="content"><div class="highlight"><pre><span class="k">structure</span><span class="w"> </span><span class="n">Stream</span><span class="w"> </span><span class="p">:&gt;</span><span class="w"> </span><span class="k">sig</span><span class="w"></span>
 204 <span class="w">   </span><span class="k">type</span><span class="w"> </span><span class="n">&#39;a</span><span class="w"> </span><span class="n">t</span><span class="w"></span>
 205 <span class="w">   </span><span class="k">val</span><span class="w"> </span><span class="n">cons</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="n">&#39;a</span><span class="w"> </span><span class="n">*</span><span class="w"> </span><span class="n">&#39;a</span><span class="w"> </span><span class="n">t</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="n">&#39;a</span><span class="w"> </span><span class="n">t</span><span class="w"></span>
 206 <span class="w">   </span><span class="k">val</span><span class="w"> </span><span class="n">get</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="n">&#39;a</span><span class="w"> </span><span class="n">t</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="p">(</span><span class="n">&#39;a</span><span class="w"> </span><span class="n">*</span><span class="w"> </span><span class="n">&#39;a</span><span class="w"> </span><span class="n">t</span><span class="p">)</span><span class="w"> </span><span class="n">Option</span><span class="p">.</span><span class="n">t</span><span class="w"></span>
 207 <span class="w">   </span><span class="k">val</span><span class="w"> </span><span class="n">Y</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="n">&#39;a</span><span class="w"> </span><span class="n">t</span><span class="w"> </span><span class="n">Tie</span><span class="p">.</span><span class="n">t</span><span class="w"></span>
 208 <span class="k">end</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="k">struct</span><span class="w"></span>
 209 <span class="w">   </span><span class="k">datatype</span><span class="w"> </span><span class="n">&#39;a</span><span class="w"> </span><span class="n">t</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">IN</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="p">(</span><span class="n">&#39;a</span><span class="w"> </span><span class="n">*</span><span class="w"> </span><span class="n">&#39;a</span><span class="w"> </span><span class="n">t</span><span class="p">)</span><span class="w"> </span><span class="n">Option</span><span class="p">.</span><span class="n">t</span><span class="w"> </span><span class="n">Promise</span><span class="p">.</span><span class="n">t</span><span class="w"></span>
 210 <span class="w">   </span><span class="k">fun</span><span class="w"> </span><span class="n">cons</span><span class="w"> </span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="w"> </span><span class="n">xs</span><span class="p">)</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">IN</span><span class="w"> </span><span class="p">(</span><span class="n">Promise</span><span class="p">.</span><span class="n">lazy</span><span class="w"> </span><span class="p">(</span><span class="k">fn</span><span class="w"> </span><span class="p">()</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="n">SOME</span><span class="w"> </span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="w"> </span><span class="n">xs</span><span class="p">)))</span><span class="w"></span>
 211 <span class="w">   </span><span class="k">fun</span><span class="w"> </span><span class="n">get</span><span class="w"> </span><span class="p">(</span><span class="n">IN</span><span class="w"> </span><span class="n">p</span><span class="p">)</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">Promise</span><span class="p">.</span><span class="n">force</span><span class="w"> </span><span class="n">p</span><span class="w"></span>
 212 <span class="w">   </span><span class="k">fun</span><span class="w"> </span><span class="n">Y</span><span class="w"> </span><span class="n">?</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">Tie</span><span class="p">.</span><span class="n">iso</span><span class="w"> </span><span class="n">Promise</span><span class="p">.</span><span class="n">Y</span><span class="w"> </span><span class="p">(</span><span class="k">fn</span><span class="w"> </span><span class="n">IN</span><span class="w"> </span><span class="n">p</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="n">p</span><span class="p">,</span><span class="w"> </span><span class="n">IN</span><span class="p">)</span><span class="w"> </span><span class="n">?</span><span class="w"></span>
 213 <span class="k">end</span><span class="w"></span>
 214 </pre></div></div></div>
 215 <div class="paragraph"><p>Note that above we make use of the <span class="monospaced">iso</span> combinator.  Here is a finite
 216 representation of an infinite stream of ones:</p></div>
 217 <div class="listingblock">
 218 <div class="content"><div class="highlight"><pre><span class="k">val</span><span class="w"> </span><span class="n">ones</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="k">let</span><span class="w"></span>
 219 <span class="w">   </span><span class="k">open</span><span class="w"> </span><span class="n">Tie</span><span class="w"> </span><span class="n">Stream</span><span class="w"></span>
 220 <span class="k">in</span><span class="w"></span>
 221 <span class="w">   </span><span class="n">fix</span><span class="w"> </span><span class="n">Y</span><span class="w"> </span><span class="p">(</span><span class="k">fn</span><span class="w"> </span><span class="n">ones</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="n">cons</span><span class="w"> </span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">ones</span><span class="p">))</span><span class="w"></span>
 222 <span class="k">end</span><span class="w"></span>
 223 </pre></div></div></div>
 224 </div>
 225 </div>
 226 </div>
 227 <div id="footnotes"><hr></div>
 228 <div id="footer">
 229 <div id="footer-text">
 230 </div>
 231 <div id="footer-badges">
 232 </div>
 233 </div>
 234 </body>
 235 </html>