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<a href="./Home">MLton 20180207</a>\r
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<h1>Fixpoints</h1>\r
</div>\r
<div id="content">\r
<div id="preamble">\r
<div class="sectionbody">\r
<div class="paragraph"><p>This page discusses a framework that makes it possible to compute\r
fixpoints over arbitrary products of abstract types.  The code is from\r
an Extended Basis library\r
(<a href="https://github.com/MLton/mltonlib/blob/master/com/ssh/extended-basis/unstable/README"><span class="monospaced">README</span></a>).</p></div>\r
<div class="paragraph"><p>First the signature of the framework\r
(<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>\r
<div class="listingblock">\r
<div class="content"><div class="highlight"><pre><span class="cm">(**</span>\r
<span class="cm"> * A framework for computing fixpoints.</span>\r
<span class="cm"> *</span>\r
<span class="cm"> * In a strict language you sometimes want to provide a fixpoint</span>\r
<span class="cm"> * combinator for an abstract type {t} to make it possible to write</span>\r
<span class="cm"> * recursive definitions.  Unfortunately, a single combinator {fix} of the</span>\r
<span class="cm"> * type {(t -&gt; t) -&gt; t} does not support mutual recursion.  To support</span>\r
<span class="cm"> * mutual recursion, you would need to provide a family of fixpoint</span>\r
<span class="cm"> * combinators having types of the form {(u -&gt; u) -&gt; u} where {u} is a</span>\r
<span class="cm"> * type of the form {t * ... * t}.  Unfortunately, even such a family of</span>\r
<span class="cm"> * fixpoint combinators does not support mutual recursion over different</span>\r
<span class="cm"> * abstract types.</span>\r
<span class="cm"> *)</span><span class="w"></span>\r
<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>\r
<span class="w">   </span><span class="k">include</span><span class="w"> </span><span class="n">ETAEXP&#39;</span><span class="w"></span>\r
<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>\r
<span class="w">   </span><span class="cm">(** The type of fixpoint witnesses. *)</span><span class="w"></span>\r
\r
<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>\r
<span class="w">   </span><span class="cm">(**</span>\r
<span class="cm">    * Produces a fixpoint combinator from the given witness.  For example,</span>\r
<span class="cm">    * one can make a mutually recursive definition of functions:</span>\r
<span class="cm">    *</span>\r
<span class="cm">    *&gt; val isEven &amp; isOdd =</span>\r
<span class="cm">    *&gt;     let open Tie in fix (function *` function) end</span>\r
<span class="cm">    *&gt;        (fn isEven &amp; isOdd =&gt;</span>\r
<span class="cm">    *&gt;            (fn 0 =&gt; true</span>\r
<span class="cm">    *&gt;              | 1 =&gt; false</span>\r
<span class="cm">    *&gt;              | n =&gt; isOdd (n-1)) &amp;</span>\r
<span class="cm">    *&gt;            (fn 0 =&gt; false</span>\r
<span class="cm">    *&gt;              | 1 =&gt; true</span>\r
<span class="cm">    *&gt;              | n =&gt; isEven (n-1)))</span>\r
<span class="cm">    *)</span><span class="w"></span>\r
\r
<span class="w">   </span><span class="cm">(** == Making New Witnesses == *)</span><span class="w"></span>\r
\r
<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>\r
<span class="w">   </span><span class="cm">(**</span>\r
<span class="cm">    * {pure} is a more general version of {tier}.  It is mostly useful for</span>\r
<span class="cm">    * computing fixpoints in a non-imperative manner.</span>\r
<span class="cm">    *)</span><span class="w"></span>\r
\r
<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>\r
<span class="w">   </span><span class="cm">(**</span>\r
<span class="cm">    * {tier} is used to define fixpoint witnesses for new abstract types</span>\r
<span class="cm">    * by providing a thunk whose instantiation allocates a mutable proxy</span>\r
<span class="cm">    * and a procedure for updating it with the result.</span>\r
<span class="cm">    *)</span><span class="w"></span>\r
\r
<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>\r
<span class="w">   </span><span class="cm">(** {id x} is equivalent to {pure (const (x, id))}. *)</span><span class="w"></span>\r
\r
<span class="w">   </span><span class="cm">(** == Combining Existing Witnesses == *)</span><span class="w"></span>\r
\r
<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>\r
<span class="w">   </span><span class="cm">(**</span>\r
<span class="cm">    * Given an isomorphism between {&#39;a} and {&#39;b} and a witness for {&#39;b},</span>\r
<span class="cm">    * produces a witness for {&#39;a}.  This is useful when you have a new</span>\r
<span class="cm">    * type that is isomorphic to some old type for which you already have</span>\r
<span class="cm">    * a witness.</span>\r
<span class="cm">    *)</span><span class="w"></span>\r
\r
<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>\r
<span class="w">   </span><span class="cm">(**</span>\r
<span class="cm">    * Dependent product combinator.  Given a witness for {&#39;a} and a</span>\r
<span class="cm">    * constructor from a {&#39;a} to witness for {&#39;b}, produces a witness for</span>\r
<span class="cm">    * the product {(&#39;a, &#39;b) Product.t}.  The constructor for {&#39;b} should</span>\r
<span class="cm">    * not access the (proxy) value {&#39;a} before it has been fixed.</span>\r
<span class="cm">    *)</span><span class="w"></span>\r
\r
<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>\r
<span class="w">   </span><span class="cm">(** {a *` b} is equivalent to {product (a, const b)}. *)</span><span class="w"></span>\r
\r
<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>\r
<span class="w">   </span><span class="cm">(**</span>\r
<span class="cm">    * Given witnesses for {&#39;a} and {&#39;b} produces a witness for the product</span>\r
<span class="cm">    * {&#39;a * &#39;b}.</span>\r
<span class="cm">    *)</span><span class="w"></span>\r
\r
<span class="w">   </span><span class="cm">(** == Particular Witnesses == *)</span><span class="w"></span>\r
\r
<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>\r
<span class="w">   </span><span class="cm">(** Witness for functions. *)</span><span class="w"></span>\r
<span class="k">end</span><span class="w"></span>\r
</pre></div></div></div>\r
<div class="paragraph"><p><span class="monospaced">fix</span> is a <a href="TypeIndexedValues">type-indexed</a> function.  The type-index\r
parameter to <span class="monospaced">fix</span> is called a "witness".  To compute fixpoints over\r
products, one uses the <span class="monospaced">*&grave;</span> operator to combine witnesses.  To provide\r
a fixpoint combinator for an abstract type, one implements a witness\r
providing a thunk whose instantiation allocates a fresh, mutable proxy\r
and a procedure for updating the proxy with the solution.  Naturally\r
this means that not all possible ways of computing a fixpoint of a\r
particular type are possible under the framework.  The <span class="monospaced">pure</span>\r
combinator is a generalization of <span class="monospaced">tier</span>.  The <span class="monospaced">iso</span> combinator is\r
provided for reusing existing witnesses.</p></div>\r
<div class="paragraph"><p>Note that instead of using an infix operator, we could alternatively\r
employ an interface using <a href="Fold">Fold</a>.  Also, witnesses are eta-expanded\r
to work around the <a href="ValueRestriction">value restriction</a>, while\r
maintaining abstraction.</p></div>\r
<div class="paragraph"><p>Here is the implementation\r
(<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>\r
<div class="listingblock">\r
<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>\r
<span class="w">   </span><span class="k">open</span><span class="w"> </span><span class="n">Product</span><span class="w"></span>\r
<span class="w">   </span><span class="k">infix</span><span class="w"> </span><span class="n">&amp;</span><span class="w"></span>\r
<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>\r
<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>\r
<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>\r
<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>\r
<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>\r
<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>\r
<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>\r
<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>\r
<span class="w">   </span><span class="k">in</span><span class="w"></span>\r
<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>\r
<span class="w">   </span><span class="k">end</span><span class="w"></span>\r
<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>\r
<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>\r
<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>\r
<span class="w">   </span><span class="k">in</span><span class="w"></span>\r
<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>\r
<span class="w">   </span><span class="k">end</span><span class="w"></span>\r
<span class="w">   </span><span class="cm">(* The rest are not primitive operations. *)</span><span class="w"></span>\r
<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>\r
<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>\r
<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>\r
<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>\r
<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>\r
<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>\r
<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>\r
<span class="w">                </span><span class="k">in</span><span class="w"></span>\r
<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>\r
<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>\r
<span class="k">end</span><span class="w"></span>\r
</pre></div></div></div>\r
<div class="paragraph"><p>Let&#8217;s then take a look at a couple of additional examples.</p></div>\r
<div class="paragraph"><p>Here is a naive implementation of lazy promises:</p></div>\r
<div class="listingblock">\r
<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>\r
<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>\r
<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>\r
<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>\r
<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>\r
<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>\r
<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>\r
<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>\r
<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>\r
<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>\r
<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>\r
<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>\r
<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>\r
<span class="w">      </span><span class="k">case</span><span class="w"> </span><span class="n">!t</span><span class="w"></span>\r
<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>\r
<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>\r
<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>\r
<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>\r
<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>\r
<span class="w">                          </span><span class="k">in</span><span class="w"></span>\r
<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>\r
<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>\r
<span class="k">end</span><span class="w"></span>\r
</pre></div></div></div>\r
<div class="paragraph"><p>An example use of our naive lazy promises is to implement equally naive\r
lazy streams:</p></div>\r
<div class="listingblock">\r
<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>\r
<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>\r
<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>\r
<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>\r
<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>\r
<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>\r
<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>\r
<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>\r
<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>\r
<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>\r
<span class="k">end</span><span class="w"></span>\r
</pre></div></div></div>\r
<div class="paragraph"><p>Note that above we make use of the <span class="monospaced">iso</span> combinator.  Here is a finite\r
representation of an infinite stream of ones:</p></div>\r
<div class="listingblock">\r
<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>\r
<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>\r
<span class="k">in</span><span class="w"></span>\r
<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>\r
<span class="k">end</span><span class="w"></span>\r
</pre></div></div></div>\r
</div>\r
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