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<a class = mltona href="Home">MLton 20061025</a>
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Fixpoints
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In a strict language, such as <a href="StandardML">Standard ML</a>, you sometimes want to provide a fixpoint combinator for an abstract type <tt>t</tt> to make it possible to write recursive definitions. While it is easy to write an ad hoc fixpoint combinator for a single abstract type, it is more challenging to provide a general purpose framework for computing fixpoints. First of all, a single combinator <tt>fix</tt> with a type of the form <tt>(t -> t) -> t</tt> does not support mutual recursion over multiple values of type <tt>t</tt>. To support mutual recursion, you might provide a family of fixpoint combinators having types of the form <tt>(u -> u) -> u</tt> where <tt>u</tt> is a type of the form <tt>t * ... * t</tt>. Unfortunately, even such a family of fixpoint combinators does not support mutual recursion over multiple different abstract types. The gist of the problem is that we need a <a href="TypeIndexedValues">type-indexed</a> family of fixpoint combinators. Below is a solution that allows for computing fixpoints over arbitrary products. The code on this page makes use of some <a href="Utilities">Utilities</a>. <p>
First the signature of the fixpoint framework:
</p>
<pre class=code>
<B><FONT COLOR="#5F9EA0">signature</FONT></B> FIX <B><FONT COLOR="#5F9EA0">=</FONT></B>
<B><FONT COLOR="#5F9EA0">sig</FONT></B>
<B><FONT COLOR="#A020F0">type</FONT></B> 'a t1
<B><FONT COLOR="#A020F0">type</FONT></B> 'a t2
<B><FONT COLOR="#A020F0">type</FONT></B> 'a t <B><FONT COLOR="#5F9EA0">=</FONT></B> 'a t1 <B><FONT COLOR="#5F9EA0">-</FONT></B><B><FONT COLOR="#5F9EA0">></FONT></B> 'a t2
<B><FONT COLOR="#A020F0">exception</FONT></B> Fix
<B><FONT COLOR="#A020F0">val</FONT></B> fix : 'a t <B><FONT COLOR="#5F9EA0">-</FONT></B><B><FONT COLOR="#5F9EA0">></FONT></B> 'a fix
<B><FONT COLOR="#A020F0">val</FONT></B> pure : ('a <B><FONT COLOR="#5F9EA0">*</FONT></B> 'a uop) thunk <B><FONT COLOR="#5F9EA0">-</FONT></B><B><FONT COLOR="#5F9EA0">></FONT></B> 'a t
<B><FONT COLOR="#A020F0">val</FONT></B> tier : ('a <B><FONT COLOR="#5F9EA0">*</FONT></B> 'a effect) thunk <B><FONT COLOR="#5F9EA0">-</FONT></B><B><FONT COLOR="#5F9EA0">></FONT></B> 'a t
<B><FONT COLOR="#A020F0">val</FONT></B> iso : ('a, 'b) emb <B><FONT COLOR="#5F9EA0">-</FONT></B><B><FONT COLOR="#5F9EA0">></FONT></B> 'b t <B><FONT COLOR="#5F9EA0">-</FONT></B><B><FONT COLOR="#5F9EA0">></FONT></B> 'a t
<B><FONT COLOR="#A020F0">val</FONT></B> <B><FONT COLOR="#5F9EA0">^</FONT></B> : 'a t <B><FONT COLOR="#5F9EA0">*</FONT></B> 'b t <B><FONT COLOR="#5F9EA0">-</FONT></B><B><FONT COLOR="#5F9EA0">></FONT></B> ('a, 'b) product t
<B><FONT COLOR="#A020F0">end</FONT></B>
</PRE>
<p>
</p>
<p>
<tt>fix</tt> is a <a href="TypeIndexedValues">type-indexed</a> function. The type parameter to <tt>fix</tt> is called a "tier". To compute fixpoints over products, one uses the <tt>^</tt> operator to combine tiers. To provide a fixpoint combinator for an abstract type, one implements a tier providing a thunk whose instantation allocates a fresh "knot" and a procedure for "tying" it. Naturally this means that not all possible ways of computing a fixpoint of a particular type are possible under this framework. The <tt>pure</tt> combinator is a generalization of <tt>tier</tt>. The <tt>iso</tt> combinator is provided for reusing existing tiers.
</p>
<p>
Note that instead of using an infix operator, we could alternatively employ an interface using <a href="Fold">Fold</a>. Also, the tiers are eta-expanded to work around the <a href="ValueRestriction">value restriction</a>, while maintaining abstraction (the signature keeps tiers abstract).
</p>
<p>
Here is an implementation:
</p>
<pre class=code>
<B><FONT COLOR="#5F9EA0">structure</FONT></B> Fix :<B><FONT COLOR="#5F9EA0">></FONT></B> FIX <B><FONT COLOR="#5F9EA0">=</FONT></B>
<B><FONT COLOR="#5F9EA0">struct</FONT></B>
<B><FONT COLOR="#A020F0">type</FONT></B> 'a t1 <B><FONT COLOR="#5F9EA0">=</FONT></B> <B><FONT COLOR="#228B22">unit</FONT></B>
<B><FONT COLOR="#A020F0">type</FONT></B> 'a t2 <B><FONT COLOR="#5F9EA0">=</FONT></B> 'a <B><FONT COLOR="#5F9EA0">*</FONT></B> 'a uop
<B><FONT COLOR="#A020F0">type</FONT></B> 'a t <B><FONT COLOR="#5F9EA0">=</FONT></B> 'a t1 <B><FONT COLOR="#5F9EA0">-</FONT></B><B><FONT COLOR="#5F9EA0">></FONT></B> 'a t2
<B><FONT COLOR="#A020F0">exception</FONT></B> Fix
<B><FONT COLOR="#A020F0"> fun </FONT></B><B><FONT COLOR="#0000FF"><B><I><FONT COLOR="#000000">fix</FONT></I></B></FONT></B> a f <B><FONT COLOR="#5F9EA0">=</FONT></B>
<B><FONT COLOR="#A020F0">let</FONT></B>
<B><FONT COLOR="#A020F0">val</FONT></B> (a, ta) <B><FONT COLOR="#5F9EA0">=</FONT></B> a ()
<B><FONT COLOR="#A020F0">in</FONT></B>
ta (f a)
<B><FONT COLOR="#A020F0">end</FONT></B>
<B><FONT COLOR="#A020F0">val</FONT></B> pure <B><FONT COLOR="#5F9EA0">=</FONT></B> id
<B><FONT COLOR="#A020F0"> fun </FONT></B><B><FONT COLOR="#0000FF"><B><I><FONT COLOR="#000000">tier</FONT></I></B></FONT></B> th () <B><FONT COLOR="#5F9EA0">=</FONT></B>
<B><FONT COLOR="#A020F0">let</FONT></B>
<B><FONT COLOR="#A020F0">val</FONT></B> (a, ta) <B><FONT COLOR="#5F9EA0">=</FONT></B> th ()
<B><FONT COLOR="#A020F0">in</FONT></B>
(a, const a <B><FONT COLOR="#A020F0">o</FONT></B> ta)
<B><FONT COLOR="#A020F0">end</FONT></B>
<B><FONT COLOR="#A020F0"> fun </FONT></B><B><FONT COLOR="#0000FF"><B><I><FONT COLOR="#000000">iso</FONT></I></B></FONT></B> (a2b, b2a) b () <B><FONT COLOR="#5F9EA0">=</FONT></B>
<B><FONT COLOR="#A020F0">let</FONT></B>
<B><FONT COLOR="#A020F0">val</FONT></B> (b, tb) <B><FONT COLOR="#5F9EA0">=</FONT></B> b ()
<B><FONT COLOR="#A020F0">in</FONT></B>
(b2a b, b2a <B><FONT COLOR="#A020F0">o</FONT></B> tb <B><FONT COLOR="#A020F0">o</FONT></B> a2b)
<B><FONT COLOR="#A020F0">end</FONT></B>
<B><FONT COLOR="#A020F0">fun</FONT></B> (a <B><FONT COLOR="#5F9EA0">^</FONT></B> b) () <B><FONT COLOR="#5F9EA0">=</FONT></B>
<B><FONT COLOR="#A020F0">let</FONT></B>
<B><FONT COLOR="#A020F0">val</FONT></B> (a, ta) <B><FONT COLOR="#5F9EA0">=</FONT></B> a ()
<B><FONT COLOR="#A020F0">val</FONT></B> (b, tb) <B><FONT COLOR="#5F9EA0">=</FONT></B> b ()
<B><FONT COLOR="#A020F0">in</FONT></B>
(a & b, <B><FONT COLOR="#A020F0">fn</FONT></B> a & b <B><FONT COLOR="#5F9EA0">=</FONT></B><B><FONT COLOR="#5F9EA0">></FONT></B> ta a & tb b)
<B><FONT COLOR="#A020F0">end</FONT></B>
<B><FONT COLOR="#A020F0">end</FONT></B>
</PRE>
<p>
</p>
<p>
Let's then take a look at some examples. First a simple tier for functions:
</p>
<pre class=code>
<B><FONT COLOR="#5F9EA0">structure</FONT></B> Fn :<B><FONT COLOR="#5F9EA0">></FONT></B>
<B><FONT COLOR="#5F9EA0">sig</FONT></B>
<B><FONT COLOR="#A020F0">val</FONT></B> Y : ('a <B><FONT COLOR="#5F9EA0">-</FONT></B><B><FONT COLOR="#5F9EA0">></FONT></B> 'b) Fix.t
<B><FONT COLOR="#A020F0">end</FONT></B> <B><FONT COLOR="#5F9EA0">=</FONT></B> <B><FONT COLOR="#5F9EA0">struct</FONT></B>
<B><FONT COLOR="#A020F0"> fun </FONT></B><B><FONT COLOR="#0000FF"><B><I><FONT COLOR="#000000">Y</FONT></I></B></FONT></B> ? <B><FONT COLOR="#5F9EA0">=</FONT></B> Fix.tier (<B><FONT COLOR="#A020F0">fn</FONT></B> () <B><FONT COLOR="#5F9EA0">=</FONT></B><B><FONT COLOR="#5F9EA0">></FONT></B>
<B><FONT COLOR="#A020F0">let</FONT></B>
<B><FONT COLOR="#A020F0">val</FONT></B> r <B><FONT COLOR="#5F9EA0">=</FONT></B> <B><FONT COLOR="#A020F0">ref</FONT></B> (fail Fix.Fix)
<B><FONT COLOR="#A020F0"> fun </FONT></B><B><FONT COLOR="#0000FF"><B><I><FONT COLOR="#000000">f</FONT></I></B></FONT></B> x <B><FONT COLOR="#5F9EA0">=</FONT></B> <B><FONT COLOR="#5F9EA0">!</FONT></B>r x
<B><FONT COLOR="#A020F0">in</FONT></B>
(f, r <\ <B><FONT COLOR="#A020F0">op</FONT></B> <B><FONT COLOR="#5F9EA0">:=</FONT></B>)
<B><FONT COLOR="#A020F0">end</FONT></B>) ?
<B><FONT COLOR="#A020F0">end</FONT></B>
</PRE>
<p>
</p>
<p>
Here is an example of a mutually recursive definition of functions:
</p>
<pre class=code>
<B><FONT COLOR="#A020F0">val</FONT></B> isEven & isOdd <B><FONT COLOR="#5F9EA0">=</FONT></B>
<B><FONT COLOR="#A020F0">let</FONT></B> <B><FONT COLOR="#A020F0">open</FONT></B> Fix Fn <B><FONT COLOR="#A020F0">in</FONT></B> fix (Y<B><FONT COLOR="#5F9EA0">^</FONT></B>Y) <B><FONT COLOR="#A020F0">end</FONT></B>
(<B><FONT COLOR="#A020F0">fn</FONT></B> isEven & isOdd <B><FONT COLOR="#5F9EA0">=</FONT></B><B><FONT COLOR="#5F9EA0">></FONT></B>
(<B><FONT COLOR="#A020F0">fn</FONT></B> 0w0 <B><FONT COLOR="#5F9EA0">=</FONT></B><B><FONT COLOR="#5F9EA0">></FONT></B> true
<B><FONT COLOR="#5F9EA0">|</FONT></B> 0w1 <B><FONT COLOR="#5F9EA0">=</FONT></B><B><FONT COLOR="#5F9EA0">></FONT></B> false
<B><FONT COLOR="#5F9EA0">|</FONT></B> n <B><FONT COLOR="#5F9EA0">=</FONT></B><B><FONT COLOR="#5F9EA0">></FONT></B> isOdd (n<B><FONT COLOR="#5F9EA0">-</FONT></B>0w1)) &
(<B><FONT COLOR="#A020F0">fn</FONT></B> 0w0 <B><FONT COLOR="#5F9EA0">=</FONT></B><B><FONT COLOR="#5F9EA0">></FONT></B> false
<B><FONT COLOR="#5F9EA0">|</FONT></B> 0w1 <B><FONT COLOR="#5F9EA0">=</FONT></B><B><FONT COLOR="#5F9EA0">></FONT></B> true
<B><FONT COLOR="#5F9EA0">|</FONT></B> n <B><FONT COLOR="#5F9EA0">=</FONT></B><B><FONT COLOR="#5F9EA0">></FONT></B> isEven (n<B><FONT COLOR="#5F9EA0">-</FONT></B>0w1)))
</PRE>
<p>
</p>
<p>
Our second example is a naive implementation of lazy promises:
</p>
<pre class=code>
<B><FONT COLOR="#5F9EA0">structure</FONT></B> Promise :<B><FONT COLOR="#5F9EA0">></FONT></B>
<B><FONT COLOR="#5F9EA0">sig</FONT></B>
<B><FONT COLOR="#A020F0">type</FONT></B> 'a t
<B><FONT COLOR="#A020F0">val</FONT></B> lazy : 'a thunk <B><FONT COLOR="#5F9EA0">-</FONT></B><B><FONT COLOR="#5F9EA0">></FONT></B> 'a t
<B><FONT COLOR="#A020F0">val</FONT></B> force : 'a t <B><FONT COLOR="#5F9EA0">-</FONT></B><B><FONT COLOR="#5F9EA0">></FONT></B> 'a
<B><FONT COLOR="#A020F0">val</FONT></B> Y : 'a t Fix.t
<B><FONT COLOR="#A020F0">end</FONT></B> <B><FONT COLOR="#5F9EA0">=</FONT></B> <B><FONT COLOR="#5F9EA0">struct</FONT></B>
<B><FONT COLOR="#A020F0">datatype</FONT></B> 'a t' <B><FONT COLOR="#5F9EA0">=</FONT></B>
Exn <B><FONT COLOR="#A020F0">of</FONT></B> exn
<B><FONT COLOR="#5F9EA0">|</FONT></B> Thunk <B><FONT COLOR="#A020F0">of</FONT></B> 'a thunk
<B><FONT COLOR="#5F9EA0">|</FONT></B> Value <B><FONT COLOR="#A020F0">of</FONT></B> 'a
<B><FONT COLOR="#A020F0">type</FONT></B> 'a t <B><FONT COLOR="#5F9EA0">=</FONT></B> 'a t' <B><FONT COLOR="#A020F0">ref</FONT></B>
<B><FONT COLOR="#A020F0"> fun </FONT></B><B><FONT COLOR="#0000FF"><B><I><FONT COLOR="#000000">lazy</FONT></I></B></FONT></B> f <B><FONT COLOR="#5F9EA0">=</FONT></B> <B><FONT COLOR="#A020F0">ref</FONT></B> (Thunk f)
<B><FONT COLOR="#A020F0"> fun </FONT></B><B><FONT COLOR="#0000FF"><B><I><FONT COLOR="#000000">force</FONT></I></B></FONT></B> t <B><FONT COLOR="#5F9EA0">=</FONT></B>
<B><FONT COLOR="#A020F0">case</FONT></B> <B><FONT COLOR="#5F9EA0">!</FONT></B>t <B><FONT COLOR="#A020F0">of</FONT></B>
Exn e <B><FONT COLOR="#5F9EA0">=</FONT></B><B><FONT COLOR="#5F9EA0">></FONT></B> <B><FONT COLOR="#A020F0">raise</FONT></B> e
<B><FONT COLOR="#5F9EA0">|</FONT></B> Thunk f <B><FONT COLOR="#5F9EA0">=</FONT></B><B><FONT COLOR="#5F9EA0">></FONT></B>
(t <B><FONT COLOR="#5F9EA0">:=</FONT></B> Value (f ())
<B><FONT COLOR="#A020F0">handle</FONT></B> e <B><FONT COLOR="#5F9EA0">=</FONT></B><B><FONT COLOR="#5F9EA0">></FONT></B> t <B><FONT COLOR="#5F9EA0">:=</FONT></B> Exn e
; force t)
<B><FONT COLOR="#5F9EA0">|</FONT></B> Value v <B><FONT COLOR="#5F9EA0">=</FONT></B><B><FONT COLOR="#5F9EA0">></FONT></B> v
<B><FONT COLOR="#A020F0"> fun </FONT></B><B><FONT COLOR="#0000FF"><B><I><FONT COLOR="#000000">Y</FONT></I></B></FONT></B> ? <B><FONT COLOR="#5F9EA0">=</FONT></B> Fix.tier (<B><FONT COLOR="#A020F0">fn</FONT></B> () <B><FONT COLOR="#5F9EA0">=</FONT></B><B><FONT COLOR="#5F9EA0">></FONT></B>
<B><FONT COLOR="#A020F0">let</FONT></B>
<B><FONT COLOR="#A020F0">val</FONT></B> r <B><FONT COLOR="#5F9EA0">=</FONT></B> lazy (fail Fix.Fix)
<B><FONT COLOR="#A020F0">in</FONT></B>
(r, r <\ <B><FONT COLOR="#A020F0">op</FONT></B> <B><FONT COLOR="#5F9EA0">:=</FONT></B> <B><FONT COLOR="#A020F0">o</FONT></B> <B><FONT COLOR="#5F9EA0">!</FONT></B>)
<B><FONT COLOR="#A020F0">end</FONT></B>) ?
<B><FONT COLOR="#A020F0">end</FONT></B>
</PRE>
<p>
</p>
<p>
An example use of our naive lazy promises is to implement equally naive lazy streams:
</p>
<pre class=code>
<B><FONT COLOR="#5F9EA0">structure</FONT></B> Stream :<B><FONT COLOR="#5F9EA0">></FONT></B>
<B><FONT COLOR="#5F9EA0">sig</FONT></B>
<B><FONT COLOR="#A020F0">type</FONT></B> 'a t
<B><FONT COLOR="#A020F0">val</FONT></B> cons : 'a <B><FONT COLOR="#5F9EA0">*</FONT></B> 'a t <B><FONT COLOR="#5F9EA0">-</FONT></B><B><FONT COLOR="#5F9EA0">></FONT></B> 'a t
<B><FONT COLOR="#A020F0">val</FONT></B> get : 'a t <B><FONT COLOR="#5F9EA0">-</FONT></B><B><FONT COLOR="#5F9EA0">></FONT></B> ('a <B><FONT COLOR="#5F9EA0">*</FONT></B> 'a t) option
<B><FONT COLOR="#A020F0">val</FONT></B> Y : 'a t Fix.t
<B><FONT COLOR="#A020F0">end</FONT></B> <B><FONT COLOR="#5F9EA0">=</FONT></B> <B><FONT COLOR="#5F9EA0">struct</FONT></B>
<B><FONT COLOR="#A020F0">datatype</FONT></B> 'a t <B><FONT COLOR="#5F9EA0">=</FONT></B> IN <B><FONT COLOR="#A020F0">of</FONT></B> ('a <B><FONT COLOR="#5F9EA0">*</FONT></B> 'a t) option Promise.t
<B><FONT COLOR="#A020F0"> fun </FONT></B><B><FONT COLOR="#0000FF"><B><I><FONT COLOR="#000000">cons</FONT></I></B></FONT></B> (x, xs) <B><FONT COLOR="#5F9EA0">=</FONT></B> IN (Promise.lazy (<B><FONT COLOR="#A020F0">fn</FONT></B> () <B><FONT COLOR="#5F9EA0">=</FONT></B><B><FONT COLOR="#5F9EA0">></FONT></B> <B><FONT COLOR="#5F9EA0">SOME</FONT></B> (x, xs)))
<B><FONT COLOR="#A020F0"> fun </FONT></B><B><FONT COLOR="#0000FF"><B><I><FONT COLOR="#000000">get</FONT></I></B></FONT></B> (IN p) <B><FONT COLOR="#5F9EA0">=</FONT></B> Promise.force p
<B><FONT COLOR="#A020F0"> fun </FONT></B><B><FONT COLOR="#0000FF"><B><I><FONT COLOR="#000000">Y</FONT></I></B></FONT></B> ? <B><FONT COLOR="#5F9EA0">=</FONT></B> Fix.iso (<B><FONT COLOR="#A020F0">fn</FONT></B> IN p <B><FONT COLOR="#5F9EA0">=</FONT></B><B><FONT COLOR="#5F9EA0">></FONT></B> p, IN) Promise.Y ?
<B><FONT COLOR="#A020F0">end</FONT></B>
</PRE>
<p>
</p>
<p>
Note that above we make use of the <tt>Fix.iso</tt> combinator. Here is a finite representation of an infinite stream of ones:
</p>
<pre class=code>
<B><FONT COLOR="#A020F0">val</FONT></B> ones <B><FONT COLOR="#5F9EA0">=</FONT></B>
<B><FONT COLOR="#A020F0">let</FONT></B>
<B><FONT COLOR="#A020F0">open</FONT></B> Fix Stream
<B><FONT COLOR="#A020F0">in</FONT></B>
fix Y (<B><FONT COLOR="#A020F0">fn</FONT></B> ones <B><FONT COLOR="#5F9EA0">=</FONT></B><B><FONT COLOR="#5F9EA0">></FONT></B> cons (1, ones))
<B><FONT COLOR="#A020F0">end</FONT></B>
</PRE>
<p>
</p>
</div>
<p>
<hr>
Last edited on 2006-08-30 09:45:25 by <span title="www-cache2.hel.fi.ssh.com"><a href="VesaKarvonen">VesaKarvonen</a></span>.
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