Chapter 23. Stream-Based Lazy Evaluation

While the core of ATS is based on call-by-value evaluation, there is also direct support in ATS for lazy (that is, call-by-need) evaluation.

There is a special language construct $delay for delaying or suspending the evaluation of an expression (by forming a thunk), and there is also a special function lazy_force for resuming a suspended evaluation (represented by a thunk). The abstract type constructor lazy of the sort (t@ype) => type forms a (boxed) type when applied to a type. Given an expression exp of type T, the value $delay(exp) of the type lazy(T) represents the suspended evaluation of exp. Given a value V of the type lazy(T) for some type T, calling lazy_force on V resumes the suspended evaluation represented by V. If the call returns, then the returned value is of type T. The interface for the function template lazy_force is given as follows:

fun{a:t@ype} lazy_force (lazyval: lazy(a)):<!laz> a

where the symbol !laz indicates a form of effect associated with lazy-evaluation. Note that the special prefix operator ! in ATS is overloaded with lazy_force.

In prelude/SATS/stream.sats, the following types stream_con and stream are declared mutually recursively for representing lazy streams:

datatype stream_con (a:t@ype+) = | stream_nil of ((*void*)) | stream_cons of (a, stream(a)) where stream (a:t@ype) = lazy (stream_con(a))

Also, a number of common functions on streams are declared in prelude/SATS/stream.sats and implemented in prelude/DATS/stream.dats.

The following code gives a standard implementation of the sieve of Eratosthenes:

// fun from (n: int): stream (int) = $delay (stream_cons (n, from (n+1))) // fun sieve ( ns: stream(int) ) :<!laz> stream(int) = $delay let // // [val-] means no warning message from the compiler // val-stream_cons(n, ns) = !ns in stream_cons (n, sieve (stream_filter_cloref<int> (ns, lam x => x mod n > 0))) end // end of [$delay let] // end of [sieve] // val thePrimes = sieve(from(2)) //

A stream is constructed consisting of all the integers starting from 2; the first element of the stream is kept and all the multiples of this element are removed from the tail of the stream; this process is then repeated on the tail of the stream recursively. Clearly, the final stream thus generated consists of all the prime numbers ordered ascendingly.

The function template stream_filter_cloref is of the following interface:

fun{a:t@ype} stream_filter_cloref (xs: stream(a), pred: a -<cloref> bool):<!laz> stream(a) // end of [stream_filter_cloref]

Given a stream and a predicate, stream_filter_cloref generates another stream consisting of all the elements in the given stream that satisfy the given predicate.

Let us see another example of lazy evaluation. The follow code demonstrates an interesting approach to computing the Fibonacci numbers:

// val _0_ = $UNSAFE.cast{int64}(0) val _1_ = $UNSAFE.cast{int64}(1) // val // the following values are defined mutually recursively rec theFibs_0 : stream(int64) = $delay (stream_cons(_0_, theFibs_1)) // fib0, fib1, ... and theFibs_1 : stream(int64) = $delay (stream_cons(_1_, theFibs_2)) // fib1, fib2, ... and theFibs_2 : stream(int64) = // fib2, fib3, fib4, ... ( stream_map2_fun<int64,int64><int64> (theFibs_0, theFibs_1, lam (x, y) => x + y) ) (* end of [val/and/and] *) //

The function template stream_map2_fun is assigned the following interface:

fun{ a1,a2:t0p}{b:t0p } stream_map2_cloref ( xs1: stream (a1), xs2: stream (a2), f: (a1, a2) -<fun> b ) :<!laz> stream (b) // end of [stream_map2_cloref]

Given two streams xs1 and xs2 and a binary function f, stream_map2_fun forms a stream xs such that xs[n] (that is, element n in xs), if exists, equals f(xs1[n], xs2[n]), where n ranges over natural numbers.

Let us see yet another example of lazy evaluation. A Hamming number is a positive natural number whose prime factors can contain only 2, 3 and 5. The following code shows a straightforward way to generate a stream consisting of all the Hamming numbers:

// val compare_int_int = lam (x1: int, x2: int): int =<fun> compare(x1, x2) // macdef merge2 (xs1, xs2) = stream_mergeq_fun<int> (,(xs1), ,(xs2), compare_int_int) // val rec theHamming : stream(int) = $delay ( stream_cons(1, merge2(merge2(theHamming2, theHamming3), theHamming5)) ) (* end of [theHamming] *) and theHamming2 : stream(int) = stream_map_fun<int><int> (theHamming, lam x => 2 * x) and theHamming3 : stream(int) = stream_map_fun<int><int> (theHamming, lam x => 3 * x) and theHamming5 : stream(int) = stream_map_fun<int><int> (theHamming, lam x => 5 * x) //

The function template stream_mergeq_fun is given the following interface:

fun{a:t0p} stream_mergeq_fun ( xs1: stream (a), xs2: stream (a), (a, a) -<fun> int ) :<!laz> stream (a) // end of [stream_mergeq_fun]

Given two streams and an ordering (represented by a function) such that the two streams are listed ascendingly according to the ordering, stream_mergeq_fun returns a stream listed ascendingly that represents the union of the two given streams such that any elements in the second stream that also occur in the first stream are dropped.

With stream-based lazy evaluation, an illusion of infinite data can be readily created. This illusion is given a simple programming interface plus automatic support for memoization, enabling a programming style that can often be both elegant and intriguing.

In general, it is difficult to estimate the time-complexity and space-complexity of a program based on lazy evaluation. This is regarded as a serious weakness. With linear stream-based lazy evalution, this weakness can essentially be removed.

Please find on-line the entirety of the code used in this chapter.