[TYPES] Combining lazy and eager evaluation of terms

Tim Sweeney Tim.Sweeney at epicgames.com
Fri Sep 3 16:30:46 EDT 2004

Thanks to everyone who responded on and off the list to this query.
Here is a summary of direct responses on combining lazy and eager
evaluation of terms:

Derek Dreyer pointed to his paper on supporting many useful forms of
recursion in a call-by-value language, framed in the context of ML:

"A Type System for Well-Founded Recursion"

In this paper I use a pure call-by-value evaluation, but I design the
type system so that it ensures that when evaluating a recursive
defintion (like val rec x = e), that x will not be *dereferenced* when
evaluating e, although it may be *referenced*.  Dereferencing means you
actually want to get at the underlying value (i.e. the value that e ends
up evaluating to), whereas referencing simply means you want a "pointer"
to that (as-yet-undefined) value.  (I say "pointer" because if the
underlying value is already a boxed object, there's no need to rebox it
in the underlying implementation.)  This is useful, for instance, for
purposes of separate compilation, where each definition in a pair of
mutually recursive definitions can be evaluated separately given a
pointer to the other one.  However, no high-level language that I'm
aware of actually makes a distinction between the notions of
"referencing" and "dereferencing" of recursive variables.

Paul Hudak pointed out some undesirable properties of the evaluation
strategy I proposed:

A desirable property of most languages is that they have some kind of a
"least fixed-point" semantics.  But yours seems to lack that.  In
particular, you say:

>> - By "strict with respect to divergence", I mean that for every
possible function or data constructor f, f(bot)=bot. Thus for lists,
Cons(bot,x)=Cons(x,bot)=bot. But Performing "lazy evaluation on
intermediate subterms" enables one to successfully express recursive
data structures like x=Cons(3,x) for an infinite list, or
r:=NewRef(Cons(3,r)) for a circular linked list. <<

But the least fixed-point of the equation x = Cons (3,x) is bottom,
according to the first bullet, i.e. not the infinite list of 3's.

Henrik Pilegaard pointed out the Hope language described at
http://www.soi.city.ac.uk/~ross/Hope/.  Hope performs eager evaluation
of functions, but supports lazy data structures.

Here are some other references that I found very useful:

"Implementation of Non-Strict Functional Languages"
Kenneth R. Traub

This 1991 book covers a broad variety of topics on practical
implementation, and describes a model known as "lenient evaluation", a
non-strict evaluation model that makes efficiency gains by not
supporting full lazyness.  In this model, parameters are be passed in to
functions unevaluated when necessary to make progress on evaluation.
This model can support cyclic data structures like "x:=Cons(x)" and
dependent data structures like "x:=Pair(2,x.first+1)", but not acyclic
infinite data structures such as "letrec f(n:int):=Cons(n,f(n+1)) in

"How much Non-strictness do Lenient Programs Require?"
Klaus Schauser, Seth Goldstein

This describes the lenient evaluation scheme of the Id90 functional
language, designed for efficient execution of code on parallel
architectures, and analyzes the expressive power of various forms of

Guy Tremblay has a number of interesting papers on lenient evaluation at

Some final thoughts on the topic:

Lenient evaluation appears to be an attractive model for future
mainstream programming languages, because it enables much of the
expressiveness of lazy evaluation (such as self-referential data
structures) without running into many of the fundamental limitations of
strictness analysis that prevent lazy languages from performing
competitively with strict languages; and because it enables a more
natural implementation of parallel evaluation than either lazy or strict

However, while implementations of lenient evaluation are widespread (in
Id90, pH, Octopus, and others), I've been unable to find an analysis of
the semantics of lenient evaluation.  This seems an important
outstanding problem because, as Paul Hudak points out, there are real
problems with a direct application of domain theory to such languages
because all functions have the property f(bottom)=bottom.  Any pointers
to papers on this topic would be quite welcome.

Another worrying property of a lenient evaluation scheme is that, though
infinite computations are prohibited, one can still construct cyclic
values like x=Cons(3,Cons(4,x)).  In a traditional eager language, it is
easy to prove that a function like Map (on lists) is convergent.  Here,
convergence depends on the list being finitary.  This also implies that
any built-in equality operation must either perform bisimulation (rather
than simple recursive comparison) or also be potentially divergent.  Has
there been any investigation into this topic?  I could imagine inserting
tests, upon each reduction of a thunk to head-normal form, to assure
that any non-finitary data structures cause divergence at creation-time,
therefore assuring that functions like Map can count on only receiving
acyclic inputs.  But this seems quite inefficient.

Tim Sweeney

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