Techniques for reasoning about extensional properties of functional programs are well understood, but methods for analysing the underlying intensional or operational properties have been much neglected. This paper begins with the development of a simple but useful calculus for time analysis of non-strict functional programs with lazy lists. One limitation of this basic calculus is that the ordinary equational reasoning on functional programs is not valid. In order to buy back some of these equational properties we develop a non-standard operational equivalence relation called cost equivalence, by considering the number of computation steps as an `observable' component of the evaluation process. We define this relation by analogy with Park's definition of bisimulation in CCS. This formulation allows us to show that cost equivalence is a contextual congruence (and thus is substitutive with respect to the basic calculus) and provides useful proof techniques for establishing cost-equivalen...

Projection analysis is a technique for finding out information about lazy functional programs. We show how the information obtained from this analysis can be used to speed up sequential implementations, and introduce parallelism into parallel implementations. The underlying evaluation model is evaluation transformers, where the amount of evaluation that is allowed of an argument in a function application depends on the amount of evaluation allowed of the application. We prove that the transformed programs preserve the semantics of the original programs. Compilation rules, which encode the information from the analysis, are given for sequential and parallel machines. 1 Introduction A number of analyses have been developed which find out information about programs. The methods that have been developed fall broadly into two classes, forwards analyses such as those based on the ideas of abstract interpretation (e.g. [9, 18, 19, 7, 17, 12, 4, 20]), and backward analyses such as those based...

. A new denotational semantics is introduced for realistic non-strict functional languages, which have a polymorphic type system and support higher order functions and user definable algebraic data types. It maps each function definition to a demand propagator, which is a higher order function, that propagates context demands to function arguments. The relation of this "higher order demand propagation semantics" to the standard semantics is explained and it is used to define a backward strictness analysis. The strictness information deduced by this analysis is very accurate, because demands can actually be constructed during the analysis. These demands conform better to the analysed functions than abstract values, which are constructed alone with respect to types like in other existing strictness analyses. The richness of the semantic domains of higher order demand propagation makes it possible to express generalised strictness information for higher order functions even ac...

We propose a method for performing backward analysis on higher--order functional programming languages based on computing inverse images of functions over abstract domains. This method can be viewed as abstract interpretation done backward. Given an abstract semantics which supports forward analysis, we can transform it into an abstract semantics which performs backward analysis. We show that if the original abstract semantics is correct and computable, then the transformed version of the abstract semantics is also correct and computable. More specifically, given a forward abstract semantics of a higher--order functional language which is expressed in terms of Scott--closed powerdomains, we derive an backward abstraction semantics which is expressed in terms of Scott--open powerdomains. The derivation is shown to be correct and the relationships between forward analysis and backward analysis is established. We apply this method to the classic strictness analysis in functional languages...

This paper presents a novel approach to strictness analysis called abstract demand propagation, approach developed for the implementation of lazy functional programming languages on parallel machines. Although some work on strictness analysis using demand propagation has been done before, the present work is original in that it gives a precise interpretation of the notions of demands and demand propagation using an exact nonstandard denotational semantics. The intuition behind this semantics is that it represents a form of inverse computation, i.e., it determines the least amount of information needed to produce at least some required (demanded) result. Viewing demand propagation as a form of inverse computation allows to establish the soundness of our non-standard semantics by formally relating it with the standard semantics. In order to define a compile-time analysis based on demand propagation, safety and termination must be ensured. This is done by defining an abstract interpreta...