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...

. Vec is a higher-order functional language of nested arrays, which includes a general folding operation. Static computation of the shape of its programs is used to support a compositional cost calculus based on a cost monad. This, in turn, is based on a cost algebra, whose operations may be customized to handle different cost regimes, especially for parallel programming. We present examples based on sequential costing and on the PRAM model of parallel computation. The latter has been implemented in Haskell, and applied to some linear algebra examples. 1 Introduction Second-order combinators such as map, fold and zip provide programmers with a concise, abstract language for writing skeletons for implicitly parallel programs, as in [Ski94], but there is a hitch. These combinators are defined for list programs (see [BW88]), but efficient implementations (which is the point of parallelism, after all) are based on arrays. This disparity becomes acute when working with nested arrays, which...

We explore some foundational issues in the development of a theory of intensional semantics. A programming language may be given a variety of semantics, differing in the level of abstraction; one generally chooses the semantics at an abstraction level appropriate for reasoning about a particular kind of program property. Extensional semantics are typically appropriate for proving properties such as partial correctness, but an intensional semantics at a lower abstraction level is required in order to reason about computation strategy and thereby support reasoning about intensional aspects of behavior such as order of evaluation and efficiency. It is obviously desirable to be able to establish sensible relationships between two semantics for the same language, and we seek a general category-theoretic framework that permits this. Beginning with an "extensional" category, whose morphisms we can think of as functions of some kind, we model a notion of computation as a comonad with certain e...

) David Sands y Department of Computing, Imperial College 180 Queens Gate, London SW7 2BZ email: ds@uk.ac.ic.doc Abstract In this paper we address the technical foundations essential to the aim of providing a semantic basis for the formal treatment of relative efficiency in functional languages. For a general class of "functional" computation systems, we define a family of improvement preorderings which express, in a variety of ways, when one expression is more efficient than another. The main results of this paper build on Howe's study of equality in lazy computation systems, and are concerned with the question of when a given improvement relation is subject to the usual forms of (in)equational reasoning (so that, for example, we can improve an expression by improving any sub-expression). For a general class of computation systems we establish conditions on the operators of the language which guarantee that an improvement relation is a precongruence. In addition, for...

Structural Operational Semantics (SOS) is a widely used formalism for specifying the computational meaning of programs, and is commonly used in specifying the semantics of functional languages. Despite this widespread use there has been relatively little work on the imetatheoryj for such semantics. As a consequence the operational approach to reasoning is considered ad hoc since the same basic proof techniques and reasoning tools are reestablished over and over, once for each operational semantics speciøcation. This paper develops some metatheory for a certain class of SOS language speciøcations for functional languages. We deøne a rule format, Globally Deterministic SOS (gdsos), and establish some proof principles for reasoning about equivalence which are sound for all languages which can be expressed in this format. More speciøcally, if the SOS rules for the operators of a language conform to the syntax of the gdsos format, then ffl a syntactic analogy of continuity holds, which rel...

Up until now classical logic has been the logic of choice in formal hardware verification. This report advances the application of intuitionistic logic to the timing analysis of digital circuits. The intuitionistic setting serves two purposes at the same time. The model-theoretic properties are exploited to handle the second-order nature of bounded delays in a purely propositional setting without need to introduce explicit time and temporal operators. The proof theoretic properties are exploited to extract quantitative timing information and to reintroduce explicit time in a convenient and systematic way. We present a natural Kripke-style semantics for intuitionistic propositional logic, as a special case of a Kripke constraint model for Propositional Lax Logic [4], in which validity is validity up to stabilization. We show that this semantics is equivalently characterized in terms of stabilization bounds so that implication oe comes out as "boundedly gives rise to." An int...

Classical logic has so far been the logic of choice in formal hardware verification. This paper proposes the application of intuitionistic logic to the timing analysis of digital circuits. The intuitionistic setting serves two purposes. The model-theoretic properties are exploited to handle the second-order nature of bounded delays in a purely propositional setting without need to introduce explicit time and temporal operators. The proof theoretic properties are exploited to extract quantitative timing information and to reintroduce explicit time in a convenient and systematic way. We present a natural Kripke-style semantics for intuitionistic propositional logic, as a special case of a Kripke constraint model for Propositional Lax Logic [15], in which validity is validity up to stabilisation, and implication oe comes out as "boundedly gives rise to." We show that this semantics is equivalently characterised by a notion of realisability with stabilisation bounds as realisers...

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Abstract. This paper investigates what is essentially a call-by-value version of PCF under a complexity-theoretically motivated type system. The programming formalism, ATR, has its first-order programs characterize the polynomial-time computable functions, and its second-order programs characterize the type-2 basic feasible functionals of Mehlhorn and of Cook and Urquhart. (The ATR-types are confined to levels 0, 1, and 2.) The type system comes in two parts, one that primarily restricts the sizes of values of expressions and a second that primarily restricts the time required to evaluate expressions. The size-restricted part is motivated by Bellantoni and Cook’s and Leivant’s implicit characterizations of polynomial-time. The time-restricting part is an affine version of Barber and Plotkin’s DILL. Two semantics are constructed for ATR. The first is a pruning of the naïve denotational semantics for ATR. This pruning removes certain functions that cause otherwise feasible forms of recursion to go wrong. The second semantics is a model for ATR’s time complexity relative to a certain abstract machine. This model provides a setting for complexity recurrences arising from ATR recursions, the solutions of which yield second-order polynomial time bounds. The time-complexity semantics is also shown to be sound relative to the costs of interpretation on the abstract machine. 1.

One emerging approach to reducing the labour and costs of software development favours the specialisation of techniques to particular application domains. The rationale is that programs within a given domain often share enough common features and assumptions to enable the incorporation of substantial support mechanisms into domain-specific programming languages and associated tools. Instead of being machine-oriented, algorithmic implementations, programs in many domain-specific languages (DSLs) are rather user-level, problem-oriented specifications of solutions. Taken further, this view suggests that the most appropriate representation of programs in many domains is diagrammatic, in a way which derives from existing design notations in the domain. This thesis conducts an investigation, using mathematical techniques and supported by case studies, of issues arising from the use of diagrammatic representations in DSLs. Its structure is conceptually divided into two parts: the first is co...