The -calculus is considered an useful mathematical tool in the study of programming languages, since programs can be identified with -terms. However, if one goes further and uses fij-conversion to prove equivalence of programs, then a gross simplification 1 is introduced, that may jeopardise the applicability of theoretical results to real situations. In this paper we introduce a new calculus based on a categorical semantics for computations. This calculus provides a correct basis for proving equivalence of programs, independent from any specific computational model. 1 Introduction This paper is about logics for reasoning about programs, in particular for proving equivalence of programs. Following a consolidated tradition in theoretical computer science we identify programs with the closed -terms, possibly containing extra constants, corresponding to some features of the programming language under consideration. There are three approaches to proving equivalence of programs: ffl T...

Abstract. A formulation of semantic theories for processes which is based on reduction relation and equational reasoning is studied. The new construction can induce meaningful theories for processes, both in strong and weak settings. The resulting theories in many cases coincide with, and sometimes generalise, observation-based formulation of behavioural equivalence. The basic construction of reduction-based theories is studied, taking a simple name passing calculus called \nu-calculus as an example. Results on other calculi are also briefly discussed.

) Davide Sangiorgi LFCS - Department of Computer Science Edinburgh University Edinburgh - EH9 3JZ - UK Abstract The use of lambda calculus in richer settings, possibly involving parallelism, is examined in terms of its effect on the equivalence between lambda terms. We concentrate here on Abramsky's lazy lambda calculus and we follow two directions. First, the lambda calculus is studied within a process calculus by examining the equivalence $ induced by Milner's encoding into the -calculus. We give exact operational and denotational characterizations for $. Secondly, we examine Abramsky's applicative bisimulation when the lambda calculus is augmented with (well-formed) operators, i.e. symbols equipped with reduction rules describing their behaviour. Then, maximal discrimination is obtained when all operators are considered; we show that this discrimination coincides with the one given by $ and that the adoption of certain non-deterministic operators is sufficient and necessary...

In this paper we introduce a variable typed logic of effects inspired by the variable type systems of Feferman for purely functional languages. VTLoE (Variable Typed Logic of Effects) is introduced in two stages. The first stage is the first-order theory of individuals built on assertions of equality (operational equivalence `a la Plotkin), and contextual assertions. The second stage extends the logic to include classes and class membership. The logic we present provides an expressive language for defining and studying properties of programs including program equivalences, in a uniform framework. The logic combines the features and benefits of equational calculi as well as program and specification logics. In addition to the usual first-order formula constructions, we add contextual assertions. Contextual assertions generalize Hoare's triples in that they can be nested, used as assumptions, and their free variables may be quantified. They are similar in spirit to program modalities in ...

Models of intuitionistic linear logic also provide models of Moggi's computational metalanguage. We use the adjoint presentation of these models and the associated adjoint calculus to show that three translations, due mainly to Moggi, of the lambda calculus into the computational metalanguage (direct, call-by-name and call-by-value) correspond exactly to three translations, due mainly to Girard, of intuitionistic logic into intuitionistic linear logic. We also consider extending these results to languages with recursion. 1. Introduction Two of the most significant developments in semantics during the last decade are Girard's linear logic [10] and Moggi's computational metalanguage [14]. Any student of these formalisms will suspect that there are significant connections between the two, despite their apparent differences. The intuitionistic fragment of linear logic (ILL) may be modelled in a linear model -- a symmetric monoidal closed category with a comonad ! which satisfies some extr...

We show how to translate the region calculus of Tofte and Talpin, a typed lambda calculus that can statically delimit the lifetimes of objects, into an extension of the polymorphic lambda calculus called F # . We give a denotational semantics of F # , and use it to give a simple and abstract proof of the correctness of memory deallocation. 1 Introduction Implementations of modern programming languages divide dynamically allocated memory into two parts. The stack is used for data that has a simple last-in, first-out lifetime determined by block structure; the other part (often called the heap) is used for data whose lifetime extends beyond the scope of program blocks. The heap is periodically "garbage collected" to reclaim memory that is no longer needed. Tofte and Talpin's region calculus [23] attempts to unify these two styles of memory management. The region calculus divides memory into regions, and provides a local scoping mechanism for those regions. Every value created by the pro...

Developing a theory of bisimulation in higher-order languages can be hard. Particularly challenging can be: (1) the proof of congruence, as well as enhancements of the bisimulation proof method with “up-to context ” techniques, and (2) obtaining definitions and results that scale to languages with different features. To meet these challenges, we present environmental bisimulations, a form of bisimulation for higher-order languages, and its basic theory. We consider four representative calculi: pure λ-calculi (call-by-name and call-byvalue), call-by-value λ-calculus with higher-order store, and then Higher-Order π-calculus. In each case: we present the basic properties of environmental bisimilarity, including congruence; we show that it coincides with contextual equivalence; we develop some up-to techniques, including up-to context, as examples of possible enhancements of the associated bisimulation method. Unlike previous approaches (such as applicative bisimulations, logical relations, Sumii-Pierce-Koutavas-Wand), our method does not require induction/indices on evaluation derivation/steps (which may complicate the proofs of congruence, transitivity, and the combination with up-to techniques), or sophisticated methods such as Howe’s for proving congruence. It also scales from the pure λ-calculi to the richer calculi with simple congruence proofs. 1

This course introduces the operational and denotational semantics of PCF and examines the relationship between the two. Topics: Syntax and operational semantics of PCF, Activity Lemma, undefinability of parallel or; Context Lemma (first principles proof) and proof by logical relations Denotational semantics of PCF induced by an interpretation; (standard) Scott model, adequacy, weak adequacy and its proof (by a computability predicate) Domain Theory up to SFP and Scott domains; non full abstraction of the standard model, definability of compact elements and full abstraction for PCFP (PCF + parallel or), properties of order-extensional (continuous) models of PCF, Milner's model and Mulmuley's construction (excluding proofs) Additional topics (time permitting): results on pure simply-typed lambda calculus, Friedman 's Completeness Theorem, minimal model, logical relations and definability, undecidability of lambda definability (excluding proof), dI-domains and stable functions Homepa...

This paper presents a typed higher-order concurrent functional programming language, based on Moggi's monadic metalanguage and Reppy's Concurrent ML. We present an operational semantics for the language, and show that a higherorder variant of the traces model is fully abstract for maytesting. This proof uses a program logic based on Hennessy-- Milner logic and Abramsky's domain theory in logical form. 1 Introduction This paper presents an operational semantics for a concurrent functional programming language, based on Reppy's [26, 27] Concurrent ML, and Moggi's [22] monadic metalanguage. CML is a concurrent extension of New Jersey ML, which adds communication primitives based on CCS [19] and CSP [11]. Reppy introduces a new type constructor of events, which can spawn concurrent processes, and communicate with them along channels. Three of the constructors for the event type are: always : a#aevent wrap : (aeventa#b)# (bevent) sync : aevent#a These are: . alwayse is an event whic...

) 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...