Studies of the mathematical properties of impredicative polymorphic types have for the most part focused on the polymorphic lambda calculus of Girard–Reynolds, which is a calculus of total polymorphic functions. This paper considers polymorphic types from a functional programming perspective, where the partialness arising from the presence of fixpoint recursion complicates the nature of potentially infinite (‘lazy’) data types. An approach to Reynolds' notion of relational parametricity is developed that works directly on the syntax of a programming language, using a novel closure operator to relate operational behaviour to parametricity properties of types. Working with an extension of Plotkin's PCF with ∀-types, lazy lists and existential types, we show by example how the resulting logical relation can be used to prove properties of polymorphic types up to operational equivalence.

A tension in language design has been between simple semantics on the one hand, and rich possibilities for side-effects, exception handling and so on on the other. The introduction of monads has made a large step towards reconciling these alternatives. First proposed by Moggi as a way of structuring semantic descriptions, they were adopted by Wadler to structure Haskell programs, and now offer a general technique for delimiting the scope of effects, thus reconciling referential transparency and imperative operations within one programming language. Monads have been used to solve long-standing problems such as adding pointers and assignment, inter-language working, and exception handling to Haskell, without compromising its purely functional semantics. The course will introduce monads, effects and related notions, and exemplify their applications in programming (Haskell) and in compilation (MLj). The course will present typed metalanguages for monads and related categorica...

Submitted for publication to Information and Computation. A summary of this paper appeared in TACS '97.

A parametric logical relation between the phrases of an Algol-like language is presented. Its definition involves the structural operational semantics of the language, but was inspired by recent denotationally-based work of O’Hearn and Reynolds on translating Algol into a predicatively polymorphic linear lambda calculus. The logical relation yields an applicative characterisation of contextual equivalence for the language and provides a useful (and complete) method for proving equivalences. Its utility is illustrated by giving simple and direct proofs of some contextual equivalences, including an interesting equivalence due to O’Hearn which hinges upon the undefinability of ‘snapback ’ operations (and which goes beyond the standard suite of ‘Meyer-Sieber ’ examples). Whilst some of the mathematical intricacies of denotational semantics are avoided, the hard work in this operational approach lies in establishing the ‘fundamental property’ for the logical relation—the proof of which makes use of a compactness property of fixpoint recursion with respect to evaluation of phrases. But once this property has been established, the logical relation provides a verification method with an attractively low mathematical overhead. 1.

Machine The semantics presented in this section is essentially Sestoft's \mark 1" abstract machine for laziness [Sestoft 1997]. In that paper, he proves his abstract machine 6 A. K. Moran and D. Sands h fx = Mg; x; S i ! h ; M; #x : S i (Lookup) h ; V; #x : S i ! h fx = V g; V; S i (Update) h ; M x; S i ! h ; M; x : S i (Unwind) h ; x:M; y : S i ! h ; M [ y = x ]; S i (Subst) h ; case M of alts ; S i ! h ; M; alts : S i (Case) h ; c j ~y; fc i ~x i N i g : S i ! h ; N j [ ~y = ~x j ]; S i (Branch) h ; let f~x = ~ Mg in N; S i ! h f~x = ~ Mg; N; S i ~x dom(;S) (Letrec) Fig. 1. The abstract machine semantics for call-by-need. semantics sound and complete with respect to Launchbury's natural semantics, and we will not repeat those proofs here. Transitions are over congurations consisting of a heap, containing bindings, the expression currently being evaluated, and a stack. The heap is a partial function from variables to terms, and denoted in an identical manner to a coll...

This tutorial paper discusses a particular style of operational semantics that enables one to give a `syntax-directed' inductive definition of termination which is very useful for reasoning about operational equivalence of programs. We restrict attention to contextual equivalence of expressions in the ML family of programming languages, concentrating on functions involving local state. A brief tour of structural operational semantics culminates in a structural definition of termination via an abstract machine using `frame stacks'. Applications of this to reasoning about contextual equivalence are given.

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

Abstract We define a typed compiler intermediate language, MIL-lite, which incorporates computational types refined with effect information. We characterise MIL-lite observational congruence by using Howe's method to prove a ciu theorem for the language in terms of a termination predicate defined directly on the term. We then define a logical predicate which captures an observable version of the intended meaning of each of our effect annotations. Having proved the fundamental theorem for this predicate, we use it with the ciu theorem to validate a number of effect-based transformations performed by the MLj compiler for Standard ML.

Abstract not found

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...