A Logic for Parametric Polymorphism

In this paper we introduce a logic for parametric polymorphism. Just as LCF is a logic for the simply-typed -calculus with recursion and arithmetic, our logic is a logic for System F. The logic permits the formal presentation and use of relational parametricity. Parametricity yields—for example—encodings of initial algebras, final co-algebras and abstract datatypes, with corresponding proof principles of induction, co-induction and simulation

Relational Parametricity for Control Considered as a Computational Effect

AbstractThis paper investigates parametric polymorphism in the presence of control operators. Our approach is to specialise a general type theory combining polymorphism and computational effects, by extending it with additional constants expressing control. By defining relationally parametric models of this extended calculus, we capture the interaction between parametricity and control. As a worked example, we show that recent results of M. Hasegawa on type definability in the second-order (call-by-name) λμ-calculus arise as special cases of general results valid for arbitrary computational effects

Selective Strictness and Parametricity in Structural Operational Semantics, Inequationally

Parametric polymorphism constrains the behavior of pure functional pro-grams in a way that allows the derivation of interesting theorems about them solely from their types, i.e., virtually for free. The formal background of such ‘free theorems’ is well developed for extensions of the Girard-Reynolds polymorphic lambda calculus by algebraic datatypes and general recursion, provided the resulting calculus is endowed with either a purely strict or a purely nonstrict semantics. But modern functional languages like Clean and Haskell, while using nonstrict evaluation by default, also provide means to enforce strict evaluation of subcomputations at will. The resulting selective strictness gives the advanced programmer explicit control over evaluation order, but is not without semantic consequences: it breaks standard parametricity results. This paper develops an operational semantics for a core calculus supporting all the language features emphasized above. Its main achievement is the characterization of observational approximation with respect to this operational semantics via a carefully constructed logical relation. This establishes the formal basis for new parametricity results, as illustrated by several example applications, including the first complete correctness proof for short cut fusion in the presence of selective strictness. The focus on observational approximation, rather than equivalence, allows a finer-grained analysis of computational behavior in the presence of selective strictness than would be possible with observational equivalence alone

Abstraction Barriers and Refinement in the Polymorphic Lambda Calculus

This thesis examines specification refinement in the setting of polymorphic type theory and a complementary logic for relational parametricity. The starting point is the specification of abstract data types as done in the discipline of algebraic specification. Here, algebras are seen to match the standard notion of data type, i.e., a data representation together with operations on that data representation. An abstract data type is then a collection of data types sharing some well-defined abstract properties. In algebraic specification, these properties are specified algebraically by axioms in some suitable logic. Specification refinement then encompasses the idea that high-level specifications may be stepwise refined to executable programs that satisfy the initial specification; all in the framework of formal language and logic. This makes certain aspects of program development amenable to formal, computer-aided proofs of correctness. On the other hand, the discipline of type theory, lambda calculus, and its semantics is the prime field for research on programming languages. This framework is capable of characterising essentially any existing sequential programming-language feature, also advanced features such as recursive types, polymorphism and class-based object orientation. Furthermore, type theory provides a powerful framework for mechanised reasoning. This thesis is a contribution to lifting the idea of algebraic specification refinement into the more powerful domain of type theory and lambda calculus, thus giving the opportunity to expand in a sensible way a traditionally first order and functional framework to a wider range of programming aspects. We take a particular account of specification refinement and express it in a type-theoretic setting consisting of the polymorphic lambda calculus and a logic for relational parametricity. Key elements of algebraic specification are internalised in the syntax, e.g., data types viz. algebras are inhabitants of existential type, the latter providing essential data abstraction. For data types with only first-order operations, this setting automatically resolves certain issues of specification refinement, such as observational equivalence, stability and input sorts. After establishing a correspondence at first order, thus implanting the idea of algebraic specification refinement into the type-theoretic setting, the scene is set for lifting the idea of algebraic specification refinement to any number of programming features. In this thesis we focus on the generalisations to higher-order functions and to polymorphism. A simulation relation between two data types is a relation between their data representations that is preserved by their respective sets of operations. Using simulation relations is a classical way of explaining data refinement and observational equivalence. This combines with specification refinement to form specification refinement up to observational equivalence. With higher-order operations, however, we encounter in the logic a phenomenon related to what happens on the semantic level, i.e., the standard notion of refinement relation in the form of logical relations does not compose and the correspondence with observational equivalence is lost. In the logic it turns out that the standard notion of simulation relation fails to take into account a certain aspect of the abstraction barrier provided by existential types. We remedy this by proposing an alternative notion of simulation relation that observes this abstraction barrier more closely. We do this in two related ways; one relates to syntactic models while the other relates to a non-syntactic PER-model more apt for interpretive investigations. In algebraic specification, there is a universal proof method for specification refinement up to observational equivalence. This method can be imported soundly into the type-theoretic setting by asserting certain axioms. At first order, showing soundness for these axioms is straight-forward w.r.t. the standard parametric PER model for the logic. At higher order there are two problems. First, these axioms seemingly do not hold in the standard model. Secondly, the axioms speak in terms of simulation relations. At higher order, it is pertinent to have versions of the axioms featuring the abstraction barrier-observing simulation relations above, and to prove soundness for these poses an additional challenge. We show that the pure higher-order aspect of this problem can be solved by giving a setoid-based semantics. For the remaining task, we continue working from the observation that standard definitions do not observe abstraction barriers closely enough. Hence, we propose an alternative interpretation into the PER-model for data types that captures the abstraction barrier provided by existential types. The main contribution of this thesis is thus in generalising a prominent account of specification refinement to higher order and polymorphism via type theory incorporating relational parametricity. We also shed light on short-comings in the logic, as well as in the standard semantics, regarding the abstraction barrier provided by existential types. Two central contributions, namely abstraction barrier-observing simulation relations and abstraction barrier-observing semantics for data types, are the result of observing these short-comings. Finally, the work in this thesis also lays a foundation on which to adapt specification refinement to an object-oriented setting, because the theoretical concepts underlying object orientation can be seen as extensions of those for abstract data types

Programming Languages and Systems

This open access book constitutes the proceedings of the 29th European Symposium on Programming, ESOP 2020, which was planned to take place in Dublin, Ireland, in April 2020, as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2020. The actual ETAPS 2020 meeting was postponed due to the Corona pandemic. The papers deal with fundamental issues in the specification, design, analysis, and implementation of programming languages and systems