Jacek Chrzaszcz

Polymorphic Subtyping Without Distributivity

The subtyping relation in the polymorphic second-order λ-calculus was introduced by John C. Mitchell in 1988. It is known that this relation is undecidable, but all known proofs of this fact strongly depend on the distributivity axiom. Nevertheless it has been conjectured that this axiom does not influence the undecidability. The paper shows undecidability of subtyping when we remove distributivity from its definition. Furthermore, the full equational axiomatisation of the corresponding equivalence relation is given. Both results follow from an analysis of rewriting-style subtyping derivations.

ML with PTIME complexity guarantees

Implicit Computational Complexity is a line of research where the possibility to inference a valid property for a program implies that the program runs in particular complexity class. Soft type systems are one of the research threads within the field. We present here a soft type system with ML-like polymorphism that enjoys decidable typechecking, type inference and typability problems and gives polynomial time computational guarantees for the running time of typed programs.

The role of polymorphism in the characterisation of complexity by soft types

Soft type assignment systems STA, STA+, and STAB characterise by means of reduction of terms the computation in complexity classes PTIME, NP, and PSPACE, respectively. All these systems are inspired by linear logic and include polymorphism similar to the one of System F. We show that the presence of polymorphism gives undecidability of typechecking and type inference. We also show that reductions in decidable monomorphic versions of these systems also capture the same complexity classes in a way sufficient for the traditional complexity theory. The reductions we propose show in addition that the monomorphic systems to serve as a programming language require some metalanguage support since the program which operates on data has form and type which depend on the size of the input.

Consistency and Completeness of Rewriting in the Calculus of Constructions

Adding rewriting to a proof assistant based on the Curry-Howard isomorphism, such as Coq, may greatly improve usability of the tool. Unfortunately adding an arbitrary set of rewrite rules may render the underlying formal system undecidable and inconsistent. While ways to ensure termination and confluence, and hence decidability of type-checking, have already been studied to some extent, logical consistency has got little attention so far. In this paper we show that consistency is a consequence of canonicity, which in turn follows from the assumption that all functions defined by rewrite rules are complete. We provide a sound and terminating, but necessarily incomplete algorithm to verify this property. The algorithm accepts all definitions that follow dependent pattern matching schemes presented by Coquand and studied by McBride in his PhD thesis. It also accepts many definitions by rewriting, containing rules which depart from standard pattern matching.

Testing of Evolving Protocols

A common assumption for the state-of-the-art methods of protocol testing is that the protocol description is precise, unambiguous and fully determined. Unfortunately, in many practical situations this assumption turns out to be unrealistic. We propose an architecture of a testing framework where different aspects and facets of protocols are separated in a clear manner so that the adaptation of the framework to amendments in protocol description is relatively straightforward. This architecture is realised in a testing framework for RCS mobile phone protocol suite we developed in cooperation with Samsung Electronics. The framework successfully went through a number of adjustments to accommodate new interpretations of RCS protocols as assimilated by the developers.