Towards sharing in lazy computation systems

Work on proving congruence of bisimulation in functional programming languages often refers to [How89,How96], where Howe gave a highly general account on this topic in terms of so-called lazy computation systems . Particularly in implementations of lazy functional languages, sharing plays an eminent role. In this paper we will show how the original work of Howe can be extended to cope with sharing. Moreover, we will demonstrate the application of our approach to the call-by-need lambda-calculus lambda-ND which provides an erratic non-deterministic operator pick and a non-recursive let. A definition of a bisimulation is given, which has to be based on a further calculus named lambda-~, since the na1ve bisimulation definition is useless. The main result is that this bisimulation is a congruence and contained in the contextual equivalence. This might be a step towards defining useful bisimulation relations and proving them to be congruences in calculi that extend the lambda-ND-calculus

Formalizing Program Equivalences in Dependent Type Theory

This brief note summarizes our formalization in a dependently typed setting of the meta-theory of several notions of program equivalences in higher-order programming languages

On well-foundedness and expressiveness of promoted tyft : being promoted makes a difference

In this paper, we solve two open problems posed by Karen L. Bernstein regarding her promoted tyft format for structured operational semantics. We show that, unlike formats with closed terms as labels, such as the tyft format, the well-foundedness assumption cannot be dropped for the promoted tyft format while preserving the congruence result. We also show that the well-founded promoted tyft format is incomparable to the tyft format with closed terms as labels, i.e., there are transition relations that can be specified by the promoted tyft format but not by the tyft format, and vice versa

Contextual Equivalences in Call-by-Need and Call-By-Name Polymorphically Typed Calculi (Preliminary Report)

This paper presents a call-by-need polymorphically typed lambda-calculus with letrec, case, constructors and seq. The typing of the calculus is modelled in a system-F style. Contextual equivalence is used as semantics of expressions. We also define a call-by-name variant without letrec. We adapt several tools and criteria for recognizing correct program transformations to polymorphic typing, in particular an inductive applicative simulation

Bisimilarity as a Theory of Functional Programming

AbstractMorris-style contextual equivalence — invariance of termination under any context of ground type — is the usual notion of operational equivalence for deterministic functional languages such as FPC (PCF plus sums, products and recursive types). Contextual equivalence is hard to establish directly. Instead we define a labelled transition system for call-by-name FPC (and variants) and prove that CCS-style bisimilarity equals contextual equivalence — a form of operational extensionality. Using co-induction we establish equational laws for FPC. By considering variations of Milner's ‘bisimulations up to ∼’ we obtain a second co-inductive characterisation of contextual equivalence in terms of reduction behaviour and production of values. Hence we use co-inductive proofs to establish contextual equivalence in a series of stream-processing examples. Finally, we consider a form of Milner's original context lemma for FPC, but conclude that our form of bisimilarity supports simpler co-inductive proofs