Relational Reasoning about Contexts

Abstract

The syntactic nature of operational reasoning requires techniques to deal with term contexts, especially for reasoning about recursion. In this paper we study applicative bisimulation and a variant of Sands’ improvement theory for a small call-by-value functional language. We explore an indirect, relational approach for reasoning about contexts. It is inspired by Howe’s precise method for proving congruence of simulation orderings and by Pitts’ extension thereof for proving applicative bisimulation up to context. We illustrate this approach with proofs of the unwinding theorem and syntactic continuity and, more importantly, we establish analogues of Sangiorgi’s bisimulation up to context for applicative bisimulation and for improvement. Using these powerful bisimulation up to context techniques, we give concise operational proofs of recursion induction, the improvement theorem, and syntactic minimal invariance. Previous operational proofs of these results involve complex, explicit reasoning about contexts.

Parametric polymorphism and operational equivalence

“…Partly for these reasons, we next develop a slightly more abstract, 'relational' treatment of PolyPCF contextual equivalence which avoids explicit use of contexts, following (Gordon 1998;Lassen 1998). This approach has the advantage of focussing on the key properties we want for a notion of semantic equality of PolyPCF programs, by taking them as the definition (Definition 2.2) and then deducing the existence of a greatest † relation with those properties (Theorem 2.3).…”

“…We define a notion of observational congruence for PolyPCF terms based upon observing convergence of evaluation in all contexts of list-type, but not of functionor ∀-type. However, instead of working with a conventional Morris-style definition of contextual equivalence, we define this observational congruence without mentioning PolyPCF contexts explicitly, making use of the 'relational' approach of (Lassen 1998). Not only does this avoid some low-level technicalities with binding in contexts, but it also makes it easier to state the relevant properties of logical relations later on.…”

Applicative May- and Should-Simulation in the Call-by-Value Lambda Calculus with AMB

“…[21,13,12,10]), to the best of our knowledge, no notion of applicative similarity for shouldconvergence has been studied. So in this paper we will make a first step to close this gap and investigate a notion of applicative similarity for should-convergence.…”

“…A further reason for analyzing the calculus with amb is that it is being studied for several decades (e.g. [18,2,19,13,11,10,14]) and for the contextual equivalence with may-and must-convergence it is a long standing open question whether a sound applicative similarity exists (see e.g. [10]).…”

Resolving Inductive Definitions with Binders in Higher-Order Typed Functional Programming

“…For example, having higher-order functions allows one to encode definitions that are parameterised by other definitions (such as the various operations on relations that occur in the relational approach to contextual equivalence of programs [12,17,25]). …”