A Complete Normal-Form Bisimilarity for State

Abstract

We present a sound and complete bisimilarity for an untyped λ-calculus with higher-order local references. Our relation compares values by applying them to a fresh variable, like normal-form bisimilarity, and it uses environments to account for the evolving store. We achieve completeness by a careful treatment of evaluation contexts comprising open stuck terms. This work improves over Støvring and Lassen's incomplete environment-based normal-form bisimilarity for the λρ-calculus, and confirms, in relatively elementary terms, Jaber and Tabareau's result, that the state construct is discriminative enough to be characterized with a bisimilarity without any quantification over testing arguments.

Compositional relational reasoning via operational game semantics

“…In order to enable computation with such names, configurations contain an environment γ, which is a mapping from the relevant names to the corresponding higher-order values or evaluation contexts. We find similar notions of such environments in eager normal-form bisimulations [8], [9] and in environmental bisimulations [10], [11], [12]. Their presence is an obstacle to naive compositional reasoning, as shown by the In order to prove the terms equivalent, we might want to prove the equivalence of inc with dec and that of get with nget separately.…”

“…This is largely an orthogonal concern to capturing the shape of potential interactions in each case. It can be handled by introducing a special class of "inconsistent" worlds that guarantee non-termination in any future, as in [16], [19], [9].…”

Complete trace models of state and control

“…Typed enf-bisimulations, for a language without store and in controlpassing style, have been introduced in [24]. Fully-abstract enf-bisimulations are presented in [7] for a language with state only, corresponding to an untyped version of HOS. Earlier works in this strand include [17,29].…”

On sequentiality and well-bracketing in the π-calculus

“…Several methods have been proposed to establish contextual equivalence of sequential programs that include higher-order and stateful computation, including the abovementioned game semantics, (step-indexed Kripke) logical relations [21], [22], dedicated forms of bisimulations designed on top of an operational semantics of the languages [32], [33], [34], [35], [36]. Works like [23] or algorithmic game semantics [37], aim at automatically establishing contextual equivalences, by relying on model-checking techniques.…”