Proving Soundness of Extensional Normal-Form Bisimilarities

Abstract

Normal-form bisimilarity is a simple, easy-to-use behavioral equivalence that relates terms in λ-calculi by decomposing their normal forms into bisimilar subterms. Moreover, it typically allows for powerful up-to techniques, such as bisimulation up to context, which simplify bisimulation proofs even further. However, proving soundness of these relations becomes complicated in the presence of η-expansion and usually relies on ad hoc proof methods which depend on the language. In this paper we propose a more systematic proof method to show that an extensional normal-form bisimilarity along with its corresponding up to context technique are sound. We illustrate our technique with three calculi: the call-by-value λ-calculus, the call-by-value λ-calculus with the delimited-control operators shift and reset, and the call-by-value λ-calculus with the abortive control operators call/cc and abort. In the first two cases, there was previously no sound up to context technique validating the η-law, whereas no theory of normal-form bisimulations for a calculus with call/cc and abort has been presented before. Our results have been fully formalized in the Coq proof assistant.Different styles of bisimilarities have been defined for the λ-calculus, including applicative bisimilarity [1], normal-form bisimilarity [21] (originally called open bisimilarity in [32]), and environmental bisimilarity [33]. Applicative and environmental bisimilarities compare terms by applying them to function arguments, while normal-form bisimilarity reduces terms to normal forms, which are then decomposed into bisimilar subterms. As we can see, applicative and environmental bisimilarities still rely on some form of quantification over arguments, which is not the case of normal-form bisimilarity. As a drawback, the latter is usually not complete w.r.t. contextual equivalence-there exist contextually equivalent terms that are not normal-form bisimilar-while the former are. Like environmental bisimilarity, normal-form bisimilarity usually allows for up-to techniques [30], relations which simplify equivalence proofs of terms by having less requirements than regular bisimilarities. For example, reasoning up to context allows to forget about a common context: to equate C[t] and C[s], it is enough to relate t and s with a bisimulation up to context.In the call-by-value λ-calculus, the simplest definition of normal-form bisimilarity compares values by equating a variable only with itself, and a λ-abstraction only with a λ-abstraction such that their bodies are bisimilar. Such a definition does not respect callby-value η-expansion, since it distinguishes x from λy.x y. A less discriminating definition instead compares values by applying them to a fresh variable, thus relating λy.v y and v for any value v such that y is not free in v: given a fresh z, (λy.v y) z reduces to v z. Such a bisimilarity, that we call extensional bisimilarity, 1 relates more contextually equivalent terms, but proving its soundness as well as proving the soundness of its up-to techniqu...

A Complete Normal-Form Bisimilarity for State

“…Progress. We define simulation using the notion of diacritical progress we developed in a previous work [2,3], which distinguishes between active and passive clauses. Roughly, passive clauses are between simulation states which should be considered equal, while active clauses are between states where actual progress is taking place.…”

“…It also allows for the definition of powerful up-to techniques, relations that are easier to use than bisimulations but still imply bisimilarity. For normal-form bisimilarity, our framework enables up-to techniques which respects η-expansion [3].…”

“…The main novelty compared to usual definitions of normal-form bisimilarity [3,11] is the set of (deferred) diverging configurations used in the stuck terms…”

“…It may be worth stressing that our congruence proof is based on the machinery we have developed before [3] and is simpler than Støvring and Lassen's one, in particular in how it accounts for the extensionality of functions.…”

“…We remind in Sect. 2.3 the notion of diacritical progress [3] and the framework our bisimilarity and its proof of soundness are based upon. We sketch the completeness proof in Sect.…”