Typed Normal Form Bisimulation

Abstract

Abstract. Normal form bisimulation is a powerful theory of program equivalence, originally developed to characterize Lévy-Longo tree equivalence and Boehm tree equivalence. It has been adapted to a range of untyped, higher-order calculi, but types have presented a difficulty. In this paper, we present an account of normal form bisimulation for types, including recursive types. We develop our theory for a continuation-passing style calculus, Jump-With-Argument (JWA), where normal form bisimilarity takes a very simple form. We give a novel congruence proof, based on insights from game semantics. A notable feature is the seamless treatment of eta-expansion. We demonstrate the normal form bisimulation proof principle by using it to establish a syntactic minimal invariance result and the uniqueness of the fixed point operator at each type.

The marriage of bisimulations and Kripke logical relations

“…From normal form (or open) bisimulations [32,21,34,22,23], we take the idea of treating unknown equivalent functions as black boxes. In particular, our expression equivalence relation E, which deals explicitly with the possibility (in its third disjunct) that related terms may get stuck by calling unknown functions, is highly reminiscent of the formulation of normal form bisimulations.…”

“…As a result, there are certain examples that have appeared in the literature on relational reasoning for ML-like languages [36,22,11], which our method cannot handle, precisely because they depend fundamentally on η-equivalence. The best-known one is the syntactic minimal invariance example [30], which demonstrates that the "infinite η-expansion" at a general recursive type (e.g., µα.unit + (α → α)) is equivalent to the identity function.…”

“…• Environmental bisimulations [36,19,33,35] [21,34,22,23] sidestep the whole question by choosing a fresh variable name x and representing equivalent arguments by the same x. As a result, these bisimulations are built over open terms, and proof obligation (1) above must be updated to account for the possibility that the evaluations of e1 and e2 get stuck trying to deconstruct the same free variable x (more about that below).…”

Open bisimulation for aspects

“…Our approach to stateful programs is in particular closely related to the concurrently and independently developed treatment of sequential control and state [63] following this approach. Furthermore, the precise relationship of this style to game-theoretic semantics of programming languages [3,26] has by now been formalized [40].…”

Transition systems over games

“…One is game semantics using pointers [10], a form of denotational semantics that has been widely adapted successfully adapted to many language features, including general references [3,27], control operators [19], exceptions [20] and polymorphism [22,23]. The other is open (aka normal form) bisimulation [28], a convenient operational technique for establishing observational equivalences in various settings [13,24,25,26,29], based on a transition system constructed from the syntax of the calculus.…”