Eager 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

“…In this respect, RTSs are again similar to normal form bisimulations [21,34], which are sometimes easier to prove congruent in their non-η-supporting formulations. There are known ways to close normal form bisimulations over η-equivalence by complicating the definition of consistency, and it is possible that we could adapt such techniques to work for RTSs.…”

“…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.…”

“…• 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).…”

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.…”

Typed Normal Form Bisimulation

“…Normal form bisimulation-also known as open (applicative) bisimulationoriginated as a coinductive way of describing Lévy-Longo tree equivalence for the lazy λ-calculus [1], and has subsequently been extended to call-by-name, call-by-value, nondeterminism, aspects, storage, and control [2][3][4][5][6][7].…”