Proof search specifications of bisimulation and modal logics for the π-calculus

Abstract

We specify the operational semantics and bisimulation relations for the finite π-calculus within a logic that contains the ∇ quantifier for encoding generic judgments and definitions for encoding fixed points. Since we restrict to the finite case, the ability of the logic to unfold fixed points allows this logic to be complete for both the inductive nature of operational semantics and the coinductive nature of bisimulation. The ∇ quantifier helps with the delicate issues surrounding the scope of variables within π-calculus expressions and their executions (proofs). We shall illustrate several merits of the logical specifications permitted by this logic: they are natural and declarative; they contain no side-conditions concerning names of variables while maintaining a completely formal treatment of such variables; differences between late and open bisimulation relations arise from familar logic distinctions; the interplay between the three quantifiers (∀, ∃, and ∇) and their scopes can explain the differences between early and late bisimulation and between various modal operators based on bound input and output actions; and proof search involving the application of inference rules, unification, and backtracking can provide complete proof systems for one-step transitions, bisimulation, and satisfaction in modal logic. We also illustrate how one can encode the π-calculus with replications, in an extended logic with induction and co-induction.

Automating Open Bisimulation Checking for the Spi Calculus

“…We prove this proposition indirectly by encoding the operational semantics of the spi-calculus into a logical framework [16], similarly to a previous work on a formalisation of the π-calculus [24]. The adequacy of the encoding can be proved similarly to the π-calculus case [24].…”

αCheck: A mechanized metatheory model checker

“…A better (proof-theoretic) way could be to relate NE to the completion by viewing logic programs as fixed points (Schroeder-Heister 1993). This view could also open the road to handle specifications that are coinductive in nature, as in concurrent calculi (Tiu and Miller 2010) or studies about program equivalence (Momigliano et al 2002). Our main contribution is showing empirically that both NAF and NE/NE − can be useful as a basis for mechanized model-checking, and the lack of answers to these questions does not detract from this contribution, but we think it would be worthwhile to study them in more detail.…”

A Lightweight Formalization of the Metatheory of Bisimulation-Up-To

“…Similarly, continuations of bound outputs (up X) are treated with the nabla-quantifier because showing that two processes, both of which can do a bound output on the same named channel, are in a given bisimulation, requires selecting a fresh new name (using the ∇ quantifier of G) and showing that the two continuations, instantiated with that name, are also in that bisimulation. The adequacy of this encoding of open bisimulation [25] for the π-calculus is shown in [27].…”

“…This theorem is proved by direct co-induction on the greatest fixpoint definitions in the Abella implementation [28] of G. It is worth remarking that such lightweight approaches to formalization have surprising consequences at times: for example, the bisimulation we define below for the π-calculus is open bisimulation. If, however, we change our formalized logic from intuitionistic to classical logic, the same specification of bisimulation becomes late bisimulation [27].…”