Robustness of Equations Under Operational Extensions

TL;DR

This paper studies when behavioral equations for open terms remain valid after extending operational semantics, providing criteria and proofs for preservation across various bisimilarity notions.

Contribution

It introduces criteria for preserving sound equations under language extensions for several bisimilarity notions, including new variations ensuring preservation.

Findings

Sound equations are preserved under label-preserving extensions for fh- and hp-bisimilarity.

New variations of fh- and hp-bisimilarity ensure preservation of all sound equations.

Syntactic criteria are provided that guarantee preservation of ci-bisimilarity.

Abstract

Sound behavioral equations on open terms may become unsound after conservative extensions of the underlying operational semantics. Providing criteria under which such equations are preserved is extremely useful; in particular, it can avoid the need to repeat proofs when extending the specified language. This paper investigates preservation of sound equations for several notions of bisimilarity on open terms: closed-instance (ci-)bisimilarity and formal-hypothesis (fh-)bisimilarity, both due to Robert de Simone, and hypothesis-preserving (hp-)bisimilarity, due to Arend Rensink. For both fh-bisimilarity and hp-bisimilarity, we prove that arbitrary sound equations on open terms are preserved by all disjoint extensions which do not add labels. We also define slight variations of fh- and hp-bisimilarity such that all sound equations are preserved by arbitrary disjoint extensions. Finally, we give two sets of syntactic criteria (on equations, resp. operational extensions) and prove each of them to be sufficient for preserving ci-bisimilarity.