Equational Characterization of Covariant-Contravariant Simulation and Conformance Simulation Semantics

TL;DR

This paper provides axiomatizations for covariant-contravariant and conformance simulation semantics, enhancing understanding of their logical properties and the boundary between axiomatizability and non-axiomatizability.

Contribution

It introduces the first axiomatizations for these new simulation relations and explores their metatheoretical implications, especially regarding axiomatizability conditions.

Findings

Axiomatizations help understand the role of covariant and contravariant behaviors.

The equivalence is axiomatizable when no mixed actions are present.

Presence of mixed actions leads to non-axiomatizability.

Abstract

Covariant-contravariant simulation and conformance simulation generalize plain simulation and try to capture the fact that it is not always the case that "the larger the number of behaviors, the better". We have previously studied their logical characterizations and in this paper we present the axiomatizations of the preorders defined by the new simulation relations and their induced equivalences. The interest of our results lies in the fact that the axiomatizations help us to know the new simulations better, understanding in particular the role of the contravariant characteristics and their interplay with the covariant ones; moreover, the axiomatizations provide us with a powerful tool to (algebraically) prove results of the corresponding semantics. But we also consider our results interesting from a metatheoretical point of view: the fact that the covariant-contravariant simulation equivalence is indeed ground axiomatizable when there is no action that exhibits both a covariant and a contravariant behaviour, but becomes non-axiomatizable whenever we have together actions of that kind and either covariant or contravariant actions, offers us a new subtle example of the narrow border separating axiomatizable and non-axiomatizable semantics. We expect that by studying these examples we will be able to develop a general theory separating axiomatizable and non-axiomatizable semantics.