A syntactic commutativity format for SOS

Abstract

Considering operators defined using Structural Operational Semantics (SOS), commutativity axioms are intuitive properties that hold for many of them. Proving this intuition is usually a laborious task, requiring several pages of boring and standard proof. To save this effort, we propose a syntactic SOS format which guarantees commutativity for a set of composition operators.

Exploiting Algebraic Laws to Improve Mechanized Axiomatizations

“…The commutativity format presented in [37] can only deal with operators that are commutative for each pair of arguments and, unlike the format that we present below, is therefore unable to detect that f is commutative in its first two arguments.…”

“…In the setting of process algebras, commutativity is defined with respect to a notion of behavioural equivalence over terms. In this section, we first present a generalized notion of commutativity that allows n-ary operators to have various sets of commutative arguments and then slightly adapt the commutativity rule format proposed in [37] to the extended setting. Moreover, we give some auxiliary definitions that will be used in the axiomatization procedure proposed in the next section.…”

“…To this end, we begin by extending the notion of commutative congruence introduced in [37] to the context of this generalized notion of commutativity.…”

“…Unlike the format for {[n]}-commutativity given in [37], the format offered below applies to generalized commutativity, in the sense of Definition 8, and is defined for TSSs whose rules can have negative premises. On the other hand, unlike ours, the format introduced in [37] applies to rules whose positive premises need not have variables as their sources and targets. This is a straightforward generalization of our format, but is not relevant for the purpose of this paper.…”

“…Our rule format for commutativity is a generalization of the rule format for commutativity presented in [37], which allows operators to have various sets of commutative arguments. Apart from being natural, such a generalization is useful in the automatic generation of ground-complete axiomatizations, as the developments in this study show.…”