Reasoning with first order nondeterministic specifications

Abstract

The paper presents a variant of first order logic for specifying nondeterministic software. Models of the logics are multialgebras, i.e. multi-sorted algebras with set-valued operations, together with multi-sorted valuations of variables. We allow empty carrier sets but the valuations are kept total. Terms are interpreted as sets and the usual set of algebraic terms is extended by an additional \({\sf let}\) construct used for limiting nondeterminism. Atomic formulae are of the form \(t_1 \to t_2\) where \(\to\) is a rewrite operator, corresponding semantically to inclusion. For the above logic, we develop two complete deduction systems in the natural deduction style: a Rasiowa-Sikorski system for sequences of formulae, and a cut-free Gentzen-style sequent calculus. We also consider the issues of determinism and partiality, proposing alternate solutions to defining the respective predicates in our logic.