Demonic Lattices and Semilattices in Relational Semigroups with Ordinary Composition

TL;DR

This paper formalizes relational reasoning about total correctness in nondeterministic programs using demonic lattice semigroups, revealing undecidability and non-finite axiomatisability results in their representability.

Contribution

It introduces a formal framework for demonic lattice semigroups in relational reasoning and proves key undecidability and non-finite axiomatisability results.

Findings

Demonic join semigroups are not finitely axiomatisable.

Representation class of demonic meet semigroups lacks finite representation property.

Representation problem for finite lattice semigroups is undecidable.

Abstract

Relation algebra and its reducts provide us with a strong tool for reasoning about nondeterministic programs and their partial correctness. Demonic calculus, introduced to model the behaviour of a machine where the demon is in control of nondeterminism, has also provided us with an extension of that reasoning to total correctness. We formalise the framework for relational reasoning about total correctness in nondeterministic programs using semigroups with ordinary composition and demonic lattice operations. We show that the class of representable demonic join semigroups is not finitely axiomatisable and that the representation class of demonic meet semigroups does not have the finite representation property for its finite members. For lattice semigroups (with composition, demonic join and demonic meet) we show that the representation problem for finite algebras is undecidable, moreover the finite representation problem is also undecidable. It follows that the representation class is not finitely axiomatisable, furthermore the finite representation property fails.