The motivation for using demonic calculus for binary relations stems from the behaviour of demonic turing machines, when modelled relationally. Relational composition (; ) models sequential runs of two programs and demonic refinement ($$\sqsubseteq $$ ⊑ ) arises from the partial order given by modeling demonic choice ($$\sqcup $$ ⊔ ) of programs (see below for the formal relational definitions). We prove that the class $$R(\sqsubseteq , ;)$$ R ( ⊑ , ; ) of abstract $$(\le , \circ )$$ ( ≤ , ∘ ) structures isomorphic to a set of binary relations ordered by demonic refinement with composition cannot be axiomatised by any finite set of first-order $$(\le , \circ )$$ ( ≤ , ∘ ) formulas. We provide a fairly simple, infinite, recursive axiomatisation that defines $$R(\sqsubseteq , ;)$$ R ( ⊑ , ; ) . We prove that a finite representable $$(\le , \circ )$$ ( ≤ , ∘ ) structure has a representation over a finite base. This appears to be the first example of a signature for binary relations with composition where the representation class is non-finitely axiomatisable, but where the finite representation property holds for finite structures.

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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.

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