Finite Representability of Semigroups with Demonic Refinement

Abstract

Composition and demonic refinement $\sqsubseteq$ of binary relations are defined by \begin{align*} (x, y)\in (R;S)&\iff \exists z((x, z)\in R\wedge (z, y)\in S) R\sqsubseteq S&\iff (dom(S)\subseteq dom(R) \wedge R\restriction_{dom(S)}\subseteq S) \end{align*} where $dom(S)=\{x:\exists y (x, y)\in S\}$ and $R\restriction_{dom(S)}$ denotes the restriction of $R$ to pairs $(x, y)$ where $x\in dom(S)$. Demonic calculus was introduced to model the total correctness of non-deterministic programs and has been applied to program verification. We prove that the class $R(\sqsubseteq, ;)$ of abstract $(\leq, \circ)$ structures isomorphic to a set of binary relations ordered by demonic refinement with composition cannot be axiomatised by any finite set of first-order $(\leq, \circ)$ formulas. We provide a fairly simple, infinite, recursive axiomatisation that defines $R(\sqsubseteq, ;)$. We prove that a finite representable $(\leq, \circ)$ 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 representations for finite representable structures property holds.