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 \(R(\sqsubseteq , ;)\) 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 \(R(\sqsubseteq , ;)\). 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.

使用恶魔演算进行二元关系的动机来自于恶魔图灵机的关系模型。关系组合（;）对两个程序的顺序运行进行建模，并且恶魔化（\（\ sqsubseteq \））来自对程序的恶魔选择（\（\ sqcup \）进行建模而给出的部分顺序（有关正式的关系定义，请参见下文） ）。我们证明，抽象\（（\ le，\ circ）\）结构的同构类\（R（\ sqsubseteq，;）\）的类同构为二元关系集，该二元关系由具有成分的恶魔细化排序，不能被任何有限集公理一阶\（（（\ le，\ circ）\）公式。我们提供了一个定义\（R（\ sqsubseteq，;）\）的相当简单，无限的递归公理。我们证明了有限可表示的\（（（le，\ circ）\）结构具有基于有限基的表示。这似乎是具有组合的二元关系的签名的第一个示例，其中表示形式类是非无限公理的，但其中有限表示属性适用于有限结构。