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Finite Representability of Semigroups with Demonic Refinement
arXiv - CS - Logic in Computer Science Pub Date : 2020-09-15 , DOI: arxiv-2009.06970
Robin Hirsch and Ja\v{s} \v{S}emrl

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.

中文翻译:

具有恶魔细化的半群的有限可表示性

二元关系的组合和恶魔精化 $\sqsubseteq$ 定义为 \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 $dom( S)=\{x:\exists y (x, y)\in S\}$ 和 $R\restriction_{dom(S)}$ 表示 $R$ 对 $(x, y)$ 对的限制,其中$x\in dom(S)$。恶魔演算被引入来模拟非确定性程序的总正确性,并已应用于程序验证。我们证明了类 $R(\sqsubseteq, ;)$ 的抽象 $(\leq, \circ)$ 结构同构于一组二元关系,该二元关系由具有组合的恶魔精修排序不能被任何有限的一阶集合公理$(\leq, \circ)$ 公式。我们提供了一个相当简单的、无限的、定义 $R(\sqsubseteq, ;)$ 的递归公理化。我们证明了有限可表示的 $(\leq, \circ)$ 结构在有限基上具有表示。这似乎是具有组合的二元关系签名的第一个示例,其中表示类是非无限公理化的,但有限表示结构属性的有限表示成立。
更新日期:2020-09-16
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