当前位置: X-MOL 学术arXiv.cs.FL › 论文详情
Our official English website, www.x-mol.net, welcomes your feedback! (Note: you will need to create a separate account there.)
A Coinductive Version of Milner's Proof System for Regular Expressions Modulo Bisimilarity
arXiv - CS - Formal Languages and Automata Theory Pub Date : 2021-08-30 , DOI: arxiv-2108.13104
Clemens Grabmayer

By adapting Salomaa's complete proof system for equality of regular expressions under the language semantics, Milner (1984) formulated a sound proof system for bisimilarity of regular expressions under the process interpretation he introduced. He asked whether this system is complete. Proof-theoretic arguments attempting to show completeness of this equational system are complicated by the presence of a non-algebraic rule for solving fixed-point equations by using star iteration. We characterize the derivational power that the fixed-point rule adds to the purely equational part $\text{Mil$^{\boldsymbol{-}}$}$ of Milner's system $\text{$\text{Mil}$}$: it corresponds to the power of coinductive proofs over $\text{Mil$^{\boldsymbol{-}}$}$ that have the form of finite process graphs with the loop existence and elimination property $\text{LEE}$. We define a variant system $\text{cMil}$ by replacing the fixed-point rule in $\text{Mil}$ with a rule that permits $\text{LEE}$-shaped circular derivations in $\text{Mil$^{\boldsymbol{-}}$}$ from previously derived equations as a premise. With this rule alone we also define the variant system $\text{CLC}$ for merely combining $\text{LEE}$-shaped coinductive proofs over $\text{Mil$^{\boldsymbol{-}}$}$. We show that both $\text{cMil}$ and $\text{CLC}$ have proof interpretations in $\text{Mil}$, and vice versa. As this correspondence links, in both directions, derivability in $\text{Mil}$ with derivation trees of process graphs, it widens the space for graph-based approaches to finding a completeness proof of Milner's system. This report is the extended version of a paper with the same title accepted for CALCO 2021.

中文翻译:

用于正则表达式模双相似性的米尔纳证明系统的共归纳版本

Milner (1984) 通过改编 Salomaa 的语言语义下正则表达式等式的完整证明系统,制定了他引入的过程解释下正则表达式的双相似性的完善证明系统。他问这个系统是否完整。由于存在使用星形迭代求解不动点方程的非代数规则,试图证明该方程系统完整性的证明理论论证变得复杂。我们表征了定点规则添加到米尔纳系统 $\text{$\text{Mil}$}$ 的纯方程部分 $\text{Mil$^{\boldsymbol{-}}$}$ :它对应于 $\text{Mil$^{\boldsymbol{-}}$}$ 上的共归纳证明的能力,该证明具有有限过程图的形式,具有循环存在和消除属性 $\text{LEE}$。我们通过将 $\text{Mil}$ 中的定点规则替换为允许 $\text{Mil$ 中的 $\text{LEE}$ 形状的圆形导数的规则来定义变体系统 $\text{cMil}$ ^{\boldsymbol{-}}$}$ 从先前导出的方程作为前提。仅凭这条规则,我们还定义了变体系统 $\text{CLC}$ 仅用于在 $\text{Mil$^{\boldsymbol{-}}$}$ 上组合 $\text{LEE}$ 形状的共归纳证明。我们证明 $\text{cMil}$ 和 $\text{CLC}$ 在 $\text{Mil}$ 中都有证明解释,反之亦然。由于这种对应关系在两个方向上将 $\text{Mil}$ 中的可推导性与过程图的推导树联系起来,它拓宽了基于图的方法来寻找 Milner 系统的完整性证明的空间。本报告是 CALCO 2021 接受的同名论文的扩展版本。
更新日期:2021-08-31
down
wechat
bug