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.
翻译:通过修改 Salomaa 的完整校正系统, 在语言语义 { 语义 { 校正 { 校正 { 校正 { 校正 (1984) 为其介绍的程序解释下常规表达式的平等性设计了一个健全的校正系统。 他询问这个系统是否完整 。 试图显示这个方程系统完整性的校正理论争论由于存在一种非以星文代谢方式解决固定点方程式的非校正规则而变得复杂 。 我们将固定点规则为纯方程式部分 $\ text{ MIL${ 校正 { 美元} 的推导力加到纯方言部分 ${ 校正 { 美元} 校正 } 美元。 我们将一个变式系统 $\ textlex c} 以美元取代固定规则 $C\\ text{ { mil} 美元。 校正 校正 校正 校正 校正 校正 校正 以 $ m * 的硬 校正 。