Milner’s Proof System for Regular Expressions Modulo Bisimilarity is Complete

Grabmayer, Clemens

doi:10.1145/3531130.3532430

Cited by 6 publications

(20 citation statements)

References 8 publications

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“…At the same time we already had in mind a possible solution to navigate this difficulty: the use of LLEE-1-chart approximations of bisimulation collapses. Indeed, we eventually described such a solution for (A) later in the summary [15] for LICS 2022 of a completeness proof of Milner's system. There, we based the development on the counterexample for LLEE-preserving bisimulation collapse that we described here in Section 5.…”

Section: Discussionmentioning

confidence: 99%

“…While we did not directly use the coinductive system cMil as a beachhead for the completeness proof later in [15], the characterization of derivability in Mil by derivability in cMil in [16] convinced us of the expediency of the underlying concepts. In particular, this result had tied derivability in Mil closely to the concept of guarded LLEE-1-charts for expressing process interpretations of regular expressions.…”

Section: Discussionmentioning

confidence: 99%

“…In particular, this result had tied derivability in Mil closely to the concept of guarded LLEE-1-charts for expressing process interpretations of regular expressions. Use of this concept, together with the fact that LLEE-1-charts are uniquely solvable modulo provability in Mil (as shown in [16], [17], [18]) were crucial stepping stones for the completeness proof of Mil that we summarize in [15].…”

Section: Discussionmentioning

confidence: 99%

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The Image of the Process Interpretation of Regular Expressions is Not Closed under Bisimulation Collapse

Grabmayer¹

2023

Preprint

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Axiomatization and expressibility problems for Milner's process semantics (1984) of regular expressions modulo bisimilarity have turned out to be difficult for the full class of expressions with deadlock 0 and empty step 1. We report on a phenomenon that arises from the added presence of 1 when 0 is available, and that brings a crucial reason for this difficulty into focus. To wit, while interpretations of 1-free regular expressions are closed under bisimulation collapse, this is not the case for the interpretations of arbitrary regular expressions.Process graph interpretations of 1-free regular expressions satisfy the loop existence and elimination property LEE, which is preserved under bisimulation collapse. These features of LEE were applied for showing that an equational proof system for 1-free regular expressions modulo bisimilarity is complete, and that it is decidable in polynomial time whether a process graph is bisimilar to the interpretation of a 1-free regular expression.While interpretations of regular expressions do not satisfy the property LEE in general, we show that LEE can be recovered by refined interpretations as graphs with 1-transitions (which are similar to silent steps for automata). This suggests that LEE can be expedient also for the general axiomatization and expressibility problems. But a new phenomenon emerges that needs to be addressed: the property of a process graph 'to can be refined into a process graph with 1-transitions and with LEE' is not preserved under bisimulation collapse. We provide a 10-vertex graph with two 1-transitions that satisfies LEE, and in which a pair of bisimilar vertices cannot be collapsed on to each other while preserving the refinement property. This implies that the image of the process interpretation of regular expressions is not closed under bisimulation collapse.

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Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

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The Image of the Process Interpretation of Regular Expressions is Not Closed under Bisimulation Collapse

Grabmayer¹

2023

Preprint

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“…We also define a theorem-equivalent system CLC ('combining LLEE-witnessed coinductive provability') with the equational coinductive proof rule alone. In the formalization of these systems we depart from the the exposition in [Gra21a,Gra21b]. There, we used a hybrid concept of formulas that included entire coinductive proofs, which then -For the proof transformation from the coinductive reformulation cMil to Milner's system Mil we show that circular coinductive proofs over the purely equational part Mil ´of Mil are admissible in Mil (see Lemma 6.10).…”

Section: G ĝmentioning

confidence: 99%

“…Relation with the completeness proof of Milner's system in [Gra22a]. The completeness proof of Milner's system Mil summarized in [Gra22a] with report [Gra22b] was finished and written only after the article [Gra21b] for CALCO 2021.…”

Section: G ĝmentioning

confidence: 99%

A Coinductive Reformulation of Milner's Proof System for Regular Expressions Modulo Bisimilarity

Grabmayer¹

2023

Logical Methods in Computer Science

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Milner (1984) defined an operational semantics for regular expressions as finite-state processes. In order to axiomatize bisimilarity of regular expressions under this process semantics, he adapted Salomaa's proof system that is complete for equality of regular expressions under the language semantics. Apart from most equational axioms, Milner's system Mil inherits from Salomaa's system a non-algebraic rule for solving single fixed-point equations. Recognizing distinctive properties of the process semantics that render Salomaa's proof strategy inapplicable, Milner posed completeness of the system Mil as an open question. As a proof-theoretic approach to this problem we characterize the derivational power that the fixed-point rule adds to the purely equational part Mil$^-$ of Mil. We do so by means of a coinductive rule that permits cyclic derivations that consist of a finite process graph with empty steps that satisfies the layered loop existence and elimination property LLEE, and two of its Mil$^{-}$-provable solutions. With this rule as replacement for the fixed-point rule in Mil, we define the coinductive reformulation cMil as an extension of Mil$^{-}$. In order to show that cMil and Mil are theorem equivalent we develop effective proof transformations from Mil to cMil, and vice versa. Since it is located half-way in between bisimulations and proofs in Milner's system Mil, cMil may become a beachhead for a completeness proof of Mil. This article extends our contribution to the CALCO 2022 proceedings. Here we refine the proof transformations by framing them as eliminations of derivable and admissible rules, and we link coinductive proofs to a coalgebraic formulation of solutions of process graphs.

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Better Automata Through Process Algebra

Cleaveland

2022

Lecture Notes in Computer Science

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Milner’s Proof System for Regular Expressions Modulo Bisimilarity is Complete

Cited by 6 publications

References 8 publications

The Image of the Process Interpretation of Regular Expressions is Not Closed under Bisimulation Collapse

The Image of the Process Interpretation of Regular Expressions is Not Closed under Bisimulation Collapse

A Coinductive Reformulation of Milner's Proof System for Regular Expressions Modulo Bisimilarity

Better Automata Through Process Algebra

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