On Star Expressions and Coalgebraic Completeness Theorems

An open problem posed by Milner asks for a proof that a certain axiomatisation, which Milner showed is sound with respect to bisimilarity for regular expressions, is also complete. One of the main difficulties of the problem is the lack of a full Kleene theorem, since there are automata that can not be specified, up to bisimilarity, by an expression. Grabmayer and Fokkink (2020) characterise those automata that can be expressed by regular expressions without the constant 1, and use this characterisation to give a positive answer to Milner’s question for this subset of expressions. In this paper, we analyse Grabmayer and Fokkink’s proof of completeness from the perspective of universal coalgebra, and thereby give an abstract account of their proof method. We then compare this proof method to another approach to completeness proofs from coalgebraic language theory. This culminates in two abstract proof methods for completeness, what we call the local and global approaches, and a description of when one method can be used in place of the other.

[1]  Jos C. M. Baeten,et al.  A characterization of regular expressions under bisimulation , 2007, JACM.

[2]  Hans Zantema,et al.  Termination Modulo Equations by Abstract Commutation with an Application to Iteration , 1997, Theor. Comput. Sci..

[3]  Hans Zantema,et al.  Basic Process Algebra with Iteration: Completeness of its Equational Axioms , 1993, Comput. J..

[4]  Robin Milner,et al.  A Complete Inference System for a Class of Regular Behaviours , 1984, J. Comput. Syst. Sci..

[5]  Dexter Kozen,et al.  Guarded Kleene Algebra with Tests: Coequations, Coinduction, and Completeness , 2021, ICALP.

[6]  Clemens Grabmayer,et al.  A Complete Proof System for 1-Free Regular Expressions Modulo Bisimilarity , 2020, LICS.

[7]  Arto Salomaa,et al.  Two Complete Axiom Systems for the Algebra of Regular Events , 1966, JACM.

[8]  Stefan Milius.  A Sound and Complete Calculus for Finite Stream Circuits , 2010, 2010 25th Annual IEEE Symposium on Logic in Computer Science.

[9]  Jan A. Bergstra,et al.  Process Algebra with Iteration and Nesting , 1994, Comput. J..

[10]  Bart Jacobs,et al.  Introduction to Coalgebra: Towards Mathematics of States and Observation , 2016, Cambridge Tracts in Theoretical Computer Science.

[11]  Dexter Kozen.  A Completeness Theorem for Kleene Algebras and the Algebra of Regular Events , 1994, Inf. Comput..

[12]  Wan Fokkink,et al.  Axiomatizations for the Perpetual Loop in Process Algebra , 1997, ICALP.

[13]  Dexter Kozen,et al.  Kleene Algebra with Tests: Completeness and Decidability , 1996, CSL.

[14]  H. Gumm.  Elements Of The General Theory Of Coalgebras , 1999 .

[15]  H. Peter Gumm,et al.  Covarieties and Complete Covarieties , 1998, CMCS.

[16]  Bart Jacobs,et al.  A Bialgebraic Review of Deterministic Automata, Regular Expressions and Languages , 2006, Essays Dedicated to Joseph A. Goguen.

[17]  H. Gumm.  Functors for Coalgebras , 2001 .

[18]  Justin Hsu,et al.  Guarded Kleene algebra with tests: verification of uninterpreted programs in nearly linear time , 2019, Proc. ACM Program. Lang..

[19]  Alexandra Silva,et al.  Sound and Complete Axiomatizations of Coalgebraic Language Equivalence , 2011, TOCL.

[20]  Peter Jipsen.  Concurrent Kleene Algebra with Tests , 2014, RAMICS.

[21]  Jan J. M. M. Rutten,et al.  Universal coalgebra: a theory of systems , 2000, Theor. Comput. Sci..

[22]  Alexandra Silva,et al.  Concurrent Kleene Algebra: Free Model and Completeness , 2017, ESOP.

[23]  J. Conway.  Regular algebra and finite machines , 1971 .

[24]  S C Kleene,et al.  Representation of Events in Nerve Nets and Finite Automata , 1951 .

[25]  Alexandra Silva,et al.  Non-Deterministic Kleene Coalgebras , 2010, Log. Methods Comput. Sci..