A Coalgebraic Approach to Kleene Algebra with Tests

Abstract Kleene Algebra with Tests is an extension of Kleene Algebra, the algebra of regular expressions, which can be used to reason about programs. We develop a coalgebraic theory of Kleene Algebra with Tests, along the lines of the coalgebraic theory of regular expressions based on deterministic automata. Since the known automata-theoretic presentation of Kleene Algebra with Tests does not lend itself to a coalgebraic theory, we define a new interpretation of Kleene Algebra with Tests expressions and a corresponding automata-theoretic presentation. One outcome of the theory is a coinductive proof principle, that can be used to establish equivalence of our Kleene Algebra with Tests expressions.

[1]  Janusz A. Brzozowski,et al.  Derivatives of Regular Expressions , 1964, JACM.

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

[3]  Jan J. M. M. Rutten,et al.  Automata and Coinduction (An Exercise in Coalgebra) , 1998, CONCUR.

[4]  D. Kozen.  Automata on Guarded Strings and Applications , 2001 .

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

[6]  Jan J. M. M. Rutten,et al.  A note on coinduction and weak bisimilarity for while programs , 1998, RAIRO Theor. Informatics Appl..

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

[8]  Jeffrey D. Ullman,et al.  Introduction to Automata Theory, Languages and Computation , 1979 .

[9]  Dexter Kozen,et al.  On Hoare logic and Kleene algebra with tests , 1999, Proceedings. 14th Symposium on Logic in Computer Science (Cat. No. PR00158).

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

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

[12]  D. Kozen.  On Hoare Logic, Kleene Algebra, and Types , 1999 .

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

[14]  D. Kozen.  Typed Kleene Algebra , 1998 .

[15]  Dexter Kozen,et al.  Kleene algebra with tests , 1997, TOPL.

[16]  Donald M. Kaplan,et al.  Regular Expressions and the Equivalence of Programs , 1969, J. Comput. Syst. Sci..