An Algebra for Kripke Polynomial Coalgebras

Several dynamical systems, such as deterministic automata and labelled transition systems, can be described as coalgebras of so-called Kripke polynomial functors, built up from constants and identities, using product, coproduct and powerset. Locally finite Kripke polynomial coalgebras can be characterized up to bisimulation by a specification language that generalizes Kleene’s regular expressions for finite automata. In this paper, we equip this specification language with an axiomatization and prove it sound and complete with respect to bisimulation, using a purely coalgebraic argument. We demonstrate the usefulness of our framework by providing a finite equational system for (non-)deterministic finite automata, la-belled transition systems with explicit termination, and automata on guarded strings.

[1]  Lutz Schröder,et al.  A finite model construction for coalgebraic modal logic , 2006, J. Log. Algebraic Methods Program..

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

[3]  Dirk Pattinson,et al.  Beyond Rank 1: Algebraic Semantics and Finite Models for Coalgebraic Logics , 2008, FoSSaCS.

[4]  Samson Abramsky,et al.  Domain Theory in Logical Form , 1991, LICS.

[5]  Alexandra Silva,et al.  A Kleene Theorem for Polynomial Coalgebras , 2009, FoSSaCS.

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

[7]  Grigore Rosu,et al.  CIRC : A Circular Coinductive Prover , 2007, CALCO.

[8]  Bart Jacobs,et al.  Many-Sorted Coalgebraic Modal Logic: a Model-theoretic Study , 2001, RAIRO Theor. Informatics Appl..

[9]  Alice Feller.  Termination , 2009 .

[10]  Dirk Pattinson,et al.  Coalgebraic modal logic: soundness, completeness and decidability of local consequence , 2003, Theor. Comput. Sci..

[11]  Luca Aceto,et al.  Termination, deadlock, and divergence , 1992, JACM.

[12]  Martin Rößiger,et al.  Coalgebras and Modal Logic , 2000, CMCS.

[13]  Marcello M. Bonsangue,et al.  Presenting Functors by Operations and Equations , 2006, FoSSaCS.

[14]  Gordon D. Plotkin,et al.  Towards a mathematical operational semantics , 1997, Proceedings of Twelfth Annual IEEE Symposium on Logic in Computer Science.

[15]  Peter Aczel,et al.  A Final Coalgebra Theorem , 1989, Category Theory and Computer Science.

[16]  Yde Venema,et al.  Coalgebraic Automata Theory: Basic Results , 2008, Log. Methods Comput. Sci..

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