Analytic Cut for Propositional Dynamic Logic

We present a Gentzen system GPDL for Propositional Dynamic Logic, deriving the classical Segerberg axiomatization from our system. The main contribution consists in proving that the system presented is equivalent to a subsystem where only Analytic Cut of a special and restricted kind is permitted. This resolves the long-standing open problem of Cut-elimination for Propositional Dynamic Logic.

[1]  Jerzy Tiuryn,et al.  Substructural logic and partial correctness , 2003, TOCL.

[2]  Mars K. Valiev.  On Axiomatization of Deterministic Propositional Dynamic Logic , 1979, MFCS.

[3]  Nikolay V. Shilov,et al.  Some Results on Propositional Dynamic Logic with Fixed Points , 1997, LFCS.

[5]  Hirokazu Nishimura,et al.  Sequential method in propositional dynamic logic , 1979, Acta Informatica.

[6]  Raymond M. Smullyan,et al.  Analytic cut , 1969, Journal of Symbolic Logic (JSL).

[7]  Vaughan R. Pratt.  A practical decision method for propositional dynamic logic (Preliminary Report) , 1978, STOC '78.

[8]  Dexter Kozen,et al.  On Kleene Algebras and Closed Semirings , 1990, MFCS.

[9]  Dov M. Gabbay,et al.  On the Logic of Information Flow , 1995, Log. J. IGPL.

[10]  Hirokazu Nishimura,et al.  Semantical Analysis of Constructive PDL , 1982 .

[11]  Vaughan R. Pratt,et al.  Dynamic algebras as a well-behaved fragment of relation algebras , 1988, Algebraic Logic and Universal Algebra in Computer Science.

[12]  Vaughan R. Pratt,et al.  Dynamic algebras: Examples, constructions, applications , 1991, Stud Logica.

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

[14]  Rajeev Goré,et al.  An Optimal On-the-Fly Tableau-Based Decision Procedure for PDL-Satisfiability , 2009, CADE.

[15]  Robert A. Bull.  Cut Elimination for Propositional Dynamic Logic without , 1992, Math. Log. Q..

[16]  Frank Wolter,et al.  Handbook of Modal Logic , 2007, Studies in logic and practical reasoning.

[17]  Bernhard Beckert,et al.  Dynamic Logic , 2007, The KeY Approach.