Termination in Modal Kleene Algebra

Modal Kleene algebras (MKAs) are Kleene algebras with forward and backward modal operators defined via domain and codomain operations. The paper formalizes and compares different notions of termination, including Lob’s formula, in MKA. It studies exhaustive iteration and gives calculational proofs of two fundamental termination-dependent statements from rewriting theory: the well-founded union theorem by Bachmair and Dershowitz and Newman’s lemma. These results are also of general interest for the termination analysis of programs and state transition systems.

[1]  Gunther Schmidt,et al.  Relations and Graphs: Discrete Mathematics for Computer Scientists , 1993 .

[2]  Georg Struth,et al.  Kleene algebra with domain , 2003, TOCL.

[3]  A. Tarski,et al.  Boolean Algebras with Operators. Part I , 1951 .

[4]  Nachum Dershowitz,et al.  Commutation, Transformation, and Termination , 1986, CADE.

[5]  Jörg H. Siekmann,et al.  8th International Conference on Automated Deduction , 1986, Lecture Notes in Computer Science.

[6]  Georg Struth,et al.  Greedy-Like Algorithms in Modal Kleene Algebra , 2003, RelMiCS.

[7]  Roland Carl Backhouse,et al.  A Calculational Approach to Mathematical Induction , 1997, Theor. Comput. Sci..

[8]  Tobias Nipkow,et al.  Term rewriting and all that , 1998 .

[9]  Georg Struth.  Non-symmetric rewriting , 1996 .

[10]  Brian F. Chellas.  Modal Logic: Normal systems of modal logic , 1980 .

[11]  Jerzy Tiuryn,et al.  Dynamic logic , 2001, SIGA.

[12]  Gunther Schmidt,et al.  Relations and Graphs , 1993, EATCS Monographs on Theoretical Computer Science.

[13]  Ernie Cohen,et al.  Separation and Reduction , 2000, MPC.

[14]  Georg Struth,et al.  Calculating Church-Rosser Proofs in Kleene Algebra , 2001, RelMiCS.

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

[16]  Robert Goldblatt.  An algebraic study of well-foundedness , 1985, Stud Logica.

[17]  Alfons Geser,et al.  Relative Termination , 1990 .

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