Modular and incremental proofs of AC-termination

Abstract Termination is a non-modular property of rewriting systems, thus it is a difficult task to discover termination proofs for rewriting systems of a large number of rules. Recently, new modular and incremental termination criteria, suitable for automation, were proposed, using an approach based on notions of termination under non-deterministic collapse and dependency pairs , which apply to hierarchical combinations of rewriting systems. We extend this approach and corresponding results to the important case of rewriting modulo associativity and commutativity.

[1]  Azuma Ohuchi,et al.  Decomposable Termination of Composable Term Rewriting Systems , 1995, IEICE Trans. Inf. Syst..

[2]  Jürgen Giesl,et al.  Termination of term rewriting using dependency pairs , 2000, Theor. Comput. Sci..

[3]  Narciso Martí-Oliet,et al.  Maude: specification and programming in rewriting logic , 2002, Theor. Comput. Sci..

[4]  Patrick Brézillon,et al.  Lecture Notes in Artificial Intelligence , 1999 .

[5]  Joseph A. Goguen,et al.  Software Engineering with Obj: Algebraic Specification In Action , 2010 .

[6]  Christiane Floyd,et al.  Theory and Practice of Software Development: Stages in a Debate , 1995, TAPSOFT.

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

[8]  Jean-Pierre Jouannaud,et al.  Rewrite Systems , 1991, Handbook of Theoretical Computer Science, Volume B: Formal Models and Sematics.

[9]  Keiichirou Kusakari.  On Proving AC-Termination by AC-Dependency Paris , 1998 .

[10]  Enno Ohlebusch,et al.  Modular Termination Proofs for Rewriting Using Dependency Pairs , 2002, J. Symb. Comput..

[11]  LEO BACHMAIR,et al.  Termination Orderings for Associative-Commutative Rewriting Systems , 1985, J. Symb. Comput..

[12]  Hélène Kirchner,et al.  Completion of a Set of Rules Modulo a Set of Equations , 1986, SIAM J. Comput..

[13]  Deepak Kapur,et al.  Proving Associative-Communicative Termination Using RPO-Compatible Orderings , 1998, FTP.

[14]  Eric Deplagne.  Sequent Calculus Viewed Modulo , 2000 .

[15]  Bernhard Gramlich,et al.  Generalized sufficient conditions for modular termination of rewriting , 1992, Applicable Algebra in Engineering, Communication and Computing.

[16]  Jean-Pierre Jouannaud,et al.  Modular Termination of Term Rewriting Systems Revisited , 1994, COMPASS/ADT.

[17]  Mark E. Stickel,et al.  Complete Sets of Reductions for Some Equational Theories , 1981, JACM.

[18]  Salvador Lucas,et al.  Modular termination of context-sensitive rewriting , 2002, PPDP '02.

[19]  Enno Ohlebusch,et al.  On the Modularity of Termination of Term Rewriting Systems , 1994, Theor. Comput. Sci..

[20]  Azuma Ohuchi,et al.  Modularity of Simple Termination of Term Rewriting Systems with Shared Constructors , 1992, Theor. Comput. Sci..

[21]  Yoshihito Toyama,et al.  Completeness of Combinations of Constructor Systems , 1991, RTA.

[22]  Claude Marché,et al.  Termination of Associative-Commutative Rewriting by Dependency Pairs , 1998, RTA.

[23]  Xavier Urbain,et al.  Automated Incremental Termination Proofs for Hierarchically Defined Term Rewriting Systems , 2001, IJCAR.

[24]  Pierre Lescanne,et al.  An Actual Implementation of a Procedure That Mechanically Proves Termination of Rewriting Systems Based on Inequalities Between Polynomial Interpretations , 1986, CADE.

[25]  Albert Rubio.  A Fully Syntactic AC-RPO , 1999, RTA.

[26]  Jürgen Giesl,et al.  Dependency Pairs for Equational Rewriting , 2001, RTA.

[27]  Enno Ohlebusch,et al.  Modular Properties of Composable Term Rewriting Systems , 1995, J. Symb. Comput..

[28]  Jürgen Giesl,et al.  Automatically Proving Termination Where Simplification Orderings Fail , 1997, TAPSOFT.

[29]  Salvador Lucas.  Context-sensitive rewriting strategies , 2002 .

[30]  Yoshihito Toyama,et al.  Counterexamples to Termination for the Direct Sum of Term Rewriting Systems , 1987, Inf. Process. Lett..

[31]  Enno Ohlebusch,et al.  Hierarchical termination revisited , 2002, Inf. Process. Lett..

[32]  Deepak Kapur,et al.  A Total, Ground path Ordering for Proving Termination of AC-Rewrite Systems , 1997, RTA.

[33]  Enno Ohlebusch,et al.  Advanced Topics in Term Rewriting , 2002, Springer New York.

[34]  Pierre Lescanne,et al.  Termination of Rewriting Systems by Polynomial Interpretations and Its Implementation , 1987, Sci. Comput. Program..

[35]  Xavier Urbain.  Approche incrémentale des preuves automatiques de terminaison , 2001 .

[36]  Albert Rubio,et al.  Monotonic AC-Compatible Semantic Path Orderings , 2003, RTA.

[37]  Masahiko Sakai,et al.  On Dependency Pair Method for Proving Termination of Higher-Order Rewrite Systems , 2001, IEICE Trans. Inf. Syst..

[38]  Claude Marché,et al.  Rewrite Systems for Natural, Integral, and Rational Arithmetic , 1997, RTA.

[39]  J. Van Leeuwen,et al.  Handbook of theoretical computer science - Part A: Algorithms and complexity; Part B: Formal models and semantics , 1990 .

[40]  Yoshihito Toyama,et al.  Argument Filtering Transformation , 1999, PPDP.

[41]  M. R. K. Krishna Rao.  Simple Termination of Hierarchical Combinations of Term Rewriting Systems , 1994, TACS.

[42]  Laurence Puel,et al.  Extension of the Associative Path Ordering to a Chain of Associative Commutative Symbols , 1993, RTA.

[43]  M.R.K. Kr ishna Rao,et al.  Modular Proofs for Completeness of Hierarchical Term Rewriting Systems , 1995, Theor. Comput. Sci..

[44]  Nachum Dershowitz,et al.  Termination of Rewriting , 1987, J. Symb. Comput..

[45]  Claude Marché,et al.  Normalized Rewriting: An Alternative to Rewriting Modulo a Set of Equations , 1996, J. Symb. Comput..