Narrowing, Abstraction and Constraints for Proving Properties of Reduction Relations

We describe in this paper an inductive proof method for properties of reduction relations. The reduction trees are simulated with proof trees generated by narrowing and an abstraction mechanism. While narrowing simulates reduction, abstraction relies on the induction principle to replace subterms by variables representing specific reduced forms that trivially satisfy the property to be proved. The induction ordering is not given a priori, but defined with ordering constraints, incrementally set during the proof. Abstraction constraints are used to control the narrowing mechanism, well-known to easily diverge. The proof method is briefly illustrated on various examples of properties.

[1]  AZEDDINE LAZREK,et al.  Tools for Proving Inductive Equalities, Relative Completeness, and omega-Completeness , 1990, Inf. Comput..

[2]  Hélène Kirchner,et al.  Termination of rewriting strategies: a generic approach , 2005, ArXiv.

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

[4]  Zhiming Liu,et al.  Theoretical Aspects of Computing - ICTAC 2004, First International Colloquium, Guiyang, China, September 20-24, 2004, Revised Selected Papers , 2005, ICTAC.

[5]  Paliath Narendran,et al.  Proof by Induction Using Test Sets , 1986, CADE.

[6]  Jean-Pierre Jouannaud,et al.  Automatic Proofs by Induction in Theories without Constructors , 1989, Inf. Comput..

[7]  Tobias Nipkow,et al.  A decidability result about sufficient-completeness of axiomatically specified abstract data types , 1983 .

[8]  Jürgen Giesl,et al.  Proving Innermost Normalisation Automatically , 1997, RTA.

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

[10]  Adel Bouhoula.  Using induction and rewriting to verify and complete parameterized specifications , 1996 .

[11]  Frank Pfenning,et al.  Logic Programming and Automated Reasoning , 1994, Lecture Notes in Computer Science.

[12]  Max Dauchet,et al.  Encompassment Properties and Automata with Constraints , 1993, RTA.

[13]  H. Comon.  Sufficient Completness, Term Rewriting Systems and Anti-Unification , 1986 .

[14]  Jürgen Giesl,et al.  Improving Dependency Pairs , 2003, LPAR.

[15]  Hélène Kirchner,et al.  System Presentation -- CARIBOO: An induction based proof tool for termination with strategies , 2002, PPDP '02.

[16]  José Meseguer,et al.  A Sufficient Completeness Reasoning Tool for Partial Specifications , 2005, RTA.

[17]  Hélène Kirchner,et al.  A Proof of Weak Termination Providing the Right Way to Terminate , 2004, ICTAC.

[18]  Bernhard Gramlich,et al.  Relating Innermost, Weak, Uniform and Modular Termination of Term Rewriting Systems , 1992, LPAR.

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

[20]  Salvador Lucas.  Termination of Rewriting With Strategy Annotations , 2001, LPAR.

[21]  H. Kirchner,et al.  Induction for termination , 1999 .

[22]  Hélène Kirchner,et al.  Outermost ground termination , 2004, WRLA.

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

[24]  Christophe Ringeissen,et al.  A Pattern Matching Compiler for Multiple Target Languages , 2003, CC.

[25]  Salvador Lucas,et al.  Termination of on-demand rewriting and termination of OBJ programs , 2001, PPDP '01.

[26]  Quang Huy Nguyen.  Compact Normalisation Trace via Lazy Rewriting , 2001, Electron. Notes Theor. Comput. Sci..

[27]  Maribel Fernández,et al.  Curry-Style Types for Nominal Terms , 2006, TYPES.

[28]  Emmanuel Kounalis,et al.  Completeness in Data Type Specifications , 1985, European Conference on Computer Algebra.

[29]  Claude Kirchner,et al.  Rewrite strategies in the rewriting calculus , 2003, WRS.

[30]  Emmanuel Kounalis,et al.  Testing for the Ground (Co-)Reducibility Property in Term-Rewriting Systems , 1992, Theor. Comput. Sci..

[31]  Jean-Pierre Jouannaud,et al.  Operational Semantics for Order-Sorted Algebra , 1985, ICALP.

[32]  Hélène Kirchner,et al.  Computing constructor forms with non terminating rewrite programs , 2006, PPDP '06.

[33]  Maribel Fernández,et al.  Proceedings of the Eighth ACM SIGPLAN Symposium on Principles and Practice of Declarative Programming , 2006 .

[34]  Eelco Visser,et al.  Stratego: A Language for Program Transformation Based on Rewriting Strategies , 2001, RTA.

[35]  Nachum Dershowitz,et al.  Chapter 9 – Rewriting , 2001 .

[36]  Alan Bundy,et al.  Constructing Induction Rules for Deductive Synthesis Proofs , 2006, CLASE.

[37]  J. Kruskal.  Well-quasi-ordering, the Tree Theorem, and Vazsonyi’s conjecture , 1960 .

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

[39]  Jürgen Giesl,et al.  Innermost Termination of Context-Sensitive Rewriting , 2002, Developments in Language Theory.

[40]  Jürgen Giesl,et al.  Automated Termination Analysis for Haskell: From Term Rewriting to Programming Languages , 2006, RTA.

[41]  Robin Milner,et al.  On Observing Nondeterminism and Concurrency , 1980, ICALP.

[42]  Bernhard Gramlich.  On Proving Termination by Innermost Termination , 1996, RTA.

[43]  Hélène Kirchner,et al.  Induction for termination with local strategies , 2001 .

[44]  Gérard P. Huet,et al.  Proofs by induction in equational theories with constructors , 1980, 21st Annual Symposium on Foundations of Computer Science (sfcs 1980).

[45]  Manfred Schmidt-Schauß,et al.  TEA: Automatically Proving Termination of Programs in a Non-strict Higher-Order Functional Language , 1997, SAS.

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

[47]  David A. Plaisted,et al.  Semantic Confluence Tests and Completion Methods , 1985, Inf. Control..

[48]  C. Kirchner,et al.  Deduction with symbolic constraints , 1990 .

[49]  Jan Willem Klop,et al.  Term Rewriting Systems: From Church-Rosser to Knuth-Bendix and Beyond , 1990, ICALP.

[50]  Salvador Lucas,et al.  Termination of Context-Sensitive Rewriting by Rewriting , 1996, ICALP.

[51]  Jürgen Giesl,et al.  Transforming Context-Sensitive Rewrite Systems , 1999, RTA.

[52]  Jean Goubault-Larrecq.  A Proof of Weak Termination of Typed lambda-sigma-Calculi , 1996, TYPES.

[53]  Bernhard Gramlich,et al.  On Termination and Confluence Properties of Disjoint and Constructor-Sharing Conditional Rewrite Systems , 1996, Theor. Comput. Sci..

[54]  Florent Jacquemard,et al.  Ground reducibility is EXPTIME-complete , 1997, Proceedings of Twelfth Annual IEEE Symposium on Logic in Computer Science.

[55]  Florent Jacquemard,et al.  Automating Sufficient Completeness Check for Conditional and Constrained TRS , 2006 .

[56]  Paliath Narendran,et al.  On sufficient-completeness and related properties of term rewriting systems , 1987, Acta Informatica.

[57]  José Meseguer,et al.  Symbolic Reachability Analysis Using Narrowing and its Application to Verification of Cryptographic Protocols , 2004, WRLA.

[58]  Alan Robinson,et al.  Handbook of automated reasoning , 2001 .

[59]  Aart Middeldorp,et al.  Completeness results for basic narrowing , 1994, Applicable Algebra in Engineering, Communication and Computing.

[60]  José Meseguer,et al.  Theoretical Computer Science: Preface , 2002 .

[61]  Albert Rubio,et al.  Orderings for Innermost Termination , 2005, RTA.