Termination of Rewriting

This survey describes methods for proving that systems of rewrite rules are terminating programs. We illustrate the use in termination proofs of various kinds of orderings on terms, including polynomial interpretations and path orderings. The effect of restrictions, such as linearity, on the form of rules is also considered. In general, however, termination is an undecidable property of rewrite systems.

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

[2]  Zohar Manna,et al.  Proving termination with multiset orderings , 1979, CACM.

[3]  J. V. Tucker,et al.  Equational specifications for computable data types: six hidden functions suffice and other sufficiency bounds : (preprint) , 1993 .

[4]  T. R. Zaloum.  J.N. Park General Electric Corporate Research and Development Schenectady, NY and , 1982 .

[5]  George E. Collins,et al.  Quantifier elimination for real closed fields by cylindrical algebraic decomposition , 1975 .

[6]  Jean-Marie Hullot,et al.  Canonical Forms and Unification , 1980, CADE.

[7]  Paul J. Cohen,et al.  Decision procedures for real and p‐adic fields , 1969 .

[8]  David A. Plaisted,et al.  The Undecidability of Self-Embedding for Term Rewriting Systems , 1985, Inf. Process. Lett..

[9]  William L. Scherlis,et al.  Expression procedures and program derivation , 1980 .

[10]  Jean-Claude Raoult,et al.  Operational and Semantic Equivalence Between Recursive Programs , 1980, JACM.

[11]  Paul Walton Purdom,et al.  Detecting Looping Simplifications , 1987, RTA.

[12]  David Haussler,et al.  Another generalization of Higman's well quasi order result on Sigma* , 1985, Discret. Math..

[13]  Yves Métivier.  Calcul de Longueurs de Chaînes de Réécriture dans le Monoïde Libre , 1985, Theor. Comput. Sci..

[14]  David Detlefs,et al.  A Procedure for Automatically Proving the Termination of a Set of Rewrite Rules , 1985, RTA.

[15]  Nissim Francez,et al.  Full-Commutation and Fair-Termination in Equational (and Combined) Term-Rewriting Systems , 1986, CADE.

[16]  David A. Plaisted,et al.  A Simple Non-Termination Test for the Knuth-Bendix Method , 1986, CADE.

[17]  Gérard Huet,et al.  On the Uniform Halting Problem for Term Rewriting Systems , 1978 .

[18]  Michaël Rusinowitch.  Path of Subterms Ordering and Recursive Decomposition Ordering Revisited , 1987, J. Symb. Comput..

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

[20]  Solomon Feferman,et al.  Systems of predicative analysis, II: Representations of ordinals , 1968, Journal of Symbolic Logic.

[21]  Joseph B. Kruskal,et al.  The Theory of Well-Quasi-Ordering: A Frequently Discovered Concept , 1972, J. Comb. Theory A.

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

[23]  C. Nash-Williams.  On well-quasi-ordering infinite trees , 1963, Mathematical Proceedings of the Cambridge Philosophical Society.

[24]  Hassan Aït-Kaci,et al.  An Algorithm for Finding A Minimal Recursive Path Ordering , 1985, RAIRO Theor. Informatics Appl..

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

[26]  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.

[27]  Nachum Dershowitz,et al.  Associative-Commutative Rewriting , 1983, IJCAI.

[28]  Pierre Lescanne.  Uniform Termination of Term Rewriting Systems: Recursive Decomposition Ordering with Status , 1984, CAAP.

[29]  Richard J. Lipton,et al.  On the Halting of Tree Replacement Systems. , 1977 .

[30]  Yoshihito Toyama,et al.  On the Church-Rosser property for the direct sum of term rewriting systems , 1984, JACM.

[31]  Stephen G. Simpson,et al.  Nichtbeweisbarkeit von gewissen kombinatorischen Eigenschaften endlicher Bäume , 1985, Arch. Math. Log..

[32]  Nachum Dershowitz,et al.  Applied Tree Enumerations , 1981, CAAP.

[33]  Nachum Dershowitz.  Termination of Linear Rewriting Systems (Preliminary Version) , 1981, ICALP.

[34]  M. Newman.  On Theories with a Combinatorial Definition of "Equivalence" , 1942 .

[35]  Isabelle Gnaedig,et al.  Proving Termination of Associative Commutative Rewriting Systems by Rewriting , 1986, CADE.

[36]  Hélène Kirchner,et al.  Construction D'un Plus Petit Odre de Simplification , 1984, RAIRO Theor. Informatics Appl..

[37]  Françoise Bellegarde,et al.  Transformation Ordering , 1987, TAPSOFT, Vol.1.

[38]  Nachum Dershowitz,et al.  A Note on Simplification Orderings , 1979, Inf. Process. Lett..

[39]  Pierre Lescanne,et al.  Some Properties of Decomposition Ordering, a Simplification Ordering to Prove Termination of Rewriting Systems , 1982, RAIRO Theor. Informatics Appl..

[40]  Renato Iturriaga.  CONTRIBUTIONS TO MECHANICAL MATHEMATICS. , 1967 .

[41]  David Haussler,et al.  On Regularity of Context-Free Languages , 1983, Theor. Comput. Sci..

[42]  Gérard P. Huet,et al.  Confluent Reductions: Abstract Properties and Applications to Term Rewriting Systems , 1980, J. ACM.

[43]  David R. Jefferson.  Type reduction and program verification , 1980 .

[44]  Nachum Dershowitz,et al.  Orderings for term-rewriting systems , 1979, 20th Annual Symposium on Foundations of Computer Science (sfcs 1979).

[45]  George E. Collins,et al.  Hauptvortrag: Quantifier elimination for real closed fields by cylindrical algebraic decomposition , 1975, Automata Theory and Formal Languages.

[46]  Michael J. O'Donnell,et al.  Computing in systems described by equations , 1977, Lecture Notes in Computer Science.

[47]  Jean-Pierre Jouannaud,et al.  Recursive Decomposition Ordering , 1982, Formal Description of Programming Concepts.

[48]  Jean-Pierre Jouannaud,et al.  On Multiset Orderings , 1982, Inf. Process. Lett..

[49]  Leo Bachmair.  Proof methods for equational theories , 1987 .

[50]  Pierre Lescanne,et al.  Computer experiments with the REVE term rewriting system generator , 1983, POPL '83.

[51]  Sophie Tison,et al.  Decidability of the Confluence of Ground Term Rewriting Systems , 1985, LICS.

[52]  Graham Higman,et al.  Ordering by Divisibility in Abstract Algebras , 1952 .

[53]  Raymond M. Smullyan.  TREES AND BALL GAMES , 1979 .

[54]  Alberto Pettorossi.  Comparing and Putting Together Recursive Path Ordering, Simplification Orderings and Non-Ascending Property for Termination Proofs of Term Rewriting Systems , 1981, ICALP.

[55]  J. Paris,et al.  Accessible Independence Results for Peano Arithmetic , 1982 .

[56]  G. Huet,et al.  Equations and rewrite rules: a survey , 1980 .

[57]  Z. Manna.  Termination of algorithms , 1968 .

[58]  R. Forgaard.  A PROGRAM FOR GENERATING AND ANALYZING TERM REWRITING SYSTEMS , 1984 .

[59]  R. Rado.  Partial well-ordering of sets of vectors , 1954 .

[60]  Françoise Bellegarde.  Rewriting systems on FP expressions that reduce the number of sequences they yield , 1984, LFP '84.

[61]  Saul Gorn,et al.  Handling the growth by definition of mechanical languages , 1967, AFIPS '67 (Spring).

[62]  T. C. Brown.  A structured design-method for specialized proof procedures , 1975 .

[63]  Jean-Marc Steyaert,et al.  On the Study Data Structures: Binary Tournaments with Repeated Keys , 1983, ICALP.

[64]  J. N. Crossley,et al.  Formal Systems and Recursive Functions , 1963 .

[65]  James R. Slagle,et al.  Automated Theorem-Proving for Theories with Simplifiers Commutativity, and Associativity , 1974, JACM.

[66]  Alberto Pettorossi,et al.  A property which guarantees termination in weak combinatory logic and subtree replacement systems , 1981, Notre Dame J. Formal Log..

[67]  Deepak Kapur,et al.  On Proving Uniform Termination and Restricted Termination of Rewriting Systems , 1983, SIAM J. Comput..

[68]  Wilhelm Ackermann.  Konstruktiver Aufbau eines Abschnitts der zweiten Cantorschen Zahlenklasse , 1951 .

[69]  O. Veblen.  Continuous increasing functions of finite and transfinite ordinals , 1908 .

[70]  D. Knuth,et al.  Simple Word Problems in Universal Algebras , 1983 .

[71]  Jan Willem Klop,et al.  A process algebra for the operational semantics of static data flow networks , 1983 .

[72]  Saul Gorn,et al.  On the conclusive validation of symbol manipulative processes (How do you know it has to work , 1973 .

[73]  Isabelle Gnaedig.  Preuves de terminaison des systèmes de réécriture associatifs commutatifs : Une méthode fondée sur la réécriture elle-même , 1986 .

[74]  Paliath Narendran,et al.  A Path Ordering for Proving Termination of Term Rewriting Systems , 1985, TAPSOFT, Vol.1.

[75]  Gerard Huet,et al.  Conflunt reductions: Abstract properties and applications to term rewriting systems , 1977, 18th Annual Symposium on Foundations of Computer Science (sfcs 1977).

[76]  Jean-Pierre Jouannaud,et al.  Termination of a Set of Rules Modulo a Set of Equations , 1984, CADE.

[77]  K. Schütte,et al.  Predicative Well-Orderings , 1965 .

[78]  Nachum Dershowitz,et al.  Rewrite Methods for Clausal and Non-Clausal Theorem Proving , 1983, ICALP.

[79]  Yves Métivier.  About the Rewriting Systems Produced by the Knuth-Bendix Completion Algorithm , 1983, Inf. Process. Lett..