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 undeeidable property of rewrite systems.

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

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

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

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

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

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

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

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

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

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

[11]  Donald E. Knuth,et al.  fundamental algorithms , 1969 .

[12]  G. Gentzen.  New version of the consistency proof for elementary number theory , 1969 .

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

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

[15]  Barry K. Rosen,et al.  Tree-Manipulating Systems and Church-Rosser Theorems , 1973, JACM.

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

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

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

[19]  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).

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

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

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

[23]  Nachum Dershowitz.  Orderings for Term-Rewriting Systems , 1979, FOCS.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[51]  G. Takeuti,et al.  On the theory of quasi-ordinal diagrams , 1985 .

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

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

[54]  Paliath Narendran,et al.  On Recursive Path Ordering , 1985, Theor. Comput. Sci..

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

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

[57]  Stephen G. Simpson,et al.  Nonprovability of Certain Combinatorial Properties of Finite Trees , 1985 .

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

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

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

[61]  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.

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

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

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

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

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

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

[68]  Bernard R. Hodgson,et al.  Extensions of Arithmetic For Proving Termination of Computations , 1989, J. Symb. Log..

[69]  Laurence Puel,et al.  Using Unavoidable Set of Trees to Generalize Kruskal's Theorem , 1989, J. Symb. Comput..

[70]  Fairness in term rewriting systems , 1985, Methods Log. Comput. Sci..

[71]  Kenneth W. Regan,et al.  Computability , 2022, Algorithms and Theory of Computation Handbook.

[72]  Zohar Manna,et al.  Mathematical Theory of Computation , 2003 .

[73]  Fedor V. Fomin,et al.  40th International Colloquium on Automata, Languages and Programming , 2015, Inf. Comput..

[74]  Claudia Biermann.  Computing In Systems Described By Equations , 2016 .