Open Problems in Rewriting

Interest in the theory and applications of rewriting has been growing rapidly, as evidenced in part by four conference proceedings (including this one) [15, 26, 41, 66]; three workshop proceedings [33, 47, 77]; ve special journal issues [5, 88, 24, 40, 67]; more than ten surveys [2, 7, 27, 28, 44, 56, 57, 76, 82, 81]; one edited collection of papers [1]; four monographs [3, 12, 55, 65]; and seven books (four of them still in progress) [8, 9, 35, 54, 60, 75, 84]. To encourage and stimulate continued progress in this area, we have collected (with the help of colleagues) a number of problems that appear to us to be of interest and regarding which we do not know the answer. Questions on rewriting and other equational paradigms have been included; many have not aged su ciently to be accorded the appellation \open problem". We have limited ourselves to theoretical questions, though there are certainly many additional interesting questions relating to applications and implementations. Previous lists of questions in this area include one distributed by Leo Marcus and one of us (Dershowitz) at the Sixth International Conference on Automated Deduction (New York, 1982), the questions posed in a set of lecture notes on \Term Rewriting Systems" by one of us (Klop) for a seminar on reduction machines (Ustica, 1985), another list by one of us (Jouannaud) in the Bulletin of the European Association for Theoretical Computer Science (Number 31, 1987), and electronic postings to the distribution list (rewriting@crin.crin.fr) maintained by Pierre Lescanne. We use primarily terminology and notation of [27].

[1]  Jieh Hsiang,et al.  Refutational Theorem Proving Using Term-Rewriting Systems , 1985, Artif. Intell..

[2]  Richard Kennaway Sequential Evaluation Strategies for Parallel-Or and Related Reduction Systems , 1989, Ann. Pure Appl. Log..

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

[4]  Michael J. O'Donnell,et al.  Equational Logic as a Programming Language , 1985, Logic of Programs.

[5]  Gregory Kucherov On Relationship Between Term Rewriting Systems and Regular Tree Languages , 1991, RTA.

[6]  Christoph M. Hoffmann,et al.  Programming with Equations , 1982, TOPL.

[7]  Nachum Dershowitz,et al.  A Rationale for Conditional Equational Programming , 1990, Theor. Comput. Sci..

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

[9]  L. Bachmair Canonical Equational Proofs , 1991, Progress in Theoretical Computer Science.

[10]  Jan Willem Klop,et al.  Unique Normal Forms for Lambda Calculus with Surjective Pairing , 1989, Inf. Comput..

[11]  Jörg H. Siekmann,et al.  Universal Unification , 1982, GWAI.

[12]  Gérard P. Huet,et al.  A Complete Proof of Correctness of the Knuth-Bendix Completion Algorithm , 1981, J. Comput. Syst. Sci..

[13]  淵 一博,et al.  Programming of future generation computers II : proceedings of the Second Franco-Japanese Symposium on Programming of Future Generation Computers, Cannes, France, 9-11, November, 1987 , 1988 .

[14]  Nachum Dershowitz,et al.  Existence, Uniqueness, and Construction of Rewrite Systems , 1988, SIAM J. Comput..

[15]  Harold T. Hodes,et al.  The | lambda-Calculus. , 1988 .

[16]  Aart Middeldorp,et al.  Sequentiality in Orthogonal Term Rewriting Systems , 1991, J. Symb. Comput..

[17]  Fernando Orejas,et al.  Clausal Rewriting , 1990, CTRS.

[18]  Henk Barendregt,et al.  The Lambda Calculus: Its Syntax and Semantics , 1985 .

[19]  Craig C. Squier,et al.  Word problems and a homological niteness condition for monoids , 1987 .

[20]  Pierre Lescanne,et al.  Well rewrite orderings , 1990, [1990] Proceedings. Fifth Annual IEEE Symposium on Logic in Computer Science.

[21]  J. Roger Hindley,et al.  Introduction to combinators and λ-calculus , 1986, Acta Applicandae Mathematicae.

[22]  Sophie Tison,et al.  Decidability of the Confluence of Finite Ground Term Rewrite Systems and of Other Related Term Rewrite Systems , 1990, Inf. Comput..

[23]  Jean-Pierre Jouannaud,et al.  Termination and Completion Modulo Associativity, Commutativity and Identity , 1992, Theor. Comput. Sci..

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

[25]  Jean-Pierre Jouannaud,et al.  Satisfiability of Systems of Ordinal Notations with the Subterm Property is Decidable , 1991, ICALP.

[26]  Roel C. de Vrijer,et al.  Extending the Lambda Calculus with Surjective Pairing is Conservative , 1989, LICS.

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

[28]  François Bronsard,et al.  Conditional Rewriting in Focus , 1990, CTRS.

[29]  Jan A. Bergstra,et al.  Conditional Rewrite Rules: Confluence and Termination , 1986, J. Comput. Syst. Sci..

[30]  Aart Middeldorp Modular Aspects of Properties of Term Rewriting Systems Related to Normal Forms , 1989, RTA.

[31]  Philippe le Chenadec Canonical forms in finitely presented algebras , 1984, Research notes in theoretical computer science.

[32]  Marisa Venturini Zilli,et al.  Reduction Graphs in the Lambda Calculus , 1984, Theor. Comput. Sci..

[33]  Claude Kirchner Computing Unification Algorithms , 1986, LICS.

[34]  Claude Kirchner,et al.  Solving Equations in Abstract Algebras: A Rule-Based Survey of Unification , 1991, Computational Logic - Essays in Honor of Alan Robinson.

[35]  Nachum Dershowitz,et al.  Inference Rules for Rewrite-Based First-Order Theorem Proving , 1987, LICS.

[36]  Michio Oyamaguchi,et al.  The Church-Rosser Property for Ground Term-Rewriting Systems is Decidable , 1987, Theor. Comput. Sci..

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

[38]  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).

[39]  Michael M. Richter,et al.  Systems of Reductions , 1987, Lecture Notes in Computer Science.

[40]  Rémi Gilleron Decision Problems for Term Rewriting Systems and Recognizable Tree Languages , 1991, STACS.

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

[42]  Maurice Nivat,et al.  Resolution of Equations in Algebraic Structures , 1989 .

[43]  Michael J. Maher Complete axiomatizations of the algebras of finite, rational and infinite trees , 1988, [1988] Proceedings. Third Annual Information Symposium on Logic in Computer Science.

[44]  Simon Kaplan,et al.  Conditional Term Rewriting Systems , 1987, Lecture Notes in Computer Science.

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

[46]  Sophie Tison,et al.  The theory of ground rewrite systems is decidable , 1990, [1990] Proceedings. Fifth Annual IEEE Symposium on Logic in Computer Science.

[47]  Jan Willem Klop,et al.  Term rewriting systems: a tutorial , 1987 .

[48]  Jean-Jacques Lévy,et al.  Minimal and Optimal Computations of Recursive Programs , 1979, J. ACM.

[49]  François Fages,et al.  Associative-Commutative Unification , 1984, CADE.

[50]  G. Bauer,et al.  n-Level Rewriting Systems , 1985, Theor. Comput. Sci..

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

[52]  N. A C H U M D E R S H O W I T Z Termination of Rewriting' , 2022 .

[53]  Satish Thatte A Refinement of Strong Sequentiality for Term Rewriting with Constructors , 1987, Inf. Comput..

[54]  Hubert Comon-Lundh,et al.  Unification et disunification : théorie et applications , 1988 .

[55]  Evelyne Contejean,et al.  A new AC unification algorithm with an algorithm for solving systems of diophantine equations , 1990, [1990] Proceedings. Fifth Annual IEEE Symposium on Logic in Computer Science.

[56]  Ronald V. Book,et al.  Rewriting Techniques and Applications , 1991, Lecture Notes in Computer Science.

[57]  Jean-Pierre Jouannaud,et al.  Introduction to Rewriting , 1993, Term Rewriting.

[58]  Jan Willem Klop,et al.  Transfinite Reductions in Orthogonal Term Rewriting Systems (Extended Abstract) , 1991, RTA.

[59]  Ralph W. Wilkerson,et al.  Complete Sets of Reductions Modulo Associativity, Commutativity and Identity , 1989, RTA.

[60]  Ralf Treinen,et al.  A New Method for Undecidability Proofs of First Order Theories , 1990, FSTTCS.

[61]  D. Plaisted Equational reasoning and term rewriting systems , 1993 .

[62]  Nachum Dershowitz,et al.  Rewrite, Rewrite, Rewrite, Rewrite, Rewrite, . . , 1991, Theor. Comput. Sci..

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

[64]  R. D. Vrijer,et al.  Extending the lambda calculus with surjective pairing is conservative , 1989, [1989] Proceedings. Fourth Annual Symposium on Logic in Computer Science.

[65]  HuetGérard Confluent Reductions: Abstract Properties and Applications to Term Rewriting Systems , 1980 .

[66]  Jan Willem Klop,et al.  Combinatory reduction systems , 1980 .

[67]  Jean-Pierre Jouannaud,et al.  Completion modulo Associativity, Commutativity and Identity (AC1) , 1990, DISCO.

[68]  M. Thomas,et al.  Editorial – Term Rewriting , 1991 .

[69]  Jean-Jacques Lévy,et al.  Minimal and optimal computations of recursive programs , 1977, JACM.