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A taste of rewrite systems

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Functional Programming, Concurrency, Simulation and Automated Reasoning

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 693))

Abstract

This survey of the theory and applications of rewriting with equations discusses the existence and uniqueness of normal forms, the KnuthBendix completion procedure and its variations, as well as rewriting-based (functional and logic) programming and (equational, first-order, and inductive) theorem proving. Ordinary, associative-commutative, and conditional rewriting are covered. Current areas of research are summarized and an extensive bibliography is provided.

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Selected bibliography

1.Rewriting

  1. Gorn, S.: Handling the growth by definition of mechanical languages. In: Spring Joint Computer Conference, Philadelphia, PA (1967) 213–224

    Google Scholar 

  2. Huet, G., Oppen, D. C.: Equations and rewrite rules: A survey. In: Book, R., ed., Formal Language Theory: Perspectives and Open Problems. Academic Press, New York (1980) 349–405

    Google Scholar 

  3. Musser, D. R., Kapur, D.: Rewrite rule theory and abstract data type analysis. In: Calmet, J., ed., Proc. of the European Computer Algebra Conference (Marseille, France), vol. 144 of Lect. Notes in Computer Sci. Springer-Verlag, Berlin (1982) 77–90

    Google Scholar 

  4. Jouannaud, J.-P., Lescanne, P.: Rewriting systems. Technology and Science of Informatics 6 (1987) 181–199

    Google Scholar 

  5. Benninghofen, B., Kemmerich, S., Richter, M. M.: Systems of Reductions, vol. 277 of Lect. Notes in Computer Sci. Springer-Verlag, Berlin (1987)

    Google Scholar 

  6. Jantzen, M.: Confluent String Rewriting, vol. 14 of EATCS Monographs on Theoretical Computer Science. Springer-Verlag, Berlin (1988)

    Google Scholar 

  7. Avenhaus, J., Madlener, K.: Term rewriting and equational reasoning. In: Banerji, R. B., ed., Formal Techniques in Artificial Intelligence: A Sourcebook. Elsevier, Amsterdam (1990) 1–41

    Google Scholar 

  8. Dershowitz, N., Jouannaud, J.-P.: Rewrite systems. In: van Leeuwen, J., ed., Handbook of Theoretical Computer Science, vol. B: Formal Methods and Semantics. North-Holland, Amsterdam (1990) 243–320

    Google Scholar 

  9. Wechler, W.: Reductions. In: Universal Algebra for Computer Scientists, vol. 25 of EATCS Monographs on Theoretical Computer Science. Springer-Verlag, Berlin (1991) 89–133

    Google Scholar 

  10. Klop, J. W.: Term rewriting systems. In: Abramsky, S., Gabbay, D. M., Maibaum, T. S. E., eds., Handbook of Logic in Computer Science, vol. 1. Oxford University Press, Oxford (1992) 1–117

    Google Scholar 

2.Termination

  1. Manna, Z., Ness, S.: On the termination of Markov algorithms. In: Proc. of the Third Hawaii International Conference on System Science, Honolulu, HI (1970) 789–792

    Google Scholar 

  2. Huet, G., Lankford, D. S.: On the uniform halting problem for term rewriting systems. Rapport laboria 283, Institut de Recherche en Informatique et en Automatique, Le Chesnay, France (1978)

    Google Scholar 

  3. Lankford, D. S.: On proving term rewriting systems are Noetherian. Memo MTP-3, Mathematics Department, Louisiana Tech. University, Ruston, LA (1979)

    Google Scholar 

  4. Dershowitz, N., Manna, Z.: Proving termination with multiset orderings. Commun. ACM 22 (1979) 465–476

    Google Scholar 

  5. Kamin, S., Lévy, J.-J.: Two generalizations of the recursive path ordering. Unpublished note, Department of Computer Science, University of Illinois, Urbana, IL (1980)

    Google Scholar 

  6. Dershowitz, N.: Orderings for term-rewriting systems. Theoretical Comput. Sci. 17 (1982) 279–301

    Google Scholar 

  7. Dershowitz, N.: Termination of rewriting. J. Symb. Comput. 3 (1987) 69–115. Corrigendum: 4, 3 (1987), 409–410

    Google Scholar 

  8. Comon, H.: Solving inequations in term algebras (Preliminary version). In: Proc. of the Fifth Annual IEEE Symposium on Logic in Computer Science, Philadelphia, PA (1990) 62–69

    Google Scholar 

  9. Jouannaud, J.-P., Okada, M.: Satisfiability of systems of ordinal notations with the subterm property is decidable. In: Albert, J. L., Monien, B., Artalejo, M. R., eds., Proc. of the Eighteenth EATCS Colloquium on Automata, Languages and Programming (Madrid, Spain), vol. 510 of Lect. Notes in Computer Sci. Springer-Verlag, Berlin (1991) 455–468

    Google Scholar 

  10. Dershowitz, N., Hoot, C.: Topics in termination. In: Kirchner, C., ed., Proc. of the Fifth International Conference on Rewriting Techniques and Applications (Montreal, Canada), Lect. Notes in Computer Sci. Springer-Verlag, Berlin (1993)

    Google Scholar 

3.Confluence

  1. Newman, M. H. A.: On theories with a combinatorial definition of ‘equivalence'. Annals of Math. 43 (1942) 223–243

    Google Scholar 

  2. Selman, A.: Completeness of calculii for axiomatically defined classes of algebras. Algebra Universalis 2 (1972) 20–32

    Google Scholar 

  3. Rosen, B.: Tree-manipulating systems and Church-Rosser theorems. J. Assoc. Comput. Mach. 20 (1973) 160–187

    Google Scholar 

  4. Staples, J.: Church-Rosser theorem for replacement systems. In: Crossley, J., ed., Algebra and Logic, vol. 450 of Lect. Notes in Mathematics. Springer-Verlag, Berlin (1975) 291–307

    Google Scholar 

  5. Bergman, G. M.: The Diamond Lemma for ring theory. Advances in Mathematics 29 (1978) 178–218

    Google Scholar 

  6. Huet, G.: Confluent reductions: Abstract properties and applications to term rewriting systems. J. Assoc. Comput. Mach. 27 (1980) 797–821

    Google Scholar 

  7. Klop, J. W.: Combinatory Reduction Systems, vol. 127 of Mathematical Centre Tracts. Mathematisch Centrum, Amsterdam (1980)

    Google Scholar 

  8. Toyama, Y.: On the Church-Rosser property for the direct sum of term rewriting systems. J. Assoc. Comput. Mach. 34 (1987) 128–143

    Google Scholar 

4.Completion

  1. Knuth, D. E., Bendix, P. B.: Simple word problems in universal algebras. In: Leech, J., ed., Computational Problems in Abstract Algebra. Pergamon Press, Oxford, U. K. (1970) 263–297

    Google Scholar 

  2. Métivier, Y.: About the rewriting systems produced by the Knuth-Bendix completion algorithm. Inf. Process. Lett. 16 (1983) 31–34

    Google Scholar 

  3. Le Chenadec, P.: Canonical Forms in Finitely Presented Algebras. Pitman-Wiley, London (1985)

    Google Scholar 

  4. Buchberger, B.: History and basic features of the critical-pair/completion procedure. J. Symb. Comput. 3 (1987) 3–38

    Google Scholar 

  5. Kapur, D., Musser, D. R., Narendran, P.: Only prime superpositions need be considered for the Knuth-Bendix procedure. J. Symb. Comput. 4 (1988) 19–36

    Google Scholar 

  6. Bachmair, L., Dershowitz, N., Plaisted, D. A.: Completion without failure. In: Aït-Kaci, H., Nivat, M., eds., Resolution of Equations in Algebraic Structures, vol. 2: Rewriting Techniques. Academic Press, New York (1989) 1–30

    Google Scholar 

  7. Dershowitz, N.: Completion and its applications. In: Aït-Kaci, H., Nivat, M., eds., Resolution of Equations in Algebraic Structures, vol. 2: Rewriting Techniques. Academic Press, New York (1989) 31–86

    Google Scholar 

  8. Lescanne, P.: Completion procedures as transition rules + control. In: Diaz, M., Orejas, F., eds., Proc. of the Conference on Theory and Practice of Software Development, vol. 351 of Lect. Notes in Computer Science. Springer-Verlag, Berlin (1989) 28–41

    Google Scholar 

  9. Dick, A. J. J.: An introduction to Knuth-Bendix completion. Comput. J. 34 (1991) 2–15

    Google Scholar 

  10. Bachmair, L.: Canonical Equational Proofs. Birkhäuser, Boston (1991)

    Google Scholar 

  11. Duffy, D. A.: Knuth-Bendix Completion. In: Principles of Automated Theorem Proving. Wiley, Chichester (1991) 147–175

    Google Scholar 

  12. Bachmair, L., Dershowitz, N.: Equational inference, canonical proofs, and proof orderings. J. Assoc. Comput. Mach. (to appear)

    Google Scholar 

5.Validity

  1. Evans, T.: On multiplicative systems defined by generators and relations, I. Proc. of the Cambridge Philosophical Society 47 (1951) 637–649

    Google Scholar 

  2. Slagle, J. R.: Automated theorem-proving for theories with simplifiers, commutativity, and associativity. J. Assoc. Comput. Mach. 21 (1974) 622–642

    Google Scholar 

  3. Huet, G.: A complete proof of correctness of the Knuth-Bendix completion algorithm. J. Computer and System Sciences 23 (1981) 11–21

    Google Scholar 

  4. Hsiang, J., Rusinowitch, M.: On word problems in equational theories. In: Ottmann, T., ed., Proc. of the Fourteenth EATCS International Conference on Automata, Languages and Programming (Karlsruhe, West Germany), vol. 267 of Lect. Notes in Computer Sci. Springer-Verlag, Berlin (1987) 54–71

    Google Scholar 

  5. Martin, U., Nipkow, T.: Ordered completion. In: Stickel, M., ed., Proc. of the Tenth International Conference on Automated Deduction (Kaiserslautern, West Germany), vol. 449 of Lect. Notes in Computer Sci. Springer-Verlag, Berlin (1990) 366–380

    Google Scholar 

  6. Peterson, G. E.: Solving term inequalities. In: Proc. of the Eighth National Conference on Artificial Intelligence, Boston, MA (1990) 258–263

    Google Scholar 

6.Satisfiability

  1. Fay, M.: First-order unification in an equational theory. In: Proc. of the Fourth Workshop on Automated Deduction, Austin, TX (1979) 161–167

    Google Scholar 

  2. Hullot, J.-M.: Canonical forms and unification. In: Kowalski, R., ed., Proc. of the Fifth International Conference on Automated Deduction (Les Arcs, France), vol. 87 of Lect. Notes in Computer Science. Springer-Verlag, Berlin (1980) 318–334

    Google Scholar 

  3. Dershowitz, N., Sivakumar, G.: Solving goals in equational languages. In: Kaplan, S., Jouannaud, J.-P., eds., Proc. of the First International Workshop on Conditional Term Rewriting Systems (Orsay, France), vol. 308 of Lect. Notes in Computer Sci. Springer-Verlag, Berlin (1987) 45–55

    Google Scholar 

  4. Nutt, W., Réty, P., Smolka, G.: Basic narrowing revisited. J. Symb. Comput. 7 (1989) 295–318

    Google Scholar 

  5. Bockmayr, A., Krischer, S., Werner, A.: An optimal narrowing strategy for general canonical systems. In: Rusinowitch, M., ed., Proc. of the Third International Workshop on Conditional Rewriting Systems (Pont-a-Mousson, France, July 1992), vol. 656 of Lect. Notes in Computer Sci. Springer-Verlag, Berlin (1993)

    Google Scholar 

7.AC-completion

  1. Peterson, G. E., Stickel, M. E.: Complete sets of reductions for some equational theories. J. Assoc. Comput. Mach. 28 (1981) 233–264

    Google Scholar 

  2. Jouannaud, J.-P., Kirchner, H.: Completion of a set of rules modulo a set of equations. SIAM J. Comput. 15 (1986) 1155–1194

    Google Scholar 

  3. Bachmair, L., Dershowitz, N.: Completion for rewriting modulo a congruence. Theoretical Comput. Sci. 67 (1989)

    Google Scholar 

  4. Zhang, H., Kapur, D.: Unnecessary inferences in associative-commutative completion procedures. Mathematical Systems Theory 23 (1990) 175–206

    Google Scholar 

8.Conditional rewriting

  1. Bergstra, J. A., Klop, J. W.: Conditional rewrite rules: Confluency and termination. J. Comput. System Sciences 32 (1986) 323–362

    Google Scholar 

  2. Kaplan, S.: Simplifying conditional term rewriting systems: Unification, termination and confluence. J. Symb. Comput. 4 (1987) 295–334

    Google Scholar 

  3. Padawitz, P.: Computing in Horn Clause Theories, vol. 16 of EATCS Monographs on Theoretical Computer Science. Springer-Verlag, Berlin (1988)

    Google Scholar 

  4. Dershowitz, N., Okada, M., Sivakumar, G.: Canonical conditional rewrite systems. In: Proc. of the Ninth Conference on Automated Deduction (Argonne, IL), vol. 310 of Lect. Notes in Computer Sci. Springer-Verlag, Berlin (1988) 538–549

    Google Scholar 

  5. Kaplan, S., Rémy, J.-L.: Completion algorithms for conditional rewriting systems. In: Aït-Kaci, H., Nivat, M., eds., Resolution of Equations in Algebraic Structures, vol. 2: Rewriting Techniques. Academic Press, Boston (1989) 141–170

    Google Scholar 

  6. Klop, J. W., de Vrijer, R. C.: Extended term rewriting systems. In: Kaplan, S., Okada, M., eds., Proc. of the Second International Workshop on Conditional and Typed Rewriting Systems (Montreal, Canada, June 1990), vol. 516 of Lect. Notes in Computer Sci. Springer-Verlag, Berlin (1991) 26–50

    Google Scholar 

  7. Padawitz, P.: Reduction and narrowing for Horn clause theories. Comput. J. 34 (1991) 42–51

    Google Scholar 

  8. Ganzinger, H.: A completion procedure for conditional equations. J. Symb. Comput. 11 (1991) 51–81

    Google Scholar 

9.Programming

  1. O'Donnell, M. J.: Computing in systems described by equations, vol. 58 of Lect. Notes in Computer Sci. Springer-Verlag, Berlin (1977)

    Google Scholar 

  2. Guttag, J. V., Horowitz, E., Musser, D. R.: Abstract data types and software validation. Commun. ACM 21 (1978) 1048–1064

    Google Scholar 

  3. Futatsugi, K., Goguen, J. A., Jouannaud, J.-P., Meseguer, J.: Principles of OBJ2. In: Reid, B., ed., Conference Record of the Twelfth Annual ACM Symposium on Principles of Programming Languages. Association for Computing Machinery, New Orleans, LA (1985) 52–66

    Google Scholar 

  4. Dershowitz, N.: Computing with rewrite systems. Information and Control 64 (1985) 122–157

    Google Scholar 

  5. O'Donnell, M. J.: Equational Logic as a Programming Language. MIT Press, Cambridge, MA (1985)

    Google Scholar 

  6. Reddy, U. S.: On the relationship between logic and functional languages. In: DeGroot, D., Lindstrom, G., eds., Logic Programming: Functions, Relations, and Equations. Prentice-Hall, Englewood Cliffs, NJ (1986) 3–36

    Google Scholar 

  7. Goguen, J. A., Meseguer, J.: Eqlog: Equality, types, and generic modules for logic programming. In: DeGroot, D., Lindstrom, G., eds., Logic Programming: Functions, Relations, and Equations. Prentice-Hall, Englewood Cliffs, NJ (1986) 295–363

    Google Scholar 

  8. Dershowitz, N., Plaisted, D. A.: Equational programming. In: Hayes, J. E., Michie, D., Richards, J., eds., Machine Intelligence 11: The logic and acquisition of knowledge. Oxford Press, Oxford (1988) 21–56

    Google Scholar 

  9. Hölldobler, S.: Foundations of Equational Logic Programming, vol. 353 of Lect. Notes in Artificial Intelligence. Springer-Verlag, Berlin (1989)

    Google Scholar 

  10. Dershowitz, N., Okada, M.: A rationale for conditional equational programming. Theoretical Comput. Sci. 75 (1990) 111–138

    Google Scholar 

  11. Huet, G., Lévy, J.-J.: Computations in orthogonal rewriting systems, I and II. In: Lassez, J.-L., Plotkin, G., eds., Computational Logic: Essays in Honor of Alan Robinson. MIT Press, Cambridge, MA (1991) 395–443

    Google Scholar 

10.Theorem proving

  1. Lankford, D. S.: Canonical inference. Memo ATP-32, Automatic Theorem Proving Project, University of Texas, Austin, TX (1975)

    Google Scholar 

  2. Lankford, D. S., Ballantyne, A. M.: The refutation completeness of blocked permutative narrowing and resolution. In: Proc. of the Fourth Workshop on Automated Deduction, Austin, TX (1979) 53–59

    Google Scholar 

  3. Musser, D. R.: On proving inductive properties of abstract data types. In: Proc. of the Seventh ACM Symposium on Principles of Programming Languages, Las Vegas, NV (1980) 154–162

    Google Scholar 

  4. Huet, G., Hullot, J.-M.: Proofs by induction in equational theories with constructors. J. Comput. System Sciences 25 (1982) 239–266

    Google Scholar 

  5. Peterson, G. E.: A technique for establishing completeness results in theorem proving with equality. SIAM J. Comput. 12 (1983) 82–100

    Google Scholar 

  6. Plaisted, D. A.: Semantic confluence tests and completion methods. Information and Control 65 (1985) 182–215

    Google Scholar 

  7. Hsiang, J.: Refutational theorem proving using term-rewriting systems. Artif. Intell. 25 (1985) 255–300

    Google Scholar 

  8. Paul, E.: Equational methods in first order predicate calculus. J. Symb. Comput. 1 (1985) 7–29

    Google Scholar 

  9. Kapur, D., Musser, D. R.: Proof by consistency. Artif. Intell. 31 (1987) 125–157

    Google Scholar 

  10. Kapur, D., Narendran, P., Zhang, H.: On sufficient completeness and related properties of term rewriting systems. Acta Inf. 24 (1987) 395–415

    Google Scholar 

  11. Jouannaud, J.-P., Kounalis, E.: Automatic proofs by induction in equational theories without constructors. Info. and Comput. 81 (1989) 1–33

    Google Scholar 

  12. Küchlin, W.: Inductive completion by ground proof transformation. In: Aït-Kaci, H., Nivat, M., eds., Resolution of Equations in Algebraic Structures, vol. 2: Rewriting Techniques. Academic Press, New York (1989) 211–244

    Google Scholar 

  13. Comon, H., Lescanne, P.: Equational problems and disunification. J. Symb. Comput. 7 (1989) 371–425

    Google Scholar 

  14. Fribourg, L.: A strong restriction of the inductive completion procedure. J. Symb. Comput. 8 (1989) 253–276

    Google Scholar 

  15. Reddy, U. S.: Term rewriting induction. In: Stickel, M., ed., Proc. of the Tenth International Conference on Automated Deduction (Kaiserslautern, West Germany), vol. 449 of Lect. Notes in Computer Sci. Springer-Verlag, Berlin (1990)

    Google Scholar 

  16. Dershowitz, N., Pinchover, E.: Inductive synthesis of equational programs. In: Proc. of the Eighth National Conference on Artificial Intelligence. AAAI, Boston, MA (1990) 234–239

    Google Scholar 

  17. Dershowitz, N.: Ordering-based strategies for Horn clauses. In: Proc. of the Twelfth International Joint Conference on Artificial Intelligence, Sydney, Australia (1991) 118–124

    Google Scholar 

  18. Kapur, D., Narendran, P., Zhang, H.: Automating inductionless induction using test sets. J. Symb. Comput. 11 (1991) 83–112

    Google Scholar 

Future research

  1. Goguen, J. A., Kirchner, C., Meseguer, J.: Concurrent term rewriting as a model of computation. In: Keller, R., Fasel, J., eds., Proc. of Graph Reduction Workshop (Santa Fe, NM), vol. 279 of Lect. Notes in Computer Sci. Springer-Verlag (1987) 53–93

    Google Scholar 

  2. Toyama, Y., Klop, J. W., Barendregt, H. P.: Termination for the direct sum of left-linear term rewriting systems. In: Dershowitz, N., ed., Proc. of the Third International Conference on Rewriting Techniques and Applications (Chapel Hill), vol. 355 of Lect. Notes in Computer Sci. Springer-Verlag, Berlin (1989) 477–491

    Google Scholar 

  3. Dick, A. J. J., Watson, P.: Order-sorted term rewriting. Comput. J. 34 (1991) 16–19

    Google Scholar 

  4. Dershowitz, N., Kaplan, S., Plaisted, D. A.: Rewrite, rewrite, rewrite, rewrite, rewrite,.... Theoretical Comput. Sci. 83 (1991) 71–96

    Google Scholar 

  5. Dershowitz, N., Jouannaud, J.-P., Klop, J. W.: Open problems in rewriting. In: Book, R., ed., Proc. of the Fourth International Conference on Rewriting Techniques and Applications (Como, Italy), vol. 488 of Lect. Notes in Computer Sci. Springer-Verlag, Berlin (1991) 445–456

    Google Scholar 

  6. Jouannaud, J.-P., Okada, M.: Executable higher-order algebraic specification languages. In: Proc. of the Sixth IEEE Symposium on Logic in Computer Science (1991)

    Google Scholar 

  7. Bachmair, L.: Associative-commutative reduction orderings. Inf. Process. Lett. 43 (1992) 21–27

    Google Scholar 

  8. Hsiang, J., Kirchner, H., Lescanne, P., Rusinowitch, M.: Automated theorem proving in the presence of equalities. J. Autom. Reas. 14 (1992) 71–100

    Google Scholar 

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  1. Department of Computer Science, University of Illinois, 61801, Urbana, IL, USA

    Nachum Dershowitz

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  1. Nachum Dershowitz
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Peter E. Lauer

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Dershowitz, N. (1993). A taste of rewrite systems. In: Lauer, P.E. (eds) Functional Programming, Concurrency, Simulation and Automated Reasoning. Lecture Notes in Computer Science, vol 693. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-56883-2_11

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