Automated Deduction with Built-In Theories Completeness Results

Automated Deduction with Built-in Theories

Completeness Results and Constraint Solving Techniques

Tesi doctoral presentada al Departament de Llenguatges i Sistemes Informàtics de la Universitat Politècnica de Catalunya

per a optar al grau de Doctor en Informàtica

per Guillem Godoy Balil

sota la direcció del doctor Robert Nieuwenhuis

Barcelona, 1 de setembre de 2001 140 CHAPTER S. ORDERING CONSTRAINTS FOR AG Chapter 9

Directions for Further Research

9.1 Open problems from Chapter 4

1. Now It becomes interesting to explore completeness results for deduction modulo equational theories E for which monotonie E-compatible total orderings do not exist, but that can be handled with the techniques described in Chapter 4. 2. It has to be more carefully explored to what extent other redundancy notions, like demodulation, are applicable in our framework for ordered paramodulation. 3. In Chapter 4 we dealt with inference systems with eager selection of negative literals, that is, a negative literal is selected whenever there is any. For the moment we know of no direct way of proving the completeness for strategies with selection of (maximal) positive literals, although in Chapter 6 a relatively simple proof transformation technique is given for the Horn case, by which the completeness is shown for arbitrary selection strategies and arbitrary paramodulation-based inference systems that are complete with constraint inheritance and eager selection of negative literals. However, there exist exam- ples showing the incompatibility of these techniques with tautology deletion (see Example 47 of Chapter 4). This makes it unlikely that a standard model construction-based proof, where such tautologies are redundant, can be found.

9.2 Open problems from Chapter 5

Regarding Knuth-Bendix completion, after the results of Chapter 5 the following interesting questions remain unanswered: 4. Does Theorem 52 still hold when simplification by rewriting with respect to the reduction ordering ^r is applied?

141 142 CHAPTERS. DIRECTIONS FOR FURTHER RESEARCH

5. Does it hold when the strict ordered paramodulation rule of Definition 28 (where inferences at topmost positions of small sides w.r.t. >- are not needed) is used?

6. Would it hold when inferences with (or on) small sides w.r.t. >~r are not com- puted at all? Or, in other words, is superposition w.r.t. arbitrary reduction orderings complete?

9.3 Open problems from Chapter 6 The results of Chapter 6 could be extended in several directions, the most interesting ones being:

• Strategies with non-eager selection of negative equations. We think our proof transformation technique could be adapted to cover also strategies with non- eager selection of negative equations, i.e., proving that if there exists any (not necessarily eager negative) complete selection strategy selecting a single literal, then arbitrary selection strategies are complete. A possible way to go could be to first transform such proofs into proofs with eager selection of negative equations. • Answer computation. In Chapter 6 we focussed on refutation completeness, but we believe that our techniques easily extend to proving the completeness for answer computation, thus obtaining similar results for this purpose as the ones of [Lyn97] for total reduction orderings and superposition. The key idea is that solutions a are preserved during our transformation process. • Non-equality constraints. Very often it is the case that an inference system works not only with equality constraints but also with other kind of constraints such as, for example, ordering constraints. But, as seen in section 6.4 of Chapter 6, the whole transformation process can be done in the same way if there exists some substitution CT that satisfies all the constraints. Note that this is a reasonable assumption for many inference systems since, in our transformation, the equations involved in the inferences of the proof by A are the same as in the proof by Af. It may be interesting to study with which other kind of constraints this transformation method preserves completeness. • General clauses. Our result is for first-order Horn clauses with equality. We think that it would not be difficult to adapt our proof transformation method to the case of general clauses, provided no factoring inferences occur in the proofs. With factoring, incompleteness already appears in the propositional case, as shown by the following counter example from [dN96]. 9.4. OPEN PROBLEMS FROM CHAPTER 7 143

Example 169 Suppose we have

2. ->9,r 3. ->r,p 4. jp,c-> 5. g,r-> 6.

This set of clauses is unsatisfiable, and factoring (in this propositional case, elimination of repeated occurrences of positive literals) is needed for obtaining the empty clause. Now, suppose we choose the following arbitrary selection:

1. -+p,q 2. ->g,r 3. ->r,p 4. p,q-> 5. q, r-> 6. r,p->-

By applying resolution involving only the selected literals we only obtain tautologies of the form A — > A, and after resolving with/on them we get clauses of the initial set. Therefore this is a counter example to the completeness of arbitrary strategies when factoring is required, even if all tautologies are kept.

9.4 Open problems from Chapter 7

The first task that has to be started is the development of an experimental implementation of the inference system, combined with some simple redundancy notions. When doing this, one should not focus on efficiency of the implementation. Rather, one should carry out the development in the context of a flexible "toolkit" system like Saturate [NN93, GNN99]. Only from experimens with such an implementation it will become clearer in which directions one should continue developing the theoretical work described in this chapter. 144 CHAPTER 9. DIRECTIONS FOR FURTHER RESEARCH 9.5 Open problems from Chapter 8 As for Chapter 7, again the first task that has to be started is the development of an experimental implementation. In fact, the implementation of the inference system described in Chapter 7 requires at least an approximation of a constraint solving mechanism like the one described in Chapter 8. Prom such a constraint solver running on more or less real-life problems, one could draw conclusions about its efficiency and possible bottlenecks and enhancements. Concerning further theoretical work, we expect that the ideas given here will provide new insights for the development of constraint solving methods for other E-compatible orderings, especially over fixed signatures. Bibliography

[Bac89] Leo Bachmair. Proof normalization for resolution and paramodulation. In Third Int. Con/, on Rewriting Techniques and Applications (RTA), LNCS 355, pages 15-28, Chapel Hill, NC, USA, 1989. Springer-Verlag.

[Bac91] Leo Bachmair. Canonical equational proofs. Birkhäuser, Boston, Mass., 1991. [BD89] Leo Bachmair and Nachum Dershowitz. Completion for rewriting modulo a congruence. Theoretical Computer Science, 2 and 3(67):173-201,1989.

[BD94] Leo Bachmair and Nachum Dershowitz. Equational inference, canonical proofs, and proof orderings. J. of the Association for Computing Machin- ery, 41(2):236-276, February 1994.

[BDH86] Leo Bachmair, Nachum Dershowitz, and Jieh Hsiang. Orderings for equational proofs. In First IEEE Symposium on Logic in Computer Science (LICS), pages 346-357, Cambridge, Massachusetts, USA, June 16-18, 1986. IEEE Computer Society Press.

[BDP89] Leo Bachmair, Nachum Dershowitz, and David Plaisted. Completion Whitout Failure. In Hassan Aït-Kaci and Maurice Nivat, editors, Resolu- tion of Equations in Algebraic Structures, volume 2: Rewriting Techniques, chapter 1, pages 1-30. Academic Press, New York, 1989.

[BG90] Leo Bachmair and Harald Ganzinger. On restrictions of ordered paramodulation with simplification. In Mark E. Stickel, editor, 10th International Conference on Automated Deduction (CADE), LNAI 449, pages 427-441, Kaiserslautern, FRG, July 24-27, 1990. Springer-Verlag.

[BG94a] Leo Bachmair and Harald Ganzinger. Associative-commutative superposition. In N. Dershowitz, editor, Proc. 5th International Workshop on Con- ditional Term Rewriting Systems, LNCS 968, pages 155-167, Jerusalem, July 11-15, 1994. Springer-Verlag.

145 146 BIBLIOGRAPHY

[BG94b] Leo Bachraair and Harald Ganzinger. Rewrite-based equational theorem proving with selection and simplification. Journal of Logic and Computa- tion, 4(3):217-247, 1994.

[BG96] David Basin and Harald Ganzinger. Complexity Analysis Based on Or- dered Resolution. In Eleventh Annual IEEE Symposium on Logic in Com- puter Science (LICS), pages 456-465, New Brunswick, New Jersey, USA, July 27-30, 1996. IEEE Computer Society Press.

[BG98] Leo Bachmair and Harald Ganzinger. Equational reasoning in saturation- based theorem proving. In W. Bibel and P. Schmitt, editors, Automated Deduction: A Basis for Applications. Kluwer, 1998.

[BGOla] Leo Bachmair and Harald Ganzinger. Resolution. In J.A. Robinson and A. Voronkov, editors, Handbook of Automated Reasoning. Elsevier Science Publishers and MIT Press, 2001.

[BGOlb] Miquel Bofill and Guillem Godoy. On the Completeness of Arbitrary Selec- tion Strategies for Paramodulation. In Automata, Languages and Program- ming, 28th Int. Colloquium (1CALP), Crete, Greece, July 2001. Springer- Verlag.

[BGLS92] Leo Bachmair, Harald Ganzinger, Christopher Lynch, and Wayne Sny- der. Basic paramodulation and superposition. In Deepak Kapur, editor, llth International Conference on Automated Deduction (CADE), LNAI 607, pages 462-476, Saratoga Springs, New York, USA, June 15-18, 1992. Springer-Verlag.

[BGLS95] L. Bachmair, H. Ganzinger, Chr. Lynch, and W. Snyder. Basic paramodulation. Information and Computation, 121(2):172-192, 1995.

[BGNR99] Miquel Bofill, Guillem Godoy, Robert Nieuwenhuis, and Albert Rubio. Paramodulation with non-monotonic orderings. In 14th IEEE Symposium on Logic in Computer Science (LICS), pages 225-233, Trento, Italy, July 2- 5, 1999.

[BP85] L. Bachmair and D. A. Plaisted. Termination orderings for associative- commutative rewriting systems. Journal of Symbolic Computation, 1:329- 349, 1985.

[Bra75] D. Brand. Proving theorems with the modification method. SI AM Journal on Computing, 4(4):412-430, December 1975. BIBLIOGRAPHY 147

[BS93] Franz Baader and Jörg Siekmann. Unification theory. In D.M. Gabbay, C.J. Hogger, and J.A. Robinson, editors, Handbook of Logic in Artificial Intelligence and Logic Programming. Oxford University Press, 1993. [BS01] Franz Baader and Wayne Snyder. Unification theory. In J.A. Robinson and A. Voronkov, editors, Handbook of Automated Reasoning. Elsevier Science Publishers and MIT Press(to appear), 2001. [Che86] Philippe Le Chenadec. Canonical Forms in Finitely Presented Algebras. Research Notes in Theoretical Computer Science. Pitman, London; Wiley, New York, Toronto, 1986. [CNNR98] Hubert Comon, Paliath Narendran, Robert Nieuwenhuis, and Michael Rusinowitch. Decision problems in ordered rewriting. In 13th IEEE Sym- posium on Logic in Computer Science (LICS), pages 410-422, Indianapolis, USA, June 27-30, 1998. [CNR95] Hubert Comon, Robert Nieuwenhuis, and Albert Rubio. Orderings, AC- Theories and Symbolic Constraint Solving. In 10th IEEE Symposium on Logic in Computer Science (LICS), pages 375-385, San Diego, USA, June 26-29, 1995. • [Com90] Hubert Comon. Solving symbolic ordering constraints. InternationalJour- nal of Foundations of Computer Science, 1(4):387-411, 1990. [CT94] Hubert Comon and Ralf Treinen. Ordering Constraints on Trees. In Proc. of Colloquium on Trees in Algebra and Programming (CAAP), LNCS 787, pages 1-14, Edinburgh, Scotland, April 1994. Springer-Verlag. [Deg79] Anatoli Degtyarev. The monotonie paramodulation strategy. In 5th All- Union Conference on Mathematical Logic, Novosibirsk, 1979. (In Russian). [Der82] Nachum Dershowitz. Orderings for term-rewriting systems. Theoretical Computer Science, 17(3):279-301, 1982. [Der91] Nachum Dershowitz. Canonical sets of Horn clauses. In J. Leach Al- bert, B. Monien, and M. Rodriguez Artalejo, editors, Proceedings of the Eighteenth International Colloquium on Automata, Languages and Pro- gramming (ICALP), LNCS 510, pages 267-278, Madrid, Spain, July 1991. Springer-Verlag. [Dev90] Hervé Dévie. Linear completion. In Stéphane Kaplan and Mitsuhiro Okada, editors, Conditional and Typed Rewriting Systems, 2nd Interna- tional Workshop, LNCS 516, pages 233-245, Montreal, Canada, June 11- 14, 1990. Springer-Verlag. 148 BIBLIOGRAPHY

[Dev92] Hervé Dévie. PhD thesis, Université de Paris-Sud, Orsay, France, 1992. [DJ90] Nachum Dershowitz and Jean-Pierre Jouannaud. Rewrite systems. In Jan van Leeuwen, editor, Handbook of Theoretical Computer Science, volume B: Formal Models and Semantics, chapter 6, pages 244-320. Elsevier Science Publishers B.V., Amsterdam, New York, Oxford, Tokyo, 1990. [DM79] Nachum Dershowitz and Zohar Manna. Proving termnation with multiset orderings. Comm. of ACM, 22(8), 1979. [DMT88] Nachum Dershowitz, Leo Marcus, and Andrzej Tarlecki. Existence, uniqueness, and construction of rewrite systems. SIAM Journal on Com- puting, 17(4):629-639,1988. [dN96] Hans de Nivelle. Ordering refinements of resolution. Dissertation, Tech- nische Universiteit Delft, Delft, 1996. [Dom92] Eric Domenjoud. A technical note on AC-unification. the number of min- imal unifiers of the equation aii + ... + axp = ßy\ -f ...+ ßyq. Journal of Automated Reasoning, 8(l):39-44, 1992. [DV86] Anatoli Degtyarev and Andrei Voronkov. Equality methods in machine theorem proving. Cybernetics, 22(3):298-307, 1986. [Gan91] Harald Ganzinger. A completion procedure for conditional equations. Jour- nal of Symbolic Computation, 11:51-81, 1991. [GdN99] Harald Ganzinger and Hans de Nivelle. A superposition decision procedure for the guarded fragment with equality. In 14th IEEE Symposium on Logic in Computer Science (LICS), pages 295-305, Trento, Italy, July 2-5, 1999. [GMV99] Harald Ganzinger, Christoph Meyer, and Margus Veanes. The two- variable guarded fragment with transitive relations. In 14th IEEE Sym- posium on Logic in Computer Science (LICS), pages 24-34, Trento, Italy, July 2-5, 1999. [GNOO] Guillem Godoy and Robert Nieuwenhuis. Paramodulation with built-in abelian groups. In 15th IEEE Symposium on Logic in Computer Science (LICS), pages 413-424, Santa Barbara, USA, June 26-29, 2000. IEEE Computer Society Press. [GNOla] Guillem Godoy and Robert Nieuwenhuis. Ordering Constraints for Deduc- tion with Built-in Abelian Semigroups, Monoids and Groups. In 16th IEEE Symposium on Logic in Computer Science (LICS), pages 38-47, Boston, USA, June 16-20, 2001. IEEE Computer Society Press. BIBLIOGRAPHY 149

[GNOlb] Guillem Godoy and Robert Nieuwenhuis. Superposition with Completely Built-in Abelian Groups. Journal of Symbolic Computation, 2001. submit- ted. [GNN99] Harald Ganzinger, Robert Nieuwenhuis, and Pilar Nivela. The Saturate System, 1999. Software and documentation available at: http://www.mpi-sb.mpg.de/SATURATE/Saturate.html. [GW96] Harald Ganzinger and Uwe Waldmann. Theorem proving in cancellative abelian monoids. In M. A. McRobbie and J. K. Slaney, editors, Thirteenth International Conference on Automated Deduction (CADE), volume 1104 of LNAI, pages 388-402, New Brunswick, 1996. Springer. [HBVL97] Thomas Hillenbrand, Arnim Buch, Roland Vogt, and Bernd Löchner. WALDMEISTER—high-performance equational deduction. Journal of Automated Reasoning, 18(2):265-270, April 1997. [HR87] J. Hsiang and M. Rusinowitch. On word problems in equational theories. In T. Ottmann, editor, Proc. 14th Int. Colloquium Automata, Languages and Programming, LNCS 267, pages 54-71, Berlin, Germany, 1987, Springer-, Verlag. [HR91] J. Hsiang and M Rusinowitch. Proving refutational completeness of theorem proving strategies: the transfinite semantic tree method. Journal of the ACM, 38(3):559-587, July 1991. [HueSO] G. Huet. Confluent reductions: abstract properties and applications to term rewriting systems. Journal of the ACM, 27(4):797-821, October 1980. [HulSO] J.M. Hullot. Compilation de Formes Canoniques dans les Teories Équationnelles. PhD thesis, Université de Paris Sud, France, 1980. [JK86] Jean-Pierre Jouannaud and Hélène Kirchner. Completion of a set of rules modulo a set of equations. SIAM Journal of Computing, 15:1155-1194, 1986. [JO91] J-P. Jouannaud and M. Okada. Satisfiability of systems of ordinal notations with the subterm property is decidable. In 18th International Colloquium Automata, Languages and Programming (ICALP), LNCS 510, pages 455- 468, Madrid, Spain, July 16-20 1991. Springer-Verlag. [Jou83] Jean-Pierre Jouannaud. Confluent and coherent equational term rewriting systems: Applications to proofs in abstract data types. In Proc. 8th Col- loquium on Trees in Algebra and Programming, LNCS 59, pages 269-283. Springer-Verlag, 1983. 150 BÍBLíOGfíAPHY

[JW86] Jean-Pierre Jouannaud and Bernard Waldmann. Reductive conditional term rewriting systems. In Proc. Third IFIP Working Conference on For- mal Description of Programming Concepts, Ebberup, Denmark, 1986. [Kap84] Stephane Kaplan. Conditional rewrite rules. Theoretical Computer Science, 33:175-193, 1984. [KB70] D.E. Knuth and P.B. Bendix. Simple word problems in universal algebras. In J. Leech, ed., Computational Problems in Abstract Algebra, pages 263- 297. Pergamon Press, Oxford, 1970. [KKR90] Claude Kirchner, Hélène Kirchner, and Michael Rusinowitch. Deduc- tion with symbolic constraints. Revue Française d'Intelligence Artificielle, 4(3):9-52, 1990. [KL80] S. Kamin and J.-J. Levy. Two generalizations of the recursive path ordering. Unpublished note, Dept. of Computer Science, Univ. of Illinois, Urbana, IL, 1980. [KN92] Deepak Kapur and Paliath Narendran. Double exponential complexity of computing complete sets of AC-unifiers. In Seventh Annual IEEE Sympo- sium on Logic in Computer Science, pages 11-21, Santa Cruz, California, USA, June 22-25, 1992. IEEE Computer Society Press. [KR91] Emmanuel Kounalis and Michael Rusinowitch. On word problems in Horn theories. Journal of Symbolic Computation, 11(1-2):113-128, January- February 1991.

[KVOO] Konstantin Korovin and Andrei Voronkov. A Decision Procedure for the Existential Theory of Term Algebras with the Knuth-Bendix Ordering. In 15th IEEE Symposium on Logic in Computer Science (LICS), pages 291-302, Santa Barbara, USA, June 26-29, 2000. IEEE Computer Society Press. [KV01] Konstantin Korovin and Andrei Voronkov. Knuth-Bendix constraint solving is NP-complete. In Automata, Languages and Programming, Inter- national Colloquium, 28th Int. colloquium, (ICALP), Crete, Greece, July 2001. Springer-Verlag. [Lan75] D. S. Lankford. Canonical inference. Technical Report ATP-32, Dept. of Mathematics and Computer Science, Univ. of Texas, Austin, TX, 1975. [LB77] D. S. Lankford and A. M. Ballantyne. Decision procedures for simple equational theories with commutative-associative axioms: Complete sets BIBLIOGRAPHY 151

of commutative-associative reductions. Technical Report Memo ATP-39, Dept. of Mathematics and Computer Science, Univ. of Texas, Austin, TX, August 1977.

[LS93] Chris Lynch and Wayne Snyder. Redundancy criteria for constrained completion. In C. Kirchner, editor, 5th International Conference on Rewriting Techniques and Applications (RTA), LNCS 690, pages 2-16, Montreal, Canada, June 16-18, 1993. Springer-Verlag.

[Lyn97] C. Lynch. Oriented equational logic programming is complete. Journal of Symbolic Computation, 23(l):23-46, January 1997.

[Mar94] Claude Marché. Normalised Rewriting and Normalised Completion. In Ninth Annual IEEE Symposium on Logic in Computer Science, Paris, Prance, July 1994. IEEE Computer Society Press, Los Alamitos, CA, USA.

[Mar96] Claude Marché. Normalized rewriting: An alternative to rewriting Modulo a set of equations. Journal of Symbolic Computation, 21(3):253-288,March 1996.

[McC94] William McCune. OTTER 3.0 Reference Manual and Guide. Technical Report ANL-94/6, Argonne National Laboratory, 1994.

[McC97a] W. McCune. 33 basic test problems: A practical evaluation of some paramodulation strategies. In Robert Veroff, editor, Automated Reasoning and its Applications: Essays in Honor of Larry Wos, pages 71-114. MIT Press, 1997.

[McC97b] W. McCune. Well behaved search and the Robbins problem. In H. Comon, editor, 8th International Conference on Rewriting Techniques and Applications (RTA), LNCS 1232, pages 1-7, Sitges, Spain, 1997. Springer-Verlag.

[McC97c] William McCune. Solution of the Robbins problem. Journal of Automated Reasoning, 19(3) :263-276, December 1997.

[MN90] Ursula Martin and Tobias Nipkow. Ordered rewriting and confluence. In Mark E. Stickel, editor, 10th International Conference on Automated De- duction (CADE), LNAI 449, pages 366-380, Kaiserslautern, FRG, July 24- 27, 1990. Springer-Verlag.

[Nie93] Robert Nieuwenhuis. Simple LPO constraint solving methods. Information Processing Letters, 47:65-69, August 1993. 152 BIBLIOGRAPHY

[Nie96] Robert Nieuwenhuis. Basic paramodulation and decidable theories. In Eleventh Annual IEEE Symposium on Logic in Computer Science, pages 473-482, New Brunswick, New Jersey, USA, July 27-30, 1996. IEEE Com- puter Society Press. [Nie98] Robert Nieuwenhuis. Decidability and complexity analysis by basic paramodulation. Information and Computation, 147:1-21, 1998. [NN91] Robert Nieuwenhuis and Pilar Nivela. Efficient deduction in equality horn logic by horn-completion. Information Processing Letters, 39(l):l-6, July 1991. [NN93] Pilar Nivela and Robert Nieuwenhuis. Practical results on the saturation of full first-order clauses: Experiments with the saturate system, (system description). In C. Kirchner, editor, 5th International Conference on Rewriting Techniques and Applications (RTA), LNCS 690, pages 436-440, Montreal, Canada, June 16-18, 1993. Springer-Verlag. [NO90] Robert Nieuwenhuis and Fernando Orejas. Clausal rewriting. In Stéphane Kaplan and Mitsuhiro Okada, editors, Conditional and Typed Rewriting Systems, 2nd International Workshop, LNCS 516, pages 246-258, Mon- treal, Canada, June 11-14, 1990. Springer-Verlag. [NR92a] Robert Nieuwenhuis and Albert Rubio. Basic superposition is complete. In B. Krieg-Briickner, editor, European Symposium on Programming, LNCS 582, pages 371-390, Rennes, France, February 26-28, 1992. Springer- Verlag. [NR92b] Robert Nieuwenhuis and Albert Rubio. Theorem proving with ordering constrained clauses. In Deepak Kapur, editor, llth International Conference on Automated Deduction (CADE), LNAI 607, pages 477-491, Saratoga Springs, New York, 1992. [NR94] Robert Nieuwenhuis and Albert Rubio. AC-Superposition with constraints: No AC-unifiers needed. In Alan Bundy, editor, 12th International Confer- ence on Automated Deduction (CADE), LNAI 814, pages 545-559, Nancy, France, June 1994. Springer-Verlag. [NR95] Robert Nieuwenhuis and Albert Rubio. Theorem Proving with Order- ing and Equality Constrained Clauses. Journal of Symbolic Computation, 19(4):321-351, April 1995. [NR97] Robert Nieuwenhuis and Albert Rubio. Paramodulation with Built-in AC- Theories and Symbolic Constraints. Journal of Symbolic Computation, 23(l):l-21,May 1997. BJBL/OGHAPHY 153

[NR99] Robert Nieuwenhuis and José Miguel Rivero. Solved forms for path ordering constraints. In P. Narendran and M. Rusinowitch, editors, Tenth In- ternational Conference on Rewriting Techniques and Applications (RTA), LNCS 1631, pages 1-15, Trento, Italy, July 2-4, 1999. Springer-Verlag. [NR01] Robert Nieuwenhuis and Albert Rubio. Paramodulation-based theorem proving. In J.A. Robinson and A. Voronkov, editors, Handbook of Auto- mated Reasoning. Elsevier Science Publishers and MIT Press, 2001.

[NRV99] P. Narendran, M. Rusinowitch, and R. Verma. RPO constraint solving is in NP. In G. Gottlob, E. Grandjean, and Katrin Seyr, editors, 12th Int. Conference of the European Association of Computer Science Logic (CSL'98), LNCS 1584, pages 385-398, Brno, Czech Republic, August 23- 28, 1999. Springer-Verlag. [Pau92] E. Paul. A general refutational completeness result for an inference procedure based on associative-commutative unification. Journal of Symbolic Computation, 14(6):577-618, 1992. [Pet90] Gerald E. Peterson. Complete sets of reductions with* constraints. In Mark E. Stickel, editor, 10th International Conference on Automated De- • auction (CADE), LNAI449, pages 381-395, Kaiserslautern, FRG, July 24- 27, 1990. Springer-Verlag. [Plo72] Gordon Plotkin. Building in equational theories. Machine Intelligence, 7:73-90, 1972. [PP91] J. Pais and G. Peterson. Using Forcing to Prove Completeness of Resolu- tion and Paramodulation. Journal of Symbolic Computation, 11(1):3-19, 1991. [PS81] G.E. Peterson and M.E. Stickel. Complete sets of reductions for some equational theories. Journal Assoc. Còmput. Mach., 28(2):233-264, 1981. [RN95] Albert Rubio and Robert Nieuwenhuis. A total AC-compatible ordering based on RPO. Theoretical Computer Science, 142(2):209-227, May 15, 1995.

[Rob65] J. A. Robinson. A machine-oriented logic based on the resolution principle. Journal of the ACM, 12(1):23-41, January 1965. [RTA01] RTA-LOOP. Int. Conf. on Rewriting Techniques and Applications, The list of open problems, 2001. Maintained at http://www.lri.fr/~rtaloop/ (by R.-Treinen). 154 BIBLIOGRAPHY

[RV95] Michael Rusinowitch and Laurent Vigneron. Automated deduction with associative commutative operators. J. of Applicable Algebra in Engineer- ing, Communication and Computation, 6(l):23-56,1995. [RW69] G. A. Robinson and L. T. Wos. Paramodulation and theorem-proving in first order theories with equality. Machine Intelligence, 4:135-150, 1969. [Sch87] Alexander Schrijver. Theory of Linear and Integer Programming. John Wiley and Sons, New York, 1987. [Sla74] James R. Slagle. Automated theorem-proving for theories with simplifiers, commutativity, and associativity. Journal of the ACM, 21(4):622-642, Oc- tober 1974. [Stu98] Jürgen Stuber. Superposition theorem proving for abelian groups repre- sented as integer modules. Theoretical Computer Science, 208(1-2):149- 177, 28 November 1998. [Stu99] Jürgen Stuber. Superposition Theorem Proving for Commutative Algebraic Theories. Dissertation, Technische Fakultät, Universität des Saarlandes, Saarbrücken, Germany, 1999. [Vig94] Laurent Vigneron. Associative Commutative Deduction with constraints. In Alan Bundy, editor, 12th International Conference on Automated De- duction (CADE), LNAI 814, pages 530-544, Nancy, France, June 1994. Springer-Verlag. [Wal97] Uwe Waldmann. Cancellative Abelian Monoids in Refutational Theorem Proving. Dissertation, Universität des Saarlandes, Saarbrücken, Germany, 1997. [Wal98] Uwe Waldmann. Superposition for divisible torsion-free abelian monoids. In Claude Kirchner and Hélène Kirchner, editors, Proceedings of the Fif- teenth International Conference on Automated Deduction (CADE-98), volume 1421 of LNAI, pages 144-159, Lindau, Germany, 1998. Springer. [Wal99] Uwe Waldmann. Cancellative superposition decides the theory of divisible torsion-free abelian groups. In Logic Programming and Automated Rea- soning, Int. Conf., LNAI 1705, pages 131-147, Tbilisi, Georgia, September 1999. Springer-Verlag. [Wec91] W. Wechler. Universal Algebra for Computer Scientists, volume 25 of EATCS Monographs on Theoretical Computer Science. Springer-Verlag, Berlin, 1991. BIBLIOGRAPHY 155

[Wer92] Ulrich Wertz. First-order theorem proving modulo equations. Technical Report MPI-I-92-216, Max-Planck-Institut für Informatik, Saarbrücken, April 1992. [Wos88] Larry Wos. Automated Reasoning: 33 Basic Research Problems. Prentice- Hall, Englewood Cliffs, 1988. [Wos96] Larry Wos. The Automation of Reasoning: An Experimenter's Notebook with OTTER Tutorial. Academic Press, 1996. [Zha88] H Zhang. Reduction, Superposition, and Induction: Automated Reasoning in an Equational Logic. PhD thesis, Renselaer Polytechnic Institute, 1988. [Zha93] Hantao Zhang. A case study of completion modulo distributivity and Abelian groups. In Claude Kirchner, editor, Proceedings of the 5th Inter- national Conference on Rewriting Techniques and Applications (RTA-93), volume 690 of LNCS, pages 32-46. Springer-Verlag, June 16-18 1993. Index

==,19 4->-ñ, 19 =Ac, 78 »-, 19 == AOi 78 ff rpo) ^0 20 ^'ei, 19 ' ocOrpoi >-c, 24 ' agrpoi -L-LO

(u ¡*-)-irreducible, 98 Xrpo, 20 (u X)-irreducible, 98 /ex on +, 78, 113 ->, 18 -, 78, 113 -»*, 18 O, 78, 113 =a 19 ->+, 18 =»19 gnd(R), 61 >s, 115 maxredn, 98 £|p,4 ndec, 132 DS, 122 síeps(P), 66 E-unification, 75 *[*]p, 17 £*, 19 í IP, 17 A*, 19 uars(í), 17 RAO, 77, 78 Ac, 25 .4, 65 Rg, 61 ,4-proof, 66 RAG, 114 Ai, 65 #(«,*), 132 ^/"-proof, 66 a, 21 "/A. 78 a, 23 n/R», 115 ft, 30 n /AXG, J, 33 >-A, 53 «S, 36 JA 74 abelian groups, 9 4-, 18 AC, 45, 111 «-»£, 19 AC-comptatibility, 45

156 INDEX 157

AC-unification, 9 E-completion, 9 ACÓ, 111 Knuth-Bendix, 5 ACl-unification, 10 modulo AC, 9 AG, 9, 75, 111 unfailing, 6 AG axioms, 78 conditional equation, 6 AG-factoring, 109 conditional rewriting, 6 AG-RPO assumptions, 127 confluence, 5, 18 AG-zero-instance, 89 congruence, 18 antecedent, 21 axioms, 4 antecedent elimination, 66 congruence axioms, 4 Argonne group, 5 constrained clause, 7, 22, 36 associate equations, 128 equality, 8 constrained empty clause, 22 basic constrained formulae, 7 paramodulation, 8 constraint, 7, 21 strategy, 8 compatible with, 63 superposition, 8 equality, 11, 21 basicness equality constrained clause, 8 restriction, 11 extended signature, 11 built-in abelian groups, 75 fixed signature, 11 built-in AG, 9 global, 32 built-in theories, 9 inheritance strategies, 8 local, 32 Church-Rosser property, 18 ordering, 11, 21 clausal rewriting, 6 satisfiable, 21 clause, 21 solution, 21 constrained, 22, 36 solving, 11, 111 constrained clause complexity, 11 equality, 8 decidability, 11 irreducibility, 8 symbolic, 11 positive, 7 tautology, 21 clauses without inheritance, 31 general, 33 convergence, 18 Horn, 23 critical pair, 6 closure substitution, 8 criteria, 6 compared by segments, 121 compatibility decisive summand, 132 AC-compatibility, 19 diophantine system, 122, 134 E-compatibility, 19 direct AG-superposition, 81, 89 compatible with constraints, 63 completion E-unification, 9 158 INDEX

E-unifier, 9 instance, 18 eager negative selection strategy, 7 interpretation, 21 empty clause, 21 equality Herbrand, 21 EQP, 5 inverse AG-superposition, 81, 89 equality constraint inverse orientation, 87 inheritance, 36 inverse reductive form, 79 equality factoring, 33 inverse splitting, 88 equality resolution, 24, 33 irreducibility, 18 equation, 19 irreducible instances, 84 conditional, 6 irreducible pair equivalence class, 115 with only constant symbols, 91 extended ^-rewriting, 9 Knuth-Bendix completion, 5, 59 factoring, 3, 4 flattened form, 45, 79, 114 Leibniz, 5 follows from, 21 lexicographic path ordering (LPO), 20 forcing, 7 lifting, 32 free symbols, 113 linear system, 115 FRPO, 114 local confluence, 18 FRPO assumptions, 118 logical consequence, 21 functional reflexivity axioms, 4 LPO, 20 fusion, 67 matching, 18 global assumptions, 117 mgu, 4, 18 ground model generation, 7, 23, 27 substitution, 18 AG term, 17 ground horn case, 82 Herbrand interpretation, 21 with variables, 91 for general clauses, 34 idempotency axiom, 12 for west orderings, 46 inference, 22 general clauses, 52 rule, 22 restricted, 48 complete, 22 monotonie, 18 correct, 22 monotonicity axioms, 4 system, 22 most general unifier, 4 general clauses 2", 33 multiset, 18 general clauses S, 36 ground Horn clauses £/, 23 narrowing, 8 Horn clauses %, 30 non-top-level reduction, 98 with selection S, 36 normal form, 5, 18 infix notation, 79, 114 number of occurrences, 132 INDEX 159 one-sided form, 84 recursively irreducible, 98 ordered paramodulation, 5, 6 reduction ordering, 19 ordered resolution, 29 reductive form, 79 ordering, 19 redundancy, 8, 55, 74 compatible, 19 refutation complete, 3 E-compatible, 19 relation, 18 reduction, 5, 19 Church-Rosser, 18 simplification, 19 confluent, 18 term, 5 inverse, 18 well-founded, 19 locally confluent, 18 ordering restrictions monotonie, 18 approximation, 44 normal form, 18 orderings reflexive-transitive closure, 18 rewrite, 19 terminating, 18 orientation, 86 transitive closure, 18 orienting, 85 well-founded, 18 Otter, 5 resolution, 3 binary, 3 paramodulation, 3, 4 rewrite ordered, 5, 6 rule, 19 path ordering system, 19 lexicographic (LPO), 20 rewrite ordering, 19 recursive (RPO), 20 rewrite system, 19 path orderings, 20 Church-Rosser, 19 position, 17 confluent, 19 above, 78 locally confluent, 19 strictly, 78 below, 78 terminating, 19 immediately below, 78 rewriting, 5, 19 strictly, 78 clausal, 6 beside, 78 conditional, 6 disjoint, 78 extended E-, 9 positive strategies, 36 modulo E, 9 positive unit strategies, 36 ordered, 5 precedence, 20, 44 rightmost inner node, 66 predicates rightmost leaf, 66 non-equality, 29 rightmost path, 66 proof orderings, 6 rightmost step, 66 proof tree, 66 Robbins conjecture, 5 Robbins problem, 11 recursive path ordering (RPO), 20 Robinson, 3 160 INDEX

RPO, 20, 44, 79 left, 24, 33 right, 24, 33 segment, 118 strict, 7 variable definition in a, 118 symetrization, 76 selected literals, 7 symmetrization, 10 selection, 35 selection strategies, 7, 49, 63 term, 17 semantic path ordering, 44 maximal, 24 signature, 17 position, 17 extended, 11 strictly maximal, 24 fixed, 11 term ordering, 5 simplification ordering, 19 term rewrite system, 19 solution of a constraint, 21 termination, 18 solved forms, 11 theories split, 85 built-in, 9 split AG-RPO assumptions, 130 top-level negative reduction, 98 split FRPO assumptions, 121 top-level positive reduction, 98 splitting, 88, 95 top-level summand, 113 splitting constraint, 86-88 totality, 19 splitting transformation, 120, 130 transfinite semantic trees, 6 - stability unfailing completion, 6 under substitutions, 19 unification, 18 strategies AC, 9 arbitrary, 63 unifier, 18 eager negative, 7, 63 positive unit, 7, 49 variable definition, 118 selection, 35 strict superposition, 7 well-constrained sets, 37 substitution, 5, 17 west ordering, 42 application, 18 word problem, 5 ground, 18 irreducible, 37 subsumption, 18 subterm, 17 property, 19 succèdent, 21 sum of variables, 114 summand, 79, 113 top-level, 113 superposition, 6, 24 basic, 8