Abstract
In resolution theorem proving as well as in its descendant, logic programming, we are frequently confronted with binary clauses causing lengthy or infinite computations because of their self-resolvents. In this paper we investigate unification modulo a binary clause, which can be used as a short cut through loops of the form L→R. For a certain class of binary clauses we show that (i) its unification problem is decidable and (ii) the unifiers can be finitely schematized. This is done by first reducing the binary clauses to a simpler form and then employing primal grammars [12]. Our work extends results obtained in the context of cycle unification.
The final version of this article was completed while visiting CRIN/INRIA Lorraine (Nancy).
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
F. Baader and J. H. 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, Oxford, UK, 1993. To appear.
W. Bibel, S. Hölldobler, and J. Würtz. Cycle unification. In D. Kapur, editor, 11th International Conference on Automated Deduction, LNAI 607, pages 94–108, Saratoga Springs, New York, USA, June 15–18, 1992. Springer-Verlag.
H. Chen and J. Hsiang. Logic programming with recurrence domains. In J. Leach Albert, B. Monien, and M. Rodriguez, editors, Automata, Languages and Programming (ICALP'91), LNCS 510, pages 20–34. Springer-Verlag, 1991.
H. Chen, J. Hsiang, and H.-C. Kong. On finite representations of infinite sequences of terms. In S. Kaplan and M. Okada, editors, Conditional and Typed Rewriting Systems, 2nd International Workshop, LNCS 516, pages 100–114, Montreal, Canada, June 11–14, 1990. Springer-Verlag.
H. Comon. On unification of terms with integer exponents. Technical Report 770, LRI, Orsay, France, 1992. To appear in: Mathematical System Theory.
D. de Schreye, M. Bruynooghe, and K. Verschaetse. On the existence of nonterminating queries for a restricted class of prolog-clauses. Artificial Intelligence, 41:237–248, 1989.
P. Devienne. Weighted graphs: a tool for studying the halting problem and time complexity in term rewriting systems and logic programming. TCS, 75:157–215, 1990.
P. Devienne, P. Lebègue, and J.-C. Routier. The emptiness problem of one binary recursive Horn clause is undecidable. In International Logic Programming Symposium '93, Vancouver, Oct. 1993. MIT Press.
P. Devienne, P. Lebègue, J.-C. Routier, and J. Würtz. One binary binary Horn clause is enough. In STACS '94, LNCS, Caen, Mar. 1994. Springer-Verlag.
R. Galbavý and M. Hermann. Unification of infinite sets of terms schematized by primal grammars. Technical Report 92-R-220, CRIN, Nancy, France, 1992.
P. Hanschke and J. Würtz. Satisfiability of the smallest binary program. IPL, 45(5):237–241, Apr. 1993.
M. Hermann. On the relation between primitive recursion, schematization, and divergence. In H. Kirchner and G. Levi, editors, Proceedings 3rd Conference on Algebraic and Logic Programming, LNCS 632, pages 115–127, Volterra (Italy), 1992. Springer-Verlag.
M. Hermann. Divergence des systèmes de réécriture et schématisation des ensembles infinis de termes. Habilitation, Université de Nancy I and CRIN-CNRS Inria Lorraine, Nancy (France), Mar. 1994.
J. Marcinkowski and L. Pacholski. Undecidability of the horn-clause implication problem. In 33rd Annual IEEE Symposium on Foundations of Computer Science, pages 354–362, Los Alamitos, 1992.
G. Salzer. Deductive generalization and meta-reasoning, or how to formalize Genesis. In Proc. 7. österreichische Artificial Intelligence Tagung, Informatik-Berichte 287, pages 103–115. Springer-Verlag, 1991.
G. Salzer. Unification of Meta-Terms. PhD thesis, Technische Universität Wien, 1991.
G. Salzer. Solvable classes of cycle unification problems. In J. Dassow, editor, International Meeting of Young Computer Scientists, Topics in Computer Science, Smolenice, Slovakia, 1992. Gordon&Breach. To appear in 1994.
G. Salzer. The unification of infinite sets of terms and its applications. In A. Voronkov, editor, Logic Programming and Automated Reasoning (LPAR'92), LNAI 624, pages 409–420, St. Petersburg, Russia, July 1992. Springer-Verlag.
M. Schmidt-Schau\. Implication of clauses is undecidable. TCS, 59:287–296, 1988.
M. E. Stickel. Automated deduction by theory resolution. JAR, 1:333–355, 1985.
J. D. Ullman and A. van Gelder. Efficient tests for top-down termination of logical rules. JACM, 35(2):345–373, 1988.
A. van Gelder. Efficient loop detection in prolog using the tortoise-and-hare technique. J. Logic Programming, 4:23–31, 1987.
J. Würtz. Unifying cycles. In European Conference on Artificial Intelligence (ECAI'92), pages 60–64, Aug. 1992.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1994 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Salzer, G. (1994). Primal grammars and unification modulo a binary clause. In: Bundy, A. (eds) Automated Deduction — CADE-12. CADE 1994. Lecture Notes in Computer Science, vol 814. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-58156-1_20
Download citation
DOI: https://doi.org/10.1007/3-540-58156-1_20
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-58156-7
Online ISBN: 978-3-540-48467-7
eBook Packages: Springer Book Archive