What you always wanted to know about clause graph resolution

• Norbert Eisinger
Graph Based Deduction
Part of the Lecture Notes in Computer Science book series (LNCS, volume 230)

Abstract

Clause graph (or connection graph) resolution was invented by Robert Kowalski in 1975. Its behaviour differs significantly from that of traditional resolution in clause sets. Standard notions like completeness do not adequately cover the new phenomena introduced by clause graph resolution and standard proof techniques do not work for clause graphs, which are the major reasons why important questions have been open for years. This paper defines a series of relevant properties in precise terms and answers several of the open questions. The clause graph inference system is refutation complete and refutation confluent. Compared to clause set resolution, clause graph resolution does not increase the complexity of simplest refutations. Many well-known restriction strategies are refutation complete, but most are not refutation confluent for clause graph resolution, which renders them useless. Exhaustive ordering strategies do not exist and contrary to a wide-spread conjecture the plausible approximations of exhaustiveness do not ensure the detection of a refutation.

Key words

Clause Graphs Connection Graphs Resolution Strategies Completeness Confluence

References

1. A68.
Andrews, P.B.: Resolution with Merging, JACM, Vol. 15, No. 3 (1968), 367–381
2. A76.
Andrews, P.B.: Refutations by Matings, IEEE Trans. Comp., Vol. C-25, No. 8 (1976), 801–807Google Scholar
3. A81.
Andrews, P.B.: Theorem Proving via General Matings, JACM, Vol. 28, No. 2 (1981), 193–214
4. AO83.
Antoniou, G., Ohlbach, H. J.: Terminator, Proc. 8th IJCAI, Karlsruhe (1983), 916–919Google Scholar
5. B75.
Bruynooghe, M.: The Inheritance of Links in a Connection Graph, Report CW2. Applied Mathematics and Programming Division, Katholieke Universiteit Leuven (1975)Google Scholar
6. B76.
Brown, F.: Notes on Chains and Connection Graphs, Personal Notes. Dept. of Computation and Logic, University of Edinburgh (1976)Google Scholar
7. B80.
Bibel, W.: A Strong Completeness Result for the Connection Graph Proof Procedure, Bericht ATP-3-IV-80, Institut für Informatik, Technische Universität, München (1980)Google Scholar
8. B81a.
Bibel, W.: On the Completeness of Connection Graph Resolution, Proc. GWAI-81, Springer Informatik Fachberichte, Vol.47, (edited by Jörg H. Siekmann), Springer, Heidelberg (1981), 246–247Google Scholar
9. B81b.
Bibel, W.: On Matrices with Connections, JACM, Vol. 28, No. 4 (1981), 633–645
10. B82a.
Bibel, W.: A Comparative Study of Several Proof Procedures, Artificial Intelligence, Vol. 18, No. 3 (1982), 269–293
11. B82b.
12. CL73.
Chang, C.-L., Lee, R. C.-T.: Symbolic Logic and Mechanical Theorem Proving, Computer Science and Applied Mathematics Series (Editor Werner Rheinboldt), Academic Press, New York (1973)Google Scholar
13. CS79.
Chang, C.-L., Slagle, J. R.: Using Rewriting Rules for Connection Graphs to Prove Theorems. Artificial Intelligence, Vol.12, No. 2 (1979), 159–178
14. E86.
Eisinger, N.: Completeness, Confluence, and Related Properties of Clause Graph Resolution, Dissertation, Fachbereich Informatik, Universität Kaiserslautern (to appear 1986)Google Scholar
15. H80.
Huet, G.: Confluent Reductions: Abstract Properties and Applications to Term Rewriting, JACM, Vol. 27, No. 4 (1980), 797–821
16. K70.
Kowalski, R.: Search Strategies for Theorem-Proving, Machine Intelligence (B. Meltzer and D. Michie, eds.), Vol. 5, Edinburgh University Press, Edinburgh (1970), 181–201Google Scholar
17. K75.
Kowalski, R.: A Proof Procedure Using Connection Graphs. JACM, Vol. 22, No. 4 (1975), 572–595
18. K79b.
Kowalski, R.: Logic for Problem Solving, Artificial Intelligence Series (Nils J. Nilsson, Editor), Vol. 7, North-Holland, New York (1979)Google Scholar
19. KK71.
Kowalski, R., Kuehner, D.: Linear Resolution with Selection Function, Artificial Intelligence, Vol. 2, No. 3–4 (1971), 227–260
20. L78.
Loveland, D.: Automated Theorem Proving: A Logical Basis, Fundamental Studies in Computer Science, Vol. 6, North-Holland, New York (1978)Google Scholar
21. N80.
Nilsson, N.: Principles of Artificial Intelligence, Tioga, Palo Alto, CA (1980)Google Scholar
22. No80.
Noll, H.: A Note on Resolution: How to Get Rid of Factoring without Losing Completeness, Proc. 5th CADE, Springer Lecture Notes in Computer Science, Vol. 87 (edited by W. Bibel and R. Kowalski), Springer, Heidelberg (1980), 250–263Google Scholar
23. S76.
Shostak, R. E.: Refutation Graphs, Artificial Intelligence, Vol. 7, No. 1 (1976), 51–64
24. S79.
Shostak, R. E.: A Graph-Theoretic View of Resolution Theorem-Proving, Report SRI International, Menlo Park, CA (1979)Google Scholar
25. S82a.
Smolka, G.: Einige Ergebnisse zur Vollständigkeit der Beweisprozedur von Kowalski, Diplomarbeit, Fakultät Informatik, Universität Karlsruhe (1982)Google Scholar
26. S82b.
Smolka, G.: Completeness of the Connection Graph Proof Procedure for Unit Refutable Clause Sets, Proc. GWAI-82, Springer Informatik Fachberichte, Vol. 58 (1982), 191–204Google Scholar
27. Si76.
Sickel, S.: A Search Technique for Clause Interconnectivity Graphs, IEEE Trans. Comp., Vol. C-25, No. 8 (1976), 823–835Google Scholar
28. SS76.
Siekmann, J., Stephan, W.: Completeness and Soundness of the Connection Graph Proof Procedure, Bericht 7/76, Fakultät Informatik, Universität Karlsruhe (1976)Google Scholar
29. SS80.
Siekmann, J., Stephan, W.: Completeness and Consistency of the Connection Graph Proof Procedure, Interner Bericht Institut I, Fakultät Informatik, Universität Karlsruhe (1980)Google Scholar
30. W84.
Wrightson, G.: An Approach to the Completeness of the Connection Graph Proof Procedure, Personal Notes, Dept. of Comp. Sc., Victoria University of Wellington, NZ (1984)Google Scholar
31. WM76.
Wilson, G. A., Minker, J.: Resolution, Refinements and Search Strategies: A Comparative Study, IEEE Trans. Comp., Vol. C-25, No. 8 (1976), 782–801Google Scholar
32. WOLB84.
Wos, L., Overbeek, R., Lusk, E., Boyle, J.: Automated Reasoning — Introduction and Applications, Prentice-Hall, Englewood Cliffs, NJ (1984)Google Scholar
33. WCR64.
Wos, L., Carson, D. F., Robinson, G. A.: The Unit Preference Strategy in Theorem Proving, Proc. AFIPS-26, Spartan Books, Washington, D.C. (1964), 615–621Google Scholar
34. WRC65.
Wos, L., Robinson, G. A., Carson, D. F.: Efficiency and Completeness of the Set of Support Strategy in Theorem Proving, JACM, Vol. 12, No. 4 (1965), 536–541
35. YRH70.
Yates, R. A., Raphael, B., Hart, T. P.: Resolution Graphs, Artificial Intelligence, Vol. 1, No. 3–4 (1970), 257–289