The Concept of Demodulation in Theorem Proving

ABSTRACT

In many fields of mathematics the richness of the underlying axiom set leads to the establishment of a number of very general equalities. For example, it is easy to prove that in groups (x-1)-1 = x and that in rings -x · - y = x · y. In the presence of such an equality, each new inference made during a proof search by a theorem-proving program may immediately yield a set of very closely related inferences. If, for example, b·a = c is inferred in the presence of (x-1)-1 = x, substitution immediately yields obviously related inferences such as (b-1)-1 · a = c. Retention of many members of each such set of inferences has seriously impeded the effectiveness of automatic theorem proving. Similar to the gain made by discarding instances of inferences already present is that made by discarding instances of repeated application of a given equality. The latter is achieved by use of demodulation. Its definition, evidence of its value, and a related rule of inference are given. In addition a number of concepts are defined the implementation of which reduces both the number and sensitivity to choice of parameters governing the theorem-proving procedures.

REFERENCES

• 1: DAVIS, MARTIN, Computability and Unsolvability, McGraw-Hill, New York, 1958, pp. 95-98.
• 2: MELTZER, B. Theorem-proving for computers: some results on resolution and renaming. Computer J. 8 (1966), 341-343.
• 3: ---- ABD POGGI, P. An improved complete strategy for theorem-proving by resolution. (Unpublished.)
• 4: J. A. Robinson, A Machine-Oriented Logic Based on the Resolution Principle, Journal of the ACM (JACM), v.12 n.1, p.23-41, Jan. 1965 [doi>10.1145/321250.321253]
• 5: ----. A review of automatic theorem-proving. Proc. Symp. Appl. Math., Vol. 19, Amer. Math. Soc. Providence, R. I., 1967.
• 6: ----. A review of automatic theorem-proving. Proc. Symp. Appl. Math., Vol. 19. Amer. Math. Soc. Providence, R. I., 1967.
• 7: James R. Slagle, Automatic Theorem Proving With Renamable and Semantic Resolution, Journal of the ACM (JACM), v.14 n.4, p.687-697, Oct. 1967 [doi>10.1145/321420.321428]
• 8: WANG, H. Formalization and automatic theorem-proving. Proc. IFIP Congr. 65, Vol. 1, pp. 51-58 Spartan Books, Washington, D. C.)
• 9: WOE, L., CARSON, D., AND ROBINSON, G. The unit preference strategy in theorem proving. Proc. AFIPS 1964 Fall Joint Comput. Conf., Vol. 26, Pt. II, pp. 615-621 (Spartan Books, Washington, D. C.).
• 10: Lawrence Wos , George A. Robinson , Daniel F. Carson, Efficiency and Completeness of the Set of Support Strategy in Theorem Proving, Journal of the ACM (JACM), v.12 n.4, p.536-541, Oct. 1965 [doi>10.1145/321296.321302]
• 11: ----, ----, and ----. Automatic generation of proofs in the language of mathematics. Proc. IFIP Congr. 65, Vol. 2, pp. 326-326 (Spartan Books, Washington, D. C.).

INDEX TERMS

Primary Classification:
  I. Computing Methodologies
  I.2 ARTIFICIAL INTELLIGENCE

Additional Classification:
  G. Mathematics of Computing
  G.2 DISCRETE MATHEMATICS

General Terms:
Measurement, Performance, Theory, Verification