Representations of the language recognition problem for a theorem prover

  • Jack Minker
  • Gordon J. VanderBrug
Article

Abstract

Two representations of the language recognition problem for a theorem prover in first-order logic are presented and contrasted. One of the representations is based on the familiar method of generating sentential forms of the language, and the other is based on the Cocke parsing algorithm. An augmented theorem prover is described which permits recognition of recursive languages. The state-transformation method developed by Cordell Green to construct problem solutions in resolution-based systems can be used to obtain the parse tree. In particular, the end-order traversal of the parse tree is derived in one of the representations. The paper defines an inference system, termed the cycle inference system, which makes it possible for the theorem prover to model the method on which the representation is based. The general applicability of the cycle inference system to state-space problems is discussed. Given an unsatisfiable setS, where each clause has at most one positive literal, it is shown that there exists an input proof. The clauses for the two representations satisfy these conditions, as do many state-space problems.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    S. Amarel, “Problem solving procedures for efficient syntactic analysis,” ACM 20th National Conference, 1965.Google Scholar
  2. 2.
    S. Amarel, “Representations of problems of reasoning about actions,” inMachine Intelligence 3, B. Meltzer and D. Michie, eds. (American Elsevier, New York, 1968), pp. 131–171.Google Scholar
  3. 3.
    S. Amarel, “On the representation of problems and goal-directed prodecures for computers,” inTheoretical Approaches to Non-Numerical Problem Solving, R. Banerji and M. D. Mesarovic, eds. (Springer-Verlag, New York, 1971), pp. 179–244.Google Scholar
  4. 4.
    C. L. Chang, “The unit proof and input proof in theorem proving,”J. ACM 17(4):698–707 (1970).Google Scholar
  5. 5.
    C. L. Chang and R. C. T. Lee,Symbolic Logic and Mechanical Theorem Proving (Academic Press, New York, 1973).Google Scholar
  6. 6.
    N. Chomsky, “On certain formal properties of grammars,”Inf. and Control 2(2): 137–167 (1959).Google Scholar
  7. 7.
    J. Cocke and J. T. Schwartz,Programming Languages and Their Compilers (Courant Institute of Mathematical Sciences, New York, 1970).Google Scholar
  8. 8.
    M. Davis, “Eliminating the irrelevant from mechanical proofs,”Proc. of Symp, in Appl. Math., Vol. 15 (1963), pp. 15–30.Google Scholar
  9. 9.
    M. Davis and H. Putnam, “A computing procedure for quantification theory,”J. ACM 7(3):201–215 (1960).Google Scholar
  10. 10.
    C. C. Green, “Application of theorem proving to problem solving,” inProc. Intern. Joint Conf. on Artificial Intelligence, D. E. Walker and L. M. Norton, eds. (Washington, D.C., 1969), pp. 219–239.Google Scholar
  11. 11.
    P. N. Hart, N. Nilsson, and B. Raphael, “A formal basis for the heuristic determination of minimum cost paths,”IEEE Trans. Syst. Sci. Cybernetics SSC-4(2):100–107 (1968).Google Scholar
  12. 12.
    J. E. Hopcroft and J. D. Ullman,Formal Languages and Their Relation to Automata (Addison-Wesley, Reading, Massachusetts, 1969).Google Scholar
  13. 13.
    D. E. Knuth,Fundamental Algorithms, Vol. 1 (Addison-Wesley, Reading, Massachusetts, 1968).Google Scholar
  14. 14.
    R. Kowalski, “Search strategies for theorem proving,” inMachine Intelligence 5, B. Meltzer and D. Michie, eds. (American Elsevier, New York, 1970), pp. 181–200.Google Scholar
  15. 15.
    R. Kowalski, “An improved theorem proving system for first-order logic,” Memo No. 65, Dept. of Computational Logic, University of Edinburgh, Edinburgh, Scotland, 1973.Google Scholar
  16. 16.
    R. Kowalski and P. Hayes, “Semantic trees in automatic theorem proving,” inMachine Intelligence 4, B. Meltzer and D. Michie, eds. (American Elsevier, New York, 1969), pp. 87–101.Google Scholar
  17. 17.
    D. Kuehner, “Some special purpose resolution systems,” inMachine Intelligence 7, B. Meltzer and D. Michie, eds. (American Elsevier, New York, 1972), pp. 117–128.Google Scholar
  18. 18.
    D. W. Loveland, “A linear format for resolution,” inProc. IRIA 1968 Symp. Auto. Demonstration, Lecture Notes on Math., No. 125, (Springer-Verlag, New York, 1970).Google Scholar
  19. 19.
    D. Luckham, “Refinement theorems in resolution theory,” inProc. IRIA 1968 Symp. Auto. Demonstration, Lecture Notes on Math., No. 125, (Springer-Verlag, New York, 1970).Google Scholar
  20. 20.
    D. Luckham and N. Nilsson, “Extracting information from resolution proof trees,”Artificial Intelligence 2(1):27–54 (1971).Google Scholar
  21. 21.
    D. Michie, “Experiments with an adaptive graph traverser,” inMachine Intelligence 5, B. Meltzer and D. Michie, eds. (American Elsevier, New York, 1970), pp. 301–320.Google Scholar
  22. 22.
    N. Nilsson,Problem-Solving Methods in Artificial Intelligence, (McGraw-Hill, New York, 1971).Google Scholar
  23. 23.
    J. A. Robinson, “A machine-oriented logic based on the resolution principle,J. ACM 12(1):23–41 (1965).Google Scholar
  24. 24.
    J. A. Robinson, “Automatic deduction with hyperresolution,”Int. J. Computer Math. 2(3):227–234 (1965).Google Scholar
  25. 25.
    J. I. Robinson and S. L. Marks, “A system for automatic analysis of english text,” RM-4654-PR, The RAND Corp., September 1965.Google Scholar
  26. 26.
    E. Sandewall, “Heuristic search: concepts and methods,” inArtificial Intelligence and Heuristic Programming, N.V. Findler and B. Meltzer, eds. (Springer-Verlag, New York, 1971), pp. 81–100.Google Scholar
  27. 27.
    J. R. Slagle and D. A. Koniver, “Finding resolution proofs and using duplicate goals in AND/OR trees,”Inf. Sciences 3(4):315–342 (1971).Google Scholar
  28. 28.
    G. J. VanderBrug and J. Minker, “State-space, problem-reduction, and theorem proving—some relationships,” TR-245, Computer Science Center, University of Maryland, College Park, Maryland, 1973; CACM, to appear.Google Scholar
  29. 29.
    D. H. Younger, “Recognition and parsing of context-free languages in time n3,”Inf. and Control 10:189–208 (1967).Google Scholar

Copyright information

© Plenum Publishing Corporation 1974

Authors and Affiliations

  • Jack Minker
    • 1
  • Gordon J. VanderBrug
    • 1
  1. 1.Department of Computer ScienceUniversity of MarylandCollege Park

Personalised recommendations