A decision procedure for unquantified formulas of graph theory

  • Louise E. Moser
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 310)


A procedure for deciding the validity of ground formulas in a theory of directed graphs is described. The procedure, which decides equality and containment relations for vertex, edge and graph terms, is based on the method of congruence closure and involves the formation of equivalence classes and their representatives. An interesting aspect of the procedure is its use of maximal, rather than minimal, normal forms as equivalence class representatives. The correctness, efficiency, and limitations of the procedure are discussed.

Key words and phrases

Directed graph decision procedure congruence closure normal form equivalence class representative 


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  1. [1]
    A. Ferro, E. G. Omodeo, and J. T. Schwartz, “Decision procedures for elementary sublanguages of set theory. I. Multilevel syllogistic and some extensions,” Comm. Pure App. Math. 33 (1980), pp. 599–608.Google Scholar
  2. [2]
    A. Ferro, E. G. Omodeo, and J. T. Schwartz, “Decision procedures for some fragments of set theory,” in: Proc. 5th Conf. on Automated Deduction, LNCS 87, (Springer-Verlag, Berlin-Heidelberg-New York, 1980), pp. 88–96.Google Scholar
  3. [3]
    J. Hsiang, “Topics in automated theorem proving and program generation,” Artificial Intelligence 25 (1985), pp. 255–300.CrossRefGoogle Scholar
  4. [4]
    J. C. C. McKinsey, “The decision problem for some classes of sentences without quantifiers,” J. Symb. Logic 8, 3 (1943), pp. 61–76.Google Scholar
  5. [5]
    L. E. Moser, “Decidability of formulas in graph theory,” conditionally accepted by Fundamenta Informaticae.Google Scholar
  6. [6]
    G. Nelson and D. C. Oppen, “Simplification by cooperating decision procedures,” ACM Trans. Prog. Lang. and Syst. 1, 2 (1979), pp. 245–257.CrossRefGoogle Scholar
  7. [7]
    G. Nelson and D. C. Oppen, “Fast decision procedures based on congruence closure,” J. ACM 27, 2 (1980), pp. 356–364.CrossRefGoogle Scholar
  8. [8]
    M. O. Rabin and M. J. Fisher, “Super-exponential complexity of theorem proving procedures,” in: Proc. AMS SIAM Symp. Appl. Math., (AMS, Providence, RI, 1973), pp. 27–42.Google Scholar
  9. [9]
    J. C. Raoult, “On graph rewritings,” Theor. Comp. Sci. 32 (1984), pp. 1–24.CrossRefGoogle Scholar
  10. [10]
    J. A. Robinson, “A machine oriented logic based on the resolution principle,” J. ACM 12 (1965), pp. 23–41.CrossRefGoogle Scholar
  11. [11]
    R. E. Shostak, “An algorithm for reasoning about equality,” Comm. ACM, 21, 7 (1978), pp. 583–585.CrossRefGoogle Scholar
  12. [12]
    R. E. Shostak, “Deciding combinations of theories,” J. ACM 31, 1 (1984), pp. 1–12.CrossRefGoogle Scholar
  13. [13]
    R. E. Shostak, R. Schwartz, and P. M. Melliar-Smith, “STP: A mechanized logic for specification and verification,” in: Proc. 6th Conf. on Automated Deduction, LNCS 138, (Springer-Verlag, Berlin-Heidelberg-New York, 1982), pp. 32–49.Google Scholar
  14. [14]
    R. E. Tarjan, “Efficiency of a good but not linear set union algorithm,” J. ACM 22, 2 (1975), pp. 215–225.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1988

Authors and Affiliations

  • Louise E. Moser
    • 1
  1. 1.Department of Mathematics and Computer ScienceCalifornia State University, HaywardHayward

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