Skip to main content
SpringerLink
Log in

Many-valued logic and mixed integer programming

  • Published:
Annals of Mathematics and Artificial Intelligence Aims and scope Submit manuscript

Abstract

We generalize prepositional semantic tableaux for classical and many-valued logics toconstraint tableaux. We show that this technique is a generalization of the standard translation from CNF formulas into integer programming. The main advantages are (i) a relatively efficient satisfiability checking procedure for classical, finitely-valued and, for the first time, for a wide range of infinitely-valued propositional logics; (ii) easy NP-containment proofs for many-valued logics. The standard translation of two-valued CNF formulas into integer programs and Tseitin's structure preserving clause form translation are obtained as a special case of our approach.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. H. Barringer, J.H. Cheng and C.B. Jones, A logic covering undefinedness in program proofs, Acta Informatica 21 (1984) 251–269.

    Google Scholar 

  2. P. Earth and A. Bockmayr, Solving 0–1 problems in CLP(PB), in:Proc. CAIA, Orlando, Florida (IEEE Press, 1993) pp. 263–269.

  3. M.G. Beavers, Automated theorem proving for Łukasiewicz logics, Manuscript of talk given at 1991 Meeting of Society for Exact Philosophy, Victoria, Canada, 1991.

  4. C. Bell, A. Nerode, R. Ng and V.S. Subrahmanian, Implementing deductive databases by mixed integer programming, in:Proc. ACM Symp. on Principles of Database Systems, 1992, pp. 283–292.

  5. C. Bell, A. Nerode, R. Ng and V.S. Subrahmanian, Mixed integer programming methods for computing nonmonotonic deductive databases, JACM (1994) to appear.

  6. D. Bjørner, M. Broy and I.V. Pottosin (eds.),Formal Methods in Programming and Their Applications, Vol. 735 of Lecture Notes in Computer Science (Springer, 1993).

  7. Th.B. de la Tour, Minimizing the number of clauses by renaming, in: M.E. Stickel (ed.),Proc. 10th Int. Conf. on Automated Deduction, Kaiserslautern, LNCS 449 (Springer, 1990) pp. 558–572.

  8. M. Broy, C. Facchi, R. Grosu, R. Hettler, H. Hussmann, D. Nazareth, F. Regensburger and K. Stølen, The requirement and design specification language Spectrum, an informal introduction, version 1.0, Technical report, Institut für Informatik, Technische Universität München (1993).

  9. W. Büttner, K. Estenfeld, R. Schmid, H.-A. Schneider and E. Tidén, Symbolic constraint handling through unification in finite algebras, Applicable Algebra in Eng. Commun. Comput. 1 (1990) 97–118.

    Google Scholar 

  10. J. Czelakowski, Algebraizability of logic and the deduction theorem, Lecture Notes Fourth Summer School on Logic, Language and Information, Colchester/England, 1992.

  11. G.B. Dantzig,Linear Programming and Extensions (Princeton University Press, 1963).

  12. M.C. Fitting,First-Order Logic and Automated Theorem Proving (Springer, New York, 1990).

    Google Scholar 

  13. R. Hähnle, Towards an efficient tableau proof procedure for multiple-valued logics, in:Proc. Workshop on Computer Science Logic, Heidelberg, Springer LNCS 533 (1990) pp. 248–260.

  14. R. Hähnle, Uniform notation of tableaux rules for multiple-valued logics, in:Proc. Int. Symp. on Multiple-valued Logic, Victoria, (IEEE Press, 1991) pp. 238–245.

  15. R. Hähnle, A new translation from deduction into integer programming, in: J. Calmet and J.A. Campbell (eds.),Proc. Int. Conf. on Artificial Intelligence and Symbolic mathematical Computing (AISMC-1), Karlsruhe, Germany, Springer LNCS 737 (1992) pp. 262–275.

  16. R. Hähnle, Automated deduction and integer programming, in:Yearbook of the Kurt-Gödel-Society (Kurt-Gödel-Society, Vienna/Austria, 1993).

    Google Scholar 

  17. R. Hähnle,Automated Deduction in Multiple-Valued Logics, Vol. 10 of International Series of Monographs on Computer Science (Oxford University Press, 1993).

  18. R. Hähnle, Short normal forms for arbitrary finitely-valued logics, in:Proc. ISMIS '93, Trondheim, Norway, Springer LNCS 689 (1993) pp. 49–58.

  19. R. Hähnle, Efficient deduction in many-valued logics, in:Proc. Int. Symp. on Multiple-Valued Logics (ISMVL '94), Boston/MA, USA (IEEE Press, Los Alamitos, 1994) pp. 240–249.

    Google Scholar 

  20. R. Hähnle and O. Ibens, Improving temporal logic tableaux using integer constraints, in:Proc. Int. Conf. on Temporal Logic, Bonn, Germany, Springer LNCS 827 (1994) pp. 535–539.

  21. R. Hähnle and W. Kernig, Verification of switch level designs with many-valued logic, in: A. Voronkov (ed.)Proc. LPAR '93, St. Petersburg, Springer LNAI 698 (1993) pp. 158–169.

  22. J.N. Hooker and C. Fedjki, Branch-and-cut solution of inference problems in prepositional logic, Ann. Math. AI 1 (1990) 123–139.

    Google Scholar 

  23. J.N. Hooker, A quantitative approach to logical inference, Decision Support Syst. 4 (1988) 45–69.

    Google Scholar 

  24. J.N. Hooker, Logical inference and polyhedral projection, in:Proc. Computer Science Logic Workshop 1991, Berne, Springer LNCS 626 (1991), pp. 184–200.

  25. J.N. Hooker, Generalized resolution for 0–1 linear inequalities, Ann. Math. AI, 6 (1992) 271–286.

    Google Scholar 

  26. J.N. Hooker, New methods for computing inferences in first order logic, Working Paper, GSIA, CMU Pittsburgh (1992).

    Google Scholar 

  27. R.G. Jeroslow, Computation-oriented reductions of predicate to prepositional logic, Decision Support Syst. 4 (1988) 183–187.

    Google Scholar 

  28. R.G. Jeroslow,Logic-Based Decision Support. Mixed Integer Model Formulation (Elsevier, Amsterdam, 1988).

    Google Scholar 

  29. R.G. Jeroslow and J. Wang, Solving propositional satisfiability problems, Ann. Math. AI l (1990) 167–187.

    Google Scholar 

  30. V. Kagan, A. Nerode and V.S. Subrahmanian, Computing definite logic programs by partial instantiation and linear programming, Draft Manuscript (1993).

  31. R.M. Karp, Reducibility among combinatorial problems, in: R.E. Miller and J.W. Thatcher (eds.),Complexity of Computer Computations (Plenum Press, 1972) 85–103.

  32. R.C.T. Lee and C.-L. Chang, Some properties of fuzzy logic, Information and Control 19 (1971) 417–431.

    Google Scholar 

  33. S.-J. Lee and D.A. Plaisted, Eliminating duplication with the hyperlinking strategy, J. Autom. Reasoning 9 (1992) 25–42.

    Google Scholar 

  34. A. Manuth, Vergleich von Normalformen zu Formeln in Łukasiewicz Logik, Studienarbeit, Universität Karlsruhe, Fakultät für Informatik (1993).

  35. R. McNaughton, A theorem about infinite-valued sentential logic, J. Symb. Logic 16 (1951) 1–13.

    Google Scholar 

  36. D. Mundici, Satisfiability in many-valued sentential logic is NP-complete, Theor. Comp. Sci. 52 (1987) 145–153.

    Google Scholar 

  37. D. Mundici, The complexity of adaptive error-correcting codes, in:Proc. Workshop Computer Science Logic 90, Heidelberg, Springer LNCS 533 (1990) 300–307.

  38. D. Mundici, Normal forms in infinite-valued logic: The case of one variable, in:Proc. Workshop Computer Science Logic 91, Berne, Springer LNCS 626 (1991) 272–277.

  39. D. Mundici, A constructive proof of McNaughton's theorem in infinite-valued logic, J. Symb. Logic 59 (1994) 596–602.

    Google Scholar 

  40. D. Mundici and M. Pasquetto, A proof of the completeness of the infinite-valued calculus of Łukasiewicz with one variable, in: E.P. Klement and U. Hoehle (eds.)Proc. Int. Conf. on Nonclassical Logics and their Applications, 1992 Linz, Austria (Kluwer, 1994).

  41. D.A. Plaisted and S.-J. Lee, Inference by clause matching, in: Z. Ras and M. Zemankova (eds.),Intelligent Systems — State of the Art and Future Directions, chap. 8 (Ellis Horwood, 1990) pp. 200–235.

  42. W.A. Pogorzelski, The deduction theorem for Łukasiewicz many-valued propositional calculi, Studia Logica 15 (1964) 7–23.

    Google Scholar 

  43. H. Rasiowa,An Algebraic Approach to Non-Classical Logics, Vol. 78 of Studies in Logic and the Foundations of Mathematics (North-Holland, Amsterdam, 1974).

    Google Scholar 

  44. K. Ries and R. Hähnle, Prädikatenlogische Beweisen mit gemischt gazzahliger Optimierung. Ein tableaubasierter Ansatz, in:Working Notes of Workshop on Artificial Intelligence and Operations Research, Berlin, published as Tech Report, Max-Planck-Institut für Informatik, Saarbrücken (1993).

    Google Scholar 

  45. H. Salkin and K. Mathur,Foundations of Integer Programming (North-Holland, 1989).

  46. J. Siekmann and G. Wrightson (eds.),Automation of Reasoning: Classical Papers in Computational Logic 1967–1970, Vol. 2 (Springer, 1983).

  47. R. Smullyan, First-Order Logic (Springer, New York, 1968).

    Google Scholar 

  48. G. Tseitin, On the complexity of proofs in propositional logics,Seminars in Mathematics, 8 (1970), reprinted in [46].

  49. R. Wójcicki,Theory of Logical Calculi (Reidel, Dordrecht, 1988).

    Google Scholar 

  50. N. Zabel, Nouvelles techniques de déduction automatique en logiques polyvalentes finies et infinies du premier ordre, Ph.D. Thesis, Institut National Polytechnique de Grenoble (1993).

Download references

Author information

Authors and Affiliations

  1. Institut für Logik, Komplexität und Deduktionssysteme, Fakultät für Informatik, Universität Karlsruhe, 76128, Karlsruhe, Germany

    Reiner Hähnle

Authors
  1. Reiner Hähnle
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Part of the research reported here was carried out while the author was supported by a grant within the DFG Schwerpunktprogramm Deduktion. Preliminary and partial versions of this paper were published as [15, 16].

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hähnle, R. Many-valued logic and mixed integer programming. Ann Math Artif Intell 12, 231–263 (1994). https://doi.org/10.1007/BF01530787

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01530787

Keywords

Search

Navigation