Journal of Automated Reasoning

, Volume 29, Issue 1, pp 91–106 | Cite as

Mathematical Programming Embeddings of Logic

  • Vivek S. Borkar
  • Vijay Chandru
  • Sanjoy K. Mitter


Can theorem proving in mathematical logic be addressed by classical mathematical techniques like the calculus of variations? The answer is surprisingly in the affirmative, and this approach has yielded rich dividends from the dual perspective of better understanding of the mathematical structure of deduction and in improving the efficiency of algorithms for deductive reasoning. Most of these results have been for the case of propositional and probabilistic logics. In the case of predicate logic, there have been successes in adapting mathematical programming schemes to realize new algorithms for theorem proving using partial instantiation techniques. A structural understanding of mathematical programming embeddings of predicate logic would require tools from topology because of the need to deal with infinite-dimensional embeddings. This paper describes the first steps in this direction. General compactness theorems are proved for the embeddings, and some specialized results are obtained in the case of Horn logic.

predicate logic mathematical programming CLP(R) 


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  1. 1.
    Andersen, K. A. and Hooker, J. N.: A linear programming framework for logics of uncertainty, Manuscript, Mathematical Institute, Århus University, 8000 Århus C,Denmark, 1992.Google Scholar
  2. 2.
    Araque, G. J. R. and Chandru, V.: Some facets of satisfiability, Technical Report CC-91-13, Institute for Interdisciplinary Engineering Studies, Purdue University, West Lafayette, IN 47907, USA, 1991.Google Scholar
  3. 3.
    Blair, C., Jeroslow, R. G. and Lowe, J. K.: Some results and experiments in programming techniques for propositional logic, Comput. Oper. Res. 13 (1988), 633–645.Google Scholar
  4. 4.
    Borkar, V. S., Chandru, V. and Mitter, S. K.: A linear programming model of first order logic, Technical Report IISc-CSA-95-5, Indian Institute of Science, 1995.Google Scholar
  5. 5.
    Chandru, V. and Hooker, J. N.: Extended horn sets in propositional logic, J. ACM 38(1) (1991), 205–221.Google Scholar
  6. 6.
    Chandru, V. and Hooker, J. N.: Optimization Methods for Logical Inference, Wiley-Interscience, 1999.Google Scholar
  7. 7.
    Conforti, M. and Cornuéjols, G.: A class of logical inference problems soluble by linear programming, J. ACM 33 (1994), 670–675.Google Scholar
  8. 8.
    Gallo, G. and Rago, G.: A hypergraph approach to logical inference for datalog formulae, working paper, Dip. di Informatica, University of Pisa, Italy, September 1990.Google Scholar
  9. 9.
    Hooker, J. N.: A mathematical programming model for probabilistic logic, working paper 05-88-89, Graduate School of Industrial Administration, Carnegie Mellon University, Pittsburgh, PA 15213, July 1988.Google Scholar
  10. 10.
    Hooker, J. N.: Input proofs and rank one cutting planes, ORSA J. Comput. 1 (1989), 137–145.Google Scholar
  11. 11.
    Hooker, J. N.: New methods for computing inferences in first order logic, Ann. Oper. Res. (1993), 479–492.Google Scholar
  12. 12.
    Jaffar, J. and Lassez, J.-L.: Constraint logic programming, Technical Report 86/73, Department of Computer Science, Monash University, 1986.Google Scholar
  13. 13.
    Jaffar, J. and Lassez, J.-L.: Constraint logic programming, in Proc. 14th Symposium on Principles of Programming Languages, Munich, Jan. 1987, pp. 111–119.Google Scholar
  14. 14.
    Jaffar, J. and Maher, M. J.: Constraint logic programming: A survey, J. Logic Programming 19/20 (1994), 503–581.Google Scholar
  15. 15.
    Jeroslow, R. G.: Computation-oriented reductions of predicate to propositional logic, Decision Support Systems 4 (1988), 183–197.Google Scholar
  16. 16.
    Jeroslow, R. G.: Logic-Based Decision Support: Mixed Integer Model Formulation, Ann. Discrete Math.40, North-Holland, Amsterdam, 1989.Google Scholar
  17. 17.
    Kagan, V., Nerode, A. and Subrahmanian, V. S.: Computing definite logic programs by partial instantiation and linear programming, Technical Report 93-15, Mathematical Sciences Institute, Cornell University, 1993.Google Scholar
  18. 18.
    Kavvadias, D. and Papadimitriou, C. H.: A linear programming approach to reasoning about probabilities, Ann. Math. Artificial Intelligence 1 (1990), 189–206.Google Scholar
  19. 19.
    Munkres, J. R.: Topology: A First Course, Prentice-Hall, 1975.Google Scholar
  20. 20.
    Nilsson, N. J.: Probabilistic logic, Artificial Intelligence 28 (1986), 71–87.Google Scholar
  21. 21.
    Rockafeller, R. T.: Convex Analysis, Princeton University Press, 1970.Google Scholar
  22. 22.
    Schrijver, A.: Theory of Linear and Integer Programming, Wiley, New York, 1986.Google Scholar
  23. 23.
    Schöning, U.: Logic for Computer Scientists, Birkhäuser, 1989.Google Scholar
  24. 24.
    Wang, J.-C. and VandeVate, J.: Question-asking strategies for Horn clause systems, Ann. Math. Artificial Intelligence 1 (1990).Google Scholar

Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  • Vivek S. Borkar
    • 1
  • Vijay Chandru
    • 2
  • Sanjoy K. Mitter
    • 3
  1. 1.Tata Institute of Fundamental ResearchMumbaiIndia
  2. 2.Indian Institute of ScienceBangaloreIndia
  3. 3.Massachusetts Institute of TechnologyCambridgeUSA

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