Canonical Inference for Implicational Systems

  • Maria Paola Bonacina
  • Nachum Dershowitz
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5195)


Completion is a general paradigm for applying inferences to generate a canonical presentation of a logical theory, or to semi-decide the validity of theorems, or to answer queries. We investigate what canonicity means for implicational systems that are axiomatizations of Moore families – or, equivalently, of propositional Horn theories. We build a correspondence between implicational systems and associative-commutative rewrite systems, give deduction mechanisms for both, and show how their respective inferences correspond. Thus, we exhibit completion procedures designed to generate canonical systems that are “optimal” for forward chaining, to compute minimal models, and to generate canonical systems that are rewrite-optimal. Rewrite-optimality is a new notion of “optimality” for implicational systems, one that takes contraction by simplification into account.


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  1. 1.
    Bachmair, L., Dershowitz, N.: Equational inference, canonical proofs, and proof orderings. Journal of the ACM 41(2), 236–276 (1994)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bertet, K., Monjardet, B.: The multiple facets of the canonical direct implicational basis. Cahiers de la Maison des Sciences Economiques b05052, Université Paris Panthéon-Sorbonne (June 2005),
  3. 3.
    Bertet, K., Nebut, M.: Efficient algorithms on the Moore family associated to an implicational system. Discrete Mathematics and Theoretical Computer Science 6, 315–338 (2004)MATHMathSciNetGoogle Scholar
  4. 4.
    Bonacina, M.P., Dershowitz, N.: Abstract canonical inference. ACM Transactions on Computational Logic 8(1), 180–208 (2007)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Bonacina, M.P., Hsiang, J.: On rewrite programs: Semantics and relationship with Prolog. Journal of Logic Programming 14(1 & 2), 155–180 (1992)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Bonacina, M.P., Hsiang, J.: Towards a foundation of completion procedures as semidecision procedures. Theoretical Computer Science 146, 199–242 (1995)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Caspard, N., Monjardet, B.: The lattice of Moore families and closure operators on a finite set: A survey. Electronic Notes in Discrete Mathematics 2 (1999)Google Scholar
  8. 8.
    Darwiche, A.: Searching while keeping a trace: the evolution from satisfiability to knowledge compilation. In: Furbach, U., Shankar, N. (eds.) IJCAR 2006. LNCS (LNAI), vol. 4130. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  9. 9.
    Dershowitz, N.: Computing with rewrite systems. Information and Control 64(2/3), 122–157 (1985)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Dershowitz, N., Huang, G.-S., Harris, M.A.: Enumeration problems related to ground Horn theories,
  11. 11.
    Dershowitz, N., Jouannaud, J.-P.: Rewrite systems. In: van Leeuwen, J. (ed.) Handbook of Theoretical Computer Science, vol. B, pp. 243–320. Elsevier, Amsterdam (1990)Google Scholar
  12. 12.
    Dershowitz, N., Marcus, L., Tarlecki, A.: Existence, uniqueness, and construction of rewrite systems. SIAM Journal of Computing 17(4), 629–639 (1988)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Dowling, W.F., Gallier, J.H.: Linear-time algorithms for testing the satisfiability of propositional Horn formulæ. Journal of Logic Programming 1(3), 267–284 (1984)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Furbach, U., Obermaier, C.: Knowledge compilation for description logics. In: Dershowitz, N., Voronkov, A. (eds.) LPAR 2007. LNCS (LNAI), vol. 4790, Springer, Heidelberg (2007)Google Scholar
  15. 15.
    Horn, A.: On sentences which are true of direct unions of algebras. Journal of Symbolic Logic 16, 14–21 (1951)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    McKinsey, J.C.C.: The decision problem for some classes of sentences without quantifiers. Journal of Symbolic Logic 8, 61–76 (1943)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Roussel, O., Mathieu, P.: Exact knowledge compilation in predicate calculus: the partial achievement case. In: McCune, W. (ed.) CADE 1997. LNCS, vol. 1249, pp. 161–175. Springer, Heidelberg (1997)Google Scholar
  18. 18.
    Sloane, N.J.A.: The On-Line Encyclopedia of Integer Sequences (1996-2006),

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Maria Paola Bonacina
    • 1
  • Nachum Dershowitz
    • 2
  1. 1.Dipartimento di InformaticaUniversità degli Studi di VeronaItaly
  2. 2.School of Computer ScienceTel Aviv UniversityRamat AvivIsrael

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