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Boolean Approximation Revisited

  • Conference paper
Abstraction, Reformulation, and Approximation (SARA 2007)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 4612))

Abstract

Most work to date on Boolean approximation assumes that Boolean functions are represented by formulas in conjunctive normal form. That assumption is appropriate for the classical applications of Boolean approximation but potentially limits wider use. We revisit, in a lattice-theoretic setting, so-called envelopes and cores in propositional logic, identifying them with upper and lower closure operators, respectively. This leads to recursive representation-independent characterisations of Boolean approximation for a large class of classes. We show that Boolean development can be applied in a representation-independent setting to develop approximation algorithms for a broad range of Boolean classes, including Horn and Krom functions.

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References

  1. Birkhoff, G.: Lattice Theory, 3rd edn. American Mathematical Society (1973)

    Google Scholar 

  2. Brace, K., Rudell, R., Bryant, R.: Efficient implementation of a BDD package. In: Proc. Twenty-seventh ACM/IEEE Design Automation Conf., pp. 40–45 (1990)

    Google Scholar 

  3. Cadoli, M., Scarcello, F.: Semantical and computational aspects of Horn approximations. Artificial Intelligence 119, 1–17 (2000)

    Article  MATH  Google Scholar 

  4. Codish, M., Søndergaard, H., Stuckey, P.J.: Sharing and groundness dependencies in logic programs. ACM Transactions on Programming Languages and Systems 21(5), 948–976 (1999)

    Article  Google Scholar 

  5. Cousot, P., Cousot, R.: Static determination of dynamic properties of recursive procedures. In: Neuhold, E.J. (ed.) Formal Description of Programming Concepts, pp. 237–277. North-Holland, Amsterdam (1978)

    Google Scholar 

  6. Cousot, P., Cousot, R.: Systematic design of program analysis frameworks. In: Proc. Sixth ACM Symp. Principles of Programming Languages, pp. 269–282. ACM Press, New York (1979)

    Google Scholar 

  7. Dechter, R., Pearl, J.: Structure identification in relational data. Artificial Intelligence 58, 237–270 (1992)

    Article  MATH  Google Scholar 

  8. del Val, A.: First order LUB approximations: Characterization and algorithms. Artificial Intelligence 162, 7–48 (2005)

    Article  Google Scholar 

  9. Giacobazzi, R.: Semantic Aspects of Logic Program Analysis. PhD thesis, University of Pisa, Italy (1993)

    Google Scholar 

  10. Halmos, P.R.: Lectures on Boolean Algebras. Springer, Heidelberg (1963)

    MATH  Google Scholar 

  11. Horiyama, T., Ibaraki, T.: Ordered binary decision diagrams as knowledge-bases. Artificial Intelligence 136, 189–213 (2002)

    Article  MATH  Google Scholar 

  12. Horiyama, T., Ibaraki, T.: Translation among CNFs, characteristic models and ordered binary decision diagrams. Inf. Processing Letters 85, 191–198 (2003)

    Article  Google Scholar 

  13. Kautz, H., Kearns, M., Selman, B.: Horn approximations of empirical data. Artificial Intelligence 74, 129–145 (1995)

    Article  MATH  Google Scholar 

  14. Kavvadias, D., Papadimitriou, C., Sideri, M.: On Horn envelopes and hypergraph transversals. In: Ng, K.W., Balasubramanian, N.V., Raghavan, P., Chin, F.Y.L. (eds.) ISAAC 1993. LNCS, vol. 762, pp. 399–405. Springer, Heidelberg (1993)

    Google Scholar 

  15. Khardon, R.: Translating between Horn representations and their characteristic models. Journal of Artificial Intelligence Research 3, 349–372 (1995)

    MATH  Google Scholar 

  16. Ore, O.: Combinations of closure relations. Ann. Math. 44(3), 514–533 (1943)

    Article  Google Scholar 

  17. Pelletier, F.J., Martin, N.M.: Post’s functional completeness theorem. Notre Dame Journal of Formal Logic 31(2) (1990)

    Google Scholar 

  18. Post, E.L.: The Two-Valued Iterative Systems of Mathematical Logic. Princeton University Press, 1941. Reprinted in Davis, M., Solvability, Provability, Definability: The Collected Works of Emil L. Post, pp. 249–374, Birkhaüser (1994)

    Google Scholar 

  19. Rudeanu, S.: Boolean Functions and Equations. North-Holland, Amsterdam (1974)

    MATH  Google Scholar 

  20. Schachte, P., Søndergaard, H.: Closure operators for ROBDDs. In: Emerson, E.A., Namjoshi, K. (eds.) VMCAI 2006. LNCS, vol. 3855, pp. 1–16. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  21. Schaefer, T.J.: The complexity of satisfiability problems. In: Proc. Tenth Ann. ACM Symp. Theory of Computing, pp. 216–226 (1978)

    Google Scholar 

  22. Selman, B., Kautz, H.: Knowledge compilation and theory approximation. Journal of the ACM 43(2), 193–224 (1996)

    Article  MATH  Google Scholar 

  23. Ward, M.: The closure operators of a lattice. Ann. Math. 43(2), 191–196 (1942)

    Article  Google Scholar 

  24. Zanuttini, B.: Approximating propositional knowledge with affine formulas. In: ECAI 2002. Proceedings of the Fifteenth European Conference on Artificial Intelligence, pp. 287–291. IOS Press, Amsterdam (2002)

    Google Scholar 

  25. Zanuttini, B.: Approximation of relations by propositional formulas: Complexity and semantics. In: Koenig, S., Holte, R.C. (eds.) SARA 2002. LNCS (LNAI), vol. 2371, pp. 242–255. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

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Authors and Affiliations

  1. NICTA Victoria Laboratory, Department of Computer Science and Software Engineering, The University of Melbourne, Vic. 3010, Australia

    Peter Schachte & Harald Søndergaard

Authors
  1. Peter Schachte
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  2. Harald Søndergaard
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Ian Miguel Wheeler Ruml

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Schachte, P., Søndergaard, H. (2007). Boolean Approximation Revisited. In: Miguel, I., Ruml, W. (eds) Abstraction, Reformulation, and Approximation. SARA 2007. Lecture Notes in Computer Science(), vol 4612. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73580-9_26

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  • DOI: https://doi.org/10.1007/978-3-540-73580-9_26

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-73579-3

  • Online ISBN: 978-3-540-73580-9

  • eBook Packages: Computer ScienceComputer Science (R0)

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