Abstract
The general problem discussed here concerns the approximation of the number of solutions of a boolean formula in conjunctive normal form F. Results previously obtained (Rouat (1999), Lerman and Rouat (1999)) are reconsidered and completed. Our method is based on the general principle “divide to resolve”. The division is achieved by cutting a seriation built on an incidence data table associated with F. In this, the independence probability concept is finely exploited. Theoretical justification and intensive experimentation validate the considerable reduction of the computational complexity obtained by our method.
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References
ANDRÉ, P. and DUBOIS, O. (1992): Utilisation de l’espérance du nombre de solutions afin d’optimiser la résolution du problème SAT. C.R. de l’Académie des Sciences, Paris, 315, 217–220.
BAILLEUX, O. and CHABRIER, J.J. (1996): Approximate resolution of hard numbering problems. In: AAAI Thirteenth National Conference on Artificial Intelligence, 169–174.
COOK, S.A. (1971): The complexity of theorem-proving procedures. In: 3rd Annual ACM Symposium on the Theory of Computing, 151–158.
DAVIS, M. and PUTNAM, H. (1960): A computing procedure for quantification theory. Journal of the ACM, vol. 7, 3, 201–215.
DUBOIS, O. (1991): Counting the number of solutions for instances of satisfiability. Theoretical Computer Science, 81, 49–64.
KARP, R.M. and LUBY, M. (1983): Monte-carlo algorithms for enumeration and reliability problems. In: 24th IEEE Symposium of Foundations of Computer Science, 56–64.
LEREDDE, H. (1979): La méthode des pôles d’attraction, la méthode des pôles d’agrégation; deux nouvelles familles d’algorithmes en classification automatique et sériation. PhD thesis, Université de Paris VI.
LERMAN, I.C. (1972): Analyse du phénomène de la “sériation” à partir d’un tableau d’incidence. Mathématiques et Sciences Humaines, 38, 39–57.
LERMAN, I.C. (1995): Statistical reduction of the satisfiability problem by means of a classification method. Data Science and its Application, Academic Press, 219–234.
LERMAN, I.C. and ROUAT, V. (1999): Segmentation de la sériation pour la résolution de #SAT. Mathématiques, Informatique et Sciences Humaines, 147, 113–134.
LOZINSKII, E.L. (1992): Counting propositional models. Information Processing Letters, 41, 327–332.
MARCOTORCHINO, F. (1987): Block seriation problems: a unified approach. Applied Stochastic Models and Data Analysis, vol. 3, 2, 73–91.
PAPADIMITRIOU, C.H. (1994): Computational complexity. Addison Wesley.
ROUAT, V. (1999): Validité de l’approche classification dans la réduction statistique de la complexité de #SAT. PhD thesis, Université de Rennes 1, 1999.
ROUAT, V. and LERMAN, I.C. (1997): Utilisation de la sériation pour une résolution approchée du problème #SAT. In: JNPC’97, résolution pratique de problèmes NP-complets, 55–60.
ROUAT, V. and LERMAN, I.C. (1998): Problématique de la coupure dans la résolution de #SAT par sériation. In: JNPC’98, résolution pratique de problèmes NP-complets, 109--114.
SIMON, J.C. and DUBOIS, O. (1989): Number of solutions of satisfiability instances — applications to knowledge bases. International Journal of Pattern Recognition and Artificial Intelligence, vol. 3, 1, 53–65.
TODA, S. (1989): On the computational power of PP and ⊕P. In: 30th Annual Symposium on Foundations of Computer Science, 514–519.
VALIANT, L.G. (1979): The complexity of enumeration and reliability problems. SIAM Journal on Computing, vol. 8, 3, 410–421.
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Lerman, I.C., Rouat, V. (2000). New Results in Cutting Seriation for Approximate #SAT. In: Gaul, W., Opitz, O., Schader, M. (eds) Data Analysis. Studies in Classification, Data Analysis, and Knowledge Organization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-58250-9_20
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DOI: https://doi.org/10.1007/978-3-642-58250-9_20
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