Conteo de modelos en la clase sintáctica 2μ-3MON

Model Counting in the 2μ -3MON Syntactic Class

Carlos Guillén, Rafael Lemuz, e Irene Ayaquica

Facultad de Ciencias de la Computación, Benemérita Universidad Autónoma de Puebla, Puebla, Mexico cguillen@cs.buap.mx, rlemuz@cs.buap.mx, ayaquica@cs.buap.mx.

Article received on 08/06/2012
Accepted on 17/06/2013.

Resumen

El problema de conteo de modelos en formulas Booleanas es un problema #P-completo, es decir, no se conocen algoritmos deterministas en el modelo clásico de computabilidad (máquinas de Turing) que realice este conteo con complejidad en tiempo polinomial. La dificultad persiste aún imponiendo condiciones mas restrictivas sobre las clases sintácticas de fórmulas Booleanas. En este artículo presentamos una familia tratable dentro de la clase sintáctica 2μ-3MON. La identificación de esta familia se hace a travos del hipergrafo asociado a estructuras simples como cadenas y ciclos. Se identifican también operadores matriciales que actúan sobre estas estructuras; estos operadores conducen a algoritmos eficientes que efectúan el conteo de modelos sobre la familia identificada en tiempo lineal con respecto al número de clausulas de la fórmula instanciada, a diferencia de los métodos basados en invariantes hipergráficos (como el ancho de árbol) que realizan este conteo en tiempo cúbico.

Palabras clave: #SAT, clase sintáctica, hipergrafo.

Abstract

The counting model problem in Boolean formulas is #P-complete, i.e., there is no known deterministic algorithm in the classical computability model (Turing machine) that makes this count in polynomial time. The difficulty persists even imposing more restrictive conditions on the syntactic classes of Boolean formulas. In this paper we present a treatable family within the syntactical class 2μ-3MON. The identification of this family is done by using the hypergraph associated with simple structures such as chains and cycles. Then, matrix operators acting over these structures are identified; these operators lead to efficient algorithms that perform the model counting on the identified family in linear time for the number of clauses in the instantiated formula; unlike hypergraphic invariant based methods (such as tree width), which perform the count in cubic time.

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Agradecimientos

Agradecemos a los revisores sus valiosas sugerencias y comentarios. Consideramos que fueron de gran utilidad para el enriquecimiento de nuestro trabajo.

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