On the Expressive Power of CNF Formulas of Bounded Tree- and Clique-Width

Abstract

Starting point of our work is a previous paper by Flarup, Koiran, and Lyaudet [5]. There the expressive power of certain families of polynomials is investigated. Among other things it is shown that polynomials arising as permanents of bounded tree-width matrices have the same expressiveness as polynomials given via arithmetic formulas. A natural question is how expressive such restricted permanent polynomials are with respect to other graph-theoretic concepts for representing polynomials over a field \(\mathbb{K}.\) One such is representing polynomials by formulas in conjunctive normal form. Here, a monomial occurs according to whether the exponent vector satisfies a given CNF formula or not. We can in a canonical way assign a graph to such a CNF formula and speak about the tree-width of the related CNF polynomial.In this paper we show that the expressiveness of CNF polynomials of bounded tree-width again gives precisely arithmetic formulas. We then study how far the approach of evaluating subclasses of permanents efficiently using a reduction to CNF formulas of bounded tree-width leads. We show that there does not exist a family of CNF polynomials of bounded tree-width which can express general permanent polynomials. The statement is unconditional. An analoguous result for CNF polynomials of bounded clique-width is given, this time under the assumption that \(\# P \not\subseteq FP/poly.\) The paper contributes to the comparison between classical Boolean complexity and algebraic approaches like Valiant’s one.KeywordsPermutation MatrixConjunctive Normal FormOutgoing EdgeTree DecompositionPrimal GraphThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.