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

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 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 6⊆ FP/poly. The paper contributes to the comparison between classical Boolean complexity and algebraic approaches like Valiant’s one.