On the expressive power of CNF formulas of bounded tree- and clique-width

https://doi.org/10.1016/j.dam.2010.09.007Get rights and content
Under an Elsevier user license
open archive

Abstract

We study representations of polynomials over a field K from the point of view of their expressive power. Three important examples for the paper are polynomials arising as permanents of bounded tree-width matrices, polynomials given via arithmetic formulas, and families of so called CNF polynomials. The latter arise in a canonical way from families of Boolean formulas in conjunctive normal form. To each such CNF formula there is a canonically attached incidence graph. Of particular interest to us are CNF polynomials arising from formulas with an incidence graph of bounded tree- or clique-width.

We show that the class of polynomials arising from families of polynomial size CNF formulas of bounded tree-width is the same as those represented by polynomial size arithmetic formulas, or permanents of bounded tree-width matrices of polynomial size. Then, applying arguments from communication complexity we show that general permanent polynomials cannot be expressed by CNF polynomials of bounded tree-width. We give a similar result in the case where the clique-width of the incidence graph is bounded, but for this we need to rely on the widely believed complexity theoretic assumption #PFP/poly.

Keywords

Expressive power of polynomials
Permanent function
Conjunctive normal form formulas
Tree- and clique-width
Valiant’s complexity theory for polynomial families

Cited by (0)