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

  • Authors:

  • Irenée Briquel;Pascal Koiran;Klaus Meer

  • Affiliations:

  • Laboratoire de l'Informatique du Parallélisme, ENS Lyon, France;Laboratoire de l'Informatique du Parallélisme, ENS Lyon, France;Lehrstuhl Theoretische Informatik, BTU Cottbus, Germany

  • Venue:
  • Discrete Applied Mathematics
  • Year:
  • 2011

Quantified Score

Hi-index 0.04

Visualization

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 #P@?FP/poly.