The Computational Complexity of ( XOR ; AND ) - Counting

We characterize the computational complexity of counting the exact number of satisfying assignments in (XOR; AND)-formulas in their RSE-representation (i.e., equivalently, polyno-mials in GF 2]]x 1 ; : : : ; x n ]). This problem refrained for some time eeords to nd a polynomial time solution and the eeorts to prove the problem to be #P-complete. Both main results can be generalized to the arbitrary nite elds GFFq]. Because counting the number of solutions of polynomials over nite elds is generic for many other algebraic counting problems, the results of this paper settle a border line for the algebraic problems with a polynomial time counting algorithms and for problems which are #P-complete. In KL 89] the couting problem for arbitrary multivariate polynomials over GFF2] has been proved to have randomized polynomial time approximation algorithms.