Learning Polynomials over GF(2) in a SAT Solver - (Poster Presentation)

One potential direction for improving the performance of SAT solvers is by using a stronger underlying proof system, e.g., [1]. We propose a step in improving the learning architecture of SAT solvers and describe a learning scheme in the polynomial calculus with resolution (PCR), a proof system that generalizes both resolution and Gaussian elimination. The scheme fits the general structure of CDCL solvers, so many of the other techniques of CDCL solvers should be reusable. The PCR proof system was introduced in [2]. In it, lines of a proof are polynomials, which are derived by summing two previous polynomials or multiplying a previous polynomial by a variable. The system also includes the axioms x−x = 0, ¬x−¬x = 0 and x + ¬x = 1 for all variables x. In our approach, we use only polynomials over GF (2). In this system, a clause (a∨b∨¬c) is expressed as the polynomial ¬a¬bc = 0. A xor-clause (a⊕b⊕¬c) is also naturally expressed, as the polynomial a+b+¬c = 0. However, neither a clause nor a xor clause can capture a general polynomial such as xy + zw + pq + 1 = 0. Note that the variables ¬x are not necessary, as they can be replaced by (1 + x) but using them can drastically reduce the number of monomials. When written as a sum of monomials, a global order on variables allows a canonical representation, unique for all equal polynomials. There is significant previous work that addresses the efficient integration of XOR (or equivalence) reasoning techniques in SAT solvers, e.g. [3, 4]. However, in these approaches, interaction between the CNF and XOR subproblems is limited to passing unit clauses from the CNF part to the XOR part and implied clauses from the XOR part to the CNF part.