On SAT representations of XOR constraints

We study the representation of systems S of linear equations over the two-element field (aka xor- or parity-constraints) via conjunctive normal forms F (boolean clause-sets). First we consider the problem of finding an "arc-consistent" representation ("AC"), meaning that unit-clause propagation will fix all forced assignments for all possible instantiations of the xor-variables. Our main negative result is that there is no polysize AC-representation in general. On the positive side we show that finding such an AC-representation is fixed-parameter tractable (fpt) in the number of equations. Then we turn to a stronger criterion of representation, namely propagation completeness ("PC") --- while AC only covers the variables of S, now all the variables in F (the variables in S plus auxiliary variables) are considered for PC. We show that the standard translation actually yields a PC representation for one equation, but fails so for two equations (in fact arbitrarily badly). We show that with a more intelligent translation we can also easily compute a translation to PC for two equations. We conjecture that computing a representation in PC is fpt in the number of equations.Comment: 39 pages; 2nd v. improved handling of acyclic systems, free-standing proof of the transformation from AC-representations to monotone circuits, improved wording and literature review; 3rd v. updated literature, strengthened treatment of monotonisation, improved discussions; 4th v. update of literature, discussions and formulations, more details and examples; conference v. to appear LATA 201

Proof Generation for CDCL Solvers Using Gauss-Jordan Elimination

Traditional Boolean satisfiability (SAT) solvers based on the conflict-driven clause-learning (CDCL) framework fare poorly on formulas involving large numbers of parity constraints. The CryptoMiniSat solver augments CDCL with Gauss-Jordan elimination to greatly improve performance on these formulas. Integrating the TBUDDY proof-generating BDD library into CryptoMiniSat enables it to generate unsatisfiability proofs when using Gauss-Jordan elimination. These proofs are compatible with standard, clausal proof frameworks.Comment: Presented at 2022 Workshop on the Pragmatics of SA

On SAT Representations of XOR Constraints

We consider the problem of finding good representations, via boolean conjunctive normal forms F (clause-sets), of systems S of XOR-constraints x_1 + ... + x_n = e, e in {0,1} (also called parity constraints), i.e., systems of linear equations over the two-element field . These representations are to be used as parts of SAT problems. The basic quality criterion is "arc consistency". We show there is no arc-consistent representation of polynomial size for arbitrary S. The proof combines the basic method by Bessiere et al. 2009 on the relation between monotone circuits and ``consistency checkers'', adapted and simplified in the underlying report , with the lower bound on monotone circuits for monotone span programs in Babai et al. 1999 . On the other side, our basic positive result is that computing an arc-consistent representation is fixed-parameter tractable in the number m of equations of S. To obtain stronger representations, instead of mere arc-consistency we consider the class PC of propagation-complete clause-sets, as introduced in Bordeaux et al 2012 . The stronger criterion is now F in PC, which requires for all partial assignments, possibly involving also the auxiliary (new) variables in F, that forced assignments can be determined by unit-clause propagation. We analyse the basic translation, which for m=1 lies in PC, but fails badly so already for m=2, and we show how to repair this