A framework for good SAT translations, with applications to CNF representations of XOR constraints

We present a general framework for “good CNF-representations” of boolean constraints, to be used for translating decision problems into SAT problems (i.e., deciding satisfiability for conjunctive normal forms). We apply it to the representation of systems of XOR-constraints (“exclusive-or”), also known as systems of linear equations over the two-element field, or systems of parity constraints, or as systems of equivalences (XOR is the negation of an equivalence). The general framework defines the notion of “representation”, and provides several methods to measure the quality of the representation, by measuring the complexity (“hardness”) needed for making implicit “knowledge” of the representation explicit (to a SAT-solving mechanism). We obtain general upper and lower bounds. Applied to systems of XOR-constraints, we show a super-polynomial lower bound on “good” representations under very general circumstances. A corresponding upper bound shows fixed-parameter tractability in the number of constraints. The measurement underlying this upper bound ignores the auxiliary variables needed for shorter representations of XOR-constraints. Improved upper bounds for special cases take them into account, and a rich picture begins to emerge, under the various hardness measurements.