On the expressive power of unit resolution

This preliminary report addresses the expressive power of unit resolution regarding input data encoded with partial truth assignments of propositional variables. A characterization of the functions that are computable in this way, which we propose to call propagatable functions, is given. By establishing that propagatable functions can also be computed using monotone circuits, we show that there exist polynomial time complexity propagable functions requiring an exponential amount of clauses to be computed using unit resolution. These results shed new light on studying CNF encodings of NP-complete problems in order to solve them using propositional satisfiability algorithms

Reified unit resolution and the failed literal rule

Unit resolution can simplify a CNF formula or detect an inconsistency by repeatedly assign the variables occurring in unit clauses. Given any CNF formula sigma, we show that there exists a satisfiable CNF formula psi with size polynomially related to the size of sigma such that applying unit resolution to psi simulates all the effects of applying it to sigma. The formula psi is said to be the reified counterpart of sigma. This approach can be used to prove that the failed literal rule, which is an inference rule used by some SAT solvers, can be entirely simulated by unit resolution. More generally, it sheds new light on the expressive power of unit resolution

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, also known as systems of linear equations over the two-element field, or systems of parity constraints. The general framework defines the notion of "representation", and provides several methods to measure the quality of the representation by 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.Comment: 67 pages; second version with extended discussion of literature. Continues arXiv:1309.306