### arXiv:1702.08392 [cs.DM]AbstractReferencesReviewsResources

#### Combining the $k$-CNF and XOR Phase-Transitions

Jeffrey M. Dudek, Kuldeep S. Meel, Moshe Y. Vardi

Published 2017-02-27Version 1

The runtime performance of modern SAT solvers on random $k$-CNF formulas is deeply connected with the 'phase-transition' phenomenon seen empirically in the satisfiability of random $k$-CNF formulas. Recent universal hashing-based approaches to sampling and counting crucially depend on the runtime performance of SAT solvers on formulas expressed as the conjunction of both $k$-CNF and XOR constraints (known as $k$-CNF-XOR formulas), but the behavior of random $k$-CNF-XOR formulas is unexplored in prior work. In this paper, we present the first study of the satisfiability of random $k$-CNF-XOR formulas. We show empirical evidence of a surprising phase-transition that follows a linear trade-off between $k$-CNF and XOR constraints. Furthermore, we prove that a phase-transition for $k$-CNF-XOR formulas exists for $k = 2$ and (when the number of $k$-CNF constraints is small) for $k > 2$.

**Comments:**Presented at The 25th International Joint Conference on Artificial Intelligence (IJCAI-16)

**Categories:**cs.DM

**Keywords:**cnf-xor formulas, xor phase-transitions, cnf formulas, runtime performance, xor constraints

**Tags:**conference paper