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** Combining the k-CNF and XOR Phase-Transitions**.

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

In * Proceedings of International Joint Conference on Artificial Intelligence (IJCAI)*, July 2016.

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.

@inproceedings{DMV16, title={Combining the k-CNF and XOR Phase-Transitions}, author={Dudek, Jeffrey and Meel, Kuldeep S. and Vardi, Moshe Y.}, bib2html_dl_pdf={../Papers/ijcai16-cnfxor.pdf}, code={http://www.cs.rice.edu/CS/Verification/Projects/CUSP/IJCAI16/}, year={2016}, month=jul, booktitle=IJCAI, bib2html_rescat={Phase Transitions}, bib2html_pubtype={Refereed Conference}, abstract={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.} }

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