Vardi, Moshe Y2017-08-022017-08-022017-052017-04-21May 2017Dudek, Jeffrey M. "Random CNF-XOR Formulas." (2017) Master’s Thesis, Rice University. <a href="https://hdl.handle.net/1911/96156">https://hdl.handle.net/1911/96156</a>.https://hdl.handle.net/1911/96156Boolean Satisfiability (SAT) is fundamental in many diverse areas such as artificial intelligence, formal verification, and biology. Recent universal-hashing based approaches to the problems of sampling and counting crucially depend on the runtime performance of specialized SAT solvers on formulas expressed as the conjunction of both k-CNF constraints and variable-width XOR constraints (known as CNF-XOR formulas), but random CNF-XOR formulas are unexplored in prior work. In this work, we present the first study of random CNF-XOR formulas. We prove that a phase-transition in the satisfiability of random CNF-XOR formulas exists for k=2 and (when the number of k-CNF constraints is small) for k>2. We empirically demonstrate that a state-of-the-art SAT solver scales exponentially on random CNF-XOR formulas across many clause densities, peaking around the empirical phase-transition location. Finally, we prove that the solution space of random CNF-XOR formulas 'shatters' at all nonzero XOR-clause densities into well-separated components.application/pdfengCopyright is held by the author, unless otherwise indicated. Permission to reuse, publish, or reproduce the work beyond the bounds of fair use or other exemptions to copyright law must be obtained from the copyright holder.Boolean SatisfiabilitySATConstrained CountingConstrained SamplingHashing-Based AlgorithmsPhase TransitionSatisfiability ThresholdThesis2017-08-02