Theoretical Foundations of Applied SAT Solving ( 14 w 5101 ) January 19 - 24 , 2014 MEALS

Speaker: Albert Atserias (Universitat Politecnica de Catalunya) Title: Mini-tutorial on semialgebraic proof systems Abstract: A variety of semialgebraic proof systems, i.e. those operating with polynomial inequalities over the reals, were defined in the last two decades to reason about optimality or near optimality in combinatorial optimization problems. In this mini-tutorial we overview their origins and also compare them to the traditional proof systems for propositional logic. From a proof complexity point of view, the main advantage that semialgebraic proof systems offer over low-level logic-based systems is the greater expressive power of their proof lines, while preserving certain good algorithmic properties for the proof-search problem. These good algorithmic properties also make them amenable to analysis by the proof-complexity methods to prove lower bounds. We will try to illustrate these points through some examples from the literature. Speaker: Paul Beame (University of Washington) Title: Exact model counting: SAT-solver based methods versus lifted inference Abstract: The best current methods for exactly computing the number of satisfying assignments, or the satisfying probability, of Boolean formulas are based on SAT solvers enhanced with component caching. These can be seen, either directly or indirectly, as building decision-DNNF (decision decomposable negation normal form) representations of their input Boolean formulas. We show that such representations, and indeed those that employ a more general form of component decomposition, can be converted into read-once branching programs (ROBPs) with only a quasi-polynomial increase in size. We use this together with new exponential lower bounds on the ROBP size of some simple natural DNF formulas associated with queries considered in probabilistic databases to derive exponential lower bounds on the sizes of these representations and, therefore, on this approach to exact model counting, including queries for which there are simple polynomial algorithms to compute these counts using ”lifted” inference methods. Joint work with Jerry Li, Sudeepa Roy, and Dan Suciu. Speaker: Chris Beck (Princeton University) Title: Strong ETH holds for regular resolution Abstract: Let ck be the infimum of real numbers so that k-SAT is solvable in time 2kpoly(n). The Strong Exponential Time Hypothesis is that the limit of the sequence ck is 1, i.e., that the difficulty of k-SAT approaches exhaustive search as k increases. This is true for the best algorithms currently known for k-SAT, as well as an empirical observation about the performance of SAT solvers on instances with large clauses. Since many SAT solvers are based on the resolution proof system, lower bounds for this system give lower bounds for large families of such solvers. We show that no algorithm formalizable in the subsystem of regular resolution can be a counter-example to SETH. More precisely, we show that there are unsatisfiable k-CNF formulas on n variables so that any regular resolution refutation requires size 21−�kn, where �k = Õ(k1/4). We also improve the lower bounds for general resolution substantially, showing that the same formulas require general resolution size at least (3/2)(1−�k)n.