| Summary: | Decision procedures for various logics are used as general-purpose solvers in computer science. A particularly popular choice is propositional logic, which is simultaneously powerful enough to model problems in many application domains, including formal verification and planning, while at the same time simple enough to be efficiently solved for many practical cases. Similarly, there are also recent interests in using QBF, an extension of propositional logic, as a modeling language to be used in a similar fashion. The hope is that QBF, being a more powerful language, can compactly encode, and in turn, be used to solve, a larger range of applications. Still, propositional logic and QBF are respectively complete for the complexity classes NP and PSPACE, thus, both can be theoretically considered intractable. A popular hypothesis is that real-world problems contain underlying structure that can be exploited by the decision procedures. In this dissertation, we study the impact of structural constraints (in the form of bounded width) and heuristics on the performance of propositional and QBF decision procedures.
The results presented in this dissertation can be seen as a contrast on how bounded-width impacts propositional and quantified problems differently. Starting with a size bound on BDDs under bounded width, we proceed to compare symbolic decision procedures against the standard DPLL search-based approach for propositional logic, as well as compare different width-based heuristics for the symbolic approaches. In general, symbolic approaches for propositional satisfiability are only competitive for a small range of problems, and the theoretical tractability for the bounded-width case rarely applies in practice. However, the picture is very different for quantified satisfiability. To that end, we start with a series of "intractability in tractability" results which shows that although the complexity of QBF with constant width and alternation is tractable, there is an inherent non-elementary blowup in the width and alternation depth such that a width-bound that is slightly above constant leads to intractability. To contrast the theoretical intractability, we apply structural heuristics to a symbolic decision procedure of QBF and show that symbolic approaches complement search-based approaches quite well for QBF.
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