Abstract
Davis-Putnam resolution is one of the fundamental theoretical decision procedures for both propositional logic and quantified Boolean formulas.
Dependency quantified Boolean formulas (DQBF) are a generalisation of QBF in which dependencies of variables are listed explicitly rather than being implicit in the order of quantifiers. Since DQBFs can succinctly encode synthesis problems that ask for Boolean functions matching a given specification, efficient DQBF solvers have a wide range of potential applications. We present a new decision procedure for DQBF in the style of Davis-Putnam resolution. Based on the merge resolution proof system, it directly constructs partial strategy functions for derived clauses. The procedure requires DQBF in a normal form called H-Form. We prove that the problem of evaluating DQBF in H-Form is NEXP-complete. In fact, we show that any DQBF can be converted into H-Form in linear time.
This research was supported by the Vienna Science and Technology Fund (WWTF) under grant number ICT19-060, and by the Austrian Science Fund (FWF) under grant number J-4361N.
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Notes
- 1.
The algorithm described by Davis and Putnam [12] also considers unit clauses and pure literals, but since these are neither necessary for completeness, nor complete on their own, we think of DP-resolution as consisting of variable elimination.
- 2.
Note that we still take the if-then-else even if the functions are compatible, and in particular also if one of the functions is undefined. This is slightly counter-intuitive at first because we could just take the union in those cases, but the if-then-else results in a more compatible strategy and is in fact necessary to ensure completeness.
- 3.
We cannot use soundness of \(\mathsf {M{\text{- }}Res}\), because our strategy compatibility notion is stronger.
- 4.
In the size of the functions, which may, inevitably, become exponential.
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Blinkhorn, J., Peitl, T., Slivovsky, F. (2021). Davis and Putnam Meet Henkin: Solving DQBF with Resolution. In: Li, CM., Manyà, F. (eds) Theory and Applications of Satisfiability Testing – SAT 2021. SAT 2021. Lecture Notes in Computer Science(), vol 12831. Springer, Cham. https://doi.org/10.1007/978-3-030-80223-3_4
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