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A Resolution Proof System for Dependency Stochastic Boolean Satisfiability

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Abstract

Dependency stochastic Boolean satisfiability (DSSAT), which generalizes stochastic Boolean satisfiability (SSAT) and dependency quantified Boolean formula (DQBF), is a new logical formalism that allows compact encoding of NEXPTIME decision problems under uncertainty. Despite potentially broad applications, a decision procedure for DSSAT remains lacking. In this work, we present the first sound and complete resolution calculus for DSSAT. The resolution system deduces the maximum satisfying probability of a DSSAT formula and provides a witnessing certificate. We also show that when the special case of SSAT formulas is considered, the DSSAT resolution calculus p-simulates a known SSAT resolution scheme. Our result may pave a theoretical foundation for further development and certification of DSSAT solvers.

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Notes

  1. An S-form DQBF is true if and only if it has a Skolem-function model for the existential variables; an H-form DQBF is false if and only if it has a Herbrand-function countermodel for the universal variables.

  2. While the rules in Sect. 2.1 provide a top-down computation for \(\textrm{Pr}[\Phi ]\), S-resolution essentially computes \(\textrm{Pr}[\Phi ]\) in a bottom-up manner. This is why the pivot has to be the rightmost quantified variable rather than the leftmost one.

  3. Although generating a formula \(\psi \) that is logically equivalent to \(\lnot \phi \) may be computationally expensive and impractical, we note that it is possible to efficiently generate \(\psi \) with the same satisfiability as \(\lnot \phi \) by Tseitin transformation [28].

  4. We note that since \(\alpha \) is a complete assignment over \(\mathcal {V}\), \(\Phi [\alpha ]\) is purely propositional and the strategy \(\vec {g}\) is empty.

  5. Since \(\vec {g}\) is unspecified for \(\mathcal {V}^{{>}i}\) and variable e can only depend on the variables \(D_e \subseteq \mathcal {V}^{{>}i}\) due to the linear extension constraint, we have that \(g_e\) is a constant function. Also, note that \(g_e=*\) is impossible because \(\vec {g}\) is a complete strategy for \(\Phi ^{{\le }i}\) by assumption. Therefore, we have either \(g_e=\top \) or \(g_e=\bot \).

  6. In this case, the local decision is done by the \(\max \)-operator.

  7. We cannot remove all dominated entries at once, since there may be some entries \(\eta \ne \eta '\in \) DP \((\Phi ,i)\) such that both \(\eta \succeq \eta '\) and \(\eta '\succeq \eta \) holds.

  8. Recall that rule R.3 of S-resolution is applied over pivots according to the prefix ordering such that the variable with the largest \(Q(v_j)\) (i.e., the smallest \({{\,\textrm{ord}\,}}(v_j)\)) in the clause is selected as the pivot. Since either \(i=0\) or \((C_0 \cup \lbrace \lnot v \rbrace )^{p_0}\) is derived from rule R.3 over a pivot variable \(v_j\) (instead of v) of order i, we have \(i<{{\,\textrm{ord}\,}}(v)\).

  9. M-Res-DP reduces universal literals (instead of existential literals) since it deals with H-form DQBF, where a formula is false if and only if there exist Herbrand functions for universal variables.

  10. M-Res-DP resolves on existential variables since it deals with H-form DQBF. See also Footnote 7.

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Acknowledgements

This work was supported in part by the Ministry of Science and Technology of Taiwan under Grant MOST 111-2221-E-002-182 and the National Science and Technology Council of Taiwan under Grant NSTC 111-2923-E-002-013-MY3.

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  1. Yun-Rong Luo and Che Cheng have contributed equally to this work.

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  1. Graduate Institute of Electronics Engineering, National Taiwan University, Taipei, 10617, Taiwan

    Yun-Rong Luo, Che Cheng & Jie-Hong R. Jiang

  2. Department of Electrical Engineering, National Taiwan University, Taipei, 10617, Taiwan

    Jie-Hong R. Jiang

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  1. Yun-Rong Luo
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Luo, YR., Cheng, C. & Jiang, JH.R. A Resolution Proof System for Dependency Stochastic Boolean Satisfiability. J Autom Reasoning 67, 26 (2023). https://doi.org/10.1007/s10817-023-09670-6

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