Reinterpreting Dependency Schemes: Soundness Meets Incompleteness in DQBF

Dependency quantified Boolean formulas (DQBF) and QBF dependency schemes have been treated separately in the literature, even though both treatments extend QBF by replacing the linear order of the quantifier prefix with a partial order. We propose to merge the two, by reinterpreting a dependency sch...

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Published in:	Journal of automated reasoning Vol. 63; no. 3; pp. 597 - 623
Main Authors:	Beyersdorff, Olaf, Blinkhorn, Joshua, Chew, Leroy, Schmidt, Renate, Suda, Martin
Format:	Journal Article
Language:	English
Published:	Dordrecht Springer Netherlands 01.10.2019 Springer Nature B.V
Subjects:	Artificial Intelligence Boolean algebra Computer Science Dependence Mapping Mathematical Logic and Formal Languages Mathematical Logic and Foundations Reduction Symbolic and Algebraic Manipulation Quantified Boolean formulas DQBF F.2.2 Nonnumerical algorithms and problems Dependency schemes
ISSN:	0168-7433, 1573-0670, 1573-0670
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Abstract	Dependency quantified Boolean formulas (DQBF) and QBF dependency schemes have been treated separately in the literature, even though both treatments extend QBF by replacing the linear order of the quantifier prefix with a partial order. We propose to merge the two, by reinterpreting a dependency scheme as a mapping from QBF into DQBF. Our approach offers a fresh insight on the nature of soundness in proof systems for QBF with dependency schemes, in which a natural property called ‘full exhibition’ is central. We apply our approach to QBF proof systems from two distinct paradigms, termed ‘universal reduction’ and ‘universal expansion’. We show that full exhibition is sufficient (but not necessary) for soundness in universal reduction systems for QBF with dependency schemes, whereas for expansion systems the same property characterises soundness exactly. We prove our results by investigating DQBF proof systems, and then employing our reinterpretation of dependency schemes. Finally, we show that the reflexive resolution path dependency scheme is fully exhibited, thereby proving a conjecture of Slivovsky.
AbstractList	Dependency quantified Boolean formulas (DQBF) and QBF dependency schemes have been treated separately in the literature, even though both treatments extend QBF by replacing the linear order of the quantifier prefix with a partial order. We propose to merge the two, by reinterpreting a dependency scheme as a mapping from QBF into DQBF. Our approach offers a fresh insight on the nature of soundness in proof systems for QBF with dependency schemes, in which a natural property called 'full exhibition' is central. We apply our approach to QBF proof systems from two distinct paradigms, termed 'universal reduction' and 'universal expansion'. We show that full exhibition is sufficient (but not necessary) for soundness in universal reduction systems for QBF with dependency schemes, whereas for expansion systems the same property characterises soundness exactly. We prove our results by investigating DQBF proof systems, and then employing our reinterpretation of dependency schemes. Finally, we show that the reflexive resolution path dependency scheme is fully exhibited, thereby proving a conjecture of Slivovsky. Dependency quantified Boolean formulas (DQBF) and QBF dependency schemes have been treated separately in the literature, even though both treatments extend QBF by replacing the linear order of the quantifier prefix with a partial order. We propose to merge the two, by reinterpreting a dependency scheme as a mapping from QBF into DQBF. Our approach offers a fresh insight on the nature of soundness in proof systems for QBF with dependency schemes, in which a natural property called 'full exhibition' is central. We apply our approach to QBF proof systems from two distinct paradigms, termed 'universal reduction' and 'universal expansion'. We show that full exhibition is sufficient (but not necessary) for soundness in universal reduction systems for QBF with dependency schemes, whereas for expansion systems the same property characterises soundness exactly. We prove our results by investigating DQBF proof systems, and then employing our reinterpretation of dependency schemes. Finally, we show that the reflexive resolution path dependency scheme is fully exhibited, thereby proving a conjecture of Slivovsky.Dependency quantified Boolean formulas (DQBF) and QBF dependency schemes have been treated separately in the literature, even though both treatments extend QBF by replacing the linear order of the quantifier prefix with a partial order. We propose to merge the two, by reinterpreting a dependency scheme as a mapping from QBF into DQBF. Our approach offers a fresh insight on the nature of soundness in proof systems for QBF with dependency schemes, in which a natural property called 'full exhibition' is central. We apply our approach to QBF proof systems from two distinct paradigms, termed 'universal reduction' and 'universal expansion'. We show that full exhibition is sufficient (but not necessary) for soundness in universal reduction systems for QBF with dependency schemes, whereas for expansion systems the same property characterises soundness exactly. We prove our results by investigating DQBF proof systems, and then employing our reinterpretation of dependency schemes. Finally, we show that the reflexive resolution path dependency scheme is fully exhibited, thereby proving a conjecture of Slivovsky.
Author	Chew, Leroy Beyersdorff, Olaf Blinkhorn, Joshua Schmidt, Renate Suda, Martin
Author_xml	– sequence: 1 givenname: Olaf surname: Beyersdorff fullname: Beyersdorff, Olaf organization: School of Computing, University of Leeds – sequence: 2 givenname: Joshua orcidid: 0000-0001-7452-6521 surname: Blinkhorn fullname: Blinkhorn, Joshua email: scjlb@leeds.ac.uk organization: School of Computing, University of Leeds – sequence: 3 givenname: Leroy surname: Chew fullname: Chew, Leroy organization: School of Computing, University of Leeds – sequence: 4 givenname: Renate surname: Schmidt fullname: Schmidt, Renate organization: School of Computer Science, University of Manchester – sequence: 5 givenname: Martin surname: Suda fullname: Suda, Martin organization: Institute of Information Systems, Vienna University of Technology
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Cites_doi	10.1016/0022-0000(80)90027-6 10.1016/j.tcs.2013.12.020 10.1006/inco.1995.1025 10.1016/j.tcs.2015.10.020 10.1145/2578043 10.1016/S0898-1221(00)00333-3 10.1016/j.artint.2016.01.004 10.1007/s10817-008-9114-5 10.2307/2273702 10.1016/j.tcs.2015.01.048 10.1007/978-3-642-33558-7_47 10.1007/978-3-319-66263-3_20 10.1007/11564751_43 10.1145/800157.805047 10.1145/800125.804029 10.1007/978-3-319-40970-2_30 10.1007/978-3-662-44465-8_8 10.1007/11814948_21 10.1007/978-3-319-09284-3_12 10.1007/978-3-319-40970-2 10.1016/B978-044450813-3/50004-7 10.1007/978-3-319-40970-2_29 10.1007/978-3-319-44953-1_7 10.1007/978-3-642-54013-4_1 10.1007/978-3-319-63390-9_25 10.7873/DATE.2015.0098 10.1007/978-3-540-89439-1_36 10.1007/978-3-319-66263-3_17 10.1007/978-3-642-31612-8_6 10.1007/978-3-319-66263-3 10.1007/978-3-642-23786-7_59
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Snippet	Dependency quantified Boolean formulas (DQBF) and QBF dependency schemes have been treated separately in the literature, even though both treatments extend QBF...
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SubjectTerms	Artificial Intelligence Boolean algebra Computer Science Dependence Mapping Mathematical Logic and Formal Languages Mathematical Logic and Foundations Reduction Symbolic and Algebraic Manipulation
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Title	Reinterpreting Dependency Schemes: Soundness Meets Incompleteness in DQBF
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Volume	63
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