On Limitations of Structured (Deterministic) DNNFs
The study of representations for propositional theories has been a central subject in knowledge compilation. Many known representations of propositional knowledge bases are restricted negation normal form circuits (NNFs) or binary decision diagrams. Sentential decision diagrams (SDDs) are one of the...
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| Published in | Theory of computing systems Vol. 64; no. 5; pp. 799 - 825 |
|---|---|
| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
New York
Springer US
01.07.2020
Springer Nature B.V |
| Subjects | |
| Online Access | Get full text |
| ISSN | 1432-4350 1433-0490 |
| DOI | 10.1007/s00224-019-09960-w |
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| Abstract | The study of representations for propositional theories has been a central subject in knowledge compilation. Many known representations of propositional knowledge bases are restricted negation normal form circuits (NNFs) or binary decision diagrams. Sentential decision diagrams (SDDs) are one of them, by definition they are restricted structured deterministic DNNFs and more general than ordered binary decision diagrams (OBDDs). In this paper limitations of structured (deterministic) DNNFs and in particular SDDs are considered. The main result in the paper is a quasipolynomial simulation of structured (deterministic) DNNFs by equivalent (unambiguous) nondeterministic OBDDs. The result is tight in the sense that there cannot be a polynomial simulation due to a known separation result. One may ask whether there are simulations in smaller size for restricted structured (deterministic) DNNFs. Answering a question posed by Bova and Szeider (
2017
) in the negative it is shown that not every SDD with disjunctions of constant fan-in can be transformed into equivalent OBDDs of polynomial size. |
|---|---|
| AbstractList | The study of representations for propositional theories has been a central subject in knowledge compilation. Many known representations of propositional knowledge bases are restricted negation normal form circuits (NNFs) or binary decision diagrams. Sentential decision diagrams (SDDs) are one of them, by definition they are restricted structured deterministic DNNFs and more general than ordered binary decision diagrams (OBDDs). In this paper limitations of structured (deterministic) DNNFs and in particular SDDs are considered. The main result in the paper is a quasipolynomial simulation of structured (deterministic) DNNFs by equivalent (unambiguous) nondeterministic OBDDs. The result is tight in the sense that there cannot be a polynomial simulation due to a known separation result. One may ask whether there are simulations in smaller size for restricted structured (deterministic) DNNFs. Answering a question posed by Bova and Szeider (2017) in the negative it is shown that not every SDD with disjunctions of constant fan-in can be transformed into equivalent OBDDs of polynomial size. The study of representations for propositional theories has been a central subject in knowledge compilation. Many known representations of propositional knowledge bases are restricted negation normal form circuits (NNFs) or binary decision diagrams. Sentential decision diagrams (SDDs) are one of them, by definition they are restricted structured deterministic DNNFs and more general than ordered binary decision diagrams (OBDDs). In this paper limitations of structured (deterministic) DNNFs and in particular SDDs are considered. The main result in the paper is a quasipolynomial simulation of structured (deterministic) DNNFs by equivalent (unambiguous) nondeterministic OBDDs. The result is tight in the sense that there cannot be a polynomial simulation due to a known separation result. One may ask whether there are simulations in smaller size for restricted structured (deterministic) DNNFs. Answering a question posed by Bova and Szeider ( 2017 ) in the negative it is shown that not every SDD with disjunctions of constant fan-in can be transformed into equivalent OBDDs of polynomial size. |
| Author | Buttkus, Matthias Bollig, Beate |
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| Cites_doi | 10.1007/s00224-012-9392-5 10.1613/jair.989 10.1609/aaai.v29i1.9423 10.1609/aaai.v30i1.10107 10.1007/s00453-015-0059-x 10.1609/aaai.v29i1.9763 10.1007/978-3-319-10428-7_7 10.1109/TC.1986.1676819 10.1137/1.9780898719789 10.1007/s00224-018-9904-z 10.1007/978-3-662-03927-4 10.1051/ita:1999108 10.1109/12.73590 10.1145/502090.502091 10.1007/978-3-319-23219-5_26 10.1145/3034786.3034787 10.1007/s00224-016-9714-0 |
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| DOI | 10.1007/s00224-019-09960-w |
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| Keywords | Ordered binary decision diagrams Complexity theory Decomposable negation normal forms Knowledge compilation Sentential decision diagrams |
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| References_xml | – reference: Pipatsrisawat, K., Darwiche, A.: New compilation languages based on structured decomposability. In: Proceedings of the Twenty-Third Conference on Artificial Intelligence, AAAI, pp 517–522 (2008) – reference: Bova, S., Szeider, S.: Circuit treewidth, sentential decision, and query compilation. In: Proceedings of the Thirty-sixth ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems, PODS, pp 233–246 (2017) – reference: BolligBLöbbingMSauerhoffMWegenerIOn the complexity of the hidden weighted bit function for various BDD modelsTheor. Inform. Appl.1999332103115170796410.1051/ita:1999108 – reference: Van den Broeck, G., Darwiche, A.: On the role of canonicity in knowledge compilation. In: Proceedings of the Twenty-Ninth Conference on Artificial Intelligence, AAAI, pp 1641–1648 (2015) – reference: DarwicheADecomposable negation normal formJ. ACM2001484608647214492410.1145/502090.502091 – reference: Wegener, I.: Branching Programs and Binary Decision Diagrams: Theory and Applications SIAM Monographs on Discrete Mathematics and Applications (2000) – reference: Bova, S.: SDDs are exponentially more succinct than OBDDs. In: Proceedings of the Thirtieth Conference on Artificial Intelligence, AAAI, pp 929–935 (2016) – reference: Razgon, I.: Quasipolynomial simulation of DNNF by a non-determinstic read-once branching program. In: Proceedings of the Twenty-first International Conference on Principles and Practice of Constraint Programming, CP, pp 367–375 (2015) – reference: Marquis, P.: Compile!. In: Proceedings of the Twenty-Ninth Conference on Artificial Intelligence, AAAI, pp 4112–4118 (2015) – reference: JhaASuciuDKnowledge compilation meets database theory: compiling queries to decision diagramsTheory Comput. Syst.2013523403440302927610.1007/s00224-012-9392-5 – reference: Razgon, I.: On OBDDs for CNFs of bounded treewidth. arXiv:1308.3829v3 (2013) – reference: Xue, Y., Choi, A., Darwiche, A.: Basing decisions on sentences in decision diagrams. In: Proceedings of the Twenty-Fourth Conference on Artificial Intelligence, AAAI (2012) – reference: RazgonIOn oblivious branching programs with bounded repetition that cannot efficiently compute CNFs of bounded treewidthTheory Comput. Syst.2017613755776368683110.1007/s00224-016-9714-0 – reference: Bova, S., Capelli, F., Mengel, S., Slivovsky, F.: Knowledge compilation meets communication complexity. In: Proceedings of the Twenty-Fifth International Joint Conference on Artificial Intelligence, IJCAI, pp 1008–1014 (2016) – reference: Oztok, U., Darwiche, A.: A top-down compiler for sentential decision diagrams. In: Proceedings of the Twenty-Fourth International Joint Conference on Artificial Intelligence, IJCAI, pp 3141–3148 (2015) – reference: BryantROn the complexity of VLSI implementations and graph representations of Boolean functions with application to integer multiplicationIEEE Trans. Comput.1991402205213109403110.1109/12.73590 – reference: Darwiche, A.: SDD: A new canonical representation of propositional knowledge bases. In: Proceedings of the Twenty-Second International Joint Conference on Artificial Intelligence, IJCAI, pp 819–826 (2011) – reference: DarwicheAMarquisPA knowledge compilation mapJ. Artif. Intell. Res.200217229264195079810.1613/jair.989 – reference: RazgonIOn the read-once property of branching programs and CNFs of bounded treewidthAlgorithmica2016752277294350619410.1007/s00453-015-0059-x – reference: Beame, P., Liew, V.: New limits for knowledge compilation and applications to exact model counting. In: Proceedings of the Thirty-First Conference on Uncertainty in Artificial Intelligence, UAI, pp 131–140 (2015) – reference: BryantRGraph-based algorithms for Boolean function manipulationIEEE Trans. Comput.198635867769110.1109/TC.1986.1676819 – reference: Beame, P., Li, J., Roy, S., Suciu, D.: Lower bounds for exact model counting and applications in probabilistic databases. In: Proceedings of the Twenty-Ninth Conference on Uncertainty in Artificial Intelligence, UAI, pp 157–162 (2013) – reference: Vollmer, H.: Introduction to Circuit Complexity - A Uniform Approach. Springer Science (1999) – reference: BolligBButtkusMOn the relative succinctness of sentential decision diagramsTheory Comput. Syst.201963612501277398433410.1007/s00224-018-9904-z – reference: Oztok, U., Darwiche, A.: On compiling CNF into decision-DNNF. In: Proceedings of the Twentieth International Conference on Principles and Practice of Constraint Programming, CP, pp 42–57 (2014) – reference: CadoliMDoniniFA survey on knowledge compilationAI Commun.1997103, 4137150 – volume: 52 start-page: 403 issue: 3 year: 2013 ident: 9960_CR15 publication-title: Theory Comput. Syst. doi: 10.1007/s00224-012-9392-5 – volume: 17 start-page: 229 year: 2002 ident: 9960_CR14 publication-title: J. Artif. Intell. Res. doi: 10.1613/jair.989 – ident: 9960_CR13 – ident: 9960_CR19 – ident: 9960_CR1 – ident: 9960_CR8 doi: 10.1609/aaai.v29i1.9423 – ident: 9960_CR5 doi: 10.1609/aaai.v30i1.10107 – volume: 75 start-page: 277 issue: 2 year: 2016 ident: 9960_CR22 publication-title: Algorithmica doi: 10.1007/s00453-015-0059-x – ident: 9960_CR16 doi: 10.1609/aaai.v29i1.9763 – ident: 9960_CR17 doi: 10.1007/978-3-319-10428-7_7 – volume: 35 start-page: 677 issue: 8 year: 1986 ident: 9960_CR9 publication-title: IEEE Trans. Comput. doi: 10.1109/TC.1986.1676819 – ident: 9960_CR26 – ident: 9960_CR25 doi: 10.1137/1.9780898719789 – volume: 63 start-page: 1250 issue: 6 year: 2019 ident: 9960_CR3 publication-title: Theory Comput. Syst. doi: 10.1007/s00224-018-9904-z – volume: 10 start-page: 137 issue: 3, 4 year: 1997 ident: 9960_CR11 publication-title: AI Commun. – ident: 9960_CR24 doi: 10.1007/978-3-662-03927-4 – volume: 33 start-page: 103 issue: 2 year: 1999 ident: 9960_CR4 publication-title: Theor. Inform. Appl. doi: 10.1051/ita:1999108 – volume: 40 start-page: 205 issue: 2 year: 1991 ident: 9960_CR10 publication-title: IEEE Trans. Comput. doi: 10.1109/12.73590 – ident: 9960_CR20 – ident: 9960_CR18 – volume: 48 start-page: 608 issue: 4 year: 2001 ident: 9960_CR12 publication-title: J. ACM doi: 10.1145/502090.502091 – ident: 9960_CR21 doi: 10.1007/978-3-319-23219-5_26 – ident: 9960_CR7 doi: 10.1145/3034786.3034787 – ident: 9960_CR6 – volume: 61 start-page: 755 issue: 3 year: 2017 ident: 9960_CR23 publication-title: Theory Comput. Syst. doi: 10.1007/s00224-016-9714-0 – ident: 9960_CR2 |
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| SubjectTerms | Canonical forms Circuit diagrams Computer Science Equivalence Knowledge bases (artificial intelligence) Knowledge representation Polynomials Simulation Theory of Computation |
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| Title | On Limitations of Structured (Deterministic) DNNFs |
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