{"type":"ScholarlyArticle","title":"Generating Models of a Matched Formula With a Polynomial Delay (Extended Abstract)","authors":[{"name":"Petr Savický","slug":"petr-savicky","given":"Petr","family":"Savický","affiliation":"Institute of Computer Science The Czech Academy of Sciences Czech Republic"},{"name":"Petr Kučera","slug":"petr-kucera","given":"Petr","family":"Kučera","affiliation":"Department of Theoretical Computer Science and Mathematical Logic Faculty of Mathematics and Physics Charles University , Czech Republic"}],"abstract":"A matched formula is a CNF formula whose incidence graph admits a matching which matches a distinct variable to every clause. Such a formula is always satisfiable. Matched formulas are used, for example, in the area of parameterized complexity. We prove that the problem of counting the number of the models (satisfying assignments) of a matched formula is #P-complete. On the other hand, we define a class of formulas generalizing the matched formulas and prove that for a formula in this class one can choose in polynomial time a variable suitable for splitting the tree for the search of the models of the formula. As a consequence, the models of a formula from this class, in particular of any matched formula, can be generated sequentially with a delay polynomial in the size of the input. On the other hand, we prove that this task cannot be performed efficiently for linearly satisfiable formulas, which is a generalization of matched formulas containing the class considered above.","doi":"10.24963/ijcai.2017/721","url":"https://ijcai.org/proceedings/2017/721","pdf":"https://ijcai.org/proceedings/2017/0721.pdf","markdown":"https://ijcai.org/proceedings/2017/721.md","bibtex":"https://ijcai.org/proceedings/2017/bibtex/721","volume":{"slug":"2017","title":"Proceedings of the Twenty-Sixth International Joint Conference on Artificial Intelligence","short_title":"IJCAI 2017","url":"https://ijcai.org/proceedings/2017/","year":2017,"location":"Melbourne, Australia","dates":"19-25 August 2017","editors":"Carles Sierra","isbn":"978-0-9992411-0-3","publisher":"International Joint Conferences on Artificial Intelligence Organization","doi_prefix":"10.24963/ijcai.2017"},"section":"Journal track","track":"Journal track","number":721,"pages":{"first":5055,"last":5059},"keywords":["Constraints and Satisfiability: Constraints and Satisfiability","Constraints and Satisfiability: Satisfiability"],"videos":[],"references":[{"position":1,"raw":"[Aceto et al., 2013] Luca Aceto, Dario Monica, Anna Ingólfsdóttir, Angelo Montanari, and Guido Sciavicco. Logic for Programming, Artificial Intelligence, and Rea- soning: 19th International Conference, LPAR-19, Stellen- bosch, South Africa, December 14-19, 2013. Proceedings, chapter An Algorithm for Enumerating Maximal Models of Horn Theories with an Application to Modal Logics, pages 1-17. Springer Berlin Heidelberg, Berlin, Heidel- berg, 2013.","title":"Logic for Programming","authors":"Luca Aceto, Dario Monica, Anna Ingólfsdóttir, Angelo Montanari, Guido Sciavicco","year":"2013","venue":"Proceedings, chapter An Algorithm for Enumerating Maximal Models of Horn Theories with an Application to Modal Logics","doi":null,"url":null,"internal_url":null},{"position":2,"raw":"[Aharoni and Linial, 1986] Ron Aharoni and Nathan Linial. Minimal non-two-colorable hypergraphs and minimal un- satisfiable formulas. Journal of Combinatorial Theory, Se- ries A, 43(2):196 -204, 1986.","title":"Minimal non-two-colorable hypergraphs and minimal unsatisfiable formulas","authors":"Linial Aharoni, Ron Aharoni, Nathan Linial","year":"1986","venue":"Journal of Combinatorial Theory, Series A","doi":null,"url":null,"internal_url":null},{"position":3,"raw":"[Coquery et al., 2012] Emmanuel Coquery, Said Jabbour, Lakhdar Sais, Yakoub Salhi, et al. A SAT-based approach for discovering frequent, closed and maximal patterns in a sequence. In Proceedings of ECAI, 2012.","title":"A SAT-based approach for discovering frequent, closed and maximal patterns in a sequence","authors":"Emmanuel Coquery, Said Jabbour, Lakhdar Sais, Yakoub Salhi","year":"2012","venue":"Proceedings of ECAI","doi":null,"url":null,"internal_url":null},{"position":4,"raw":"[Creignou and Hébrard, 1997] Nadia Creignou and J-J Hébrard. On generating all solutions of generalized satis- fiability problems. Informatique théorique et applications, 31(6):499-511, 1997.","title":"On generating all solutions of generalized satisfiability problems","authors":"Nadia Creignou, J-J Hébrard","year":"1997","venue":"Informatique théorique et applications","doi":null,"url":null,"internal_url":null},{"position":5,"raw":"[Creignou et al., 2011] Nadia Creignou, Frédéric Olive, and Johannes Schmidt. Theory and Applications of Satisfia- bility Testing -SAT 2011: 14th International Conference, SAT 2011, Ann Arbor, MI, USA, June 19-22, 2011. Pro- ceedings, chapter Enumerating All Solutions of a Boolean CSP by Non-decreasing Weight, pages 120-133. Springer Berlin Heidelberg, Berlin, Heidelberg, 2011.","title":"Theory and Applications of Satisfiability Testing -SAT 2011: 14th International Conference, SAT 2011","authors":"Nadia Creignou, Frédéric Olive, Johannes Schmidt","year":"2011","venue":null,"doi":null,"url":null,"internal_url":null},{"position":6,"raw":"[Dechter and Itai, 1992] Rina Dechter and Alon Itai. Finding all solutions if you can find one. In AAAI-92 Workshop on Tractable Reasoning, pages 35-39, 1992.","title":"Finding all solutions if you can find one","authors":"Rina Dechter, Alon Itai","year":"1992","venue":"AAAI-92 Workshop on Tractable Reasoning","doi":null,"url":null,"internal_url":null},{"position":7,"raw":"[Fleischner et al., 2002] Herbert Fleischner, Oliver Kull- mann, and Stefan Szeider. Polynomial-time recognition of minimal unsatisfiable formulas with fixed clause-variable difference. Theoretical Computer Science, 289(1):503 - 516, 2002.","title":"Polynomial-time recognition of minimal unsatisfiable formulas with fixed clause-variable difference","authors":"Herbert Fleischner, Oliver Kullmann, Stefan Szeider","year":"2002","venue":"Theoretical Computer Science","doi":null,"url":null,"internal_url":null},{"position":8,"raw":"[Flum and Grohe, 2006] Jörg Flum and Martin Grohe. Pa- rameterized complexity theory, volume 3 of Texts in The- oretical Computer Science. An EATCS Series. Springer- Verlag Berlin Heidelberg, 1st edition, 2006.","title":"Parameterized complexity theory","authors":"Grohe Flum, Jörg Flum, Martin Grohe","year":"2006","venue":"3 of Texts in Theoretical Computer Science. An EATCS Series","doi":null,"url":null,"internal_url":null},{"position":9,"raw":"[Franco and Van Gelder, 2003] John Franco and Allen Van Gelder. A perspective on certain polynomial-time solvable classes of satisfiability. Discrete Appl. Math., 125(2-3):177-214, 2003.","title":"A perspective on certain polynomial-time solvable classes of satisfiability","authors":"Van Franco, John Gelder, Allen Franco, Van Gelder","year":"2003","venue":"Discrete Appl. Math","doi":null,"url":null,"internal_url":null},{"position":10,"raw":"[Jabbour et al., 2014] Said Jabbour, Jerry Lonlac, Lakhdar Sais, and Yakoub Salhi. Extending modern SAT solvers for models enumeration. In IEEE 15th International Con- ference on Information Reuse and Integration (IRI), 2014, pages 803-810. IEEE, 2014.","title":"Extending modern SAT solvers for models enumeration","authors":"Said Jabbour, Jerry Lonlac, Lakhdar Sais, Yakoub Salhi","year":"2014","venue":"IEEE 15th International Conference on Information Reuse and Integration (IRI)","doi":null,"url":null,"internal_url":null},{"position":11,"raw":"[Johnson et al., 1988] David S. Johnson, Mihalis Yan- nakakis, and Christos H. Papadimitriou. On generating all maximal independent sets. Information Processing Let- ters, 27(3):119 -123, 1988.","title":"On generating all maximal independent sets","authors":"David S Johnson, Mihalis Yannakakis, Christos H Papadimitriou","year":"1988","venue":"Information Processing Letters","doi":null,"url":null,"internal_url":null},{"position":12,"raw":"[Kang and Park, 2005] Hyeong-Ju Kang and In-Cheol Park. SAT-based unbounded symbolic model check- ing. Computer-Aided Design of Integrated Circuits and Systems, IEEE Transactions on, 24(2):129-140, Feb 2005.","title":"SAT-based unbounded symbolic model checking","authors":"Hyeong-Ju Kang, In-Cheol Park","year":"2005","venue":"Computer-Aided Design of Integrated Circuits and Systems, IEEE Transactions on","doi":null,"url":null,"internal_url":null},{"position":13,"raw":"[Kavvadias et al., 2000] Dimitris J. Kavvadias, Martha Sideri, and Elias C. Stavropoulos. Generating all maximal models of a Boolean expression. Information Processing Letters, 74(3-4):157-162, 2000.","title":"Generating all maximal models of a Boolean expression","authors":"Dimitris J Kavvadias, Martha Sideri, Elias C Stavropoulos","year":"2000","venue":"Information Processing Letters","doi":null,"url":null,"internal_url":null},{"position":14,"raw":"[Kullmann, 2000] Oliver Kullmann. Investigations on autark assignments. Discrete Applied Mathematics, 107(1-3):99 -137, 2000.","title":"Investigations on autark assignments","authors":"Oliver Kullmann","year":"2000","venue":"Discrete Applied Mathematics","doi":null,"url":null,"internal_url":null},{"position":15,"raw":"[Kullmann, 2003] Oliver Kullmann. Lean clause-sets: gen- eralizations of minimally unsatisfiable clause-sets. Dis- crete Applied Mathematics, 130(2):209 -249, 2003. The Renesse Issue on Satisfiability.","title":"Lean clause-sets: generalizations of minimally unsatisfiable clause-sets","authors":"Oliver Kullmann","year":"2003","venue":"Discrete Applied Mathematics","doi":null,"url":null,"internal_url":null},{"position":16,"raw":"[McMillan, 2002] Ken L. McMillan. Computer Aided Verifi- cation: 14th International Conference, CAV 2002 Copen- hagen, Denmark, July 27-31, 2002 Proceedings, chapter Applying SAT Methods in Unbounded Symbolic Model Checking, pages 250-264. Springer Berlin Heidelberg, Berlin, Heidelberg, 2002.","title":"Proceedings, chapter Applying SAT Methods in Unbounded Symbolic Model Checking","authors":"Ken L Mcmillan","year":"2002","venue":null,"doi":null,"url":null,"internal_url":null},{"position":17,"raw":"[Morgado and Marques-Silva, 2005] A Morgado and J Marques-Silva. Algorithms for propositional model enumeration and counting. Technical report, Instituto de Engenharia de Sistemas e Computadores, Investigac ¸ão e Desenvolvimento, Lisboa, February 2005.","title":"Algorithms for propositional model enumeration and counting","authors":"Marques-Silva ; A Morgado, J Morgado, Marques-Silva","year":"2005","venue":null,"doi":null,"url":null,"internal_url":null},{"position":18,"raw":"[Savický and Kučera, 2016] Petr Savický and Petr Kučera. Generating models of a matched formula with a polyno- mial delay. Journal of Artificial Intelligence Research, 56:379-402, 2016.","title":"Petr Savický and Petr Kučera. Generating models of a matched formula with a polynomial delay","authors":"Kučera Savický","year":"2016","venue":"Journal of Artificial Intelligence Research","doi":null,"url":null,"internal_url":null},{"position":19,"raw":"[Szeider, 2003] Stefan Szeider. Minimal unsatisfiable for- mulas with bounded clause-variable difference are fixed- parameter tractable. In Tandy Warnow and Binhai Zhu, ed- itors, Computing and Combinatorics, volume 2697 of Lec- ture Notes in Computer Science, pages 548-558. Springer Berlin Heidelberg, 2003.","title":"Minimal unsatisfiable formulas with bounded clause-variable difference are fixedparameter tractable","authors":"Stefan Szeider","year":"2003","venue":"itors, Computing and Combinatorics","doi":null,"url":null,"internal_url":null},{"position":20,"raw":"[Szeider, 2005] Stefan Szeider. Generalizations of matched CNF formulas. Annals of Mathematics and Artificial In- telligence, 43(1-4):223-238, 2005.","title":"Generalizations of matched CNF formulas","authors":"Stefan Szeider","year":"2005","venue":"Annals of Mathematics and Artificial Intelligence","doi":null,"url":null,"internal_url":null},{"position":21,"raw":"[Szeider, 2007] Stefan Szeider. Matched formulas and back- door sets. In João Marques-Silva and Karem A. Sakallah, editors, Theory and Applications of Satisfiability Testing -SAT 2007, volume 4501 of Lecture Notes in Computer Science, pages 94-99. Springer Berlin Heidelberg, 2007.","title":"Matched formulas and backdoor sets","authors":"Stefan Szeider","year":"2007","venue":"Theory and Applications of Satisfiability Testing -SAT 2007","doi":null,"url":null,"internal_url":null},{"position":22,"raw":"[Tovey, 1984] Craig A. Tovey. A simplified NP-complete satisfiability problem. Discrete Applied Mathematics, 8(1):85 -89, 1984.","title":"A simplified NP-complete satisfiability problem","authors":"Craig A Tovey","year":"1984","venue":"Discrete Applied Mathematics","doi":null,"url":null,"internal_url":null},{"position":23,"raw":"[Valiant, 1979] L.G. Valiant. The complexity of computing the permanent. Theoretical Computer Science, 8(2):189 - 201, 1979.","title":"The complexity of computing the permanent","authors":"L G Valiant","year":"1979","venue":"Theoretical Computer Science","doi":null,"url":null,"internal_url":null}],"license_note":"Copyright International Joint Conferences on Artificial Intelligence."}