Parameterized and exact algorithms for finding a read-once resolution refutation in 2CNF formulas

In this paper, we discuss algorithms for the problem of finding read-once resolution refutations of unsatisfiable 2CNF formulas within the resolution refutation system. Broadly, a read-once resolution refutation is one in which each constraint (input or derived) is used at most once. Read-once resol...

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Veröffentlicht in:	Annals of mathematics and artificial intelligence Jg. 90; H. 1; S. 3 - 29
Hauptverfasser:	Subramani, K., Wojciechowski, Piotr
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	Cham Springer International Publishing 01.01.2022 Springer Springer Nature B.V
Schlagworte:	Algorithms Analysis Artificial Intelligence Boolean Canonical forms Complex Systems Computer Science Design parameters Logic programs Mathematics Parameterization Polynomials Exact exponential algorithms Read-once resolution 09 27 2CNF 90
ISSN:	1012-2443, 1573-7470
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Abstract	In this paper, we discuss algorithms for the problem of finding read-once resolution refutations of unsatisfiable 2CNF formulas within the resolution refutation system. Broadly, a read-once resolution refutation is one in which each constraint (input or derived) is used at most once. Read-once resolution refutations have been widely studied in the literature for a number of constraint system-refutation system pairs. For instance, read-once resolution has been analyzed for boolean formulas in conjunctive normal form (CNF) and read-once cutting planes have been analyzed for polyhedral systems. By definition, read-once refutations are compact, and hence valuable in applications that place a premium on visualization. The satisfiability problem (SAT) is concerned with finding a satisfying assignment for a boolean formula in CNF. While SAT is NP-complete in general, there exist some interesting subclasses of CNF formulas, for which it is decidable in polynomial time. One such subclass is the class of 2CNF formulas, i.e., CNF formulas in which each clause has at most two literals. The existence of efficient algorithms for satisfiability checking in 2CNF formulas (2SAT), makes this class useful from the perspective of modeling selected logic programs. The work in this paper is concerned with the read-once refutability problem (under resolution) in this subclass. Although 2SAT is decidable in polynomial time, the problem of finding a read-once resolution refutation of an unsatisfiable 2CNF formula is NP-complete . We design non-trivial, parameterized and exact exponential algorithms for this problem. Additionally, we study the computational complexity of finding a shortest read-once resolution refutation of a 2CNF formula.
AbstractList	In this paper, we discuss algorithms for the problem of finding read-once resolution refutations of unsatisfiable 2CNF formulas within the resolution refutation system. Broadly, a read-once resolution refutation is one in which each constraint (input or derived) is used at most once. Read-once resolution refutations have been widely studied in the literature for a number of constraint system-refutation system pairs. For instance, read-once resolution has been analyzed for boolean formulas in conjunctive normal form (CNF) and read-once cutting planes have been analyzed for polyhedral systems. By definition, read-once refutations are compact, and hence valuable in applications that place a premium on visualization. The satisfiability problem (SAT) is concerned with finding a satisfying assignment for a boolean formula in CNF. While SAT is NP-complete in general, there exist some interesting subclasses of CNF formulas, for which it is decidable in polynomial time. One such subclass is the class of 2CNF formulas, i.e., CNF formulas in which each clause has at most two literals. The existence of efficient algorithms for satisfiability checking in 2CNF formulas (2SAT), makes this class useful from the perspective of modeling selected logic programs. The work in this paper is concerned with the read-once refutability problem (under resolution) in this subclass. Although 2SAT is decidable in polynomial time, the problem of finding a read-once resolution refutation of an unsatisfiable 2CNF formula is NP-complete. We design non-trivial, parameterized and exact exponential algorithms for this problem. Additionally, we study the computational complexity of finding a shortest read-once resolution refutation of a 2CNF formula. Keywords 2CNF * Read-once resolution * NP-hard * Exact exponential algorithms Mathematics Subject Classification (2010) 90C27 * 90C09 In this paper, we discuss algorithms for the problem of finding read-once resolution refutations of unsatisfiable 2CNF formulas within the resolution refutation system. Broadly, a read-once resolution refutation is one in which each constraint (input or derived) is used at most once. Read-once resolution refutations have been widely studied in the literature for a number of constraint system-refutation system pairs. For instance, read-once resolution has been analyzed for boolean formulas in conjunctive normal form (CNF) and read-once cutting planes have been analyzed for polyhedral systems. By definition, read-once refutations are compact, and hence valuable in applications that place a premium on visualization. The satisfiability problem (SAT) is concerned with finding a satisfying assignment for a boolean formula in CNF. While SAT is NP-complete in general, there exist some interesting subclasses of CNF formulas, for which it is decidable in polynomial time. One such subclass is the class of 2CNF formulas, i.e., CNF formulas in which each clause has at most two literals. The existence of efficient algorithms for satisfiability checking in 2CNF formulas (2SAT), makes this class useful from the perspective of modeling selected logic programs. The work in this paper is concerned with the read-once refutability problem (under resolution) in this subclass. Although 2SAT is decidable in polynomial time, the problem of finding a read-once resolution refutation of an unsatisfiable 2CNF formula is NP-complete. We design non-trivial, parameterized and exact exponential algorithms for this problem. Additionally, we study the computational complexity of finding a shortest read-once resolution refutation of a 2CNF formula. In this paper, we discuss algorithms for the problem of finding read-once resolution refutations of unsatisfiable 2CNF formulas within the resolution refutation system. Broadly, a read-once resolution refutation is one in which each constraint (input or derived) is used at most once. Read-once resolution refutations have been widely studied in the literature for a number of constraint system-refutation system pairs. For instance, read-once resolution has been analyzed for boolean formulas in conjunctive normal form (CNF) and read-once cutting planes have been analyzed for polyhedral systems. By definition, read-once refutations are compact, and hence valuable in applications that place a premium on visualization. The satisfiability problem (SAT) is concerned with finding a satisfying assignment for a boolean formula in CNF. While SAT is NP-complete in general, there exist some interesting subclasses of CNF formulas, for which it is decidable in polynomial time. One such subclass is the class of 2CNF formulas, i.e., CNF formulas in which each clause has at most two literals. The existence of efficient algorithms for satisfiability checking in 2CNF formulas (2SAT), makes this class useful from the perspective of modeling selected logic programs. The work in this paper is concerned with the read-once refutability problem (under resolution) in this subclass. Although 2SAT is decidable in polynomial time, the problem of finding a read-once resolution refutation of an unsatisfiable 2CNF formula is NP-complete . We design non-trivial, parameterized and exact exponential algorithms for this problem. Additionally, we study the computational complexity of finding a shortest read-once resolution refutation of a 2CNF formula.
Audience	Academic
Author	Wojciechowski, Piotr Subramani, K.
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Cites_doi	10.1016/j.tcs.2018.02.002 10.1007/s10601-016-9257-7 10.1016/0890-5401(92)90056-L 10.1016/0020-0190(79)90002-4 10.1007/978-3-030-24658-7 10.1002/9781118033166 10.1145/322047.322059 10.1016/0743-1066(84)90014-1 10.1007/978-3-642-16533-7 10.1007/BFb0029974 10.1016/0743-1066(94)90033-7 10.1145/2043174.2043195 10.1016/0304-3975(85)90144-6 10.1051/ita/2010018 10.1016/j.tcs.2005.10.005 10.1109/DSD.2010.111 10.1007/978-3-030-29007-8_9 10.1007/978-3-030-18126-0_9 10.1109/DSD.2011.50 10.1007/11814948_6 10.1007/3-540-45744-5_13
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Snippet	In this paper, we discuss algorithms for the problem of finding read-once resolution refutations of unsatisfiable 2CNF formulas within the resolution...
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SubjectTerms	Algorithms Analysis Artificial Intelligence Boolean Canonical forms Complex Systems Computer Science Design parameters Logic programs Mathematics Parameterization Polynomials
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Title	Parameterized and exact algorithms for finding a read-once resolution refutation in 2CNF formulas
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