## On the Boolean connectivity problem for Horn relations (2007)

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Venue: | IN PROC. 10 TH INTL. CONFERENCE ON THEORY AND APPLICATIONS OF SATISFIABILITY TESTING (SAT), 2007 |

Citations: | 2 - 0 self |

### BibTeX

@INPROCEEDINGS{Makino07onthe,

author = {Kazuhisa Makino and Suguru Tamaki and Masaki Yamamoto},

title = {On the Boolean connectivity problem for Horn relations},

booktitle = {IN PROC. 10 TH INTL. CONFERENCE ON THEORY AND APPLICATIONS OF SATISFIABILITY TESTING (SAT), 2007},

year = {2007},

pages = {187--200},

publisher = {}

}

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### Abstract

### Citations

10921 |
Computers and Intractability: A Guide to the Theory of NP-Completeness
- Garey, Johnson
(Show Context)
Citation Context ...llary 1 that the complement of Conn(S) for Horn relations S belongs to N P. To show the N P-hardness, we reduce to it 3Uniform Hypergraph 2-Colorability, which is known to be N P-complete (see SP4 in =-=[16]-=-). Let H = (V, E) be a 3-uniform hypergraph, where a hypergraph is called 3-uniform if |E| = 3 holds for all E ∈ E. From H with E = {E1, · · · ,Ek}, we construct a 3-CNF Horn formula ϕH over a variabl... |

549 |
Combinatorial Optimization: Polyhedra and Efficiency
- Schrijver
- 2003
(Show Context)
Citation Context ...here n denotes the number of variables, ϕ denotes the corresponding 2-CNF formula, and T(n) denotes the time needed to compute the transitive closure of a directed graph of n vertices 3 . It is known =-=[31]-=- that T(n) = O(n ω ), where ω ≤ 2.376. We also investigate a tractable aspect of the intractable side (i.e., Horn and dual Horn relations). We consider the semantic (i.e., model-based) representation ... |

409 |
The complexity of satisfiability problems
- Schaefer
- 1978
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Citation Context ...o characteristic sets. 1 Introduction The Boolean satisfiability problem (satisfiability problem for short) is one of the central problems in the computational complexity theory. Schaefer proposed in =-=[30]-=- a framework for expressing variants of the satisfiability problem, and showed a dichotomy theorem: the satisfiability problem for certain classes of Boolean formulas is in P while it is N P-complete ... |

160 |
A linear-time algorithm for testing the truth of certain quantified boolean formulas
- Aspvall, Plass, et al.
- 1979
(Show Context)
Citation Context ...rmula ϕ is not Horn, then we construct a Horn formula ψ(x) = ϕ(x ⊕ a) by computing a satisfying assignment a for ϕ. Since ψ(0 n ) = 1, we can see that ψ is Horn, and it can be computed in linear time =-=[1]-=-. In what follows, we assume that a given 2-CNF formula ϕ is satisfiable and Horn without unit clauses, which is equivalent to the condition that ϕ is satisfied by 0 n . We now present a notion of cor... |

143 | On the complexity of dualization of monotone disjunctive normal forms - Fredman, Khachiyan - 1996 |

139 | Complexity classifications of Boolean constraint satisfaction problems - Creignou, Khanna, et al. - 2001 |

79 | Reasoning with models
- Khardon, Roth
- 1996
(Show Context)
Citation Context ..., instead of the traditional syntactic (i.e., formula-based) one. The model-based representation has been proposed as an alternative form of representing and accessing a logical knowledge base, e.g., =-=[10, 11, 12, 20, 21, 22, 24, 25]-=-. In contrast to the formula-based representation, if we have the model-based representation, that is, if we are given the characteristic set of Horn relations, the connectivity problem is solvable in... |

77 | Structure identification in relational data
- Dechter, Pearl
- 1992
(Show Context)
Citation Context ..., instead of the traditional syntactic (i.e., formula-based) one. The model-based representation has been proposed as an alternative form of representing and accessing a logical knowledge base, e.g., =-=[10, 11, 12, 20, 21, 22, 24, 25]-=-. In contrast to the formula-based representation, if we have the model-based representation, that is, if we are given the characteristic set of Horn relations, the connectivity problem is solvable in... |

74 |
Complexity of generalized satisfiability counting problems
- Creignou, Hermann
- 1996
(Show Context)
Citation Context ... XSat (or Exact Sat) are all N P-complete 1 . Since then, dichotomies or trichotomies have been established for several aspects of the satisfiability problem such as optimization [6, 8, 27], counting =-=[7]-=-, inverse satisfiability [26], minimal satisfiability [19], unique satisfiability [18], 3-valued satisfiability [4] and propositional abduction [9]. Very recently, Gopalan et al. studied in [17] conne... |

69 | The approximability of constraint satisfaction problems
- Khanna, Sudan, et al.
- 2001
(Show Context)
Citation Context ...t-All-Equal Sat), and XSat (or Exact Sat) are all N P-complete 1 . Since then, dichotomies or trichotomies have been established for several aspects of the satisfiability problem such as optimization =-=[6, 8, 27]-=-, counting [7], inverse satisfiability [26], minimal satisfiability [19], unique satisfiability [18], 3-valued satisfiability [4] and propositional abduction [9]. Very recently, Gopalan et al. studied... |

63 |
A dichotomy theorem for constraints on a three-element set
- Bulatov
- 2002
(Show Context)
Citation Context ...eral aspects of the satisfiability problem such as optimization [6, 8, 27], counting [7], inverse satisfiability [26], minimal satisfiability [19], unique satisfiability [18], 3-valued satisfiability =-=[4]-=- and propositional abduction [9]. Very recently, Gopalan et al. studied in [17] connectivity properties of the solution-space of Boolean formulas, and investigated complexity issues on connectivity pr... |

63 |
A dichotomy theorem for maximum generalized satisfiability problems
- Creignou
(Show Context)
Citation Context ...t-All-Equal Sat), and XSat (or Exact Sat) are all N P-complete 1 . Since then, dichotomies or trichotomies have been established for several aspects of the satisfiability problem such as optimization =-=[6, 8, 27]-=-, counting [7], inverse satisfiability [26], minimal satisfiability [19], unique satisfiability [18], 3-valued satisfiability [4] and propositional abduction [9]. Very recently, Gopalan et al. studied... |

45 |
The decision problem for some classes of sentences without quantifiers
- McKinsey
- 1943
(Show Context)
Citation Context ...⊆ {0,1} n denote Cl∧(M) the closure of M under ∧ . Then, for every M ⊆ {0,1} n , it holds that M is the set of satisfying assignments for some Horn formula ϕ (i.e, M = Vϕ) if and only if 9M = Cl∧(M) =-=[29]-=- (see e.g., [10, 24] for proofs). Namely, the set of satisfying assignments of a Horn formula is closed under the intersection ∧ , and any set of assignments which is closed under the intersection can... |

44 | On the solution-space geometry of random constraint satisfaction problems
- Achlioptas, Ricci-Tersenghi
- 2006
(Show Context)
Citation Context ...n right, since they shed light on the structure of the solution-space, and moreover, structural studies on the solution-space are important to analyze the satisfiability problem and algorithms for it =-=[2]-=-. They [17] established a dichotomy for the st-connectivity problem: It is polynomially solvable if given Boolean relations are tight, while it is PSPACE-complete in all other cases. This reveals that... |

41 |
Extended Horn sets in propositional logic
- Chandru, Hooker
- 1991
(Show Context)
Citation Context ... forms (DNFs) were studied by 1 There are many classes of formulas that do not fit Schaefer’s framework, but can be solved in polynomial time. Such examples include renamable Horn [28], extended Horn =-=[5]-=-, and q-Horn [3], for example. 1 ISSN 1433-8092Ekin et al. [14]. The connectivity problem (Conn) is to decide if the solutions of a given Boolean formula ϕ on n variables induce a connected subgraph ... |

37 | Reasoning With characteristic models
- Kautz, Kearns, et al.
- 1993
(Show Context)
Citation Context ..., instead of the traditional syntactic (i.e., formula-based) one. The model-based representation has been proposed as an alternative form of representing and accessing a logical knowledge base, e.g., =-=[10, 11, 12, 20, 21, 22, 24, 25]-=-. In contrast to the formula-based representation, if we have the model-based representation, that is, if we are given the characteristic set of Horn relations, the connectivity problem is solvable in... |

35 |
Renaming a set of clauses as a Horn set
- Lewis
- 1978
(Show Context)
Citation Context ...f disjunctive normal forms (DNFs) were studied by 1 There are many classes of formulas that do not fit Schaefer’s framework, but can be solved in polynomial time. Such examples include renamable Horn =-=[28]-=-, extended Horn [5], and q-Horn [3], for example. 1 ISSN 1433-8092Ekin et al. [14]. The connectivity problem (Conn) is to decide if the solutions of a given Boolean formula ϕ on n variables induce a ... |

34 | On Horn envelopes and hypergraph transversals
- Kavvadias, Papadimitriou, et al.
- 1993
(Show Context)
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33 | Horn approximations of empirical data
- Kautz, Kearns, et al.
- 1995
(Show Context)
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33 | The inverse satisfiability problem
- Kavvadias, Sideri
- 1998
(Show Context)
Citation Context ... N P-complete 1 . Since then, dichotomies or trichotomies have been established for several aspects of the satisfiability problem such as optimization [6, 8, 27], counting [7], inverse satisfiability =-=[26]-=-, minimal satisfiability [19], unique satisfiability [18], 3-valued satisfiability [4] and propositional abduction [9]. Very recently, Gopalan et al. studied in [17] connectivity properties of the sol... |

24 | Translating between Horn representations and their characteristic models - Khardon - 1995 |

23 |
A Complexity Index for Satisfiability Problems
- Boros, Crama, et al.
- 1994
(Show Context)
Citation Context ...re studied by 1 There are many classes of formulas that do not fit Schaefer’s framework, but can be solved in polynomial time. Such examples include renamable Horn [28], extended Horn [5], and q-Horn =-=[3]-=-, for example. 1 ISSN 1433-8092Ekin et al. [14]. The connectivity problem (Conn) is to decide if the solutions of a given Boolean formula ϕ on n variables induce a connected subgraph of the n-dimensi... |

22 | The complexity of minimal satisfiability problems
- Kirousis, Kolaitis
- 2003
(Show Context)
Citation Context ... dichotomies or trichotomies have been established for several aspects of the satisfiability problem such as optimization [6, 8, 27], counting [7], inverse satisfiability [26], minimal satisfiability =-=[19]-=-, unique satisfiability [18], 3-valued satisfiability [4] and propositional abduction [9]. Very recently, Gopalan et al. studied in [17] connectivity properties of the solution-space of Boolean formul... |

21 | Computational aspects of monotone dualization: A brief survey - Eiter, Makino, et al. - 2006 |

19 | A complete classification of the complexity of propositional abduction
- Creignou, Zanuttini
- 2006
(Show Context)
Citation Context ...ty problem such as optimization [6, 8, 27], counting [7], inverse satisfiability [26], minimal satisfiability [19], unique satisfiability [18], 3-valued satisfiability [4] and propositional abduction =-=[9]-=-. Very recently, Gopalan et al. studied in [17] connectivity properties of the solution-space of Boolean formulas, and investigated complexity issues on connectivity problems in Schaefer’s framework [... |

19 |
On connected Boolean functions
- Ekin, Hammer, et al.
- 1999
(Show Context)
Citation Context ...las that do not fit Schaefer’s framework, but can be solved in polynomial time. Such examples include renamable Horn [28], extended Horn [5], and q-Horn [3], for example. 1 ISSN 1433-8092Ekin et al. =-=[14]-=-. The connectivity problem (Conn) is to decide if the solutions of a given Boolean formula ϕ on n variables induce a connected subgraph of the n-dimensional hypercube, while the st-connectivity proble... |

15 | The connectivity of Boolean satisfiability: Computational and structural dichotomies
- Gopalan, Kolaitis, et al.
- 2006
(Show Context)
Citation Context ...to, 606-8501, Japan. tamak@kuis.kyoto-u.ac.jp 3 Graduate School of Informatics, Kyoto University, Kyoto, 606-8501, Japan. masaki.yamamoto@kuis.kyoto-u.ac.jp Abstract Gopalan et al. studied in ICALP06 =-=[17]-=- connectivity properties of the solution-space of Boolean formulas, and investigated complexity issues on the connectivity problems in Schaefer’s framework. A set S of logical relations is Schaefer if... |

11 | Dichotomy theorem for the generalize unique satisfiability problem - Juban - 1999 |

9 | Computing intersections of Horn theories for reasoning with models
- Eiter, Ibaraki, et al.
- 1999
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8 |
Defaults and relevance in model-based reasoning
- Khardon, Roth
- 1997
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5 | Generating all abductive explanations for queries on propositional Horn theories
- Eiter, Makino
- 2006
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