## Propositional Semantics for Disjunctive Logic Programs (1994)

Venue: | Annals of Mathematics and Artificial Intelligence |

Citations: | 145 - 2 self |

### BibTeX

@ARTICLE{Ben-eliyahu94propositionalsemantics,

author = {Rachel Ben-eliyahu and Rina Dechter},

title = {Propositional Semantics for Disjunctive Logic Programs},

journal = {Annals of Mathematics and Artificial Intelligence},

year = {1994},

volume = {12},

pages = {53--87}

}

### Years of Citing Articles

### OpenURL

### Abstract

In this paper we study the properties of the class of head-cycle-free extended disjunctive logic programs (HEDLPs), which includes, as a special case, all nondisjunctive extended logic programs. We show that any propositional HEDLP can be mapped in polynomial time into a propositional theory such that each model of the latter corresponds to an answer set, as defined by stable model semantics, of the former. Using this mapping, we show that many queries over HEDLPs can be determined by solving propositional satisfiability problems. Our mapping has several important implications: It establishes the NP-completeness of this class of disjunctive logic programs; it allows existing algorithms and tractable subsets for the satisfiability problem to be used in logic programming; it facilitates evaluation of the expressive power of disjunctive logic programs; and it leads to the discovery of useful similarities between stable model semantics and Clark's predicate completion. 1 Introduction ...

### Citations

1423 |
A Logic for Default Reasoning
- Reiter
- 1980
(Show Context)
Citation Context ...They then generalized stable model semantics for the class of extended logic programs. One advantage of stable model semantics is that it is closely related to the semantics of Reiter's default logic =-=[Rei80]-=-, and within the framework of default logic an extended logic program may be viewed as a default theory with special features. This relationship implies that insights, techniques, and analytical resul... |

992 | Depth first search and linear graph algorithms
- Tarjan
- 1972
(Show Context)
Citation Context ...odes such that, for each subset C in the partition and for each x; y 2 C, there are directed paths from x to y and from y to x in G. The strongly connected components can be identified in linear time =-=[Tar72]-=-. Thus, once again we realize that if the HEDLP is acyclic, we do not need any indexing. The above ideas are summarized in the following theorem, which is a restricted version of Theorem 2.8 for the c... |

960 |
Negation as failure
- Clark
- 1978
(Show Context)
Citation Context ...we deal with a special case of recursive disjunctive logic programs, namely, disjunctive logic programs that use rules that are not head-cycle-free. 4.6 Relation to Clark's predicate completion Clark =-=[Cla78] made one -=-of the first attempts to give meaning to logic programs with negated atoms in a rule's body ("normal programs"). He shows how each normal program \Pi can be associated with a first-order the... |

943 | Temporal constraint networks
- Dechter, Meiri, et al.
- 1991
(Show Context)
Citation Context ...at is mapped into a tractable subset of satisfiability is tractable as well. Among other possibilities, we can apply techniques developed in the constraints satisfaction literature (for a survey, see =-=[Dec92]-=-) to solve satisfiability and to identify tractable classes. A constraint satisfaction problem (CSP) consists of a set of n variables X 1 ; :::; X n , their respective domain values R 1 ; :::; R n , a... |

847 | Classical negation in logic programs and disjunctive databases
- Gelfond, Lifschitz
- 1991
(Show Context)
Citation Context ...ntroduction Stable model semantics for logic programs [BF87, Fin89, GL91] successfully bridges the gap between two lines of research --- default reasoning and logic programming. Gelfond and Lifschitz =-=[GL91]-=- pointed out the need for explicit representation of negated information in logic programs and accordingly definedsextended logic programs as those that use classical negation in addition to the negat... |

610 | Towards a theory of declarative knowledge - Apt, Blair, et al. - 1988 |

389 |
Network-based heuristics for constraint-satisfaction problems
- Dechter, Pearl
- 1987
(Show Context)
Citation Context ...ful case is when the constraint graph is a tree (no cycles). In this case, we can find whether there is a solution in time O(nk 2 ) where k is the maximal domain size and n is the number of variables =-=[DP88]-=-. The efficiency by which a tree problem can be solved serves as the basis for identifying many topologically based tractable classes of CSPs. It was shown that various other graph parameters are indi... |

337 | Graph algorithms - Even - 1979 |

327 |
Complexity of finding embeddings in a k-tree
- Arnborg, Corneil, et al.
- 1987
(Show Context)
Citation Context .... Consequently, the structuring process in tree-clustering is equipped with heuristics for generating clusters that are as small possible. Finding the optimal clustering scheme is known to be NP-hard =-=[ACP87]-=-, but good tractable approximations are available. Step 1 works by embedding the constraint graph of the problem within a chordal graph, because the maximal cliques of a chordal graph interact in a tr... |

271 |
Enhancement schemes for constraint processing: backjumping, learning, and cutset decomposition
- Dechter
- 1990
(Show Context)
Citation Context ... ahead of time. In this paper we choose to demonstrate the effectiveness of graph-based methods through two algorithms: one known as tree-clustering [DP89] and the other as cycle-cutset decomposition =-=[Dec90]-=-. The algorithms use different approaches for extending the class of tree-like problems. The following two subsections briefly describe the algorithms and quote relevant results. To avoid duplicating ... |

251 |
Tree Clustering for Constraint Networks
- Dechter, Pearl
- 1989
(Show Context)
Citation Context ...e parameters can be used for assessing tractability ahead of time. In this paper we choose to demonstrate the effectiveness of graph-based methods through two algorithms: one known as tree-clustering =-=[DP89]-=- and the other as cycle-cutset decomposition [Dec90]. The algorithms use different approaches for extending the class of tree-like problems. The following two subsections briefly describe the algorith... |

164 | A Sufficient Condition for Backtrack-bounded Search - Freuder - 1985 |

142 | Consistency of clark's completion and existence of stable models
- Fages
- 1994
(Show Context)
Citation Context ... completion of the program can be extended to be a model for this theory. The above example can easily be generalized to a proof of the following theorem, which was also proved independently by Fages =-=[Fag92]-=-: Theorem 4.19 Let \Pi be a normal acyclic propositional logic program. Then M is a model for COMP (\Pi) iff fI P jP 2 Mg is a model for T \Pi . 2 Proof: (sketch) Let \Pi be an acyclic normal logic pr... |

67 | problems for simple default logics - Hard - 1991 |

64 | On the declarative and procedural semantics of logic programs - Przymusinski - 1995 |

55 | A fixpoint semantics for disjunctive logic programs - Minker, Rajasekar - 1990 |

43 |
A rational reconstruction of nonmonotonic truth maintenance systems
- Elkan
- 1990
(Show Context)
Citation Context ...esults established in this paper with respect to disjunctive logic programs also apply to disjunctive default programs. 10 Note the appearance of E in the condition. 29 6 Related Work Elkan has shown =-=[Elk90]-=- that stable models of normal logic programs can be represented as models of propositional logic. Our results extend his work to a more expressive class of logic programs and provide an explicit propo... |

40 | Minimalism subsumes default logic and circumscription - Bidoit, Froidevaux - 1987 |

37 |
The relationship between logic program semantics and non{monotonic reasoning
- Marek, Subrahmanian
(Show Context)
Citation Context ...ong to any of the answer sets of \Pi (as defined by stable model semantics) iff COMP (\Pi) j= :P . 2 It is already known that each stable model for a normal logic program is a model of its completion =-=[MS89]-=- and that if an atom is implied by the completion of a locally stratified normal program, then it belongs to its (unique) answer set [ABW88, Prz89]. We believe that the above observations are new beca... |

31 | Complexity results for disjunctive logic programming and application to nonmonotonic logics
- Eiter, Gottlob
- 1993
(Show Context)
Citation Context ...ble model semantics for the class of all extended disjunctive logic programs (the class EDLPs) can be expressed in propositional logic has been regarded as an open problem. However, Eiter and Gottlob =-=[EG92]-=- have shown that the existence and set-membership problems for the class EDLPs is \Sigma P 2 complete and that set entailment for this class is \Pi P 2 complete. Therefore, Eiter and Gottlob conclude:... |

31 | C.: WFS + Branch and Bound = Stable Models - Subrahmanian, Nau, et al. - 1995 |

27 |
It’s not my default: The complexity of membership problems in restricted propositional default logics
- Stillman
- 1990
(Show Context)
Citation Context ...s techniques and complexity results obtained for default logic to be applied to logic programming, and vice versa. For example, the complexity results obtained by Kautz and Selman [KS91] and Stillman =-=[Sti90]-=- for default logic show that the satisfiability problem is polynomially reducible to deciding answer set existence and membership in a subset of extended logic programs 7 and that entailment in propos... |

24 | Default logic, propositional logic and constraints - Ben-Eliyahu, Dechter - 1991 |

14 | and Miroslaw Truszczyński. Autoepistemic logic - Marek - 1991 |

12 | The justification of negation as failure - Fine - 1989 |

11 | Computation and implementation of non-monotonic deductive databases - Bell, Nerode, et al. - 1991 |

5 |
Miroslaw Truszczy'nski. Computing intersection of autoepistemic expansions
- Marek
- 1991
(Show Context)
Citation Context ...isms and how it can be used to transfer computational techniques from propositional satisfiability to logic programs. An approach similar to ours was developed independently by Marek and Truszczynski =-=[MT91b]-=- in the context of autoepistemic logic. They have shown how questions of membership in expansions of an autoepistemic theory can be reduced to propositional provability. In this paper we provide an ex... |

4 |
Miroslaw Truszczy'nski. Disjunctive defaults
- Gelfond, Przymusinska, et al.
- 1991
(Show Context)
Citation Context ...ressive than nondisjunctive rules? Can we find a nondisjunctive theory for each disjunctive theory such that they have the same answer sets/extensions? This question has been raised by Gelfondset al. =-=[GPLT91]-=-. They consider translating a disjunctive logic program \Pi into a nondisjunctive program \Pi 0 by replacing each rule of the form (1) (see page 3 above) with k new rules: L 1 /\GammaL k+1 ; :::; L k+... |

2 | Inference in inheritance networks using propositional logic and constraints networks techniques - Ben-Eliyahu, Dechter - 1992 |