## Probabilistic Deduction with Conditional Constraints over Basic Events (1999)

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Venue: | J. Artif. Intell. Res |

Citations: | 45 - 30 self |

### BibTeX

@ARTICLE{Lukasiewicz99probabilisticdeduction,

author = {Thomas Lukasiewicz},

title = {Probabilistic Deduction with Conditional Constraints over Basic Events},

journal = {J. Artif. Intell. Res},

year = {1999},

volume = {10},

pages = {380--391}

}

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### Abstract

We study the problem of probabilistic deduction with conditional constraints over basic events. We show that globally complete probabilistic deduction with conditional constraints over basic events is NP-hard. We then concentrate on the special case of probabilistic deduction in conditional constraint trees. We elaborate very efficient techniques for globally complete probabilistic deduction. In detail, for conditional constraint trees with point probabilities, we present a local approach to globally complete probabilistic deduction, which runs in linear time in the size of the conditional constraint trees. For conditional constraint trees with interval probabilities, we show that globally complete probabilistic deduction can be done in a global approach by solving nonlinear programs. We show how these nonlinear programs can be transformed into equivalent linear programs, which are solvable in polynomial time in the size of the conditional constraint trees. 1. Introduction Dealing wit...

### Citations

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Citation Context ...and Haddawy (1994) discuss the use of inference rules for deduction in probabilistic logic. Recent work on inference rules concentrates on integrating probabilistic knowledge into description logics (=-=Heinsohn, 1994-=-) and on analyzing the interplay between taxonomic and probabilistic deduction (Lukasiewicz 1998a, 1999a). We now summarize the main characteristics of the global and the local approach. The global ap... |

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Citation Context ...t al. (1991) identify special cases of probabilistic satisfiability that can be solved in polynomial time. Other work on the global approach concentrates on reducing the number of linear constraints (=-=Luo et al. 1996-=-) and the number of variables (Lukasiewicz, 1997). Finally, Fagin et al. (1992) present a sound and complete axiom system for reasoning about probabilities that are expressed by linear inequalities ov... |

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Citation Context ...listic satisfiability that can be solved in polynomial time. Other work on the global approach concentrates on reducing the number of linear constraints (Luo et al. 1996) and the number of variables (=-=Lukasiewicz, 1997-=-). Finally, Fagin et al. (1992) present a sound and complete axiom system for reasoning about probabilities that are expressed by linear inequalities over propositional events. They show that the sati... |

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- Lukasiewicz
- 1999
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Citation Context ...cent work on inference rules concentrates on integrating probabilistic knowledge into description logics (Heinsohn, 1994) and on analyzing the interplay between taxonomic and probabilistic deduction (=-=Lukasiewicz 1998-=-a, 1999a). We now summarize the main characteristics of the global and the local approach. The global approach can be performed within quite rich probabilistic languages (Fagin et al., 1992). Cruciall... |

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Citation Context ...ction technique of Section 4 can easily be generalized to conditional constraint trees (B; KB) that satisfy only the restriction u 1 ? 0 iff v 1 ? 0 for all (BjA)[u 1 ; u 2 ]; (AjB)[v 1 ; v 2 ] 2 KB (=-=Lukasiewicz, 1996-=-). The restriction that for each query 9(F jE)[x 1 ; x 2 ], all paths from a basic event in E to a basic event in F have at least one basic event in common is crucial for the deduction technique of Se... |

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Citation Context ...the deduced results are obtained. Mainly to overcome these deficiencies, researchers started to work on local techniques based on inference rules. The local approach (see, for example, [7], [9], [2], =-=[8]-=-, [28], [11], [15], and [18]) is generally performed within more restricted probabilistic languages. The iterative application of inference rules is very rarely and only within very restricted languag... |

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Citation Context ...ained. Mainly to overcome these deficiencies, researchers started to work on local techniques based on inference rules. The local approach (see, for example, [7], [9], [2], [8], [28], [11], [15], and =-=[18]-=-) is generally performed within more restricted probabilistic languages. The iterative application of inference rules is very rarely and only within very restricted languages globally complete (see [1... |

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Citation Context ...to model default reasoning with imprecise numerical and fuzzy quanti ers. For this reason, subsequent research on inference rules especially aims at analyzing patterns of human commonsense reasoning (=-=Dubois et al. 1990-=-, 1993; Amarger et al. 1991; Thone, 1994; Thone et al. 1995). Frisch and Haddawy (1994) discuss the use of inference rules for deduction in probabilistic logic. Recent work on inference rules concentr... |

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