## Anytime Deduction for Probabilistic Logic (1994)

Venue: | Artif. Intell |

Citations: | 62 - 1 self |

### BibTeX

@ARTICLE{Frisch94anytimededuction,

author = {Alan Frisch and Peter Haddawy},

title = {Anytime Deduction for Probabilistic Logic},

journal = {Artif. Intell},

year = {1994},

volume = {69},

pages = {93--122}

}

### Years of Citing Articles

### OpenURL

### Abstract

This paper proposes and investigates an approach to deduction in probabilistic logic, using as its medium a language that generalizes the propositional version of Nilsson's probabilistic logic by incorporating conditional probabilities. Unlike many other approaches to deduction in probabilistic logic, this approach is based on inference rules and therefore can produce proofs to explain how conclusions are drawn. We show how these rules can be incorporated into an anytime deduction procedure that proceeds by computing increasingly narrow probability intervals that contain the tightest entailed probability interval. Since the procedure can be stopped at any time to yield partial information concerning the probability range of any entailed sentence, one can make a tradeoff between precision and computation time. The deduction method presented here contrasts with other methods whose ability to perform logical reasoning is either limited or requires finding all truth assignments consistent ...

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Citation Context ...on of rule (xx) can be found in a paper by Dubois [11]. Rule (xxv) for introducing conjunction is also presented by Ursic [34]. Rules (xxiii) and (xxiv) accomplish a transformation similar to Pearl's =-=[29]-=- clustering technique for removing loops in Bayes nets. Rules (viii) and (ix) are respectively the the generalized Bayes' rule 10 and the rule of quantified syllogism, presented by Amarger, et.al. [2]... |

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Citation Context ... the procedure "anytime deduction" here because our principal focus is the ability to provide partial information at any time, regardless of whether convergence is ultimately attained. Dean =-=and Boddy [9]-=- coined the term anytime algorithm in the context of work that explored issues in time-dependent planning. They defined an anytime algorithm as one that i) lends itself to preemptive scheduling techni... |

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Citation Context ...t we have some notion of the marginal improvement in an answer as a function of a marginal increase in computation time. Since then researchers have presented anytime algorithms for other AI problems =-=[10; 37]-=-. The concept of an algorithm that produces approximate answers that improve with time has also appeared in the literature on data base theory [33; 8; 6; 36] and real-time systems [22; 21; 7]. In the ... |

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Citation Context ...lans, allowing a planning system to adapt flexibly under time constraints. Guidance in applying anytime deduction procedures to decision problems could be provided by Horvitz, Cooper, and Heckerman's =-=[20]-=- framework for reasoning about actions under time constraints. Their work explores how to optimally control methods of computing probability bounds in Bayes nets, showing how knowledge of the rate of ... |

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Citation Context ...ocedure. They further extend inferno's network approach with rules for introducing new nodes. These new nodes are limited to conjunctions and disjunctions of nodes in the network. Ng and Subrahmanian =-=[25; 24]-=- formulate a sound and complete inference system based on SLD deduction (as used in logic programming) for a class of probabilistic entailment problems that is almost totally disjoint from those addre... |

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Citation Context ... entailment. Bundy's approach, like Nilsson's, requires performing an initial impractical computation. Another approach to computing probabilistic entailment is Quinlan's network propagation approach =-=[31]-=-. This method has its own limitations. Quinlan's method works with a static network representing a fixed set of formulas and relationships among them. Because it is unable to form new formulas as is d... |

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Citation Context ...light generalizations of rules that appear in the literature in the sense that we have added a fixed conditioning proposition ffi. Rules (xx) and (xxi) are presented by Garvey, Lowrance, and Fischler =-=[13]-=-. A derivation of rule (xx) can be found in a paper by Dubois [11]. Rule (xxv) for introducing conjunction is also presented by Ursic [34]. Rules (xxiii) and (xxiv) accomplish a transformation similar... |

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Citation Context ...lanation-based learning algorithms. These drawbacks are inherited by subsequently developed elaborations of Nilsson's basic approach [35; 32]. A related approach to computing probabilities is Bundy's =-=[4]-=- incidence calculus. This approach was proposed prior to the publication of Nilsson's probabilistic logic and so was not described in terms of his semantics. But if viewed in terms of the semantics of... |

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Citation Context ...entions some of the subsequent research on this topic. 2 For an earlier investigation of probabilistic entailment for both propositional and first-order logic see the article by Adams and Levine [1]. =-=[2]-=- extend Quinlan's approach with rules for the derivation of conditional probabilities, as well as means of introducing new nodes. However, the new nodes are limited to conjunctions and disjunctions of... |

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Citation Context ...inlan's [31] inferno system is also an anytime deduction procedure but he did not present or discuss it as such. 13 In 1986 Haddawy [15] presented an inference system based on Michalski and Winston's =-=[23]-=- Variable Precision Logic that could vary the precision of its inference to produce an answer to a given query within a specified amount of time. That system, however, lacks the important property tha... |

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Citation Context ...t we have some notion of the marginal improvement in an answer as a function of a marginal increase in computation time. Since then researchers have presented anytime algorithms for other AI problems =-=[10; 37]-=-. The concept of an algorithm that produces approximate answers that improve with time has also appeared in the literature on data base theory [33; 8; 6; 36] and real-time systems [22; 21; 7]. In the ... |

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Citation Context ...esented anytime algorithms for other AI problems [10; 37]. The concept of an algorithm that produces approximate answers that improve with time has also appeared in the literature on data base theory =-=[33; 8; 6; 36]-=- and real-time systems [22; 21; 7]. In the literature on real-time systems, such algorithms are known as imprecise, monotone algorithms. Concurrent with Dean and Boddy's work, Horvitz [19] presented a... |

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Citation Context ...line of reasoning involving formulas other than those built into the network. The network can only propagate probability constraints among its existing formulas. Amarger, Dubois and Prade 1 Nilsson's =-=[26]-=- retrospective on the paper mentions some of the subsequent research on this topic. 2 For an earlier investigation of probabilistic entailment for both propositional and first-order logic see the arti... |

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Citation Context ... is a closed interval. This property does not hold for point probabilities; though the sentences in \Psi may all have point probabilities, the tight entailment of (OE j ��) may not be a point. Gro=-=sof [14]-=- also observed this closure property in relation to his Type-1-ui theories. This property is important for the formulation of inference rules since the rules must be applicable to the sentences they d... |

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Citation Context ...esented anytime algorithms for other AI problems [10; 37]. The concept of an algorithm that produces approximate answers that improve with time has also appeared in the literature on data base theory =-=[33; 8; 6; 36]-=- and real-time systems [22; 21; 7]. In the literature on real-time systems, such algorithms are known as imprecise, monotone algorithms. Concurrent with Dean and Boddy's work, Horvitz [19] presented a... |

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Citation Context ...s on the probabilities of the associated sentences. This algorithm fits our definition of an anytime deduction procedure and is shown to be both sound and complete for unconditional sentences of L PL =-=[5]-=-. The major drawback of Bundy's approach is the lack of a general mechanism for assigning the initial incidences to a set of sentences, given their probabilities. The difficulty arises because each po... |

10 |
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Citation Context ...per mentions some of the subsequent research on this topic. 2 For an earlier investigation of probabilistic entailment for both propositional and first-order logic see the article by Adams and Levine =-=[1]-=-. [2] extend Quinlan's approach with rules for the derivation of conditional probabilities, as well as means of introducing new nodes. However, the new nodes are limited to conjunctions and disjunctio... |

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Citation Context ...uch proofs are useful as explanations to humans and as input to explanation-based learning algorithms. These drawbacks are inherited by subsequently developed elaborations of Nilsson's basic approach =-=[35; 32]-=-. A related approach to computing probabilities is Bundy's [4] incidence calculus. This approach was proposed prior to the publication of Nilsson's probabilistic logic and so was not described in term... |

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Citation Context ...hese assignments are then used to set up a system of linear equations that expresses the constraints among the probability distribution over these assignments and the probabilities of LC(\Psi). Paass =-=[28]-=- generalizes this method to allow \Psi to contain conditional probability sentences with arbitrary probability intervals. The target sentence must still be an unconditional probability. This more gene... |

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Citation Context ...e sense that we have added a fixed conditioning proposition ffi. Rules (xx) and (xxi) are presented by Garvey, Lowrance, and Fischler [13]. A derivation of rule (xx) can be found in a paper by Dubois =-=[11]-=-. Rule (xxv) for introducing conjunction is also presented by Ursic [34]. Rules (xxiii) and (xxiv) accomplish a transformation similar to Pearl's [29] clustering technique for removing loops in Bayes ... |

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Citation Context ...xx) and (xxi) are presented by Garvey, Lowrance, and Fischler [13]. A derivation of rule (xx) can be found in a paper by Dubois [11]. Rule (xxv) for introducing conjunction is also presented by Ursic =-=[34]-=-. Rules (xxiii) and (xxiv) accomplish a transformation similar to Pearl's [29] clustering technique for removing loops in Bayes nets. Rules (viii) and (ix) are respectively the the generalized Bayes' ... |

1 |
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Citation Context ...ince P(ff j fi) = P(ffsfi)=P(fi), he suggests using the matrix 12 Quinlan's [31] inferno system is also an anytime deduction procedure but he did not present or discuss it as such. 13 In 1986 Haddawy =-=[15]-=- presented an inference system based on Michalski and Winston's [23] Variable Precision Logic that could vary the precision of its inference to produce an answer to a given query within a specified am... |

1 |
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Citation Context ...ocedure. They further extend inferno's network approach with rules for introducing new nodes. These new nodes are limited to conjunctions and disjunctions of nodes in the network. Ng and Subrahmanian =-=[25; 24]-=- formulate a sound and complete inference system based on SLD deduction (as used in logic programming) for a class of probabilistic entailment problems that is almost totally disjoint from those addre... |

1 | Probabilities that imply certainties - Shvaytser - 1990 |