## System Z: a natural ordering of defaults with tractable applications to default reasoning (1990)

Citations: | 173 - 0 self |

### BibTeX

@INPROCEEDINGS{Pearl90systemz:,

author = {Judea Pearl},

title = {System Z: a natural ordering of defaults with tractable applications to default reasoning},

booktitle = {},

year = {1990},

pages = {121--135},

publisher = {Morgan Kaufmann}

}

### Years of Citing Articles

### OpenURL

### Abstract

Recent progress towards unifying the probabilistic and preferential models semantics for non-monotonic reasoning has led to a remarkable observation: Any consistent system of default rules imposes an unambiguous and natural ordering on these rules which, to emphasize its sim-ple and basic character, we term "Z-ordering. " This ordering can be used with various levels of refinement, to prioritize conflicting arguments, to rank the degree of abnormality of states of the world, and to define plausible consequence relationships. This paper defines the Z-ordering, briefly mentions its semantical origins, and iUustrates two simple entailment relationships in-duced by the ordering. Two extensions are then described, maximum-entropy and conditional entailment, which trade in computational simplicity for semantic refinements. 1. Description We begin with a set of rules R = {r: %. ~ 6,-} where % and [~r are propositional formulas over a finite alphabet of literals, and--o denotes a new connective to be given default interpretations later on. A truth valuation of the fiterals in the language will be called a model. A model M is said to verify a rule ot ~ ifM ~ot ^ [3(i.e., o~and ~ are both true in M), and to falsify ot ~ ~ifM ~A ~ 13. Given a set R of such rules, we first define the relation of toleration.

### Citations

7493 |
Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference
- Pearl
- 1988
(Show Context)
Citation Context ..._ R there is at least one rule that is tolerated by all the others, i.e., VR" c R, ~ r" ~ R ", such that T (r' I R' - r') (1) This definition, named p -consistent in [Adams 1975] and e-consistent in [=-=Pearl 1988-=-], assures the existence of an admissible probability assignment when rules are given a probabilistic interpretation. In other words, if each rule t~--4 [3 is interpreted as a statement of high condit... |

241 | What does a conditional knowledge base entail - Lehmann - 1992 |

219 |
Probability and the logic of conditionals
- Adams
- 1966
(Show Context)
Citation Context ... for every non-empty subset R ' c_ R there is at least one rule that is tolerated by all the others, i.e., VR" c R, ~ r" ~ R ", such that T (r' I R' - r') (1) This definition, named p -consistent in [=-=Adams 1975-=-] and e-consistent in [Pearl 1988], assures the existence of an admissible probability assignment when rules are given a probabilistic interpretation. In other words, if each rule t~--4 [3 is interpre... |

145 |
Default Reasoning, Nonmonotonic Logics, and the Frame Problem
- Hanks, McDermott
- 1986
(Show Context)
Citation Context ...~ b and a ~ c. The failure to respond to causal information (see Pearl [1988, pp. 463, 519] and Hunter [1989]) prevents the ME approach from properly handling tasks such as the Yale shooting problem [=-=Hanks and McDermott 1986-=-], where rules of causal character are given priority over other rules. This weakness may perhaps be overcome by introducing causal operators into the ME formulation, similar to the way causal operato... |

95 |
Default reasoning: causal and conditional theories
- Geffner
- 1992
(Show Context)
Citation Context ...entropy approach [Goldszmidt and Pearl 1989c] where each model is ranked by the sum of weights on the rules falsified by that model Another refinement is provided by Geffner's conditional entailment [=-=Geffner 1989-=-], where the priority of rules induces a partial order on models. These two refinements will be summarized next. 4. The Maximum Entropy Approach The maximum-entropy (ME) approach [Pearl 1988] is motiv... |

89 |
Probabilistic semantics for nonmonotonic reasoning: A survey
- Pearl
- 1989
(Show Context)
Citation Context ...t preferred models of tx [Lehmann and Magidor 1988]. A slightly more elaborate definition of consistency applies to databases containing mixtures of defeasible and nondefeasible rules [Goldszmidt and =-=Pearl 1989-=-a]. Note that the condition of consistency is stronger than that of mere satisfiability. For example, the two rules a ~ b and a ~ ---, b are satisfiable (if a is false) but not consistent. Intuitively... |

79 |
A maximum entropy approach to nonmonotonic reasoning
- Goldszmidt, Morris, et al.
- 1993
(Show Context)
Citation Context ... in all the most preferred models of tx [Lehmann and Magidor 1988]. A slightly more elaborate definition of consistency applies to databases containing mixtures of defeasible and nondefeasible rules [=-=Goldszmidt and Pearl 1989-=-a]. Note that the condition of consistency is stronger than that of mere satisfiability. For example, the two rules a ~ b and a ~ ---, b are satisfiable (if a is false) but not consistent. Intuitively... |

30 |
On the consistency of defeasible databases
- Goldszmidt, Pearl
- 1991
(Show Context)
Citation Context ... in all the most preferred models of tx [Lehmann and Magidor 1988]. A slightly more elaborate definition of consistency applies to databases containing mixtures of defeasible and nondefeasible rules [=-=Goldszmidt and Pearl 1989-=-a]. Note that the condition of consistency is stronger than that of mere satisfiability. For example, the two rules a ~ b and a ~ ---, b are satisfiable (if a is false) but not consistent. Intuitively... |

23 | Chronological ignorance: Time, nonmonotonicity, necessity and causal theories - Shoham - 1986 |

17 | Causality and maximum entropy updating - Hunter - 1989 |

5 |
Rational logics and their models; A study in cumulative logics
- Lehmann, Magidor
- 1988
(Show Context)
Citation Context ...so assures the existence of an admissible preference ranking on models, when each rule o~ ~ 13 is given a preferential model interpretation, namely, 13 is true in all the most preferred models of tx [=-=Lehmann and Magidor 1988-=-]. A slightly more elaborate definition of consistency applies to databases containing mixtures of defeasible and nondefeasible rules [Goldszmidt and Pearl 1989a]. Note that the condition of consisten... |

1 |
On the relation between rational closure
- Goldszmidt, Pearl
- 1989
(Show Context)
Citation Context ... in all the most preferred models of tx [Lehmann and Magidor 1988]. A slightly more elaborate definition of consistency applies to databases containing mixtures of defeasible and nondefeasible rules [=-=Goldszmidt and Pearl 1989-=-a]. Note that the condition of consistency is stronger than that of mere satisfiability. For example, the two rules a ~ b and a ~ ---, b are satisfiable (if a is false) but not consistent. Intuitively... |