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.

Graphical representations for probabilistic relationships have recently received considerable attention in A1. Qualitative probabilistic networks abstract from the usual numeric representations by encoding only qualitative relationships, which are inequality constraints on the joint probability distribution over the variables. Although these constraints are insufficient to determine probabilities uniquely, they are designed to justify the deduction of a class of relative likelihood conclusions that imply useful decision-making properties. Two types of qualitative relationship are defined, each a probabilistic form of monotonicity constraint over a group of variables. Qualitative influences describe the direction of the relationship between two variables. Qualitative synergies describe interactions among influences. The probabilistic definitions chosen justify sound and efficient inference procedures based on graphical manipulations of the network. These procedures answer queries about qualitative relationships among variables separated in the network and determine structural properties of optimal assignments to decision variables.

This paper investigates the appropriateness of formal dialectics as a basis for non-monotonic reasoning and defeasible reasoning that takes computational limits seriously. Rules that can come into conflict should be regarded as policies, which are inputs to deliberative processes. Dialectical protocols are appropriate for such deliberations when resources are bounded and search is serial. AI, it is claimed here, is now perfectly positioned to correct many misconceptions about reasoning that have resulted from mathematical logic's enormous success in this century: among them, (1) that all reasons are demonstrative, (2) that rational belief is constrained, not constructed, (3) that process and disputation are not essential to reasoning. AI mainly provides new impetus to formalize the alternative (but older) conception of reasoning, and AI provides mechanisms with which to create compelling formalism that describes the control of processes. The technical contributions here are: the partial justification of dialectic based on controlling search; the observation that non-monotonic reasoning can be subsumed under certain kinds of dialectics; the portrayal of inference in knowledge bases as policy reasoning; the review of logics of dialogue and proposed extensions; and the pre-formal and initial formal discussion of aspects and variations of dialectical systems with non-demonstrative reasons. 1. ARGUMENTS AND DEMONSTRATION

this paper: default reasoning. In recent years, a number of different semantics for defaults have been proposed, such as preferential structures, ffl-semantics, possibilistic structures, and -rankings, that have been shown to be characterized by the same set of axioms, known as the KLM properties. While this was viewed as a surprise, we show here that it is almost inevitable. In the framework of plausibility measures, we can give a necessary condition for the KLM axioms to be sound, and an additional condition necessary and sufficient to ensure that the KLM axioms are complete. This additional condition is so weak that it is almost always met whenever the axioms are sound. In particular, it is easily seen to hold for all the proposals made in the literature. Categories and Subject Descriptors: F.4.1 [Mathematical Logic and Formal Languages]:

Given a knowledge base KB containing first-order and statistical facts, we consider a principled method, called the random-worlds method, for computing a degree of belief that some formula ' holds given KB . If we are reasoning about a world or system consisting of N individuals, then we can consider all possible worlds, or first-order models, with domain f1; : : : ; Ng that satisfy KB , and compute the fraction of them in which ' is true. We define the degree of belief to be the asymptotic value of this fraction as N grows large. We show that when the vocabulary underlying ' and KB uses constants and unary predicates only, we can naturally associate an entropy with each world. As N grows larger, there are many more worlds with higher entropy. Therefore, we can use a maximum-entropy computation to compute the degree of belief. This result is in a similar spirit to previous work in physics and artificial intelligence, but is far more general. Of equal interest to the result itself are...

We describe a new approach to default reasoning, based on a principle of indifference among possible worlds. We interpret default rules as extreme statistical statements, thus obtaining a knowledge base KB comprised of statistical and first-order statements. We then assign equal probability to all worlds consistent with KB in order to assign a degree of belief to a statement '. The degree of belief can be used to decide whether to defeasibly conclude '. Various natural patterns of reasoning, such as a preference for more specific defaults, indifference to irrelevant information, and the ability to combine independent pieces of evidence, turn out to follow naturally from this technique. Furthermore, our approach is not restricted to default reasoning; it supports a spectrum of reasoning, from quantitative to qualitative. It is also related to other systems for default reasoning. In particular, we show that the work of [ Goldszmidt et al., 1990 ] , which applies maximum entropy ideas t...

It is suggested that taking into account considerations that traditionally fall within the scope of computer science in general. and artificial intelligence in particular, sheds new light on the subject of causation. It is orgued that adopting causal nations can be viewed as filling a computational need: They allow reasoning with incomplete information, facilitate economical representations, and afford relatively efficient methods for reasoning about those representations. Specif-ically, it is proposed that causal reasoning is intimately bound to nonmonotonic reasoning. An account of causation is offered that relies upon this connection, and compares this proposal to previous accounts within philosophy and artificial intelligence. 1.

Conditional logics, introduced by Lewis and Stalnaker, have been utilized in artificial intelligence to capture a broad range of phenomena. In this paper we examine the complexity of several variants discussed in the literature. We show that, in general, deciding satisfiability is PSPACE-complete for formulas with arbitrary conditional nesting and NP-complete for formulas with bounded nesting of conditionals. However, we provide several exceptions to this rule. Of particular note are results showing that (a) when assuming uniformity (i.e., that all worlds agree on what worlds are possible), the decision problem becomes EXPTIME-complete even for formulas with bounded nesting, and (b) when assuming absoluteness (i.e., that all worlds agree on all conditional statements), the decision problem is NP-complete for formulas with arbitrary nesting. 1 INTRODUCTION The study of conditional statements of the form "If : : : then : : :" has a long history in philosophy [Sta68, Lew73, Che80, Vel8...

We present an approach to reasoning from statistical and subjective knowledge, which is based on a combination of probabilistic reasoning from conditional constraints with approaches to default reasoning from conditional knowledge bases. More precisely, we introduce the notions of -, lexicographic, and conditional entailment for conditional constraints, which are probabilistic generalizations of Pearl's entailment in system , Lehmann's lexicographic entailment, and Geffner's conditional entailment, respectively. We show that the new formalisms have nice properties. In particular, they show a similar behavior as referenceclass reasoning in a number of uncontroversial examples. The new formalisms, however, also avoid many drawbacks of reference-class reasoning. More precisely, they can handle complex scenarios and even purely probabilistic subjective knowledge as input. Moreover, conclusions are drawn in a global way from all the available knowledge as a whole. We then show that the new formalisms also have nice general nonmonotonic properties. In detail, the new notions of -, lexicographic, and conditional entailment have similar properties as their classical counterparts. In particular, they all satisfy the rationality postulates proposed by Kraus, Lehmann, and Magidor, and they have some general irrelevance and direct inference properties. Moreover, the new notions of - and lexicographic entailment satisfy the property of rational monotonicity. Furthermore, the new notions of -, lexicographic, and conditional entailment are proper generalizations of both their classical counterparts and the classical notion of logical entailment for conditional constraints. Finally, we provide algorithms for reasoning under the new formalisms, and we analyze its computational com...

Although the usefulness of belief networks for reasoning under uncertainty is widely accepted, obtaining numerical probabilities that they require is still perceived a major obstacle. Often not enough statistical data is available to allow for reliable probability estimation. Available information may not be directly amenable for encoding in the network. Finally, domain experts may be reluctant to provide numerical probabilities. In this paper, we propose a method for elicitation of probabilities from a domain expert that is non-invasive and accommodates whatever probabilistic information the expert is willing to state. We express all available information, whether qualitative or quantitative in nature, in a canonical form consisting of (in)equalities expressing constraints on the hyperspace of possible joint probability distributions. We then use this canonical form to derive second-order probability distributions over the desired probabilities.