Stereotypical Reasoning: Logical Properties

Stereotypical reasoning assumes that the situation at hand is one of a kind and that it enjoys the properties generally associated with that kind of situation. It is one of the most basic forms of nonmonotonic reasoning. A formal model for stereotypical reasoning is proposed and the logical properties of this form of reasoning are studied. Stereotypical reasoning is shown to be cumulative under weak assumptions.Comment: Presented at Fourth Workshop on Logic, Language, Information and Computation, Fortaleza (Brasil), August 199

The logical meaning of Expansion

The Expansion property considered by researchers in Social Choice is shown to correspond to a logical property of nonmonotonic consequence relations that is the {\em pure}, i.e., not involving connectives, version of a previously known weak rationality condition. The assumption that the union of two definable sets of models is definable is needed for the soundness part of the result.Comment: 9 pages. Unpublishe

A presentation of Quantum Logic based on an "and then" connective

When a physicist performs a quantic measurement, new information about the system at hand is gathered. This paper studies the logical properties of how this new information is combined with previous information. It presents Quantum Logic as a propositional logic under two connectives: negation and the "and then" operation that combines old and new information. The "and then" connective is neither commutative nor associative. Many properties of this logic are exhibited, and some small elegant subset is shown to imply all the properties considered. No independence or completeness result is claimed. Classical physical systems are exactly characterized by the commutativity, the associativity, or the monotonicity of the "and then" connective. Entailment is defined in this logic and can be proved to be a partial order. In orthomodular lattices, the operation proposed by Finch (1969) satisfies all the properties studied in this paper. All properties satisfied by Finch's operation in modular lattices are valid in Hilbert Space Quantum Logic. It is not known whether all properties of Hilbert Space Quantum Logic are satisfied by Finch's operation in modular lattices. Non-commutative, non-associative algebraic structures generalizing Boolean algebras are defined, ideals are characterized and a homomorphism theorem is proved.Comment: 28 pages. Submitte

Some properties of n-party entanglement under LOCC operations

Nielsen characterized in full those 2-party quantum protocols of local operations and classical communication that transform, with probability one, a pure global initial state into a pure global final state. The present work considers the generalization of Nielsen's characterization to n-party protocols. It presents a sweeping generalization of the only if part of Nielsen's result. The result presented here pertains also to protocols that do not generate a final state for sure, it considers arbitrary mixed initial states instead of pure states and n-party protocols for an arbitrary n. In this very general setting, local operations and classical communication can never decrease the expected spectra of the local mixed states in the majorization ordering. In other terms, the local states can only become purer (weakly) in expectation. The proof also provides an improvement on Nielsen's. The if part of Nielsen's characterization does not generalize. This is shown by studying the entanglement of three qubits. It is shown that one can find pure states of a system of three qubits that are not equivalent under unitary local operations but define local mixed states on all subparts of the system that have the same spectra. Neither equivalence of pure states under local unitary operations nor accessibility under LOCC operations among a system of three qubits can be characterized by properties of the spectra of the local mixed states.Comment: 21 pages. Compare with Theorem 12 of Nielsen and Vidal QIC 1 (1) 200

Measurements and majorization

Majorization is an outstanding tool to compare the purity of mixed states or the amount of information they contain and also the degrees of entanglement presented by such states in tensor products. States are compared by their spectra and majorization defines a partial order on those. This paper studies the effect of measurements on the majorization relation among states. It, then, proceeds to study the effect of local measurements on the agents sharing an entangled global state. If the result of the measurement is recorded, Nielsen and Vidal showed that the expected spectrum after any P.O.V.M. measurement majorizes the initial spectrum, i.e., a P.O.V.M. measurement cannot, in expectation, reduce the information of the observer. A new proof of this result is presented and, as a consequence, the only if part of Nielsen's characterization of LOCC transformations is generalized to n-party entanglement. If the result of a bi-stochastic measurement is not recorded, the initial state majorizes the final state, i.e., no information may be gained by such a measurement. This strengthens a result of A. Peres. In the n-party setting, no local trace preserving measurement by Alice can change the local state of another agent.Comment: 10 page

Another perspective on Default Reasoning

The lexicographic closure of any given finite set D of normal defaults is defined. A conditional assertion "if a then b" is in this lexicographic closure if, given the defaults D and the fact a, one would conclude b. The lexicographic closure is essentially a rational extension of D, and of its rational closure, defined in a previous paper. It provides a logic of normal defaults that is different from the one proposed by R. Reiter and that is rich enough not to require the consideration of non-normal defaults. A large number of examples are provided to show that the lexicographic closure corresponds to the basic intuitions behind Reiter's logic of defaults.Comment: Presented at Workshop on Logical Formalizations of Commense Sense, Austin (Texas), January 199

Connectives in Quantum and other Cumulative Logics

Cumulative logics are studied in an abstract setting, i.e., without connectives, very much in the spirit of Makinson's early work. A powerful representation theorem characterizes those logics by choice functions that satisfy a weakening of Sen's property alpha, in the spirit of the author's "Nonmonotonic Logics and Semantics" (JLC). The representation results obtained are surprisingly smooth: in the completeness part the choice function may be defined on any set of worlds, not only definable sets and no definability-preservation property is required in the soundness part. For abstract cumulative logics, proper conjunction and negation may be defined. Contrary to the situation studied in "Nonmonotonic Logics and Semantics" no proper disjunction seems to be definable in general. The cumulative relations of KLM that satisfy some weakening of the consistency preservation property all define cumulative logics with a proper negation. Quantum Logics, as defined by Engesser and Gabbay are such cumulative logics but the negation defined by orthogonal complement does not provide a proper negation.Comment: 21 page

Ultra valuations

This paper proposes an original exchange property of valuations.This property is shown to be equivalent to a property described by Dress and Terhalle in the context of discrete optimization and matroids and shown there to characterize the valuations for which the demand oracle can be implemented by a greedy algorithm. The same exchange property is also equivalent to a property described independently by Reijnierse, van Gellekom and Potters and by Lehmann, Lehmann and Nisan and shown there to be satisfied by substitutes valuations. It studies the family of valuations that satisfy this exchange property, the ultra valuations. Any substitutes valuation is an ultra valuation, but ultra valuations may exhibit complementarities. Any symmetric valuation is an ultra valuation. Substitutes valuations are exactly the submodular ultra valuations. Ultra valuations define ultrametrics on the set of items. The maximum of an ultra valuation on $n$ items can be found in $O(n^2)$ steps. Finding an efficient allocation among ultra valuations is NP-hard.Comment: 31 pages, preprint. This is a fourth version. The first version has been enlarged and much improved. The relation to extant work and the overall presentation are much improved. Some new results are included. The fourth version has a better comparison with the choice-language properties and an added appendix on the topi

Generalized Qualitative Probability: Savage Revisited

Preferences among acts are analyzed in the style of L. Savage, but as partially ordered. The rationality postulates considered are weaker than Savage's on three counts. The Sure Thing Principle is derived in this setting. The postulates are shown to lead to a characterization of generalized qualitative probability that includes and blends both traditional qualitative probability and the ranked structures used in logical approaches.Comment: Appears in Proceedings of the Twelfth Conference on Uncertainty in Artificial Intelligence (UAI1996

Combinatorial Auctions with Decreasing Marginal Utilities

In most of microeconomic theory, consumers are assumed to exhibit decreasing marginal utilities. This paper considers combinatorial auctions among such submodular buyers. The valuations of such buyers are placed within a hierarchy of valuations that exhibit no complementarities, a hierarchy that includes also OR and XOR combinations of singleton valuations, and valuations satisfying the gross substitutes property. Those last valuations are shown to form a zero-measure subset of the submodular valuations that have positive measure. While we show that the allocation problem among submodular valuations is NP-hard, we present an efficient greedy 2-approximation algorithm for this case and generalize it to the case of limited complementarities. No such approximation algorithm exists in a setting allowing for arbitrary complementarities. Some results about strategic aspects of combinatorial auctions among players with decreasing marginal utilities are also presented.Comment: To appear in GEB. Preliminary version appeared in EC'0