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Daniel Lehmann

Publications and source records attributed to Daniel Lehmann.

At least 55 records · Page 3Linked to original sources
arXiv2002

Representing and Aggregating Conflicting Beliefs

We consider the two-fold problem of representing collective beliefs and aggregating these beliefs. We propose modular, transitive relations for collective beliefs. They allow us to represent conflicting opinions and they have a clear semantics. We compare them with the quasi-transitive relations often used in Social Choice. Then, we describe a way to construct the belief state of an agent informed by a set of sources of varying degrees of reliability. This construction circumvents Arrow's Impossibility Theorem in a satisfactory manner. Finally, we give a simple set-theory-based operator for combining the information of multiple agents. We show that this operator satisfies the desirable invariants of idempotence, commutativity, and associativity, and, thus, is well-behaved when iterated, and we describe a computationally effective way of computing the resulting belief state.

cs.AI↗
arXiv2002

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.

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arXiv2002

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.

cs.AI↗
arXiv2002

Deductive Nonmonotonic Inference Operations: Antitonic Representations

We provide a characterization of those nonmonotonic inference operations C for which C(X) may be described as the set of all logical consequences of X together with some set of additional assumptions S(X) that depends anti-monotonically on X (i.e., X is a subset of Y implies that S(Y) is a subset of S(X)). The operations represented are exactly characterized in terms of properties most of which have been studied in Freund-Lehmann(cs.AI/0202031). Similar characterizations of right-absorbing and cumulative operations are also provided. For cumulative operations, our results fit in closely with those of Freund. We then discuss extending finitary operations to infinitary operations in a canonical way and discuss co-compactness properties. Our results provide a satisfactory notion of pseudo-compactness, generalizing to deductive nonmonotonic operations the notion of compactness for monotonic operations. They also provide an alternative, more elegant and more general, proof of the existence of an infinitary deductive extension for any finitary deductive operation (Theorem 7.9 of Freund-Lehmann).

cs.AI↗
arXiv2002

Expected Qualitative Utility Maximization

A model for decision making that generalizes Expected Utility Maximization is presented. This model, Expected Qualitative Utility Maximization, encompasses the Maximin criterion. It relaxes both the Independence and the Continuity postulates. Its main ingredient is the definition of a qualitative order on nonstandard models of the real numbers and the consideration of nonstandard utilities. Expected Qualitative Utility Maximization is characterized by an original weakening of von Neumann-Morgenstern's postulates. Subjective probabilities may be defined from those weakened postulates, as Anscombe and Aumann did from the original postulates. Subjective probabilities are numbers, not matrices as in the Subjective Expected Lexicographic Utility approach. JEL no.: D81 Keywords: Utility Theory, Non-Standard Utilities, Qualitative Decision Theory

cs.GT↗
arXiv2002

Classes of service under perfect competition and technological change: a model for the dynamics of the Internet?

Certain services may be provided in a continuous, one-dimensional, ordered range of different qualities and a customer requiring a service of quality q can only be offered a quality superior or equal to q. Only a discrete set of different qualities will be offered, and a service provider will provide the same service (of fixed quality b) to all customers requesting qualities of service inferior or equal to b. Assuming all services (of quality b) are priced identically, a monopolist will choose the qualities of service and the prices that maximize profit but, under perfect competition, a service provider will choose the (inferior) quality of service that can be priced at the lowest price. Assuming significant economies of scale, two fundamentally different regimes are possible: either a number of different classes of service are offered (DC regime), or a unique class of service offers an unbounded quality of service (UC regime). The DC regime appears in one of two sub-regimes: one, BDC, in which a finite number of classes is offered, the qualities of service offered are bounded and requests for high-quality services are not met, or UDC in which an infinite number of classes of service are offered and every request is met. The types of the demand curve and of the economies of scale, not the pace of technological change, determine the regime and the class boundaries. The price structure in the DC regime obeys very general laws.

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arXiv2002

Nonstandard numbers for qualitative decision making

The consideration of nonstandard models of the real numbers and the definition of a qualitative ordering on those models provides a generalization of the principle of maximization of expected utility. It enables the decider to assign probabilities of different orders of magnitude to different events or to assign utilities of different orders of magnitude to different outcomes. The properties of this generalized notion of rationality are studied in the frameworks proposed by von Neumann and Morgenstern and later by Anscombe and Aumann. It is characterized by an original weakening of their postulates in two different situations: nonstandard probabilities and standard utilities on one hand and standard probabilities and nonstandard utilities on the other hand. This weakening concerns both Independence and Continuity. It is orthogonal with the weakening proposed by lexicographic orderings.

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arXiv2002

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.

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arXiv2002

Nonmonotonic inference operations

A. Tarski proposed the study of infinitary consequence operations as the central topic of mathematical logic. He considered monotonicity to be a property of all such operations. In this paper, we weaken the monotonicity requirement and consider more general operations, inference operations. These operations describe the nonmonotonic logics both humans and machines seem to be using when infering defeasible information from incomplete knowledge. We single out a number of interesting families of inference operations. This study of infinitary inference operations is inspired by the results of Kraus, Lehmann and Magidor on finitary nonmonotonic operations, but this paper is self-contained.

cs.AI↗
arXiv2002

Optimal Solutions for Multi-Unit Combinatorial Auctions: Branch and Bound Heuristics

Finding optimal solutions for multi-unit combinatorial auctions is a hard problem and finding approximations to the optimal solution is also hard. We investigate the use of Branch-and-Bound techniques: they require both a way to bound from above the value of the best allocation and a good criterion to decide which bids are to be tried first. Different methods for efficiently bounding from above the value of the best allocation are considered. Theoretical original results characterize the best approximation ratio and the ordering criterion that provides it. We suggest to use this criterion.

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arXiv2002

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.

cs.AI↗
arXiv2002

Covariance Plasticity and Regulated Criticality

We propose that a regulation mechanism based on Hebbian covariance plasticity may cause the brain to operate near criticality. We analyze the effect of such a regulation on the dynamics of a network with excitatory and inhibitory neurons and uniform connectivity within and across the two populations. We show that, under broad conditions, the system converges to a critical state lying at the common boundary of three regions in parameter space; these correspond to three modes of behavior: high activity, low activity, oscillation.

cs.NE↗
arXiv2002

Nonmonotonic Reasoning, Preferential Models and Cumulative Logics

Many systems that exhibit nonmonotonic behavior have been described and studied already in the literature. The general notion of nonmonotonic reasoning, though, has almost always been described only negatively, by the property it does not enjoy, i.e. monotonicity. We study here general patterns of nonmonotonic reasoning and try to isolate properties that could help us map the field of nonmonotonic reasoning by reference to positive properties. We concentrate on a number of families of nonmonotonic consequence relations, defined in the style of Gentzen. Both proof-theoretic and semantic points of view are developed in parallel. The former point of view was pioneered by D. Gabbay, while the latter has been advocated by Y. Shoham in. Five such families are defined and characterized by representation theorems, relating the two points of view. One of the families of interest, that of preferential relations, turns out to have been studied by E. Adams. The "preferential" models proposed here are a much stronger tool than Adams' probabilistic semantics. The basic language used in this paper is that of propositional logic. The extension of our results to first order predicate calculi and the study of the computational complexity of the decision problems described in this paper will be treated in another paper.

cs.AI↗
arXiv2002

What does a conditional knowledge base entail?

This paper presents a logical approach to nonmonotonic reasoning based on the notion of a nonmonotonic consequence relation. A conditional knowledge base, consisting of a set of conditional assertions of the type "if ... then ...", represents the explicit defeasible knowledge an agent has about the way the world generally behaves. We look for a plausible definition of the set of all conditional assertions entailed by a conditional knowledge base. In a previous paper, S. Kraus and the authors defined and studied "preferential" consequence relations. They noticed that not all preferential relations could be considered as reasonable inference procedures. This paper studies a more restricted class of consequence relations, "rational" relations. It is argued that any reasonable nonmonotonic inference procedure should define a rational relation. It is shown that the rational relations are exactly those that may be represented by a "ranked" preferential model, or by a (non-standard) probabilistic model. The rational closure of a conditional knowledge base is defined and shown to provide an attractive answer to the question of the title. Global properties of this closure operation are proved: it is a cumulative operation. It is also computationally tractable. This paper assumes the underlying language is propositional.

cs.AI↗
arXiv2002

Distance Semantics for Belief Revision

A vast and interesting family of natural semantics for belief revision is defined. Suppose one is given a distance d between any two models. One may then define the revision of a theory K by a formula a as the theory defined by the set of all those models of a that are closest, by d, to the set of models of K. This family is characterized by a set of rationality postulates that extends the AGM postulates. The new postulates describe properties of iterated revisions.

cs.AI↗
arXiv2002

Preferred History Semantics for Iterated Updates

We give a semantics to iterated update by a preference relation on possible developments. An iterated update is a sequence of formulas, giving (incomplete) information about successive states of the world. A development is a sequence of models, describing a possible trajectory through time. We assume a principle of inertia and prefer those developments, which are compatible with the information, and avoid unnecessary changes. The logical properties of the updates defined in this way are considered, and a representation result is proved.

cs.AI↗
arXiv2002

Linear Programming helps solving large multi-unit combinatorial auctions

Previous works suggested the use of Branch and Bound techniques for finding the optimal allocation in (multi-unit) combinatorial auctions. They remarked that Linear Programming could provide a good upper-bound to the optimal allocation, but they went on using lighter and less tight upper-bound heuristics, on the ground that LP was too time-consuming to be used repetitively to solve large combinatorial auctions. We present the results of extensive experiments solving large (multi-unit) combinatorial auctions generated according to distributions proposed by different researchers. Our surprising conclusion is that Linear Programming is worth using. Investing almost all of one's computing time in using LP to bound from above the value of the optimal solution in order to prune aggressively pays off. We present a way to save on the number of calls to the LP routine and experimental results comparing different heuristics for choosing the bid to be considered next. Those results show that the ordering based on the square root of the size of the bids that was shown to be theoretically optimal in a previous paper by the authors performs surprisingly better than others in practice. Choosing to deal first with the bid with largest coefficient (typically 1) in the optimal solution of the relaxed LP problem, is also a good choice. The gap between the lower bound provided by greedy heuristics and the upper bound provided by LP is typically small and pruning is therefore extensive. For most distributions, auctions of a few hundred goods among a few thousand bids can be solved in practice. All experiments were run on a PC under Matlab.

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