10307056
Intransitivity and Vagueness
There are many examples in the literature that suggest that indistinguishability is intransitive, despite the fact that the indistinguishability relation is typically taken to be an equivalence relation (and thus transitive). It is shown that if the uncertainty perception and the question of when an agent reports that two things are indistinguishable are both carefully modeled, the problems disappear, and indistinguishability can indeed be taken to be an equivalence relation. Moreover, this model also suggests a logic of vagueness that seems to solve many of the problems related to vagueness discussed in the philosophical literature. In particular, it is shown here how the logic can handle the Sorites Paradox.
152014
Causality, Responsibility, and Blame: A Structural-Model Approach
This talk provides an overview of work that I have done with Hana Chockler, Orna Kupferman, and Judea Pearl (H. Chockler and J. Y. Halpern, 2004), (H. Chockler, et al., 2003), (J. Y. Halpern and J. Pearl, 2005), on defining notions such as causality, explanation, responsibility, and blame. I first review the Halpern-Pearl definition of causality, what it means that A is a cause of B, and show how it handles well some standard problems of causality. This definition of causality (like most in the literature) views causality as an all-or-nothing concept. Either A is a cause of B or it is not. I show how it can be extended to take into account the degree of responsibility of A for B. For example, if someone wins an election 11-0, each person is less responsible for his victory than if he had won 6-5. Finally, I show how this notion of degree of responsibility can be used to provide insight into model checking notions such as coverage
215949262
Knowledge, probability, and adversaries
What should it mean for an agent to know or believe an assertion is true with probability 9.99? Different papers [2, 6, 15] give different answers, choosing to use quite different probability space...
15233558
A new look at fault tolerant network routing
Consider a communication network G in which a limited number of link and/or node faults F might occur. A routing &rgr; for the network (a fixed path between each pair of nodes) must be chosen without any knowledge of which components might become faulty. Choosing a good routing corresponds to bounding the diameter of the surviving route graph R(G,&rgr;)/F, where two nonfaulty nodes are joined by an edge if there are no faults on the route between them. We prove a number of results concerning the diameter of surviving route graphs. We show that if &rgr; is a minimal length routing, then the diameter of R(G,&rgr;)/F can be on the order of the number of nodes of G, even if F consists of only a single node. However, if G is the n-dimensional cube, the diameter of R(G,&rgr;)/F≤3 for any minimal length routing &rgr; and any set of faults F with |F|<n. We also show that if F consists only of edges and does not disconnect G, then the diameter of R(G,&rgr;)/F is ≤ 3|F|+1, while if F consists only of nodes and does not disconnect G, then the diameter of R(G,&rgr;)/F is ≤ the sum of the degrees of the nodes in F, where in both cases &rgr; is an arbitrary minimal length routing. We conclude with one of the most important contributions of this paper: a list of interesting and apparently difficult open problems.
14995360
A model-theoretic analysis of knowledge
A general semantic model of knowledge is introduced, to allow reasoning about statements such as «He knows that I know whether or not she knows whether or not it is raining». This approach more naturally models a state of knowledge than previous proposals (including Kripke structures). Using this notion of model, a model theory for knowledge is developed. This theory enables one to interpret the notion of a «finite amount of information»
47316226
Equations Between Regular Terms and an Application to Process Logic
Regular terms with the Kleene operations $ \cup $, ; and $ * $ can be thought of as operators on languages, generating other languages. An equation $\tau _1 = \tau _2 $ between two such terms is said to be satisfiable just in case languages exist which make this equation true. We show that the satisfiability problem even for $ * $-free regular terms is undecidable. Similar techniques are used to show that a very natural extension of the Process Logic of Harel, Kozen and Parikh is undecidable.
2990183
Performing work efficiently in the presence of faults
We consider a system of t synchronous processes that communicate only by sending messages to one another, and together the processes must perform n independent units of work. Processes may fail by crashing; we want to guarantee that in every execution of the protocol in which at least one process survives, all n units of work will be performed. We consider three parameters: the number of messages sent, the total number of units of work performed (including multiplicities), and time. We present three protocols for solving the problem. All three are work optimal, doing O(n+t) work. The rst has moderate costs in the remaining two parameters, sends O(t p t) messages, and takes O(n+t) time. This protocol can be easily modied to run in any completely asynchronous system equipped with a failure detection mechanism. The second sends only O(tlogt) messages, but its running time is large (O(t 2 (n+t)2 n+t )). The third is essentially time optimal in the (usual) case in which there are no failures, and its time complexity degrades gracefully as the number of failures increases.
8145017
Characterizing the Common Prior Assumption
Abstract Logical characterizations of the common prior assumption (CPA) are investigated. Two approaches are considered. The first is called frame distinguishability , and is similar in spirit to the approaches considered in the economics literature. Results similar to those obtained in the economics literature are proved here as well, namely, that we can distinguish finite spaces that satisfy the CPA from those that do not in terms of disagreements in expectation. However, it is shown that, for the language used here, no formulas can distinguish infinite spaces satisfying the CPA from those that do not. The second approach considered is that of finding a sound and complete axiomatization. Such an axiomatization is provided; again, the key axiom involves disagreements in expectation. The same axiom system is shown to be sound and complete both in the finite and the infinite case. Thus, the two approaches to characterizing the CPA behave quite differently in the case of infinite spaces. Journal of Economic Literature Classification Numbers: C70, D80.
7572747
Set-theoretic completeness for epistemic and conditional logic
The standard approach to logic in the literature in philosophy and mathematics, which has also been adopted in computer science, is to define a language (the syntax), an appropriate class of models together with an interpretation of formulas in the language (the semantics), a collection of axioms and rules of inference characterizing reasoning (the proof theory), and then relate the proof theory to the semantics via soundness and completeness results. Here we consider an approach that is more common in the economics literature, which works purely at the semantic, set-theoretic level. We provide set-theoretic completeness results for a number of epistemic and conditional logics, and contrast the expressive power of the syntactic and set-theoretic approaches.
1225089
Defining Explanation in Probabilistic Systems
As probabilistic systems gain popularity and are coming into wider use, the need for a mechanism that explains the system's findings and recommendations becomes more critical. The system will also need a mechanism for ordering competing explanations. We examine two representative approaches to explanation in the literature-- one due to Gardenfors and one due to Pearl--and show that both suffer from significant problems. We propose an approach to defining a notion of "better explanation" that combines some of the features of both together with more recent work by Pearl and others on causality.
10886930
Belief Revision with Unreliable Observations
Research in belief revision has been dominated by work that lies firmly within the classic AGM paradigm, characterized by a well-known set of postulates governing the behavior of "rational" revision functions. A postulate that is rarely criticized is the success postulate: the result of revising by an observed proposition ? results in belief in ?. This postulate, however, is often undesirable in settings where an agent's observations may be imprecise or noisy. We propose a semantics that captures a new ontology for studying revision functions, which can handle noisy observations in a natural way while retaining the classical AGM model as a special case. We present a characterization theorem for our semantics, and describe a number of natural special cases that allow ease of specification and reasoning with revision functions. In particular, by making the Markov assumption, we can easily specify and reason about revision.