8649178
Asymptomatic conditional probabilities for first-order logic
Motivated by problems that arise in computing degrees of belief, we consider the problem of computing asymptotic conditional probabilities for first-order formulas. That is, given first-order formulas &fgr; and &thgr;, we consider the number of structures with domain {1,…,N} that satisfy &thgr;, and compute the fraction of them in which &fgr; is true. We then consider what happens to this probability of first-order formulas, except that now we are considering asymptotic conditional probabilities. Although work has been done on special cases of asymptotic conditional probabilities, no general theory has been developed. This is probably due in part to the fact that it has been known that, if there is a binary predicate symbol in the vocabulary, asymptotic conditional probabilities do not always exist. We show that in this general case, almost all the questions one might want to ask (such as deciding whether the asymptotic probability exists) are highly undecidable. On the other hand, we show that the situation with unary predicates only is much better. If the vocabulary consists only of unary predicate and constant symbols, it is decidable whether the limit exists, and if it does, there is an effective algorithm for computing it. The complexity depends on two parameters: whether there is a fixed finite vocabulary or an infinite one, and whether there is a bound on the depth of quantifier nesting.
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.
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.
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...
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»
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
35307334
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 first has moderate costs in the remaining two parameters, sends $O(t\sqrt{t})$ messages, and takes O(n+t) time. This protocol can be easily modified to run in any completely asynchronous system equipped with a failure detection mechanism. The second sends only O(t log t) messages, but its running time is large (O(t2(n+t) 2n+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.
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.
10093090
A decision-theoretic approach to reliable message delivery
Summary. We argue that the tools of decision theory should be taken more seriously in the specification and analysis of systems. We illustrate this by considering a simple problem involving reliable communication, showing how considerations of utility and probability can be used to decide when it is worth sending heartbeat messages and, if they are sent, how often they should be sent.
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.
16360228
Least expected cost query optimization: an exercise in utility
We identify two unreasonable, though standard, assumptions made by database query optimizers that can adversely affect the quality of the chosen evaluation plans. One assumption is that it is enough to optimize for the expected case---that is, the case where various parameters (like available memory) take on their expected value. The other assumption is that the parameters are constant throughout the execution of the query. We present an algorithm based on the ``System R''-style query optimization algorithm that does not rely on these assumptions. The algorithm we present chooses the plan of the least expected cost instead of the plan of least cost given some fixed value of the parameters. In execution environments that exhibit a high degree of variability, our techniques should result in better performance.