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.
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.
42142945
Likelihood, Probability, and Knowledge
The modal logic LL was introduced by Halpern and Rabin [HR] as a means of doing qualitative reasoning about likelihood. Here the relationship between LL and probability theory is examined. It is shown that there is a way of translating probability assertions into LL in a sound manner, so that LL in some sense can capture the probabilistic interpretation of likelihood. However, the translation is subtle; several more obvious attempts are shown to lead to inconsistencies. We also extend LL by adding modal operators for knowledge. The propositional version of the resulting logic LLK is shown to have a complete axiomatization and to be decidable in exponential time, provably the best possible.
205008749
The Complexity of Reasoning about Knowledge and Time. I. Lower Bounds
Abstract We study the propositional model logic of knowledge and time for distributed systems. We consider a number of logics (ninety-six in all!), which vary according to the choice of language and the assumptions made on the underlying system. The major parameters in the language are whether there is a common knowledge operator, whether we reason about the knowledge of one or more than one processor, and whether our temporal operators are branching or linear. The assumptions on distributed systems that we consider are: whether or not processors forget, whether or not processors learn, whether or not time is synchronous, and whether or not there is a unique initial state in the system. We completely characterize the complexity of the validity problem for all the logics we consider. This paper focuses on lower bounds; a sequel will deal with the corresponding upper bounds. Typical results include a ∏-completeness result for the language with common knowledge with respect to systems where processors do not forget, and a corresponding non-elementary-time result for the language without common knowledge. It is shown that, in general, the assumption that processors do not forget or do not learn greatly increases the complexity of reasoning about knowledge and time.
11653369
Axiomatic definitions of programming languages, II
Sufficient conditions are given for partial correctness assertions to determine the input-output semantics of quite general classes of programming languages. This determination cannot be unique unless states which are indistinguishable by predicates in the assertions are identified. Even when indistinguishable states are identified, partial correctness assertions may not suffice to determine program semantics.
2298547
Modelling Knowledge and Action in Distributed Systems
We present a formal model that captures the subtle interaction between knowledge and action in distributed systems. We view a distributed system as a set of runs, where a run is a function from time to global states and a global state is a tuple consisting of an environment state and a local state for each process in the system. This model is a generalization of those used in many previous papers. Actions in this model are associated with functions from global states to global states. A protocol is a function from local states to actions. We extend the standard notion of a protocol by defining knowledge-based protocols, ones in which a process’ actions may depend explicitly on its knowledge. Knowledge-based protocols provide a natural way of describing how actions should take place in a distributed system. Finally, we show how the notion of one protocol implementing another can be captured in our model.
2908780
Irrelevance and Conditioning in First-Order Probabilistic Logic
First-order probabilistic logic is a powerful knowledge representation language. Unfortunately, deductive reasoning based on the standard semantics for this logic does not support certain desirable patterns of reasoning, such as indifference to irrelevant information or substitution of constants into universal rules. We show that both these patterns rely on a first-order version of probabilistic independence, and provide semantic conditions to capture them. The resulting insight enables us to understand the effect of conditioning on independence, and allows us to describe a procedure for determining when independencies are preserved under conditioning. We apply this procedure in the context of a sound and powerful inference algorithm for reasoning from statistical knowledge bases.