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A logic for reasoning about probabilities. (English) Zbl 0811.03014
Summary: We consider a language for reasoning about probability which allows us to make statements such as “the probability of \(E_ 1\) is less than \({1\over 3}\)” and “the probability of \(E_ 1\) is at least twice the probability of \(E_ 2\)”, where \(E_ 1\) and \(E_ 2\) are arbitrary events. We consider the case where all events are measurable (i.e., represent measurable sets) and the more general case, which is also of interest in practice, where they may not be measurable. The measurable case is essentially a formalization of (the propositional fragment of) Nilsson’s probabilistic logic. As we show elsewhere, the general (nonmeasurable) case corresponds precisely to replacing probability measures by Dempster-Shafer belief functions. In both cases, we provide a complete axiomatization and show that the problem of deciding satisfiability is NP-complete, no worse than that of propositional logic. As a tool for proving our complete axiomatizations, we give a complete axiomatization for reasoning about Boolean combinations of linear inequalities, which is of independent interest. This proof and others make crucial use of results from the theory of linear programming. We then extend the language to allow reasoning about conditional probability and show that the resulting logic is decidable and completely axiomatizable, by making use of the theory of real closed fields.

MSC:
03B48 Probability and inductive logic
68T27 Logic in artificial intelligence
68Q25 Analysis of algorithms and problem complexity
03B25 Decidability of theories and sets of sentences
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