Decidability and Expressiveness for First-Order Logics of Probability (1989)
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| Venue: | Information and Computation |
| Citations: | 36 - 5 self |
BibTeX
@ARTICLE{Abadi89decidabilityand,
author = {Mart'in Abadi and Joseph Y. Halpern},
title = {Decidability and Expressiveness for First-Order Logics of Probability},
journal = {Information and Computation},
year = {1989},
volume = {112},
pages = {1--36}
}
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Abstract
We consider decidability and expressiveness issues for two first-order logics of probability. In one, the probability is on possible worlds, while in the other, it is on the domain. It turns out that in both cases it takes very little to make reasoning about probability highly undecidable. We show that when the probability is on the domain, if the language contains only unary predicates then the validity problem is decidable. However, if the language contains even one binary predicate, the validity problem is \Pi 2 1 complete, as hard as elementary analysis with free predicate and function symbols. With equality in the language, even with no other symbol, the validity problem is at least as hard as that for elementary analysis, \Pi 1 1 hard. Thus, the logic cannot be axiomatized in either case. When we put the probability on the set of possible worlds, the validity problem is \Pi 2 1 complete with as little as one unary predicate in the language, even without equality. With equalit...
