A logic for reasoning about probabilities

Fagin, Ronald; Halpern, Joseph Y.; Megiddo, Nimrod

doi:10.1016/0890-5401(90)90060-u

Cited by 458 publications

(375 citation statements)

References 12 publications

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“…The corresponding models were Kripke models with two families of measures that were subadditive and superadditive, respectively, and monotone with respect to the order of the worlds. Our approach is more similar to [4,5,9,10,11]. However, as the basic logic in those papers is classical propositional logic, there are formulas that are valid in the mentioned logics, but not in ours.…”

Section: Resultsmentioning

confidence: 89%

“…However, as the basic logic in those papers is classical propositional logic, there are formulas that are valid in the mentioned logics, but not in ours. For example, as discussed in Section 1, P ≥1 ((p → q) ∨ (q → p)) is valid in logics from [4,5,9,10,11] (q → p)). Moreover, we can construct a model in which both p → q and q → p will have probability 1/n, by simply adding n − 3 linearly ordered new worlds below w 0 in M .…”

Section: Resultsmentioning

confidence: 99%

“…The classical (boolean) logic, on the other hand, may be regarded as starting from a position of omniscience, in so far as it considers that each proposition must be either true or false. Therefore, it is not surprising that combining these two approaches, e. g. by adding probability operators to classical propositional logic as it was done in [4,5,9,10,11], would be sometimes difficult.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A probabilistic extension of intuitionistic logic

Marković

Ognjanović²,

Rašković

2003

Mathematical Logic Qtrly

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We introduce a probabilistic extension of propositional intuitionistic logic. The logic allows making statements such as P ≥s α, with the intended meaning "the probability of truthfulness of α is at least s". We describe the corresponding class of models, which are Kripke models with a naturally arising notion of probability, and give a sound and complete infinitary axiomatic system. We prove that the logic is decidable.

show abstract

Section: Resultsmentioning

confidence: 89%

Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A probabilistic extension of intuitionistic logic

Marković

Ognjanović²,

Rašković

2003

Mathematical Logic Qtrly

View full text Add to dashboard Cite

show abstract

“…One, as noted in [13], is that polynomials arise naturally from conditional probability. If we would like to include linear combinations of conditional probability expressions in the language, we find that this motivates a generalization to polynomial combinations of probability expressions.…”

Section: Discussionmentioning

confidence: 99%

Undecidable Cases of Model Checking Probabilistic Temporal-Epistemic Logic (Extended Abstract)

Meyden

Patra

2016

Electron. Proc. Theor. Comput. Sci.

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We investigate the decidability of model-checking logics of time, knowledge and probability, with respect to two epistemic semantics: the clock and synchronous perfect recall semantics in partially observed discrete-time Markov chains. Decidability results are known for certain restricted logics with respect to these semantics, subject to a variety of restrictions that are either unexplained or involve a longstanding unsolved mathematical problem. We show that mild generalizations of the known decidable cases suffice to render the model checking problem definitively undecidable. In particular, for a synchronous perfect recall, a generalization from temporal operators with finite reach to operators with infinite reach renders model checking undecidable. The case of the clock semantics is closely related to a monadic second order logic of time and probability that is known to be decidable, except on a set of measure zero. We show that two distinct extensions of this logic make model checking undecidable. One of these involves polynomial combinations of probability terms, the other involves monadic second order quantification into the scope of probability operators. These results explain some of the restrictions in previous work.

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“…In [1], the reader can find a rather extensive bibliography, together with motivation for such studies. In the present paper we introduce a generalization of the widely accepted language proposed in [2] (working with so-called polynomial weight formulas) by allowing meta-variables (in other words, parameters) in propositions under the probability sign, and quantifiers bounding these variables.…”

Section: Introductionmentioning

confidence: 99%

Quantification over propositional formulas in probability logic: decidability issues

Speranski

2011

Algebra Logic

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A language for reasoning about probability is generalized by adding quantifiers over propositional formulas to the language. Then relevant decidability issues are considered. In particular, the results presented demonstrate that a rather weak fragment of the new language has an undecidable validity problem. On the other hand, it is stated that a restricted version of the validity problem is decidable for ∀∃-sentences.

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A logic for reasoning about probabilities

Cited by 458 publications

References 12 publications

A probabilistic extension of intuitionistic logic

A probabilistic extension of intuitionistic logic

Undecidable Cases of Model Checking Probabilistic Temporal-Epistemic Logic (Extended Abstract)

Quantification over propositional formulas in probability logic: decidability issues

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