7,641 research outputs found
Belief as Willingness to Bet
We investigate modal logics of high probability having two unary modal
operators: an operator expressing probabilistic certainty and an operator
expressing probability exceeding a fixed rational threshold . Identifying knowledge with the former and belief with the latter, we may
think of as the agent's betting threshold, which leads to the motto "belief
is willingness to bet." The logic for has an
modality along with a sub-normal modality that extends
the minimal modal logic by way of four schemes relating
and , one of which is a complex scheme arising out of a theorem due to
Scott. Lenzen was the first to use Scott's theorem to show that a version of
this logic is sound and complete for the probability interpretation. We
reformulate Lenzen's results and present them here in a modern and accessible
form. In addition, we introduce a new epistemic neighborhood semantics that
will be more familiar to modern modal logicians. Using Scott's theorem, we
provide the Lenzen-derivative properties that must be imposed on finite
epistemic neighborhood models so as to guarantee the existence of a probability
measure respecting the neighborhood function in the appropriate way for
threshold . This yields a link between probabilistic and modal
neighborhood semantics that we hope will be of use in future work on modal
logics of qualitative probability. We leave open the question of which
properties must be imposed on finite epistemic neighborhood models so as to
guarantee existence of an appropriate probability measure for thresholds
.Comment: Removed date from v1 to avoid confusion on citation/reference,
otherwise identical to v
Real-time and Probabilistic Temporal Logics: An Overview
Over the last two decades, there has been an extensive study on logical
formalisms for specifying and verifying real-time systems. Temporal logics have
been an important research subject within this direction. Although numerous
logics have been introduced for the formal specification of real-time and
complex systems, an up to date comprehensive analysis of these logics does not
exist in the literature. In this paper we analyse real-time and probabilistic
temporal logics which have been widely used in this field. We extrapolate the
notions of decidability, axiomatizability, expressiveness, model checking, etc.
for each logic analysed. We also provide a comparison of features of the
temporal logics discussed
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Logics of Imprecise Comparative Probability
This paper studies connections between two alternatives to the standard probability calculus for representing and reasoning about uncertainty: imprecise probability andcomparative probability. The goal is to identify complete logics for reasoning about uncertainty in a comparative probabilistic language whose semantics is given in terms of imprecise probability. Comparative probability operators are interpreted as quantifying over a set of probability measures. Modal and dynamic operators are added for reasoning about epistemic possibility and updating sets of probability measures
Modelling default and likelihood reasoning as probabilistic
A probabilistic analysis of plausible reasoning about defaults and about likelihood is presented. 'Likely' and 'by default' are in fact treated as duals in the same sense as 'possibility' and 'necessity'. To model these four forms probabilistically, a logic QDP and its quantitative counterpart DP are derived that allow qualitative and corresponding quantitative reasoning. Consistency and consequence results for subsets of the logics are given that require at most a quadratic number of satisfiability tests in the underlying propositional logic. The quantitative logic shows how to track the propagation error inherent in these reasoning forms. The methodology and sound framework of the system highlights their approximate nature, the dualities, and the need for complementary reasoning about relevance
Computing Quantiles in Markov Reward Models
Probabilistic model checking mainly concentrates on techniques for reasoning
about the probabilities of certain path properties or expected values of
certain random variables. For the quantitative system analysis, however, there
is also another type of interesting performance measure, namely quantiles. A
typical quantile query takes as input a lower probability bound p and a
reachability property. The task is then to compute the minimal reward bound r
such that with probability at least p the target set will be reached before the
accumulated reward exceeds r. Quantiles are well-known from mathematical
statistics, but to the best of our knowledge they have not been addressed by
the model checking community so far.
In this paper, we study the complexity of quantile queries for until
properties in discrete-time finite-state Markov decision processes with
non-negative rewards on states. We show that qualitative quantile queries can
be evaluated in polynomial time and present an exponential algorithm for the
evaluation of quantitative quantile queries. For the special case of Markov
chains, we show that quantitative quantile queries can be evaluated in time
polynomial in the size of the chain and the maximum reward.Comment: 17 pages, 1 figure; typo in example correcte
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