40,805 research outputs found

    Probability Measures and projections on Quantum Logics

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    The present paper is devoted to modelling of a probability measure of logical connectives on a quantum logic (QL), via a GG-map, which is a special map on it. We follow the work in which the probability of logical conjunction, disjunction and symmetric difference and their negations for non-compatible propositions are studied. We study such a G G -map on quantum logics, which is a probability measure of a projection and show, that unlike classical (Boolean) logic, probability measure of projections on a quantum logic are not necessarilly pure projections. We compare properties of a GG-map on QLs with properties of a probability measure related to logical connectives on a Boolean algebra

    A note on many valued quantum computational logics

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    The standard theory of quantum computation relies on the idea that the basic information quantity is represented by a superposition of elements of the canonical basis and the notion of probability naturally follows from the Born rule. In this work we consider three valued quantum computational logics. More specifically, we will focus on the Hilbert space C^3, we discuss extensions of several gates to this space and, using the notion of effect probability, we provide a characterization of its states.Comment: Pages 15, Soft Computing, 201

    Non-classical conditional probability and the quantum no-cloning theorem

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    The quantum mechanical no-cloning theorem for pure states is generalized and transfered to the quantum logics with a conditional probability calculus in a rather abstract, though simple and basic fashion without relying on a tensor product construction or finite dimension as required in other generalizations.Comment: 6 page

    Belief as Willingness to Bet

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    We investigate modal logics of high probability having two unary modal operators: an operator KK expressing probabilistic certainty and an operator BB expressing probability exceeding a fixed rational threshold c≥12c\geq\frac 12. Identifying knowledge with the former and belief with the latter, we may think of cc as the agent's betting threshold, which leads to the motto "belief is willingness to bet." The logic KB.5\mathsf{KB.5} for c=12c=\frac 12 has an S5\mathsf{S5} KK modality along with a sub-normal BB modality that extends the minimal modal logic EMND45\mathsf{EMND45} by way of four schemes relating KK and BB, 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 c=12c=\frac 12. 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 c≠12c\neq\frac 12.Comment: Removed date from v1 to avoid confusion on citation/reference, otherwise identical to v

    Characterizing and Reasoning about Probabilistic and Non-Probabilistic Expectation

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    Expectation is a central notion in probability theory. The notion of expectation also makes sense for other notions of uncertainty. We introduce a propositional logic for reasoning about expectation, where the semantics depends on the underlying representation of uncertainty. We give sound and complete axiomatizations for the logic in the case that the underlying representation is (a) probability, (b) sets of probability measures, (c) belief functions, and (d) possibility measures. We show that this logic is more expressive than the corresponding logic for reasoning about likelihood in the case of sets of probability measures, but equi-expressive in the case of probability, belief, and possibility. Finally, we show that satisfiability for these logics is NP-complete, no harder than satisfiability for propositional logic.Comment: To appear in Journal of the AC

    Formal verification of higher-order probabilistic programs

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    Probabilistic programming provides a convenient lingua franca for writing succinct and rigorous descriptions of probabilistic models and inference tasks. Several probabilistic programming languages, including Anglican, Church or Hakaru, derive their expressiveness from a powerful combination of continuous distributions, conditioning, and higher-order functions. Although very important for practical applications, these combined features raise fundamental challenges for program semantics and verification. Several recent works offer promising answers to these challenges, but their primary focus is on semantical issues. In this paper, we take a step further and we develop a set of program logics, named PPV, for proving properties of programs written in an expressive probabilistic higher-order language with continuous distributions and operators for conditioning distributions by real-valued functions. Pleasingly, our program logics retain the comfortable reasoning style of informal proofs thanks to carefully selected axiomatizations of key results from probability theory. The versatility of our logics is illustrated through the formal verification of several intricate examples from statistics, probabilistic inference, and machine learning. We further show the expressiveness of our logics by giving sound embeddings of existing logics. In particular, we do this in a parametric way by showing how the semantics idea of (unary and relational) TT-lifting can be internalized in our logics. The soundness of PPV follows by interpreting programs and assertions in quasi-Borel spaces (QBS), a recently proposed variant of Borel spaces with a good structure for interpreting higher order probabilistic programs
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