219 research outputs found

    An introduction to the special issue on question processing

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    On Resource-bounded versions of the van Lambalgen theorem

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    The van Lambalgen theorem is a surprising result in algorithmic information theory concerning the symmetry of relative randomness. It establishes that for any pair of infinite sequences AA and BB, BB is Martin-L\"of random and AA is Martin-L\"of random relative to BB if and only if the interleaved sequence A⊎BA \uplus B is Martin-L\"of random. This implies that AA is relative random to BB if and only if BB is random relative to AA \cite{vanLambalgen}, \cite{Nies09}, \cite{HirschfeldtBook}. This paper studies the validity of this phenomenon for different notions of time-bounded relative randomness. We prove the classical van Lambalgen theorem using martingales and Kolmogorov compressibility. We establish the failure of relative randomness in these settings, for both time-bounded martingales and time-bounded Kolmogorov complexity. We adapt our classical proofs when applicable to the time-bounded setting, and construct counterexamples when they fail. The mode of failure of the theorem may depend on the notion of time-bounded randomness

    Reasoning in non-probabilistic uncertainty: logic programming and neural-symbolic computing as examples

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    This article aims to achieve two goals: to show that probability is not the only way of dealing with uncertainty (and even more, that there are kinds of uncertainty which are for principled reasons not addressable with probabilistic means); and to provide evidence that logic-based methods can well support reasoning with uncertainty. For the latter claim, two paradigmatic examples are presented: Logic Programming with Kleene semantics for modelling reasoning from information in a discourse, to an interpretation of the state of affairs of the intended model, and a neural-symbolic implementation of Input/Output logic for dealing with uncertainty in dynamic normative context
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