32 research outputs found

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    On the relative asymptotic expressivity of inference frameworks

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    Let σ\sigma be a first-order signature and let Wn\mathbf{W}_n be the set of all σ\sigma-structures with domain {1,…,n}\{1, \ldots, n\}. By an inference framework we mean a class F\mathbf{F} of pairs (P,L)(\mathbb{P}, L), where P=(Pn:n=1,2,3,…)\mathbb{P} = (\mathbb{P}_n : n = 1, 2, 3, \ldots) and Pn\mathbb{P}_n is a probability distribution on Wn\mathbf{W}_n, and LL is a logic with truth values in the unit interval [0,1][0, 1]. An inference framework F′\mathbf{F}' is asymptotically at least as expressive as another inference framework F\mathbf{F} if for every (P,L)∈F(\mathbb{P}, L) \in \mathbf{F} there is (P′,L′)∈F′(\mathbb{P}', L') \in \mathbf{F}' such that P\mathbb{P} is asymptotically total-variation-equivalent to P′\mathbb{P}' and for every φ(xˉ)∈L\varphi(\bar{x}) \in L there is φ′(xˉ)∈L′\varphi'(\bar{x}) \in L' such that φ′(xˉ)\varphi'(\bar{x}) is asymptotically equivalent to φ(xˉ)\varphi(\bar{x}) with respect to P\mathbb{P}. This relation is a preorder and we describe a partial order on the equivalence classes of some inference frameworks that seem natural in the context of machine learning and artificial intelligence. Several previous results about asymptotic (or almost sure) equivalence of formulas or convergence in probability can be formulated in terms of relative asymptotic strength of inference frameworks. We incorporate these results in our classification of inference frameworks and prove two new results. Both concern sequences of probability distributions defined by directed graphical models that use ``continuous'' aggregation functions. The first considers queries expressed by a logic with truth values in [0,1][0, 1] which employs continuous aggregation functions. The second considers queries expressed by a two-valued conditional logic that can express statements about relative frequencies.Comment: 52 page

    Crisp bi-G\"{o}del modal logic and its paraconsistent expansion

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    In this paper, we provide a Hilbert-style axiomatisation for the crisp bi-G\"{o}del modal logic \KbiG. We prove its completeness w.r.t.\ crisp Kripke models where formulas at each state are evaluated over the standard bi-G\"{o}del algebra on [0,1][0,1]. We also consider a paraconsistent expansion of \KbiG with a De Morgan negation ¬\neg which we dub \KGsquare. We devise a Hilbert-style calculus for this logic and, as a~con\-se\-quence of a~conservative translation from \KbiG to \KGsquare, prove its completeness w.r.t.\ crisp Kripke models with two valuations over [0,1][0,1] connected via ¬\neg. For these two logics, we establish that their decidability and validity are PSPACE\mathsf{PSPACE}-complete. We also study the semantical properties of \KbiG and \KGsquare. In particular, we show that Glivenko theorem holds only in finitely branching frames. We also explore the classes of formulas that define the same classes of frames both in K\mathbf{K} (the classical modal logic) and the crisp G\"{o}del modal logic \KG^c. We show that, among others, all Sahlqvist formulas and all formulas ϕ→χ\phi\rightarrow\chi where ϕ\phi and χ\chi are monotone, define the same classes of frames in K\mathbf{K} and \KG^c

    Towards a logical foundation of randomized computation

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    This dissertation investigates the relations between logic and TCS in the probabilistic setting. It is motivated by two main considerations. On the one hand, since their appearance in the 1960s-1970s, probabilistic models have become increasingly pervasive in several fast-growing areas of CS. On the other, the study and development of (deterministic) computational models has considerably benefitted from the mutual interchanges between logic and CS. Nevertheless, probabilistic computation was only marginally touched by such fruitful interactions. The goal of this thesis is precisely to (start) bring(ing) this gap, by developing logical systems corresponding to specific aspects of randomized computation and, therefore, by generalizing standard achievements to the probabilistic realm. To do so, our key ingredient is the introduction of new, measure-sensitive quantifiers associated with quantitative interpretations. The dissertation is tripartite. In the first part, we focus on the relation between logic and counting complexity classes. We show that, due to our classical counting propositional logic, it is possible to generalize to counting classes, the standard results by Cook and Meyer and Stockmeyer linking propositional logic and the polynomial hierarchy. Indeed, we show that the validity problem for counting-quantified formulae captures the corresponding level in Wagner's hierarchy. In the second part, we consider programming language theory. Type systems for randomized \lambda-calculi, also guaranteeing various forms of termination properties, were introduced in the last decades, but these are not "logically oriented" and no Curry-Howard correspondence is known for them. Following intuitions coming from counting logics, we define the first probabilistic version of the correspondence. Finally, we consider the relationship between arithmetic and computation. We present a quantitative extension of the language of arithmetic able to formalize basic results from probability theory. This language is also our starting point to define randomized bounded theories and, so, to generalize canonical results by Buss

    Automated Reasoning

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    This volume, LNAI 13385, constitutes the refereed proceedings of the 11th International Joint Conference on Automated Reasoning, IJCAR 2022, held in Haifa, Israel, in August 2022. The 32 full research papers and 9 short papers presented together with two invited talks were carefully reviewed and selected from 85 submissions. The papers focus on the following topics: Satisfiability, SMT Solving,Arithmetic; Calculi and Orderings; Knowledge Representation and Jutsification; Choices, Invariance, Substitutions and Formalization; Modal Logics; Proofs System and Proofs Search; Evolution, Termination and Decision Prolems. This is an open access book

    Mathematical Logic: Proof Theory, Constructive Mathematics (hybrid meeting)

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    The Workshop "Mathematical Logic: Proof Theory, Constructive Mathematics" focused on proofs both as formal derivations in deductive systems as well as on the extraction of explicit computational content from given proofs in core areas of ordinary mathematics using proof-theoretic methods. The workshop contributed to the following research strands: interactions between foundations and applications; proof mining; constructivity in classical logic; modal logic and provability logic; proof theory and theoretical computer science; structural proof theory

    K + K = 120 : Papers dedicated to László Kálmán and András Kornai on the occasion of their 60th birthdays

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    Computer Science Logic 2018: CSL 2018, September 4-8, 2018, Birmingham, United Kingdom

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