Interpolation theorem for Marcinkiewicz spaces with applications to Lorentz gamma spaces

AbstractAnalogues of Scott’s isomorphism theorem, Karp’s theorem as well as results on lack of compactness and strong completeness are established for infinitary propositional relevant logics. An “interpolation theorem” (of a particular sort introduced by Barwise and van Benthem) for the infinitary quantificational boolean logic L∞ω holds. This yields a preservation result characterizing the expressive power of infinitary relevant languages with absurdity using the model-theoretic relation of relevant directed bisimulation as well as a Beth definability property.

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Leon Henkin. An extension of the Craig-Lyndon interpolation theorem. The journal of symbolic logic, vol. 28 no. 3 (for 1963, pub. 1964), pp. 201–216.

Journal of Symbolic Logic ◽

10.2307/2270596 ◽

1965 ◽

Vol 30 (1) ◽

pp. 98-99

Author(s):

M. A. Taitslin

Keyword(s):

Interpolation Theorem ◽

Symbolic Logic

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Banach–Saks property in Marcinkiewicz spaces

Journal of Mathematical Analysis and Applications ◽

10.1016/j.jmaa.2007.03.040 ◽

2007 ◽

Vol 336 (2) ◽

pp. 1231-1258 ◽

Cited By ~ 9

Author(s):

S.V. Astashkin ◽

F.A. Sukochev

Keyword(s):

Marcinkiewicz Spaces

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A normal form theorem for Lω1p, with applications

Journal of Symbolic Logic ◽

10.2307/2273591 ◽

1982 ◽

Vol 47 (3) ◽

pp. 605-624 ◽

Cited By ~ 8

Author(s):

Douglas N. Hoover

Keyword(s):

Normal Form ◽

Probability Model ◽

Random Variables ◽

Interpolation Theorem ◽

Complete Theory ◽

De Finetti’S Theorem ◽

Form Theorem ◽

Normal Form Theorem ◽

De Finetti's Theorem

AbstractWe show that every formula of Lω1P is equivalent to one which is a propositional combination of formulas with only one quantifier. It follows that the complete theory of a probability model is determined by the distribution of a family of random variables induced by the model. We characterize the class of distribution which can arise in such a way. We use these results together with a form of de Finetti’s theorem to prove an almost sure interpolation theorem for Lω1P.

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Randomized feasible interpolation and monotone circuits with a local oracle

Journal of Mathematical Logic ◽

10.1142/s0219061318500125 ◽

2018 ◽

Vol 18 (02) ◽

pp. 1850012 ◽

Cited By ~ 1

Author(s):

Jan Krajíček

Keyword(s):

Lower Bounds ◽

Cutting Planes ◽

Communication Complexity ◽

Interpolation Theorem ◽

Proof System ◽

Symbolic Logic ◽

Approximate Counting ◽

Small Width ◽

Randomized Protocols ◽

Monotone Circuits

The feasible interpolation theorem for semantic derivations from [J. Krajíček, Interpolation theorems, lower bounds for proof systems, and independence results for bounded arithmetic, J. Symbolic Logic 62(2) (1997) 457–486] allows to derive from some short semantic derivations (e.g. in resolution) of the disjointness of two [Formula: see text] sets [Formula: see text] and [Formula: see text] a small communication protocol (a general dag-like protocol in the sense of Krajíček (1997) computing the Karchmer–Wigderson multi-function [Formula: see text] associated with the sets, and such a protocol further yields a small circuit separating [Formula: see text] from [Formula: see text]. When [Formula: see text] is closed upwards, the protocol computes the monotone Karchmer–Wigderson multi-function [Formula: see text] and the resulting circuit is monotone. Krajíček [Interpolation by a game, Math. Logic Quart. 44(4) (1998) 450–458] extended the feasible interpolation theorem to a larger class of semantic derivations using the notion of a real communication complexity (e.g. to the cutting planes proof system CP). In this paper, we generalize the method to a still larger class of semantic derivations by allowing randomized protocols. We also introduce an extension of the monotone circuit model, monotone circuits with a local oracle (CLOs), that does correspond to communication protocols for [Formula: see text] making errors. The new randomized feasible interpolation thus shows that a short semantic derivation (from a certain class of derivations larger than in the original method) of the disjointness of [Formula: see text], [Formula: see text] closed upwards, yields a small randomized protocol for [Formula: see text] and hence a small monotone CLO separating the two sets. This research is motivated by the open problem to establish a lower bound for proof system [Formula: see text] operating with clauses formed by linear Boolean functions over [Formula: see text]. The new randomized feasible interpolation applies to this proof system and also to (the semantic versions of) cutting planes CP, to small width resolution over CP of Krajíček [Discretely ordered modules as a first-order extension of the cutting planes proof system, J. Symbolic Logic 63(4) (1998) 1582–1596] (system R(CP)) and to random resolution RR of Buss, Kolodziejczyk and Thapen [Fragments of approximate counting, J. Symbolic Logic 79(2) (2014) 496–525]. The method does not yield yet lengths-of-proofs lower bounds; for this it is necessary to establish lower bounds for randomized protocols or for monotone CLOs.

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