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September 2005 Weak cardinality theorems
Till Tantau
J. Symbolic Logic 70(3): 861-878 (September 2005). DOI: 10.2178/jsl/1122038917

Abstract

Kummer's Cardinality Theorem states that a language A must be recursive if a Turing machine can exclude for any n words w₁, …, wn one of the n + 1 possibilities for the cardinality of {w₁,…,wn} ∩ A. There was good reason to believe that this theorem is a peculiarity of recursion theory: neither the Cardinality Theorem nor weak forms of it hold for resource-bounded computational models like polynomial time. This belief may be flawed. In this paper it is shown that weak cardinality theorems hold for finite automata and also for other models. An explanation is proposed as to why recursion-theoretic and automata-theoretic weak cardinality theorems hold, but not corresponding ‘middle-ground theorems’: The recursion- and automata-theoretic weak cardinality theorems are instantiations of purely logical weak cardinality theorems. The logical theorems can be instantiated for logical structures characterizing recursive computations and finite automata computations. A corresponding structure characterizing polynomial time computations does not exist.

Citation

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Till Tantau. "Weak cardinality theorems." J. Symbolic Logic 70 (3) 861 - 878, September 2005. https://doi.org/10.2178/jsl/1122038917

Information

Published: September 2005
First available in Project Euclid: 22 July 2005

zbMATH: 1092.03020
MathSciNet: MR2155269
Digital Object Identifier: 10.2178/jsl/1122038917

Rights: Copyright © 2005 Association for Symbolic Logic

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Vol.70 • No. 3 • September 2005
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