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Dimension in team semantics

Published online by Cambridge University Press:  12 March 2024

Lauri Hella Open the ORCID record for Lauri Hella [Opens in a new window] ,
Kerkko Luosto Open the ORCID record for Kerkko Luosto [Opens in a new window]  and
Jouko Väänänen Open the ORCID record for Jouko Väänänen [Opens in a new window]
Lauri Hella
Affiliation:
Tampere University, Tampere, Finland
Kerkko Luosto
Affiliation:
Tampere University, Tampere, Finland
Jouko Väänänen*
Affiliation:
University of Helsinki, Helsinki, Finland
*
Corresponding author: Jouko Väänänen; Email: [email protected]
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Abstract

We introduce three measures of complexity for families of sets. Each of the three measures, which we call dimensions, is defined in terms of the minimal number of convex subfamilies that are needed for covering the given family. For upper dimension, the subfamilies are required to contain a unique maximal set, for dual upper dimension a unique minimal set, and for cylindrical dimension both a unique maximal and a unique minimal set. In addition to considering dimensions of particular families of sets, we study the behavior of dimensions under operators that map families of sets to new families of sets. We identify natural sufficient criteria for such operators to preserve the growth class of the dimensions. We apply the theory of our dimensions for proving new hierarchy results for logics with team semantics. To this end we associate each atom with a natural notion or arity. First, we show that the standard logical operators preserve the growth classes of the families arising from the semantics of formulas in such logics. Second, we show that the upper dimension of $k+1$-ary dependence, inclusion, independence, anonymity, and exclusion atoms is in a strictly higher growth class than that of any k-ary atoms, whence the $k+1$-ary atoms are not definable in terms of any atoms of smaller arity.

Keywords

team semanticsdependence logicdimensiongrowth class
Type
Special Issue: Logic and Complexity
Information
Mathematical Structures in Computer Science , Volume 34 , Special Issue 5: Logic and Complexity , May 2024 , pp. 410 - 454
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Copyright
© The Author(s), 2024. Published by Cambridge University Press

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Footnotes

a

Supported by the Academy of Finland (grant No 322795) and the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant No 101020762).

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