Large finite structures with few Lk-types

Authors:
Martin Grohe
Affiliations:
Laboratory for Foundations of Computer Science, Division of Informatics, University of Edinburgh, Edinburgh EH9 3JZ, Scotland, United Kingdom
Venue:
Information and Computation - Special issue: LICS'97
Year:
2002

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Abstract

For any k ≥ 3, we prove the following results on theories in the k -variable fragment Lk of first-order logic: (1) The so-called Lk-invariant is not recursively invertible. (2) There is no recursive bound on the size of the minimal model of a complete Lk-theory in terms of its k-size, that is, the number of its Lk-types. (3) Assume that NP ⊈ P/poly. Then there is no P-computable canonization function for Lk-equivalence. Our results answer questions of Dawar [3,4] and Dawar, Lindell, and Weinstein [5].