Positive existential definability in finite structures

For a finite structure M on the domain of k≥2 elements consider the description of the Galois closures InvEnd(M) and InvAut(M) (the set of relations on the domain of M that are invariant to all endomorphisms, resp. automorphisms of M) via positive existential formulae over the signature of M. It is...

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Bibliographic Details
Published in:Discrete Applied Mathematics Vol. 366; pp. 210 - 215
Main Author: Romov, Boris A.
Format: Journal Article
Language:English
Published: Elsevier B.V 15.05.2025
Subjects:
Algorithmic complexity
Finite structure
Galois closure
Positive existential formula
Positive existential formula
Algorithmic complexity
Finite structure
Galois closure
ISSN:0166-218X
Online Access:Get full text
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Summary:For a finite structure M on the domain of k≥2 elements consider the description of the Galois closures InvEnd(M) and InvAut(M) (the set of relations on the domain of M that are invariant to all endomorphisms, resp. automorphisms of M) via positive existential formulae over the signature of M. It is shown that for every relation R from the set InvEnd(M) (R from InvAut(M)) there exists a positive existential formula over M (resp., over M and the inequality predicate) with no more than (k-1)m (m is the size of R) existential quantifiers that defines R. This implies that every first order n-ary (n≥1) predicate over M (i.e., with n free variables) is equivalent to a positive existential formula over M and inequality with at most (k-1)(kn-1) existential quantifiers and this estimate does not depend on the signature of M. Next, for a finite M with a finite signature there is a cubic algorithm that for every R on the domain of M it recognizes properties: R belongs to InvAut(M) (that is, R is defined by some first order predicate over M), as well as R belongs to InvEnd(M). Moreover, for such structure M there is a quadratic algorithm that for every R from InvAut(M) (R from InvEnd(M)) it produces a positive existential formula over the unnested atomic predicates from M and the inequality x≠y (resp., over unnested atomic predicates from M), which defines R in mO(n2) steps, where n is the arity of R.
ISSN:0166-218X
DOI:10.1016/j.dam.2025.01.001