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Definability of Group Theoretic Notions
, 2000
"... We consider logical definability of the grouptheoretic notions of simplicity, nilpotency and solvability on the class of finite groups. On one hand, we show that these notions are definable by sentences of deterministic transitive closure logic DTC. These results are based on known grouptheoretic ..."
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We consider logical definability of the grouptheoretic notions of simplicity, nilpotency and solvability on the class of finite groups. On one hand, we show that these notions are definable by sentences of deterministic transitive closure logic DTC. These results are based on known grouptheoretic results. On the other hand, we prove that simplicity, nilpotency and the normal closure of a subset of a group are not definable by single sentences of first order logic FO. In addition, we show that an isomorphism between two arbitrary finite abelian groups can be expressed by a sentence of DTC(I), where DTC(I) is DTC enhanced with the equicardinality quantifier I, and that it is not expressible by a sentence of L ! 1! . 1 Introduction Descriptive complexity theory is a branch of Finite model theory in which expressive power of various logics is studied on the class of finite models. Descriptive complexity theory and computational complexity theory have a close connection. For example o...
Descriptive complexity of finite abelian groups
, 2007
"... We investigate the descriptive complexity of finite abelian groups. Using EhrenfeuchtFraïssé games we find upper and lower bounds on quantifier depth, quantifier alternations, and number of variables of a firstorder sentence that distinguishes two finite abelian groups. Our main results are the fo ..."
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We investigate the descriptive complexity of finite abelian groups. Using EhrenfeuchtFraïssé games we find upper and lower bounds on quantifier depth, quantifier alternations, and number of variables of a firstorder sentence that distinguishes two finite abelian groups. Our main results are the following. Let G1 and G2 be a pair of nonisomorphic finite abelian groups, and let m be a number that divides one of the two groups ’ orders. Then the following hold: (1) there exists a firstorder sentence ϕ that distinguishes G1 and G2 such that ϕ is existential, has quantifier depth O(log m), and has at most 5 variables and (2) if ϕ is a sentence that distinguishes G1 and G2 then ϕ must have quantifier depth Ω(log m). These results are applied to (1) get bounds on the firstorder distinguishability of dihedral groups, (2) to prove that on the class of finite groups both cyclicity and the closure of a single element are not firstorder definable, and (3) give a different proof for the firstorder undefinability of simplicity, nilpotency, and the normal closure of a single element on the class of finite groups (their undefinability were shown by A. Koponen and K. Luosto in an unpublished paper). Keywords: EhrenfeuchtFraïssé games, groups, descriptive complexity, definability, expressibility 1