Finite automata presentable abelian groups

Nies, André; Semukhin, Pavel

doi:10.1016/j.apal.2009.07.006

Cited by 16 publications

(11 citation statements)

References 6 publications

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“…Most of the material in this subsection is from [21]. We give examples of abelian groups that are torsion-free, indecomposable (or even rigid) and of rank 2 or higher.…”

Section: More Examplesmentioning

confidence: 99%

“…The second method is via amalgams of abelian groups. (In [21] we actually use amalgams of commutative semigroups with the cancellation property to prove the formal details of Example 3.5, since it is easier to work with the presentation of R +,0 k from the proof of Proposition 3.1 than with a presentation of R k .) Proposition 3.4.…”

Section: More Examplesmentioning

confidence: 99%

“…However, a counterexample can be derived from recent work of Akiyama et al [1]: for each prime q > 1, there is an FA-presentation of the abelian group R q = {zq −i : z ∈ Z, i ∈ Z} where the representations of integers do not form an FA-recognizable subset (see [22] for more details). Recently, in [21] an FA-presentation of Z × Z was given where no non-trivial cyclic subgroup is FA-recognizable.…”

Section: 2mentioning

confidence: 99%

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Describing Groups

Nies¹

2007

Bull. symb. log.

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Two ways of describing a group are considered. 1. A group is finite-automaton presentable if its elements can be represented by strings over a finite alphabet, in such a way that the set of representing strings and the group operation can be recognized by finite automata. 2. An infinite f.g. group is quasi-finitely axiomatizable if there is a description consisting of a single first-order sentence, together with the information that the group is finitely generated. In the first part of the paper we survey examples of FA-presentable groups, but also discuss theorems restricting this class. In the second part, we give examples of quasi-finitely axiomatizable groups, consider the algebraic content of the notion, and compare it to the notion of a group which is a prime model. We also show that if a structure is bi-interpretable in parameters with the ring of integers, then it is prime and quasi-finitely axiomatizable.

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“…Most of the material in this subsection is from [21]. We give examples of abelian groups that are torsion-free, indecomposable (or even rigid) and of rank 2 or higher.…”

Section: More Examplesmentioning

confidence: 99%

Section: More Examplesmentioning

confidence: 99%

Section: 2mentioning

confidence: 99%

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Describing Groups

Nies¹

2007

Bull. symb. log.

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“…However, we leave this to further investigations. We note that in [25], an amalgamated products of abelian automatic groups is studied. In the paper, automatic groups are the groups in which the graph of the group operation is FA recognizable.…”

Section: Theorem 18 (With Nies) For Every Finite Group G the Restrimentioning

confidence: 99%

Automatic Structures and Groups

Khoussainov

2011

Language and Automata Theory and Applications

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“…One can use these techniques to give decision algorithms for Presburger arithmetic, the real numbers under addition, and atomless Boolean algebras, and there have even been applications to decision problems in chess [BHS12]. The study of automatic structures, and especially of automatic groups, has been particularly fruitful; see, e.g., [Tsa11,BS11,NS09, ECH + 92, KKM14, JKS + 17, JKS18].…”

Section: Introductionmentioning

confidence: 99%

Automatic and Polynomial-Time Algebraic Structures

Bazhenov¹,

Harrison-Trainor²,

Kalimullin³

et al. 2019

J. symb. log.

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A structure is automatic if its domain, functions, and relations are all regular languages. Using the fact that every automatic structure is decidable, in the literature many decision problems have been solved by giving an automatic presentation of a particular structure. Khoussainov and Nerode asked whether there is some way to tell whether a structure has, or does not have, an automatic presentation. We answer this question by showing that the set of Turing machines that represent automata-presentable structures is ${\rm{\Sigma }}_1^1 $-complete. We also use similar methods to show that there is no reasonable characterisation of the structures with a polynomial-time presentation in the sense of Nerode and Remmel.

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Finite automata presentable abelian groups

Cited by 16 publications

References 6 publications

Describing Groups

Describing Groups

Automatic Structures and Groups

Automatic and Polynomial-Time Algebraic Structures

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