Describing Groups

Nies, André

doi:10.2178/bsl/1186666149

Cited by 67 publications

(56 citation statements)

References 28 publications

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“…However it is finite-word automatic with oracle #2#3#4 · · · . This is based on the idea, independently found by Frank Stephan and Joe Miller and reported in [114], that there is a presentation of ([0, 1) ∩ Q, +) by finite words in which + is regular, but the domain is not: every rational in [0, 1) can be expressed as n i=2 ai i! for a unique sequence of natural numbers a i satisfying 0 ≤ a i < i.…”

Section: Consider An Expansion ∆mentioning

confidence: 99%

“…(i) If a group (G, ·) is word-automatic, then every finitely generated subgroup is virtually Abelian (has an Abelian subgroup of finite index). In particular, a finitely generated group is in S-AutStr if, and only if, it is virtually Abelian [116,114].…”

Section: Growth Of Generationsmentioning

confidence: 99%

“…The rest of the proof uses powerful theorems of Gromov and Ershov (see [114] for a survey of word-automatic groups).…”

Section: Growth Of Generationsmentioning

confidence: 99%

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Automata-based presentations of infinite structures

Bárány¹,

Grädel²,

Rubin³

2011

Finite and Algorithmic Model Theory

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The model theory of finite structures is intimately connected to various fields in computer science, including complexity theory, databases, and verification. In particular, there is a close relationship between complexity classes and the expressive power of logical languages, as witnessed by the fundamental theorems of descriptive complexity theory, such as Fagin's Theorem and the ImmermanVardi Theorem (see [78, Chapter 3] for a survey).However, for many applications, the strict limitation to finite structures has turned out to be too restrictive, and there have been considerable efforts to extend the relevant logical and algorithmic methodologies from finite structures to suitable classes of infinite ones. In particular this is the case for databases and verification where infinite structures are of crucial importance [130]. Algorithmic model theory aims to extend in a systematic fashion the approach and methods of finite model theory, and its interactions with computer science, from finite structures to finitely-presentable infinite ones.There are many possibilities to present infinite structures in a finite manner. A classical approach in model theory concerns the class of computable structures; these are countable structures, on the domain of natural numbers, say, with a finite collection of computable functions and relations. Such structures can be finitely presented by a collection of algorithms, and they have been intensively studied in model theory since the 1960s. However, from the point of view of algorithmic model theory the class of computable structures is problematic. Indeed, one of the central issues in algorithmic model theory is the effective evaluation of logical formulae, from a suitable logic such as, for instance, first-order logic (FO), monadic second-order logic (MSO), or a fixed point logic like LFP or the modal µ-calculus. But on computable structures, only the quantifier-free formulae generally admit effective evaluation, and already the existential fragment of first-order logic is undecidable, for instance on the computable structure (N, +, · ).This leads us to the central requirement that for a suitable logic L (depending on the intended application) the model-checking problem for the class C of finitely presented structures should be algorithmically solvable. At the very 1

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Section: Consider An Expansion ∆mentioning

confidence: 99%

Section: Growth Of Generationsmentioning

confidence: 99%

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Automata-based presentations of infinite structures

Bárány¹,

Grädel²,

Rubin³

2011

Finite and Algorithmic Model Theory

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“…In modern terminology (N, +) is (finite-word) automatic. Also, the rational group (Q, +) is finite-set interpretable in a decidable expansion of ω (see [19]). …”

Section: Why Do Interpretations In Trees Matter?mentioning

confidence: 99%

Interpretations in Trees with Countably Many Branches

Rabinovich

Rubin

2012

2012 27th Annual IEEE Symposium on Logic in Computer Science

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Abstract-We study the expressive power of logical interpretations on the class of scattered trees, namely those with countably many infinite branches. Scattered trees can be thought of as the tree analogue of scattered linear orders. Every scattered tree has an ordinal rank that reflects the structure of its infinite branches. We prove, roughly, that trees and orders of large rank cannot be interpreted in scattered trees of small rank. We consider a quite general notion of interpretation: each element of the interpreted structure is represented by a set of tuples of subsets of the interpreting tree. Our trees are countable, not necessarily finitely branching, and may have finitely many unary predicates as labellings. We also show how to replace injective set-interpretations in (not necessarily scattered) trees by 'finitary' set-interpretations.Index Terms-Composition method, finite-set interpretations, infinite scattered trees, monadic second order logic.

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“…The problem of determining the isomorphism type of a finitely generated field from its first-order theory and especially our eventual solution through a biinterpretation with the natural numbers is closely related to the problem of quasifinite axiomatizability for finitely generated groups considered by A. Nies [13]. Indeed, in work independent from our own, A. Khelif employs biinterpretations with the natural numbers in order to relatively axiomatize certain finitely generated groups and rings.…”

Section: Introductionmentioning

confidence: 99%

Infinite finitely generated fields are biinterpretable with ℕ

Scanlon

2008

J. Amer. Math. Soc.

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Using the work of several other mathematicians, principally the results of Poonen refining the work of Pop that algebraic independence is definable within the class of finitely generated fields and of Rumely that the ring of rational integers is uniformly interpreted in global fields, and a theorem on the definability of valuations on function fields of curves, we show that each infinite finitely generated field considered in the ring language is parametrically biinterpretable with N \mathbb {N} . As a consequence, for any finitely generated field there is a first-order sentence in the language of rings which is true in that field but false in every other finitely generated field and, hence, Pop’s conjecture that elementarily equivalent finitely generated fields are isomorphic is true.

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Describing Groups

Cited by 67 publications

References 28 publications

Automata-based presentations of infinite structures

Automata-based presentations of infinite structures

Interpretations in Trees with Countably Many Branches

Infinite finitely generated fields are biinterpretable with ℕ

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