The additive group of the rationals does not have an automatic presentation

Tsankov, Todor

doi:10.2178/jsl/1318338853

Cited by 38 publications

(25 citation statements)

References 14 publications

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“…These results are interesting given that it is known that addition over rationals is not expressible using automata over finite words [18]. Recent work by Abu Zaid et al demonstrates the surprising result that if for any chosen representation for real numbers addition is ω−automatic then multiplication is not, and vice-versa [19].…”

Section: A Elementary Operationsmentioning

confidence: 99%

Regular Real Analysis

Chaudhuri

Sankaranarayanan

Vardi

2013

2013 28th Annual ACM/IEEE Symposium on Logic in Computer Science

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Abstract-We initiate the study of regular real analysis, or the analysis of real functions that can be encoded by automata on infinite words. It is known that ω-automata can be used to represent relations between real vectors, reals being represented in exact precision as infinite streams. The regular functions studied here constitute the functional subset of such relations.We show that some classic questions in function analysis can become elegantly computable in the context of regular real analysis. Specifically, we present an automatatheoretic technique for reasoning about limit behaviors of regular functions, and obtain, using this method, a decision procedure to verify the continuity of a regular function. Several other decision procedures for regular functions-for finding roots, fixpoints, minima, etc.-are also presented. At the same time, we show that the class of regular functions is quite rich, and includes functions that are highly challenging to encode using traditional symbolic notation.

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Section: A Elementary Operationsmentioning

confidence: 99%

Regular Real Analysis

Chaudhuri

Sankaranarayanan

Vardi

2013

2013 28th Annual ACM/IEEE Symposium on Logic in Computer Science

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“…However, the additive group of the rationals, (Q, +), is not automatic [136]. In fact, Tsankov shows that no torsion free Abelian group that is p-divisible for infinitely many primes p is automatic.…”

Section: Example 38 (Groups) (I)mentioning

confidence: 99%

“…The group of rationals (Q, +) has recently been shown to have no word-automatic presentation [136]. However it is finite-word automatic with oracle #2#3#4 · · · .…”

Section: Consider An Expansion ∆mentioning

confidence: 99%

“…Its restriction to the set of elements represented by ultimately periodic ω-words is isomorphic to the additive group of the rationals (Q, +). Tsankov [136] has shown that there is no automatic divisible torsion-free Abelian group (DTAG). Hence the theory of DTAGs is another example of a first-order theory having an uncountable ω-automatic model but no countable (ω-)automatic models.…”

Section: Löwenheim-skolemmentioning

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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“…This is a part of a larger theme where the goal is to classify tree and wordautomatic structures. Much work has already been done on the classification of automatic structures in certain classes such as linear orders, Boolean algebras, abelian groups [6] [13] [15] [16] [21] [26], [3]. Recent results by Kuske, Lohrey Supported in part by NUS grants R252-000-420-112 and C252-000-087-001.…”

Section: Introductionmentioning

confidence: 99%

Tree-automatic scattered linear orders

Jain

Khoussainov

Schlicht

et al. 2016

Theoretical Computer Science

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Abstract. We study tree-automatic linear orders on regular tree languages. We first show that there is no tree-automatic scattered linear order, and in particular no well-order, on the set of all finite labeled trees. This also follows from results of Gurevich-Shelah [8] and Carayol-Löding [4]. We then show that a regular tree language admits a tree-automatic scattered linear order if and only if all trees are included in a subtree of the full binary tree with finite tree-rank. As a consequence of this characterization, we obtain an algorithm which, given a regular tree language, decides if the tree language can be well-ordered by a tree automaton. Finally, we connect tree automata with automata on ordinals and determine sharp lower and upper bounds for tree-automatic well-orders on natural examples of regular tree languages. Our proofs use elementary techniques of automata theory.

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The additive group of the rationals does not have an automatic presentation

Cited by 38 publications

References 14 publications

Regular Real Analysis

Regular Real Analysis

Automata-based presentations of infinite structures

Tree-automatic scattered linear orders

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