LOGICAL CHARACTERIZATION OF RECOGNIZABLE SETS OF POLYNOMIALS OVER A FINITE FIELD

The ring of integers and the ring of polynomials over a finite field share a lot of properties. Using a bounded number of polynomial coefficients, any polynomial can be decomposed as a linear combination of powers of a non-constant polynomial P playing the role of the base of the numeration. Having in mind the theorem of Cobham from 1969 about recognizable sets of integers, it is natural to study P-recognizable sets of polynomials. Based on the results obtained in the Ph.D. thesis of the second author, we study the logical characterization of such sets and related properties like decidability of the corresponding first-order theory.

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PRESBURGER ARITHMETIC, RATIONAL GENERATING FUNCTIONS, AND QUASI-POLYNOMIALS

Journal of Symbolic Logic ◽

10.1017/jsl.2015.4 ◽

2015 ◽

Vol 80 (2) ◽

pp. 433-449 ◽

Cited By ~ 10

Author(s):

KEVIN WOODS

Keyword(s):

Generating Functions ◽

Counting Function ◽

Order Theory ◽

Characteristic Functions ◽

Presburger Arithmetic ◽

First Order ◽

First Order Theory ◽

Rational Generating Function ◽

Rational Generating Functions

AbstractPresburger arithmetic is the first-order theory of the natural numbers with addition (but no multiplication). We characterize sets that can be defined by a Presburger formula as exactly the sets whose characteristic functions can be represented by rational generating functions; a geometric characterization of such sets is also given. In addition, ifp= (p1, . . . ,pn) are a subset of the free variables in a Presburger formula, we can define a counting functiong(p) to be the number of solutions to the formula, for a givenp. We show that every counting function obtained in this way may be represented as, equivalently, either a piecewise quasi-polynomial or a rational generating function. Finally, we translate known computational complexity results into this setting and discuss open directions.

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Global quantification in Zermelo-Fraenkel set theory

Journal of Symbolic Logic ◽

10.2307/2274215 ◽

1985 ◽

Vol 50 (2) ◽

pp. 289-301

Author(s):

John Mayberry

Keyword(s):

Set Theory ◽

Order Theory ◽

Prima Facie ◽

Local Theory ◽

Axiom Schema ◽

First Order ◽

First Order Theory ◽

Naturally Occurring ◽

Universe Of Sets

My aim here is to investigate the role of global quantifiers—quantifiers ranging over the entire universe of sets—in the formalization of Zermelo-Fraenkel set theory. The use of such quantifiers in the formulas substituted into axiom schemata introduces, at least prima facie, a strong element of impredicativity into the thapry. The axiom schema of replacement provides an example of this. For each instance of that schema enlarges the very domain over which its own global quantifiers vary. The fundamental question at issue is this: How does the employment of these global quantifiers, and the choice of logical principles governing their use, affect the strengths of the axiom schemata in which they occur?I shall attack this question by comparing three quite different formalizations of the intuitive principles which constitute the Zermelo-Fraenkel system. The first of these, local Zermelo-Fraenkel set theory (LZF), is formalized without using global quantifiers. The second, global Zermelo-Fraenkel set theory (GZF), is the extension of the local theory obtained by introducing global quantifiers subject to intuitionistic logical laws, and taking the axiom schema of strong collection (Schema XII, §2) as an additional assumption of the theory. The third system is the conventional formalization of Zermelo-Fraenkel as a classical, first order theory. The local theory, LZF, is already very strong, indeed strong enough to formalize any naturally occurring mathematical argument. I have argued (in [3]) that it is the natural formalization of naive set theory. My intention, therefore, is to use it as a standard against which to measure the strength of each of the other two systems.

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Julia Robinson numbers

International Journal of Number Theory ◽

10.1142/s1793042119500908 ◽

2019 ◽

Vol 15 (08) ◽

pp. 1565-1599

Author(s):

Pierre Gillibert ◽

Gabriele Ranieri

Keyword(s):

Infinite Family ◽

Order Theory ◽

Ring Of Integers ◽

First Order ◽

First Order Theory ◽

Infinite Degree ◽

Totally Real

We construct an infinite family of totally real algebraic extensions of [Formula: see text] whose ring of integers has a Julia Robinson number distinct from [Formula: see text] and [Formula: see text]. In fact, the set of Julia Robinson numbers obtained is unbounded. This gives new examples of algebraic extensions of [Formula: see text] of infinite degree whose ring of integers has undecidable first-order theory.

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A Characterization of Strongly Dependent Ordered Abelian Groups

Revista Colombiana de Matemáticas ◽

10.15446/recolma.v52n2.77154 ◽

2018 ◽

Vol 52 (2) ◽

pp. 139-159 ◽

Cited By ~ 1

Author(s):

Alfred Dolich ◽

John Goodrick

Keyword(s):

Abelian Groups ◽

Order Theory ◽

First Order ◽

First Order Theory

We characterize all ordered Abelian groups whose ﬁrst-order theory in the language {+, <, 0} is strongly dependent. The main result of this note was obtained independently by Halevi and Hasson [7] and Farré [5].

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NIP FOR THE ASYMPTOTIC COUPLE OF THE FIELD OF LOGARITHMIC TRANSSERIES

Journal of Symbolic Logic ◽

10.1017/jsl.2016.59 ◽

2017 ◽

Vol 82 (1) ◽

pp. 35-61 ◽

Cited By ~ 1

Author(s):

ALLEN GEHRET

Keyword(s):

Order Theory ◽

Exchange Property ◽

Independence Property ◽

First Order ◽

Valued Field ◽

First Order Theory ◽

Order Language ◽

Complete Survey ◽

Image Position

AbstractThe derivation on the differential-valued field Tlog of logarithmic transseries induces on its value group ${{\rm{\Gamma }}_{{\rm{log}}}}$ a certain map ψ. The structure ${\rm{\Gamma }} = \left( {{{\rm{\Gamma }}_{{\rm{log}}}},\psi } \right)$ is a divisible asymptotic couple. In [7] we began a study of the first-order theory of $\left( {{{\rm{\Gamma }}_{{\rm{log}}}},\psi } \right)$ where, among other things, we proved that the theory $T_{{\rm{log}}} = Th\left( {{\rm{\Gamma }}_{{\rm{log}}} ,\psi } \right)$ has a universal axiomatization, is model complete and admits elimination of quantifiers (QE) in a natural first-order language. In that paper we posed the question whether Tlog has NIP (i.e., the Non-Independence Property). In this paper, we answer that question in the affirmative: Tlog does have NIP. Our method of proof relies on a complete survey of the 1-types of Tlog, which, in the presence of QE, is equivalent to a characterization of all simple extensions ${\rm{\Gamma }}\left\langle \alpha \right\rangle$ of ${\rm{\Gamma }}$. We also show that Tlog does not have the Steinitz exchange property and we weigh in on the relationship between models of Tlog and the so-called precontraction groups of [9].

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