scholarly journals A NEW DP-MINIMAL EXPANSION OF THE INTEGERS

2019 ◽  
Vol 84 (02) ◽  
pp. 632-663 ◽  
Author(s):  
ERAN ALOUF ◽  
CHRISTIAN D’ELBÉE
Keyword(s):  
Analogous Result ◽  
Abelian Groups ◽  
Quantifier Elimination ◽  
Alternative Proof ◽  
Order Structure ◽  
First Order ◽  
Adic Valuation ◽  
Image Position ◽  
Natural Expansion

AbstractWe consider the structure $({\Bbb Z}, + ,0,|_{p_1 } , \ldots ,|_{p_n } )$, where $x|_p y$ means $v_p \left( x \right) \leqslant v_p \left( y \right)$ and vp is the p-adic valuation. We prove that this structure has quantifier elimination in a natural expansion of the language of abelian groups, and that it has dp-rank n. In addition, we prove that a first order structure with universe ${\Bbb Z}$ which is an expansion of $({\Bbb Z}, + ,0)$ and a reduct of $({\Bbb Z}, + ,0,|_p )$ must be interdefinable with one of them. We also give an alternative proof for Conant’s analogous result about $({\Bbb Z}, + ,0, < )$.

The diversity of quantifier prefixes

1973 ◽  
Vol 38 (1) ◽  
pp. 79-85 ◽  
Author(s):  
H. Jerome Keisler ◽  
Wilbur Walkoe
Keyword(s):  
Positive Integer ◽  
Analogous Result ◽  
Predicate Logic ◽  
Finite Sequence ◽  
Arithmetical Hierarchy ◽  
First Order ◽  
Free Variables ◽  
The Standard Model ◽  
Image Position ◽  
Identity Symbol

The Arithmetical Hierarchy Theorem of Kleene [1] states that in the complete theory of the standard model of arithmetic there is for each positive integer r a Σr0 formula which is not equivalent to any Πr0 formula, and a Πr0 formula which is not equivalent to any Πr0 formula. A Πr0 formula is a formula of the formwhere φ has only bounded quantifiers; Πr0 formulas are defined dually.The Linear Prefix Theorem in [3] is an analogous result for predicate logic. Consider the first order predicate logic L with identity symbol, countably many n-placed relation symbols for each n, and no constant or function symbols. A prefix is a finite sequenceof quantifier symbols ∃ and ∀, for example ∀∃∀∀∀∃. By a Q formula we mean a formula of L of the formwhere v1, …, vr are distinct variables and φ has no quantifiers. A sentence is a formula with no free variables. The Linear Prefix Theorem is as follows.Linear Prefix Theorem. Let Q and q be two different prefixes of the same length r. Then there is a Q sentence which is not logically equivalent to any q sentence.Moreover, for each s there is a Q formula with s free variables which is not logically equivalent to any q formula with s free variables.For example, there is an ∀∃∀∀∀∃ sentence which is not logically equivalent to any ∀∃∃∀∀∃ sentence, and vice versa. Recall that in arithmetic two consecutive ∃'s or ∀'s can be collapsed; for instance all ∀∃∀∀∀∃ and ∀∃∃∀∀∃ formulas are logically equivalent to Π40 formulas. But the Linear Prefix Theorem shows that in predicate logic the number of quantifiers in each block, as well as the number of blocks, counts.

scholarly journals DEFINABLE TOPOLOGICAL DYNAMICS

2017 ◽  
Vol 82 (3) ◽  
pp. 1080-1105 ◽  
Author(s):  
KRZYSZTOF KRUPIŃSKI
Keyword(s):  
Topological Dynamics ◽  
Connected Components ◽  
Order Structure ◽  
Topological Dynamic ◽  
Bohr Compactification ◽  
First Order ◽  
Ellis Group ◽  
Fixed Set ◽  
Image Position

AbstractFor a group G definable in a first order structure M we develop basic topological dynamics in the category of definable G-flows. In particular, we give a description of the universal definable G-ambit and of the semigroup operation on it. We find a natural epimorphism from the Ellis group of this flow to the definable Bohr compactification of G, that is to the quotient ${G^{\rm{*}}}/G_M^{{\rm{*}}00}$ (where G* is the interpretation of G in a monster model). More generally, we obtain these results locally, i.e., in the category of Δ-definable G-flows for any fixed set Δ of formulas of an appropriate form. In particular, we define local connected components $G_{{\rm{\Delta }},M}^{{\rm{*}}00}$ and $G_{{\rm{\Delta }},M}^{{\rm{*}}000}$, and show that ${G^{\rm{*}}}/G_{{\rm{\Delta }},M}^{{\rm{*}}00}$ is the Δ-definable Bohr compactification of G. We also note that some deeper arguments from [14] can be adapted to our context, showing for example that our epimorphism from the Ellis group to the Δ-definable Bohr compactification factors naturally yielding a continuous epimorphism from the Δ-definable generalized Bohr compactification to the Δ-definable Bohr compactification of G. Finally, we propose to view certain topological-dynamic and model-theoretic invariants as Polish structures which leads to some observations and questions.

A definability result for compact complex spaces

2004 ◽  
Vol 69 (1) ◽  
pp. 241-254 ◽  
Author(s):  
Dale Radin
Keyword(s):  
Algebraic Geometry ◽  
Complex Space ◽  
Analogous Result ◽  
Order Structure ◽  
Holomorphic Map ◽  
Complex Spaces ◽  
Compact Complex Space ◽  
Image Position

AbstractA compact complex space X is viewed as a 1-st order structure by taking predicates for analytic subsets of X, X x X, … as basic relations. Let f: X → Y be a proper surjective holomorphic map between complex spaces and set Xy ≔ f−1(y). We show that the setis analytically constructible, i.e.. is a definable set when X and Y are compact complex spaces and f: X → Y is a holomorphic map. The analogous result in the context of algebraic geometry gives rise to the definability of Morley degree.

Definability and decidability issues in extensions of the integers with the divisibility predicate

1996 ◽  
Vol 61 (2) ◽  
pp. 515-540 ◽  
Author(s):  
Patrick Cegielski ◽  
Yuri Matiyasevich ◽  
Denis Richard
Keyword(s):  
Computational Complexity ◽  
Alternative Proof ◽  
Order Structure ◽  
Prime Divisors ◽  
Additive Theory ◽  
First Order ◽  
Decision Method ◽  
Definable Relations ◽  
Positive Integers ◽  
Number Of Prime Divisors

AbstractLet be a first-order structure; we denote by DEF() the set of all first-order definable relations and functions within . Let π be any one-to-one function from ℕ into the set of prime integers.Let ∣ and • be respectively the divisibility relation and multiplication as function. We show that the sets DEF(ℕ, π, ∣) and DEF(ℕ, π, •) are equal. However there exists function π such that the set DEF(ℕ, +, ∣), or, equivalently, DEF(ℕ, π, •) is not equal to DEF(ℕ, +, •). Nevertheless, in all cases there is an {π, •}-definable and hence also {π, |}-definable structure over π which is isomorphic to 〈ℕ, +, •〉. Hence theories TH(ℕ, π, ∣) and TH(ℕ, π, •) are undecidable.The binary relation of equipotence between two positive integers saying that they have equal number of prime divisors is not definable within the divisibility lattice over positive integers. We prove it first by comparing the lower bound of the computational complexity of the additive theory of positive integers and of the upper bound of the computational complexity of the theory of the mentioned lattice.The last section provides a self-contained alternative proof of this latter result based on a decision method linked to an elimination of quantifiers via specific tables.

Hyper-regular lattice-ordered groups

1993 ◽  
Vol 58 (4) ◽  
pp. 1342-1358 ◽  
Author(s):  
Daniel Gluschankof ◽  
François Lucas
Keyword(s):  
Abelian Groups ◽  
General Context ◽  
Lattice Ordered Group ◽  
Regular Lattice ◽  
Ordered Group ◽  
First Order ◽  
Elementary Class ◽  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Image Position

It is a well-known fact that the notion of an archimedean order cannot be formalized in the first-order calculus. In [12] and [18], A. Robinson and E. Zakon characterized the elementary class generated by all the archimedean, totally-ordered abelian groups (o-groups) in the language 〈+,<〉, calling it the class of regularly ordered or generalized archimedean abelian groups. Since difference (−) and 0 are definable in that language, it is immediate that in the expanded language 〈 +, −, 0, < 〉 the definable expansion of the class of regular groups is also the elementary class generated by the archimedean ones. In the more general context of lattice-ordered groups (l-groups), the notion of being archimedean splits into two different notions: a strong one (being hyperarchimedean) and a weak one (being archimedean). Using the representation theorem of K. Keimel for hyperarchimedean l-groups, we extend in this paper the Robinson and Zakon characterization to the elementary class generated by the prime-projectable, hyperarchimedean l-groups. This characterization is also extended here to the elementary class generated by the prime-projectable and projectable archimedean l-groups (including all complete l-groups). Finally, transferring a result of A. Touraille on the model theory of Boolean algebras with distinguished ideals, we give the classification up to elementary equivalence of the characterized class.We recall that a lattice-ordered group, l-group for short, is a structure

scholarly journals THERE ARE NO INTERMEDIATE STRUCTURES BETWEEN THE GROUP OF INTEGERS AND PRESBURGER ARITHMETIC

2018 ◽  
Vol 83 (1) ◽  
pp. 187-207 ◽  
Author(s):  
GABRIEL CONANT
Keyword(s):  
Order Structure ◽  
Presburger Arithmetic ◽  
First Order ◽  
Image Position

AbstractWe show that if a first-order structure ${\cal M}$, with universe ℤ, is an expansion of (ℤ,+,0) and a reduct of (ℤ,+,<,0), then ${\cal M}$ must be interdefinable with (ℤ ,+,0) or (ℤ ,+,<,0).

The order indiscernibles of divisible ordered abelian groups

1984 ◽  
Vol 49 (1) ◽  
pp. 151-160
Author(s):  
David Rosenthal
Keyword(s):  
Abelian Group ◽  
Model Theory ◽  
Algebraic Structure ◽  
Abelian Groups ◽  
Quantifier Elimination ◽  
Simple Condition ◽  
Algebraic Invariants ◽  
Image Position ◽  
And Algebra ◽  
Simplified Form

There has been much work in developing the interconnections between model theory and algebra. Here we look at a particular example, the divisible ordered abelian groups, and show how the indiscernibles are related to the algebraic structure. Now a divisible ordered abelian group is a model of Th (Q, +, 0, <) and so is linearly ordered by <. Thus the theory is unstable and has a large number of models. It is therefore unrealistic to expect that a simple condition will completely determine a model. Instead we would just like to obtain nice algebraic invariants.Definition. A subset C of a divisible ordered abelian group is a set of (order) indiscernibles iff for every sequence of integers n1,…,nk and for every c1 < … < ck and d1 < … < dk in CNote that this simplified form of indiscernibility is an immediate consequence of quantifier elimination for the theory. The above definition could be formulated in the language of +, 0, < but we have used subtraction as a matter of convenience. Similarly we may also use rational coefficients. Also note that a set of order indiscernibles is usually defined with respect to some external order. But in this case there are only two possibilities: the ordering inherited from or its reverse. So we will always assume that a set of order indiscernibles has the ordering inherited from . We may sometimes refer to a set of indiscernibles as a sequence of indiscernibles if we want to explicitly mention the ordering associated with the set.

scholarly journals Model Completeness, Uniform Interpolants and Superposition Calculus

2021 ◽  
Author(s):  
Diego Calvanese ◽  
Silvio Ghilardi ◽  
Alessandro Gianola ◽  
Marco Montali ◽  
Andrey Rivkin
Keyword(s):  
Automated Reasoning ◽  
Quantifier Elimination ◽  
Effective Solution ◽  
Model Completeness ◽  
Present Contribution ◽  
First Order ◽  
Reduction Strategies ◽  
Infinite State Systems ◽  
Infinite State ◽  
Superposition Calculus

AbstractUniform interpolants have been largely studied in non-classical propositional logics since the nineties; a successive research line within the automated reasoning community investigated uniform quantifier-free interpolants (sometimes referred to as “covers”) in first-order theories. This further research line is motivated by the fact that uniform interpolants offer an effective solution to tackle quantifier elimination and symbol elimination problems, which are central in model checking infinite state systems. This was first pointed out in ESOP 2008 by Gulwani and Musuvathi, and then by the authors of the present contribution in the context of recent applications to the verification of data-aware processes. In this paper, we show how covers are strictly related to model completions, a well-known topic in model theory. We also investigate the computation of covers within the Superposition Calculus, by adopting a constrained version of the calculus and by defining appropriate settings and reduction strategies. In addition, we show that computing covers is computationally tractable for the fragment of the language used when tackling the verification of data-aware processes. This observation is confirmed by analyzing the preliminary results obtained using the mcmt tool to verify relevant examples of data-aware processes. These examples can be found in the last version of the tool distribution.

Resolution of T. Ward's Question and the Israel–Finch Conjecture: Precise Analysis of an Integer Sequence Arising in Dynamics

2014 ◽  
Vol 24 (1) ◽  
pp. 195-215
Author(s):  
JEFFREY GAITHER ◽  
GUY LOUCHARD ◽  
STEPHAN WAGNER ◽  
MARK DANIEL WARD
Keyword(s):  
Weighted Average ◽  
Analytic Number Theory ◽  
Combinatorial Structure ◽  
Asymptotic Growth ◽  
First Order ◽  
Integer Sequence ◽  
Precise Analysis ◽  
Formula Group ◽  
Fourth Author ◽  
Image Position

We analyse the first-order asymptotic growth of \[ a_{n}=\int_{0}^{1}\prod_{j=1}^{n}4\sin^{2}(\pi jx)\, dx. \] The integer an appears as the main term in a weighted average of the number of orbits in a particular quasihyperbolic automorphism of a 2n-torus, which has applications to ergodic and analytic number theory. The combinatorial structure of an is also of interest, as the ‘signed’ number of ways in which 0 can be represented as the sum of ϵjj for −n ≤ j ≤ n (with j ≠ 0), with ϵj ∈ {0, 1}. Our result answers a question of Thomas Ward (no relation to the fourth author) and confirms a conjecture of Robert Israel and Steven Finch.

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