Admits Elimination Of Quantifiers Research Articles

Quantifier elimination on some pseudo-algebraically closed valued fields

In this article, some theories of pseudo-algebraically closed non-trivially valued fields are shown to admit quantifier elimination in the language obtained by adjoining to the language of rings the function symbols for splitting coefficients, the function symbols for relative p-coordinate functions, and the division predicate for a valuation.

The asymptotic couple of the field of logarithmic transseries

The derivation on the differential-valued field Tlog of logarithmic transseries induces on its value group Γlog a certain map ψ. The structure (Γlog,ψ) is a divisible asymptotic couple. We prove that the theory Tlog=Th(Γlog,ψ) admits elimination of quantifiers in a natural first-order language. All models (Γ,ψ) of Tlog have an important discrete subset Ψ:=ψ(Γ∖{0}). We give explicit descriptions of all definable functions on Ψ and prove that Ψ is stably embedded in Γ.

Some Notes on Elimination Properties for The Theory of Riesz MV-Chains

Abstract Riesz MV-algebras are a variety of algebras strongly connected to Riesz spaces. In this short article we investigate some elimination properties of the first-order theory RMV of linearly ordered Riesz MV-algebras and show that RMV admits elimination of quantifiers and uniform elimination of imaginary elements. In the process, we also prove several other results such as modelcompleteness, o-minimality, definability of Skolem functions, and a version of the Di Nola Representation Theorem for Riesz MV-algebras.

Quantifier elimination in quasianalytic structures via non-standard analysis

The paper is a continuation of our earlier article where we developed a theory of active and non-active infinitesimals and intended to establish quantifier elimination in quasianalytic structures. That article, however, did not attain full generality, which refers to one of its results, namely the theorem on an active infinitesimal, playing an essential role in our non-standard analysis. The general case was covered in our subsequent preprint, which constitutes a basis for the approach presented here. We also provide a quasianalytic exposition of the results concerning rectilinearization of terms and of definable functions from our earlier research. It will be used to demonstrate a quasianalytic structure corresponding to a Denjoy-Carleman class which, unlike the classical analytic structure, does not admit quantifier elimination in the language of restricted quasianalytic functions augmented merely by the reciprocal function. More precisely, we construct a plane definable curve, which indicates both that the classical theorem by J. Denef and L. van den Dries as well as \L{}ojasiewicz's theorem that every subanalytic curve is semianalytic are no longer true for quasianalytic structures. Besides rectilinearization of terms, our construction makes use of some theorems on power substitution for Denjoy-Carleman classes and on non-extendability of quasianalytic function germs. The last result relies on Grothendieck's factorization and open mapping theorems for (LF)-spaces. Note finally that this paper comprises our earlier preprints on the subject from May 2012.

Quantifier Elimination for a Class of Intuitionistic Theories

From classical, Fraïissé-homogeneous, ( ≤ ω )-categorical theories over finite relational languages, we construct intuitionistic theories that are complete, prove negations of classical tautologies, and admit quantifier elimination. We also determine the intuitionistic universal fragments of these theories.

An incremental algorithm for DLO quantifier elimination via constraint propagation

The first-order logical theory of dense linear order has long been known to admit quantifier elimination. This paper develops an explicit algorithm that yields an equivalent quantifier free form of its input formula. This algorithm performs existential quantifier elimination via constraint propagation. The result is computed incrementally using functional programming techniques. This approach may be of interest in implementing query languages for constraint databases.

Randomizing a Model

A randomization of a first order structure M is a new structure with certain closure properties whose universe is a setKof “random elements” of M. Randomizations assign probabilities to sentences of the language of M with new constants fromK. Our main theorem shows that all randomizations of M are models of the same first order theoryT, which has a nice set of axioms and admits elimination of quantifiers. Moreover, the class of substructures of models ofTis characterized by a natural setVof universal axioms ofT, so thatTis the model completion ofV.

Elimination of algorithmic quantifiers for ordered differential algebra

In [2], Singer proved that the theory of ordered differential fields has a model completion, i.e, the theory of closed ordered differential fields, CODF. As a result, CODF admits elimination of quantifiers. In this paper we give an algorithm to eliminate the quantifiers of CODF-formulas.

Morphisms and almost-periodicity

We give a criterion, using oriented graphs, to decide whether an infinite word generated as fixed point of an expanding morphism on a finite alphabet is (effectively) almost-periodic. We present a non-almost-periodic system of almost-periodic predicates and deduce from it an example of structure 〈 N; S, <, P〉, where P is a set of unary predicates, whose first-order theory is decidable but which does not admit quantifier elimination.

A note on valuation definable expansions of fields

In this note, we consider models of the theories of valued algebraically closed fields and convexly valued real closed fields, their reducts to the pure field or ordered field language respectively, and expansions of these by predicates which are definable in the valued field. We show that, in terms of definability, there is no structure properly between the pure (ordered) field and the valued field. Our results are analogous to several other definability results for reducts of algebraically closed and real closed fields; see [9], [10], [11] and [12]. Throughout this paper, definable will mean definable with parameters.Theorem A. Let ℱ = (F, +, ×, V) be a valued, algebraically closed field, where V denotes the valuation ring. Let A be a subset ofFndefinable in ℱv. Then either A is definable in ℱ = (F, +, ×) or V is definable in.Theorem B. Let ℛv = (R, &lt;, +, ×, V) be a convexly valued real closed field, where V denotes the valuation ring. Let Abe a subset ofRndefinable in ℛv. Then either A is definable in ℛ = (R, &lt;, +, ×) or V is definable in.The proofs of Theorems A and B are quite similar. Both ℱv and ℛv admit quantifier elimination if we adjoin a definable binary predicate Div (interpreted by Div(x, y) if and only if v(x) ≤ v(y)). This is proved in [14] (extending [13]) in the algebraically closed case, and in [4] in the real closed case. We show by direct combinatorial arguments that if the valuation is not definable then the expanded structure is strongly minimal or o-minimal respectively. Then we call on known results about strongly minimal and o-minimal fields to show that the expansion is not proper.

Stable structures homogeneous for a finite binary language

LetL be a finite relational language andH(L) denote the class of all countable stableL-structuresM for which Th(M) admits elimination of quantifiers. ForM ∈H(L) define the rank ofM to be the maximum value of CR(p, 2), wherep is a complete 1-type over O and CR(p, 2) is Shelah’s complete rank. IfL has only unary and binary relation symbols there is a uniform finite bound for the rank ofM ∈H(L). This theorem confirms part of a conjecture of the first author. Intuitively it says that for eachL there is a finite bound on the complexity of the structures inH(L).

Partial orderings of fixed finite dimension: Model companions and density

The model companions of the theories of n-dimensional partial orderings and n-dimensional distributive lattices are found for each finite n. Each model companion is given as the theory of a structure which is specified. The model companions are model completions only for n = 1. The structure of the model companion of the theory of n-dimensional partial orderings is a lattice only for n = 1. Each of the model companions is seen to be finitely axiomatizable, and a set of basic formulas, each of which is existential, is specified for each model companion. Finally a topolo-gically natural notion of dense n-dimensional partial ordering is introduced and shown to have a finitely axiomatizable undecidable theory.In this paragraph we shall define the notion of model companion (cf. [4]) and indicate the way in which we shall demonstrate that one theory is the model companion of another in this paper. For T and T* theories in a common language, T* is called a model companion of T if and only if the following two conditions are satisfied: first, Tand T* are mutually model consistent, which means that every model of either is embeddable in some model of the other; secondly, T* is model complete, which means that if and are both models of T* and is a substructure of , then is an elementary substructure of . A definition of model completion may be obtained by strengthening the notion of model companion to also require that T* admit elimination of quantifiers. In all of our examples the model companion will have only one countable model. Although the ℵ0-categoricity of the model companions follows from Saracino [8], we give specific proofs since these proofs fit so naturally in our analyses.

Rings which admit elimination of quantifiers

AbstractWe say that a ring admits elimination of quantifiers, if in the language of rings, {0, 1, +, ·}, the complete theory of R admits elimination of quantifiers.Theorem 1. Let D be a division ring. Then D admits elimination of quantifiers if and only if D is an algebraically closed or finite field.A ring is prime if it satisfies the sentence: ∀x∀y∃z (x =0 ∨ y = 0∨ xzy ≠ 0).Theorem 2. If R is a prime ring with an infinite center and R admits elimination of quantifiers, then R is an algebraically closed field.Let be the class of finite fields. Let be the class of 2 × 2 matrix rings over a field with a prime number of elements. Let be the class of rings of the form GF(pn)⊕GF(pk) such that either n = k or g.c.d. (n, k) = 1. Let be the set of ordered pairs (f, Q) where Q is a finite set of primes and such that the characteristic of the ring f(q) is q. Finally, let be the class of rings of the form ⊕q ∈ Qf(q), for some (f, Q) in .Theorem 3. Let R be a finite ring without nonzero trivial ideals. Then R admits elimination of quantifiers if and only if R belongs to.Theorem 4. Let R be a ring with the descending chain condition of left ideals and without nonzero trivial ideals. Then R admits elimination of quantifiers if and only if R is an algebraically closed field or R belongs to.In contrast to Theorems 2 and 4, we haveTheorem 5. If R is an atomless p-ring, then R is finite, commutative, has no nonzero trivial ideals and admits elimination of quantifiers, but is not prime and does not have the descending chain condition.We also generalize Theorems 1, 2 and 4 to alternative rings.