Questions tagged [quantifier-elimination]

Quantifier elimination is the removal of all quantifiers (universal and existential) from a quantified formula in order to produce an equivalent quantifier-free formula.

Filter by
Sorted by
Tagged with
0
votes
1answer
108 views

Exercise 3.4.3 in David Marker's “Model Theory”

While studying Model Theory for my exam, I came across the following question: a) Show that the theory of $(\mathbb{Z}, s)$ has quantifier elimination where $s(x) = x + 1.$ b) Show that the ...
6
votes
2answers
264 views

Why is quantifier elimination desirable for a given theory?

We say that a given theory $T$ admits QE in a language $\mathcal{L}$ if for every $\mathcal{L}$-formula, there is an equivalent quantifier free $\mathcal{L}$-formula. That is for every $\mathcal{L}$-...
5
votes
0answers
69 views

How much arithmetic is required to formalize quantifier elimination in Presburger arithmetic?

As we know, Presburger arithmetic can be proved decidable by demonstrating that it admits quantifier elimination, i.e. that there is an algorithm that reduces any sentence in the language to some ...
1
vote
1answer
31 views

Quantificational formatting and going from using logic with words, to using it for math proofs

Background My grasp of propositional logic is pretty good, as is my ability to translate it to and from symbols. Quantificational logic however is a little more challenging. Mostly because I get ...
2
votes
1answer
46 views

Complete theory with quantifier elimination has finite boolean algebra

I have the following problem written down: If $\mathcal{L}$ has a finite signature, with no functions, and $T$ is a complete theory with quantifier elimination, then the boolean algebra of $\mathcal{...
0
votes
0answers
129 views

Show that the theory of $ (\mathbb{Z}, s)$ has quantifier elimination.

I am trying to solve Exercise 3.4.3 from Marker's Model Theory: An introduction which states a) Show that the Theory of $(\mathbb{Z},s)$ has quantifier elimination, where $s(x)=x+1$. Show that this ...
1
vote
1answer
52 views

Proof concerning quantifier elimination and substructures

I'm new to everything involving languages and I'm having trouble proving properties with quantifier elimination. In particular what's below. Suppose we know the following: $Lemma$ (L1): Let $N \leq ...
1
vote
1answer
167 views

Universal instantiation with universal quantifiers within scope

If I have a formula in FOL of this kind $\forall x, [[\forall y, \neg P(x) \vee \neg S(x, y)] \vee [\forall z, A(z) \wedge G(x,z)]]$ I want to remove the universal quantifiers. What's the intuition ...
1
vote
1answer
62 views

Is (o-)minimality preserved by elementary equivalence?

Consider two structures which are elementary equivalent. If one is minimal/o-minimal, can the same be said about the other? The pooint is that one cannot quantify over sets (in first-order logic), ...
1
vote
2answers
72 views

Can every theory $T$ be expanded to a theory with quantifier elimination in the same language?

Consider a theory $T$ in a language $L$. there are several examples of theories which expand to theories in the same language which admit quantifier elimination: ring theory to $ACF_0$, Boolean ...
7
votes
1answer
149 views

Elimination of quantifiers for $\exists x\ x^2=y$

Consider the formula $\exists x\ x^2=y$ with free variable $y$. We know that it is equivalent in $Th(\mathbb R,+,0,\cdot,1, \geq)$ (the complete theory of the ordered field $\mathbb R$) to $y\geq 0$. ...
1
vote
1answer
198 views

Quantifier Elimination in the quantifier free type [closed]

Show that if in the theory of $T$ every quantifier-free type has a unique extension to a complete type, then $T$ has quantifier elimination.
0
votes
0answers
83 views

Lemma for quantifier elimination over Presburger arithmetic.

I want to proof the following lemma taken from the book "The calculus of Computation": Let F be a quantifier-free $\sum_{\mathbb{Z}}$-formula with $free(F) = \{y\}$ representing the set $$S: \{n \...
1
vote
1answer
60 views

clause “elimination” using assumptions in a propositional formula

There are a couple of weeks that I am trying to understand how I can eliminate/remove from the analysis some clauses in a propositional formula. For example, I have this formula: $\phi_m = x \wedge (\...
2
votes
1answer
83 views

Verification of simple quantifier elimination in $T_{acf}$

I'm doing some model theory, and just want to make sure I'm not making a mistake here. We consider $T_{acf}$, the theory of algebraically closed fields. I'm asked to find a $T_{acf}$ equivalent, ...
0
votes
2answers
30 views

Logical equivalence of a given formula

Is it true that $$ a(x) \Rightarrow \forall{y} \varphi(x,y) \equiv \forall{y} \left( a(x) \Rightarrow \varphi(x,y) \right), $$ where $a(x)$ is a quantifier free formula with only one free variable $...
1
vote
2answers
59 views

Unique minimal model for set of quantifier-free sentences

We'll use a language $L$ that has at least one constant symbol. We have a set of quantifier-free sentences ($\Gamma$). We'll say that an $L$-structure is minimal if it has no proper substructure (...
0
votes
1answer
292 views

Can we eliminate quantifiers in predicate logic

If we have multiple quantifiers on a predicate say something like $ \exists x \forall y \forall z P(x,y,z)$, can we eliminate $ \forall z$ first using the universal elimination or do we have to start ...
3
votes
1answer
178 views

Decidability of quantifier-free formulae in Peano- and True Arithmetic

It is well-known that validity in Peano Arithmetic is undecidable. It is less well-known that validity is already undecidable in True Arithmetic (the theory of the standard model of Peano Arithmetic). ...
1
vote
1answer
258 views

Concerning substitution and existential elimination in classic natural deduction using sequents

I am trying to prove $\exists x(P\lor Q)\vdash \exists x P \lor \exists x Q$, so I have: $$\begin{array}{r l l} (1) ~&~~ \exists x (P \lor Q) ~&~ \mbox{[premise]} \\ (2) ~&~ \quad (P \...
1
vote
1answer
311 views

Elimination of quantifiers for the theory of equivalence relations with two infinite classes by back-and-forth

As I said in an earlier question, I'm trying to understand how to obtain elimination sets by way of back-and-forth arguments. Since I'm not totally sure I understood how it works, I wanted to check my ...
4
votes
1answer
371 views

Does the theory of equivalence relations have quantifier elimination?

I am aware that the theory of equivalence relations with infinitely many classes, all of which infinite, has quantifier elimination, as can be seen from the answer to this question. However, does the ...
1
vote
0answers
73 views

Quantifier Elimination Tree

I found this example in "A Course in Model Theory", but don't seem to figure out why it is true. Let $L$ be a language having a unary predicate $P_s$ for each (finite) binary string $s \in \{0,1\}^*$ ...
6
votes
1answer
268 views

Are algorithms for elimination of quantifiers over the reals practical?

I wanted to find the semialgebraic set in the $(a_0,a_1,a_2,a_3)$ space that guarantees that there exists at least one real root of the general polynomial equation of degree 4. For that purpose, ...
0
votes
0answers
22 views

Composing relation with identity yields original relation

Let $\phi: F\rightarrow G$ be a relation and $id_F$ be the identity relation on $F$. Then $\phi\circ id_F = \phi$ . Attempted proof: $$\phi\ \circ id_F = \{(f,g)\in F\times G: \exists f_2\in F((f_1,...
1
vote
1answer
87 views

A test for quantifier eliumination

In David Marker's "Model Theory: An Introduction" book I was trying to prove Corollary 3.1.12 which is left for the reader, but I couldn't reach any solution. The aforementioned corollary states: ...
4
votes
1answer
105 views

If $\mathcal{T}_1$ and $\mathcal{T}_2$ admit quantifier elimination, does $\mathcal{T}$ admit quantifier elimination?

Let $\mathcal{T}_1$ and $\mathcal{T}_2$ be theories with disjoint signatures $\mathcal{L}_1, \mathcal{L}_2$. Form a new language $\mathcal{L} = \mathcal{L}_1 \cup \mathcal{L}_2 \cup \{P_1, P_2\}$, ...
4
votes
1answer
270 views

Hodges exercise 2.7.1: Quantifier elimination in dense linear orderings

In Hodges' A Shorter Model Theory, exercise 2.7.1 tells you to prove theorem 2.7.1, which says that the following five formulas are an elimination $\Phi$ set for the class of all dense linear ...
5
votes
1answer
237 views

Elimination of quantifiers

What does it mean that a theory admits constructive elimination of quantifiers? A theory admits elimination of quantifiers when each formula of the theory is equivalent to a quanifier-free formula, ...
2
votes
1answer
185 views

Simplifying theories with quantifier elimination

Let $\Sigma$ be a theory that has quantifier elimination. I'm trying to show that there is then an equivalent theory $\Sigma^*$, with each $\sigma\in\Sigma^*$ of the form $\forall x\psi(x)$ or $\...
4
votes
1answer
428 views

Quantifier elimination for theory of equivalence relations

Let $\mathcal{L}=\{\sim\}$ and $\Sigma_\infty$ be the set of axioms stating that: (i) $\sim$ is an equivalence relation (ii) Every equivalence class is infinite (iii) there are infinitely many ...
0
votes
0answers
62 views

Quantifier elimination in the structure of exponential sums

We consider the language $L=\{+, -, ' , T, 0, 1\}$ Let $\text{Exp}(\mathbb{C})$ (the exponential sums) be the structure in that we interpret $L$. We define $\text{Exp}(\mathbb{C})$ as the set of ...
2
votes
1answer
479 views

I don't understand how the theory of algebraically closed fields admits quantifier elimination

I was reading the wiki page about quantifier elimination and it says that the theory of algebraically closed fields is decidable using quantifier elimination, what I understand by this is that all ...
3
votes
1answer
72 views

Elimination of quantifiers in the strucure of polynomials and in the structure of exponentials

I am looking at the elimination of quantifiers. In my notes there is the following: $L=\{+, ' , T, 0, 1\}$ ($"="$ is meant to be included in $L$) First-order Logic: $Q_1 x_1 \dots Q_m x_m \ \ [\...
1
vote
2answers
98 views

Real closed field with the restricted exponential function

Is the theory of real closed fields augmented with the restricted exponential function decidable? If so, can someone explain that decision procedure?
1
vote
1answer
334 views

About the proof of a test for quantifier elimination.

I've been reading D. Marker's book on Model theory. In the part dealing with quantifier elimination there's a corollary I've been trying to prove without any luck: Corollary 3.1.6 Let $T$ be an $L$-...
1
vote
1answer
209 views

Show every boolean combination of $\mathcal{L}$-formula is equivalent one with quantifiers.

This is part 2 of a question I asked here: Prove this claim about language and structures. The setting is that suppose $\phi_1,\ldots,\phi_n$ are $\mathcal{L}$-formulas and $\psi$ is a Boolean ...
2
votes
1answer
385 views

Model complete theories without quantifier elimination

As we know, if a theory $T$ admits quantifier elimination, then $T$ is model complete. What are the simplest examples that show that the converse is not true?
2
votes
1answer
295 views

How is quantifier elimination accomplished in second and higher order logic?

In first order logic we can eliminate existential quantifiers using a second order equivalence relation: $\forall$x$\exists$y P(x, y) $\iff$ $\exists$f$\forall$x P(x, f(x)) Dropping the existential ...
3
votes
0answers
94 views

Ordering of $\mathbb{R}$ not quantifier-free definable in $L_{R}$

I'm reading David Marker's book "Model Theory: An Introduction" and I'm trying to solve Exercise 3.4.24 which is stated as follows: Let $x$ and $y$ be algebraically independent over $\mathbb{R}$. a) ...
3
votes
1answer
930 views

Why is Skolem normal form equisatisfiable while the second order form equivalent?

I asked in another question when is it appropriate to de-Skolemize a statement. The answer, I'm not sure I'm satisfied with yet, relies on a second order logical equivelance, but Skolem normal form ...
1
vote
2answers
140 views

Trying to understand negation of quantifiers

Trying to understand the negation of the following: For this: ∀x~P(x) I have this as negation: ~∃xP(x) For this: ~∃x(∀yP(y) Λ Q(x)) I have this: ∀x(~∃yP(y) V ~Q(x)) Are these correct? If not please ...
6
votes
2answers
464 views

Proper definition of quantifier elimination

I study Marker book "Model Theory, An Introduction". Definition 3.1.1 on page 72 defines "theory T has quantifier elimination". A theory $T$ has quantifier elimination if for every formula $\phi$ ...
5
votes
2answers
3k views

How is a quantifier-free formula actually interpreted?

My understanding is that a quantifier-free formula in FOL is simply a formula that contains no quantifiers, just possibly free variables. How is such a formula interpreted? My understanding is that if ...
2
votes
1answer
101 views

Why Quantifier Free Formulas define Linear Functions.

How do you prove that functions definable by quantifier-free formulas must be linear? I am interested for the structure $\langle Q,+,0\rangle$.
0
votes
2answers
52 views

Resultant$(f,g)$ says when there exist $\phi,\psi$ such that $\psi f + \phi g = 0$. How do I actually find them?

If $f$ and $g \in k[X]$ are two polynomials such that $\textrm{Res }(f,g)=0$ how do I find $\phi$ and $\psi$ with $\deg \phi < \deg f$ and $\deg \psi < \deg g$ such that $$\psi f +\phi g =0$$
7
votes
1answer
357 views

Characterization of the First Order Theory of Ordered Abelian Groups via Quantifier Elimination

In the paper "Elimination of Quantifiers in Algebraic Structures" Macintyre, McKenna and van den Dries, proved that every field (ordered field) whose theory admits quantifier elimination in the ...
1
vote
2answers
444 views

Herbrand Logic-Fitch System

Given $$\forall x.(p(x) \implies q(x))\quad and \quad p(a)$$ use the Fitch system to prove q(a) I have started: $$\\$$ $$1) \forall X.(p(X) \implies q(X)) \qquad (Premise)$$ $$2) p(a) \qquad (...
2
votes
2answers
446 views

Proof of Robinson's test

I have been working with Tent and Ziegler's Model Theory. I am on the Quantifier elimination chapter, and there they mention Robinson's test. It says that, for an $L$-theory $T$ three statements ...
3
votes
1answer
201 views

Quantifier Elimintion of $(\Bbb{Q},+,0)$

I want to prove that the structure $(\Bbb{Q},+,0)$ has Quantifier Elimination. I can prove it for some simple basic formulas, but what if i get a formula which says that i have a linear combination, ...