Quantifier elimination is the removal of all quantifiers (universal and existential) from a quantified formula in order to produce an equivalent quantifier-free formula.
Questions tagged [quantifier-elimination]
119 questions
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Software for solving geometry questions
When I used to compete in Olympiad Competitions back in high school, a decent number of the easier geometry questions were solvable by what we called a geometry bash. Basically, you'd label every angle in the diagram with the variable then use a…
Casebash
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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 $\phi$ there is a quantifier-free formula $\psi$ such that…
Aki Sakaguchi
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How is a quantifier-free formula actually interpreted?
A quantifier-free formula in FOL is simply a formula:
-that contains no quantifiers.
-it possibly has free variables.
How is such a formula interpreted?
My understanding is that if we're interested in satisfiability, we agree to take the existential…
Motorhead
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Understanding Quantifier Elimination
I am struggling to understand Quantifier Elimination as it is treated in Hodges' "A Shorter Model Theory".
The relevant definitions are :
Definition: Take $K$ to be a class of $L$-structures, for $L$ a first-order language. An elimination set for…
EarlyGame
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Quantifier elimination for $\mathbb Z$ as a group?
Question 1: What definable sets should one add to the language to obtain quantifier elimination for the theory of $(\mathbb Z, +)$, i.e. the integers as a group (short of simply Morleyizing)?
An $\omega$-saturated model is given by $\hat{\mathbb Z}…
tcamps
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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 $\mathcal{L}$-formula $\phi(x)$, where $x$ is a free variable,…
quanticbolt
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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, installing QEPCAD on Ubuntu 14, I tried to eliminate…
user64540
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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 language of rings (ordered rings) must be…
Requiem
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How to show that $\mathbb{R}$ in the language of rings does not admit quantifier elimination?
I am studying for an exam in an introductory mathematical logic course. Suppose we are working in the language of rings $\mathcal{L}=(+,-,\cdot,0,1)$. Then $\mathbb{R}$ does not admit quantifier elimination.
First of all I am a bit unsure of what it…
Jarne Renders
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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 $y\geq 0$. Now I have been told that such elimination of…
W.Rether
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uniformity in quantifier elimination
Let $T_0$ be a consistent theory. Let ${\cal T}$ be the set of all complete theories that contain $T_0$.
Is it true that if each $T\in{\cal T}$ eliminates quantifiers so does $T_0$?
Note. The point is about uniformity. Assume for every formula…
Escherica
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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 equivalent quantifier-free sentence (which in turn…
Morgan Sinclaire
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Want to show Quantifier elimination and completeness of this set of axioms...
Let $\Sigma_\infty$ be a set of axioms in the language $\{\sim\}$ (where $\sim$ is a binary relation
symbol) that states:
(i) $\sim$ is an equivalence relation;
(ii) every equivalence class is infinite;
(iii) there are infinitely many equivalence…
Jmaff
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Need a FOL derivation validated. Unclear on universal generalization restriction.
I'm currently working through Suppes' Introduction to Logic, and while the text is excellent, the lack of solutions to the exercises can be frustrating at times. This is one of those times. I've racked my brain, but I can't seem to get best…
Paul Evans
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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, right?
But what is meant when we use the term…
Mary Star
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