All Questions
Tagged with model-theory decidability
21
questions
3
votes
0
answers
97
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Decidability of theory of modules over a ring of finite representation type
I have read from Mike Prest's model theory for modules (London lecture note series) chapter 17 that a Ring of finite representation type has a decidable theory of modules. Here decidability was ...
- 31
7
votes
1
answer
237
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What is known about first order logic of $\mathbb{N}$ with + and a unary predicate?
In "Weak Second-Order Arithmetic and Finite Automata", Büchi claims that the first order theory of $\mathbb{N}$ with + and a predicate for recognizing powers of 2 ($Pw_2$) is expressively ...
- 429
4
votes
2
answers
645
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Tarski's original proof of quantifier elimination in algebraically closed fields
I am currently helping teach a course about foundations of mathematics, which has thus far focused mostly on propositional and first-order logic. As part of the course, the students are each required ...
- 741
4
votes
2
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278
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Quantifier elimination in $S^1$
Does quantifier elimination (by cylindrical decomposition) work for systems of polynomial equations and inequalities where some or all of the variables are complex numbers of unit modulus, rather than ...
- 19.3k
12
votes
1
answer
928
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Testing whether $e^x+ax^2+bx+c$ has a zero
What is the simple test with exponential polynomials to determine whether
$$f(x)=e^x+ax^2+bx+c$$ has a positive zero?
This was prompted by the question about discriminants here. We have an ineffective ...
user44143
12
votes
2
answers
551
views
Decidability of a first-order theory of hyperreals
The theory of real closed fields is decidable. The hyperreals satisfy that theory, so we can interpret statements in the theory of real closed fields as being about hyperreals.
If we add a unary ...
- 6,298
0
votes
1
answer
135
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Linear programming with exponential inequalities and rational variables
If we are given a set of real linear inequalities then using elimination theory or just linear programming we can decide. If the program also has inequalities of form $2^x\leq g$ in addition to linear ...
- 1,826
7
votes
1
answer
263
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Is $\mathbb{F}_{p}(t)^{h}$ an elementary substructure of/existentially closed in $\mathbb{F}_{p}((t))$?
It is a well-known fact that the Henselization of the function field $\mathbb{F}_{p}(t)$ in regard to the $t$-adic valuation is $\mathbb{F}_{p}(t)^{alg} \cap \mathbb{F}_{p}((t))$, so of course $\...
- 311
6
votes
1
answer
245
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Example of applying real quantifier elimination algorithm for polynomials
Sorry if any of this is unclear, or doesn't make much sense, I'm still trying to figure it out, a practical example such as this would likely help me understand better than anything else. I have read ...
- 61
2
votes
0
answers
217
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The elementary theory of finite commutative rings
I have wondered the decidability of elementary theory of finite commutative rings. Since we know that the elementary theory of finite fields is decidable shown by J.Ax (The Elementary Theory of Finite ...
- 151
6
votes
1
answer
270
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Deciding isomorphism between graphs which interpret in the pure set
I am interested in the following decision problem:
Given descriptions of two graphs $G,H$ which interpret in the pure set $\mathbb N=(\{0,1,2,\ldots\},=)$, decide whether $G$ and $H$ are isomorphic....
- 618
16
votes
1
answer
583
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Is this theory decidable?
It is well-known that both Presburger arithmetic (by contrast with Peano arithmetic) and Tarski geometry are decidable. I was in the shower this morning and wondered whether there exists an elegant ...
- 12.3k
0
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0
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196
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Is the positive existential theory undecidable?
Could you tell if the positive existential theory of $\mathbb{C}[e^{\mu x} \mid \mu \in \mathbb{C}]$ is undecidable in the language $\{+, \cdot , \frac{d}{dx} , 0, 1, e^x\}$ ?
How can we prove the (...
- 299
6
votes
1
answer
480
views
Show that the positive existential theory is undecidable
To show that the positive existential theory of $\mathbb{C}[t, e^{\lambda t} \mid \lambda \in \mathbb{C}]$ in the language $\{+, \cdot , ' , 0 , 1, t\}$ is undecidable we have to prove the following: $...
- 299
15
votes
2
answers
1k
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Decidability of decidability
The questions I'm going to ask are non formal because they concern decidability of decidability, and I couldn't find any references on that after some quick searches. I hope that this thread is still "...
- 438