All Questions
Tagged with decidability computability-theory
36 questions
2
votes
0
answers
78
views
Is this variant of post correspondence problem undecidable?
The post correspondence problem, as defined by wikipedia, is undecidable. The problem is defined as follows.
Let $A$ be an alphabet with at least two symbols. The input of the problem consists of ...
- 21
8
votes
1
answer
1k
views
Why don't Zeilberger and Gosper's algorithms contradict Richardson's theorem?
Richardson's theorem proves that whether an expression A is equal to zero is undecidable. A is in this case an expression, constructed from $x,e^x,\sin(x)$ and the constant function $\pi$ and $\ln(2)$ ...
- 157
2
votes
1
answer
176
views
How can Kőnig's Lemma be expressed in Monadic Second-Order Logic of 2 Successors?
I read the following on Wikipedia's page on Monadic Second-Order Logic of Two Successors (MS2S):
Weak S2S (WS2S) requires all sets to be finite (note that finiteness
is expressible in S2S using Kőnig'...
- 123
9
votes
2
answers
954
views
What theories are larger than the real closed field but still decidable?
It's well known that sentences about the real closed field can be decided by algorithm and the complexity of this is about $d^{2^{O(n)}}$ where $d$ is the product of the degrees of polynomials in the ...
- 2,689
2
votes
1
answer
145
views
Computability of fillability of unit cube in $\mathbb{R}^n$ by $k$ $\varepsilon$-balls
Let $\mathbb{N}$ denote the set of positive integers. We define a relation $R \subseteq \mathbb{N}^4$ in the following way:
$(p,q,n,s)\in R$ if and only if there is $S\subseteq [0,1]^n$ with $|S| = s$...
- 48.8k
2
votes
0
answers
73
views
Is parquetability decidable?
Let $T\neq \emptyset$ be a finite subset of $\mathbb{Z}\times\mathbb{Z}$. We say that $\mathbb{Z}^2 = \mathbb{Z}\times\mathbb{Z}$ is parquettable by $T$ if there is a partition $\frak P$ of $\mathbb{Z}...
- 48.8k
2
votes
1
answer
154
views
Axiomatization of S2S
What is a reasonable axiomatization of S2S?
S2S is the monadic second order theory with two successors (Wikipedia link). It has finite binary strings, operations $s→s0$ and $s→s1$ on strings, and ...
- 7,525
9
votes
1
answer
372
views
Decidable theories with arbitrary complexity
Are there complete finitely axiomatizable first order theories (with equality) with arbitrarily high computational complexity?
Here, arbitrarily high (computational) complexity means that for every ...
- 7,525
13
votes
3
answers
834
views
Undecidable infinite analogs of NP-complete problems?
In the paper Some undecidable problems involving edge-coloring of graphs, Burr proves that a certain k-coloring problems for certain infinite graphs (however, with finite descriptions - here "...
21
votes
1
answer
1k
views
Is "almost-solvability" of Diophantine equations decidable?
Say that a Diophantine equation is almost-satisfiable iff for each $n\in\mathbb{N}$ it has a solution mod $n$. Trivially genuine satisfiability over $\mathbb{N}$ implies almost-satisfiability, but the ...
- 21.1k
6
votes
1
answer
334
views
Given some recursive function, can we effectively associate it a polynomial as in the DPRM theorem?
I'm interested in the following assertion about the Davis-Putnam-Robinson-Matijasevich theorem
Given a recursive function $f:\mathbb{N}\rightarrow\mathbb{N}$, i.e. its index, we can effectively get ...
- 75
7
votes
0
answers
274
views
Is decidability reducible to unique decidability (perhaps in multilinear polynomial situations)?
Given a Diophantine equation it is not decidable if it has integer solution.
I. Is there a Diophantine set $\mathcal D_{unique}$ satisfying the properties
every member in $\mathcal D_{unique}$ is a ...
- 13.9k
3
votes
0
answers
116
views
Variation in decidability of diophantine equations with field extension
Let $O_k$ be the ring of integers in a subfield $k$ of $\overline{\mathbb{Q}}$. Let's call an equation $p(x_1, \dots, x_n) = 0$ where $p$ is a polynomial in $n$-variables $x_1, \dots, x_n$ with ...
- 31
4
votes
2
answers
278
views
Checking for finite fibers in hash functions
Let $\{0,1\}^{<\omega}$ denote the collection of finite binary sequences. By a hash function we mean a computable map $$h: \{0,1\}^{<\omega} \to \{0,1\}^n$$ for some fixed $n\in\omega$. Define $\...
- 48.8k
3
votes
0
answers
122
views
Post correspondence problem: Busy beaver variant
The Post correspondence problem (Wikipedia link) is to decide for $k$ pairs of strings $$(a_1,b_1), (a_2, b_2), ..., (a_k,b_k),$$ if there exists a finite sequence of numbers $c_j, 0\le j\le j_\max $ ...
- 2,811
1
vote
1
answer
199
views
decidability of regularity of a language depending on representation
It is well known that many decision problems for regular languages are decidable. However, the proofs seem to rely on a witness of the regularity of said language, be it an automaton, a grammar, a ...
2
votes
1
answer
223
views
Decidability of S2S with real numbers
Is the theory of natural numbers and functions $ℕ → ℝ$ decidable under:
- for natural numbers: $\mathrm{succ1}(n) = 2n+1$; $\mathrm{succ2}(n) = 2n+2$; equality
- for functions: pointwise addition and ...
- 7,525
4
votes
0
answers
164
views
Subgroup membership problem for Noetherian groups
I am interested in the status of the subgroup membership problem (MP) for finitely presented Noetherian groups. That is, given a finite presentation $\langle X,R\rangle$ for a Noetherian group,
\begin{...
- 343
5
votes
0
answers
356
views
minimum size of undecidable quadratic diophantine problems
According to Matiyasevich, the existence of integer solutions of systems of polynomial equations with integer coefficients is undecidable. By introducing additional variables substituting factors of ...
- 3,873
2
votes
0
answers
218
views
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
1
vote
1
answer
299
views
Is Calculus of Constructions type inhabitance semi-decideable?
I'm wondering if type inhabitance for the calculus of constructions is semi-decideable. I know the following:
System F inhabitance and, correspondingly, second-order unification are semi-decideable
...
- 31
2
votes
0
answers
91
views
Matrix (geometric sum) orbit problem
Is the following algorithmic problem known to be decidable/undecidable?
Input: an element $\mathbf{v} \in \mathbb{Z}^n$, a matrix $\mathbf{A} \in GL_n(\mathbb{Z})$, and a subgroup $H \leqslant \...
- 343
36
votes
3
answers
2k
views
Is it decidable to check if an element has finite order or not?
Suppose we have a finitely presented group $G$ with decidable word problem. Is it decidable to check whether a given element $x\in G$ has finite order or infinite?
- 1,281
1
vote
1
answer
183
views
Understanding the paper: "Guarded Fixed Point Logic"
This question is specifically about the paper "Guarded Fixed Point Logic" by Gradel and Walukiewicz. Among other things they prove the decidability of the satisfiability problem for Fixpoint Loosely ...
- 111
3
votes
1
answer
156
views
Relationship between computational undecidability and axiomatic undecidability
On surface, these seem two completely different class of problems. One class represent statements which can't be proved or disproved in an axiomatic theory. For example
One can write down a ...
- 31
6
votes
1
answer
281
views
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
0
votes
0
answers
197
views
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 (...
- 309
9
votes
1
answer
753
views
List of finitely presented groups with undecidable word problem
Is there any reasonably updated list of (representative) examples of finitely presented groups with undecidable word problem?
By "representative" I mean "avoiding obvious redundancy", i.e. examples ...
- 343
5
votes
1
answer
243
views
Undecidability of the existential theory
Do you know if I can find the proof that the existential theory of $\mathbb{Z}$ with the structure of addition , divisibility and the relation $(\exists s \in \mathbb{Z})m=np^s$ is undecidable, ...
- 309
0
votes
2
answers
284
views
Undecidable set of problems [closed]
Is there some set of problems, for which determining if given problem is decidable or not is itself undecidable?
- 69
-1
votes
1
answer
550
views
Are limits decidable? Should definitions be decidable? [closed]
This question is about the Turing computability of the $\epsilon-N$ definition of a limit of an infinite sequence $S$. First, a proposition:
There cannot exist a Turing Machine $M$ which, given a ...
- 718
17
votes
0
answers
808
views
Decidability of $x^3+y^3+z^3 = c$
I wondering if it is known whether the following problem is algorithmically decidable or undecidable by Turing machines: given an integer c, determine if there are integers $(x,y,z)$ such that $x^3+y^...
- 171
10
votes
2
answers
455
views
Is equivalence of functions built from nested exponentiations a decidable problem?
Let $\mathcal{E}$ be the minimal set of symbolic expressions (without any predefined meaning) such that
The symbol $x$ is in $\mathcal{E}$, and
If expressions $P,Q\in\mathcal{E}$, then the ...
- 1,699
10
votes
2
answers
1k
views
Decidability of periodic tilings of the plane
I'm interested in tilings of the plane by squares, with labels on the edges. It's well known that (1) the question "can one tile the plane with the following finite set of tiles?" is undecidable, and (...
- 2,519
17
votes
7
answers
2k
views
Non-constructive proofs of decidability?
Are there examples of sets of natural numbers that are proven to be decidable but by non-constructive proofs only?
29
votes
3
answers
3k
views
Is the theory of categories decidable?
There are a lot of theorems in basic homological algebra, such as the five lemma or the snake lemma, that seem like they'd be more easily proven by computer than by hand. This led me to consider the ...