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13
votes
1answer
322 views

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 ...
0
votes
0answers
143 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 ...
2
votes
1answer
82 views

Complexity of Deciding Feasibility of a system of linear inequalities over restricted variables

I am working out an interesting problem and would like some help with this particular sub problem: Suppose we have a matrix $ M =\left\lbrace a_{ij}\right\rbrace $ of size $n\times m$ where $ ...
4
votes
1answer
291 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: ...
19
votes
3answers
612 views

Is being rational decidable?

Given a real number uniquely defined by a finite system of equations and inequalities with rational coefficients involving the standard elementary functions only. Is it decidable whether this number ...
5
votes
0answers
142 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 ...
7
votes
0answers
207 views

Proving Richardson's theorem for constants

(I asked this a little over 3 months ago on math.SE, and when I initially re-asked here, no one had responded there. $\:$ After I re-asked here, Eric Towers responded there, since I had forgotten to ...
1
vote
1answer
89 views

Recurrence relations with polynomial coefficients: an undecidable problem

I read once (somewhere) that solving a recurrence relation with polynomial coefficients, in the general case, is an undecidable problem. I can't remember the exact reference and I've been trying to ...
4
votes
0answers
109 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, ...
2
votes
0answers
84 views

Positive existential theory of $(\mathbb{Z}; +, |_n)$

I am reading a paper and there is the following theorem: Let $n$ be a fixed integer, and $n >1$. Denote divisibility in $\mathbb{Z}[\frac{1}{n}]$ by $|_n$, thus for all $x, y \in ...
12
votes
0answers
175 views

Is tightness decidable?

Given a contact structure on a three-manifold, is there an algorithm to decide whether or not it tight? For concreteness' sake, let's agree to represent the given contact three-manifold via an ...
0
votes
2answers
145 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?
-1
votes
1answer
243 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 ...
1
vote
0answers
30 views

Relation between indexed languages (OI-macro or context-free tree) and scattered context languages

I'm not sure about the relation between indexed languages (generated by indexed grammars--Aho) and scattered context languages (generated by scattered context grammars--J Hopcroft). I think that ...
7
votes
1answer
183 views

Decidability of an Algebraic System in Real Numbers

Is there an algorithm to decide whether an algebraic system \begin{gathered} {f_1}({x_1}, \ldots ,{x_n}) = 0 \hfill \\ \vdots \hfill \\ {f_m}({x_1}, \ldots ,{x_n}) = 0 \hfill \\ ...
2
votes
2answers
163 views

Pseudo-decision procedures for first order arithmetic

I was reading this paper http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.117.2911&rep=rep1&type=pdf in which the author describes an algorithm, based on Groebner basis, ...
11
votes
0answers
377 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 ...
20
votes
2answers
911 views

What do you do if you believe a problem is undecidable?

While the title of this question is subjective, I hope to make what I'm looking for quite concrete. The first, and main question is this: If you believe that a problem you are working on is formally ...
2
votes
0answers
43 views

Decidability of first order theory of subclasses of posets

Is the first order theory of finite posets known to be undecidable? Does anyone know a survey about such results?
3
votes
1answer
221 views

Is the equational theory of commutative vN regular rings decidable?

I wanted to check whether $A(x,y):=\frac{xy}{x+y}$ is an associative operation in every commutative vN regular ring. Now $A(-1,A(1,1))=A(-1,\frac{1}{2})=1\neq 0 =A(0,1)=A(A(-1,1),1)$. On the other ...
11
votes
2answers
935 views

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 ...
6
votes
2answers
166 views

Algorithm for determining when polynomial iteration is bounded?

Let $f: \mathbb{Q}\to \mathbb{Q}$ be a polynomial map with rational coefficients. Let $p\in \mathbb{Q}^n$. Is there a known algorithm that given this data determines whether or not the iterates ...
8
votes
2answers
533 views

Undecidable puzzles

There are plenty of popular NP-hard puzzles, for example, generalized Sudoku ($n^2 \times n^2$-board), Flow (I cannot give a source for this), Minesweeper, etc. Recently, I read a bit about aperiodic ...
1
vote
0answers
71 views

What is the generic complexity of First Order Predicate Calculus?

I suspect that it should be the same as that of the Turing machine halting problem, which wikipedia gives as GenP and attributes this result to Hamkins and Miasnikov, but I am not sure. Is the generic ...
4
votes
1answer
271 views

Quantifier elimination vs decidability

Quantifier elimination is used as a technique to get decidability (e.g. $Th( \mathbb{N}, +)$ ) of theories, but typically one has to go over to some expansion. Are there examples of theories which are ...
1
vote
0answers
84 views

unique types and decidability

Suppose $\mathcal{M}$ is an infinite structure which has the property that every type that is realised is realised uniquely. Also assume that every element of $\mathcal{M}$ is definable but there is ...
2
votes
0answers
132 views

What is the general feeling for Hilbert's 10th problem for Q?

We know that Hilbert's 10th problem for $\mathbb{Z}$ is undecidable. I was wondering whether there is a strong opinion in the mathematical community on the decidability of Hilbert's 10th for ...
4
votes
1answer
149 views

Decidability of a matrix product being the identity

Given a finite set $S$ of $n\times n$ integer matrices, it is known that for $k\geq 3$ it is undecidable whether some product of them (allowing repetitions) is the zero matrix (called the mortality ...
4
votes
0answers
160 views

The theory of two finite linear orders

My colleague Matthias Baaz is looking for a reference for the following question (or possibly theorem): Let T be the "theory of pairs of finite linear orders". That is, consider all finite ...
5
votes
2answers
188 views

Decidability of differential equations

Is there anything well-known about the algorithmic decidability of the satisfiability of an ODE $\dot{x}=f(x)$, $x: [0,1]\to R^n$ with an initial condition $x(0)=x_0$, given that $f(x)$ belongs to ...
0
votes
1answer
161 views

Can aperiodic tilings be non-hierarchical? and confusion over domino problem

Anyone experienced with the undecidability of aperiodic tiling? It's related to the halting problem which Turing proved was undecidable in the 30's and basically superimposes tiles onto other tiles ...
10
votes
2answers
365 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 ...
21
votes
0answers
410 views

Decidability of equality of elementary expressions

In the following definition the term expression is to be understood as a finite tree built from formal symbols without any predefined meaning assigned to them. Define the set $\mathcal{E}$ of ...
27
votes
3answers
684 views

Is the field of constructible numbers known to be decidable?

By the field of constructible numbers I mean the union of all finite towers of real quadratic extensions beginning with $\mathbb{Q}$. By decidable I mean the set of first order truths in this field, ...
4
votes
1answer
781 views

Is compass and straight edge geometry complete?

Euclid's first three postulates are the basis of compass and straight edge constructions which are as complex as arithmetic. The constructions themselves may be expressed as a formula with each of ...
2
votes
1answer
136 views

Recognizing parallelogram tilings from their vertex set

Suppose I have a tiling of the plane with parallelograms where the sides of the parallelograms come from a specified finite set of vectors. If I only have access to the vertices of this tiling I may ...
8
votes
1answer
1k views

Is the first-order theory (with =) of real numbers with addition and multiplication complete and decidable?

Due to Andreas Blass's answer to my question "Is the feasibility of a system of nonlinear, non-convex equations (inequalities) decidable?", i have now investigated real closed fields (RCF), because i ...
2
votes
1answer
176 views

Is the feasibility of a system of nonlinear, non-convex equations (inequalities) decidable?

I would like to know whether the following problem is decidable. Is the system $x^T Q_i x + r_i = 0 \mbox{ for } i = 1, ..., k$ $x^T Q_j x + r_j \neq 0 \mbox{ for } j = k+1, ..., t$ feasible, ...
2
votes
0answers
105 views

Extend Lowenheim's decidability result to fragment of second-order logic

Since relational monadic first-order logic has finite model property, its SAT problem is decidable. In H.Behmann's paper, he extended this result to fragment of SOL where all predicates, free and ...
3
votes
2answers
260 views

Decide the limit of a decreasing sequence of real number

Given a bounded decreasing sequence of rational number $a_0,a_1,\cdots$, then we know that it has limit $a$. Suppose this sequence satisfy that $|a_k-a|<1/2^k$ for any $k$. The sequence is given ...
7
votes
1answer
722 views

Could the Jacobian conjecture be undecidable?

Most of us know the Jacobian conjecture. Here's a version below for fixed positive integers $d$ and $n$: $J(d,n)$: If $f: C^n \rightarrow C^n$ is a polynomial map of degree $d$, and if the Jacobian ...
27
votes
1answer
463 views

Decidability of equality of expressions built using 1,+,-,*,/,^

Consider expressions built using number $1$, arithmetical operators $+, -, *, /$ and exponentiation ^ (in case of multiple values, the principal value is assumed, the same way as it implemented in ...
16
votes
3answers
466 views

Is the isomorphism problem for amenable groups decidable?

Is it algorithmically decidable if two finitely presented amenable groups are isomorphic? Or slightly different: Does there exist a family of amenable groups (indexed by natural numbers) for which ...
7
votes
1answer
321 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 ...
0
votes
1answer
197 views

Can one always Decide whether a Systems of Nonlinear Equations with Bilinear terms is Feasible?

I have come to a point in my PhD research were i need to prove that a particular decision procedure is decidable or not. And if i can solve the sub-problem described below, i shall have proved it. The ...
0
votes
0answers
50 views

Is the question whether a FO formula F has a model of size k (k is a finite number) decidable?

Hi all, can someone please tell me whether the question whether a FO formula F has a model of size k (where k is a finite number) is decidable? Thanks in advance! TL
5
votes
2answers
303 views

(Un)Decidability of the root existence problem for functions with bounded domain

The problem whether a real function $f$ has a root or not is undecidable, given that $f$ is from a class of functions including polynomials and the sine function ...
31
votes
3answers
3k views

For which Millennium Problems does undecidable -> true?

Three good answers were received — by Alex Gavrilov, Bjørn Kjos-Hanssen, and Terry Tao — and the bounty has been awarded (somewhat arbitrarily) to Alex Gavrilov. The answers ...
16
votes
4answers
842 views

Proving univariate polynomials (defined by sums, binomial coeffs, etc.) are nonnegative: is it 'routine'?

My colleagues and I are working on a project related to an old paper of C. Borell and we have boiled it down to the following problem: Show, for all integers $1 \leq i \leq k$, that the univariate ...
8
votes
1answer
276 views

Algorithmic decidability of equality in the ring of periods

Suppose two elements of the ring of periods are given by their systems of polynomial inequalities with rational coefficients. Is there a known algorithm deciding their equality? Is it known if their ...