Questions about Turing computability and recursion theory, including the halting problem and other unsolvable problems. Questions about the resources required to solving particular problems should be tagged (computational-complexity).

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5
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2answers
216 views

Can it be decidable for any polynomials to have the intersecting point?

Give system of polynomials$$P_1(x_1,x_2,\dots,x_n)=0,$$$$\vdots,$$$$P_k(x_1,x_2,\dots,x_n)=0$$ Can it be decidable for thoses polynomials to have the intersecting point ?
0
votes
2answers
91 views

If all infinite r.e. languages have an infinite recursive subset, then do co-r.e. languages not have such subsets?

If all infinite r.e. languages have an infinite recursive subset, then can we take it to be true that co-r.e. languages do not have such subsets by complemence?
0
votes
1answer
46 views

Is the following statement true ? If $L$ is a decidable language and $L' \subseteq \; L$, then $L'$ is also decidable ? Prove your answer is correct [closed]

Is the following statement true ? If $L$ is a decidable language and $L' \subseteq \; L$, then $L'$ is also decidable ? Prove your answer is correct I can't figure out this question. Any tips ?
1
vote
1answer
53 views

Are well-orders of the same recursive length recursively isomorphic?

If the ordinal length of $A$ and $B$ is the same recursive ordinal, does it follow that there is a recursive one-one order-preserving correspondence between $A$ and $B$?
2
votes
0answers
46 views

How to find the shortest path of a graph in a turing machine

I'm reading about Turing machine and I saw some examples as: Let $M_{1}$ a Turing Machine and the language $B = \{w\#w \vert w \in \{0,1\}^{*}\}$, We want $M_{1}$ to accept if its input is a member of ...
1
vote
1answer
20 views

Computable function that enumerates the primitive recursive functions

I'm trying to construct a computable function $f:\omega^2\to\omega$ such that For all $e\in\omega$, $x\mapsto f(e,x)$ is primitive recursive. If $g:\omega\to\omega$ is primitive recursive, then ...
7
votes
0answers
135 views

Fixed points in computability and logic

I asked this question on CS.SE, too: http://cstheory.stackexchange.com/questions/27322/fixed-points-in-computability-and-logic I would like to understand better the relation between fixed point ...
4
votes
2answers
158 views

Is it possible to deduce Godel's first incompleteness theorem from Chaitin's incompleteness theorem?

I want to ask if it is possible to deduce Godel's first incompleteness theorem from Chaitin's incompleteness theorem. I am reading the following AMS-Notice article. The authors claim that: The ...
0
votes
1answer
131 views

Is Bell's Notion of “Abstract Set” Flawed?

Consider the following definition of "abstract set" given by John L. Bell (who wrote the book "Set Theory: Boolean-valued Models and Independence Proofs") from his preprint "Abstract and Variable ...
21
votes
5answers
2k views

Given any computable number, is there any algorithm to decide whether it is transcendental?

Given any computable number $a_c$, is there any algorithm to decide whether it is transcendental? Definition of “computable number”: According to Ming Li and Vitanyi, a real number $x=0.x_1x_2\ldots$ ...
2
votes
2answers
150 views

Different models of ZF disagree on equality of explicit recursively enumerable sets

Assuming that ZF is consistent, are there two recursively enumerable sets defined by explicit enumerators that are the same in one model of ZF+Con(ZF) but different in another model of ZF+Con(ZF)? If ...
2
votes
0answers
19 views

Kleene normal form : elementary?

The Kleene normal form explains there are primitive recursive functions $T$ (a predicate indeed) and $U$ such that for any computable function $\phi_n$, and for any $x\in\mathbb N$ : ...
0
votes
0answers
33 views

How to track unboundedly many changes?

Suppose that I have a piece of paper with 0 on it (and nothing else). Suppose that, at each instant, I can either replace what is on the paper by writing either 0 or 1. I say that I change the value ...
3
votes
0answers
93 views

Whats the connection between Turing machine and First order logic?

Today in my Computing class i came across the theorem which states that., If language $L$ and $\Sigma^*\setminus L$ are recursively enumerable then L is recursive (total turing machine). Which looks ...
1
vote
0answers
77 views

$n^{\text{th}}$ digit of $\sqrt{2}$ decimal representation is primitive recursive function

An exercise from Maltsev's "Algorithms and recursive functions". Problem: Let $\sqrt{2} = a_0,a_1a_2\dots a_n\dots$ be the decimal representation of $\sqrt{2}$. Show that the function $f(n) = a_n$ ...
0
votes
1answer
57 views

Partial / Total / Primitive recursive functions and recursive enumerability

After having compiled several sources from handbooks or the web, and read some answers posted here, I'm still confused with the question of non recursive enumerability of total recursive functions, ...
1
vote
0answers
25 views

Question about Computability

Q:Suppose $U(n,x)$ is Gödel Universal Function, show that there is $n$ such that $U(n,x)=n+x$ for all $x$ I did some proof but I am mot sure if I am right. Let's consider a computable binary function ...
0
votes
1answer
42 views

show that inverses $\pi_{1},\pi_{2}$ are recursive?

show that one can define inverses $\pi_{1},\pi_{2}$ for $ \langle.,.\rangle$ with$\pi_{1}(\langle m,n \rangle)=m,\pi_{2}(\langle m,n \rangle)=n\ \ \forall n,m$ wich are also recursive?
3
votes
1answer
42 views

Is there a recursive injective and surjective function f:N→PRF?

It is well known and easy to see that it is possible to effectively number Turing Machine codes. That is, there is an injective and surjective recursive mapping $g:\mathbb N\to {\rm TM}$: each Turing ...
1
vote
7answers
143 views

Text books on computability

I collected the following "top eight" text books on computability (in alphabetical order): Boolos et al., Computability and Logic Cooper, Computability Theory Davis, Computability and unsolvability ...
0
votes
0answers
31 views

It is undecidable if The Intersection between Context Free Language and Context Sensitive Language is the empty set

I'm trying to show that the following problem is undecidable: The intersection between a Context Free Language (CFL) and a Context Sensitive Language(CSL) is the empty set. I know that is undecidable ...
0
votes
1answer
28 views

Proving non-regularity of a language

How can I prove $L = (01^n2^n | n\geq 0)$ is not regular? Would it be sufficient to say that $01^p2^p$ is in $L$ and by pumping lemma, $01^p2^p$ can be written as $xyz$ such that $|y|>0, ...
3
votes
1answer
112 views

Undefinable Real Numbers

Disclamer: I'm sure my definition of "definable" may be different than the/a established mathematical one, I am more than interested in learning why/how this is so, but that is not my question Part ...
2
votes
1answer
34 views

Non-decidable $\Pi^0_1$ (effectively closed) classes

Are there non-decidable $\Pi^0_1$ (effectively closed) classes? According to a draft of Effectively closed sets by Cenzer and Remmel, the class $$ P = \{ 0^n1^\omega \mid n \in B\} $$ is a ...
0
votes
0answers
49 views

Proving the intersection and union of two simple sets is simple.

Question: Suppose $A$ and $B$ are simple. Prove that $A \cap B$ is simple and $A \cup B$ is either simple or cofinite. I need to verify that $A \cap B$ and $A \cup B$ are computably enumerable ...
2
votes
0answers
84 views

Challenge on Some Definition on Formal Language & Recursive & Automata

We know set A is countable if A is finite or in a one-to-one mapping to natural numbers. Suppose $\Sigma$ be an arbitrary finite alphabet. I summarize my inference: a) Each arbitrary Language on ...
1
vote
1answer
29 views

Gathering nonconsecutive 1's by a Turing machine

S. Barry Cooper comments his output convention for $\mathbb{N}\rightarrow\mathbb{N}$ Turing machines like this: Outputting $n$ as $n$ possibly nonconsecutive $1$'s is very natural. [...] We can ...
0
votes
2answers
102 views

Input and output of a Turing machine

For some machine models of computation there is no question what their input and output is: it's just the contents of some specific "cells", e.g. on a "tape" isomorphic to $\mathbb{N}$. Consider for ...
0
votes
0answers
49 views

Existence of a basis in constructive vector spaces

As I was trying to review forgotten knowledge on Vector Spaces in wikipedia, I read that the existence of a basis follows from Zorn lemma, hence equivalently from the axiom of choice. Actually, the ...
0
votes
1answer
41 views

What mathematical structure best entails self-modifying programs?

If a program description can be represented as a sequence, then what is the best structure to entail program descriptions which self-modify? There must exist a relationship between the structure in ...
1
vote
1answer
31 views

Show that whether or not an arbitrary Turing machine ever executes a particular one of its instructions is unsolvable

Show that whether or not an arbitrary Turing machine ever executes a particular one of its instructions is unsolvable. (This is the same as the problem of detecting unreachable code in a program.)
0
votes
1answer
22 views

What do $A \upharpoonright x$ and $\mu s \ge x$ denote?

I am reading Computability Theory by Cooper and I do not understand the notation in the definition on the page 230: Let $\{A^s\}_{s \ge 0}$ be a $\Delta_2$-approximating sequence for $A \in ...
1
vote
0answers
50 views

An algorithmic approach to constructing the real numbers

To specify a real number, we can describe a rule which, given any rational number, tells you whether it's Too Big or Too Small. The rule should be self-consistent, in the sense that if $a$ is Too Big ...
1
vote
0answers
25 views

Decidability involving functions

I'm trying to figure out how to resolve this exercise. $$ \Sigma = \{a,b\} $$ is a set while $$ \mathcal{P}(\Sigma^*) $$ is the partition of sigma star. I have a function f: $$ f: ...
1
vote
1answer
17 views

An $n+1$-c.e. set which is not $n$-c.e.

A set $X\subseteq \mathbb{N}$ is $n$-c.e. if there is a total recursive guessing procedure $g(x,s)$ so that $$ g(x,0) = 0,\ \lim_s g(x,s) = X(x) $$ and the number of times $g$ changes its mind on a ...
0
votes
1answer
54 views

Why is this relation recursive?

A relation $R \subset \mathbb{N}^d$ is called recursive if there exists a primitive recursive function f with $$ (x_1 ,\dots,x_d) \in R \Leftrightarrow f(x_1,\dots,x_d)=0.$$ In Kurt Gödel's article ...
1
vote
1answer
42 views

Is the function $f(n)=\begin{cases} 0,& \text{If $CH$} \\ 1,& \text{If $\lnot CH$} \end{cases}$ $\mu$-recursive?

Using a Turing machine model of computation one can show that the function $f:\mathbb{N}\rightarrow \mathbb{N}$, given by: $$f(n)=\begin{cases} 0,& \text{If $CH$} \\ 1,& \text{If $\lnot CH$} ...
0
votes
0answers
42 views

countable subset of surreal games

Surreal numbers are the largest possible structure to have a complete order. Games are an extension of the Surreals which only admits a partial order. Along with being larger, smaller or equal to each ...
2
votes
3answers
87 views

Showing that a certain recursive set cannot exist?

I'm having a lot of trouble with problem 17.2 of Computability and Logic (Boolos, Burgess, Jeffrey). Here's the problem: Let $T$ be a consistent, axiomatizable theory (in the language of ...
0
votes
1answer
96 views

Turing Machine Problem

We know, A Turing machine is a hypothetical device that manipulates symbols on a strip of tape according to a table of rules I Draw a TM for input $x=(0+1)^*$ i want to implement ...
2
votes
1answer
167 views

Turing machines that compute $\pi$

For each $K > 0$ there is a brut force Turing machine $\pi_K$ that "computes" the first $K$ digits of $\pi$ starting on the blank tape (all $b$s) with $K+1$ states $S \in \mathsf{S} = ...
0
votes
0answers
50 views

Arithmetic Turing machines

Consider the family $T_{1}$ of Turing machines with two tape symbols $b,1$ ($b$ the blank symbol). The family $T_{1}$ is Turing complete. Identify the tape with $\mathbb{Z}$ and let $0\in \mathbb{Z}$ ...
0
votes
1answer
42 views

infinitely long input for a turing machine

I have a question about Turing machines. Is it allowed to give them infinitely long input? Can I give a Turing machine for example all of natural numbers as input?
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0answers
41 views

Proof that Finite Turing Machine is reducible to Regular Turing Machine

I know that Finite Turing Machine and Regular Turing Machine are undecidable through Rice's theorem, but I may find a reduction among them? Finite TM = {< M > | L(M) is finite on {a}} Regular TM ...
3
votes
2answers
38 views

Number of $1$s in the binary representation of $n$

Trying to define the function $b(n)$ which counts the number of $1$s in the binary representation of $n$ arithmetically I came up with the following definition: $$b(n)=m :\equiv (\exists k_1\dots ...
0
votes
1answer
37 views

TOTAL is not Recursively Enumerable

$\overline{HALT}=$ { (M, w) : M does not halt on w } $TOTAL=$ { M : M halts on every input } The following is the proof from Hopcoft that TOTAL is not R.E. Let R(x) be the following machine: ...
-1
votes
1answer
72 views

Countable Set & Formal Grammar

We know set A is countable if A is finite or in a one-to-one mapping to natural numbers. I try to summarize my though. I think the following proposition is true. suppose $\Sigma$ is arbitrary ...
2
votes
1answer
51 views

Binary representation of real numbers without dots

How can I represent a real number using only 0's and 1's? I do not want to use any extra symbol like '.' to separate the integer part and the mantissa.
1
vote
1answer
67 views

In general, are subsets of recursively enumerable sets recursive sets?

I recently became interested in the solution to Hilbert's tenth problem, in reading about the succession of results that lead up to the proof I came across the notion of recursive sets and ...
0
votes
1answer
99 views

What's time complexity of algorithm for “Word Break”?

Word Break(Dynamic Programming) Given a string s and a dictionary of words dict, add spaces in s to construct a sentence where each word is a valid dictionary word. Return all such possible ...