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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how to do that function $n \rightarrow h(f(n),g(n))$ is URM [on hold]

if $f:N\rightarrow N$ ,$g:N\rightarrow N$ , $h:N^{2}\rightarrow N$ are URM computable then so is the function $n \rightarrow h(f(n),g(n))$.
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0answers
26 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 ...
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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 ...
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41 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 $f(n) = a_n$ ...
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1answer
41 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, ...
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9 views

Compute the composition of functions

I got wrong in this very odd question for my assignment. Can somebody help me with the answer of this question provided an explantion? Thank you a lot in advance!
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0answers
21 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 ...
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1answer
40 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?
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1answer
38 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 ...
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7answers
116 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 ...
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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 ...
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25 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, ...
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1answer
93 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 ...
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0answers
111 views

indices set and halting problem in computation course

I ran into a multiple choice question that confused me with this notation. anyone could help me? this is adapted from an old class quiz in Calgary. Suppose A is be indices (i think index set) of type ...
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1answer
32 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 ...
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0answers
22 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 ...
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0answers
80 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 ...
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1answer
28 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 ...
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2answers
81 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 ...
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0answers
39 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 ...
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1answer
30 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 ...
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1answer
28 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.)
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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 ...
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0answers
38 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 ...
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23 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: ...
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1answer
15 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 ...
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1answer
50 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 ...
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1answer
37 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 $\lnot CH$} ...
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0answers
24 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 ...
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3answers
83 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 ...
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1answer
73 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 ...
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1answer
145 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} = ...
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0answers
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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 $0\in \mathbb{Z}$ ...
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1answer
26 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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31 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 ...
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2answers
37 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 ...
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1answer
28 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: ...
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1answer
66 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 ...
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1answer
47 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.
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1answer
29 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 ...
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1answer
40 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 ...
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Generalization of standard technique for proving that an undecidable language is unrecognizable

Suppose $L = \{P:P(x) \; outputs \; x^2 \;for\; all\; x\}$ Then $\bar L = \{P: P(x)\; does\; not\; output\; x^2 for\; all\; x \}$. By Rice's Theorem or by reduction from the Halting Problem, let's say ...
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1answer
36 views

$L \in RE$ Question in Computation [closed]

Let L be a language. Suppose a TM exists that halts on all words in L. Which of the following statements is true? a) if L is r.e we have such TM. b) if L is r.e and complement of L is r.e then we ...
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2answers
128 views

What if a conjecture were provably unprovable?

Suppose we found a proof that "The Twin Prime Conjecture cannot be proven", without any conclusion as to the conjecture itself being true or false. Is it then possible for the conjecture to be true? ...
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2answers
187 views

The mother of all undecidable problems

It is usual to show that a problem P is undecidable by showing that the halting problem reduces to P. Is it the case that the halting problem is the mother of all undecidable problems in the sense ...
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1answer
20 views

Algorithm that takes input desc. of two PDAs and outputs intersection of langs. recognized by two PDAs

Does there exist an algorithm which takes as input the descriptions of two pushdown automata, $P1$ and $P2$, and prints the description of another pushdown automaton which recognizes the intersection ...
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1answer
19 views

Deciding TM which fails to halt whenever the length of its input string is a prime number

I have the following Statement: "A TM called $A$ which fails to halt (i.e runs forever) whenever the length of its input string is a prime number, and eventually halts for all other input strings" ...
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1answer
25 views

disproving union of infinitely many regular languages

I want to disprove the following statement: "if $L$ is the union of infinitely many regular languages, then $L$ is guaranteed to be a regular language." I don't know where to start. Any hint will be ...
4
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0answers
93 views

Is it decidable whether the iterates of a polynomial map are bounded?

Let $f:\mathbb{Q}^n\to \mathbb{Q}^n$ 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 ...
4
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0answers
49 views

predicate logic with assumption NP $\neq$ CO-NP?

Anyone could describe why: Set of All Tautology in propositional logic with assumption NP $\neq$ CO-NP is CO-NP Complete. Thanks. I ask it here before: Is the language of tautologies NP-complete? ...