This tag is suited for questions involving Turing machines. Not to be confused with finite state machines and finite automata.

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Tag systems to cyclic tag systems and turing completeness

Consider the 2-tag system Alphabet: {a,b,c} Production rules: a --> bc b --> a c --> aaa and stating words aaa...a halts. on ...
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Are these functions computable - Understanding computable functions

There is a theorem in computability theory which states: B.Cooper: If $A\subseteq N$ is computable, then $A$ is also computably enumerable. In the proof of this theorem -which is an ...
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Deterministic Turing Machines

Let's say that M is a deterministic Turing Machine, can I say that for a certain input I will have the same output? How can I demonstarte this?
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19 views

A Question About Recursive Functions

We want to find a recursive function $f(x,y)$ in order to have this equality: $$ \mathbf \varphi_{f(x,y)} = \varphi_x + \varphi_y$$ I know we should use "s-m-n" theorem, but I can't find the ...
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22 views

How to derive Turing Machine language?

Say you're given a TM (Turing Machine) $M = (Q, \Sigma, \Gamma, \vdash, \sqcup, \Diamond)$ and given the partial $\delta$: $$\begin{array}{c|cc} ...
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Are the derivatives of the Busy Beaver function positive?

Let $BB:\mathbb Z_{\ge1}\to\mathbb Z$ be the Busy Beaver sequence, usually called the Busy Beaver function, as defined in terms of Turing machines in Section 1.3 of this text of Aaronson and Yedidia. ...
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Equivalence of Turing Machines and Lambda Calculus

Based on the Church Turing Thesis, we conjecture that Turing Machines are the "correct," model of computation. It is well known that they are equivalent to the Lambda Calculus, another model of ...
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43 views

Induction with Turing machines.

how would I go about proving by induction that the Turing Machine pictured below, that if it is started with a blank tape, after 10n+6 steps the machine will be in state [3] with the tape reading . . ...
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1answer
37 views

What is the smallest number $n$ for which $bb(n)>f_{\epsilon_0}(5)$ is known?

It is known that $bb(23)$>Graham's number (I do not remember exactly, but $bb(21)$ could already be larger). But what is the smallest number $n$, such that $bb(n)>f_{\epsilon_0}(5)$ is known ? ...
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1answer
31 views

Turing Machine Diagram, one Solved Problem ?!

The following Diagram Gets binary number $x$ and produce $x+1$. complete it: the book solution is says first line is the answer. any hint or idea for completing this TM?
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1answer
15 views

Demonstrate that a language is semi-decidable

I need some help to demonstrate that this set below is decidable, semi-decidable, or undecidable. Here's the set: H = {p| |Images(fp)| >= 10} explanation: an ...
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1answer
895 views

Showing that Turing-recognizable languages are closed under union

I'm reading "Theory of Computation" by Michael Sipser and I've encountered a solution (provided by the book) that I don't understand. The question: Show that the collection of Turing-recognizable ...
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1answer
24 views

Algorithm with undecidable input set?

I am interested in "Relative Decision Problems" in the following sense: Let $\mathbb{N} \supseteq U \supseteq S$. Is there an algorithm such that on a given input $u \in U$ decides whether $u \in S$? ...
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$K(xy)\leq K(x)+K(y) +c$?

Could anyone show that for any $c$, some strings $x$ and $y$ exist, where $K(xy)>K(x)+K(y)+c$? Here $K(x)$ is the Kolmogorov complexity. I already know that $K(xy) \leq 2K(x) + K(y) +c$ and $K(xy) ...
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1answer
36 views

Is the problem decidable with Turing machine M that inputs x,y,z does M halts on these 3 instances

Is the following problem is decidable? Given a Turing machine M inputs x,y,z does M halts on these 3 instances? Hint: make y and z any two artificial inputs that the program stops with these inputs. ...
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1answer
34 views

Using reductions of turing machines properly

I recently learned about reductions of Turing machines (here after TM), and here is a solution to a problem using reduction (showing L is undecidable, as defined bellow). I have given the reduction, ...
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Is the Antikythera Mechanism Turing-Complete?

Is the following Turing-complete? https://en.wikipedia.org/wiki/Antikythera_mechanism As in, it possible to perform all of the operations of a Turing-machine, albeit with finite memory, with this ...
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simple questions on $TM$s runs lengths

Is it possible that the number of running steps in $TM$ that runs on word $w$ will be $0$? Is it possible that the number of running steps in $TM$ that runs on the empty word $\epsilon$ will be ...
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38 views

Prove that a certain intrinsic property of Turing machines is not decidable

Can anyone help me to prove that the following language is nod decidable? $$ A=\{\langle\,M,w,q\,\rangle\mid M \text{ is a $TM$ , $w$ is a word, $q$ is a state in $M$ and while $M$ runs on $w$ it ...
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42 views

Classifying languages

I'm working on understanding what kind of languages are decidable, recognizable, and co-recognizable. I came across this problem that I think will really help me but I'm still quite unsure of how to ...
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1answer
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Is the union of undecidable languages not Turing-recognizable?

The question is as follows: Let us define $$L := \{w \mbox{ | either }w = 1x \mbox{ for some } x \mbox{ ∈ $A_{TM}$ or } \mbox{$w$ = 0$y$ for some $y$ ∈ $\overline {A_{TM}}$}\}.$$Prove that neither ...
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1answer
52 views

Design Turing Machine

Design a single-tape Turing machine with input alphabet {0, 1} to decide the language $$\{ x\in\{0,1\}^* \mid \#(0,x)=2\cdot\#(1,x)\}.$$ Could someone give me clarification on how to approach and ...
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35 views

proof that languages are/are not in RE (probably with mapping reduce)

Given $2$ languages: Let $u \in \Sigma^*$ (constant word). $A_u=\{<M> \big{|}\,\, u\in L(M) \text{ and M is TM }\}$ $B_u=\{<M> \big{|}\,\, L(M)=\{u\} \text{ and M is TM }\}$ I ...
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Prove uncountability of set L that L and L' neither of which is recursively enumerable.

How do I prove that the set of all languages L on alphabet {0,1} that neither L or L' are recursively enumerable, is uncountable? Proving uncountability can be done through diagonalization like the ...
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1answer
14 views

Is each string decideble?

Is it possible to prove that there exist for every string a Turing Machine that decides that string? I think it is provable that for every string you can build a TM that recognises that string, but I ...
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turing machine decidable description for the language

L = { | R is a regular expression that produces at least one word in {a, b} * which contains a symbol exactly 3 times} ...
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1answer
36 views

turing machine decidability language

I must show that this language is decidable but I think it's not {D, Ρ} | D is a DFA and P is a ΡDA which L(D) ∩ L(Ρ) = ∅ } Here what I think I give a reduction from E(TM). I suppose that this ...
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1answer
110 views

Turing Machine That Accepts Machines With Undecidable Languages

So I'm reviewing my Computability notes for my final, and I understand how reduction arguments work, but I'm having trouble framing one for the following Turing machine: Undecidable TM = { ⟨M⟩ | L(M) ...
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2answers
35 views

Automatic proof verification by a Turing machine

Is it possible to automatically verify a mathematical proof? Or is it proven that this cannot be done by a Turing machine? Thank you very much Kevin
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Turing machine macro notation

This is an example from the book Automata, Computability and Complexity by Elaine Rich. Macro language is defined as follows: (screenshot from the book) And these are the steps mentioned : Scan ...
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1answer
86 views

Prove claims about disjoint union and decidable/undecidable languages

Let $L\subseteq\Sigma^*$ decidable language and $A\subseteq\Sigma^*$. Let $B=A\sqcup L$ (a disjoint union). Prove: $1$. $B\in RE \Rightarrow A\in RE$ $2$. $B\in R \Rightarrow A\in R$ Thanks!
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Prove by printing turing machine that RE closed under iteration

I do not know what is the formal name of printing turing machine in english, maybe "counter machine". This machine prints a whole language without any input. for example: counter machine that counts ...
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1answer
72 views

non deterministic turing machine for concatenation

Let $L_1, L_2$ decidable languages on deterministic single-tape TM $M_1$ and $M_2$. How can I build non-deterministic TM that decides $L_1L_2$? What should be the formal definition of $\delta$ (the ...
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1answer
25 views

Drawing a state diagram for a Turing Machine

I'm a bit rusty since it's been a couple weeks since I've last done this, but I could really use some help with starting out the Turing Machine for {a^i b^j c^k | i = j + k} I'm confused on how I can ...
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2answers
78 views

Why is it so hard to translate some proves into machine-readable form?

I have just read a topic on mathoverflow about man vs. machine in mathematics. The topic was inspired by the recent victory of Alpha Go over the World Go Champion, Lee Sedol. It reminded me of an ...
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Is there a universal Turing machine on arbitrary number of input variables?

I know that for every $n \geq 1$ there is a partial recursive (p.r.) function $\phi^{(n+1)}_{z_n}(e,x_1,...,x_n)$ such that $\phi_{z_n}^{(n+1)}=\phi_e^{(n)}(x_1,...,x_n)$, where $\phi_e^{(n)}$ is the ...
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1answer
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Cannot create algorithm for decidable language

L2 = {<M> : M is a TM and there exists an input string w such that M halts within 10 steps on input w} Hi. I am creating an algorithm to show above L2 is ...
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1answer
44 views

Cannot understand solution (Turing Machine & Reduction)

Photo of my problem that I don't understand About question above in photo, I just can't understand its solution provided. We know the complement of Atm = {...
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1answer
67 views

Problems understanding proof of s-m-n Theorem using Church-Turing thesis

I am reading Barry Cooper's Computability Theory and he states the following as the s-m-n theorem: Let $f:\mathbb{N}^2\mapsto\mathbb{N}$ be a (partial) recursive function. Then there exists a ...
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1answer
47 views

Reduction to/from REC and RE language?

Let $X$ be a recursive language and $Y$ be a recursively enumerable but not recursive language. Let $W$ and $Z$ be two languages such that $\overline{Y}$ reduces to $W$, and $Z$ reduces to ...
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1answer
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Space-Hierarchy Theorem in Theoretical CS

Sipser has a proof this theorem that goes like this: $$D = \text{"On input } w$$ $$1. \text{Let } n \text{ be the length of } w$$ $$2. \text{Compute } f(n) \\ \text{using space constructibility and ...
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1answer
32 views

Which language is decidable

Just been at the Math-exam. One question I was really unsure about, was this question - so I didn't answer it, as you get minus point if the answer is wrong. Does somebody know, what the right answer ...
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1answer
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How is the non-existence of a solution proven?

I've been wondering how an argument that a solution to a particular problem doesn't exist is put together. For instance "Pour-El and Richards found an ordinary differential equation ...
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122 views

Read-only Turing machine recognizes only regular languages?

Show that the Turing machines, which have a read only input tape and constant size work tape, recognize precisely the class of regular languages. According to wiki : A read-only Turing machine or ...
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1answer
58 views

Showing set is undecidable with Turing Machines

I'm given the set $T = \{\langle M, w\rangle : M $ is a Turing Machine that accepts $w$ reversed whenever it accepts $w \}$ and I want to show it's undecidable but recognizable. (I'm using the bracket ...
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Is there a way to prove that a Turing machine computes the function we designed it to?

Say we design a simple Turing machine that adds two numbers together. Is there any way to formally prove that the machine actually computes the function we 'know' it does? Is there a general method ...
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1answer
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Savitch theorem and its assumption

famous Savitch theorem states: For any function $f\in\Omega(\log(n)), \text{NSPACE}(f(n)) \subseteq > \text{DSPACE}((f(n))^2).$ Why we need an assumption that $f\in\Omega(\log(n))$? Thank ...
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1answer
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Abstract machines that compute primitive recursive functions

What it the simplest (least powerful) abstract machine that can compute primitive recursive sets, i.e. sets whose characteristic or indicator function is primitive recursive? ...
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1answer
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Turing recognizable - $B = \{a^n b^n c^n \}$

Question: My answer is no, because it loops forever. But I am a bit unsure if this is the right answer.