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

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28
votes
4answers
905 views

Why do we believe the Church-Turing Thesis?

The Church-Turing Thesis, which says that the Turing Machine model is at least as powerful as any computer that can be built in practice, seems to be pretty unquestioningly accepted in my exposure to ...
15
votes
3answers
535 views

How can Busy beaver($10 \uparrow \uparrow 10$) have no provable upper bound?

This wikipedia article claims that the number of steps for a $10 \uparrow \uparrow 10$ state (halting) Turing Machine to halt has no provable upper bound: "... in the context of ordinary ...
12
votes
4answers
2k views

How large is the set of all Turing machines?

How large is the set of all Turing machines? I am confident it is infinitely large, but what kind of infinitely large is its size?
10
votes
0answers
195 views

What turmite runs the longest before becoming predictable?

When looking at 2D Turing machines, many of them eventually become predictable. For example, Langton's Ant, the champion 2-color 1-state turmite, develops a highway after 10,000 steps. Predictable ...
9
votes
2answers
221 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 ...
7
votes
4answers
467 views

Is the set of PA theorems the same as the set of solvable halting problems?

I am not sure if this is a trivial question. By Post's theorem we know that every PA (first order logic) theorem is equivalent to stating that a given input C in a given Turing machine halts or ...
7
votes
2answers
163 views

Are there known natural problems of intermediate degrees of unsolvability?

I know there exist intermediate degrees of unsolvability, i.e. there are undecidable problems which can be reduced to the Halting Problem, but not vice versa. Are there any "natural" problems known or ...
6
votes
1answer
79 views

An Undecidable but not Universal Turing Machine?

I have seen many examples of universal Turing machines, all of which are undecidable due to the undecidability of the halting problem. I have also seen proofs that certain really small Turing ...
5
votes
2answers
471 views

Why is propositional logic not Turing complete?

According to 1 (probably not the most relevant source), propositional logic is not Turing complete. Aren't all computations in computers performed using logic gates, which can be represented as ...
5
votes
1answer
129 views

Lower bounds for bb(7) and bb(8) wanted

The busy beaver function $bb(n)$ is not known for $n \geq 5$. Does Anyone know suitable lower bounds for $bb(7)$ and $bb(8)$? Remark: $bb(6)$ as a trivial lower bound does not count as a suitable ...
4
votes
1answer
124 views

How many digits of Chaitin's $\Omega$ constant would we know if we had a $\Sigma_1$-Oracle?

According to Wikipedia (and it seems intuitive from the definition itself), $\Omega$ is Turing equivalent to the halting problem and thus at level $\Delta_2^0$ of the arithmetical hierarchy. Do this ...
4
votes
0answers
84 views

Designing a turing machine for primality check.

I am designing some turing machines, so far I have made Binary Addition and subtraction. Now I've been thinking that what if turing machine can check if the number is prime or not. Lets suppose we ...
3
votes
1answer
553 views

How can a turing machine solve the element distinctness problem?

I am reading an example in Sipser's famous book on the theory of computation. In this example, Sipser creates a turing machine M to solve the element distinctness problem. M is given a list of strings ...
3
votes
1answer
53 views

the time required to decide $L$

Suppose that a language $L$ is decided in space $S(n)$ by a DTM with alphabet  $\Sigma$ and set of states $\Gamma$. What upper bound can you give for the time required to decide $L$?
3
votes
1answer
136 views

Explain why if the language A is recursive, then A is reducible to 0*1*

I'm in a theory of computation class and there is a problem that I think I am way overthinking. Can anyone point me in the right direction with the following: Give a short justification of the fact ...
3
votes
1answer
60 views

Axioms defining a Turing machine

I have found the following characterisation in axiomatical terms of a Turing machine: $Q_0(q)\rightarrow T(q)$ $S_0(x)\rightarrow S(x)$ $C(x)\rightarrow S(x)$ $Q_0(q)\land T(qx)\land ...
3
votes
1answer
62 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 ...
3
votes
1answer
73 views

A constructive algorithm for a jump of a low set.

Suppose we have an oracle Turing machine which, with $K$ (the halting problem) as an oracle, computes a low set $A$. ($A$ is low if $A'\equiv_T K$) Is there an algorithmic way of obtaining a Turing ...
3
votes
1answer
60 views

Are all constrtuctively describable functions continuous? Do they necessarily come with a topology?

In the paper "An injection from $\mathbb{N}^\mathbb{N}$ to $\mathbb{N}$" by @AndrejBauer, about the question whether there exists an injection $\mathbb{N}^\mathbb{N}\to\mathbb{N}$, we writes ...
3
votes
1answer
122 views

Why do complex grammars require powerful algorithms?

I am reading a fabulous book on Formal Languages and in the book it says: As the rewrite rules of a grammar become more complex, the algorithm for recognizing the associated language becomes ...
3
votes
1answer
116 views

Rice's theorem_Theory of computation

Is there any body tell me, where is wrong in this proof Problem: The set of number of turing machine that has 5 state is decidable or not? Answer: The set is obviously 'Set of partial computable ...
3
votes
1answer
134 views

Question about $\Sigma_n$-soundness

According to wikipedia (http://en.wikipedia.org/wiki/%CE%A9-consistent_theory#Definition): "$\Sigma_n$-soundness has the following computational interpretation: if the theory proves that a program C ...
3
votes
1answer
645 views

Baker-Gill-Solovay theorem

I have been trying to understand the proof of Baker-Gill-Solovay theorem as described in Complexity Theory: Modern Approach. I think I do understand most of it, but what troubles me is that let's say ...
3
votes
1answer
112 views

Using Extended Rice's Theorem to Prove Decidability

I have a Turing Machine M. Let L be the set of all strings representing the encoding of M that has input alphabet {1,2}, where M accepts infinitely many strings that start with 1 and finitely many ...
3
votes
0answers
52 views

Do there exist a pair of 'orthogonal' non-halting Turing machines?

I'll explain what I mean by orthogonal, which is probably a poor choice of words on my part. Given two Turing machines $\lambda $ and $\tau$,and two inputs $i$ and $j$. lets say $\tau(i) \preceq ...
3
votes
2answers
53 views

NP-Complete and Poly Time Reduction Problems [closed]

I Took Some Priminlairity Learning Method on Complexity Theory. I get trouble with some definition. anyone could help me, Why the mentioned statement is True? if a Problem A can be reducible to ...
3
votes
0answers
64 views

Show that minimal CFG is undecidable (Sipser 5.36)

Question: Say that a CFG (context-free grammar) is minimal if none of its rules can be removed without changing the language generated. Let $MIN_{\text{CFG}}$ = $\{\, \langle G \rangle$ | $G$ is a ...
3
votes
0answers
107 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 ...
3
votes
1answer
149 views

Challenge on Some Language and PDA

Suppose We have Some language as follows: $L_1=\{w^* | w=x \text{ and } x \in \Sigma^*\}$ $L_2=\{ww^R ww^R | w \in ( \Sigma + \Sigma)^*\}$ $L_3=\{w | w=xy, x,y \in \Sigma^*, y \text{ is a ...
3
votes
2answers
49 views

Turing Machine Decidability

I have been working on this problem for few hours, but haven't been able to come up with a solution : Is the following problem decidable? Given a TM M, whether there is a w such that M enters each of ...
3
votes
0answers
163 views

Proving a language is not recognizable

I have the following question that I just want to verify I have done correctly. Let $L$, $L_1$, $L_2$ $\subseteq \Sigma^*$ such that $L = L_1 \cup L_2$, and $L_2$ is decidable. Prove that if $L$ is ...
2
votes
1answer
96 views

Showing this language is not decidable by rice theorem or reduction

Consider this language: L = {<M1,M2> : M1 and M2 are TMs and L(M1) contained in L(M2) contained in {1}*} Intuition says that it's undecidable, though can ...
2
votes
2answers
1k views

Why is showing a language is Turing recognizable trickier than showing Turing decidable?

I have written a proof to show that a Turing Decidable languages are closed under union (amongst other things). Later, I have written a proof to show that Turing Recognizable languages are closed ...
2
votes
1answer
60 views

Zermelo–Fraenkel Set Theory

So I'll try keeping this real short and simple. Assume that language $L$ is defined as $\{ x\in \{0,1\}^* \}$ (finite binary strings) such that $x$ encodes a proof in ZFC that 4 is a prime number. I ...
2
votes
1answer
81 views

Unknown symbol '#' in set

I am reading a text on Complexity theory. There is a set whose notation I cannot understand: "Let $\sum$ = {0,1,#}" From the context, and given that the book is used computer science courses, it ...
2
votes
1answer
405 views

Is this undecidable language recognizable?

Is this language: $L = \{\langle M\rangle : \text{$M$ is a Turing machine and $L(M)$ is decidable}\}$ which I know that is undecidable, turing-recognizable? Is its complement recognizable? ...
2
votes
1answer
56 views

an strange set $ \Xi_A =$ {$ n \in N | \exists k^2 \in A $ s.t $ k^2 \leq n$} is decidable ?, an Interview questions?

We are some student that had an Interview for M.sc Entrance Exam. This interview has two part and one multiple choice question. We see 1 strange question that some definition is so strange for us, we ...
2
votes
1answer
33 views

Can a halting turing machine write any combination on a tape before halting?

Assume, a halting turing machine uses $n$ items of the tape. Can it write every possible combination on this $n$ items before halting ? We start with a blank tape. Example $n=2$ , alphabet $0,1$ ...
2
votes
1answer
21 views

Turing machine that goes left on first symbol

I have a turing machine with transitions given by the following table I'm inputting the string aaaa. So if I look at the first symbol "a" in state A, it says to replace it with an X, go into state ...
2
votes
1answer
1k views

Build a deterministic turing machine to decide L = { ww }

As the title says. w is in {a, b}^*.Note that I am not looking for the non-deterministic one. Use a Turing machine of one tape and "pointer". An idea: I thought that I would do something like ...
2
votes
1answer
53 views

Given a single taped deterministic turing machine what's the least amount of calculations needed in order to receive the language

Given a single taped deterministic turing machine what's the least amount of calculations needed in order to receive the language $L_k=${$0,1$}$^*0${$0,1$}$^{k-1}$. My intuition says that i'll need ...
2
votes
1answer
75 views

Reducing A$_\text{TM}$ to REGULAR$_\text{TM}$

We can solve A$_\text{TM}$ problem using REGULAR$_\text{TM}$. Assume $R$ is a Turing machine that decides REGULAR$_\text{TM}$. We construct $S$ to decide A$_\text{TM}$ as follows On input ...
2
votes
1answer
175 views

Turing Machine for comparing, copying, and operating

If one wants to design a Turing Machine for a function such as this: Where $x>0,y>0$ and are both integers represented in unary, so an example movement in this TM on the read-write head would ...
2
votes
1answer
58 views

Undecidability of REGULAR_TM

In case you have Sipser's Introduction to the Theory of Computation 3rd edition, I am asking specifically about the proof of theorem 5.3, how the language REGULAR_TM is undecidable. \begin{equation} ...
2
votes
3answers
3k views

Proving that a Turing Machine that only accepts even length strings is undecidable

I need to prove that a Turing Machine that only accepts even length strings in undecidable. The proof I was thinking is explaining the following: Given an input that contains even length strings, if ...
2
votes
1answer
115 views

Where is the flaw in the following proof?

Where is the flaw in the following proof, that if a language is Turing recognizable then we can enumerate it? Proof Let $TM1$ be a Turing machine for language $L$. We can create an enumerator $E$ ...
2
votes
1answer
33 views

How was the busy beaver candidate for 6 states calculated?

The current busy beaver candidate on 6 states, with the original binary alphabet configuration, produces about 10^18267 1's, according to the wiki page on Busy Beaver. I could not find any working ...
2
votes
1answer
89 views

Construct a Turing-Machine for Factorial(unary)

I am designing a turing machine which calculates the factorial of any given input for example, $3! = 3.2.1$, on tape it will look like this $Blank|1|1|1|Blank$ What I have done so far is that, I made ...
2
votes
1answer
200 views

Designing a Turing machine for Binary Multiplication

I need help designing a turing machine that will compute the following f(x,y) = x*y How to approach this problem in binary base? This is a assignment so I don't ...
2
votes
1answer
43 views

Designing a Turing machine for $a^n$ $b^{2n}$ $a^n$

I am new to theory of automata, and i have a little knowledge of designing a turing machine, I am stuck in this question which is given to us as an assignment, $\{$$a^n$ $b^{2n}$ $a^n$ : ...