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

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26
votes
4answers
752 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
405 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 ...
9
votes
2answers
176 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 ...
9
votes
0answers
175 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 ...
7
votes
4answers
457 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 ...
6
votes
2answers
143 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
0answers
49 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
1answer
116 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
2answers
316 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 ...
4
votes
1answer
110 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 ...
3
votes
1answer
354 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
52 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
115 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
66 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
93 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
118 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
19 views

$L=${$a^nb^nc^n : n \geq 0 $} CFG Recognizing

Suppose $L=${$a^nb^nc^n : n \geq 0 $} and I. $h(L), h(a)=a, h(b)=bb, h(c)=b$ II. $L^R$ III. $L^*$ IV. $h(L), h(a)=a, h(b)=bb, h(c)=a$ Why just I is a CFG and other is not? anyone can help me to ...
3
votes
2answers
30 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
117 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
92 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
922 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
75 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
322 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
81 views

Designing a turing machine

Suppose you have a tape that has a block of $a$ strokes followed by a space, followed by a block of $b$ strokes, followed by a space, followed by a block of $c$ strokes, and otherwise blank. ...
2
votes
1answer
48 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
51 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
95 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
40 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
1answer
110 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 ...
2
votes
1answer
104 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
59 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
60 views

Challenge on Property of Complement in Language

We have: and M be a finite automata. Suppose d(M) be a deterministic automata that equivalence to M. if $M_1$ and $M_2$ be two finite automata, $M_1 + M_2$ is the finite automata such that the ...
2
votes
1answer
81 views

Many to one Reducible & Polynomial time

we know that If $A \le_p B$, then $A$ can be reduced to $B$ in polynomial time. we know that If $A \le_m B$, then $A$ is many to one reduction to $B$ . can we deduce that: if $A \le_m B$ then $A ...
2
votes
1answer
88 views

Is it decidable: is there an input for which turing machine will move its head left?

$L=\{\langle M \rangle | M $is a Turing machine and $\exists$ input $x$ such that in $M(x)$ running $M$ moves its head left at least once $\}$ Is $L$ decidable?
2
votes
1answer
30 views

What's the difference between a non-deterministic turing machine and a deterministic turing machine?

From what I understood, it seems that the difference is that a NTM can have 2 inputs for which there is a different output or direction. For example, state A for input 1 can result in output 0 and ...
2
votes
1answer
50 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 ...
2
votes
1answer
128 views

Number of possible configurations in a Turing Machine

While studying for an up-and-coming exam, I stumbled upon the following question: $$ L=\{\langle M \rangle,\langle w\rangle:M\text{ is single-tape and }M \text{ running on }w \text{ doesn't go over ...
2
votes
1answer
61 views

Prove that RE is closed under reduction

Prove that the class RE is closed under reduction. Definitions: A language $ A \subseteq \Sigma^*$ is called reducible to $ B \subseteq \Gamma^*$ ( denoted by $A \leq B$) if there is a computable ...
2
votes
1answer
51 views

Concrete universal turing machine

I read about universal turing machines in the internet, but I did not find a concrete listing of a universal turing machine and a descreption, how a specific turing machine has to be coded that the ...
2
votes
1answer
228 views

Show that the language TOT={<M> | M is a Turing Machine that halts with all inputs} is not recursively enumerable nor its complement.

I've been thinking about how to show this but I'm stuck. I'm required to prove this: "Show that the language TOT={#M# | M is a Turing Machine that halts with all inputs} is not recursively ...
2
votes
3answers
2k 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
468 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 ...
2
votes
1answer
31 views

Pushdown Automata and Challenge in Grammar

I read one old-midterm exam on Automata. consider: the language that accepted by above pushdown automata is not generated by which of the following grammar? 1) S->aBaa|a$\epsilon$ ...
2
votes
1answer
40 views

A CFG Grammar for One Language

Suppose : $w_1,w_2 \in \{a,b\}^∗$ and $ L=\{w_1w_2 \mid w_1,w_2 \in \{a,b\}^* \land n_a(w_1)=n_b(w_2)\}$ $n_a$ is number of $a$'s and $n_b$ is number of $b$'s. This is a Entrance Exam question. I ...
2
votes
0answers
24 views

Strange Turing Machine Definition [closed]

i prepare for Autotmata Course Final Exam. in one of lecture, our professor draw this Turing Machine, and wrote DELTA is Neutral element of TM. it'w wrote: Language of this TM is: {$W \in ...
2
votes
0answers
52 views

Big Challenge in TM & R.E Set [closed]

we know that Halting problem {(M.w) | M halts on input w} is r.e but not recursive. i see the following sentence in one book. "the set of {(M.w) | M halts on input w and M is a TM}} is not r.e" ...
2
votes
0answers
43 views

Maths, esp. Godel, and poetry

At the risk of being an interloper: I'm a poet with a bit of mathematical training. Right now I've got a grant from the Arts Council of Northern Ireland to write a collection (loosely) based on the ...
2
votes
0answers
73 views

Prove that $\overline{L}$ is not recognizable by showing that $B_{TM} \le_m L$

$\textbf{Problem}:$ $L$ = $\{\langle M \rangle$ | $M$ is a Turing machine over $\{0, 1\}$ such that for some $x \in \{0,1\}^*$, $M$ does not halt on input $x\}$. $B_{TM}$ = $\{ \langle M \rangle$ | ...
2
votes
0answers
62 views

Blanks in the Tape of a Turing Machine

I used to have a lot of trouble with Turing Machines, primarily because I thought that in-between input symbols on the tape, one could have an arbitrary number of blanks, so every input would need to ...
2
votes
1answer
157 views

Injection from computable numbers into natural numbers

Each Turing machine which writes an infinite sequence of 1 and 0 can be regarded as representing a (computable) real number (and of course each Turing machine represents a natural number by its ...