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

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What is meant by “finite algorithm” in Turing's definition of the computable numbers?

In a comment thread on SlateStarCodex, I made a philosophical point the "realness" of the reals, in the process of which I attempted to summarize the definition of Turing-computable numbers: ...
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Is there a turing machine for which halting is equivalent to the Axiom of Choice or its negation?

As seen in "A Turing machine for which halting is outside ZFC", Gödel's incompletness theorem can that there a turing machines for which halting can not be decided. My question is, is there a turing ...
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Can Incompleteness be Computable?

Chaitin's incompleteness theorem says no sufficiently strong theory of arithmetic can prove $K(x) > L$ where $K(x)$ is the Kolmogorov complexity of natural number $x$ and $L$ is a sufficiently ...
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Prove $K_4-Cover$ is NP-Complete

I'm studying for a computational theory exam, and as part of my studying I'm trying to solve previous years' exams. I have come across this problem and I'm having some difficulty with it: Let $ G ...
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1answer
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Is PA+ TM doesnt halts consistent?

Suppose there isnt a proof in PA whether some TM halts or not. Suppose further that TM doesnt halt and PA is consistent. Is PA+TM halts necesserely consistent? Is PA+TM doesnt halt necesserely ...
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Reduction to Halting problem

I have the following problem. Prove by reduction to the halting problem, that $L:=\left \{ <M> | L(M)=\emptyset \right \}$ is undecidable. L(M) is the defined as: $L(M) = \left \{ w \:| M ...
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Number of finite-state machines with $n$ states, output alphabet size $a$, and binary input

How many FSMs are there where the machine has $n$ states, reads a binary symbol at each time-step, and may or may not output a symbol from an alphabet of size $a$ after each transition?
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5answers
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Where does this argument showing there are uncountably many TMs fail?

This argument comes up once every while on Lambda the Ultimate. I want to know where the flaw is. Take a countable number of TMs all generating different bitstreams. Construct a Cantor TM which runs ...
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DTM that halts on a minimum one input

I am trying to show that this is Turing Recognizable. The language A, that is a DTM that halts on a minimum of one input. My justification is that of course it is because, you just need to check ...
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1answer
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How to simulate a 3-stack automaton with a 2-stack automaton?

Since a 2-stack automaton is Turing-equivalent, it is possible to simulate a 3-stack automaton with just a 2-stack automaton. But how so? How it is normally done?
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Generalizing equal turing machine problem

I know that $EQ_{TM} = \{<M_1,M_2> | L(M_1)=L(M_2)\} \notin RE \cup CO-RE$ Can I generalize and say that $L' = \{<M> | L(M) = C \} \notin RE \cup CO-RE$ Where C is the language of any ...
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Turing Machine Membership problem and how to prove its undecidable

ATM = {$<m, w>$ | M is a Turing Machine that accepts string w}. How can I prove that ATM is undecidable? Here's what I have so far: Any decidable problem is accepted by a Turing Machine. It ...
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Are there any Turing-undecidable problems whose undecidability is independent of the Halting problem?

To be more specific, does there exist a decision problem $P$ such that given an oracle machine solving $P$, the Halting problem remains undecidable, and given an oracle machine solving the Halting ...
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inverse proof of closed under union

I am being posed the following question: given are languages $L_1, L_2, L_3 $ and $ L_4$. $L_1$ is a decidable language and $L_2$ and $L_4$ are both recognizable languages. Considering $L_1 = L_2 ...
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34 views

Direct Reduction, Turing machine and a DFA

I have been reading and I am trying to understand the reduction when it comes to truing machine. This is how I understand it: it means that it reduces problem A into problem C. But I am not quite sure ...
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Prove that exists undecidable subset of $\{1\}*$

Hello my dear friends! I have following problem: Prove that exists undecidable subset of $\{1\}*$ The problem is that I don't know how to start. In real I don't what does it mean undecidable set ?
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Prove that language $X$ is not decidable

$$X =\{\langle M, w\rangle \mid\text{$M$ has one tape and never modifies portion of the}$$ $$\text{tape that contains the input $w$}\}$$ And my proposition: Let $@$ will be character such that there ...
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1answer
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prove that language is not deciable by reduction

I show you my approach to one problem, and try to assess it. Show that following language is not decidable: $L=\{\langle M\rangle|\text{M is Turing Machine and M has one or more unreachable state} \}$ ...
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prove that language is not decidable (string and reverse)

Prove that $T=\{\langle M\rangle\mid M \text{ is TM that accepts $w^R$ iff it accepts $w$}\} $ is not decidable. I have no idea how to start. Help me, please
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1answer
21 views

Deterministic Turing machine for a duplicate concatenation of a string

What's the best approach for building a deterministic Turing machine for the language $$L = \{vv : v \in \{a,b\}^+ \}$$ where there is no midpoint marker in the string? How can we determine where ...
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1answer
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What breaks the Turing Completeness of simply typed lambda calculus?

On the Wikipedia page about Turing Completeness, we can read that: Although (untyped) lambda calculus is Turing-complete, simply typed lambda calculus is not. I am curious as to what exactly ...
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What Constitutes a Pattern

Mathematics is often referred to as the "study of patterns." What I'm wondering is whether there is somehow a technical way to describe a pattern. For the length of this question let's assume that ...
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1answer
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Turing machine that accepts even length strings

Can someone help me with some tips on how to create a turing machine that only accepts even length strings with an input alphabet of {0,1}?
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2answers
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Show that the question “Is there life beyond earth?” is decidable

I was given a question to prove that there exists a turing machine that solves the question Is there life beyond earth? and is decidable. I actually don't understand how to show a turing ...
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Why does a certain turing machine only accept regular languages?

why does a single tape turing machine, which can not write on the part of its tape containing the input, only accept regular languages? I am asking this because this question is being opposed to me ...
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Example of recursively enumerable languages that under intersection are $\emptyset$

I am trying to think about an example of a recursively enumerable languages $L_1,L_2 \in RE $ and $L_1,L_2 \notin R $ that satisfy: $L_1 \cap L_2 \in R $ I know that it will be probably something to ...
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Is it possible to show that a particular theorem or its negation is provable, without knowing which of the two is true?

I've been thinking about this for a while: as far as we know, is it possible that for a particular statement $\sigma$ of $\textsf{ZFC}$, we can prove that $(\textsf{ZFC} \vdash \sigma) \vee ...
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1answer
20 views

Languages in coNP

if a language $L \in$ coNP, i.e. it's complement is in NP, then does L have a deterministic turing machine that decides it? i think that this is false, but am unsure how to show it? my guess is using ...
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17 views

determining whether a turing machine

For $i\in \mathbb{N}$, define $L_i :=${$ ⟨M⟩ |$ On input $101$ M halts after at most i steps} For any fixed i the language $L_i$ is decidable as if there is no end state up to the i$^{th}$ position ...
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25 views

To prove a language is not recursive

Prove the language $$L_1=\{\sigma\in\{0,1\}^*|\sigma \text{ codes a TM which accepts at least one word }\}$$ is not recursive. I think it has something to do with $$L=\{\sigma\in\{0,1\}^*|\sigma ...
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Simple Turing machine problems [duplicate]

I'm trying to go over some review problems regarding Turing Machine recognizability, and am still pretty confused about the following problems. This is the only information we are given in the problem ...
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114 views

Turing Machine recognizability

I'm trying to go over some review problems regarding Turing Machine recognizability, and am still pretty confused about the following problems. This is the only information we are given in the problem ...
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Can you solve the halting problem for a single, non-universal Turing machine?

So, I'm familiar with the halting problem and its proof. However, I also understand that the proof is for any universal machine $U$; that is, the set ...
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1answer
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Designing a Turing Machine - low level transitions

I couldn't figure out how to proceed with this question. Preparing for the finals, can someone explain how to do this step by step? Design a TM, write low level transitions for $\{a^i b^j :i ≤ j ≤ ...
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Why are every finite language decidable?

I don't understand why every finite language is decidable. For example, if I have an infinite set of strings L over an alphabet E, why is there a Turing Machine M that decides L? I understand the ...
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1answer
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Let $LOOP_{TM}$ be descriptor language of all touring machines that won't halt for any input. Show reduction of $HALT_{TM}$ to $LOOP_{TM}$

Question from Homework that I'm having difficult to answer on: Let $LOOP_{TM} = \{\langle M\rangle \mid \text{M is a TM that does not halt on any input w}\}$ Let $LOOP_{TM}$ be descriptor ...
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1answer
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Finding regular expressions for which a given Turing machine halts and accepts, halts and rejects, and diverges.

Consider the Turing machine M = (Q,Σ,Γ,δ,q,F) F = {t} Q = {q,r,s,t,v,x} Σ = {a,b,c} Γ = {B,a,b,c} δ = [q,a,q,a,R] [q,b,q,b,R] [q,c,v,b,R] [q,B,r,B,L] [r,a,s,B,L] [r,b,s,B,L] [r,c,s,B,L] ...
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Examples of undecidable languages contained in 1*?

I've been given the following question Show that there is an undecidable language contained in $1^*$. But I can't think of any undecidable languages that are contained! Can someone please lend a ...
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Is the problem of determining the complement of of a given language decidable?

By which I mean, given Turing machines M and N, is determining whether L(N) is the complement of L(M) decidable or not? My instinct is that it is undecidable, but I'm unsure of how to make a formal ...
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1answer
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What is turing machine for $a^i b^j c^k$ where $i=j$ or $j=k$

I am trying to construct turing machine for $a^ib^jc^k$ where $i=j$ or $j=k$. Every time I come up with solution its getting fail for some other string.
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1answer
63 views

Turing Machine Halting problem

I have come across this halting problem question during my exam preparation and can't come up with a solid proof for the following question. Question: Let L be { Ti does not halt on input i} Show ...
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What's the significance of the Church-Turing Thesis?

My understanding is that the thesis is essentially a definition of the term "computable" to mean something that is computable on a Turing Machine. Is this really all there is to it? If so, what makes ...
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Computability: is there an alternative method to decide this language?

For my computability revision I am trying to decide the language, $$L = \{ \text{all binary strings containing the pattern 001 (not necessarily in consecutive places)} \}.$$ I believe that I can do ...
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$ E_{\text{TM}} = \{ \langle M \rangle \mid L(M) = \varnothing \} $ is undecidable.

In this proof, we need to convert the input from $ \langle M,w \rangle $ to $ M_{1} $ as $ E_{\text{TM}} $’s input is only a Turing Machine. However, I couldn’t understand the construction of $ M_{1} ...
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Show the following languages are not recursive

Show that the language $$L = \{ M : M \text{ is a Turing Machine that halts on input $M$ } \} $$ is not recursive. Show that the language $$ L = \{M : M \text{ is a Turing Machine such that $L(M)$ ...
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Problem with tracing computation from Turing Machine

Transition Function of TM M start_state: q final_state: qf R denotes move right L denotes move left S denotes stop I traced the computation but not sure if it is exactly correct my problem is ...
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1answer
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determining recognizable or decidable (TM that accepts a TM)

I'm having an issue determining whether certain languages are decidable, recognizable or neither. The specific languages I'm referring to are of the following form L = {<M> | for every w, M accepts ...
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1answer
90 views

is differ between distributive lattice vs semi-lattice on Turing Degrees

We know a Posed Closed under suprema but not necessarily under infima is an upper semi-lattice. We now r.e set forms a distributive lattice. But my question is why following statement is hold? I ...
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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 ...
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Are there any known noncomputability proofs that do not rely on the halting problem?

I have looked around and thought of this for a while, and I have not found or been able to construct any proof that a problem is not decidable, without said proof being fundamentally equivalent to ...