Questions tagged [turing-machines]

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

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Two tape turing machine for L = {a∗wwb∗ : w ∈{a,b}∗} without any bruteforce

I don't even know how to approach this problem. I was thinking to use two tape turing machine. First tape would be the input string and second tape for guessing w. And even if I have guessed w ...
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Rice theorem and trivial properties for decidability proof

I'm going to have a complexity theory exam and i understood the importance of Rice theorem in proving if given a language $L_{p}=(L|L\space satisfies\space the \space property\space\space p)$, is ...
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Regular Languages

i am stuck in this problem. Prove that shuffle of 2 Context-free Languages is Recursive and Recursive-enumerable. Also prove that this new language is not necessarily Context-Free. I am able to do the ...
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Turing Equivalence counterexample

Given that A is Turing reducible to B, what would the set B need to look like such that B is not Turing reducible to A? I've been having a hard time with this idea and I would appreciate some examples ...
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I'm stuck on this FLAT question, could someone help? [closed]

Let $\Sigma_1$ and $\Sigma_2$ be finite alphabets such that $Σ_1∩Σ_2=∅$. Let $L_1 \subseteq \Sigma_1^*$, $L_2 \subseteq \Sigma_2^*$ and define $L \subseteq \Sigma^*$ as follows: A string in $\Sigma^*...
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On the proof of the unsolvability of the word problem in semigroups

I'm trying to understand the following proof of the unsolvability of the word problem in semigroups. I tried to reproduce the proof from some kind of personal communication, so I'm not sure everything ...
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Quadratic lower bound for one-tape Turing machine to decide $\{ww\mid w\in\{0,1\}^*\}$

It's somewhat well-known that any one-tape Turing machine that decides $L=\{ww^R\mid w\in\{0,1\}^*\}$ requires $\Omega(|w|^2)$ time via crossing sequence arguments. But consider the language $L=\{ww\...
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What does the following notation for something in the language of a turing machine mean?

So I see the following being used in turing machines: a configuration is $Q \times \{ y\sqcup^\omega | y \in \Gamma^\ast \} \times \Bbb N$ What is $y\sqcup^\omega$ supposed to mean and what is the ...
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Proving that there exists a Recursively Enumerable language that has a special property

In an assignment about mapping reductions, we're required to prove the following claim: Prove that there exists $L_2\notin RE$ such that $\overline L_2 \leq_m L_2$. We also got the following guidance: ...
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Will this function grow faster than Busy Beavers as $n \to \infty$

Consider the following function: $$f(x)=x \uparrow ^{x} x$$ Where the notation $\uparrow$ is Knuth's up-arrow notation and $\uparrow ^{n}$ means $n$ number of up-arrows. For example, $2\uparrow ^{4}...
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Existence of infinite sets of a certain property

I've been thinking about this problem for a long time, but I can't come up with a solution. It must be proved that there exists an infinite family of infinite pairwise disjoint subsets of $\mathbb{N}$,...
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Turing degrees of subsets of Kleene $\mathcal{O}$ which are ordinal notations of subsets of the set of recursive ordinals

An ordinal $\alpha$ is said to be recursive if there is a recursive well-ordering of a subset of the natural numbers having the order type $\alpha$. The smallest ordinal that is not recursive is ...
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Prove that $A^{(\omega)}\nleq_T A^{(n)}$

I am trying to solve Exercise 7.1.24 (i) of Computability Theory by Rebecca Weber. $A^{(n)}$ denotes the $n$-th Turing jump and $A^{(\omega)}=\{\langle x,n\rangle: x\in A^{(n)}\}$ the $\omega$-jump. ...
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Can we guarantee that the computation on the Turing machine in our case will never stop?

Assume that we have a Turing machine which on an empty input, with $n > 100$ states worked $n^{ n^n}$ steps. Can we guarantee that in this case the computation will never stop?
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Could Turing machines + random “do something” Turing machines cannot?

A machine G that outputted a bit randomly could decide any language (by guessing luckily). Formally, for all languages L, for ...
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Is there an undecidable question that doesn't ask something about Turing machines (or something Turing-complete)?

Is there an undecidable question that doesn't ask something about Turing machines (or something Turing-complete)? The halting problem asks, given a Turing machine, ...
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Prove uncountability of R using an algorithm?

Let's say I want to prove uncountability of $\mathbb{R}$ using an algorithm (I will use Python). I will consider reals $0 \le x \lt 1$ and represent the decimal development of $x$ with a generator. ...
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1answer
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The class of primitive recursive functions

Usually the scheme of primitive recursion is defined as follows: $$ h(x, 0)=f(x) \\ h(x, y+1)=g(x,y,h(x,y)) $$ I was wondering whether the class of primitive recursive functions would be smaller if we ...
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Halt Turing Machines

Imagine that a friend Sheikh is capable of solving HALT. Given any instance of to the Halting problem, we can query Sheikh, if ∈ HALT, and he will instantaneously give us the right answer. He is ...
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59 views

Given FOL=Turing machines, why is SOL different than FOL?

[1] Every SOL (second order logic axiom system) has a corresponding Turing machine that verifies SOL statements, given a proof and axioms. (If this weren't the case, how could we be sure that our SOL ...
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53 views

Is FOL equivalent to Turing Machines?

Does every FOL (axiom set expressible in First Order Logic) have a corresponding Turing machine? The proved FOL statements would be strings the Turing machine accepts. Does every Turing machine have ...
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Infinite Time Turing Machines and the continuum hypothesis

Maybe I am misunderstanding something, but the problem described in this answer at Math.SE seems to be representable as a program for Infinite Time Turing Machines (under the assumption that any ...
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53 views

How exactly do diophantine polynomial equations map to turing machines?

From a wikipedia page: One can write down a concrete polynomial p ∈ Z[x1, ..., x9] such that the statement "there are integers m1, ..., m9 with p(m1, ..., m9) = 0" can neither be proven nor ...
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A multiple choice on formal languages, Turing machines and automata

$ \newcommand{\lang}{\mathcal L} $ I have been given the following claims about formal languages, that are either true or false: If a language $\lang$ can be accepted with a finite automaton, it can ...
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Is a language $L$ recursive, if it and its complement $L^c$ are both recursively enumerable?

$ \newcommand{\lang}{L} \newcommand{\Nset}{\mathbb N} \newcommand{\Lset}{\mathcal L} \newcommand{\Rec}{\mathcal R} \newcommand{\RecEnum}{\Rec_\Nset} \newcommand{\accept}{\mathbf{a}} \newcommand{\...
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What are some theorems or hypotheses that can be tested by a two-symbol low-state Turing machine?

I'm looking for relatively interesting examples of theorems or hypotheses that could be proven (or disproven) by running at most BB(n) [S(n)] operations - on a $n-$state, two-symbol TM , where $n$ is ...
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Turing decidability

Consider intermediate chess board configuration. L={w|w represents a board configuration, and white is guaranteed to win if it is white’s move and white plays optimally} Is L decidable, recognizable ...
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enumerator and turning machine mapping reduction?

Let, E = enumerator M = Turing machine the language SAMEe,tm = {⟨E,M⟩ | E is an enumerator, M is a TM, and L(E) ∩ L(M) != ∅}. Can someone give me a hint on how to prove the complement of SAMEe,tm is ...
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The problem of determining whether a given context-free grammar generates the empty string

I was unsure if you were to formulate this problem as a language would it be correct to say is ECFG = { | G is a CFG with L(G) = ∅ }? Otherwise im unsure how you would write it as a language.
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How can I read the following Turing machine?

I'm struggling to undersand the following Turing machine. When I see a,b->R, I read that as if you're at a, relace the ...
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Could the halting problem be computed in GlooP?

In Douglas Hofstadter's book Gödel, Escher, Bach, he uses 3 theoretical programming languages to describe computation. BlooP represents primitive recursive programs. FlooP represents general or ...
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Prove that if L is decidable then half(L) is decidable too

Let L be decidable language, and let half(L) be: half(L)={u∣uv∈L s.t.|u|=|v|}. Prove that if L is decidable then half(L) is decidable too. I tried to build a Turing Machine to decide half(L) but none ...
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Derivation from a Turing machine

In one of my questions for my work, I have been asked to give a derivation for the word ABBA with respect to my Turing machine. I have done research into how I would do this but am unable to find ...
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Constructing multitape Turing machine L={a^k#y }

I'm having problems with construction of Turing multitape machine construction for the L= a^k#y, where y is the input data, for example abbaba and I need to check that in the y there is no a^k. ...
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Show that $L1/L2$ is semi-decidable

I am stuck at the following exercise: Let $L_1$ be semi-decidable and $L_2$ be a decidable language over $\{0,1\}$. Show that $$L_1/L_2 := \{w \mid \exists y \in L_2 \text{ such that } wy \in ...
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28 views

Finding an algorithm that after removing k edges we get an acyclic graph

Assuming there's an algorithm that can decide belonging to ACYCLIC in polynomial time. How can I use this algorithm in another algorithm that upon the input of a directed graph and a positive number k,...
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minimal vertex cover and P=NP

could someone please explain to me why the following occurs? let function f be a function that finds the minimal vertex cover. meaning: f(G,v)=minimal vertex cover that v belongs to (the graph is ...
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Proving existence of algorithm based on existence of algorithm to decide acyclic language

I am struggling to understand(actually can't understand) how to prove the following: If there exists an algorithm that runs in polynomial time and decides belonging to ACYCLIC(the language of all ...
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1answer
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proving why can't a P-Complete language exist in log-logarithmic space

I am struggling to understand why can't such a language exist(a P-Complete language in log-logarithmic space), according to the question details: Defining a new kind of reduction: a reduction in log-...
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Constructing Turing machine for $L=\{a^n b^m, m\%n=0\}$

As it says in the title I'm struggling with a problem: construct a Turing machine that decides the language $$L= \{a^n b^m |n,m ∈ \mathbb{N} \text{ and }m\%n=0 \text{ (n divisor of m)}\} $$ It's ...
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Problem understanding Turing Halting problem

As I understand it in simple language the proof of this goes Take a program (called oracle below) that will stop if the program it is examining never halts and never halts if the program it is ...
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Quantifying the “Complexity” or “Strength” of Axiomatic Systems

The halting Problem states that there is no Turing Maschine that is able to decide whether an arbitrary other Turing Machine will halt. In 2016 Adam Yedidia and Scott Aaronson presented a turing ...
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partial recursive functions

Since the partial recursive functions are those that can be computed by a Turing machine, it seems that there ought to be a simple set of restrictions that can be placed on them to get the subset of ...
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Proof: Time complexity of deterministic vs nondeterministic Turing machines

I am stuck trying to understand a proof in my book for why, given a nondeterministic single-tape Turing machine $N$ that runs in time $t(n) \geq n$, the deterministic single-tape Turing machine $D$ ...
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3answers
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Turing machine to the languge $\{ a^ib^j | j=2^i \}$

I need to build a deterministic turing machine to the languge: $L = \{ a^ib^j | j=2^i \}$ I figured that I need to delete one $a$ for every ${2^i}$ $b$ until the tape contains no $a$ or $b$ or both. ...
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Question about Friedberg’s original proof of the Friedberg-Muchnik Theorem

This is Richard Friedberg’s original 1957 proof of the Friedberg-Muchnik Theorem, the origin of the ground-breaking priority argument. It is actually surprisingly understandable once you get past the ...
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What Turing degrees can truth in $\mathbb{N}$ have for different languages?

Tarski’s theorem implies that set of Gödel numbers of statements in the language of Peano arithmetic which are true in $\mathbb{N}$, the standard model of arithmetic, is not a recursive set. In fact ...
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A function that is computable but not weakly time constructible

I know that f(n) = n is a computable, weakly time constructible function but NOT a time constructible function. But I can't think of any computable function that is not weakly time constructible. Can ...
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Are the Turing degrees of truth sets cofinal?

Godel’s Completeness Theorem, in Henkin’s version at least, says that every consistent countable set of first-order sentences has a model. Based on that, let us call a set of natural numbers a truth ...
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What is the highest ordinal that can’t be obtained from Kleene’s O with oracles?

Kleene’s $O$ is a way to use natural numbers as notations for recursive ordinals. $0$ is a notation for $0$. If $i$ is a notation for $\alpha$, then $2^i$ is a notation for $\alpha+1$. And if $\...

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