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Questions tagged [computability]

Questions related to computability theory, a.k.a. recursion theory

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Does solving all Halting problem instances 'in the limit' imply we solve an undecidable problem?

The recent Arxiv paper "Learning the undecidable from networked systems" attempts to construct a network of $N$ Turing machines$^1$ that can solve the Halting problem for any program of size $O(\log N)...
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1answer
37 views

Can the notation for polynomial reduction, A ≤p B be reversed in computability theory?

I don't know this is a proper question on this forum but I was reading about computability theory and I saw the reduction concept and its notation like this: $A \le_pB$. I just wanted to know is this ...
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4answers
3k views

What does Turing mean by this statement?

I encountered below statement by Alan M. Turing at here: "The view that machines cannot give rise to surprises is due, I believe, to a fallacy to which philosophers and mathematicians are ...
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0answers
42 views

Turing Machine the undecidable language

Consider the undecidable language $$\mathrm{ALLTM} = \{\langle M\rangle \mid M\text{ is a Turing machine with }L(M) = \Sigma^*\}.$$ Suppose you had an oracle for the halting problem (that is, you have ...
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Question regarding A/D Converters and calculating voltage [on hold]

You are comparing different A/D converters that will have 3.2 V and -3.2 V as their positive and negative reference voltages, respectively. Out of the following, __(List below)____ converter can ...
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1answer
47 views

Proving $L = \{\langle M, w, n \rangle$ : $M$ accepts $w$ within $n$ steps $\}$ is decidable

Show the following problem is decidable: Given $w\in \Sigma^{*}$, $n\in \mathbb{N}$, and a Turing machine $M$, does $M$ on $w$ halt within $n$ steps. My Thoughts: I am new to proving results like ...
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0answers
51 views

Can hypercomputation compute the impossible? [on hold]

There are things which are illogical/logically impossible (like saying that 2+2=4 and 2+2=5. Without changing anything in the axioms of mathematics or logic, this would be a contradiction and would be ...
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1answer
31 views

Construct a decidable set $B$ such that $B \neq A_w$ for any $w \in \Sigma^\star$

I've been stuck on this problem for a while. Any hints would be appreciated! Let $A \subseteq \Sigma^\star$ be decidable. Given $w \in \Sigma^\star$, define $$A_w = \{x \in \Sigma^\star\:|\: \langle ...
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Proving $E_{DFA}$ is decidable by running $A_{DFA}$ several times

I am trying to prove that language $E_{DFA}$ is decidable using multiple executions of $A_{DFA}$ (not using the proof in Sipser's book "Introduction to the Theory of Computation"). Can i just use ...
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1answer
32 views

Finding a mapping reduction from $A_{TM}$ to $\overline{CF_{TM}}$

I am trying to find a mapping reduction from $A_{TM}$ to $\overline{CF_{TM}}$, but I can't seem to find one. Definitions: $$\begin{align*} CF_{TM} &= \left\{ \langle M \rangle \mid \text{$M$ is a ...
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2answers
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Is the language {<p,n> | p and n are natural numbers and there's no prime number in [p,p+n]} belongs to NP class?

I was wondering if the following language belongs to NP class and if its complimentary belongs to NP class: \begin{align} C=\left\{\langle p,n\rangle\mid\right.&\ \left. p \text{ and $n$ are ...
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1answer
19 views

Does $E_{TM}$ accpets the empty word $\varepsilon$?

Let $L = E_{TM} = \left\{ \left<M \right> | M \text{ is a TM and L(M)} = \emptyset \right\}$. Does $L$ accepts the empty word $\varepsilon$? In other words, is $$\varepsilon \in L$$ I'm ...
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0answers
24 views

Is the proof for the undecidability of $A_{TM}$ still valid if we change certain parts?

i have a question based on a question i saw exists on the site, but with wrong information in it and no answer there, so i am reposting it with valid information(cited wrong from the book). on page ...
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1answer
29 views

Language decidable or not?

I have just began with my course of complexity and computability and I need your experience to help me progress !! Something is not clear for me: It is asked to determine if L1 is decidable or not, ...
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2answers
90 views

Can a TM recognize whether another TM recognizes a non-empty language?

Let $$L_1 = \{\langle M\rangle\mid M \text{ is a Turing Machine and }L(M)\ne\emptyset\}.$$ Is $L_1$ recognizable? If so, can you give me a pseudo-algorithm? My attempt: I wanted to study ...
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1answer
18 views

Enumerators and recognizers

I got confused by this a bit... the words of any recursively enumerable language $\mathcal{L}_{RE}$ can be enumerated by an enumerator $E$, i.e. there is an effective procedure (using lexicographic ...
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2answers
41 views

Rice's Theorem - usage on $DFA$ or $LBA$

I have read about Rice's Theorem on Sipser's book, and I think I understand it quite well. I understand that it can be used to show that a language is not decidable. However I am not sure about one ...
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3answers
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Are $CF_{TM}$, $\space $ $\overline{CF_{TM}}$ Turing-recognizable?

I have searched the site well through, and also using Google and notes and couldn't find an answer to a question I am wondering about. Given: $$CF_{TM} =\{ \langle M \rangle \mid \text{$M$ is a TM ...
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1answer
52 views

is it possible to reduce $HALT_{TM}$ to $E_{TM}$?

I am wondering, if it is even possible: is it possible to reduce $HALT_{\text{TM}}$ to $E_{\text{TM}}$? $HALT_{\text{TM}}=\{\langle M,w\rangle\mid M\text{ is a }TM\text{ and }M\text{ halts on input }...
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Proving language K is undecidable using the diagonalization method

I have a problem proving the following properties of given language K: $K = \{< M > | M\ accepts < M >\}$ I am trying to prove that language K is Turing-recognizable but undecidable ...
3
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2answers
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One counter automaton as a function

We can associate a one counter finite automaton with a function $f:\Sigma^* \to \mathbb{N} \times \{0,1\}$, where $f(x)=(n,b)$ describes the state where the automaton terminates when fed an input word ...
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1answer
78 views

Finding two languages satisfying conditions

Let $$E_{TM} = \left \{ \langle M\rangle \mid L(M) = \emptyset \right\}$$ Prove that there are two languages $L_1, L_2$ such that $L_1, L_2 $ are infinite. $L_1 \cup L_2 = E_{TM}$ $...
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1answer
59 views

What doest it mean: “computer is an intelligence amplifier”?

There is one example in Kolmogorov complexity books and related articles: Consider we have a monkey at a typewriter and a monkey at a computer keyboard. If the monkey types at random on a typewriter,...
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1answer
28 views

Do all DFA's containing an “accepting path containing a cycle” accept infinite languages?

So I've seen this claim being made on different questions: Do self-loops in DFA cause infinite languages? I'd like to find formal proof for this. I think I should also note that an "accepting path ...
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0answers
31 views

Effects of changes in the proof that $A_{\text{TM}}$ is undecidable

In the proof that $A_{\text{TM}}$ undecidable we use the following machine: $D =$ On input $\langle M, w \rangle$: Simulate $M$ on input $w$. If $M$ ever enters its accept state, accept. ...
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0answers
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Is the set of NFAs that recognize a finite language decidable? [duplicate]

Question: Let $L$ be the set of NFAs that recognize a finite language, is $L$ decidable?
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2answers
74 views

What are the other language models of computation similar to lambda calculus?

I hope this question makes sense, but I was wondering if there are other models of computation similar to lambda calculus that you can use to build up axiomatic mathematical and logical fundamentals ...
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1answer
68 views

What makes a function computable?

It seems to me, at the moment, that I don't understand where the line is drawn between "computability" and being incomputable. The main question is, is it the ability to be "solved" only by trying out ...
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1answer
25 views

checking whether a language is turing recognizable

After reading about it in the textbook and in the web, i was wondering about the "turing recognizable" concept. so for instance, if i take a simple language like:"L = {< M > | M ACCEPTS < M >}",...
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1answer
33 views

Constructing a reduction between two languages about pairs of Turing machines

I'm curious about a potential relation between the following two languages. $L_1 := \{\langle M_1, M_2 \rangle : L(M_1) \cap L(M_2) \ne \emptyset \}$. $L_2 := \{\langle M_1, M_2 \rangle : L(M_1) \...
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0answers
56 views

Is it a bad idea to require a correctness proof as part of a computable real number?

At 30:42 of Norman Wildberger's Difficulties with real numbers as infinite decimals (II) lecture, he raises the question whether "certificates of boundedness" (of the runtime of the algorithm to ...
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1answer
52 views

Are there decision problems outside of NP?

Consider any problem in NP-hard, then it has a polynomial reduction from a problem in NP in polynomial time. Though, it isn't clear by this definition whether there are decision problems in NP-hard ...
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0answers
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Analyzing the encoding of a DFA by a deterministic Turing machine [duplicate]

Define $L=\{\langle D,R \rangle \mid D$ is a DFA, $R$ is a regular expression, $L(D)=L(R) \}$ Is $L$ decidable? Is $L$ decidable in polynomial time ($L \in P$)? I am trying to ask: can a TM analyze ...
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1answer
40 views

How to judge the searching precision of Particle Swarm Optimization?

As the title mentioned, how can I judge the searching precision of PSO? Is this depending on the velocity of the particles? I would like to give an example to clarify my question: For a 2-D searching, ...
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1answer
70 views

Decide if a string is in a language without simulating the automata accepting the languge

Is it possible for a Turing machine with input of a DFA that accepts a finite language and a string to decide whether the string is in the language without "fully simulating" the DFA on the string? ...
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0answers
22 views

Is there a formal way of defining a Zeno Machine?

The idea of a Zeno machine is pretty interesting to me, but I can't seem to find a formal definition for how a Zeno machine would work. I can find a couple of definitions around but they are all ...
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1answer
37 views

Is this language is Context-free language or not?

Is anybody can help me please to determine is this language is Context-free language or not? L={wvw | w,v∈{a,b,c}+} for example: part of the language: acbac, abcab, bbcbb not part of the language:...
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0answers
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Is McCarthy Formalism first ever formalism for defining functions recursively in computer science?

McCarthy formalism is a formalism for defining functions recursively, first introduced in classic paper Recursive Functions of Symbolic Expressions and Their Computation by Machine, Part I (1960). ...
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1answer
101 views

Is it decidable whether a Turing machine M will reach state q on input s?

Given a turing machine $M$, one of its states $q$ and an input word $w$, will $M$ ever reach $q$ on $w$? As we are not given anything about the word length, I assume that we have a finite length word....
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1answer
51 views

Prove the languages |L<M>| = 2 and |L<M>| $\not=$ 2 to be non-Turing recognizable or non-recursively enumerable

I am trying to prove the non-recursively enumerable property of two languages. $L_2 = \{\langle M \rangle: |L\langle M \rangle| = 2\}$ and $L_{\not=2} = \{\langle M \rangle: |L\langle M \rangle| \not=...
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1answer
24 views

Reading elements of a countable set with Turing machine

I have a basic question about the behavior of a potential Turing machine. Suppose that $S$ is a countable set of binary strings, so that we can enumerate $S$ as $(s_i)_{n\in \mathbb{N}}$. Suppose ...
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Can we see all of mathematics as an attempt to simplify computations?

This is a rather strong claim, and therefore likely to be incorrect, but hear me out. Firstly, when I talk of “computations”, I mean this in a broader sense than normally used, because I am including ...
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2answers
54 views

Proof of Space Hierarchy Theorem incompatible with Linear Speed Up Theorem for time

In this proof of the Space Hierarchy Theorem the following langugae is defined $$ L = \{ (\langle M \rangle, 10^k) : M \mbox{ does not accept } (\langle M \rangle, 10^k) \mbox{ using space } \le f(|\...
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2answers
202 views

Is the set of language decidable by some Turing machine computing in some given computable time bound decidable

Let $T : \mathbb N \to \mathbb N$ be some computable function. Then by $\mathcal C_T$ we denote the class of languages decidable by a deterministic Turing machine in at most $T(|w|)$ steps for an ...
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1answer
277 views

What's after EXPSPACE?

As far as I'm aware, EXPSPACE is the most inclusive computational complexity class. I was wondering if/how people conceptualize supersets of EXPSPACE. Thinking about this question, I came up with a ...
4
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1answer
307 views

Given a total recursive function, can you always compute its fixed-point?

We know from Kleene's recursion theorem that if $f$ is total recursive, there must be an integer $n$ for which $\varphi_n=\varphi_{f(n)}$. My question is: for every $f$ total recursive, is there a ...
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1answer
42 views

How do you prove that $A = \{ x \in \mathbb{N} | W_{x} = [0..x]\}$ is a productive set through functional reduction?

As the title states, how do you prove that $A$ is productive? With $W_{x}$ I mean the set of points in which the turing machine with index $x$ halts. The standard approach that I follow is functional ...
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2answers
86 views

Proving a language as undecidable without using reductions

Let's say our Σ is 0 and 1. I want to disprove the following: There can be Turing Machines that accept only 1's, i.e. 1, 11, 111, etc. Therefore, all languages that have strings of 1's are ...
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1answer
53 views

How does one use the Nerode-Myhill theorem to prove that a language is regular?

Showing that a language is not regular is straight-forward, because all one needs to do is find an infinite set of inputs which has an injective mapping to the set of equivalence classes which compose ...
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2answers
59 views

Would any continuous model of the universe have/be based on hypercomputational laws?

I've read that when Turing-Church thesis is applied to the universe and physics, one of the three interpretations that we can use and is defended by some important physicists is that: "The universe ...