Questions tagged [computability]

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

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What computational model supports arbitrarily sized integers?

I want to do some research, but I don't think it's important the number of bits it takes to represent the integer input and arithmetic on the abstract machine. So what is the model that addresses ...
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Union of a decidable language with complement of a recursively enumerable language

So the question wants to prove or disprove that 'a Union of a decidable/recursive(i understand them to be the same) language and the complement of a recursively enumerable language is a recursive/...
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Can a computable program have an infinite output?

Can a computable program produce an infinite output from its (presumably finite) input? *I wouldn't think so for similar argument as to why it can't compute over an infinite input. A follow on (...
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if there is no reduction from A to B

I'm facing the following question : If there is no $𝐴\leq_𝑚𝐵$ reduction, does this necessarily mean that A is not decidable? for any choice of B. thanks in advance.
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Undecidability of language involving two TMs

I am currently browsing the lecture notes on computability/decidability and I have encountered the following exercise I am unable to solve. Given $M_1$, $M_2$ Turing machines, is it true that for ...
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Use the Rice's theorem to prove that the following property of a Recursive Enumerable language L is undecidable

This exercise was taken from the book "Languages and Machines: An Introduction to the Theory of Computation" by Thomas Sudkamp. It refers to exercise 12 (b) chapter 12. Given a language L which is ...
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Compare two complexity functions having the same asymptotic complexity

For a certain problem two solution algorithms (A1 and A2) with the following execution times have been found: $A1: T_{A1}(n)=4n^2 +7log(n^2)$ $A2: T_{A2}(n) = 4T(n/2) + log(n)$ Say, technically ...
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Proof by reduction that the Universal Language is not recursive using the complement of the Diagonalization language

I have the following proof which I don't fully understand. L D/ is the complement of the Diagonalizaton Language. L U is the Universal language. Assume U* is a TM for Lu which always halts. ...
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Are Turing unrecognizable and undecidable languages, recognized and decided by hyper computation?

Do the hyper computing machines/models that are supposed to be more powerful than Turing machines, capable of recognizing and deciding the languages that are not recognizable/decidable by Turing ...
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In the reduction from HALT to ALLHALT, why does the constructed Turing machine loop indefinitely when the inputted Turing machine rejects?

Let HALT be the language $\{\langle M, w\rangle : M\text{ is a TM that halts on }w \}$. Let ALLHALT be the language $\{\langle M\rangle : M\text{ is a TM that halts on all inputs}\}$. Use a reduction ...
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Do $PR/Poly$ solve the halting problem of Turing Machine?

I know that $R/Poly$ solves the halting problem, as we can have the program that runs for longest time as advice and check which halts earlier. But what if we weaken the $R$ into something, like $PR$?
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Would models of computation in other conceivable universes be Turing complete?

I'm interested in gathering some references that discuss the topic of the relationship between computation and physics. Specifically, I'm interested in investigating two points of view: our current ...
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Can a subset of indexes of a partial recursive function be recursively enumerable but not recursive?

Consider a Gödelian numbering of partial recursive functions. Consider the set of indexes corresponding to some function $\phi$. It may be seen that this set is not recursively enumerable. Does this ...
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Grammar with fewest variables

I am looking over a past exam for a theory of computation class I am taking, and unfortunately no solutions are provided. I am stuck on this question, and would greatly appreciate any help or hints. ...
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Is checking if the length of a C program that can generate a string is less than a given number decidable?

I was given this question: Komplexity(S) is the length of the smallest C program that generates the string S as an output. Is the question "Komplexity(S) < K" decidable? With respect to ...
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Restriction: polynomial time decision of instance is why needed to “decision Problem”?

I am reading book "combinatorial optimization 3rd edition(Bernhard Korte、 Jens Vygen)". (latest version is sixth.) There are some discriptions in this book that I don't understand Not all binary ...
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Proving the existence of a $\Pi_1$-sentence in True Arithmetic that is independent of Peano Arithmetic

I am trying to wrap my head around how to prove the following statement: There exists some $\Pi_1$-sentence $A$ such that $A \in \textbf{TA}$ but $\{A, \neg A\} \cap \textbf{PA} = \emptyset$. $\...
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Why can't we prove decidability of $L= \{ \langle M \rangle : M$ accepts $ \epsilon \}$ with a configurations graph?

Since every deterministic Turing Machine can be translated to a graph of configurations such that $M$ accepts a word $w$ iff there is a path from the initial configuration that matches $w$ to an ...
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Is there a model of ZF¬C where some program always terminates but has no loop variant?

Wikipedia has a proof that every loop that terminates has a loop variant—a well-founded relation on the state space such that each iteration of the loop results in a state that is less than the ...
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Analogue of the topology-computability correspondence for computational complexity

There is an interesting correspondence between notions of topology and notions of computability theory originating from the ingenious idea of Dana Scott to identify computable functions with ...
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Are there any problems that reduce to the halting problem?

I'm reading through sipser and there is a lot of computability problems that the halting problem reduces to, i.e. if $A_{TM} = \{\langle M,w\rangle : M$ accepts input $w\}$ then $A_{TM} \leq P$ where ...
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Why does such reductions work [duplicate]

In class we saw examples of reductions like from Independent Set (IS) to Longest common subsequence (arbitrary number of sequences) (LCS) $V = \{v_1,\ldots,v_n\} E =\{ e_1,\ldots, e_m \}$ The ...
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1answer
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Reductions from non decision problems

I want to show a minimization problem $Y$ has no approximation factor of 1.36. To be more specific the problem $Y$ is the exemplar distance problem between two genomes. Could I reduce from the min ...
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Turing machine that can read and write simultaneously equivalence proof

How would I go about proving that a TM that can both read and write at the same time is equivalent to a typical TM? For constructing a read+write TM from normal TMs, my idea is to split the state of ...
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How undecidable is it whether a given Turing machine runs in polynomial time?

The proof of Theorem 1 that PTime is not semi-decidable in this recent preprint effectively shows that it is $\mathsf{R}\cup\mathsf{coR}$-hard. The proof itself is similar to undecidability proofs at ...
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Union of infinitely many recursive sets that is not recursive but recursively enumerable?

I am looking for an example such that the union of infinitely many recursive sets is not recursive but recursively enumerable. I was thinking taking $S_i=\{i\}$ and then taking the union of these ...
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Turing machine with k-tape, tape of output

Consider a Turing machine with input alphabet $\{a,b\}$ that computes the following function: $$ f(w, v) = \begin{cases} w & \text{if } \operatorname{length}(w) > \operatorname{length}(v), \\ ...
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How to proof a function is not computale [duplicate]

I wish to undestand how to proof a function is/is not computable. I found this example online (without solution) beacuse I was thinking was easy to understand, but I am stuck in understanding how to ...
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On the computable function of a problem that halts

Let's say program $P$ with given input $i$ is found to halt (or doesn’t halt) by a Turing machine. Is it true that the same program $P$ with input $F(i)$ also halts (or not, respectively), where $F$ ...
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Deciding whether set of running times is infinite

I have a language $\mathrm{Count}(M)$, defined below, and a finite number $k$. \begin{align} \mathrm{Count}(M)= \{k \in \mathbb{N} \mid \text{there exists some input on which $M$ halts after exactly ...
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Is my formal definition of programming language correct?

I found this formal definition of a programming language in the 1973 paper Formal definition of programming languages by Terrence Pratt. PL is a formal language endowed with two structures: a ...
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If base 2 looks like a square wave, and base 3 looks like this, what does base e look like?

In computing, base 2 is a 0 and a 1 / on and off in a transistor. Base 3 is -1, 0 and 1. Electronically/in computing how would/could base e be represented visually?
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Why do intuitionists accept the nonconstructive proof that the halting problem is undecidable? [duplicate]

On the intuitionism page at Stanford Encyclopedia of Philosophy (SEP), it's said in Section 3.3 that Because of the finiteness of a natural number in contrast to, for example, a real number, many ...
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Designing a PDA that keeps track of the stack size

Would it be possible/legal to design a PDA that can use the stack as a way to keep track of the number of inputs seen? (i.e the size of the stack would act as some sort of counter). What I was ...
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Is this language Recursively Enumerable or Not RE?

$L = \{\langle M, k\rangle : M\;\text{is a Turing Machine and } |\{w \in L(M) : w \in a^*b^*\}| \geq k \}$ My Interpretation of language is that $L$ is a language which contains Turing machine ...
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Turing Machine to return all prime numbers

My task is to design Turing Machine that ignores its input and returns all the prime numbers. I have some basic idea how to do that but I am not completely sure whether my approach is correct or not. ...
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Decidable questions of undecidable problems

Even if there is no general algorithm to decide if any program will halt, but there could be properties or meta-questions about the programs that is decidable. For example, given program $A$ and a ...
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I want to solve this question for algorithm, please [closed]

Write an algorithm that calculates the monthly payment of a bank loan with a fixed interest-rate. Given the principal amount, the fixed interest rate, the number of years to pay the loan, you can ...
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1answer
45 views

Showing the following language is decidable

Let $BAL_{DFA} = \{<M> \mid M \text{ is a DFA that accepts some string containing an equal number of 0's and 1's } \}$ Show that $BAL_{DFA}$ is decidable. Generally such questions seem to be ...
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Deciding whether $f(x) = f(y)$ is beyond RE and coRE

I would like to prove that the following subset is outside both RE and coRE: $$A = \{ (p, (d_1, d_2,\dots, d_k)) \mid \text{for each } 1 \le i,j \le k, \; [p]d_i = [p]d_j \}, $$ where $p$ is a ...
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Are models of computation closed under composition?

It's common to ask whether a particular class of languages $\mathcal{C} \subseteq \mathcal{P}(\Sigma^*)$, for some alphabet $\Sigma$, is closed under complement, or union, or intersection, or ...
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Busy-Beaver-like question for WHILE-Programs (Theoretical CS)

So this is exam-task is called "Busy WHILE-Programs" In our lecture it was proven that WHILE-Programs are turing-complete. In short a WHILE-Program only allows the following: ...
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Expressing partial decidability using existential quantification

def. A predicate M(x,y) is partially decidable if the function f given by " f(x,y) = 1(if M(x,y) holds), f(x,y) = undefined(otherwise) " is computable. Thm. If M(x,y) is partially decidable, then so ...
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What Is the Complexity Class of Deciding Whether a Problem Is in NP? Is It Decidable?

Title says it all, but to clarify: Define a problem, called $IsInNP$, as follows: Given a Turing Machine $M$ that always halts, $IsInNP$ is the problem of deciding if the problem that $M$ ...
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What does B compute in Recursion theorem

I am reading Michael Sipser's book for this theorem Recursion theorem Let T be a Turing machine that computes a function t : Σ* × Σ* → Σ* . There is a Turing machine R that computes a function ...
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Does Types and Programming Languages use a recursive equation to define a recursive type or its generator?

In Types and Programming Languages by Pierce et al: The recursive equation specifying the type of lists of numbers is similar to the equation specifying the recursive factorial function on page 52: ...
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Doubt in definition of closure under concatenation operation in Recursive Enumerative languages

I recently started studying theory of computation. Recusive enumerable language – closed under concatenation. Sir, I have a doubt regarding understanding of this. Please Note - RE shortform i am ...
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An example of a computable problem that is not in P

I am trying to find a simple example of a problem that is computable but not in P, I know very well that it would be enough to get one in NEXTIME-complete however the problems that I find in this set ...
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1answer
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Is this set computable?

Let be $B$ a Busy Beaver function and set $W=\{\langle M \rangle :\text{$M$ stops in less than $B(10^{1000})$ steps on an empty tape}\}$. Is this set computable? I'm not sure how to approach this ...
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What is known about the sets enumerated by primitive recursive functions?

Let's say that a set of natural numbers $S \subseteq \mathbb{N}$ is primitive recursively enumerable if there exists some primitive recursive function $f$ such that $S$ is the range of $f$. That is, ...