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

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Is there a more intuitive proof of the halting problem's undecidability than diagonalization?

I understand the proof of the undecidability of the halting problem (given for example in Papadimitriou's textbook), based on diagonalization. While the proof is convincing (I understand each step of ...
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1answer
58 views

Does every infinite recursive language contain an infinite regular subset? [duplicate]

My intuition is telling me that this is not the case. But I am having trouble formulating a proof for this.How do I prove it ?
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0answers
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How do we explain this in Dovetailing [on hold]

A=Turing Machines that accept only 1 string. A= Not recursively enumerable. A'= Not recursively enumerable. Question is how can we prove it. Explanation if other than dovetailing concept would be ...
2
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1answer
60 views

Power of variants of Turing machines

I'm having trouble with this problem as I haven't discovered a good way to determine the power of a Turing machine. I was under the impression that if a Turing machine can perform the same actions ...
1
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1answer
33 views

What is a standard way to construct a turing machine for any function to compute

I am new to turing machines, I am having problems with mapping a function to a turing maching that computes that particular function. for example: f(x) = 2x + 3 n>= 0 MIN(x,y) leaves the smallest ...
12
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2answers
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In what sense is the Mandelbrot set “computable”?

The Mandelbrot set is a beautiful creature in Mathematics. There are a lot of beautiful images of this set created with high precision, so obviously this set is "computable" in some sense. However, ...
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0answers
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Exam question: Computability and tractability [closed]

I am revising for my exams and I found this question: Explain the terms computability and tractability and give one example for each concept to support your explanations. I am very confused it is ...
1
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1answer
41 views

m-functions in Turing's paper “On Computable Numbers and applications…”

I was reading Alan Turing's paper "On Computable Numbers with an Application to the Entscheidungsproblem". I was reading well until I encountered "4. Abbreviated Tables", page 235-236, where Turing ...
0
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0answers
33 views

Is this equivalent to Turing Completeness [closed]

This is a definition I came up with A Turing complete finite language is a finite language where there exists a Turing machine such that for any given computable number there is an element of the ...
0
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1answer
37 views

Prove Undecidability: TM M enters each of its states on Input W?

Consider the following problem: given a Turing Machine $M$ and an input string $w$, does $M$ enter each of its states during its computation on input $w$? How to prove that the problem is ...
1
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1answer
20 views

Proving a language to be Recursively Enumerable?

I know to prove a language to be Recursively Enumerable, it is ideal to represent a Turing machine for it. Let L be set of strings which have alphabet {u,d,l,r}, where u is up 1, d is down 1, etc. L ...
1
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2answers
38 views

Curry Howard correspondence and Church-Turing thesis

Curry-Howard correspondence states the equivalence between logic/deduction and types/programs. The Church-Turing thesis states the equivalence of some models of computation. Specifically, all ...
3
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1answer
24 views

Why is it true that the relation R and its negation are not semi decidable?

An example given for a relation R where its negation and itself are not semi-decidable was: $R(x,y)$ holds iff $y = 0$ then $R_{HALT}(x)$ holds, otherwise $y = 1$ and $R_{HALT}(x)$ does not hold. ...
3
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0answers
249 views

Is the logarithm of $\aleph_0$ infinite? [migrated]

In classical mathematics $2^{\aleph_0}=\aleph_1$, right? So if $2^x=\aleph_0$, what does $x$ equal? In other words, can we define a logarithm for $\aleph_0$, and what should it be. Is it infinite? ...
3
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1answer
47 views

Are there BBP formulas for all computable numbers?

A BBP algorithm is a formula with the following form where $b \in \mathbb Z$, $b \ge 2$, and $p(k)$ and $q(k)$ are polynomials in $k$ $$\sum_{k=0}^{\infty}\frac{1}{b^k}\frac{p(k)}{q(k)}$$ There are ...
7
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3answers
163 views

Is Deciding Decidability Decidable?

I am wondering if deciding the decidability of problem is a decidable problem. I am guessing not, but after initial searches I cannot find any literature on this problem.
1
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1answer
56 views

Can you write a program that alters another program semantically?

It is definitely possible to write a program that takes another program as an input and changes it syntactically, but does not alter the meaning. But is it possible to write a general program $alter$ ...
2
votes
1answer
218 views

Can a recursive language be uncountable?

Does there exist a recursive language $L$ whose cardinality is uncountable? I would like to have an explanation whether Turing Machine can encode uncountable languages and whether we can use this to ...
2
votes
2answers
57 views

Recursive language subtracted from recursively enumerable language

This is a homework problem but I am awfully confused. The problem reads as follows: If $L_1$ is recursively enumerable but not recursive, and $L_2$ is recursive, then which of the following is the ...
2
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1answer
80 views

Is deciding whether the language of a TM contains all strings of length 4 computable?

I was going through some Halting Problem reduction and I found the following problem: Given a semi-decider TM $M$, does the language $L(M)$ contain all strings of exactly length $4$? The ...
6
votes
1answer
78 views

How can a universal Turing machine simulate “bigger” ones?

I'm trying to find the answers of two questions about the Universal Turing machine. How can the Universal Turing machine simulate a Turing machine if the one that is being simulated has a bigger ...
10
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1answer
275 views

Is it decidable whether a pushdown automata recognizes a given regular language?

The problem whether two pushdown automata recognize the same language is undecidable. The problem whether a pushdown automata recognizes the empty language is decidable, hence it is also decidable ...
2
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2answers
31 views

Is the question of whether the language of a DFA/CFG is equal to a particular set of string decidable?

Suppose I have a set of strings $S$ that is generated from the alphabet. Suppose I have a DFA $D$ and a CFG $G$, are the questions of $\{D\mid D\text{ is a DFA and }L(D) = S\}$ and $\{G\mid G\text{ ...
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1answer
15 views

Efficient algorithms for finding the limit of a sub-sequence [closed]

Given a sequence $A_N={a_1,a_2,a_3...,a_N}$ of real numbers, and given that there exist some sub-sequence which generated from some deterministic converging sequence. Are there any efficient ...
0
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1answer
43 views

Turing Machines and Algorithm for Language Acceptance

Is there an algorithm to decide if any two Turing machines accept the same language? I can't find a definite answer to this. My guess is that there isn't, because then we would be able to decide if ...
5
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0answers
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Calculating with regexes

We use a regex engine (say, PCRE) that allows grouping subexpressions with parentheses and recalling the value they match in the search / replace expressions (backreferences, denoted by \i for ...
0
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1answer
26 views

If the language $A$ is decidable and the language $B$ is recognizable, then the language $A \cap B$ is recognizable?

I am discussing with a friend the following question: If the language $A$ is decidable and the language $B$ is recognizable, Then the language $A \cap B$ is recognizable? I believe it is. My point ...
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0answers
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Decidable language properties, L1 - L2

Given two decidable languages, L1 and L2, I have to show whether or not L1 - L2 results in a decidable language. I am not sure how to proceed, I am aware of the closure properties of decidable ...
0
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0answers
16 views

Forward jump turing machine and r.e languages [duplicate]

I was going through some exercises I found online and I am really stuck at this problem: Consider Turing Machines with the following restriction: they are only allowed forward jumps, i.e. if ...
3
votes
4answers
561 views

Do we need recursion in programming language to solve any problem?

My question is simple: If we want to be able to solve every problem, that we can solve using recursions, do we need programming language to allow us use recursions? Assuming we are allowed to use: ...
1
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1answer
40 views

If A is decidable and B is decidable, then A is Turing Reducible to B

The statement seems intuitively true but is it? If so, how can I prove this?
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1answer
46 views

Why does my answer sheet say the set of computable functions is uncountable?

I'm trying to understand why I can't find room for the set of computable functions in the hotel of the Hilbert's Hotel Paradox. I was thinking that, because Gödel numbering, I could consider the set ...
1
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1answer
37 views

Trying to understand the proof of the halting problem presented in Sipser textbook

I'm having some problems to understand the classic proof of the halting problem. The Proof: $A_{tm} = ${$<M,w>$ | $M$ is a $TM$ and $M$ accepts $w$}. We assume that $A_{tm}$ is decidable and ...
2
votes
0answers
69 views

A complete catalog of 2-state Turing machines?

For educational purposes, I'm about to start a research project that involves creating a complete database documenting and classifying all 2-state, 2-symbol Turing machines, according to a ...
3
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2answers
92 views

Halting Problem and Turing Degree and Reduction? [closed]

This is a Local Olympiad question on computation and computer science on 2013. How can explain it and says some hint for understanding such an example question. for $ A \subseteq \mathbb{N}$ we ...
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3answers
64 views

Language is recursive, hence recursively enumerable

I was going through a book of proof and I read: If L is recursive, L is r.e. And the proof goes: Let L be recursive, hence there is a TM that decides it Convert an halt state to a normal state ...
6
votes
2answers
247 views

Decidability of “Is this regular expression prefix-free?”

Say that string $x$ is a prefix of a string $y$ if there exists a string $z$ such that $xz = y$, and say that $x$ is a proper prefix of $y$ if in addition $x \not= y$. A language is prefix-free if it ...
2
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1answer
99 views

Turing recognizable -decidable languages-

I was wondering how to prove that $C$ (which is a language) is Turing-recognizable iff a decidable language $D$ exists such that $C = \{x \mid \exists y \;(\langle x, y\rangle \in D)\}$. I do not ...
2
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1answer
44 views

Can a probabilistic Turing Machine compute an uncomputable number?

Can a probabilistic Turing Machine compute an uncomputable number? My question probably does not make sense, but, that being the case, is there a reasonably simple formal explanation for it. I should ...
2
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2answers
112 views

Does “contains only” imply “contains”?

Written in English, does "the set S contains only members of set T" imply that S does contain some member of set T? How would this relationship be written formally?
6
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2answers
311 views

Are these sets of indices also index sets?

An index set is a set of all indices of some family of computably enumerable sets. It is known that the empty set is an index set and that $K = \{e \mid e \in W_e\}$ is not an index set. The ...
0
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0answers
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Can we compute the fastest algorithm for a given total function? [duplicate]

First let $f$ be the description of a partial function. Let $\operatorname{optimize}(f)$ be a function that returns a description of the "fastest" Turing machine that computes $f$ for some sensible ...
0
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1answer
69 views

Algorithm to compute a recursive function on a given set [closed]

I am working on a property of a given set of natural numbers and it seems difficult to compute. There is a function 'fun' which takes two inputs, one is the cardinal value and another is the set. If ...
0
votes
1answer
56 views

Recursive algorithm to compute a sum of product like function

I am working on a recursive formula associated with discrete mathematics which seems very difficult to compute. The formula is as follows $F_{i,j}(m)=\sum_{t=j}^{m}\left [ ...
4
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2answers
149 views

Unprovable Post correspondence problem instance

Since there is no algorithm for the post correspondence problem, there exists an instance of this problem such that we can neither prove that the instance is positive nor prove that the instance is ...
4
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1answer
119 views

Primitive Recursion and course-of-values recursion - examples?

I ran into examples that I not trivially understand on course-of-values recursion, In defining a function by primitive recursion, the value of the next argument $f(n+1)$ depends only on the ...
0
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1answer
52 views

Is a partial function Turing-computable?

From my understanding for a function to be considered Turing-computable the Turing machine which computes it must terminate for all inputs (according to this http://planetmath.org/turingcomputable and ...
6
votes
3answers
125 views

Unreachable Real Numbers - Randomness & Computability

I've recently read that there were many real numbers that would never be reachable by humanity. The explanation itself says that we can write as many programs as integers which is infinite, but there ...
0
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1answer
68 views

Does Church's Thesis include artificial intelligence?

By Church's Thesis it is impossible to design an algorithm to decide halting problem. I would like to know the word algorithm in this context includes artificial intelligence or not? I mean is it ...
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1answer
67 views

Are finitely many statements resp. variables sufficient to compute every function?

I prepare for local complexity contest and review some old Interview questions banks. I get stuck in one problem and no idea how we can solve it. please share your idea or help with this question: ...