Tagged Questions

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

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-2
votes
2answers
34 views

Is emptiness of the intersection of the languages of two TMs decidable? [duplicate]

Let $\qquad \mathrm{DISJOINT} = \{ \langle M_1,M_2 \rangle : M_1, M_2 \text{ are TMs and } L(M_1) \cap L(M_2) = \emptyset\}$. How do I know if this language is decidable or not? And How do I prove ...
0
votes
0answers
14 views

Partial recursive characteristic function for finite sets

In class we were told that, for every finite subset $X$ of the natural numbers, it is possible to find a partial recursive function $g(x)$ that outputs $1$ if $x\in X$ and $0$ if ...
-1
votes
0answers
26 views

Is it decidable whether a TM modifies the input more than once?

I am having trouble with this problem: Let $P = \{\langle M,w\rangle \mid M \text{ on input } w \text{ modifies the input at most once}\}$. Is P decidable?
0
votes
1answer
46 views

Decide whether DFA have useless states

A useless state in a DFA is one that is never entered on any input string. Consider the problem of determining whether a DFA has any useless states. Formulate this problem as a language and show that ...
0
votes
1answer
34 views

Function is recursive iff its graph is recursively enumerable

So I understand that a function is recursive if there exist a Turing Machine that accepts it and halts on every input, since function is defined everywhere. But how to prove that function is recursive ...
-1
votes
2answers
110 views

Is the infinite union of computable sets computable? [duplicate]

My intuition is telling me that this is untrue. But I am having trouble formulating a proof for this. Can anyone point me in the right direction? I've seen a proof by contradiction involving the union ...
-1
votes
1answer
72 views

If all infinite r.e. languages have an infinite recursive subset, then do co-r.e. languages not have such subsets?

If all infinite r.e. languages have an infinite recursive subset, then can we logically take co-r.e. languages to not have such subsets by complemence?
-1
votes
0answers
36 views

Is the following statement true ? If L is a decidable language and L′⊆L, then L′ is also decidable ? Prove your answer is correct [duplicate]

Is the following statement true ? If L is a decidable language and L′⊆L, then L′ is also decidable ? Prove your answer is correct I can't figure out this question. Any tips ?
61
votes
9answers
12k views

Why, really, is the Halting Problem important?

I don't understand why the Halting Problem is so often used to dismiss the possibility of determining whether a program halts. The Wikipedia article correctly explains that a deterministic machine ...
-1
votes
1answer
9 views

Cocurrent programming language being Turing-equivalent and difference between Turing-complete and equivalent

In Is concurrent language CCS or CSP turing-equivalent in language power?, the answer says that CCS or CSP is Turing-complete. But that does not seem to answer whether CCS or CCP is Turing-equivalent. ...
0
votes
1answer
68 views

Is the identity function a many-one reduction from a language to super-set?

I need help with a question. Prove or disprove the following claim: Let $f\colon \Sigma^* \to \Sigma^*$ be the identity function, i.e., $f(w) = w$ for all $w \in \Sigma^*$. Let $L_1$ and $L_2$ be ...
2
votes
1answer
36 views

Is concurrent language CCS or CSP turing-equivalent in language power?

Does the concurrent language CSP (or CCS, $pi$-calculus) model interacting machines? Is CSP (or CCS, $pi$-calculus) Turing-equivalent to other programming languages like C?
1
vote
1answer
95 views

Prove Σ* is decidable

I see that Σ* is claimed to be decidable in many documents, but I have never seen an example or easy demostration that it is decidable. What is the proof that Σ* is decidable?
1
vote
1answer
15 views

Can we obtain a state diagram of a single Turing machine

When illustrating what states are in Turing machine, often the examples of programs, like a checker that checks an input number is even number, are given. But different programs seem to have different ...
3
votes
1answer
173 views

Could two decidable languages ever not have a mapping reduction?

Is it ever the case that two decidable languages $L_1$ and $L_2$ that cannot be reduced to one another (in either or both directions)? Intuitively, I would not expect there to be, but rigorously, are ...
0
votes
2answers
62 views

What is the definition of a problem

In computation theory, when talking about the computability and complexity of a problem, what is the definition of a problem? How specific should a problem be? For example, can the followings all ...
2
votes
2answers
50 views

Is Newton's Method to compute the zeros of a function an algorithm?

Looking for Newton's method in Wikipedia, I read the following: In numerical analysis, Newton's method (also known as the Newton-Raphson method), named after Isaac Newton and Joseph Raphson, ...
3
votes
1answer
24 views

Why can't we search lexicographicaly ordered programs to compute Kolmogrov complexity?

Kolmogrov complexity is known to be uncomputable. Why can't we enumerate all programs of size i = 0 in lexicographical order - if any produce string s, that is the Kolmogrov complexity; if not, ...
5
votes
1answer
215 views

Is a secondary TM sufficient to detect all loops?

Procedure: Start a secondary TM in parallel with the first, but have the second perform exactly 1 step each 2 steps the first TM performs (i.e. it runs at half speed). If the second machine ever ...
3
votes
2answers
142 views

Finite number of Turing machines running concurrently on multi-tapes Turing-machine-equivalent?

So basically, there are several (finite number of) Turing machines being able to read off and write to the same set of tapes (the number of tapes is finite, but each tape may have infinite tape ...
2
votes
3answers
82 views

Is $AlwaysHalt$ recursively enumerable?

I was doing some complexity theory exercices and I came over this one: $AlwaysHalt = \{R(M) | M$ halts with all $x \in \{0,1\}^*\}$ Is $AlwaysHalt$ recursively enumerable? I would say YES and ...
2
votes
2answers
91 views

Is it possible to obtain a total function by composition of partial functions?

This statement is Theorem 1.1 (page 39) of Computability, Complexity and languages by Martin Davis: If function $h$ is obtained from the (partially) computable functions $f$, $g_1$, $g_2$, ..., ...
-1
votes
0answers
20 views

primitive recursive( course of values recursion)

If $f(n)$ is any function, we write $f(0)=1,f(n)=[f(0),f(1),...f(n-1)]$ if $n\ne 0$ and let $f(n)=g(n,f(n))$ for all $n$. Show that if $g$ is recursive so is $f$. I don't want anybody solve this ...
0
votes
0answers
49 views

Language accepted by a RAM

Show that any language accepted by a RAM can be accepted by a RAM without indirect addressing. Could you give me some hints what I could do??
2
votes
1answer
67 views

Is there a class of formal grammars that generate Recursive Languages only?

Is there a class of formal grammars that generate Recursive Languages only? (ie with which it is not possible to generate non recursive languages.) If so what kind of production rules/restrictions do ...
0
votes
0answers
20 views

How to show {n:U(n,x) is defined for all x} is not enumerable

U(n,x) is Gödel universal function, and we need to show {n:U(n,x) is defined for all x} is not enumerable. I do not have any clue right now. Anyone can give me some hint about this question.
1
vote
1answer
42 views

Satisfiability of first-order logic is undecidable?

I struggle with understanding why the satisfiability in the first-order logic is undecidable. Could you explain it with some examples? I've also seen that satisfiability in some first-order formulas ...
2
votes
2answers
257 views

Is this variant of ATM decidable?

Ok so I understand how $\mathrm{ATM} = \{\langle M,w \rangle \mid \text{$M$ is a TM and $M$ accepts $w$}\}$ is undecidable. Is this because $w$ is a variable? What if the parameter is fixed? ...
1
vote
2answers
82 views

Computation is effectively computable in theory and in practice

My big question is the following: What is the meaning of a computation being "effectively computable" (EC) in theory and in practice? In trying to understand these concepts further, I have a couple ...
3
votes
1answer
133 views

Are there more partially recursive functions than and recursive functions?

Is the cardinality of the set of partially recursive functions greater than the cardinality of the set of recursive functions ?
6
votes
0answers
53 views

Is There a Complete Problem for the Class of Turing Decidable Problems?

Languages such as $\text{HALT}_{TM}$ are $\textsf{RE-complete}$ under many-one reductions. It is trivial to see that $\text{co-RE}$ has complete problems, too. S. Schmitz [1] considers some classes ...
0
votes
2answers
90 views

Decidability of empty intersection of two languages accepted by Turing machines

I am really struggling with determining the decidability of languages and cant figure out whether this problem is decidable or not. I have a language $\qquad\displaystyle L = \{ (R(M_1), R(M_2)) ...
0
votes
2answers
106 views

Can languages with infinite strings be recursively enumerable?

I am not 100% sure about the definition of recursively enumarable languages. Yes I know how are they defined: There has to exist a Turing machine that accepts all wrods of the language and halts but ...
1
vote
5answers
115 views

Function whose implementation is difficult (computationaly) to work out

Let's say I've got a function $f$ that takes a single number and returns a number. And I have another function $\mathrm{verify}f$ which takes the input I gave to $f$ and the number returned by $f$ ...
-1
votes
2answers
85 views

Extension of Rice's theorem

How can one prove that every nontrivial property of pairs of semi-decidable sets is undecidable? (This is an extension of Rice's theorem that "Every nontrivial property of the r.e. sets is ...
2
votes
1answer
42 views

How is Rice's theorem applicable to this decision problem?

I recently had a test in introduction to computability and I got the following question wrong. The question Input: A classical Turing machine $M$ with a 2-dimensional tape. output: Does there ...
2
votes
1answer
55 views

Decide if a specific Turing machine halts on a specific string

Can you always decide if a specific Turing machine accepts a specific string? I started thinking about this after reading an answer to this question, Rice's theorem vs Turing completeness, which ...
1
vote
2answers
64 views

A recursively enumerable language and a recursively enumerable set

I am confused between these two terminologies: recursively enumerable language, recursively enumerable set. Do they have the exactly same meaning?
2
votes
1answer
41 views

Converting decision problems to grammars?

I'm struggling to understand some concepts related to the relationship between language and computability theories. Can we convert decision problems to the corresponding grammars describing the ...
0
votes
1answer
46 views

Clarification of Hopcroft's proof that “deciding whether a program halts on all inputs” is not R.E

$DoesNotHaltOn\_w=\{(M, w) : M$ does not halt on input w$\}$ $AlwaysHalt =\{ M : M$ halts on all inputs x $\}$ Hopcroft gives the following proof that $AlwaysHalt$ is not R.E. 1) Given an input ...
1
vote
5answers
270 views

How is Turing's Solution to the Halting Problem Not Simply “Failure By Design”?

I'm having a hard time viewing Turing's solution to the Halting Problem as a logician, rather than as an engineer. Here is my understanding of the Halting Problem: Let $M$ be the set of all ...
3
votes
2answers
70 views

Does “output” always imply halting in computability?

$L = \{P : P(n)$ outputs $n^2$ for all $n \in N \}$ In questions of this nature, are we supposed to assume that "outputs" means "halts and outputs"? In modern programming languages, I can ...
-2
votes
1answer
16 views

$L \in RE$ Question [closed]

I see a sentence in one final exam on automaton course. I have one problem: if we want to have a TM that halts for all word in L, it's enough to have L be R.E? or we should have R be R.E and ...
0
votes
2answers
106 views

The image of a recursive language under a computable function

Let $f:\Sigma^{*}\to\Sigma^{*}$ be a computable function and let $L$ be a recursive language. Is $f(L):=\left \{{f(w)|w\in L} \right\}$ recursive? Here, I see clearly, that $f^{-1}(L)$ is recursive ...
4
votes
2answers
211 views

Halting problem without self-reference

In the halting problem, we are interested if there is a Turing machine $T$ that can tell whether a given Turing machine $M$ halts or not on a given input $i$. Usually, the proof starts assuming such a ...
-1
votes
1answer
70 views

are there any decidable problems not verifiable in polynomial time?

As I understand it NP requires a solution to be verifiable in polynomial time. Can you provide examples of solvable problems not verifiable in polynomial time ?
6
votes
1answer
72 views

Is Post's Correspondence Problem decidable with fixed word size?

So, it's known that PCP is undecidable even when we fix the number of tiles to $n \geq 7$. I'm wondering, can anything similar be said for when there is a fixed word length? To be precise, here's ...
3
votes
1answer
45 views

Is there an undecidable decision problem that computable algorithm for it leads to an algorithm for halting problem?

Suppose, to the contrary, that there exists a computable algorithm for some undecidable decision problem. Would this mean that halting problem would be solved by a computable algorithm? I know that ...
-1
votes
3answers
117 views

If Halting problem is decidable, are all RE languages decidable? [closed]

Assume the halting problem was decidable. Is then every recursively enumerable languagerecursive?
1
vote
2answers
97 views

Can an intersection of two context-free languages be an undecidable language?

I'm trying to prove that $\exists L_1, L_2 : L_1$ and $L_2$ are context-free languages $\land\;L_1 \cap L_2 = L_3$ is an undecidable language. I know that context-free languages are not closed ...