Linked Questions

0 votes
1 answer
96 views

Do proofs of $HALT$'s undecidability make it clear that it's practically relevant?

The proof of $HALT$'s undecidability usually goes like this: we assume the existence of a halting decider and incorporate it into a machine $D$ that takes a TM as input, runs it on its own encoding ...
CuriosityScream's user avatar
18 votes
5 answers
4k views

Regular languages that seem irregular

I'm trying to find examples of languages that don't seem regular, but are. A reference to where such examples may be found is also appreciated. So far I've found two. One is $L_1=\{a^ku\,\,|\,\,u\in \{...
user6767509's user avatar
0 votes
2 answers
76 views

Why is $\{ w \in \Sigma^* : M_w[\epsilon]\downarrow \land |w| \leq 7\}$ decidable?

I get that the argument for this set $\{ w \in \Sigma^* : M_w[\epsilon]\downarrow \land |w| \leq 7\}$ to be decidable is that $|w|\leq7$ meaning it is a finite set and therefore it can be decided. ...
linuxxx's user avatar
2 votes
2 answers
903 views

Are functions with a finite domain and codomain always computable?

I apologise if my following reasoning is flawed, but I cannot find the "bug" in it. Consider two finite subsets of $\mathbb{N}$, namely $A$ and $B$. The set of all functions $f:A\rightarrow ...
olinarr's user avatar
  • 394
1 vote
2 answers
541 views

(Un)computability of a restricted Halting Problem

Before I start with my question, I want to state some notation I am using. I fix some arbitrary but fixed enumeration of Turing Machines (TMs) and denote with $\Phi_i : \mathbb{N}\to\mathbb{N}$ the ...
hetzi's user avatar
  • 131
1 vote
1 answer
86 views

Does there exist a undecidable infinite language with only a finite undecidable subset?

I know that there's no such thing as a finitely sized undecidable language. However, does there exist an undecidable language where a finitely sized set of undecidable elements are 'hiding among' an ...
orlp's user avatar
  • 13.8k
1 vote
1 answer
129 views

Why Right-Division of regular language with RE\E language is regualr?

I think I can't understand the meaning of language being decidable. The next case makes no sense to me: Considering I have language L1 which is regular, and language L2 which is in RE\R (in ...
Ella 's user avatar
  • 109
2 votes
1 answer
120 views

Decide if a language has a word of a given size

Suppose that $L$ is some language over the alphabet $\Sigma$. I was asked to show that the following languages is decidable: $$L' = \{w \in \Sigma^* | \text{ there exists a word } w'\in L \text{ ...
Sam's user avatar
  • 23
2 votes
1 answer
621 views

Can we find a Turing machine such that there is no Turing machine to decide whether it halts on $\epsilon$?

The halting problem states that there is no Turing machine that can determine whether an arbitrary Turing machine halts on $\epsilon$. But I try to ask something different, can we find a specific ...
user183748292's user avatar
0 votes
1 answer
182 views

Some questions about the Computability of Turing Machines

I'm learning for a test and I have some important questions about Computability of deterministic and non deterministic Turing Machines. Consider we have the partial functions $f,g,h,t: \mathbb{N} \...
katarina's user avatar
1 vote
2 answers
190 views

Decidable problems for which no concrete decision procedure is known

I am looking for an example of decidable problems the decision procedures of which are unknown. I believe someone mentioned one to me once, and I also have read somewhere, but my memory is corrupted. ...
Jason Hu's user avatar
  • 632
9 votes
2 answers
674 views

Are there any problems in $P$ which we do not know any $P$ algorithms?

A problem in $P$ is one that can be solved in polynomial time (or faster) on a deterministic Turing machine. Now if I am correct, there is nothing here referring to the algorithms - which can ...
Quantum spaghettification's user avatar
6 votes
1 answer
14k views

Show that a language is decidable iff some enumerator enumerates the language in lexicographic order

The proof is given in the below: If $A$ is decidable, the enumerator operates by generating the strings in lexicographic order and testing each in turn for membership in $A$ using the decider. Those ...
Idonotknow's user avatar
0 votes
1 answer
80 views

Are there noncomputable functions with a finite search space? [closed]

The top rated answer to Why, really, is the Halting Problem so important? lists a few examples for a noncomputable problem. However, these mostly involve an infinite search space. Are there ...
Karsten's user avatar
  • 101
64 votes
6 answers
26k views

If everyone believes P ≠ NP, why is everyone sceptical of proof attempts for P ≠ NP?

Many seem to believe that $P\ne NP$, but many also believe it to be very unlikely that this will ever be proven. Is there not some inconsistency to this? If you hold that such a proof is unlikely, ...
pafnuti's user avatar
  • 739

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