Linked Questions

5 votes
2 answers
3k views

Union of R.E. and Non R.E. language

Let \begin{align*} L_1 &=\{\langle M,w\rangle \mid M\text{ halts on }w\}\\ L_2 &=\{\langle M,w\rangle \mid M\text{ does not halt on }w\}\,. \end{align*} Here $M$ represents encoding ...
Zephyr's user avatar
  • 993
0 votes
2 answers
3k views

Is P decidable?

It seems correct that any single given algorithm must either have polynomial runtime or not. Is there a specific algorithm that (does or does not actually lie in $P$, but) can neither be proven nor ...
heinzelotto's user avatar
2 votes
1 answer
87 views

Is a finite Solomonoff learner worse than human learning?

An elegant program for a bitstring is the shortest program on a universal Turing machine that outputs this bitstring. According to Kolmogorov complexity, the length of the elegant program is ...
yters's user avatar
  • 1,427
4 votes
3 answers
670 views

To what extent is my interpretation of computable numbers correct?

Interpretation: Consider the comic strip below, where a person tries to prevent a robot from dismembering them by asking the robot to compute $\pi$ - the robot quickly produces an algorithm to ...
Chill2Macht's user avatar
4 votes
1 answer
795 views

Finite languages are Turing decidable - contradiction [duplicate]

Let's say that I define the language $L$ over the alphabet $\{0, 1\}$ to be a language containing only one word, $w$, where: $$ w = \begin{cases} 1 & \text{if the continuum hypothesis is ...
SebiSebi's user avatar
  • 143
26 votes
4 answers
5k views

Is the halting problem decidable for pure programs on an ideal computer?

It's fairly simple to understand why the halting problem is undecidable for impure programs (i.e., ones that have I/O and/or states dependent on the machine-global state); but intuitively, it seems ...
Jules's user avatar
  • 632
7 votes
1 answer
3k views

Undecidable vs Unsolvable?

In decidability theory, I understand that if a problem is labeled "decidable", then we can construct a Turing Machine that definitively tells us whether an input is valid or invalid. My question is ...
Jenna Maiz's user avatar
2 votes
2 answers
242 views

Is it possible to design a programming task that is unsolvable?

Can a problem (described by a set of inputs and accepted answers) be designed such that for all programs which produce an answer in finite time for a (countably) infinite number of inputs, at least ...
Daniel McIntosh's user avatar
10 votes
3 answers
286 views

Constructive version of decidability?

Today at lunch, I brought up this issue with my colleagues, and to my surprise, Jeff E.'s argument that the problem is decidable did not convince them (here's a closely related post on mathoverflow). ...
G. Bach's user avatar
  • 2,019
18 votes
5 answers
6k views

How long does the Collatz recursion run?

I have the following Python code. ...
9bi7's user avatar
  • 305
13 votes
2 answers
15k views

How to prove P$\neq$NP?

I am aware that this seems a very stupid (or too obvious to state) question. However, I am confused at some point. We can show that P $=$ NP if and only if we can design an algorithm that solves any ...
padawan's user avatar
  • 1,445
32 votes
5 answers
9k views

Proof that dead code cannot be detected by compilers

I'm planning to teach a winter course on a varying number of topics, one of which is going to be compilers. Now, I came across this problem while thinking of assignments to give throughout the quarter,...
thomas's user avatar
  • 421
36 votes
8 answers
11k views

What are the simplest examples of programs that we do not know whether they terminate?

The halting problem states there is no algorithm that will determine if a given program halts. As a consequence, there should be programs about which we can not tell whether they terminate or not. ...
MaiaVictor's user avatar
  • 4,137
17 votes
5 answers
7k views

Are there any compression algorithms based on PI?

What we know is that π is infinite and quite likely it contains every possible finite string of digits (disjunctive sequence). I've seen recently some prototype of πfs which assume that every file ...
kenorb's user avatar
  • 275
8 votes
1 answer
3k views

If the Halting Problem was solvable, and we solved it, what would be its implications?

Perhaps a way to better understand the Halting Problem's importance is to know what would happen or what could be possible if this was solved. What would be the Halting Problem's implications in today'...
Zaenille's user avatar
  • 191

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