660 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 ...
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 ...
• 149
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 ...
• 993
669 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 ...
• 281
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'...
• 191
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 ...
285 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). ...
• 2,019
1 vote
2k views

### Why every finite set is computable?

According to wikipedia, every finite set is computable. Definition: set $S \subset N$ is computable if there exists an algorithm which defines in finite time if a given number $n$ is in Set. ...
• 1,075
883 views

• 151
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 ...
• 255
218 views

### Semi-decidable problems with linear bound

Take a semi-decidable problem and an algorithm that finds the positive answer in finite time. The run-time of the algorithm, restricted to inputs with a positive answer, cannot be bounded by a ...