Questions tagged [halting-problem]

Given a program and the input for it, does it halt or run forever?

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14
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2answers
4k views

What is the “nearest” problem to the Collatz conjecture that has been successfully resolved?

I am interested in the "nearest" (and "most complex") problem to the Collatz conjecture that has been successfully solved (which Erdos famously said "mathematics is not yet ripe for such problems"). ...
48
votes
3answers
2k views

Is there a sensible notion of an approximation algorithm for an undecidable problem?

Certain problems are known to be undecidable, but it is nevertheless possible to make some progress on solving them. For example, the halting problem is undecidable, but practical progress can be ...
29
votes
1answer
2k views

Halting problem, uncomputable sets: common mathematical proof?

It is known that with a countable set of algorithms (characterised by a Gödel number), we cannot compute (build a binary algorithm which checks belonging) all subsets of N. A proof could be ...
31
votes
4answers
5k views

What is the smallest Turing machine where it is unknown if it halts or not?

I know that the halting problem is undecidable in general but there are some Turing machines that obviously halt and some that obviously don't. Out of all possible turing machines what is the smallest ...
16
votes
2answers
855 views

Collatz Conjecture & Grammars / Automata

I was wondering if there is a good bibliography of attempts to investigate the Collatz conjecture as a formal grammar? (or any other attempts in the CS community to deal with this class of generative ...
-2
votes
1answer
735 views

Is there any proof that a network made of Turing machines can't solve the halting problem? [closed]

My question points to the fact that Turing machines are isolated by definition. But what if they can send and receive information from/to other Turing machines? What if they can be "interrupted" at ...