This tag is suited for questions involving Turing machines. Not to be confused with finite state machines and finite automata.
Questions tagged [turing-machines]
1004 questions
61
votes
2 answers
Are there any Turing-undecidable problems whose undecidability is independent of the Halting problem?
To be more specific, does there exist a decision problem $P$ such that
given an oracle machine solving $P$, the Halting problem remains undecidable, and
given an oracle machine solving the Halting problem, $P$ remains undecidable?
David Zhang
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43
votes
5 answers
Why do we believe the Church-Turing Thesis?
The Church-Turing Thesis, which says that the Turing Machine model is at least as powerful as any computer that can be built in practice, seems to be pretty unquestioningly accepted in my exposure to computer science. Why? Do we have any more…
GMB
- 4,256
31
votes
5 answers
Exactly when and why can a Turing-machine not solve the halting problem?
I perfectly understand and accept the proof that a Turing-machine cannot solve the halting problem.
Indeed, this is not one of those questions that challenges the proof or result.
However, I feel that there is still something left to be explained…
Bram28
- 103,721
22
votes
2 answers
How do uncomputable numbers relate to uncomputable functions?
All the online resources that I've seen on uncomputable numbers assume that they're all irrational. But this doesn't seem to be required by the definition. Wikipedia, for example, says that "[un]computable numbers are the real numbers that can [not]…
Isaac King
- 427
21
votes
4 answers
How can we know we're not accidentally talking about non-standard integers?
This question is mostly from pure curiosity.
We know that any formal system cannot completely pin down the natural numbers. So regardless of whether we're reasoning in PA or ZFC or something else, there will be nonstandard models of the natural…
N. Virgo
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21
votes
5 answers
How can Busy beaver($10 \uparrow \uparrow 10$) have no provable upper bound?
This wikipedia article claims that the number of steps for a $10 \uparrow \uparrow 10$ state (halting) Turing Machine to halt has no provable upper bound:
"... in the context of ordinary mathematics, neither the value nor any upper-bound of…
ronno
- 12,914
21
votes
1 answer
What breaks the Turing Completeness of simply typed lambda calculus?
On the Wikipedia page about Turing Completeness, we can read that:
Although (untyped) lambda calculus is Turing-complete, simply typed lambda calculus is not.
I am curious as to what exactly makes a simply typed lambda calculus not Turing…
Erwan Aaron
- 927
17
votes
3 answers
Why did nobody prove undecidability by the "too many problems" argument?
To the best of my knowledge, the two major breakthroughs in regarding negative
results in computation/recursion theory came in 1936 by Church and Turing.
They both prove that some variant of the Halting problem was undecidable.
However, a very…
Ainsley H.
- 965
16
votes
0 answers
What turmite runs the longest before becoming predictable?
When looking at 2D Turing machines, many of them eventually become predictable. For example, Langton's Ant, the champion 2-color 1-state turmite, develops a highway after 10,000 steps. Predictable behavior includes Traps (as in Worm Trails),…
Ed Pegg
- 21,868
16
votes
4 answers
How large is the set of all Turing machines?
How large is the set of all Turing machines? I am confident it is infinitely large, but what kind of infinitely large is its size?
Kevin
- 483
15
votes
3 answers
A seemingly contradictory function - where's the issue?
I have constructed a function of seemingly contradictory nature.
Let $f$ be a function which, given an input $n\in \mathbb{N}$, lexicographically searches through all strings and finds the $n$th pair $\langle T, P\rangle$ where $T$ is a valid Turing…
volcanrb
- 3,114
13
votes
2 answers
The mother of all undecidable problems
It is usual to show that a problem P is undecidable by showing that the halting problem reduces to P.
Is it the case that the halting problem is the mother of all undecidable problems in the sense that it reduces to any undecidable problem? If the…
Bob
- 1,598
13
votes
3 answers
How do we know that the P versus NP problem is an NP problem itself?
I have been doing some research on the P versus NP problem. On multiple occasions, I have seen people say that the problem itself is an NP problem. I have been curious about how we know this. If we know that the problem is NP, then has anyone…
Bobjoesmith
- 165
11
votes
4 answers
What is the difference between a shuffle and a permutation?
A shuffle is defined (at least in my class) as:
Let $x = x_1x_2 \cdots x_k$ and $y = y_1 y_2 \cdots y_k$ be words over the same alphabet $\Sigma$. We say that $x$ is a shuffle of $y$ (denoted $x \sim y$) if there exists a permutation $\pi$ of…
Kitalda
- 213
11
votes
1 answer
How are weakly universal Turing machines actually defined?
For what I know, the definition of a universal Turing machine is something along the lines of the following (of course, details might vary from source to source):
A Turing machine $M$ is called universal if there are computable functions $f:\Bbb…
Wojowu
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