This tag is suited for questions involving Turing machines. Not to be confused with finite state machines and finite automata.
Questions tagged [turing-machines]
1001 questions
61
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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
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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
- 4,266
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
- 104,111
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
22
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
- 13,054
21
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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 Complete?…
Erwan Aaron
- 937
20
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1 answer
Is the Collatz map algorithmically decidable?
Question: Has it been proven that the following decision problem is algorithmically decidable? $$ P_\text{Collatz} := \text{Given $n \in \mathbb{N}_+$, does the Collatz-Iteration of $n$ eventually reach $1$?} $$ Obviously, we strongly suspect that…
Vincent
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17
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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.
- 967
16
votes
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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
- 22,188
16
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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
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15
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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…
volcanrb
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14
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
- 1,620
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 come…
Bobjoesmith
- 165
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Are Turing machines still Turing complete if restricted to only moving to adjacent states?
The Turing machine is used as a model for computation because of its simplicity. I'm curious whether it's a minimal model. Starting from the TM over the alphabet ${0, 1}$ (with a potentially unlimited amount of states), the one thing that strikes me…
Kotlopou
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