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Results tagged with computational-complexity
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user 4600
This is a branch that includes: computational complexity theory; complexity classes, NP-completeness and other completeness concepts; oracle analogues of complexity classes; complexity-theoretic computational models; regular languages; context-free languages; Komolgorov Complexity and so on.
25
votes
Accepted
Languages beyond enumerable
Yes, for starters there is the arithmetical hierarchy, where enumerable = $\Sigma^0_1$ and it continues $\Pi^0_1$, $\Delta^0_2$, $\Sigma^0_2$ etc.
See also the Computability Menagerie.
- 24.8k
13
votes
How did the Baker-Gill-Solovay paper come to be?
Apparently, we would have gotten at least half of the BGS result without any of the three named authors and also without any of the 4 people they credit, all we needed was Dekhtiar. 😊
The Annals of t …
- 24.8k
11
votes
Can We Decide Whether Small Computer Programs Halt?
You're right that such a project is possible. Calude et al. (http://www.emis.de/journals/EM/expmath/volumes/11/11.3/Calude361_370.pdf) have some results in this direction.
- 24.8k
10
votes
How many Complexity Classes do you know?
The top part of the Computability Zoo (r.e., recursive, and beyond) is covered in more detail in the Computability Menagerie.
- 24.8k
9
votes
0
answers
2k
views
Weighted Hamming distance
Basically my question is, what kind of geometry do we get if we use a "weighted" Hamming distance. This is combinatorics but similar things come up sometimes in theoretical computer science, for examp …
- 24.8k
7
votes
Complexity of Turing Machine behavior
If you restrict attention to TMs that always halt, then:
One measure of complexity of a Turing machine is its running time, the maximum number of steps taken before it halts on inputs of length $n$, …
- 24.8k
6
votes
powers in strings
Regarding the 3rd question, I will show this:
Theorem. For a random binary word of length $n$, the expected number of $h$th powers is
$$
\sim \frac{n}{2^{h-1}-1}.
$$
Proof. A basic event about occurr …
- 24.8k
6
votes
Accepted
What is the probability a random Turing machine is isomorphic to a DFA?
The set of possible answers to this question is a countable dense subset of (0,1), because it depends on your choice of universal Turing machine.
- 24.8k
4
votes
Accepted
How to define the input of computable function or Turing machine over real numbers
A good place to start learning about different representations of reals and their computability- and complexity-theoretic consequences is Weihrauch's book Computable Analysis.
- 24.8k
4
votes
Connections between algebraic semantics and computational complexity of a logic?
The example you gave extends as follows:
SAT for arbitrary lattices (meaning, is a given formula satisfiable in some lattice) is polynomial-time decidable
SAT for modular lattices is Turing undecida …
- 24.8k
3
votes
Computational complexity of solution of Pell equation and more
The problem of finding $x$ and $y$ in a given Pell equation $x^2-ny^2=1$ is not known to be solvable in polynomial time, see Wikipedia.
- 24.8k
3
votes
Accepted
The definition of computational complexity or complexity measure of computing reals
This is interesting. I think a number could be of low complexity in terms of approximating it to within smaller and smaller $\epsilon $, while of high complexity in terms of finding its binary represe …
- 24.8k
3
votes
Accepted
The link and equivalence between variant definition of computation model and computational c...
The following models are probably the two most well known, and they are not equivalent at the level of computability.
BCSS
standard/Grzegorczyk (same as in Weihrauch's book)
In fact, the function
…
- 24.8k
3
votes
Accepted
If the set of the output of a computable function is finite, is the sequence periodic eventu...
Regarding the 2nd question, the set of output sequences of an autonomous finite automaton consists of ultimately periodic sequences.
- 24.8k
3
votes
Is the Kolmogorov complexity of at least one string of a given length equal to its length?
It depends on the universal machine. Consider length 0, the empty string could have complexity 455, say.
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