22
votes
Accepted
Did Alan Turing's student Robin Gandy assert that Charles Babbage had no notion of a universal computing machine?
No, the opposite. This quote of Gandy's is not referring to Babbage, but to some intervening proposals for universal-style computing between Babbage and Turing. Gandy says those proposals did not have ...
- 7,555
12
votes
Accepted
Function that is guaranteed to be one-way if one-way functions exist?
Yes, such a function was found by Levin himself, published somewhat recently:
The tale of one-way functions. Problems of Information Transmission (= Problemy Peredachi Informatsii), 39(1):92-103, ...
- 4,465
9
votes
Accepted
Construction of arbitrary functions with exponential-size $MODp \circ MODq$ circuits
$\def\M#1{\mathrm{MOD}_{#1}}\def\F#1{\mathbb F_{#1}}$I don’t know of a reference, but here is one way how to prove the result. I’ll do it in three stages, each using one new idea: (1) multilinear ...
- 16.5k
8
votes
Accepted
theorems for universal set of quantum gates for SU(d)
I'm not aware of any proof that the Clifford group + any non-Clifford element gives a universal set of quantum gates. The closest related result that I know is that the Clifford group + any non-...
- 743
7
votes
Accepted
Terminology about computation and Finite algebra
Such algebras are called functionally complete. Also, what you call terms are actually called polynomials. In standard terminology, term operations have a more restricted definition that allows ...
- 16.5k
7
votes
Can we not output the Kolmogorov complexity?
The question can be rephrased as whether or not $\lim \inf_{\vert x \vert \rightarrow \infty}{\vert T(x) - K(x) \vert} = 0$, and as Denis points out in the comments this is false for some encodings. ...
- 548
6
votes
Justification of log f in DTIME hierarchy theorem
For a fixed number of tapes greater than one, $\mathrm{Time}(o(f)) ⊊ \mathrm{Time}(O(f)$) for time-constructible $f$. The logarithmic overhead comes from the tape reduction theorem, where any number ...
- 2,537
3
votes
Computation with cellular automata in practice
To be more precise, I would want that the runtime scaling for the universal CA of each task is the same as for the best CA specifically designed for that task.
Game of life is intrinsically universal ...
- 181
3
votes
Can we not output the Kolmogorov complexity?
I think the following works. I'll use $C(x)$ for the Kolmogorov complexity
Give $U$ a time bound $t$ (say, some exponential function of the length of the input program), and call the result $U^t$. ...
- 459
2
votes
Is this variant of bitwise cyclic tag Turing-complete?
It is. There's a fairly simple construction compiling from CT to CT2.
First, consider that it's possible to double every command in a CT program without producing any behaviour (that is, ...
- 121
2
votes
Accepted
Universal Boolean Formulas
A $O(2^n/\log n)$-size depth-3 universal Boolean formula was constructed in
O.B. Lupanov. Complexity of the universal parallel-series network of
depth 3. Trudy Matem. Inst. Steklov, 133:127-131, ...
2
votes
Accepted
Are single hidden-layered neural networks at least as good as multi hidden-layered neural networks?
Short answer: Not necessarily. Likely nothing fishy is going on.
Longer answer:
The Universal Approximation Theorem (UAT) says nothing about an individual network's capacity to approximate a function....
- 203
1
vote
Can exponential-size depth-2 $CC^0[m]$ circuits with generalized $MOD_m$ gates compute arbitrary functions from $Z/mZ$ to $Z/2Z$?
This is not a complete answer; just partial progress.
For certain output sets of the output gate (the only gate on the second layer), we can prove that there are functions uncomputable by this type of ...
- 1,124
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