All Questions
48 questions
3
votes
0
answers
146
views
Lower Bound of Solutions to P=NP?
Do we at least know that simulating polynomial time non-deterministic Turing machines requires more than a linear slowdown? That is, do we know there is some non-deterministic Turing machine with ...
- 3,029
-2
votes
1
answer
181
views
What is the computational complexity to verify a P solution with a deterministic Turing machine? [closed]
As we know, NP (nondeterministic polynomial time) is a complexity class used to classify decision problems. NP is the set of decision problems for which the problem instances, where the answer is &...
- 3,094
4
votes
0
answers
214
views
Computational complexity of zeros of an analytic function
The work of Friedman and Ko, page 342, Corollary 4.3.1
states that all zeros of analytic polynomial time computable function are polynomial time computable, but for me that is not clear how it could ...
- 81
5
votes
0
answers
192
views
Complexity implications on computability
Are there any known links between complexity theory and computability theory by which I mean non-trivial theorems of the form: If NP $\neq$ co-NP then there is no strong minimal pair of r.e. sets or ...
- 3,029
9
votes
2
answers
954
views
What theories are larger than the real closed field but still decidable?
It's well known that sentences about the real closed field can be decided by algorithm and the complexity of this is about $d^{2^{O(n)}}$ where $d$ is the product of the degrees of polynomials in the ...
- 2,689
9
votes
1
answer
372
views
Decidable theories with arbitrary complexity
Are there complete finitely axiomatizable first order theories (with equality) with arbitrarily high computational complexity?
Here, arbitrarily high (computational) complexity means that for every ...
- 7,545
6
votes
1
answer
1k
views
MIP^*=RE and quantum computation
I recently learned about the MIP^*=RE result. I have to admit that I don't understand big parts of this paper and I am barely familiar with quantum physics. I hope my questions below make sense.
I ...
- 2,882
3
votes
0
answers
186
views
Decidable equality for computable functions $\mathbb{N}\to\mathbb{N}$
Suppose we have two computable functions $f, g:\mathbb{N}\to\mathbb{N}$. When is $f=g$ algorithmically decidable?
For example it is decidable if $f$ and $g$ are polynomials of a priori known degree.
- 31
0
votes
1
answer
267
views
Algorithmically decide if an algorithm has optimal time complexity [closed]
Is there an algorithm with the following input and output?
INPUT: an algorithm computing a function $\mathbb{N}\to\mathbb{N}$. The algorithm is guaranteed to halt on all inputs.
OUTPUT: "YES"...
- 1
5
votes
0
answers
246
views
Does $\mathsf{Q}$ have any interesting provably recursive functions?
This question was asked and bountied at MSE without success.
For an appropriate theory $T$, say that an $n$-ary $T$-provably recursive function is a $\Sigma_1$ formula $\varphi$ with $n+1$ free ...
- 21.2k
15
votes
2
answers
2k
views
How did the Baker-Gill-Solovay paper come to be?
How did the Baker-Gill-Solovay paper come to be? Why were those three people talking together about "Relativizations of the $P=?NP$" question, and what was their collaboration like for the ...
user44143
3
votes
2
answers
331
views
Time functions of non-deterministic Turing machines (a better question)
This is a more precise version of that question.
Let $M$ be a non-deterministic Turing machine which recognizes a language $L$, that is, for every input word $u$ there is an accepting computation ...
user6976
113
votes
11
answers
18k
views
On mathematical arguments against Quantum computing
Quantum computing is a very active and rapidly expanding field of research. Many companies and research institutes are spending a lot on this futuristic and potentially game-changing technology. Some ...
6
votes
1
answer
216
views
A "dense" extension of the set of primitive recursive functions
Let $\mathcal{PR}$ be the set of primitive recursive functions. Let $\mathcal{PR}(f)$ be $\mathcal{PR}$ which we have amplified by adding (a recursive) $f$ the in the set of initial functions. To make ...
user39297
5
votes
0
answers
106
views
Collapsing the Exponential time Hierarchy with a complete language as oracle
It is known that $\mathsf{P^A=NP^A}$ is true for every $\mathsf{EXP}$ complete language $\mathsf{A}$. The question is the whether the similar things hold for Exponential time Hierarchy.
Is there ...
- 1,689
2
votes
0
answers
90
views
Recursion theoretical Characterization of time complexity classes
Is there any known Recursion theoretical Characterization of time complexity classes like $\mathsf{DTIME(n^k)}$ or $\mathsf{NTIME(n^k)}$ for some fixed $k$?
Thanks.
- 1,689
6
votes
1
answer
214
views
Finite-variable fragments of $\Delta_0$-formulas
Consider sets definable in the usual structure of arithmetic $(\mathbb{N},0,1,+,\times)$ by $\Delta_0$-formulas, i.e., formulas with bounded quantifiers. The quantifier alternation hierarchy has been ...
- 211
5
votes
1
answer
345
views
Decoding a Remark of Gödel on Complexity Theory
In Gödel's Collected Works (Vol 2), there is a discussion of von Neumann which was brought about by a query, made to Gödel, concerning the existence of a Turing machine which is so complex that its ...
- 314
1
vote
1
answer
183
views
Understanding the paper: "Guarded Fixed Point Logic"
This question is specifically about the paper "Guarded Fixed Point Logic" by Gradel and Walukiewicz. Among other things they prove the decidability of the satisfiability problem for Fixpoint Loosely ...
- 111
17
votes
4
answers
3k
views
Languages beyond enumerable
A language is a set of finite-length strings from some finite alphabet $\Sigma$.
It is no loss of generality (for my purposes) to take $\Sigma=\{0,1\}$; so a language is a set of bit-strings.
...
- 151k
2
votes
2
answers
624
views
Time Hierarchy Theorem and P vs NP
One obvious strategy for proving P not equal to NP would be to show that there is some problem in NP which is hard for a time class strictly containing P (the origin of this question is the recent ...
- 99
2
votes
1
answer
287
views
Effectively non-recursiveness of some sets
A set $A$ is completely productive if there exists a computable function $f$ such that for every $e$, $f(e)\in (A-W_e)\cup (W_e-A)$. A set is effectively non-recursive if it is r.e. and its ...
- 870
15
votes
0
answers
425
views
Complexity classes for BSS machines
Given a first-order structure $\mathcal{S}$, a Blum-Shub-Smale machine on $\mathcal{S}$ is essentially a Turing machine where
Cells on the tape can hold arbitrary elements of $\mathcal{S}$.
The ...
- 21.2k
7
votes
1
answer
258
views
Oracle queries asked in parallel
Definition: Assume that $\phi(q)$ is of the form $\exists y \leq 2^{p(n)} \varphi(q,y)$, where $p$ is a polynomial and $n = |q|$ (i.e. $n$ is the length of the binary representation of $q$). Then a ...
2
votes
3
answers
987
views
An established proof in Wang Tile which I doubt
When I was reading the paper:
Wang, Hao. "Notes on a class of tiling problems." Fundamenta Mathematicae 82.4 (1975): 295-305.
from http://matwbn.icm.edu.pl/ksiazki/fm/fm82/fm82119.pdf
I could not ...
- 867
1
vote
1
answer
631
views
relationship between corner tile and edge tile of wang tile
It is clear that any corner type of Wang Tile could be converted to edge type of Wang Tile by defining the edge color according to the corner color.
However, could we convert edge type of Wang Tile ...
- 867
2
votes
0
answers
163
views
Graph theoretical representation of Wang Tile
We note that for one dimensional tiling problem of Wang Tile could be represented by a graph. Each cycle on the graph represents a periodic solution.
However, is there a well established counter-part ...
- 867
3
votes
1
answer
509
views
Application of Combinatorics, Logic and computability theory in physical science: Tiling of Wang Tile with proportionality
The original problem of Domino Tiling and Wang Tile has great theoretical interest on computability theory... However, the great emerging problem on application of Wang Tile in material science and ...
- 867
3
votes
2
answers
297
views
Conjecture of a subset of Wang tile which might be decidable
From the two papers proving the undecidability of Wang tile in 1966 by Berger and in 1971 by RM Robinson, the tiles used in proving undecidability has a general common feature:
The left color and ...
- 867
5
votes
6
answers
2k
views
practical algorithms for np complete problems
Inspired by:
Conjecture on NP-completeness of tesselation of Wang Tile up to finite size
And the practicality of this topic (solving tessellation on a lattice):
coloring in lattice
Computational ...
- 867
2
votes
0
answers
123
views
What are natural examples of non-relativizable proofs? [duplicate]
As I understand it, a proof that P=NP or P≠NP would need to be non-relativizable (as in recursion theory oracles).
Virtually all proofs seem to be relativizable, though.
What are good examples of ...
- 179
27
votes
10
answers
4k
views
Can We Decide Whether Small Computer Programs Halt?
The undecidability of the halting problem states that there is no general procedure for deciding whether an arbitrary sufficiently complex computer program will halt or not.
Are there some large $n$ ...
- 711
4
votes
0
answers
568
views
About "natural proof" of Razborov and Rudich
The famous "Natural Proof" paper ,http://www.cs.umd.edu/~gasarch/BLOGPAPERS/natural.pdf , of Razborov and Rudich gives a barrier for any proof that try to separate P and NP. It mainly shows that if ...
- 781
5
votes
3
answers
1k
views
Is There An Algorithmic Complexity Of A Random Distribution
Has anyone studied an equivalent to algorithmic complexity for probability distributions?
This would be a measure which was similar to Kolmogorov complexity but look at the complexity of a (discreet ...
24
votes
1
answer
1k
views
Are sums of sequences decidable?
Suppose that $f,g$ are rational functions with integer coefficients such that $\sum_{n=0}^{\infty}f(n)$ and $\sum_{n=0}^{\infty}g(n)$ both converge. Is it decidable whether
$\sum_{n=0}^{\infty}f(n)=\...
1
vote
1
answer
292
views
Can all programs reducible to ones with only arithmetic operations on inputs be simulated with polynomial overhead by arithmetic machine?
I failed to get an answer at https://math.stackexchange.com/questions/364061/can-all-programs-reducible-to-ones-with-only-arithmetic-operations-on-inputs-be, so I am asking here.
In https://math....
- 11
17
votes
1
answer
960
views
Polynomial-time algorithm to compare numbers in Conway chained arrow notation
I am looking for a polynomial-time algorithm which, given a character string containing two numbers in Conway's chained arrow notation for large numbers, indicates whether the first number is less ...
- 171
6
votes
3
answers
1k
views
computational complexity of primitive recursive functions
If we have a rewrite system for primitive recursive functions, which simplifies each term according to how the function was defined, then what is the computational complexity of this calculation? That ...
- 63
3
votes
1
answer
445
views
Diagonalization and classes of computable functions
Fix a standard effective listing $(\phi_e)_{e\in\omega}$ of the partial computable functions from $\omega$ to $\omega$. Let $\mathcal{C}$ be a class of computable total functions $\omega\rightarrow \...
- 21.2k
4
votes
1
answer
830
views
Infinite monkeys computing ... triangle area?
I wonder if it is possible to specialize the question:
(a) What is the probability that a random Turing Machine program
will halt?, to: (b) What is the probability that a random Turing Machine
...
- 151k
11
votes
2
answers
668
views
What is the computational-complexity-theoretic analogue of computable inseparability? For example, if P is not NP, are there disjoint NP sets with no separation in P?
Disjoint sets $A$ and $B$ are computably inseparable, if there
is no computable separating set, a computable set $C$ containing $A$ and disjoint from $B$. The
existence of c.e. computably inseparable ...
- 236k
13
votes
1
answer
973
views
Does every feasible partial order relation on the natural numbers extend to a feasible linear order relation?
It is well known that every partial order on a set can be extended
to a linear order on that set. That is, for every partial order
$\lhd$ on a set $X$, there is a linear order $\prec$ on $X$ such
that ...
- 236k
3
votes
1
answer
266
views
A question about the "information-content" of a very simple type of Turing machine.
All the Turing machines we consider have (1) a two-way infinite tape (2) one and only one halting
state (3) an alphabet of exactly two symbols-"1" and " "(or "blank"). Let n be any positive integer.
...
- 6,215
18
votes
2
answers
1k
views
Is there a name for sets for which it is easier to test membership than to find members---and vice versa?
This is a question my son Bob asked me. For some sets it is relatively easy
to test for membership but a lot more difficult to find members, and for others
the reverse is true. Here is an elementary ...
- 15.3k
7
votes
3
answers
2k
views
Decidable but nonrecursive sets
Until recently, I believed that recursive=decidable,
subscribing to this Wikipedia quote:
"In computability theory, a set is decidable, computable, or recursive if there
is an algorithm that ...
- 151k
16
votes
2
answers
2k
views
Structure theorems for Turing-decidable languages?
Languages decidable by weak models of computation often have certain necessary characteristics, e.g. the pumping lemma for regular languages or the pumping lemma for context-free languages. Such ...
- 23k
31
votes
3
answers
3k
views
Given a polynomial-time algorithm, can we compute an explicit polynomial time bound just from the program?
Question. Given a Turing-machine program $e$, which
is guaranteed to run in polynomial time, can we computably
find such a polynomial?
In other words, is there a
computable function $e\mapsto p_e$, ...
- 236k
29
votes
3
answers
3k
views
Is the theory of categories decidable?
There are a lot of theorems in basic homological algebra, such as the five lemma or the snake lemma, that seem like they'd be more easily proven by computer than by hand. This led me to consider the ...