# Questions tagged [computational-complexity]

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.

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### Computational complexity of topological K-theory

I am a novice with K-theory trying to understand what is and what is not possible. Given a finite simplicial complex $X$, there of course elementary ways to quickly compute the cohomology of $X$ with ...
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### Do we know how to determine the $2^{2020}$ decimal of $\sqrt{2}$?

In the case of $\dfrac{1}{7^{800}}$ it's easy, to find the $2^{2020}$ decimal, but what about the simplest of the irrational numbers. Question: Do we know how to determine the $2^{2020}$ decimal of ...
368 views

### Straight-line drawing of regular polyhedra

Find the minimum number of straight lines needed to cover a crossing-free straight-line drawing of the icosahedron $(13\dots 15)$ and of the dodecahedron $(9\dots 10)$ (in the plane). For example, ...
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### Checking Mertens and the like in less than linear time or less than $\sqrt{x}$ space

Say you want to check that $|\sum_{n\leq x} \mu(n)|\leq \sqrt{x}$ for all $x\leq X$. (I am actually interested in checking that $\sum_{n\leq x} \mu(n)/n|\leq c/\sqrt{x}$, where $c$ is a constant, and ...
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### Reference request: Parallel processor theorem of William Thurston

Sometime in the 1980's or 1990's, Bill Thurston proved a theorem regarding the existence of a universal parallel processing machine, using a certain class for such machines having finite deterministic ...
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### Complexity of a Fibonacci numbers discrete log variation

In my work I encountered the following FIBMOD PROBLEM: Given $k,m$ in binary, decide if there exists $n$ such that $\, F_n = k \,$ (mod $m$). Here $F_n$ is a Fibonacci number. This is a variation ...
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### 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 ...
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### Are there any natural theories T for which P=NP implies T proves P=NP?

The qualifier "natural" is meant to exclude examples like "PA + P=NP" or "PA + True $\Pi_1$". For concreteness, let's say that "natural" = sound, computably enumerable, with a feasible proof-checker. ...
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### Splay trees and Thompson's group $F$

( I apologize for only indicating some easy to find references, but new users are not allowed to link more than five). This is very speculative, but: Question: Is there a reformulation of the Dynamic ...
669 views

### Why should Algebraic Geometers and Representation Theorists care about Geometric Complexity Theory?

Geometric Complexity Theory has demonstrated that Complexity Theorists should care about Algebraic Geometry and Representation Theory, but, why should Algebraic Geometers and Representation Theorists ...
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### Razborov's response to Almost Natural Proofs

This post is about Natural Proofs barrier in computational complexity. There are two recent papers related to this. They are: Amplifying lower bounds by means of self-reducibility by Eric Allender ...
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### Does the Angel have to be really smart?

My question is about the computational complexity of the Angel's strategy in the Angels and Devils game, tl;dr does the Angel have a polynomial time strategy. I'm a big Conway fan, so as you can ...
315 views

### Geometric complexity theory for finite fields

Geometric complexity theory (GCT) is an approach via algebraic geometry and representation theory towards the P vs. NP problems and related problems Ketan D. Mulmuley. More precisely, the idea is to ...
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### Collapsing the Linear Time Hierarchy and finite axiomatizability of bounded arithmetic

It is well known that if ${\bf T_2}$ (or $I\Delta_0+\Omega_1$) is finitely axiomatizable, then the Polynomial Hierarchy collapses. Q. Is there any similar relation between $I\Delta_0$ and Linear ...
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### When is a reduction not a reduction?

Every mathematician understands the concept of reducing a complicated problem to a simpler problem. "Without loss of generality, we may assume…" However, I've noticed that some kinds of "...
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### Feasible Type Theories

I am looking for references about efficient type theories, efficiency in the sense of computational complexity, and type theory in the sense of Martin-Lof's type theories. Has there been any studies ...
925 views

### Computational complexity of multiplication in a nilpotent group?

What is the computational complexity of multiplication in a Carnot group ? Background: A Carnot group is a real nilpotent Lie group $N$ whose Lie algebra $Lie(N)$ admits a direct sum decomposition ...
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### On earlier references for $P=BPP$ and Kolmogorov's possible view on modern breakthroughs involving randomness?

Kolmogorov and Uspenskii in this paper 'http://epubs.siam.org/doi/pdf/10.1137/1132060' speculate $P=BPP$ in $1986$. They do this without getting into circuit lower bounds and from a different view ...
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### Complexity of scissors congruence?

Where is the complexity of the problem 'Given two bounded compact convex integral polyhedra in $\mathbb R^n$ presented by $O(poly(n))$ integer valued halfspaces given by linear inequalities with ...
I've seen a bunch of similar MO questions, yet hopefully this is not a complete duplicate. Consider $n$ halfplanes in $\mathbb{R}^2$ with their borders in general position, that is, no point of $\... 0answers 214 views ### Multidimensional hook length formula A well-known hook length formula states that the number of ways to arrange the elements of$[n]$in a Young tableau with$n$cells so that all columns and rows are increasing is$\frac{n!}{\prod_c h(c)...
Given an $m$ by $n$ matrix $M$ with $m \leq n$ and elements from $\{-1,1\}$, let us define: $$S_M = |\{Mx : x \in \{-1,1\}^n\}.$$ It is NP-hard to compute $S_M$ exactly I believe by applying the ...