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If we knew a specific computable language $L$ such that we could prove $L\in\mathrm P\iff\mathrm P\ne\mathrm{NP}$, this would make $\mathrm P\ne\mathrm{NP}$ equivalent to a $\Sigma^0_2$ sentence. Whil …

This example is a bit lower in the hierarchy than what Kaveh asks for, but it is an open problem whether the soundness of the uniform $\mathrm{TC}^0$ algorithms for integer division and iterated multi …

For a fixed alphabet $\Sigma$, the blow-up is at most polynomial. First, given a regular expression $r$, it is straightforward to construct an expression $\tilde r$ using the operators $a\in\Sigma$, $ …

$\def\M#1{\mathrm{MOD}_{#1}}\def\F#1{\mathbb F_{#1}}$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 expa …

When you want to define (fully) uniform versions of circuit classes, log-space or poly-time uniformity is only sensible for classes of circuits whose power exceeds log-space or poly-time, respectively …

I’ll comment on why a relation as in the question $$ (2^n)! = \sum_{k=0}^{m-1} a_k b_k^{c_k} $$ (for every $n$) helps factoring. I can’t quite finish the argument, but maybe someone can. The first ob …

The revised conjecture is true, even under relaxed constraints on $S$ and $t$—they may be arbitrary integer vectors (as long as the set $S$ is finite). Notice that if we arrange the vectors from $S$ i …

Tseitin tautologies are unsatisfiable systems of linear equations over $\mathbb F_2$, and as such they can be refuted just by summing all the equations together (possibly after reconstructing the equa …

Since the other answer makes it sound as if it were not obvious, let me point out that the problem is computable in PSPACE. First, we observe: Lemma. For any NFA $A$ with $n$ states, the following ar …

Here is a measure-free proof which works for affine manifolds over an arbitrary infinite field $\mathbb F$ (the result is false for finite fields). By induction on $n\ge0$, we will show that an affin …

There are no known NP-complete problems whose input would consist of primes (or, say, $k$-tuples of primes, or even more complicated structures as long as they contain at least one prime of length $\g …

(This is shameless self-promotion.) If you don’t mind either assuming the generalized Riemann hypothesis (for $L$-functions of quadratic Dirichlet characters) or using randomized polynomial time, then …

By Proposition 13 in Benedikt and Hrushovski, the theory of finite fields has nonelementary complexity (it is harder than $k$-times iterated exponential time for all constants $k$). Apparently, the th …

I will give two upper bounds. Let $A$ and $B$ be the sets given to Alice and Bob, respectively, and put $a=|A|$, $b=|B|$, $c=|A\cap B|$. First, there is a randomized protocol that, given $d>0$ and $ …

Yes, any unsatisfiable Horn CNF has a tree-like resolution refutation with a linear number of clauses. Consider the standard poly-time Horn-SAT algorithm, which works as follows. First, set all variab …