# Tagged Questions

computable sets and functions, Turing degrees, c.e. degrees, models of computability, primitive recursion, oracle computation, models of computability, decision problems, undecidability, Turing jump, halting problem, notions of computable randomness, computable model theory, computable equivalence ...

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### Groups whose word problem can be solved in constant time

Given a finitely generated group $G$, define an encoding of $G$ to be a one-to-one function $\Phi:G\to \bigcup_n \{0,1\}^n$ that sends each group element to a unique finite word. For $a,b\in G$, ...
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### Efficient deterministic algorithms of factorizing

My question is about efficient deterministic algorithms of factorizing polynomials of degree $n$ over $\mathbb{F}_q$. Are there such algorithms that use poly$(n, \log q)$ bit operations? I know ...
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### An efficient isomorphism between finite fields

Let $p$ be a prime number. Let $f$ and $g$ be irreducible polynomials over $\mathbb{F}_p$, both of degree $n$. We know that factor-rings $\mathbb{F}_p[x]/(f)$ and $\mathbb{F}_p[x]/(g)$ are isomorphic ...
356 views

### Consistent sentences with no arithmetically definable models

I've seen a construction of a sentence of first order logic that is consistent, but has no models with underlying set $\mathbb{N}$ and recursive functions and relations. Do there also exist consistent ...
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### Computable function [closed]

Let $f(n)$ be $$f(n)=\begin{cases} 1,&\small{\text{if there are digits 1 in the constant \textit{e} \textit{n} times in a row}}\\ 0,&\small{\text{otherwise.}}\\ \end{cases}$$ Is it true ...
369 views

### What Turing degree is this function?

Over at http://www.scottaaronson.com/blog/?p=2725#comment-1089004 we had a discussion of intermediate Turing degrees. The following function came up: Take Chaitin’s constant, and rearrange its ...
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### Turing and Many one reductions in computability versus complexity

What are some non-trivial (please exclude poly time definitional difference) differences between Turing versus Many-one reductions in computability theory and those that occur in complexity theory?
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### When do wide initial segments ruin admissibility?

Suppose $\alpha$ is admissible and $\beta<\alpha$. We know that $L_\alpha$ is an admissible set (by definition), but adding subsets of $\beta$ to $L_\alpha$ might break admissibility: while set ...
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### Comparing product of positive affine functions over integers

Problem Let $f_i: \mathbb Z^n \mapsto \mathbb Z$ and $g_i: \mathbb Z^n \mapsto \mathbb Z$ affine functions and $\mathcal D \subseteq \mathbb Z^n$ a set on which they are all positive. Let $P$ and $Q$ ...