Related papers: Turing Degrees of Hyperjumps

The Jump Inversion Theorem says that for every real $A \ge_T 0'$ there is a real $B$ such that $A \equiv_T B' \equiv_T B \oplus 0'$. A known refinement of this theorem says that we can choose $B$ to be a member of any special $\Pi^0_1$…

We define the bounded jump of A by A^b = {x | Exists i <= x [phi_i (x) converges and Phi_x^[A|phi_i(x)](x) converges} and let A^[nb] denote the n-th bounded jump. We demonstrate several properties of the bounded jump, including that it is…

The Turing degree of a real measures the computational difficulty of producing its binary expansion. Since Turing degrees are tailsets, it follows from Kolmogorov's 0-1 law that for any property which may or may not be satisfied by any…

Recently we have introduced a new model of infinite computation by extending the operation of ordinary Turing machines into transfinite ordinal time. In this paper we will show that the infinite time Turing machine analogue of Post's…

It is a commonplace to say that `one can search through the natural numbers', by which is meant the following: For a property, decidable in finite time and which is not false for all natural numbers, checking said property starting at zero,…

Bernstein's classical inequality asserts that given a trigonometric polynomial $T$ of degree $n\geq1$, the sup-norm of the derivative of $T$ does not exceed $n$ times the sup-norm of $T$. We present various approaches to prove this…

The Turing degree spectrum of a countable structure $\mathcal{A}$ is the set of all Turing degrees of isomorphic copies of $\mathcal{A}$. The Turing degree of the isomorphism type of $\mathcal{A}$, if it exists, is the least Turing degree…

If $\mathfrak{n}$ is a $\mathbb{Z}^d_+$-graded nilpotent finite dimensional Lie algebra over a field of characteristic zero, it is well known that $\dim H^{\ast }(\mathfrak{n})\geq L(p) $ where $p$ is the polynomial associated to the…

A Turing degree d bounds a principle P of reverse mathematics if every computable instance of P has a d-computable solution. P admits a universal instance if there exists a computable instance such that every solution bounds P. We prove…

Assuming that $0^#$ exists, we prove that there is a structure that can effectively interpret its own jump. In particular, we get a structure $\mathcal A$ such that \[ Sp({\mathcal A}) = \{{\bf x}':{\bf x}\in Sp ({\mathcal A})\}, \] where…

We construct an increasing $\omega$-sequence $(a_n)$ of Turing degrees which forms an initial segment of the Turing degrees, and such that each~$a_{n+1}$ is diagonally noncomputable relative to $a_n$. It follows that the~$\mathsf{DNR}$…

Models of computation operating over the real numbers and computing a larger class of functions compared to the class of general recursive functions invariably introduce a non-finite element of infinite information encoded in an arbitrary…

For a ring $R$, Hilbert's Tenth Problem $HTP(R)$ is the set of polynomial equations over $R$, in several variables, with solutions in $R$. We view $HTP$ as an operator, mapping each set $W$ of prime numbers to $HTP(\mathbb Z[W^{-1}])$,…

Let $\mathfrak M=(M,\mathcal X)$ be a model of $\mathsf{RCA}_0+\text{$\Sigma^0_2$-bounding}$ in which $\Sigma^0_2(A)$-induction fails for some $A\in\mathcal X$. We show that (i) if $\mathfrak M$ is a model of the combinatorial principle…

A real number $\alpha\in [0, 1)$ is a jump for an integer $r\ge 2$ if there exists $c>0$ such that no number in $(\alpha , \alpha + c)$ can be the Tur\'an density of a family of $r$-uniform graphs. A classical result of Erd\H os and Stone…

In the early 1980s, Selman's seminal work on positive Turing reductions showed that positive Turing reduction to NP yields no greater computational power than NP itself. Thus, positive Turing and Turing reducibility to NP differ sharply…

Lusin's Theorem states that, for every Borel-measurable function $\bf{f}$ on $\mathbb R$ and every $\epsilon>0$, there exists a continuous function $\bf{g}$ on $\mathbb R$ which is equal to $\bf{f}$ except on a set of measure $<\epsilon$.…

We say that $\alpha\in [0,1)$ is a jump for an integer $r\geq 2$ if there exists $c(\alpha)>0$ such that for all $\epsilon >0 $ and all $t\geq 1$ any $r$-graph with $n\geq n_0(\alpha,\epsilon,t)$ vertices and density at least…

Let $ a,b \in {\bf Z}^+$, $r=a + b$, and let $T$ be a tree with parts $U = \{u_1,u_2,\dots,u_s\}$ and $V = \{v_1,v_2,\dots,v_t\}$. Let $U_1, \dots ,U_s$ and $V_1, \dots, V_t$ be disjoint sets, such that {$|U_i|=a$ and $|V_j|=b$ for all…

Martin's Conjecture states that every definable function on the Turing degrees is either constant or increasing, and that every increasing function is an iterate of the Turing jump. This classification has already been corroborated for the…