Related papers: Upper bound on some hightness notions

A critical milestone on the path to useful quantum computers is quantum supremacy - a demonstration of a quantum computation that is prohibitively hard for classical computers. A leading near-term candidate, put forth by the Google/UCSB…

Claims about recursive self-improvement in AI often slide from repeated internal revision to the possibility of qualitatively stronger capability without clearly distinguishing the underlying computational regimes. This paper gives a formal…

Adapting a result of Bazhenov, Kalimullin, and Yamaleev, we show that if a Turing degree $\textbf{d}$ is the degree of categoricity of a computable structure $\mathcal{M}$ and is not the strong degree of categoricity of any computable…

We show that a question of Miller and Solomon -- that whether there exists a coloring $c:d^{<\omega}\rightarrow k$ that does not admit a $c$-computable variable word infinite solution, is equivalent to a natural, nontrivial combinatorial…

Consider a discrete-time martingale $\{X_t\}$ taking values in a Hilbert space $\mathcal H$. We show that if for some $L \geq 1$, the bounds $\mathbb{E} \left[\|X_{t+1}-X_t\|_{\mathcal H}^2 \mid X_t\right]=1$ and $\|X_{t+1}-X_t\|_{\mathcal…

In this paper, we compute the tightest possible bounds on the probability that the optimal value of a combinatorial optimization problem in maximization form with a random objective exceeds a given number, assuming only knowledge of the…

We prove an NP upper bound on a theory of integer-indexed integer-valued arrays that extends combinatory array logic with an ordering relation on the index set and the ability to express sums of elements. We compare our fragment with seven…

As inductive inference and machine learning methods in computer science see continued success, researchers are aiming to describe ever more complex probabilistic models and inference algorithms. It is natural to ask whether there is a…

We introduce a new conjecture on the computational hardness of detecting random lifts of graphs: we claim that there is no polynomial-time algorithm that can distinguish between a large random $d$-regular graph and a large random lift of a…

In this first of two papers, strong limits on the accuracy of physical computation are established. First it is proven that there cannot be a physical computer C to which one can pose any and all computational tasks concerning the physical…

Can a probabilistic gambler get arbitrarily rich when all deterministic gamblers fail? We study this problem in the context of algorithmic randomness, introducing a new notion -- almost everywhere computable randomness. A binary sequence…

The third author noticed in his 1992 PhD Thesis [Sim92] that every regular tree language of infinite trees is in a class $\Game (D\_n({\bf\Sigma}^0\_2))$ for some natural number $n\geq 1$, where $\Game$ is the game quantifier. We first give…

In the first of this pair of papers, it was proven that that no physical computer can correctly carry out all computational tasks that can be posed to it. The generality of this result follows from its use of a novel definition of…

Quantum theory imposes fundamental limitations to the amount of information that can be carried by any quantum system. On the one hand, Holevo bound rules out the possibility to encode more information in a quantum system than in its…

We give a survey on the theory of representation-finite and certain minimal representation-infinite algebras.The main goals are the existence of multiplicative bases and of coverings with good properties. Both are attained via…

Cryptographic protocols are often based on the two main resources: private randomness and private key. In this paper, we develop a relationship between these two resources. First, we show that any state containing perfect, directly…

In this paper, we determine the sharp threshold for universality of cokernels of random matrices over finite fields. More precisely, we prove the following: given any constant $c>1$, let $A(n)$ be a random $n \times n$ matrix over…

A century ago, discoveries of a serious kind of logical error made separately by several leading mathematicians led to acceptance of a sharply enhanced standard for rigor within what ultimately became the foundation for Computer Science. By…

Many constructions in computability theory rely on "time tricks". In the higher setting, relativising to some oracles shows the necessity of these. We construct an oracle~$A$ and a set~$X$, higher Turing reducible to~$X$, but for which…

We survey results on the formalization and independence of mathematical statements related to major open problems in computational complexity theory. Our primary focus is on recent findings concerning the (un)provability of complexity…