Related papers: On low for speed oracles
Consider an oracle which takes a point $x$ and returns the minimizer of a convex function $f$ in an $\ell_2$ ball of radius $r$ around $x$. It is straightforward to show that roughly $r^{-1}\log\frac{1}{\epsilon}$ calls to the oracle…
The kernel method is a potential approach to analyzing structured data such as sequences, trees, and graphs; however, unordered trees have not been investigated extensively. Kimura et al. (2011) proposed a kernel function for unordered…
The {\em diagonalization technique} was invented by Georg Cantor to show that there are more real numbers than algebraic numbers and is very crucial in {\em theoretical computer science}. In this work, we enumerate all of the…
An important theorem in classical complexity theory is that LOGLOGSPACE=REG, i.e. that languages decidable with double-logarithmic space bound are regular. We consider a transfinite analogue of this theorem. To this end, we introduce…
Convertibility checking - determining whether two lambda-terms are equal up to reductions - is a crucial component of proof assistants and dependently-typed languages. Practical implementations often use heuristics to quickly conclude that…
The history of computability theory and and the history of analysis are surprisingly intertwined since the beginning of the twentieth century. For one, \'Emil Borel discussed his ideas on computable real number functions in his introduction…
TLA is a popular temporal logic for writing stuttering-invariant specifications of digital systems. However, TLA lacks higher-order features useful for specifying modern software written in higher-order programming languages. We use…
A typical oracle problem is finding which software program is installed on a computer, by running the computer and testing its input-output behaviour. The program is randomly chosen from a set of programs known to the problem solver. As…
The Ku\v{c}era-G\'acs theorem is a landmark result in algorithmic randomness asserting that every real is computable from a Martin-L\"of random real. If the computation of the first $n$ bits of a sequence requires $n+h(n)$ bits of the…
We consider the quantum complexities of the following three problems: searching an ordered list, sorting an un-ordered list, and deciding whether the numbers in a list are all distinct. Letting N be the number of elements in the input list,…
We introduce a concept of efficiency for which we can prove that it applies to all paddable languages, but still does not conflict with potential worst case intractability. Note that the family of paddable languages apparently includes all…
We show the following hold, unconditionally unless otherwise stated, relative to a random oracle: - There are NP search problems solvable by quantum polynomial-time machines but not classical probabilistic polynomial-time machines. - There…
The speed of open-weights large language models (LLMs) and its dependency on the task at hand, when run on GPUs, is studied to present a comparative analysis of the speed of the most popular open LLMs.
Shor's and Grover's famous quantum algorithms for factoring and searching show that quantum computers can solve certain computational problems significantly faster than any classical computer. We discuss here what quantum computers_cannot_…
Causality serves as an abstract notion of time for concurrent systems. A computation is causal, or simply valid, if each observation of a computation event is preceded by the observation of its causes. The present work establishes that this…
Even if Google AI's Sycamore processor is efficient for the particular task it has been designed for it fails to deliver universal computational capacity. Furthermore, even classical devices implementing transverse homoclinic orbits realize…
We construct an oracular (i.e., black box) problem that can be solved exponentially faster on a quantum computer than on a classical computer. The quantum algorithm is based on a continuous time quantum walk, and thus employs a different…
Frequently, the burgeoning field of black-box optimization encounters challenges due to a limited understanding of the mechanisms of the objective function. To address such problems, in this work we focus on the deterministic concept of…
For over a decade, researchers in formal methods tried to create formalisms that permit natural specification of systems and allow mathematical reasoning about their correctness. The availability of fully-automated reasoning tools enables…
Quantum Merlin-Arthur proof systems are believed to be stronger than both their classical counterparts and ``stand-alone'' quantum computers when Arthur is assumed to operate in $\Omega(\log n)$ space. No hint of such an advantage over…