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We explore several concepts for analyzing the intuitive notion of computational irreducibility and we propose a robust formal definition, first in the field of cellular automata and then in the general field of any computable function f…
Involving only the measurements of commuting observables - the problem-setting and the corresponding solution - quantum algorithms should be subject to classical logic. This would allow flanking their customary quantum description with a…
Landauer's Principle that information loss from a computation implies entropy increase can be rigorously proved from mathematical physics. However, carefully examining its detailed formulation reveals that the traditional identification of…
We analyze and compare the optimality of approximate and probabilistic universal programmable quantum processors. We define several characteristics how to quantify the optimality and we study in detail performance of three types of…
In distributed applications, Brewer's CAP theorem tells us that when networks become partitioned (P), one must give up either consistency (C) or availability (A). Consistency is agreement on the values of shared variables; availability is…
We investigate the connections between computability theory and Nonstandard Analysis. In particular, we investigate the two following topics and show that they are intimately related. (T.1) A basic property of Cantor space $2^{\mathbb{N}}$…
This paper concludes a three-Part series on the limits the laws of physics place on the sustained performance of reversible computers. Part I concerned aggregate performance in terms of computational operations per unit time, but neglected…
We prove the following conjecture, raised by Aaronson and Ambainis in 2008: Let $f:\{-1,1\}^n \rightarrow [-1,1]$ be a multilinear polynomial of degree $d$. Then there exists a variable $x_i$ whose influence on $f$ is at least…
Large reasoning models have shown strong performance through extended chain-of-thought reasoning, yet their computational cost remains significant. Probably approximately correct (PAC) reasoning provides statistical guarantees for efficient…
Computational pseudorandomness studies the extent to which a random variable $\bf{Z}$ looks like the uniform distribution according to a class of tests $\cal{F}$. Computational entropy generalizes computational pseudorandomness by studying…
A parallel program can be represented as a directed acyclic graph. An important performance bound is the time to execute the critical path through the graph. We show how this performance metric is related to Amdahl speedup and the degree of…
The coalgebraic $\mu$-calculus provides a generic semantic framework for fixpoint logics over systems whose branching type goes beyond the standard relational setup, e.g. probabilistic, weighted, or game-based. Previous work on the…
An enduring challenge in computer science is reducing the runtime required to solve computational problems. Quantum computing has attracted significant attention due to its potential to deliver asymptotically faster solutions to certain…
We present a logic that extends CTL (Computation Tree Logic) with operators that express synchronization properties. A property is synchronized in a system if it holds in all paths of a certain length. The new logic is obtained by using the…
Scaling laws, a defining feature of deep learning, reveal a striking power-law improvement in model performance with increasing dataset and model size. Yet, their mathematical origins, especially the scaling exponent, have remained elusive.…
Inspired by connections to two dimensional quantum theory, we define several models of computation based on permuting distinguishable particles (which we call balls), and characterize their computational complexity. In the quantum setting,…
The ultraproduct construction is generalized to $p$-ultramean constructions ($1\leqslant p<\infty$) by replacing ultrafilters with finitely additive measures. These constructions correspond to the linear fragments $\mathscr L^p$ of…
A real number is called left-computable if there exists a computable increasing sequence of rational numbers converging to it. In this article we are investigating a proper subset of the left-computable numbers. We say that a real number…
Poisson's equation is fundamental to the study of Markov chains, and arises in connection with martingale representations and central limit theorems for additive functionals, perturbation theory for stationary distributions, and average…
We study in-network computation on general network topologies. Specifically, we are given the description of a function, and a network with distinct nodes at which the operands of the function are made available, and a designated sink where…