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In Monoidal Computer I, we introduced a categorical model of computation where the formal reasoning about computability was supported by the simple and popular diagrammatic language of string diagrams. In the present paper, we refine and…
Classic algorithms and machine learning systems like neural networks are both abundant in everyday life. While classic computer science algorithms are suitable for precise execution of exactly defined tasks such as finding the shortest path…
Quantum computers can solve many number theory problems efficiently. Using the efficient quantum algorithm for order finding as an oracle, this paper presents an algorithm that computes the Carmichael function for any integer $N$ with a…
The ``impossibility theorem'' -- which is considered foundational in algorithmic fairness literature -- asserts that there must be trade-offs between common notions of fairness and performance when fitting statistical models, except in two…
Graded posets frequently arise throughout combinatorics, where it is natural to try to count the number of elements of a fixed rank. These counting problems are often $\#\textbf{P}$-complete, so we consider approximation algorithms for…
Approximation algorithms for classical constraint satisfaction problems are one of the main research areas in theoretical computer science. Here we define a natural approximation version of the QMA-complete local Hamiltonian problem and…
In the literature on Kleene algebra (KA), a number of variants have been proposed such as Kleene algebra with tests, commutative KA, bi-KA, and concurrent KA. The equational theories of some of these structures have then been studied in the…
A phoneme-retrieval technique is proposed, which is due to the particular way of the construction of the network. An initial set of neurons is given. The number of these neurons is approximately equal to the number of typical structures of…
We study Martin-L\"{o}f random (ML-random) points on computable probability measures on sample and parameter spaces (Bayes models). We consider variants of conditional randomness defined by ML-randomness on Bayes models and those of…
We begin by reiterating that common neural network activation functions have simple Bayesian origins. In this spirit, we go on to show that Bayes's theorem also implies a simple recurrence relation; this leads to a Bayesian recurrent unit…
Genetic algorithms are considered as an original way to solve problems, probably because of their generality and of their "blind" nature. But GAs are also unusual since the features of many implementations (among all that could be thought…
We present a quantum version of the classical probabilistic algorithms $\grave{a}$ la Rabin. The quantum algorithm is based on the essential use of Grover's operator for the quantum search of a database and of Shor's Fourier transform for…
Symmetry of information establishes a relation between the information that x has about y (denoted I(x : y)) and the information that y has about x (denoted I(y : x)). In classical information theory, the two are exactly equal, but in…
Data Science and Machine learning have been growing strong for the past decade. We argue that to make the most of this exciting field we should resist the temptation of assuming that forecasting can be reduced to brute-force data analytics.…
There exists a well established differential topological theory of singularities of ordinary differential equations. It has mainly studied scalar equations of low order. We propose an extension of the key concepts to arbitrary systems of…
Theorems from universal algebra such as that of Murski\u{i} from the 1970s have a striking similarity to universal approximation results for neural nets along the lines of Cybenko's from the 1980s. We consider here a discrete analogue of…
In this article we demonstrate how algorithmic probability theory is applied to situations that involve uncertainty. When people are unsure of their model of reality, then the outcome they observe will cause them to update their beliefs. We…
In this paper the elicitation of probabilities from human experts is considered as a measurement process, which may be disturbed by random 'measurement noise'. Using Bayesian concepts a second order probability distribution is derived…
We discuss the role and merits of symmetry methods for the analysis of biological systems. In particular, we consider systems of first order ordinary differential equations and provide a comprehensive review of the geometrical foundations…
Prior knowledge on properties of a target model often come as discrete or combinatorial descriptions. This work provides a unified computational framework for defining norms that promote such structures. More specifically, we develop…