Related papers: Representation Independent Decompositions of Compu…
This article is intended to an introductory lecture in material physics, in which the modern computational group theory and the electronic structure calculation are in collaboration. The effort of mathematicians in field of the group…
We show that every finite semilattice can be represented as an atomized semilattice, an algebraic structure with additional elements (atoms) that extend the semilattice's partial order. Each atom maps to one subdirectly irreducible…
Compositionality is a key property for dealing with complexity, which has been studied from many points of view in diverse fields. Particularly, the composition of individual computations (or programs) has been widely studied almost since…
Functional decomposition is the process of breaking down a function $f$ into a composition $f=g(f_1,\dots,f_k)$ of simpler functions $f_1,\dots,f_k$ belonging to some class $\mathcal{F}$. This fundamental notion can be used to model…
Algorithmicists are well-aware that fast dynamic programming algorithms are very often the correct choice when computing on compositional (or even recursive) graphs. Here we initiate the study of how to generalize this folklore intuition to…
We give an algorithm that learns a representation of data through compression. The algorithm 1) predicts bits sequentially from those previously seen and 2) has a structure and a number of computations similar to an autoencoder. The…
We extend a previously introduced semi-analytical representation of a decomposition of CA dynamics in arbitrary dimensions and neighborhood schemes via the use of certain universal maps in which CA rule vectors are derivable from the…
Recently, we have endowed various categories of groups with topologies. The purpose of this paper is to introduce on these categories others topologies which are statistically more suitable to study well-known problems in groups theory. We…
We present an algorithm which computes a cylindrical algebraic decomposition of a semialgebraic set using projection sets computed for each cell separately. Such local projection sets can be significantly smaller than the global projection…
Symmetries impose structure on the Hilbert space of a quantum mechanical model. The mathematical units of this structure are the irreducible representations of symmetry groups and I consider how they function as conceptual units of…
We design various logics for proving hyper properties of iterative programs by application of abstract interpretation principles. In part I, we design a generic, structural, fixpoint abstract interpreter parameterized by an algebraic…
A study of assisted problem solving formalized via decompositions of deterministic finite automata is initiated. The landscape of new types of decompositions of finite automata this study uncovered is presented. Languages with various…
Higher-dimensional automata constitute a very expressive model for concurrent systems. In this paper, we discuss "topological abstraction" of higher-dimensional automata, i.e., the replacement of HDAs by smaller ones that can be considered…
In this paper, we propose a graph classification approach for automatically determining whether to use a monolithic or a decomposition-based solution method. In this approach, an optimization problem is represented as a graph that captures…
This is the first of a series of papers in which we study deep computations (ultracomputations) and deep iterates, formalizing the ideas of "asymptotic limit" of computations and compositional iterates, respectively. In this first paper of…
Decomposition, statically dividing a program into multiple units, is a common programming technique for realizing parallelism and refining programs. The decomposition of a sequential program into components is tedious, due to the…
This paper focuses on the derivations and automorphism groups of certain finite-dimensional associative algebras over the field of complex numbers. Using classification results for algebras of dimensions two, three, and four, along with…
We develop a package using the computer algebra system GAP for computing the decomposition of a representation $\rho$ of a finite group $G$ over $\mathbb{C}$ into irreducibles, as well as the corresponding decomposition of the centraliser…
Learning abstractions directly from data is a core challenge in robotics. Humans naturally operate at an abstract level, reasoning over high-level subgoals while delegating execution to low-level motor skills -- an ability that enables…
The study of representations is of fundamental importance to any form of communication, and our ability to exploit them effectively is paramount. This article presents a novel theory -- Representational Systems Theory -- that is designed to…