Related papers: Leveraging the Information Contained in Theory Pre…
We give an introduction to the topics of our forthcoming work, in which we introduce and study new mathematical objects which we call "higher theories" of algebras, where inspiration for the term comes from William Lawvere's notion of…
This work proposes a complete algebraic model for classical information theory. As a precursor the essential probabilistic concepts have been defined and analyzed in the algebraic setting. Examples from probability and information theory…
As the twin movements of open science and open source bring an ever greater share of the scientific process into the digital realm, new opportunities arise for the meta-scientific study of science itself, including of data science and…
This paper explores the application of automated planning to automated theorem proving, which is a branch of automated reasoning concerned with the development of algorithms and computer programs to construct mathematical proofs. In…
We outline a program in the area of formalization of mathematics to automate theorem proving in algebra and algebraic geometry. We propose a construction of a dictionary between automated theorem provers and (La)TeX exploiting syntactic…
This dissertation focuses on the design and the implementation of domain-specific compilers for linear algebra matrix equations. The development of efficient libraries for such equations, which lie at the heart of most software for…
We propose to address the problem of how to know students' knowledge in an entirely new approach called ?epistemography? which is, roughly, an attempt to describe the structure of this knowledge. We claim that what is to be known is made of…
We consider the task of automated theorem proving, a key AI task. Deep learning has shown promise for training theorem provers, but there are limited human-written theorems and proofs available for supervised learning. To address this…
Large formal mathematical libraries consist of millions of atomic inference steps that give rise to a corresponding number of proved statements (lemmas). Analogously to the informal mathematical practice, only a tiny fraction of such…
Linear Software Models is a systematic effort to formulate a theory of software systems neatly based upon standard mathematics, viz. linear algebra. It has appeared in a series of papers dealing with various aspects of the theory. But one…
Recent years have seen tremendous growth in the amount of verified software. Proofs for complex properties can now be achieved using higher-order theories and calculi. Complex properties lead to an ever-growing number of definitions and…
Theory of representations of universal algebra is a natural development of the theory of universal algebra. In the book, I considered representation of universal algebra, diagram of representations and examples of representation. Morphism…
In the last decades, several objects such as grammars, economical agents, laws of physics... have been defined as algorithms. In particular, after Brouwer, Heyting, and Kolomogorov, mathematical proofs have been defined as algorithms. In…
One hallmark of human language is its combinatoriality -- reusing a relatively small inventory of building blocks to create a far larger inventory of increasingly complex structures. In this paper, we explore the idea that combinatoriality…
Some notions of algebraic geometry can be defined for arbitrary varieties of algebras. This leads to universal algebraic geometry. The main idea of the presented theory is to consider interactions between algebra, logic and geometry in…
This work proposes an algebraic model for classical information theory. We first give an algebraic model of probability theory. Information theoretic constructs are based on this model. In addition to theoretical insights provided by our…
We describe Active Libraries, which take an active role in compilation. Unlike traditional libraries which are passive collections of functions and objects, Active Libraries may generate components, specialize algorithms, optimize code,…
Class algebra provides a natural framework for sharing of ISA hierarchies between users that may be unaware of each other's definitions. This permits data from relational databases, object-oriented databases, and tagged XML documents to be…
A finite-dimensional unital and associative algebra over $\mathbb{R}$, or what we shall call simply "an algebra" in this paper for short, generalities the construction by which we derive the complex numbers by "adjoining an element $i$" to…
The traditional foundation of science lies on the cornerstones of theory and experiment. Theory is used to explain experiment, which in turn guides the development of theory. Since the advent of computers and the development of…