Related papers: Can Computer Algebra be Liberated from its Algebra…
This paper introduces abstractions that are meaningful for computers and that can be built and used according to computers' own criteria, i.e., computable abstractions. It is analyzed how abstractions can be seen to serve as the building…
Cylindrical algebraic decomposition (CAD) is a core algorithm within Symbolic Computation, particularly for quantifier elimination over the reals and polynomial systems solving more generally. It is now finding increased application as a…
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…
The Eureqa symbolic regression program has recently received extensive press praise. A representative quote is "There are very clever 'thinking machines' in existence today, such as Watson, the IBM computer that conquered Jeopardy! last…
Constructing complex computation from simpler building blocks is a defining problem of computer science. In algebraic automata theory, we represent computing devices as semigroups. Accordingly, we use mathematical tools like products and…
We consider relationships between cubic algebras and implication algebras. We first exhibit a functorial construction of a cubic algebra from an implication algebra. Then we consider an collapse of a cubic algebra to an implication algebra…
Cylindrical Algebraic Decomposition (CAD) is a key tool in computational algebraic geometry, best known as a procedure to enable Quantifier Elimination over real-closed fields. However, it has a worst case complexity doubly exponential in…
The concept of a universal algorithm is discussed. Examples of this kind of algorithms are presented. Software implementations of such algorithms in C++ type languages are discussed together with means that provide for computations with an…
Making new methods for quantum problems often relies on using basic operations in linear algebra. Often these routines are hidden behind well-known libraries that have been optimized over decades. Attempting to improve on those basic…
In recent years there has been increased use of machine learning (ML) techniques within mathematics, including symbolic computation where it may be applied safely to optimise or select algorithms. This paper explores whether using…
For some research questions that involve Spin(p, q) representation theory, using symbolic algebra based techniques might be an attractive option for simplifying and manipulating expressions. Yet, for some such problems, especially as they…
Our Chapter in the upcoming Volume I: Computer Science and Software Engineering of Computing Handbook (Third edition), Allen Tucker, Teo Gonzales and Jorge L. Diaz-Herrera, editors, covers Algebraic Algorithms, both symbolic and numerical,…
Programming languages tend to evolve over time to use more and more concepts from theoretical computer science. Still, there is a gap between programming and pure mathematics. Not all theoretical results have realized their promising…
Symbolic integration deals with the evaluation of integrals in closed form. We present an overview of Risch's algorithm including recent developments. The algorithms discussed are suited for both indefinite and definite integration. They…
Drori et al. (2022) report that "A neural network solves, explains, and generates university math problems by program synthesis and few-shot learning at human level ... [It] automatically answers 81\% of university-level mathematics…
A general overview of the existing difference ring theory for symbolic summation is given. Special emphasis is put on the user interface: the translation and back translation of the corresponding representations within the term algebra and…
In this work, a novel quaternary algebra has been proposed that can be used to implement an arbitrary quaternary logic function in more than one systematic ways. The proposed logic has evolved from and is closely related to the Boolean…
The syntactic nature of logic and computation separates them from other fields of mathematics. Nevertheless, syntax has been the only way to adequately capture the dynamics of proofs and programs such as cut-elimination, and the finiteness…
Probabilistic logic programming is increasingly important in artificial intelligence and related fields as a formalism to reason about uncertainty. It generalises logic programming with the possibility of annotating clauses with…
This survey describes probabilistic algorithms for linear algebra computations, such as factorizing matrices and solving linear systems. It focuses on techniques that have a proven track record for real-world problem instances. The paper…