scholarly journals Conditional Independence in Categories

10.29007/tg3g ◽  
2018 ◽  
Author(s):  
Alex Simpson
Keyword(s):  
Probability Theory ◽  
Computer Science ◽  
Conditional Independence ◽  
Computability Theory ◽  
Separation Logic ◽  
General Category ◽  
Topos Theory ◽  
Monoidal Structure ◽  
Standard Notion ◽  
Nominal Sets

In this talk I shall discuss a general category-theoretic structure for modelling conditional independence. The standard notion of conditional independence in probability theory provides a motivating example. But other rather different examples arise in many contexts: computability theory, nominal sets (used to model `names' in computer science), separation logic (used to reason about heap memory in computer science), and others.Category-theoretic structure common to these examples can be axiomatized by the notion of a category with local independent products, which combines fibrational and symmetric monoidal structure in a somewhat particular way. In the talk I shall expound this notion, and I shall present several illustrative examples of such structure. If time permits, I may also describe some curious connections with topos theory.

scholarly journals Computational Complexity Theory and the Philosophy of Mathematics†

2019 ◽  
Vol 27 (3) ◽  
pp. 381-439
Author(s):  
Walter Dean
Keyword(s):  
Computational Complexity ◽  
Computer Science ◽  
Complexity Theory ◽  
Philosophy Of Mathematics ◽  
Computability Theory ◽  
Computational Complexity Theory ◽  
Mathematical Problems ◽  
Np Problem

Abstract Computational complexity theory is a subfield of computer science originating in computability theory and the study of algorithms for solving practical mathematical problems. Amongst its aims is classifying problems by their degree of difficulty — i.e., how hard they are to solve computationally. This paper highlights the significance of complexity theory relative to questions traditionally asked by philosophers of mathematics while also attempting to isolate some new ones — e.g., about the notion of feasibility in mathematics, the $\mathbf{P} \neq \mathbf{NP}$ problem and why it has proven hard to resolve, and the role of non-classical modes of computation and proof.

Computing with Functionals—Computability Theory or Computer Science?

2006 ◽  
Vol 12 (1) ◽  
pp. 43-59 ◽  
Author(s):  
Dag Normann
Keyword(s):  
Computer Science ◽  
Computability Theory ◽  
Theoretical Computer Science ◽  
Theoretical Computer ◽  
Active Subject ◽  
Higher Types ◽  
History Of

AbstractWe review some of the history of the computability theory of functionals of higher types, and we will demonstrate how contributions from logic and theoretical computer science have shaped this still active subject.

scholarly journals The Polynomial Complexity of Vector Addition Systems with States

2020 ◽  
pp. 622-641 ◽  
Author(s):  
Florian Zuleger
Keyword(s):  
Computer Science ◽  
Time Complexity ◽  
Theoretical Computer Science ◽  
Initial Configuration ◽  
Computational Time ◽  
Theoretical Computer ◽  
Vector Addition ◽  
Standard Notion ◽  
Vector Addition System ◽  
Asymptotic Complexity

AbstractVector addition systems are an important model in theoretical computer science and have been used in a variety of areas. In this paper, we consider vector addition systems with states over a parameterized initial configuration. For these systems, we are interested in the standard notion of computational time complexity, i.e., we want to understand the length of the longest trace for a fixed vector addition system with states depending on the size of the initial configuration. We show that the asymptotic complexity of a given vector addition system with states is either $$\varTheta (N^k)$$ Θ ( N k ) for some computable integer k, where N is the size of the initial configuration, or at least exponential. We further show that k can be computed in polynomial time in the size of the considered vector addition system. Finally, we show that $$1 \le k \le 2^n$$ 1 ≤ k ≤ 2 n , where n is the dimension of the considered vector addition system.

Models of the U.S. Economy

1977 ◽  
Vol 70 (2) ◽  
pp. 102-110
Author(s):  
Samuel L. Dunn ◽  
Lawrence W. Wright
Keyword(s):  
Probability Theory ◽  
Computer Science ◽  
Linear Algebra ◽  
The U.S

The field of economics provides many opportunities for applying mathematics. The quantification of economics that has occurred in the last thirty years has made it necessary that economists be trained in the uses of higher mathematics. Algebra, geometry, calculus, probability theory and statistics, higher analysis, linear algebra, and computer science are some of the tools being used in contemporary approaches to economics.

scholarly journals Preface

2017 ◽  
Vol 28 (3) ◽  
pp. 338-339
Author(s):  
Ekaterina Fokina
Keyword(s):  
Computer Science ◽  
Computability Theory ◽  
Special Issue ◽  
Mathematical Structures ◽  
Conference Series ◽  
The World ◽  
Huge Impact

This special issue of Mathematical Structures in Computer Science is dedicated to the memory of Barry Cooper (October 9, 1943–October 26, 2015). Barry's life in research had a huge impact on the development of the world of computability. Aside from his enormous achievements in classical computability theory, he contributed immensely to the promotion of interdisciplinary developments related to computability, in particular, by founding the Association for Computability in Europe (CiE) and its annual, highly successful conference series.

scholarly journals SOME IMPORTANT APPLICATIONS OF SEMIGROUPS

2021 ◽  
Vol 2 (2) ◽  
pp. 317-321
Author(s):  
Iqra Liaqat ◽  
Wajeeha Younas
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Probability Theory ◽  
Markov Process ◽  
Computer Science ◽  
Finite Automata ◽  
Theoretical Computer Science ◽  
Theoretical Computer ◽  
Regular Semigroups ◽  
Finite Semigroups

This Paper deals with the some important applications of semigroups in general and regular semigroups in particular.The theory of finite semigroups has been of particular importance in theoretical computer science since the 1950s because of the natural link between finite semigroups and finite automata via the syntactic monoid. In probability theory, semigroups are associated with Markov process. In section 2 we have seen different areas of applications of semigroups. We identified some Applications in biology, Partial Differential equation, Formal Languages etc whose semigroup structures are nothing but regular.

Review of The Foundations of Computability Theory (Second Edition) by Borut Robič

2021 ◽  
Vol 52 (2) ◽  
pp. 7-9
Author(s):  
Erick Galinkin
Keyword(s):  
Computer Science ◽  
Finite Automata ◽  
Computability Theory ◽  
Theoretical Computer Science ◽  
Turing Machines ◽  
Theoretical Computer ◽  
Self Study ◽  
First Results ◽  
Data Structures And Algorithms ◽  
Pushdown Automata

Computability theory forms the foundation for much of theoretical computer science. Many of our great unsolved questions stem from the need to understand what problems can even be solved. The greatest question of computer science, P vs. NP, even sidesteps this entirely, asking instead how efficiently we can find solutions for the problems that we know are solvable. For many students both at the undergraduate and graduate level, a first exposure to computability theory follows a standard sequence on data structures and algorithms and students often marvel at the first results they see on undecidability - how could we possibly prove that we can never solve a problem? This book, in contrast with other books that are often used as first exposures to computability, finite automata, Turing machines, and the like, focuses very specifically on the notion of what is computable and how computability theory, as a science unto itself, fits into the grander scheme. The book is appropriate for advanced undergraduates and beginning graduate students in computer science or mathematics who are interested in theoretical computer science. Robič sidesteps the standard theoretical computer science progression - understanding finite automata and pushdown automata before moving into Turing machines - by setting the stage with Hilbert's program and mathematical prerequisites before introducing the Turing machine absent the usual prerequisites, and then introducing advanced topics often absent in introductory texts. Most chapters are relatively short and contain problem sets, making it appropriate for both a classroom text or for self-study.

scholarly journals Limit theorems for Floyd's triangle: a new approach to not a new problem

2021 ◽  
Vol 62 ◽  
pp. 22-27
Author(s):  
Igoris Belovas
Keyword(s):  
Probability Theory ◽  
Computer Science ◽  
Limit Theorems ◽  
Science Students ◽  
Mathematical Apparatus ◽  
New Approach ◽  
Proof Schemes ◽  
Study Programs ◽  
Proof Techniques

Floyd's triangle is often presented to computer science students as an exercise or example to illustrate the concepts of text formatting and loop constructs. The paper proposes to look at an object from a different angle and to examine limit theorems for the numbers of generalized Floyd's triangles. Tasks of this type can be used as exercises in study programs of mathematics and informatics (couses of probability theory and combinatorics). It would help to master the appropriate proof techniques and mathematical apparatus. The article proposes a series of possible problems and their proof schemes.

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