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.

On the algorithmic complexity of crystals

2014 ◽  
Vol 78 (2) ◽  
pp. 415-435 ◽  
Author(s):  
S. V. Krivovichev
Keyword(s):  
Computer Science ◽  
Coordination Polymers ◽  
Finite Automata ◽  
Theoretical Computer Science ◽  
Algorithmic Complexity ◽  
Inorganic Materials ◽  
Important Class ◽  
Theoretical Computer ◽  
Deterministic Finite Automata ◽  
Orthogonal Networks

AbstractThe concept of the algorithmic complexity of crystals is developed for a particular class of minerals and inorganic materials based on orthogonal networks, which are defined as networks derived from the primitive cubic net (pcu) by the removal of some vertices and/or edges. Orthogonal networks are an important class of networks that dominate topologies of inorganic oxysalts, framework silicates and aluminosilicate minerals, zeolites and coordination polymers. The growth of periodic orthogonal networks may be modelled using structural automata, which are finite automata with states corresponding to vertex configurations and transition symbols corresponding to the edges linking the respective vertices. The model proposed describes possible relations between theoretical crystallography and theoretical computer science through the theory of networks and the theory of deterministic finite automata.

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 Optimization Over the Boolean Hypercube Via Sums of Nonnegative Circuit Polynomials

2021 ◽  
Author(s):  
Mareike Dressler ◽  
Adam Kurpisz ◽  
Timo de Wolff
Keyword(s):  
Computer Science ◽  
Affine Transformation ◽  
Optimization Problems ◽  
Polynomial Optimization ◽  
Theoretical Computer Science ◽  
Sums Of Squares ◽  
Theoretical Computer ◽  
Complexity Bounds ◽  
Polynomial Optimization Problems ◽  
Transformation Of Variables

AbstractVarious key problems from theoretical computer science can be expressed as polynomial optimization problems over the boolean hypercube. One particularly successful way to prove complexity bounds for these types of problems is based on sums of squares (SOS) as nonnegativity certificates. In this article, we initiate optimization problems over the boolean hypercube via a recent, alternative certificate called sums of nonnegative circuit polynomials (SONC). We show that key results for SOS-based certificates remain valid: First, for polynomials, which are nonnegative over the n-variate boolean hypercube with constraints of degree d there exists a SONC certificate of degree at most $$n+d$$ n + d . Second, if there exists a degree d SONC certificate for nonnegativity of a polynomial over the boolean hypercube, then there also exists a short degree d SONC certificate that includes at most $$n^{O(d)}$$ n O ( d ) nonnegative circuit polynomials. Moreover, we prove that, in opposite to SOS, the SONC cone is not closed under taking affine transformation of variables and that for SONC there does not exist an equivalent to Putinar’s Positivstellensatz for SOS. We discuss these results from both the algebraic and the optimization perspective.

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