Transformation semigroups

Introduction

Published in John N. Mordeson, Davender S. Malik, Fuzzy Automata and Languages, 2002

The concepts of transformation semigroup, covering, cascade product, and wreath product play a prominent role in the study of automata [92]. In this chapter, we examine these ideas for fuzzy finite state machines. Some severe and interesting complications arise when introducing these ideas to the fuzzy setting. One of the concepts we introduce to overcome some of the problems that arise is that of a polysemigroup. Let P be a nonempty set and P(P) be the power set of P. Let * be a function of P×P into P(P)∖{∅}. Then (P,*) is called a polysemigroup, [39, 102], if and only if x*(y*z)=(x*y)*z∀x,y,z∈P. The obvious abuse of notation is explained as follows: If x∈P and A,B⊂P, then x*A denotes {x}*A, A*x denotes A*{x}, and A*B=∪a∈A,b∈Ba*b.

When is deep learning better and when is shallow learning better: qualitative analysis

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Journal Information

Published in International Journal of Parallel, Emergent and Distributed Systems, 2022

Salvador Robles Herrera, Martine Ceberio, Vladik Kreinovich

In the deep learning case, we have classes that are closed under composition. Such classes are known as transformation semigroups. An important particular case is transformation groups – they describe the case when all transformations are reversible, and the inverse transformations also belongs to the same class F.

Transformation semigroups

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Introduction

When is deep learning better and when is shallow learning better: qualitative analysis