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# Generalized Nets (World Scientific)

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**"Generalized Nets"** is a book by Krassimir Atanassov, published in 1991 by World Scientific Publishing house under ISBN 978-981-02-0598-0.

In the book, definitions and the basic properties of generalized nets are given. The GNs extensions and reductions are discussed.

GNs, which describe the functioning and results of the work of different types of Petri nets, different types of finite automata and of Turing machines, are given. Over the GNs are defined different operations, relations and operators.

Many open problems in the GNs theory are given.

This book is continued by the book "Applications of Generalized Nets", published in 1993.

## Table of contents

- Chapter 0. Generalized Nets (GNs) — retrospection, present, perspective and applications for modelling of real processes
- Chapter 1. On the Concept GN

- 1.1. A definition of the concept GN
- 1.2. On the GNs's functioning
- 1.3. Basic GN properties
- 1.4. Analysis of the results of the functioning of a GN

- Chapter 2. Reduced GNs

- 2.1. A definition of the concept "reduced GN"
- 2.2. A class of reduced GNs, which represent [math]\displaystyle{ \Sigma }[/math]

- Chapter 3. Conservative Extensions of GNs

- 3.1. First type intuitionistic fuzzy GNs
- 3.2. Second type intuitionistic fuzzy GNs
- 3.3. Colour GNs
- 3.4. GNs with interval time for activation
- 3.5. GNs with a complex structure
- 3.6. GNs with global memory
- 3.7. GNs with optimization components
- 3.8. GNs with additional clocks

- Chapter 4. GNs and Other Objects

- 4.1. GNs and the other types PN modifications
- 4.2. Dynamical properties of GNs
- 4.3. On the unique GN, which represents the functioning of all PNs and its modifications
- 4.4. GNs, finite automata and Turing machines

- Chapter 5. Algebraic Aspect of the Theory of GNs

- 5.1. Operations and relations defined over GN's transitions
- 5.2. Operations and relations defined over GNs
- 5.3. Inductive definition of the concept GN and theorem for equivalence of both GN-definitions

- Chapter 6. Topological Aspect of the Theory of GNs

- 6.1. Complexity operators over GNs and topological constructions generated by them
- 6.2. Graph-operators and related complexity operators

- Chapter 7. Logical Aspect of the Theory of GNs
- Chapter 8. Operator Aspect of the Theory of GNs

- 8.1. Global operators
- 8.1.1 Structural global operators
- 8.1.2 Temporal global operators
- 8.1.3 Dynamical global operators
- 8.1.4 Global operators, modifying the auxiliary components of GNs
- 8.1.5 Relations between different operators

- 8.2. Local operators
- 8.2.1 Temporal local operators
- 8.2.2 Matrix local operators
- 8.2.3 Other local operators
- 8.2.4 Relations between different operators

- 8.3. Hierarchical operators
- 8.4. Reducing operators
- 8.5. Extending operators
- 8.5.1 Fuzzying operators
- 8.5.2 Interval-temporal operators
- 8.5.3 Colouring operators

- 8.6. Dynamical operators

- Chapter 9. Other Extensions of GNs

- 9.1. Universal GNs
- 9.2. Self-modifying GNs

- Chapter 10. Metodological Aspect of the Theory of GNs
- Chapter 11. Open Problems

- Appendix 1: Generalized Index Matrices
- Appendix 2: Intuitionistic Fuzzy Sets
- Subject Index