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Network Algebra considers the algebraic study of networks and their behaviour. It contains general results on the algebraic theory of networks, recent results on the algebraic theory of models for parallel programs, as well as results on the algebraic theory of classical control structures. The results are presented in a unified framework of the calculus of flownomials, leading to a sound understanding of the algebraic fundamentals of the network theory. The term 'network' is used in a broad sense within this book, as consisting of a collection of interconnecting cells, and two radically different specific interpretations of this notion of networks are studied. One interpretation is additive, when only one cell is active at a given time - this covers the classical models of control specified by finite automata or flowchart schemes. The second interpretation is multiplicative, where each cell is always active, covering models for parallel computation such as Petri netsor dataflow networks. More advanced settings, mixing the two interpretations are included as well. Network Algebra will be of interest to anyone interested in network theory or its applications and provides them with the results needed to put their work on a firm basis. Graduate students will also find the material within this book useful for their studies.
List of contents
I. An introduction to Network Algebra.- Brief overview of the key results.- 1. Network Algebra and its applications.- II. Relations, flownomials, and abstract networks.- 2. Networks modulo graph isomorphism.- 3. Algebraic models for branching constants.- 4. Network behaviour.- 5. Elgot theories.- 6. Kleene theories.- 7. Flowchart schemes.- 8. Automata.- 9. Process algebra.- 10. Data-flow networks.- 11. Petri nets.- IV. Towards an algebraic theory for software components.- 12. Mixed Network Algebra.- Related calculi, closing remarks.- Appendix B: Lifting BNA from connections to networks.- Appendix C: Demonic relation operators.- Appendix D. Generating congruences.- Appendix E: Automata, complements.- Appendix F: Data-flow networks; checking NA axioms.- Appendix G: Axiomatizing mixed relations.- Appendix H: Discats as sysecats.- Appendix I: Decomposing morphisms in discats.- Appendix J: Plans as free discats.- List of tables.- List of figures.
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"Interesting exercises and problems accompany most of sections. Also some open questions related to this subject are present." (Daniela Marinescu, zbMATH 0956.68002, 2022)
