Enhanced Formal Reasoning for Algebraic Network Theory

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Diagrammatic languages are used in diverse fields of science to specify and study systems based on interacting components. Graphics outperforms textual information in highlighting connectivity and resource-exchange between parts of a system. This makes diagrammatic languages particularly effective in the analysis of subtle interactions as those found in cyber-physical, concurrent and quantum systems.In recent years "algebraic network theory" emerged as a unifying mathematical framework in which diagrammatic languages are given a completely formal status and are studied using the compositional methods of algebraic program semantics. Nowadays the algebraic approach founds application in fields as diverse as quantum theory, linguistics, concurrency theory and the analysis of signal processing, electrical and digital circuits. This proposal aims at making diagrammatic reasoning within this framework more scalable, mathematically robust and easier to implement. Our approach is based on the integration of rewriting techniques, which emphasise the algorithmic aspects of diagrammatic reasoning, and modular techniques, which are more adapted to prove appealing mathematical properties. The resulting technology - which intends to take the best from the two worlds - will be then applied to timely areas of application for algebraic network theory, including diagrammatic calculi for quantum processes, signal processing circuits and Bayesian networks.