Category Theory Foundation For Engineering Modelling

Category theory provides a formal foundation for engineering modelling, as well as, mathematics and science. Both structure and behaviour, as they occur in engineering models for manufactured products and biomedicine, can be embedded as axiom sets within a mathematical formalism, called Algos. The Algos language is a two sorted first order Horn clause theory based on topos language constructions. An Algos theory, generated by Horn clause axioms is an elementary topos. The Horn clause formalism lends itself to automated reasoning. Algos has both a linear syntax and a graphical syntax based on the engineering modelling language SysML. The use of Algos for axiom development is illustrated with axioms for two classes of engineering models, one called Structure Descriptions and the other called Composite Structure models. An example of a Structure Description is the class of 2-amino acids. The problem exhibits common issues of constraining realizations of descriptions to have a specific graph theoretic structure. Algos contains the language of a Description Logic and generalizes several formalisms which have been used for modelling structure descriptions. Composite Structure models represent systems which have behaviour, as well as component structure. Following the topos lead, the terminal object of these models have a time structure. State machines, as well as equations representing physical laws, can be represented and are used to axiomatize these models. An example of a vehicle test system illustrates how behaviour is represented. A description of the formalism including soundness and decidability results for restricted axiom sets is presented, together with comparisons to other logic based formalisms.