Categorical equational systems : algebraic models and equational reasoning

Declaration This dissertation is the result of my own work done under the guidance of my supervisor, and includes nothing which is the outcome of work done in collaboration except where specifically indicated in the text. This dissertation is not substantially the same as any that I have submitted or will be submitting for a degree or diploma or other qualification at this or any other University. This dissertation does not exceed the regulation length of 60,000 words, including tables and footnotes. 5 Summary We introduce two abstract notions of equational algebraic system, called Equational System (ES) and Term Equational System (TES), in order to achieve sufficient expressivity as needed in modern applications in computer science. These generalize the classical concept of (enriched) algebraic theory of Kelly and Power [1993]. We also develop a theory for constructing free algebras for ESs and a theory of equational reasoning for TESs. In Part I, we introduce the general abstract, yet practical, concept of equational system and develop finitary and transfinitary conditions under which we give an explicit construction of free algebras for ESs. This free construction extends the well-known construction of free algebras for ω-cocontinuous endofunctors to an equational setting, capturing the intuition that free algebras consist of freely constructed terms quotiented by given equations and congruence rules. We further show the monadicity and cocom-pleteness of categories of algebras for ESs under the finitary and transfinitary conditions. To illustrate the expressivity of equational systems, we exhibit various examples including two modern applications, the Σ-monoids of Fiore et al. [1999] and the π-algebras of Stark [2005]. In Part II, we introduce the more concrete notion of term equational system, which is obtained by specializing the concept of equational system, but remains more general than that of enriched algebraic theory. We first develop a sound logical deduction system, called Term Equational Logic (TEL), for equational reasoning about algebras of TESs. Then, to pursue a complete logic, we give an internal completeness result, from which together with the explicit construction of free algebras one can typically synthesize sound and complete rewriting-style equational logics. To exemplify this scenario, we give two applications: multi-sorted algebraic theories and nominal equational theories of Clouston and Pitts [2007] and of Gabbay and Mathijssen [2007]. 7 Acknowledgements First and foremost, I am deeply grateful to my Ph.D. supervisor Marcelo Fiore for his patient guidance and support, without which this thesis would …