The purpose of this paper is threefold: to present a general abstract, yet practical, notion of equational system; to investigate and develop the finitary and transfinite construction of free algebras for equational systems; and to illustrate the use of equational systems as needed in modern applications.

We introduce an abstract general notion of system of equations between terms, called Term Equational System, and develop a sound logical deduction system, called Term Equational Logic, for equational reasoning. Further, we give an analysis of algebraic free constructions that together with an internal completeness result may be used to synthesise complete equational logics. Indeed, as an application, we synthesise a sound and complete nominal equational logic, called Synthetic Nominal Equational Logic, based on the category of Nominal Sets.

Our view of computation is still evolving. The concrete theories for specific computational phenomena that are emerging encompass three aspects: specification and programming languages for describing computations, mathematical structures for modelling computations, and logics for reasoning about properties of computations. To make sense of this complexity, and also to compare and/or relate different concrete theories, meta-theories have been built. These metatheories are used for the study, formalisation, specification, prototyping, and testing of concrete theories. Our main concern here is the investigation of meta-theories to provide systems that better support the formalisation of concrete theories. Thereby we propose a research programme based on the development of mathematical models of computational languages, and the systematic use of these models to synthesise formal deduction systems for reasoning and computation. Specifically, we put forth a mathematical methodology for the synthesis of equational and rewriting logics from algebraic meta-theories. The synthesised logics are guaranteed to be sound with respect to a canonical model theory, and we provide a framework for analysing completeness that typically leads to canonical logics. Our methodology can be used to rationally reconstruct the traditional equational logic of universal algebra and its multi-sorted version from first principles. As for modern applications, we have synthesised: (1) a nominal equational logic for specifying and reasoning about languages with name-binding operators, and (2) a second-order equational logic for specifying and reasoning about simple type theories. Overall, we aim at incorporating into the research programme further key features of modern languages, as e.g. type dependency, linearity, sharing, and graphical structure.

We provide an extension of universal algebra and its equational logic from first to second order. Conservative extension, soundness, and completeness results are established.

Birkhoff [1935] initiated the general study of algebraic structure. Importantly for our concerns here, his development was from (universal) algebra to (equational) logic. Birkhoff’s starting point was the informal conception of algebra based on familiar concrete examples. Abstracting from these, he introduced the concepts of signature and equational presentation,

Fiore and Hur [18] recently introduced a novel methodology—henceforth referred to as Sol—for the Synthesis of equational and rewriting logics from mathematical models. In [18], Sol was successfully applied to rationally reconstruct the traditional equational logic for universal algebra of Birkhoff [3] and its multi-sorted version [26], and also to synthesise a new version of the Nominal Algebra of Gabbay and Mathijssen [41] and the Nominal Equational Logic of Clouston and Pitts [8] for reasoning about languages with name-binding operators. Based on these case studies and further preliminary investigations, we contend that Sol can make an impact in the problem of engineering logics for modern computational languages. For example, our proposed research on secondorder equational logic will provide foundations for designing a second-order extension of the Maude system [37], a first-order semantic and logical framework used in formal software engineering for specification and programming. Our research strategy can be visualised as follows: (I)