Presentable signatures and initial semantics

Abstract

We present a device for specifying and reasoning about syntax for datatypes,programming languages, and logic calculi. More precisely, we study a notion of"signature" for specifying syntactic constructions. In the spirit of Initial Semantics, we define the "syntax generated by asignature" to be the initial object -- if it exists -- in a suitable categoryof models. In our framework, the existence of an associated syntax to asignature is not automatically guaranteed. We identify, via the notion ofpresentation of a signature, a large class of signatures that do generate asyntax. Our (presentable) signatures subsume classical algebraic signatures (i.e.,signatures for languages with variable binding, such as the pure lambdacalculus) and extend them to include several other significant examples ofsyntactic constructions. One key feature of our notions of signature, syntax, and presentation is thatthey are highly compositional, in the sense that complex examples can beobtained by gluing simpler ones. Moreover, through the Initial Semanticsapproach, our framework provides, beyond the desired algebra of terms, awell-behaved substitution and the induction and recursion principles associatedto the syntax. This paper builds upon ideas from a previous attempt by Hirschowitz-Maggesi,which, in turn, was directly inspired by some earlier work ofGhani-Uustalu-Hamana and Matthes-Uustalu. The main results presented in the paper are computer-checked within theUniMath system