Extended Initiality for Typed Abstract Syntax

Abstract

Initial Semantics aims at interpreting the syntax associated to a signatureas the initial object of some category of 'models', yielding induction andrecursion principles for abstract syntax. Zsid\'o proves an initiality resultfor simply-typed syntax: given a signature S, the abstract syntax associated toS constitutes the initial object in a category of models of S in monads.However, the iteration principle her theorem provides only accounts fortranslations between two languages over a fixed set of object types. Wegeneralize Zsid\'o's notion of model such that object types may vary, yieldinga larger category, while preserving initiality of the syntax therein. Thus weobtain an extended initiality theorem for typed abstract syntax, in whichtranslations between terms over different types can be specified via theassociated category-theoretic iteration operator as an initial morphism. Ourdefinitions ensure that translations specified via initiality are type-safe,i.e. compatible with the typing in the source and target language in theobvious sense. Our main example is given via the propositions-as-typesparadigm: we specify propositions and inference rules of classical andintuitionistic propositional logics through their respective typed signatures.Afterwards we use the category--theoretic iteration operator to specify adouble negation translation from the former to the latter. A second example isgiven by the signature of PCF. For this particular case, we formalize thetheorem in the proof assistant Coq. Afterwards we specify, via thecategory-theoretic iteration operator, translations from PCF to the untypedlambda calculus