Extended Initiality for Typed Abstract Syntax

Abstract

Abstract. Initial Semantics aims at interpreting the syntax associated to a signature as the initial object of some category of "models", yielding induction and recursion principles for abstract syntax. Zsidó [Zsi10, Chap. 6] proves an initiality result for simply-typed syntax: given a signature S, the abstract syntax associated to S constitutes the initial object in a category of models of S in monads.However, the iteration principle her theorem provides only accounts for translations between two languages over a fixed set of object types. We generalize Zsidó's notion of model such that object types may vary, yielding a larger category, while preserving initiality of the syntax therein. Thus we obtain an extended initiality theorem for typed abstract syntax, in which translations between terms over different types can be specified via the associated category-theoretic iteration operator as an initial morphism. Our definitions ensure that translations specified via initiality are type-safe, i.e. compatible with the typing in the source and target language in the obvious sense.Our main example is given via the propositions-as-types paradigm: we specify propositions and inference rules of classical and intuitionistic propositional logics through their respective typed signatures. Afterwards we use the category-theoretic iteration operator to specify a double negation translation from the former to the latter.A second example is given by the signature of PCF. For this particular case, we formalize the theorem in the proof assistant Coq. Afterwards we specify, via the category-theoretic iteration operator, translations from PCF to the untyped lambda calculus.

Initiality for Typed Syntax and Semantics

“…1-signatures are defined in [2]. There, we associate a category Rep(S, Σ) of representations to any 1-signature (S, Σ), and show that the types and terms freely generated by (S, Σ) form an initial object in this category.…”

“…There, we define a notion of 2-signature for untyped syntax with semantics in form of reduction rules and show that its associated category of representations has an initial object. In the present work, we define inequations over typed 1-signatures as defined in [2]. Given a set A of inequations over a 1-signature (S, Σ), the representations of (S, Σ) that satisfy each inequation of A, form a full subcategory of Rep ∆ (S, Σ), which we call the category of representations of (S, Σ, A).…”

“…Two important constructions on modules over monads of [2], derivation and fibre modules, carry over to modules over monads on ∆ T . Intuitively, derivation corresponds to considering terms in an extended context, whereas the fibre corresponds to picking terms of a specific object type.…”

“…We combine the techniques of [2] and [1] in order to obtain an initiality result for simple type systems with reductions on the term level. As an example, we specify, via the iteration principle stemming from the universal property, a semantically faithful translation from PCF with its usual reduction relation to the untyped lambda calculus with beta reduction.…”

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We give an algebraic characterization of the syntax and semantics of a class of simply-typed languages, such as the language PCF: we characterize simply-typed binding syntax equipped with reduction rules via a universal property, namely as the initial object of some category. For this purpose, we employ techniques developed in two previous works: in [2], we model syntactic translations between languages over different sets of types as initial morphisms in a category of models. In [1], we characterize untyped syntax with reduction rules as initial object in a category of models.

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