Contextual Modal Type Theory with Polymorphic Contexts

Abstract

Modal types—types that are derived from proof systems of modal logic—have been studied as theoretical foundations of metaprogramming, where program code is manipulated as first-class values. In modal type systems, modality corresponds to a type constructor for code types and controls free variables and their types in code values. Nanevski et al. have proposed contextual modal type theory, which has modal types with fine-grained information on free variables: modal types are explicitly indexed by contexts—the types of all free variables in code values.This paper presents $$\lambda _{\forall []}$$ λ ∀ [ ] , a novel extension of contextual modal type theory with parametric polymorphism over contexts. Such an extension has been studied in the literature but, unlike earlier proposals, $$\lambda _{\forall []}$$ λ ∀ [ ] is more general in that it allows multiple occurrence of context variables in a single context. We formalize $$\lambda _{\forall []}$$ λ ∀ [ ] with its type system and operational semantics given by $$\beta $$ β -reduction and prove its basic properties including subject reduction, strong normalization, and confluence. Moreover, to demonstrate the expressive power of polymorphic contexts, we show a type-preserving embedding from a two-level fragment of Davies’ $$\lambda _{\bigcirc }$$ λ ◯ , which is based on linear-time temporal logic, to $$\lambda _{\forall []}$$ λ ∀ [ ] .

Linear Contextual Metaprogramming and Session Types

“…Types include the unit ( * ) linear type, the linear arrow type T → U, and the linear boxed contextual type τ. A contextual type τ of the form (τ ⊢ T ) represents code of type T , parameterized on variables typed by a list of contextual types τ, called contexts [8,11].…”

“…Metaprogramming manipulates code in order to generate and evaluate code at runtime, allowing in particular to explore the availability of certain arguments to functions in order to save computational effort. In this paper we are interested in programming languages where the code produced is typed by construction and where code may refer to a context providing types for the free variables, commonly known as contextual typing [8,11,12,13]. On an orthogonal axis, session types have been advocated as a means to discipline concurrent computations, by accurately describing protocols for the channels used to exchange messages between processes [1,5,6,7,15,16,17].…”

“…An alternative starting point would have been the Fitch-or Kripke-style formulation, providing for the Lisp quote/unquote, where typing contexts are viewed as stacks modeling the different stages of computation [11]. It seemed to us that the let-box approach would simplify the extension to the linear setting, and then to session types.…”