A graded dependent type system with a usage-aware semantics

Abstract

Graded Type Theory provides a mechanism to track and reason about resource usage in type systems. In this paper, we develop GraD, a novel version of such a graded dependent type system that includes functions, tensor products, additive sums, and a unit type. Since standard operational semantics is resource-agnostic, we develop a heap-based operational semantics and prove a soundness theorem that shows correct accounting of resource usage. Several useful properties, including the standard type soundness theorem, non-interference of irrelevant resources in computation and single pointer property for linear resources, can be derived from this theorem. We hope that our work will provide a base for integrating linearity, irrelevance and dependent types in practical programming languages like Haskell.

Graded Modal Dependent Type Theory

“…zero-unique) 1 = 0; (positivity) ∀r, s. r + s = 0 =⇒ r = 0 ∧ s = 0; (zero-product) ∀r, s. r * s = 0 =⇒ r = 0 ∨ s = 0.These axioms 4 ensure that a 0-grade in a quantitative semiring represents irrelevant variable use. This notion has recently been proved for computational use by Choudhury et al[13] via a heap-based semantics for grading (on computations) and the same result applies here. Conversely, in a quantitative semiring any grade other than 0 denotes relevance.…”

“…Choudhury et al [13] give a system closely related (but arguably simpler) to Qtt called GraD. One key difference is that rather than annotating type usage with 0, grades are simply ignored in types.…”

“…In the context of an implementation like Gerty, this amounts to using type signatures to restrict dataflow. This section shows the power of tracking dataflow in types via grades, going beyond Qtt [6] and GraD [13]. For 'quantitative' semirings, a 0 type-grade means that we can recover simply-typed reasoning (Section 3.3) and distinguish computational functions from type-parameter functions for parametric reasoning (Section 3.4), embedding a grade-restricted subset of Grtt into System F.…”

Languages With Potential

“…To implement proper type checking, the syntax of dλ-Amor would have to be changed such that every variable binder includes a type annotation. This is a heavy burden on the programmer: annotating binders with types is tedious, error prone, and generally uninteresting 13 . Instead, biλ-Amor adopts bidirectional type checking, a technique pioneered by Pierce and Turner [62] which trades off some of the generality of full type inference for added ergonomics over standard type checking.…”

“…Luckily, the equational theory of types is simpler than that of a simply-typed lambda calculus, since the type-level 13 As Pierce [61] notes: "The more interesting your types get, the less fun it is to write them down!" lambda in dλ-Amor λi ∶ S.τ ranges over index terms, not types.…”