Bounded Linear Types in a Resource Semiring

Abstract

Abstract. Bounded linear types have proved to be useful for automated resource analysis and control in functional programming languages. In this paper we introduce a bounded linear typing discipline on a general notion of resource which can be modeled in a semiring. For this type system we provide both a general type-inference procedure, parameterized by the decision procedure of the semiring equational theory, and a (coherent) categorical semantics. This could be a useful type-theoretic and denotational framework for resource-sensitive compilation, and it represents a generalization of several existing type systems. As a nontrivial instance, motivated by hardware compilation, we present a complex new application to calculating and controlling timing of execution in a (recursion-free) higher-order functional programming language with local store.

Effect Systems Revisited—Control-Flow Algebra and Semantics

“…Coeffect structures tend to comprise some form of resource semiring with a semiring-graded comonad in the semantics [8].…”

“…Conversely, start with an effect system, say the one in the introduction for music. An effect-graded semantics then requires semantic operations annotated by each of the effect operations used, 8 with a structure that reflects that of the effect system. In this way, graded approaches aid a kind of co-design process between analysis and semantics.…”

Combining effects and coeffects via grading

“…Dual to effect systems, for fine-grained effect information, are coeffect systems for resource requirements, which have been recently introduced [8,15,36,[39][40][41]. For example, the reuse bounds in Bounded Linear Logic are an instance of a coeffect system which precisely tracks the usage requirements on variables.…”

“…A natural way to recover this information is by adding it explicitly to the type system. We can do this in a uniform way for a large class of coeffectful computations by following Brunel et al [8], Ghica and Smith [15], Petricek et al [41] and using a coeffect-graded comonad which provides a type constructor Dr indexed by elements r ∈ R of a preordered semiring (R, ≤, 0, +, 1, * ). In the concrete case of resource analysis we take the standard natural numbers semiring N, which gives an indexed !…”

“…Coeffects Similarly, we follow the approach by Petricek et al [40], Brunel et al [8], and Ghica and Smith [15], identifying coeffects with the elements of a semiring.…”