Resourceful Program Synthesis from Graded Linear Types

Jack Hughes,Dominic Orchard

doi:10.1007/978-3-030-68446-4_8

Abstract

Linear types provide a way to constrain programs by specifying that some values must be used exactly once. Recent work on graded modal types augments and refines this notion, enabling fine-grained, quantitative specification of data use in programs. The information provided by graded modal types appears to be useful for type-directed program synthesis, where these additional constraints can be used to prune the search space of candidate programs. We explore one of the major implementation challenges of a synthesis algorithm in this setting: how does the synthesis algorithm efficiently ensure that resource constraints are satisfied throughout program generation? We provide two solutions to this resource management problem, adapting Hodas and Miller’s input-output model of linear context management to a graded modal linear type theory. We evaluate the performance of both approaches via their implementation as a program synthesis tool for the programming language Granule, which provides linear and graded modal typing.

Highlights

Type-directed program synthesis is a long-studied technique rooted in automated theorem proving [29]
A type-directed synthesis algorithm can be constructed as an inversion of type checking, starting from a type and inductively synthesising well-typed subterms, pruning the search space via typing
Linearity introduces a new dimension to proof search and program synthesis: how do we inductively generate terms whilst pruning the search space of those which violate linearity? For example, consider the following inductive synthesis rule, mirroring Gentzen’s sequent calculus [15], which synthesises a term of type A ⊗ B : c The Author(s) 2021 M

Summary

Introduction

Type-directed program synthesis is a long-studied technique rooted in automated theorem proving [29]. A type-directed synthesis algorithm can be constructed as an inversion of type checking, starting from a type and inductively synthesising well-typed subterms, pruning the search space via typing. Via the Curry-Howard correspondence [21], we can view this as proof search in a corresponding logic, where the goal type is a proposition and the synthesised program is its proof. Automated proof search techniques have been previously adapted to linear logics, accounting for resource-sensitive reasoning [7–9,20,31]. Linearity introduces a new dimension to proof search and program synthesis: how do we inductively generate terms whilst pruning the search space of those which violate linearity? Consider the following inductive synthesis rule, mirroring Gentzen’s sequent calculus [15], which synthesises a term of type A ⊗ B : Linearity introduces a new dimension to proof search and program synthesis: how do we inductively generate terms whilst pruning the search space of those which violate linearity? For example, consider the following inductive synthesis rule, mirroring Gentzen’s sequent calculus [15], which synthesises a term of type A ⊗ B :

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