Monadic and Comonadic Aspects of Dependency Analysis

TL;DR

This paper introduces a new dependency calculus inspired by category theory that unifies monadic and comonadic approaches, addressing limitations of the Dependency Core Calculus (DCC) and $\\lambda^{\circ}$, and provides a novel correctness proof technique.

Contribution

It develops a categorical framework for dependency analysis that subsumes existing calculi and offers a new method for proving their correctness.

Findings

The new calculus unifies monadic and comonadic dependency analyses.

It explains the nonstandard rules of DCC using standard categorical concepts.

A novel technique for proving correctness of dependency analyses is introduced.

Abstract

Dependency analysis is vital to several applications in computer science. It lies at the essence of secure information flow analysis, binding-time analysis, etc. Various calculi have been proposed in the literature for analysing individual dependencies. Abadi et. al., by extending Moggi's monadic metalanguage, unified several of these calculi into the Dependency Core Calculus (DCC). DCC has served as a foundational framework for dependency analysis for the last two decades. However, in spite of its success, DCC has its limitations. First, the monadic bind rule of the calculus is nonstandard and relies upon an auxiliary protection judgement. Second, being of a monadic nature, the calculus cannot capture dependency analyses that possess a comonadic nature, for example, the binding-time calculus, $\lambda^{\circ}$, of Davies. In this paper, we address these limitations by designing an alternative dependency calculus that is inspired by standard ideas from category theory. Our calculus is both monadic and comonadic in nature and subsumes both DCC and $\lambda^{\circ}$. Our construction explains the nonstandard bind rule and the protection judgement of DCC in terms of standard categorical concepts. It also leads to a novel technique for proving correctness of dependency analysis. We use this technique to present alternative proofs of correctness for DCC and $\lambda^{\circ}$.