Monadic Expressions and their Derivatives [extended version]

TL;DR

This paper introduces a monadic framework for derivatives of regular expressions, generalizing data structures and extending automaton construction, with applications in parsing and weighted operations, demonstrated through a Haskell implementation.

Contribution

It presents a novel monadic interpretation of derivatives, generalizes automaton construction for extended expressions, and applies category theory to unify these concepts.

Findings

A new derivation technique based on the graded module monad.

Generalization of automaton construction for extended expressions.

Implementation of the framework in Haskell with a web interface.

Abstract

We propose another interpretation of well-known derivatives computations from regular expressions, due to Brzozowski, Antimirov or Lombardy and Sakarovitch, in order to abstract the underlying data structures (e.g. sets or linear combinations) using the notion of monad. As an example of this generalization advantage, we first introduce a new derivation technique based on the graded module monad and then show an application of this technique to generalize the parsing of expression with capture groups and back references. We also extend operators defining expressions to any n-ary functions over value sets, such as classical operations (like negation or intersection for Boolean weights) or more exotic ones (like algebraic mean for rational weights). Moreover, we present how to compute a (non-necessarily finite) automaton from such an extended expression, using the Colcombet and Petrisan categorical definition of automata. These category theory concepts allow us to perform this construction in a unified way, whatever the underlying monad. Finally, to illustrate our work, we present a Haskell implementation of these notions using advanced techniques of functional programming, and we provide a web interface to manipulate concrete examples.