A unified implementation of automata and expression structures, and of the associated algorithms using enriched categories

In this document, we propose a description, via a Haskell implementation, of a generalization of the notion of regular expression allowing us to group the definitions and the methods of (tree or word) automata constructions over one generic structure, based on enriched category theory tools. We first recall several methods of conversion from expressions to automata, enlightening the similarities between the words and trees cases. We then produce an original study of the power of enriched category theory applied 1) to automata and expressions implementation, and 2) to the study of associated algorithms, using advanced concepts of functional programming, while simultaneously constructing a Haskell implementation of notions of enriched category theory and associated automata. More precisely, the Haskell implementation and the algebraic definition of the generic automaton structure are based on the following ideas: - enriched categories, enriched functors, enriched monads, etc. can be implemented in Haskell; - Type level programming can be used to properly encode function arity; - monoids (word structure) and operads (tree structure) can be encoded as monoid objects; - tree and word automata can be represented by the same algebraic structure, via enriched categories. This generalization leads to surprising remarks. As an example, some classical algorithms (determinization, completion, conversion from alternating to deterministic automaton) can be regrouped in only one function. We will then define a notion of generalized expressions based on the notion of monoidal tensor product. Haskell sources are available at: http://ludovicmignot.free.fr/HDR/src-HDR.zip