Monads in functional programming: a formalized proof of their equivalence with Kleisli triples
“Being a language, mathematics may be used not only to inform but also, among other things, to seduce.”
“There is no branch of mathematics, however abstract, which may not some day be applied to phenomena of the real world.”
─Nikolai Ivanovich Lobachevsky
“Category theory is a relatively young branch of pure mathematics that most computer scientists would consider esoteric”
─Benjamin C. Pierce
This repository contains my work for obtain my computer scientist degree. My thesis is here.
My work consisted in to formalize basic concepts of Category Theory in Coq, and realize a formal verification of a proof of equivalence between monads and Kleisli triples.
A few history
The notion of monad was invented by Roger Godement in 1958 under the name "standard construction." In the 1960s and 1970s, many people used the name "triple." The now standard term "monad" is due to Saunders Mac Lane. (See Monad)
Eugenio Moggi was the first to explicitly link the monad of category theory to functional programming. In earlier work, several computer scientists had advanced using category theory to provide semantics for the lambda calculus. Several others popularized and built on this idea, including Philip Wadler and Simon Peyton Jones, both of whom were involved in the specification of Haskell. At first, programming with monads was largely confined to Haskell and its derivatives, but as functional programming has influenced other paradigms, many languages have incorporated a monad pattern (in spirit if not in name). Formulations now exist in Scheme, Perl, Python, Racket, Clojure, Scala, F#, and have also been considered for a new ML standard. (See Monad (functional programming))
The more common definition for a monad in functional programming is actually based on a Kleisli triple rather than category-theory's standard definition. The two constructs turn out to be mathematically equivalent. (See Kleisli Triple)
In 1991, Moggi give us a small and informal proof of the mathematical equivalence between monads and Kleisli triples. (See Moggi, E. (1991). Notions of computation and monads). In 2007, Gammon shows a detailed and extended formal proof of the mathematical equivalence between monads and Kleisli triples. (See Gammon, S. C. (2007). Notions of category theory in functional programming (Doctoral dissertation, University of British Columbia).) Finally, in 2016, I formalized Gammon's proof in the proof assistant Coq.
I used CoqIde 8.6 to run my script.
I use type classes to define category theory concepts (see A Gentle Introduction to Type Classes and Relations in Coq) and inspiration given from Matthieu Sozeau's work.
My script contains:
* Category definition * Category examples * Functor definition * Identity functor definition * Functors composition definition * Natural Transformation definition * Monad definition * Kleisli triple definition * Formalized proof of the mathematical equivalence between monads and Kleisli triples