Declarative Combinatorics: Isomorphisms, Hylomorphisms and Hereditarily Finite Data Types in Haskell

Abstract: This paper is an exploration in a functional programming framework of {\em isomorphisms} between elementary data types (natural numbers, sets, multisets, finite functions, permutations binary decision diagrams, graphs, hypergraphs, parenthesis languages, dyadic rationals, primes, DNA sequences etc.) and their extension to hereditarily finite universes through {\em hylomorphisms} derived from {\em ranking/unranking} and {\em pairing/unpairing} operations. An embedded higher order {\em combinator language} provides any-to-any encodings automatically. Besides applications to experimental mathematics, a few examples of ``free algorithms'' obtained by transferring operations between data types are shown. Other applications range from stream iterators on combinatorial objects to self-delimiting codes, succinct data representations and generation of random instances. The paper covers 59 data types and, through the use of the embedded combinator language, provides 3540 distinct bijective transformations between them. The self-contained source code of the paper, as generated from a literate Haskell program, is available at \url{http://logic.csci.unt.edu/tarau/research/2008/fISO.zip}. {\bf Keywords}: Haskell data representations, data type isomorphisms, declarative combinatorics, computational mathematics, Ackermann encoding, G\"{o}del numberings, arithmetization, ranking/unranking, hereditarily finite sets, functions and permutations, encodings of binary decision diagrams, dyadic rationals, DNA encodings