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

Isomorphic Data Encodings in Haskell and their Generalization to Hylomorphisms on Hereditarily Finite Data Types Paul Tarau Department of Computer Science and Engineering University of North Texas E-mail: tarau@cs.unt.edu Abstract. This paper is an exploration in a functional programming framework of isomorphisms between elementary data types (natural num- bers, sets, multisets, finite functions, permutations binary decision di- agrams, graphs, hypergraphs, parenthesis languages, dyadic rationals, primes, DNA sequences etc.) and their extension to hereditarily finite universes through hylomorphisms derived from ranking/unranking and pairing/unpairing operations. An embedded higher order combinator language provides any-to-any en- codings 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 combina- torial 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 http://logic.csci.unt.edu/tarau/ research/2008/fISO.zip. A short, 5 page version of the paper, published as [1] describes the idea of organizing various data transformations as encodings to sequences of natural numbers and gives a few examples of hylomorphisms that lift the encodings to related hereditarily finite universes. Keywords: Haskell data representations, data type isomorphisms, declar- ative combinatorics, computational mathematics, Ackermann encoding, G¨odel numberings, arithmetization, ranking/unranking, hereditarily fi- nite sets, functions and permutations, encodings of binary decision dia- grams, dyadic rationals, DNA encodings 1 Introduction Analogical/metaphorical thinking routinely shifts entities and operations from a field to another hoping to uncover similarities in representation or use [2]. arXiv:0808.2953v4 [cs.PL] 19 Jan 2009 Compilers convert programs from human centered to machine centered rep- resentations - sometime reversibly. Complexity classes are defined through compilation with limited resources (time or space) to similar problems [3, 4]. Mathematical theories often borrow proof patterns and reasoning techniques across close and sometime not so close fields. A relatively small number of universal data types are used as basic building blocks in programming languages and their runtime interpreters, correspond- ing to a few well tested mathematical abstractions like sets, functions, graphs, groups, categories etc. A less obvious leap is that if heterogeneous objects can be seen in some way as isomorphic, then we can share them and compress the underlying informa- tional universe by collapsing isomorphic encodings of data or programs whenever possible. Sharing heterogeneous data objects faces two problems: – some form of equivalence needs to be proven between two objects A and B before A can replace B in a data structure, a possibly tedious and error prone task – the fast growing diversity of data types makes harder and harder to recognize sharing opportunities. Besides, this rises the question: what guaranties do we have that sharing across heterogeneous data types is useful and safe? The techniques introduced in this paper provide a generic solution to these problems, through isomorphic mappings between heterogeneous data types, such that unified internal representations make equivalence checking and sharing pos- sible. The added benefit of these “shapeshifting” data types is that the functors transporting their data content will also transport their operations, resulting in shortcuts that provide, for free, implementations of interesting algorithms. The simplest instance is the case of isomorphisms – reversible mappings that also transport operations. In their simplest form such isomorphisms show up as en- codings to some simpler and easier to manipulate representation, for instance natural numbers. Such encodings can be traced back to G¨odel numberings [5, 6] associated to formulae, but a wide diversity of common computer operations, ranging from data compression and serialization to wireless data transmissions and crypto- graphic codes qualify. Encodings between data types provide a variety of services ranging from free iterators and random objects to data compression and succinct representations. Tasks like serialization and persistence are facilitated by simplification of reading or writing operations without the need of special purpose parsers. Sensitivity to internal data representation format or size limitations can be circumvented without extra programming effort. 2 An Embedded Data Transformation Language We will start by designing an embedded transformation la

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