“Everything Is Everything” Revisited: Shapeshifting Data Types with Isomorphisms and Hylomorphisms (original) (raw)
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We introduce a logic programming framework for data type transformations based on isomorphisms between elementary data types (natural numbers, finite functions, sets and permutations, digraphs, hypergraphs, etc.) and automatically derived extensions to hereditarily finite universes through ranking/unranking operations. An embedded higher order combinator language provides anyto-any encodings automatically. Applications range from stream iterators on combinatorial objects and uniform generation of random instances to succinct data representations and serialization of Prolog terms.
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2008
Using specializations of unfold and fold on a generic tree data type we derive unranking and ranking functions providing natural number encodings for various Hereditarily Finite datatypes. In this context, we interpret unranking operations as instances of a generic anamorphism and ranking operations as instances of the corresponding catamorphism. Starting with Ackerman's Encoding from Hereditarily Finite Sets to Natural Numbers we define pairings and finite tuple encodings that provide building blocks for a theory of Hereditarily Finite Functions. The more difficult problem of ranking and unranking Hereditarily Finite Permutations is then tackled using Lehmer codes and factoradics. 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/fFUN.zip.
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