Ranking and Unranking of Hereditarily Finite Functions and Permutations

Details Ranking and Unranking of Hereditarily Finite Functions and Permutations Ranking and Unranking of Hereditarily Finite Functions and Permutations

2008-08-05

arxiv.org/abs/0808.0554v1

Computer Science

Logic in Computer Science

unpublished draft

Scientific paper

Prolog's ability to return multiple answers on backtracking provides an elegant mechanism to derive reversible encodings of combinatorial objects as Natural Numbers i.e. {\em ranking} and {\em unranking} functions. Starting from a generalization of Ackerman's encoding of Hereditarily Finite Sets with Urelements and a novel tupling/untupling operation, we derive encodings for Finite Functions and use them as building blocks for an executable theory of {\em Hereditarily Finite Functions}. The more difficult problem of {\em ranking} and {\em unranking} {\em Hereditarily Finite Permutations} is then tackled using Lehmer codes and factoradics. The paper is organized as a self-contained literate Prolog program available at \url{http://logic.csci.unt.edu/tarau/research/2008/pHFF.zip}

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