A Functional Hitchhikers Guide to Hereditarily Finite Sets, Ackermann Encodings and Pairing Functions

A Functional Hitchhiker’s Guide to Hereditarily Finite Sets, Ackermann Encodings and Pairing Functions – unpublished draft – Paul Tarau Department of Computer Science and Engineering University of North Texas tarau@cs.unt.edu Abstract The paper is organized as a self-contained literate Haskell program that implements elements of an executable fi- nite set theory with focus on combinatorial generation and arithmetic encodings. The code, tested under GHC 6.6.1, is available at http://logic.csci.unt.edu/tarau/ research/2008/fSET.zip. We introduce ranking and unranking functions generaliz- ing Ackermann’s encoding to the universe of Hereditarily Fi- nite Sets with Urelements. Then we build a lazy enumerator for Hereditarily Finite Sets with Urelements that matches the unranking function provided by the inverse of Ackermann’s encoding and we describe functors between them resulting in arithmetic encodings for powersets, hypergraphs, ordinals and choice functions. After implementing a digraph repre- sentation of Hereditarily Finite Sets we define decoration functions that can recover well-founded sets from encodings of their associated acyclic digraphs. We conclude with an en- coding of arbitrary digraphs and discuss a concept of duality induced by the set membership relation. Keywords hereditarily finite sets, ranking and unrank- ing functions, executable set theory, arithmetic encodings, Haskell data representations, functional programming and computational mathematics 1. Introduction While the Universe of Hereditarily Finite Sets is best known as a model of the Zermelo-Fraenkel Set theory with the Axiom of Infinity replaced by its negation (Takahashi 1976; Meir et al. 1983), it has been the object of renewed practical Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. WXYZ ’08 date, City. Copyright c⃝2008 ACM [to be supplied]...$5.00 interest in various fields, from representing structured data in databases (Leontjev and Sazonov 2000) to reasoning with sets and set constraints in a Logic Programming framework (Dovier et al. 2000; Piazza and Policriti 2004; Dovier et al. 2001). The Universe of Hereditarily Finite Sets is built from the empty set (or a set of Urelements) by successively applying powerset and set union operations. A surprising bijection, discovered by Wilhelm Ackermann in 1937 (Ackermann 1937; Abian and Lamacchia 1978; Kaye and Wong 2007) from Hereditarily Finite Sets to Natural Numbers, was the original trigger for our work on building in a mathematically elegant programming language, a concise and executable hereditarily finite set theory. The arbitrary size of the data objects brought in the need for arbitrary length integers. The focus on potentially infinite enumerations brought in the need for lazy evaluation. These have made Haskell a natural choice. We will describe our constructs in a subset of Haskell (Peyton Jones 2002, 2003a,b) seen as a concrete syntax for a generic lambda calculus based functional language1. We will only make the assumptions that non-strict func- tions (higher order included), with call-by-need evaluation and arbitrary length integers are available in the language. While our code will conform Haskell’s type system, we will do that without any type declarations, by ensuring that the types of our functions are all inferred. This increases chances that the code can be ported, through simple syntax transfor- mations, to any programming language that implements our basic assumptions. The paper is organized as follows: section 2 introduces the reader to combinatorial generation with help of a bit- string example, section 3 introduces Ackermann’s encod- 1 As a courtesy to the reader wondering about the title, the author confesses being a hitchhiker in the world of functional programming, coming from the not so distant galaxy of logic programming but still confused by recent hitchhiking trips in the exotic worlds of logic synthesis, foundations of mathematics, natural language processing, conversational agents and virtual reality. And not being afraid to go boldly where . . . a few others have already been before. arXiv:0808.0754v1 [cs.MS] 6 Aug 2008 ing in the more general case when urelements are present and shows an encoding for hypergraphs as a particular case. Section 4 gives examples of transporting common set and natural number operations from one side to the other. After discussing some classic pairing functions, section 5 intro- duces new pairing/unpairing on natural numbers. Section 6 discusses graph representations and decoration functions on Hereditarily Finite Sets (6.1), and provides encodings for di- rected acycli

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