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Datatype defining rewrite systems for naturals and integers
arXiv - CS - Logic in Computer Science Pub Date : 2016-08-22 , DOI: arxiv-1608.06212
Jan A. Bergstra; Alban Ponse

A datatype defining rewrite system (DDRS) is a ground-complete term rewriting system, intended to be used for the specification of datatypes. First we define two concise DDRSes for the ring of integers, each comprising twelve rewrite rules, and prove their ground-completeness. Then we introduce natural number and integer arithmetic specified according to unary view, that is, arithmetic based on a postfix unary append constructor (a form of tallying) or on the successor function. Then we specify arithmetic based on two other views: binary and decimal notation. The binary and decimal view have as their characteristic that each normal form resembles common number notation, that is, either a digit, or a string of digits without leading zero, or the negated versions of the latter. Integer arithmetic in binary and decimal notation is based on (postfix) digit append functions. For each view we define a DDRS, and in each case the resulting datatype is a canonical term algebra that extends a corresponding canonical term algebra for natural numbers. Then, for each view, we consider an alternative DDRS based on tree constructors that yield comparable normal forms, which for that view admits expressions that are algorithmically more involved. These DDRSes are incorporated because they are closer to existing literature. For all DDRSes considered, ground-completeness is either proved, or references to a proof are provided.
更新日期:2020-01-14

 

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