Specifying Real Numbers in CASL

We present a weak theory BasicReal of the real numbers in the rst order speciication language Casl. The aim is to provide a datatype for practical purposes, including the central notions and results of basic analysis. BasicReal captures for instance e and ; as well as the trigonometric and other standard functions. Concepts such as continuity, diierentiation and integration are shown to be deenable and tractable in this setting; Newton's Method is presented as an example of a numerical application. Finally, we provide a proper connection between the speciied datatype BasicReal and speciications of the real numbers in higher order logic and various set theories. c Springer-Verlag LNCS 1827 There seem to be two obstacles to the speciication of the real numbers as a datatype within rst order logic: The real numbers form an uncountable set, which may lead to the conclusion that they cannot be speciied in rst order logic due to the theorem of LL owenheim and Skolem. More importantly, the usual axiomatizations and constructions of the real numbers use higher-order concepts. As a consequence, rst order algebraic speciication languages up to now either do not provide such a datatype at all or ooer only a speciication of a oating point system as an approximation. Both solutions are unsatisfactory: The rst leaves a great deal of applications out of scope. The second is of course better, but nevertheless problematic: It forces the speciier to deal with implemention details even on the level of a requirement speciication, and the development of numerical algorithms cannot be described. In this paper we present an axiomatization BasicReal of a weak theory of the real numbers in the rst order speciication language Casl, the Common Algebraic Speciication Language CoF99]. Casl has been designed by CoFI, the international Common Framework Initiative for algebraic speciication and development CoF]. It is based on a critical selection of concepts and constructs that have already proven successful in various algebraic speciication frameworks. Its features include rst-order predicate logic, partial functions, and subsorts as well as structured and architectural speciications. The speciication presented here can easily be translated into any speciication language that supports partial subsorted rst order logic with sort generation constraints.