J. A. Bergstra
Abstract
We give an equational specification of the field operations on the rational numbers under initial algebra semantics using just total field operations and 12 equations. A consequence of this specification is that 0−1 = 0, an interesting equation consistent with the ring axioms and many properties of division. The existence of an equational specification of the rationals without hidden functions was an open question. We also give an axiomatic examination of the divisibility operator, from which some interesting new axioms emerge along with equational specifications of algebras of rationals, including one with the modulus function. Finally, we state some open problems, including: Does there exist an equational specification of the field operations on the rationals without hidden functions that is a complete term rewriting system?
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Index Terms
The rational numbers as an abstract data type
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Reviews
Markus Wolf
Conditional equational specifications of abstract data types have the advantage that, in principle, the specification is executable via a conditional term rewriting system. Hence, it is important to understand what can be modeled with conditional equational specification. It is a long-standing question whether there exists a conditional equational specification without hidden function symbols, the initial algebra of which is the rational numbers. In this paper, the authors extend this idea to arrive at a solution: it is possible to give a conditional equational specification of the rational numbers without hidden functions. The introduction gives a historical perspective, with attention to the question of the specification of the rational numbers. Next, a short overview on algebraic specification that introduces the necessary axioms needed to model the rational numbers is presented. In this section, and the one that follows, the problem of finding a conditional equational specification is elaborated, and two algebraic specifications of the rational numbers without hidden functions are presented. With a clever use of Lagrange's theorem, an equation is constructed that shows that a certain axiom set together with this equation indeed models the rational numbers without hidden function symbols. However, the trick with Lagrange's theorem is a bit hard to swallow; hence, in the fourth section the authors provide a simpler specification involving a rather useful additional modulus operation. In the fifth section, the authors explore the algebra that results from the replacement of the general axiom for the multiplicative inverse with a weaker strictly conditional axiom. This section is more or less a byproduct, but exhibits a certain mathematical beauty. The conclusion lists a number of open problems. The paper is well written, detailed, and understandable, and makes a definite contribution to the theoretical underpinnings of computer science. The ingenuity of the paper lies in the use of Lagrange's theorem. All other techniques are fairly straightforward algebraic specifications, and one is left wondering why it took so long to arrive at this point. Online Computing Reviews Service
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