Proving Fixed Points

We propose a method to characterize the fixed points described in Tarski’s theorem for complete lattices. The method is deductive: the least and greatest fixed points are ”proved” in some inference system defined from deduction rules. We also apply the method to two other fixed point theorems, a generalization of Tarski’s theorem to chain-complete posets and Bourbaki-Witt’s theorem. Finally, we compare the method with the traditional iterative method resorting to ordinals and the original impredicative method used by Tarski. We are interested in fixed points of maps defined over partially ordered sets (abbreviated as posets). Consider Tarski’s fixed point theorem. Asserting the existence of fixed points under certain conditions, it is proved according to one of these two methods. In the impredicative method, the fixed points are characterized by a property (expressing extremality) using a quantification over a domain that includes the fixed point itself. In the iterative method, the fixed points are iteratively computed by using a transfinite induction. The impredicative method corresponds to a logical specification that is essentially not constructive. As for the iterative method, it seems to be more constructive, in that it corresponds to an iteration. However, first, it assumes the machinery of ordinals and second, it therefore requires a specific computation for limit ordinals, the next value being then computed from an infinite set of preceding values. Is there a proof method that not only is not impredicative, but also does not resort to ordinals? In this paper, we positively answer by proposing an alternative method, where fixed points are (inductively) proved in inference systems: this is a deductive method. It corresponds to a proof construction, using forward chaining for deduction rules, as used in logic programming, for instance in Datalog, a query language for deductive databases. The paper is organized as follows. After recalling Tarski’s theorem, the first section deals with inference systems, and their interpretations, either inductive or coinductive. In the second section, we introduce the deductive method, and apply it to different fixed point theorems: Tarski’s theorem, its generalization to chain-complete posets and Bourbaki-Witt’s theorem. Finally, we compare the deductive method with the iterative and impredicative methods. 1 Induction and Coinduction for Inference Systems Generally speaking, following Aczel’s classical presentation [1], an inference system over a set U of judgments is a set of deduction rules. A deduction rule is an ordered pair (A,c), where A⊆ U is the set of premises or antecedents and c ∈ U is the conclusion. A rule is usually written as follows: A . c Its intuitive interpretation is that the judgment c can be deduced from the set of judgments A. ∗An extended abstract of this article will appear in the proceedings of the 7th Workshop on Fixed Points in Computer Science (FICS 2010, Brno, Czech Republic, August 21-22 2010). In a common sense: we do not specifically refer here to constructive mathematics.