Temporal Algebra

We develop temporal logic from the theory of complete lattices, Galois connections, and xed points. In particular, we prove that all seventeen axioms of Manna and Pnueli's sound and complete proof system for linear temporal logic can be derived from just two postulates, namely that (+; e ?) is a Galois connection and that (?; +) is a perfect Galois connection. For the branching time logic CTL we obtain a similar result. A surprising insight is that most of the theory can be developed without the use of negation. In eeect, we are studying intuitionistic temporal logic. Several examples of such structures occuring in computer science are given. Finally, we show temporal algebra at work in the derivation of a simple graph-theoretic algorithm. This paper is tutorial in style and there are no diicult technical results. To the experts in temporal logics, we hope to convey the simplicity and beauty of algebraic reasoning as opposed to the machine-orientedness of logical deduction. To those familiar with the calculational approach to programming, we want to show that their methods extend easily and smoothly to temporal reasoning. For anybody else, this text may serve as a gentle introduction to both areas.