We develop temporal logic from the theory of complete lattices,
Galois connections and
fixed 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 ([oplus ], &[ominus ]tilde;) is a Galois connection and that
([ominus ], [oplus ])
is a perfect Galois connection. We also obtain a similar result for
the branching time logic CTL.
A surprising insight is that most of the theory can be developed without
the use of
negation. In effect, we are studying intuitionistic temporal logic. Several
examples of such
structures occurring in computer science are given. Finally, we show temporal
work in the derivation of a simple graph-theoretic algorithm.
This paper is tutorial in style and there are no difficult technical
To the experts in
temporal logics, we hope to convey the simplicity and beauty of algebraic
opposed to the machine-orientedness of logical deduction. To those familiar
calculational approach to programming, we want to show that their methods
and smoothly to temporal reasoning. For anybody else, this text may serve
as a gentle
introduction to both areas.