Abstract
Hybridization is a method invented by Arthur Prior for extending the expressive
power of modal languages. Although
developed in interesting ways by Robert Bull, and the Sofia school (notably,
George Gargov, Valentin Goranko, Solomon Passy
and Tinko Tinchev), the method remains little known. In our view this has
deprived temporal logic of a valuable tool. The
aim of the paper is to explain why hybridization is useful in temporal logic.
We make two major points, the first
technical, the second conceptual. Technically, we show that hybridization gives
rise to well-behaved logics that exhibit an
interesting synergy between modal and classical ideas. This synergy, obvious
for hybrid languages with full first-order
expressive strength, is demonstrated for three weaker local languages, all of
which are capable of defining the Until
operator; we provide simple minimal axiomatizations for all three systems, and
show that in a wide range of temporally
interesting cases, extended completeness results can be obtained automatically.
Conceptually, we argue that the idea of
sorted atomic symbols which underpins the hybrid enterprise can be developed
much further. To illustrate this, we discuss
the advantages and disadvantages of a simple hybrid language which can quantify
over paths. This is the original version of
a paper which was accepted for publication in a special issue of the Journal of
the IGPL on temporal logic. Unfortunately,
the length of the article meant that it had to be drastically cut, and only a
shorter version will appear. While the short
version covers one of the most elegant results (@-driven completeness results)
and is slightly more up to date in certain
respects, the long version is probably the most detailed discussion of the
completeness theory of local hybrid languages
around. The long version also contains many lengthy footnotes. These outline
the history of hybrid languages in
considerable detail, and contain many remarks on philosophical, methodological,
and technical issues.
