The Computational Complexity of Hybrid Temporal Logics
Areces, C.; Blackburn, P.; Marx, M.
In their simplest form, hybrid languages are propositional modal
languages which can refer to states. They were introduced by Arthur
Prior, the inventor of tense logic, and played an important role in
his work: because they make reference to specific times possible, they
remove the most serious obstacle to developing modal approaches to
temporal representation and reasoning. However very little is known
about the computational complexity of hybrid temporal logics.
In this paper we analyze the complexity of the satisfiability problem
of a number of hybrid temporal logics: the basic hybrid language over
transitive frames; nominal tense logic over transitive frames, strict
total orders, and transitive trees; nominal _Until_ logic and
referential interval logic. We discuss the effects of including
nominals, the @ operator, the somewhere modality \udi, and the
difference operator \diff. Adding nominals to tense logic leads for
several frame--classes to an increase in complexity of the
satisfiability problem from PSPACE to EXPTIME. On transitive trees,
however, the satisfiability problem for this language can be decided
in PSPACE.
Along the way we make a detour through hybrid propositional dynamic
logic: we establish upper bounds for a number of temporal logics by
generalizing results due to Passy and Tinchev and De Giacomo. We
conclude with some remarks on the relevance of our results to
description logic, and draw attention to the utility of the spypoint
technique for proving upper and lower bounds.
Keyword(s): Hybrid Logic, Computational Complexity, Modal and Temporal
Logics, Propositional Dynamic Logic, Description Logic, At Operator,
Difference Operator, Universal Modality.