Abstract:
Abstract
This thesis takes an algorithmic perspective on the correspondence between modal and hybrid
logics on the one hand, and first-order logic on the other. The canonicity of formulae, and by
implication the completeness of logics, is simultaneously treated.
Modal formulae define second-order conditions on frames which, in some cases, are equiv-
alently reducible to first-order conditions. Modal formulae for which the latter is possible
are called elementary. As is well known, it is algorithmically undecidable whether a given
modal formula defines a first-order frame condition or not. Hence, any attempt at delineating
the class of elementary modal formulae by means of a decidable criterium can only consti-
tute an approximation of this class. Syntactically specified such approximations include the
classes of Sahlqvist and inductive formulae. The approximations we consider take the form
of algorithms.
We develop an algorithm called SQEMA, which computes first-order frame equivalents for
modal formulae, by first transforming them into pure formulae in a reversive hybrid language.
It is shown that this algorithm subsumes the classes of Sahlqvist and inductive formulae, and
that all formulae on which it succeeds are d-persistent (canonical), and hence axiomatize
complete normal modal logics.
SQEMA is extended to polyadic languages, and it is shown that this extension succeeds
on all polyadic inductive formulae. The canonicity result is also transferred.
SQEMA is next extended to hybrid languages. Persistence results with respect to discrete
general frames are obtained for certain of these extensions. The notion of persistence with
respect to strongly descriptive general frames is investigated, and some syntactic sufficient
conditions for such persistence are obtained. SQEMA is adapted to guarantee the persistence
with respect to strongly descriptive frames of the hybrid formulae on which it succeeds, and
hence the completeness of the hybrid logics axiomatized with these formulae. New syntactic
classes of elementary and canonical hybrid formulae are obtained.
Semantic extensions of SQEMA are obtained by replacing the syntactic criterium of nega-
tive/positive polarity, used to determine the applicability of a certain transformation rule, by
its semantic correlate—monotonicity. In order to guarantee the canonicity of the formulae on
which the thus extended algorithm succeeds, syntactically correct equivalents for monotone
formulae are needed. Different version of Lyndon’s monotonicity theorem, which guarantee
the existence of these equivalents, are proved. Constructive versions of these theorems are
also obtained by means of techniques based on bisimulation quantifiers.
Via the standard second-order translation, the modal elementarity problem can be at-
tacked with any second-order quantifier elimination algorithm. Our treatment of this ap-
proach takes the form of a study of the DLS-algorithm. We partially characterize the for-
mulae on which DLS succeeds in terms of syntactic criteria. It is shown that DLS succeeds
in reducing all Sahlqvist and inductive formulae, and that all modal formulae in a single
propositional variable on which it succeeds are canonical.