14 research outputs found
Algorithmic correspondence and completeness in modal logic. I. The core algorithm SQEMA
Modal formulae express monadic second-order properties on Kripke frames, but
in many important cases these have first-order equivalents. Computing such
equivalents is important for both logical and computational reasons. On the
other hand, canonicity of modal formulae is important, too, because it implies
frame-completeness of logics axiomatized with canonical formulae.
Computing a first-order equivalent of a modal formula amounts to elimination
of second-order quantifiers. Two algorithms have been developed for
second-order quantifier elimination: SCAN, based on constraint resolution, and
DLS, based on a logical equivalence established by Ackermann.
In this paper we introduce a new algorithm, SQEMA, for computing first-order
equivalents (using a modal version of Ackermann's lemma) and, moreover, for
proving canonicity of modal formulae. Unlike SCAN and DLS, it works directly on
modal formulae, thus avoiding Skolemization and the subsequent problem of
unskolemization. We present the core algorithm and illustrate it with some
examples. We then prove its correctness and the canonicity of all formulae on
which the algorithm succeeds. We show that it succeeds not only on all
Sahlqvist formulae, but also on the larger class of inductive formulae,
introduced in our earlier papers. Thus, we develop a purely algorithmic
approach to proving canonical completeness in modal logic and, in particular,
establish one of the most general completeness results in modal logic so far.Comment: 26 pages, no figures, to appear in the Logical Methods in Computer
Scienc
Elementary Canonical Formulae: A Survey on Syntactic, Algorithmic, and Modeltheoretic Aspects
In terms of validity in Kripke frames, a modal formula expresses a universal monadic second-order condition. Those modal formulae which are equivalent to first-order conditions are called elementary. Modal formulae which have a certain persistence property which implies their validity in all canonical frames of modal logics axiomatized with them, and therefore their completeness, are called canonical. This is a survey of a recent and ongoing study of the class of elementary and canonical modal formulae. We summarize main ideas and results, and outline further research perspectives
Algorithmic correspondence and completeness in modal logic
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
Taming "McKinsey-like" formula: An Extended Correspondence and Completeness Theory for Hybrid Logic H(@)
In the present article, we extend the fragment of inductive formulas for the
hybrid language L(@) in [8] including a McKinsey-like formula, and show that
every formula in the extended class has a first-order correspondent, by
modifying the algorithm hybrid-ALBA in [8]. We also identify a subclass of this
extended inductive fragment, namely the extended skeletal formulas, which
extend the class of skeletal formulas in [8], each formula in which axiomatize
a complete hybrid logic. Our proof method here is proof-theoretic, following
[10, 19] and [3, Chapter 14], in contrast to the algebraic proof in [8].Comment: arXiv admin note: text overlap with arXiv:2207.0128
Elementary canonical formulae: extending Sahlqvist’s theorem
AbstractWe generalize and extend the class of Sahlqvist formulae in arbitrary polyadic modal languages, to the class of so called inductive formulae. To introduce them we use a representation of modal polyadic languages in a combinatorial style and thus, in particular, develop what we believe to be a better syntactic approach to elementary canonical formulae altogether. By generalizing the method of minimal valuations à la Sahlqvist–van Benthem and the topological approach of Sambin and Vaccaro we prove that all inductive formulae are elementary canonical and thus extend Sahlqvist’s theorem over them. In particular, we give a simple example of an inductive formula which is not frame-equivalent to any Sahlqvist formula. Then, after a deeper analysis of the inductive formulae as set-theoretic operators in descriptive and Kripke frames, we establish a somewhat stronger model-theoretic characterization of these formulae in terms of a suitable equivalence to syntactically simpler formulae (‘primitive regular formulae’) in the extension of the language with reversive modalities. Lastly, we study and characterize the elementary canonical formulae in reversive languages with nominals, where the relevant notion of persistence is with respect to discrete frames