ISSN 0032-9460, Problems of Information Transmission, 2007, Vol. 43, No. 3, pp. 255–262.
Pleiades Publishing, Inc., 2007.
Original Russian Text
I.B. Shapirovsky, 2007, published in Problemy Peredachi Informatsii, 2007, Vol. 43, No. 3, pp. 97–104.
Modal Logics of Some Geometrical Structures
I. B. Shapirovsky
Kharkevich Institute for Information Transmission Problems, RAS, Moscow
Received June 19, 2007
Abstract—We study modal logics of regions in a real space ordered by the inclusion and
compact inclusion relations. For various systems of regions, we propose complete ﬁnite modal
axiomatizations; the described logics are ﬁnitely approximable and PSPACE-complete.
One of intensively developing ﬁelds in theoretical computer science is the study of the so-called
spatial logics, used for the description of geometrical and topological relations and operations, and in
particular, investigation of semantic and algorithmic properties of these logics. This development is
due to, one the one hand, applied problems (such as pattern recognition or design of geoinformation
systems) and on the other hand, issues of axiomatization of various areas of mathematics and
mathematical physics (for instance, topology and theory of relativity).
In particular, logical calculi that axiomatize relations between regions (i.e., sets of certain type in
a topological space) are actively being investigated nowadays. For these purposes, various systems
have been proposed in classical ﬁrst-order languages, as well as in languages of modal logic.
As examples of relations between regions, one may consider ⊆ (inclusion), (compact inclusion:
A B ⇔ CA ⊆ IB,whereI and C are the interior and the closure operators, respectively),
their converse ⊇ and , (disconnectedness: A B ⇔ A ∩ B = ∅), (partial overlapping :
A B ⇔ IA ∩ IB = ∅ ∧ A B ∧ B A), etc.
If we consider a set of regions with relations as a Kripke-frame, then the question on its modal
logic arises. Recently, it was shown in  that modal logics of regions with several relations can
be undecidable or even not recursively enumerable (in , among others, modal logics of regular
sets ordered by the eight Egenhofer-Franzosa relations RCC-8 [2,3] were considered). In  it was
shown that there is a number of monomodal logics of regions which are not ﬁnitely axiomatiz-
able. This leads to the problem to ﬁnd expressive modal systems with “good” properties (ﬁnite
axiomatizability, ﬁnite approximability, appropriate computational complexity).
In this paper we consider frames of the form (W,R), where R is one of the relations ⊆, , ⊇,or
and W is a nonempty set of n-regions;byann-region we mean the closure of a domain in R
Logics of these frames were described in  for the cases where W consists of all (convex) n-regions,
n-dimensional balls, or bricks. In this paper we generalize this result: we deﬁne a special class of
saturated sets of regions and describe the logics of (W,R) for any saturated W; in particular, any
nonempty set of convex regions closed under homotheties is saturated (Theorems 2 and 3). We also
axiomatize logics of some bimodal frames with the additional universal relation W×W (Theorem 5).
All described logics are ﬁnitely axiomatizable, ﬁnitely approximable, and PSPACE-complete.
Supported in part by the Russian Foundation for Basic Research, project no. 06-01-72555, and RFBR-
NWO, project no. 047.011.2004.04.