This paper was presented at the workshop on spatial and temporal reasoning, IJCAI93 , Chambery.
Modal and Non Modal Qualitative Spatial Logics A G Cohn3
Abstract. In this paper we review previous work on non modal spatial logics and explore a corresponding modal spatial logic. Furthermore we present an initial classi cation of kinds of spatially indexed propositions.
a pure rst order calculus of qualitative space and then we brie y sketch how a corresponding modal logic might be developed. We then use this formulation to present a classi cation of spatially indexed propositions.
1
2
Introduction
Although the use of interval temporal logics has been an active research area in AI for some time, the analogous development of ontologies for space and spatial logics based on regions has only relatively recently started to become a serious research activity (eg Pribbenow and Schlieder 1992, Narayan 1992). Various approaches have been promulgated; for example one can simply use Allen's (1983) temporal relations on each of the cartesian axes to specify the qualitative relationship between two regions (eg Hernandez 1990, Mukerjee and Joe 1990), but this has the disadvantage of either requiring knowledge about the absolute orientation of the two regions or their orientation relative to a xed viewpoint. For many applications one might only have local information available. Qualitative orientation representation and reasoning has however been explicitly investigated by Freksa(1992) and Mukerjee and Joe (1990) amongst others. Given a qualitative spatial knowledge there are various kinds of reasoning that have been investigated or are desirable. For example one can perform a qualitative spatial simulation (Cui, Cohn and Randell 1992), or reason spatially about physical systems such as a force pump (Randell, Cohn and Cui 1992a), or the heart (Gotts et al 1989). Another obvious application area is natural language understanding (see eg Vieu 1991, Aurnague 1991). Other work can be found in, for example (Pribbenow and Schlieder 1992, or Narayannan 1992). The structure of the rest of the paper is as follows: rst we brie y review our previous work in
Previous Work
The basic ontological entity we consider1 is a region; note that boundaries, lines and points are not regions.2. Regions in the theory support either a spatial or temporal interpretation, though we will only consider the spatial interpretation here. Informally, these regions may be thought to be potentially in nite in number, and any degree of connection between them is allowed in the intended model, from external contact to identity in terms of mutually shared parts. The formalism supports two or three dimensional interpretations (or higher dimensions!). The basic part of the formalism assumes one primitive dyadic relation: C(x; y) read as `x connects with y'. The relation C(x; y) is axiomatised to be re exive and symmetric. We can give a topological model to interpret the theory, namely that C(x; y) holds when the topological closures of regions x and y share a common point.3 Using C(x; y), a basic set of dyadic relations can be formally de ned: `DC(x; y)' (`x is disconnected from y'), `P(x; y)' (`x is a part of y'), `PP(x; y) (`x is a proper part of y'), `x = y' (`x is identical with y'), `O(x; y)' (`x overlaps y'), `DR(x; y)' (`x
1 Most of the material in this section can be found in (Randell, Cui and Cohn 1992) but we review the most relevant notions here for convenience and to make this paper more self contained. However, regrettably we do not have space for the logical de nitions of the predicates and functions described here. This work is based on that of Clarke(1981,1985). 2 However we believe that from a modelling point of view, such mathematical abstractions are not necessary and one can use special kinds of regions such as skins and atoms { see (Randell, Cui and Cohn 1992) 3 Division of AI, School of Computer Studies, Univ. of 3 In Clarke's theory and in our original theory (Randell Leeds, LS2 9JT, England;
[email protected]. The sup- and Cohn 1989, 1992), when two regions x and y connect, port of the SERC under grant no. GR/G36852 is gratefully they are said to share a point in common; thus the interacknowledged. This work has also been partially supported pretation of the connects relation here and in (Randell, Cui by a CEC ESPRIT basic research action, MEDLAR. and Cohn 1992) is weaker.
3 A MODAL SPATIAL LOGIC NTPP
a a
a
a
b
b
b
a
DC
EC
PO
b
b
b
TPP
a
b
b
a
=
a TPP
-1
NTPP -1
Figure 1: A pictorial representation of the base relations and their direct topological transitions. is discrete from y') `PO(x; y)' (`x partially overlaps y'), `EC(x; y)' (`x is externally connected with y)', `TPP(x; y)' (`x is a tangential proper part of y') and `NTPP(x; y)' (`x is a nontangential proper part of y'). The relations: P,PP,TPP and NTPP being non-symmetrical support inverses. For the inverses we use the notation 801 , where 8 2 fP,PP,TPP and NTPPg. Of the de ned relations, DC,EC,PO,=,TPP,NTPP and the inverses for TPP and NTPP are provably mutually exhaustive and pairwise disjoint. A pictorial representation of the base relations de ned above is given in gure ??. This gure also depicts the direct (i.e. `continuous') possible transitions between the base relations (cf Freksa's (1992) conceptual neighbourhoods). These transitions are alternatively expressed as `envisioning axioms' (Randell 1991)and are used as the basis of the qualitative simulation program in (Cui, Cohn and Randell 1992). The Boolean functions are: `sum(x; y)' which is read as `the sum of x and y', `us' as `the universal (spatial) region', `compl(x)' as `the complement of x', `prod(x; y)' as `the product (i.e. the intersection of x and y' and `di (x; y)' as `the di erence of x and y'. The functions: `compl(x)', `prod(x; y)' and `di (x; y)' are partial but are made total in our sorted logic by specifying sorts restrictions and by introducing a new sort called NULL. The sorts NULL and REGION are disjoint. An additional axiom is also required which stipulates that every region has a nontangential proper part.4 A primitive function `conv(x)' (`the convex-hull of x') is de ned and axiomatised. We use conv to de ne a further set of relations: `INSIDE(x; y)' (x 4
A consequence of this axiom is that there can be no
atomic regions, i.e. regions which contain no subparts. For
a discussion of how such regions can be introduced into the language, see Randell, Cui and Cohn (1992).
2 is inside y'), `P-INSIDE(x; y)' (`x is partially inside y') and `OUTSIDE(x; y)' (`x is outside y'), each of which also has an inverse. Two functions capturing the concept of the inside and the outside of a particular region are also de nable (where `inside(x)' is read as `the inside of x', and `outside(x)' as `the outside of x' respectively. This particular set of relations re nes DR(x; y) in the basic theory. In (Randell, Cui and Cohn 1992, Randell, Cohn, Cui 1992) we generated a pairwise disjoint and mutually exhaustive set of relations by taking the relations given above, their inverses, and the set of relations that result from non-empty intersections. The set of base relations for this particular set were then nally generated by de ning a EC and DC variant for each of these relations. This gives a total of 22 base relations instead of the original 8. Cohn, Randell and Cui (1993) further develop the theory to a set of 95 pairwise disjoint and mutually exhaustive relations and discuss some criteria on which to base the choice of grain size for such relations.
3
A Modal Spatial Logic
Although there has been a long history of both modal and non modal temporal logics, only recently have spatial logics been investigated at all. These seem to have been entirely built on classical rst order logic as is the case in our spatial logic sketched above. This logic was built from one primitive relation, C (x,y) which is true if the spatial region x is connected to the region y (i.e. if there is no distance between x and y). C is axiomatised to be both re exive and symmetric. In seeking to develop a modal spatial logic this binary relation seems a natural choice for an accessibility relation: each spatial region is interpreted as a possible world at which propositions may hold and the relation C de nes the accessible worlds/regions. We will write the modal operators as 2 and 3. The semantics of 2 is given by ; w1 j= 2 i 8w2C (w1; w2) ) ; w2 j= and similarly for 3. Such a logic appears to be a KTB system (Chellas 1980), or Brouwerian system (Hughes and Cresswell, 1968). However we de ned many more binary spatial relations above, for example NTPP (proper part), EC (externally connects) etc. Each such relation can have a pair of corresponding modal connectives; thus for example we obtain NTPP 2 , NTPP 3 , EC EC 2 , 3 etc. The semantics of each such modal operator is given in an analogous way to the way in C
C
C
C
C
3 de nes the subsumption relationships between the predicates naturally speci es a collection of axioms: each link in the hierarchy corresponds to an axiom of the form, e.g. 2 ! 2 . Moreover, the transitivity or composition tables (Randell and Cohn 1992) also naturally give rise to a set of axioms. For example in the table below, it can be seen that NTPP (a,b) and NTPP (b,c) implies NTPP (a,c); from this one can obtain the modal axiom ^ 2 ! 2 2 . However, since the relational lattice and the composition table do not characterise the entire set of theorems, these axioms are not likely to be complete.
4 CLASSIFYING SPATIALLY INDEXED PROPOSITIONS
which the semantics for 2 and 3 was given above. In the classical rst order calculus presented previously we de ned some function symbols as well as binary predicates; it is also possible to de ne modal operators corresponding to these function symbols. For example a 2 operator for the binary function symbol sum can be semantically de ned thus: ; w1 j= 2 i 8(w2; w3) w1 = sum (w2 ; w3) ) ; w2 j= & ; w3 j= and similarly for 3 . Analogous operators can of course be de ned for the other Boolean function symbols. One could even schematise such connective de nitions to allow arbitrarily complicated connectives such as (2 ) which would ( ) be semantically de ned thus: ; w1 j= 2 i 8(w2; w3; w4) w1 = sum (prod (w2; w3); w4) ) ; w2 j= & ; w3 j= & ; w4 j= As mentioned above, the rst order spatial calculus can also be extended by an additional primitive function conv (x) which then allows many more binary predicates on regions to be de ned. First we can de ne the modal operators for conv : ; w1 j= 2 i 8w2 w2 = conv (w1 ) ) ; w2 j= . A 3 operator can also be de ned but this is semantically equivalent to 2 since conv is a one place function. It should be obvious how to de ne the connectives for the relations de ned in terms of conv (eg INSIDE , OUTSIDE etc). However we might point out here that some of the modal accessibility relations we have de ned here are non serial; for example TPP is non serial for open regions (as we allowed in (Randell and Cohn 1989)); or PP for atomic regions; or PO , P P 01 , EC for the universal region; or INSIDE for convex regions. Our presentation of a spatial modal logic here has been semantic rather than axiomatic as is perhaps more traditional. Of course a semantic speci cation may well be the rather more computationally useful speci cation since there are now several ecient theorem proving techniques available for modal logics in this format { e.g. (Gent 1992; Ohlbach 1991; Frisch & Scherl 1991). However it would still be interesting, from a theoretical standpoint at least, if we could come up with an axiomatic speci cation. One possibility for automatically generating the required axioms would be if the SCAN algorithm (Gabbay and Ohlbach 1992) could be applied in reverse. Meanwhile it is clear that many axioms can be extracted from our earlier papers analysing the purely rst order logic and the theorems we proved there. For example, the relational lattice depicted in gure ?? which C
C
sum
sum
sum prod ;
sum prod ;
conv
conv
conv
P
NTPP
4
C
NTPP NTPP
Classifying spatially indexed propositions
Just as in the temporal case where temporally indexed propositions can be classi ed depending on whether they, for example, hold continuously throughout the interval or just in at least one sub interval, so in the spatial case when propositions are spatially indexed we can distinguish di erent propositions according to their spatial modalities. Suppose that holds at a spatial region/world, r. A non exhaustive list of the possibilities follows: 1. holds in all sub regions of r, i.e. ! P P2 2. does not hold in all sub parts, i.e. ! :PP20 3. does not hold true anywhere else apart from r (i.e. r is maximal): ! 2 : 01
1
DR
4. is locally maximal: ! 2 : EC
5. identi es a \skin": ! : PP3 0 6. de nes an interior: ! : PP3 7. does not hold in any sub region of (i.e. r 0 is locally minimal): ! :PP3 Items 1 and 2 above correspond to the standard temporal classi cations o whether propositions hold throughout an interval or merely during some sub interval; e.g. \I slept all last night" v.\I wrote this paper last week". In the spatial case the English rendering might be \the raspberry is red" v. \the Financial Times is pink". Of course given a more expressive ontology of spatial relations such as those de nable by the additional primitive function conv , (e.g. INSIDE, P-INSIDE, OUTSIDE NT
01
T
1
1
C
5 FINAL COMMENTS
DR
4
O -1 P
P
PP-1
PP
PO
TPP
=
NTPP
-1 TPP
-1 NTPP
EC
DC
PO(a,b) TPP(a,b) NTPP(a,b) a=b NTPP-1(a,b)TPP-1(a,b) EC(a,b) DC(a,b) a b a a b b a b a b ab b b a a
Figure 2: The relational lattice of binary spatial relations. .
H H
R2(b,c) R1(a,b)
DC
EC
PO
TPP
NTPP
TPP
-1
NTPP
-1
=
H H
DC
no.info
EC
DR,PO,PP
DR,PO,PP
DR,PO,PP DR,PO,PP
-1 DR,PO -1 DR,PO,PP EC,PO,PP TPP,TP -1 -1 DR,PO,PP DR,PO,PP no.info PO,PP
PO
DR,PO,PP DC
DC
DC
PO,PP
DR
DC
EC
PO,PP
-1 DR,PO,PP
DR,PO -1 PP DR,PO -1 PP no.info
TPP
DC
DR
DR,PO,PP PP
NTPP
DR,PO -1 TPP,TP
NTPP
DC
DC
DR,PO,PP NTPP
NTPP
DR,PO,PP
PO,PP
-1 PP
TPP
-1
NTPP =
-1
-1 -1 EC,PO,PP PO,PP
PO,TPP,TP
-1
-1 PO,PP
-1 PO,PP
DR,PO,PP -1
DR,PO,PP DC
EC
-1 PO,PP PO
TPP
-1
O NTPP
NTPP
NTPP TPP
-1
-1
NTPP NTPP
PO TPP NTPP
-1
-1 TPP
-1
-1 NTPP
-1
=
Table 1: Transitivity table for the 8 basic relations. If R1 (a,b) and R2(b,c), it follows that R3(a,c) where R3 is looked up
in the table. \no info." means that no base relation is excluded. Multiple entries in a cell are interpreted as disjunctions. Note that DR stands for DC and EC, PP for TPP and NTPP, PP01 for TPP01 and NTPP01 , TP01 for TPP01 and =, and O for PO, TPP, NTPP, TPP01 , NTPP01 , and =.
etc) we could classify spatially indexed propositions in a more detailed way, for example by whether holds in the \inside" of the region r. It is interesting to note that Fleck (1986) in her work on vision, uses the idea of locally maximal regions with respect to a particular property; it is precisely this notion which we have formalised in our classi cation above (4).
5
Final comments
There are clearly many ways in which the work presented here could be extended; for example we could further re ne the taxonomy by introducing more distinctions, or we could investigate more computational mechanisms (such as those outlined
in (Cui, Cohn and Randell 1992)), or integrate this purely qualitative formalism with more geometric, metric and analogical notions. Of course the modal logic sketched here requires much more work both on the formal side and in evaluating its utility and comparative expressiveness. It is clear that the modal spatial logic is less expressive than the rst order formulation but the availability of ecient modal logic theorem provers for modal logics may well make such formulations an attractive point in the continuum of spatial logics. We plan to work on these tasks amongst others in a new SERC funded project (GR/H 78955) \Logical and Computational Aspects of Spatial Reasoning". Finally, we take the opportunity to point out that we have also consider temporal aspects in our pre-
REFERENCES
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