A relation â algebraic approach to the region connection calculus
A relation – algebraic approach to the region connection calculus Ivo Düntsch, Hui Wang, Steve McCloskey School of Information and Software Engineering, University of Ulster, Newtownabbey, BT 37 0QB, N.Ireland {I.Duentsch,H.Wang,S.McCloskey}@ulst.ac.uk
Abstract We explore the relation–algebraic aspects of the region connection calculus (RCC) of Randell et al. (1992a). In particular, we present a refinement of the RCC8 table which shows that the axioms provide for more relations than are listed in the present table. We also show that each RCC model leads to a Boolean algebra. Finally, we prove that a refined version of the RCC5 table has as models all atomless Boolean algebras with the natural ordering as the “part – of” relation, and that the table is closed under first order definable relations iff is homogeneous.
1 Introduction Qualitative reasoning (QR) has its origins in the exploration of properties of physical systems when numerical information is not sufficient – or not present – to explain the situation at hand (Weld and Kleer, 1990). Furthermore, it is a tool to represent the abstractions of researchers who are constructing numerical systems which model the physical world. Thus, it fills a gap in data modeling which often leaves out the researcher as an active component in the modelling process. If we follow the description of data modelling presented by Gigerenzer (1981) which is pictured in Figure 1, then the two places where QR resides are at the level of the empirical model, and in including the intentions and actions of the researcher as part of the process. Conceptually, QR can be called a form of soft computing, in particular related to the philosophy of rough set data analysis (Pawlak, 1982, 1991) as presented in Düntsch and Gediga (1997) (see also Cohn, 1997, p.1, footnote 1, which points in the same direction). A special area of QR, qualitative spatial reasoning (QSR), has evolved in the last decade which is concerned with the qualitative aspects of representing – and reasoning about – spatial entities as opposed to the earlier emphasis on one–dimensional situations.
Address from May 2002: Department of Computer Science, Brock University, St. Catherines, Ontario, Canada, L2S 3AI, [email protected]
Fig. 1: The data modeling process
Domain
Representation
of
interest
Researcher
system
Empirical system
“The challenge of QSR then is to provide calculi which allow a machine to represent and reason with spatial entities of higher dimension, without resorting to the traditional quantitative techniques prevalent in, for example, the computer graphics or computer vision communities.” (Cohn, 1997) Applications of QSR can be found in geographical information systems (Worboys, 1998), spatial query languages (Clementini et al., 1994), natural languages (Asher and Vieu, 1995) and many other fields. Evidence that QSR is now firmly established in AI are numerous presentations, workshops and tutorials which have been given at important AI conferences, most recently at COSIT’97, KR’98 and ECAI’98. We invite the reader to consult Cohn (1996) and its updated version Cohn (1997) for an introduction and an overview of current trends. In the wider context of formal ontology, the special edition of the International Journal of Human–Computer Studies 43 (1995) exhibits the width and depth of the area. The basis of QSR are “part – of” and “connection”, respectively, “contact” relations. The formalization of the “part – of” relationship goes back to the mereology of Le´sniewski (1886 – 1939), developed from 1915 onwards. One of the main concerns of Le´sniewski was to build a paradox–free foundation of Mathematics, one pillar of which was mereology, or, as it was originally called, the general theory of manifolds or collective sets (Le´sniewski, 1928). Mereology was subsequently taken up by Leonard and Goodman (1940) (though, for a somewhat different purpose). Formally, Le´sniewski’s mereology and the calculus of Leonard and Goodman – the classical mereology (CM) – are the same. Based on classical mereology, and the work of Whitehead (1929) on the relation “! is extensionally connected with " ”, Clarke (1981) presents an axiom system for a “Calculus of individuals” based on a “connected – with” relation # . The intended domain is such that “ . . . we may interpret the individual variables as ranging over spatial–temporal regions and the two–place primitive ‘! is connected with " ’ as a rendering of ‘! and " share a common point.” (Clarke, 1981, p.205).
2
Suppose that $ is a nonempty set of regions, and # a binary relation on % ; we let #&!('*)+"-,-$ !/#0"21 . If 3546$ , and !7,8$ , then ! is the fusion of 3 , if
.
#0!9';: #&" ?
(1.1)
Clarke assumes are two mereological axioms: A 1. # is reflexive and symmetric, A 2. If #0!9'@#&" , then !A'B" , and one axiom concerning the fusion: A 3. If 3C46$
is nonempty, then the fusion of 3
exists in $
..
If a region ! is not connected to every other region, then the complement DE! of ! is defined as the fusion of all regions F which are not connected to ! . In other words,
#HG DI!/JK'
(1.2)
: #VFW? L MONQPSRUT
Biacino and Gerla (1991) show that the domains satisfying A 1 – A 3 are exactly the complete orthocomplemented lattices in which
] !/#0"YXHZ5!\[ DE" ?
(1.3)
]
Here, is the lattice ordering; the fusion is just the lattice join. It may be worthy to point out that, although Clarke calls his operations “quasi–Boolean”, the models for his calculus are not necessarily (quasi–) Boolean algebras. Clarke (1985) adds another axiom, the purpose of which is to define a ‘point’ within his calculus. Unfortunately, the full system collapses to classical mereology, as Biacino and Gerla (1991) show, and they suggest a modification of Clarke’s system calculus: “The new system should still admit as models the class of the nonempty regular open sets of a topological space1 . . . But in these models the connection relation should be as follows: (1.4)
!S#&"HXHZ
!Y^ "7'` [ _ .”
Here, ! is the topological closure of ! . Such a system, the “region connection calculus” (RCC), was presented by Randell et al. (1992a,b), and has since received prominence in spatial reasoning (see Cohn et al., 1997, for an overview). The differences to Clarke’s system are that only the existence of the fusion of finite sets is postulated, and different notion of “complement”. Already in the early RCC presentation of Randell et al. (1992a), the importance of relational transitivity tables for qualitative reasoning about regions was recognized; recently, Bennett et al. (1997) 1
These are the models of classical mereology.
3
have raised several questions regarding the expressiveness of relational reasoning, in particular with respect to the RCC. Relational reasoning as algebraic manipulation of relations has a long-standing tradition, going back to A. De Morgan, C.S. Peirce, and E. Schröder (cf. Anellis and Houser, 1991). From the 1940s onwards, A. Tarski (who, incidentally, was Le´sniewski’s only doctoral student) and his colleagues have continued the work on the calculus of relations which eventually led to an algebraization of first order logic via cylindric algebras (Henkin et al., 1971, 1985), and its finite fragments, in particular, first order logic with three variables via relation algebras (cf. Tarski and Givant, 1987). In this paper we shall explore the relation – algebraic aspects of the RCC relations, and suggest some modifications. We also hope to answer some of the questions raised in Bennett et al. (1997). The paper is structured as follows: We first introduce the necessary machinery of relation algebras; based on these, we will then discuss some aspects of Bennett et al. (1997) from a relation – algebraic point of view. Section 4 introduces the RCC and lists some of its properties. We show that the algebraic part of the RCC leads to quasi – Boolean operations, and present a refined (weak) composition table which contains additional definable relations which do not appear in the original RCC. Finally, we investigate a reduced set of RCC relations (RCC5).
2 Relations and their algebras A relation algebra (RA)
acbedfgdihUd
awvrdxvrd+mndkjldkjldxvrd+mndkjpt is a structure of type
acb|dfgdihUd
1. 2.
D acb|dopdq}d+m s t
dkjld+mt
D
dkjld+mndopdqrd+m sut
which satisfies for all y
dxz dk{ ,
b ,
is a Boolean algebra (BA). is an involuted monoid, i.e.
acb|dopd+m s t ms (a) q~q d is aI o semigroup z q znqo with q identity , (b) y 'y Gcy J ' y .
3. The following conditions are equivalent: (2.1)
oz +h { jld qo{ hz jld {oEznq h j cG y J ' Gcy J ' G J H y ' ?
In the sequel, we will usually identify algebras with their base set.
a dd d d d dopdq d+m s t The full algebra of binary relations is a structured%Vd +G VJ on a set d , where ^ D _ ( %V+G VJ is the set of all binary relations on , ^ D areo the usual set theoretic operations, q _ are, respectively,q the empty and the universal relation, is relational composition, the a d t a d t ms b relational converse (i.e. ') ! " . " ! ,Y1 ), and is the identity relation on . A subset of %V G VJ which is closed under the distinguished operations of %V G VJ and contains the distinguished constants is called an algebra of binary relations (BRA) on . It is a subalgebra of %V+G VJ , a fact which we 4
b
]
a by t %V+G VJ . a If )%., t 194@%VG 0J , we denote the BRA generated by )%.,1 denote by % .Q, . If , % .&, , we say that is RA – definable by )% .&,W1 . If the set of generators is understood, we shall usually omit mentioning it, and just say that is RA definable. b A relation algebra is called representable if it is isomorphic to a subalgebra of a product of full algebras of binary relations. The logic of RAs is a fragment of first order logic, and the following fundamental result is due to A. Tarski (see Tarski and Givant, 1987):
d
d
a
d
d
t
d t relations on % is thea setd ofd all binary Proposition 2.1. If %& ?i?i? %&,%V+G VJ , then % ?i?i? which are definable in the (language of the) relational structure %& ?i?i? & % by first order formulas using at most three variables. b
ms
b
b
a d t b = An RA is called integral, if is an atom of . If ' % a is d a BRA, t then is integral if and only if no proper nonempty subset of is definable in the model % = by a formula with at
most three variables (see Andréka det al., 1995b). b b d d a d t ] and ! " F8, we usually write !Q0" if ! " , , and Suppose that %V G VJ . For , !QV" F means !Q0" and " F . With some abuse of notation, we let %&!5. ')+" ,¡ ¢.! %&"21 be the range of ! w.r.t. R. a d t a d t
b group of , and ¥,B£I¤ ; we will write ¥ Let £I¤ be the symmetric under ¥ is denoted by %V¨ , i.e. The image of %§,
! "
a d t a d t % ¨ ')¥ ! " . ! " ,©%g1~?
(2.2)
instead of ¥G¦!/J ¥G¦"J .
b
If %V¨©'ª% b , we call % invariant under ¥ . The permutation ¥ is called ab base automorphism b¬« of , if every %§, b« is invariant under ¥ . The set of b all base automorphisms b« of is denoted by ; itd is easy to see that is a subgroup of £ ¤ . We call b permutational, if is transitive, i.e. for all ! "8,\ b« there is some ¥, such that ¥G¦!/J' a " d . Ift is permutational, then no proper nonempty subset of is first order definable in the model % d =® . Conversely, if ¯ is a subgroup of £¬¤ and !
",- , we set d d ¯ Tn° < ')¥G¦! "WJE.r¥\,(¯|1 d and let ¯&± be the BRA on generated by )¯ Tn° < .r! "©, e1 . Observe that the sets ¯ Tn° < are just the b ¯&± , and orbits of the action of ¯ on &² , and hence a partition of ² . Indeed, each ¯ Tn° < is an atom of every atom of ¯&b¬ ± « has this b form. The assignments ³ and ´ form a Galois connection, and is called Galois closed if ±µ' (see Jónsson, 1984, Börner and Pöschel, 1991, Andréka et al., 1995a). b
b b first order closed if every first order definable binary relation in the language A BRA on is called of is an element of . The next result is most likely known, and its easy proof is left to the reader: b
Lemma 2.2. If
b«
is integral and first order closed, then
is transitive.
The following result gives a connection between Galois closure and first order closure (Jónsson, 1991, Andréka and Németi, 1991): 5
b Proposition 2.3. If
b
'
b «
± iff
b
is finite, then is closed under every permutation invariant operation on binary relations.
We shall need this in our discussion of RCC5 in Section 4.4. The concept of residuation will be of importance in our later considerations. It will turn out that many theorems of the mereological part of spatial relations are consequences of the residual operators, since the “part of” relation turns out to bedxthe residual of the “connected to” b z right b o relation. z o z
Suppose that is an RA, and that y , . Even though z the equations z¬¸ y !©' and ! y¶' do not always have a solution, there are always elements y|· and z y , called, respectively, the right and left residual of by y such that
o
y o !
z
]
zd
]
y& z XHZC! z¹¸ · ] ! y XHZC! y2? z The residuals can be expressed as RA terms in y and by z qo z d (2.3) y0¸ · 'ºDeGcy zKo D q J z¹ (2.4) yµ'ºDeG D y J? z If y' , we shall only speak of the right (left) residual of y . These residuals have the following properties: Lemma 2.4.
¸
1. y0·y and y
2. If y is reflexive, then y0·y
]
]
y are reflexive and transitive.
3. If y is symmetric, then y&·y
]
y . y if and only if GcyV·yWJ
qo
Gcy0·yWJ
]
y .o
Proof. A proof ofm s1.o can be found in Pratt (1990). For 2., o o the monotony of ] ] imply that DIyµ' DIy y D¬y . Thus, yV·yµ'§DeGcy DIylJ y . Suppose that y is symmetric. Then, one line implies the next:
and the reflexivity of y
] yV y o o ·y ] Gch yV·yWJ o DIy D y o jy I GGcyV·yWJ DI D Gcy h DIyWJ»' j qyWo J e GGcy0·yWJ cG yV·yWJJ DIyµ' ? q o q\h j ] qom suppose j Gcy©·gyWJ Gcy·Yo yWJ h~ym s , andj assume that Gcy©·gyWJ D y¼' [ . Then, I Conversely, that s h GGcyV·yWJ J DIy7' [ , and hence, GcyV·y DIyWJ ' [ . On the other hand, qo qo h j o h j ] cG y&·yWJ y0·y yµXHZ½GGcyV·yWJ y0·ylJ yY' XHZ;GcyV·y DIylJ yV · yH' ? (2.5) Reflexivity of y0·y implies that (2.6)
Gcy0·y
o
DIylJ
hpm s ]
GcyV·y
which proves our claim. 6
o
h
DIyWJ yH'
jld
If %
d
¡,%V+G VJ , then the residuals are given by the conditions d
! Gc%6¸ ·IJ"µXHZ¿GÀ FpJ GcF~%&!©Á5Fp¾"J d ¾ !¾G %eJ"µXHZ¿GÀ FpJ G¦"l%0FgÁC!SÂFpJ (see e.g. Jónsson, 1991). We also use the following conditions (Jipsen, 1992):
b
Lemma 2.5. Let
ms
be an RA and y
oz
,
{
z
b
j
·) 1 . Then, qo{
] ] , then y . (Integral lemma) z z o{ K z o {Ãq ] ] 2. If is an atom and y , then . (Atom lemma) y b bÄ b d d Suppose that is a complete and atomic RA with b atoms b bÄ relational b { G b JY'Å)yÆ ?i?idx?z y}ÇQ1 . { Then, ] composition can be interpreted as a mapping È7. oEz ÁÊÉ8b¬G Ä b J withbÈËÄ Gcy b JE'Ì) , G J. y 1 . Indeed, it is sufficient G J. b to look d at the restriction of È to d G Ï J 1. If
]
q»o
dxz dk{
y
y , and y
a list of all atoms below Theo composition table of is an G¦Í ÍÎJ –matrix where entry G¦ J contains b y ynÐ ; an example is shown in Table 1. A complete and atomic RA is completely determined by
o Table 1:Ñ A composition Ѿq Ò tableÓ Ò Ñ Ñ Ò Ó Ñ¾q ÑÔdcÑÂqd+m s DѾq Ò Ñ¾qdkÓ Ò Ò Ò Ò Ò DÒ Ó ÑÔdkÓ Ó Ò D bÄ b m s m s table of its b set of atoms G J . When writing such a table, we will omit the relational composition column and row
, if
is an atom of .
In our construction of RAs we have been aided by the RA Scratchpad, designed and written by Peter Jipsen (1992). For other properties of relations and their algebras see Jónsson (1982, 1991) and Andréka et al. (1998); we recommend Grätzer (1978) as a reference text for lattice theory, and Koppelberg (1989) for Boolean algebras.
3 Weak composition A more general view of composition is taken in Randell et al. (1992a) and Bennett et al. (1997); there, a compositiona table d t (CT) is just a daÖlmapping t ÈA.2ÕA©ÕÁCÉ8G Ö ÕËJ , where Õ is a set of relational symbols. AÖ model of Õ È is a pair , where is a set and .IÕ@Á %V G VJ is a mapping such that ) GcyWJ.~yA, ÕÎ1 is a partition of - . The model is called consistent if
{
(3.1)
,©ÈËGcy
dxz
Ö
oÖ z Ö { JXHZ½G GcyWJ G J JË^ G JV'` [ _r?
7
dxzdk{
for all y ,×Õ . In the sequel, we will call such a table a weak composition (table) to distinguish it from the usual relational composition. Bennett et al. (1997) call a weak composition table extensional, if
Ö
(3.2)
GcyWJ
oÖ z G J'
Ö { : G J? Ø = ÙMÛÚi° ÜR
Then, they write “One might . . . conjecture that by refining relations in a set Rels one can always arrive at a set Rels’ which is more expressive than Rels and whose CT can be interpreted extensionally.”
a d t a dÖlt Ö { { If Õ È has a model at all, Ö then, { in { case ) G J. ,§ÕÎ1 is closed under converse, and the identity is a union of elements of ) G J0. 6 , ÕÎ1 , its elements are the atoms of a relation algebra iff the table is extensional (see e.g. Jónsson, 1984). Given a partition Ý of @8 , the relations in Ý will always generate a relation algebra which has an extensional table in case it is atomic and closed under arbitrary unions. A weak composition table is called complete w.r.t. a theory Þ whose language contains Õ ßnà á if, “. . . whenever a set of (ground) facts involving only relations in Õ/ßnà á and constants is inconsistent, this can be detected by reference to the table.” (Bennett et al., 1997) Bennett et al. then conjecture that “. . . a CT is complete w.r.t. a theory Þ iff Þ extensional interpretation of composition.”
implies all formulae corresponding to the
ms
j s following is a simple b The counterexample: Consider the RA with the two atoms (identity) and Ï d Ï on a three element set. Let b Þ say that there are four (diversity), and let j s be itsd representation ]6â ¬' [ 1 . Then, Þ is not satisfiable in , but each triangle is. elements, e.g. Þ@')+!2 ! Ð .~
4 The region connection calculus The Region Connection Calculus (RCC) introduced in Randell et al. (1992a) is a system similar to the mereological and quasi–Boolean part of Clarke’s calculus of individuals, the difference being the definition of complementation. The base relation is a connection # , and further relations are obtained
8
from # by the relational operators as in Clarke’s system:
'@#6·E# m s V'q^-o D ã ' q ã ã ' ^-D|Gc o J ä V'V^ GcåY# åYe # J æ ä ä V'V^-D V ã åg#'@#6^-D ç #'ºD#
part of proper part of overlap partially overlap tangential proper part non–tangential proper part externally connected disconnected ?
4.1 RCC axioms
æ
d
æ
A model for the RCC consists of a base set è'é% , where % are disjoint,f a distinguished ê ,©% , a unary operation h D`.}%&¬Á5%& , where ê %&&.r'%6·¬) 1 , a binary operation .}§ % ©%Áë% , æ , and a binary relation # on % . another binary operation .~%§8%Á5% There are 8 axioms for the RCC: RCC 1. GÀ !(d ,©%eJ!S#&!
RCC 2. GÀ !
",%eJ ì !/#0"µZC"#&!2í RCC 3. GÀ !(,©%eJd !S# ê RCC 4. GÀ !(,© a % d "At ,©%&J ,
æ ä
D " t ,(#X¶Zëî! V0" (a) a ! d E ã (b) d ! d E D " , a d X¶Zë f ît ! 0"
RCC 5. GÀ ! d " d F¶,©%eJ ì a ! d " h Ft ,(#XHZï!/#0" or !S#0F RCC 6. GÀ ! d " ¶ F ,©%eJ ì h ! " F ,8#X¶Z½Gðrñº,©%eJ G¦ñ0" and ñVF and !/#0ñ&J í RCC 7. GÀ !
",%eJ ì !
"A,©%XHZ5!
ã
"pí
RCC 8. If ! 0" and "r0! , then !A'B" . We have added RCC 8, since this is what is intended, but it does not seem to follow from the other axioms. By the definition of , Lemma 2.4.1, and RCC 8, we see that is a partial order. The original RCC system contains another axiom:
æ ä
d
“. . . A rather deep theorem of the theory is given by the formula ÀQ!Sðr"/ì VòG¦" !SJ í which was demonstrated by informal argument in Randell, Cohn, and Cui (1992a). Because we have so far not been able to give a fully formal proof of this theorem we often regard the formula as an additional axiom of the theory” (Cohn et al., 1997). Below follows a simple proof of this property:
9
Lemma 4.1.
æ ä
GÀQ!7,©%eJ Gðr"A,©%eJ"
(4.1)
VV! æ ä
V0! ; By RCC 4a, this Proof. Assume that there is some !§,% such that for all "@,% , îÂ" o t is the largest relation Ä on R with implies that "W#ªD¡a ! d for t all ",% . Since ó'ôo #a ·V# d – i.e. ] ] # a¦Äd t # –, and ! DE! ,© [ , we obtain that # ) ! DE! 19[ # . Hence, there is some ,8% such DE! ,7 [ # , a contradiction. According to Randell et al. (1992a), the weak composition table has the form given in Tab. 2. Since there are eight base relations, the system is called RCC8. Table 2: RCC8 weak composition table C
õ ö
O
DR DC
PP EC
PO
DC EC
1 DR,PO,PP
PO TPP
DR,PO,PP DC
NTPP TPP
DC DR,PO,PP
÷
DC EC,PO,PP
÷
DR,PO,PP
÷
PO,PP
÷
NTPP
÷
DR,PO,PP 1’,DR,PO, TPP TPP DR,PO,PP DR
÷
÷
÷ ÷
NTPP
TPP
PP
÷
DR,PO,PP EC,PO,PP
DR,PO,PP PO,PP
DC DR
1 DR,PO,PP
PO,PP PP
PO,PP NTPP
DR,PO,PP PO,PP
NTPP 1’,PO, TPP,TPP PO,PP
NTPP PO,PP
DR,PO,PP 1’,DR,PO, TPP,TPP DR,PO,PP PP
PO,PP
÷
÷
÷
ø
O 1’
o ù
÷
DC DC
÷
÷
DR,PO,PP DR,PO, PP
÷
1 NTPP
÷
NTPP
÷ NTPP
DR,PO,PP DR,PO,PP
÷
÷
TPP
÷
÷
÷
÷ o ù
ç
NTPP
that the weak compositiono åg#ú' Recall is to be interpreted as minimalo inclusion, e.g. # ã ã ç ç ç ç % V , and # åY# intersects each relation on % V means that # åg#ª4 the right hand side. The table is, in a way, a minimal requirement: It follows from the RCC axioms that each RCC model must satisfy the condition given in the table. Michael Winter has pointed out, that Table 2 has an extensional interpretation, namely, the closed circle algebra introduced in Düntsch et al. (1999b). There, the domain of regions is the collection of closed circles in the Euclidean plane, and !/#0"YXHZ !µ^9"7'@ [ _.
4.2 RCC models are Boolean algebras As in Clarke’s system, the operations of the RCC axioms are called “quasi–Boolean”. In contrast to Clarke’s operations – which define the more general orthocomplemented lattices –, our next result shows that RCC operations indeed define a Boolean algebra, if we extend them and the relation j æ over the set ') 1 in a natural way2 . Proposition 4.2. Suppose that
2
æ
j
j
') 1 , and let %Vû '% ) 1 . Then, a dfgdihUd dkjld t ü ê ' % û D
The same result has been independently obtained by John Stell (1997).
10
is an atomless Boolean algebra with natural order .
j
jld j ê h jþ ý ÿ such a way that 0 in ! for all !f , %Vû ; furthermore, we set D ' D ' ê , Proof. We extend ã and also, ! " ' XHZ î¾! " . Finally, we extend in the obvious way. It is clear that all these
extensionsa can d be t reversed uniquely, so that we can always return to the original structure. Note that %
is atomless by Lemma 4.1. The claim follows now from the statements below:
1. GÀ !(,©%eJ! ê , and ê is the only element with this property. 2. ! 3. !
f
h
" is the supremum of ! and " w.r.t. . " is the infimum of ! and " w.r.t. .
4. DI! is the unique complement of ! .
a
5. The lattice % û
dfgdihUd
dkjld t ê is modular. D Ö
Ö Ö 1. The first part follows immediately from RCC 3 and theÖ definition of . Suppose that !Q for all ê ê ê !(,8% ; then, in particular, by RCC 8. a d f t . Since , we have ' a d f t " ,7 [ . Then, there is some F9,7% sucha that d f Fp#0! t and F ! " ,[ # . This 2. Assume that ! ! o F , and that ñ ! " ,¡# . By RCC 5, we can contradicts RCC 5. Now, suppose that ! VF and "re a f d that t ñ0#&! . Now, ñ0#&! VF , and # ô4§# implies ñ0#0F . The definition of now assume w.l.o.g. " F ,© . h gives us ! a d h t
d td¹a F d ! t " o,6# . By RCC 6, there a ish dsome t ñ¼,¡ a % h such d t that 3. Suppose w.l.o.g that ! "\,¡a % .d Let ñV! ñV" , andd Fp#&ñ d , and thus, F ! F " ,(# *46# .a This d h shows t ! " ! , , ! " " ,© . Now, let F¶,©% F~V! F~V" , and ñ0#0F ; we need to show that ñ ! " ,(# . This follows immediately from RCC h j a d 6. t ã
, then î¾!Q0! f by RCC 4, a contradiction. Thus, ! DI!' by RCC 7. Let If ! DE! , F,6% , and assume that î\Fp#HG¦! DI!/J . Then, q»byo RCC 5, îFr#&! , î\Fp#ºD! , and RCC a d 4t implies ã ã æ ä æ ä æ ä æ ä that F V0! and F @D! . Now, V V V*4 , and it follows that ! DI! , ,a contradiction. a d t
4.
Next, we show that D|G DE!/JK'B! which will be needed for the uniqueness proof. Since DeG DI!/J DE! , [ ã ] ] , we obtain DeG DI!/J ! . For the converse, assume that ![ DeG DI!/J . By definition of , there is æ ä some o ñ½, % such that ñ&#&! and q o î¾ñ&# DºG DI!/J , i.e. ñ V D! . Now, ! åg#¢DB! , and thus æ ä æ ä ç !QåY# V F ; however, åY# Vª4 # , contradicting !/#0ñ . f d h j
It remains to show that DE! is the unique complement of ! . Suppose that ! "µ' ê ! "H' . Then, ã ] ] î¾! " by RCC 7, and h RCC j 4b implies that " DI! . Assume that DE!B[ " ; then, again d by RCC 4b, ã æ ä æ ä DE! D(f" , i.e. DE! DE" . With (4.1) choose some ñº,© a % d such t that ñ `D7! ñ V VD7" . æ ä ê Since ! "µ' , let w.l.o.g ñ�! . By RCC 4a we have ñ DE! , [ V , a contradiction.
]
]
We also note that ! " implies DI" D ! : Assume not; then, there is someo F¶,8% such that Fp#D7" . E æ ä æ ä æ ä æ ä ] i.e. îÂF V0" , and îKFp#@D! , i.e. F 0! . Since ! e " and e ¢4 V , we obtain æ ä F V0" , a contradiction. 11
It follows that D also fulfills the De Morgan condition
]
“ ”: Since !
]
f !
]
f
" and " ]
a d! t " D G DE! e “ ”: We show ,[ % . contradicting ! DE! ©
D|G¦! f
!
"J»'ºDE!
]
"J
DE"9. f
DE! and D6G¦!
]
"J
DE" ?
] DI"WJ . Assume w.l.o.g. that !B[ e D G DE!
5. To show modularity, it is enough to prove
d
If F}0!
(4.2)
h
" , we have f
D|G¦! h
f
f
then F
h
G¦!
h
"WJK'!
f
G¦"
FpJ
h
DE"WJ ; then, !
ã
G DE!
h
DE"WJ ,
d
see Grätzer (1978), Lemma I.4.12.
]
“ ”: In any lattice we have
f F
(4.3)
]
see Grätzer (1978), Lemma I.4.9. F “ ”: We first show
Ä
(4.4)
Ä
If
!
h "
]
f
GcF
Ä
DIF and
]
f
!/J GcF f
! implies F
]
h
"J
d
!A'! , and the claim follows. f "
d F
Ä then
]
"Q?
ã
ã
Proof. Assume not;Ä then, D\" by RCC 4b. By (4.1) and theÄ definition of there is some ñÌ,-% æ ä æ ä ç ç such that Ä ñ V and ñ e D" . It Ä followsf that ñ&# , ñ " , and ñ F , the latter because æ äIä ] ] ñ DIF . On the other hand, since " F , we have #0" or n#0F for any with n# , a contradiction. Next, we need (4.5) Proof. Since !
h "
]
h
! and !
!
h
"
f
!
h
DE"H'B!Î? f h ] ] D " I ! , we have ! " ! DE" ! . h
Conversely,
! " # %$
&' %$( &) %$ &' %$
æ ä
and & %$+*
æ ä
f
Thus, if this is true, there some ñ such that ñ V0! and ñ V¶G DI! DE"J , and from (4.4) we ã ã ] obtain that ñ DE" ; it follows that ! D " . Similarly, we see that ! " , a contradiction.
h
f
]
Now, assume that ! G¦" FpJ0[ F is some ñ,©% such that
f
!
h
" , and w.l.o.g. !
12
h
G¦"
f
FpJ,©% . Then, by definition of , there
1. ñ0#HG¦!
h
2. î8ñ0#HGcF
G¦" f
f
FpJJ , and h ! "J . Ä
The first condition says with RCC 6 that there is a ,©% such that
Ä
]
(4.6)
! and
Ä
]
Ä " and & # ñe?
The second condition tells us with RCC 5, RCC 4a, and RCC 6 that ñ
æ äIä
DF , and
] ! and " imply î8ñ&# ií? Ä h Î j Ä d æ ä D F . If ' , then î #0ñ , since ñ ¬ DF , a contradiction. Otherwise, e,8% V Set ' ] " , the latter by (4.4). Now, and Ä ÄÂh f×ÄÎh by (4.5) ñ&# X¶Zïñ&#HÄÂG h F D¬ ľFph J by RCC 5 X¶Zïñ&# ÄÂh F or ñ&# DIF d æ ä since ñ X¶Zïñ&# DIF VDF X¶Zïñ&# GÀ,nJ ì-
(4.7)
]
]
! ,
which contradicts (4.7).
4.3 Refining the RCC table It is pointed out in Bennett et al. (1997) that the RCC axioms do not take into account that the largest Ï RCC m table to take care of this fact. Set @'`% , region ê isd definable. Our next task will be to refined the ] ê ê d Ï Î¶ m Ð for ¾¬') 1 ¹Æ¹'%(·¾) 1 , and ¾ Ð ' ; it is easy to check that for all base relations ] of the RCC (listed on p. 8), and ,
¾ Ð 4 or Î Ð ^\'`_r? Now,
ms
ms
o
¾m s ¬' m s ^-DeìUGcV d V . ' m s ^o D& Âo& m s d ÆÆ' . o ² o . d ¾Æ'  o ² o ÆÆ d Æ ' ÆÆ ²
q
ms d J ^ í S
which shows that all ¾ Ð and . are RA definable. The equation which tells us that ê is the largest element with respect to now is (4.8)
Æ 046E? 13
d Thus, in the sequel, we shall restrict the relations to %§·Y) ê 1 . In order to show that the defining ã ä equations on p. 8 and the axioms still hold, it suffices to prove it for V , and the axiom RCC 6, since all other definitions, respectively axioms, are universal, and thus carry over to substructures. This is straightforward, and is left to the reader; note that complementation of relations is restricted to %6·) ê 1|©%6·¬) ê 1 .
Let / be the incomparability relation, i.e. ª / 'ºDeGc åg# by
å # Y æ åY# and
ã
by
q
J . We extend the original RCC8 by replacing
o
q» qo d 'ºDeGcV V d V V|J ç 'åg#^7DIåg#
ç
q¹o o q d ã æ ' /@^-GcV q¹o V 0 |JË^-Gce V o J q ã ç '0/@^-GcV |JË^7D|GcV V V J ?
Then,
d
ç
Q! åY# "µXHZ5!A'ºDE" f d æ ê !QåY# µ " XHZ5!Qåg#0d " and " '[ f h ! jld d ã æ ê !Q " XHZ5!,/g" d ! h "7' [ jld ! f µ " '[ ã ç !Q " XHZ5!,/g" ! "7' [ µ ! " ' ê ? H This gives us 10 base relations, and we call the resulting system RCC10. The extended weak composition can be found in Table 3 on the following page. For cells containing ' , the RCC axioms together with general RA properties such as Lemma 2.4 or the equations (2.1) imply that strict composition (i.e. equality) holds; for cells containing ' [ , there is a model in which the composition is strictly smaller than the cell entry. For cell entries which can be shown to be below the weak composition we use the superscript 1 . In this way, we indicate in which cells the composition is extensional, and when it need not be. In computing the table, we have used the RA scratchpad, which in turn uses Lemma 2.5 and the equations (2.1); we are grateful to Michael Winter who spotted and corrected several inaccuracies. We have also used the following properties, which may be interesting in their own right.
æ
ä "HXHZC! V`D " . f ç ä FpJ . 2. If ! #VF , then ! VòG¦! f æ ä æ ä æ ä 3. ! VVF and " VVFgXHZ½G¦! "WJ VF . V h æ ä ä 4. If ! eVF , then DE! F VVF .
Lemma 4.3.
1. ! åg#
14
23 4
4
4
4 576
8
4
4
15
TPP, NTPP, PON, POD, ECN, ECD ( ), POD TPP, NTPP, PON, ECN, DC
4
ECN
4 4
4
4
4
4
4
4
4
4
4
8
8
4
8
TPP, NTPP,= NTPP, =
4
DC DC
8
POD TPP, NTPP, PON, POD, ECN, ECD, DC,=
576 4
TPP ,= TPP, NTPP, PON, ECN, DC, =
1’, TPP, TPP , PON, ECN, DC
1’,= NTPP, =
TPP, =
4
4
ECN,= DC =
4
4
TPP , NTPP ( ), PON, ECN, DC NTPP ,= 1’, TPP, TPP , NTPP, NTPP , PON, ECN, DC
NTPP ,= 4
TPP, NTPP, PON ( ), ECN, DC ( ), PON,= TPP, NTPP, PON, ECN, DC,=
4
TPP , NTPP ( )
TPP , NTPP
4
TPP, NTPP, PON, POD
1’, TPP, TPP , NTPP, NTPP , PON, POD,= TPP, NTPP ( )
4
TPP , NTPP , PON, POD, ECN, ECD, DC,= DC, =
PON, =
4
TPP , NTPP , PON, ECN, DC,
4
POD
TPP, NTPP, PON, POD, =
TPP , NTPP , PON, POD, ECN, ECD, DC, = TPP , NTPP , PON, ECN, DC, =
DC
TPP , NTPP , PON, ECN, DC
DC, =
DC
4
TPP , NTPP , PON, POD( ), ECN, ECD( ) ECN, DC
4
POD
4
1’, TPP, TPP , NTPP, NTPP , PON, POD, ECN, ECD, DC,= TPP , NTPP , PON, POD,=
4
POD
4
TPP , NTPP , PON, ECN, DC,=
POD
TPP , NTPP , PON, POD
DC, =
POD
ECN, =
ECD
4
8
ECD DC
576
TPP , NTPP , PON, ECN, DC
8
TPP , NTPP , PON, POD, =
4
4
TPP, NTPP( ), PON( ), POD
4 4
PON
4
1’, TPP, TPP , NTPP, NTPP , PON, POD, = TPP( ), NTPP, PON, POD( )
8
NTPP , =
576
TPP , NTPP , PON, POD
4
TPP, NTPP, PON, POD, ECN, ECD, DC,= POD
4
TPP , NTPP , PON, POD, ECN, ECD, DC,=
ECN, DC
ECN
4
NTPP
576
1’, TPP, TPP , NTPP, NTPP , PON, ECN, DC, = NTPP
8
NTPP
4
TPP, NTPP, PON, ECN, DC,
TPP, NTPP, PON, POD ( ), ECN, ECD ( ) POD
POD
8
NTPP, =
8
TPP, NTPP ( ), PON ( ), ECN, DC ( ) TPP , NTPP ( ), PON ( ), POD TPP, NTPP, PON, ECN, DC, =
PON
4
NTPP
TPP, NTPP, PON, POD
4
TPP , NTPP
8
TPP , NTPP ( ), PON, ECN, DC, NTPP , =
NTPP 8
1’, TPP, TPP , PON, POD
NTPP, =
NTPP 8
TPP
4
1’, TPP, TPP , PON, ECN, DC,
TPP
4
TPP, NTPP
TPP 4
TPP
Table 3: The RCC 10 weak composition table
4 4
4
576
8
4
4
8
4
4
4 8
576
8
4
4
8
8
f
"7' [ ê . Then, h lj d æ ! åg# "HXHZï! "µ' d !S#&" XHZï! 9@DE" d !/#0" æ ä XHZï! 9@DE" !¾G D |JÂD-" e ä XHZï! V@D " ?
Proof. 1: Suppose that !'º [ DE" , so that !
f f f # F ; then, F:9CDE! and !¾G D#|J F . Since 0 !/#*D6! , and DE!'óF DeG¦! FpJ , we have æ ä !/#@D6G¦! FpJ by RCC 5.d Thus, !¾G D VgJ G¦! FrJ by f RCC 4a. f æ ä æ ä æ ä 3: “ Z ”: Let ! VVF " VVF and assume îG¦! "J VVF . Then, by RCC 4a, G¦! "J#0F , æ ä VVF , a contradiction. and RCC 5 implies that w.l.o.g. !S#0F . RCC 4a now implies îÂ! f f æ ä æ ä “ X ”: Let G¦! "J eVF , and assume that î¾! VVF . Then, !/#VF , which, together with îEG¦! "J#0F contradicts RCC 5. h æ ä VgJ! and 3 by setting "µ'ºDI! F . 4: This follows from !¾G D 2: Let ! f
ç
There is no relation algebra which is a model of the RCC10 table. In the standard model of nonempty proper regular open sets of a regular connected topological space, there are at least 25 atoms Düntsch et al. (1999a).
4.4 A reduced set of RCC relations
m sd
d
ç
ã
d
d
q
% V V 1 of RCC relations has received some attention, and is usually The subset ) ã ä called RCC5. It arises from disregarding the split of # into and åg# , and e into V and ã æ ä V ; in other words, one adds the additional axiom #¢' . If one takes the weak composition induced by the RCC8 table, one arrives at Table 4. o ù DR PO PPq PP
Table 4: RCC5 weak composition table C=O DR PO PP 1 q DR, PO, PP DR q DR, PO, PP
DR, PO, PP 1 DR, PO, PP q PO, PP
DR, PO, PP PO, PP PP
D
ç
%
q PP DR q DR, PO, PP 1q PP
If we take into consideration that the largest region ê is RA definable and work within %6·¬) ê 1 , then, ã ã æ ã ç ç çòæ çòç similar to the RCC10 table, splits into and , and % splits into and , where ç9ç çòæ ç çòç is the complement, and . The composition induced by the RCC10 table is ' %`^D given in Table 5.
jld+m The next proposition shows that the RCC7 table – and thus RCC5 – has a very simple interpretation ü ü ü ·e) 1; (Düntsch et al., 1998). Suppose that is an atomless Boolean algebra, and that ¶' 16
Table 5: RCC7 table DR ;
DN DD PON POD PP PP @
also, let §' ü on :
O
DN
DD
ACB
Abstract We explore the relation–algebraic aspects of the region connection calculus (RCC) of Randell et al. (1992a). In particular, we present a refinement of the RCC8 table which shows that the axioms provide for more relations than are listed in the present table. We also show that each RCC model leads to a Boolean algebra. Finally, we prove that a refined version of the RCC5 table has as models all atomless Boolean algebras with the natural ordering as the “part – of” relation, and that the table is closed under first order definable relations iff is homogeneous.
1 Introduction Qualitative reasoning (QR) has its origins in the exploration of properties of physical systems when numerical information is not sufficient – or not present – to explain the situation at hand (Weld and Kleer, 1990). Furthermore, it is a tool to represent the abstractions of researchers who are constructing numerical systems which model the physical world. Thus, it fills a gap in data modeling which often leaves out the researcher as an active component in the modelling process. If we follow the description of data modelling presented by Gigerenzer (1981) which is pictured in Figure 1, then the two places where QR resides are at the level of the empirical model, and in including the intentions and actions of the researcher as part of the process. Conceptually, QR can be called a form of soft computing, in particular related to the philosophy of rough set data analysis (Pawlak, 1982, 1991) as presented in Düntsch and Gediga (1997) (see also Cohn, 1997, p.1, footnote 1, which points in the same direction). A special area of QR, qualitative spatial reasoning (QSR), has evolved in the last decade which is concerned with the qualitative aspects of representing – and reasoning about – spatial entities as opposed to the earlier emphasis on one–dimensional situations.
Address from May 2002: Department of Computer Science, Brock University, St. Catherines, Ontario, Canada, L2S 3AI, [email protected]
Fig. 1: The data modeling process
Domain
Representation
of
interest
Researcher
system
Empirical system
“The challenge of QSR then is to provide calculi which allow a machine to represent and reason with spatial entities of higher dimension, without resorting to the traditional quantitative techniques prevalent in, for example, the computer graphics or computer vision communities.” (Cohn, 1997) Applications of QSR can be found in geographical information systems (Worboys, 1998), spatial query languages (Clementini et al., 1994), natural languages (Asher and Vieu, 1995) and many other fields. Evidence that QSR is now firmly established in AI are numerous presentations, workshops and tutorials which have been given at important AI conferences, most recently at COSIT’97, KR’98 and ECAI’98. We invite the reader to consult Cohn (1996) and its updated version Cohn (1997) for an introduction and an overview of current trends. In the wider context of formal ontology, the special edition of the International Journal of Human–Computer Studies 43 (1995) exhibits the width and depth of the area. The basis of QSR are “part – of” and “connection”, respectively, “contact” relations. The formalization of the “part – of” relationship goes back to the mereology of Le´sniewski (1886 – 1939), developed from 1915 onwards. One of the main concerns of Le´sniewski was to build a paradox–free foundation of Mathematics, one pillar of which was mereology, or, as it was originally called, the general theory of manifolds or collective sets (Le´sniewski, 1928). Mereology was subsequently taken up by Leonard and Goodman (1940) (though, for a somewhat different purpose). Formally, Le´sniewski’s mereology and the calculus of Leonard and Goodman – the classical mereology (CM) – are the same. Based on classical mereology, and the work of Whitehead (1929) on the relation “! is extensionally connected with " ”, Clarke (1981) presents an axiom system for a “Calculus of individuals” based on a “connected – with” relation # . The intended domain is such that “ . . . we may interpret the individual variables as ranging over spatial–temporal regions and the two–place primitive ‘! is connected with " ’ as a rendering of ‘! and " share a common point.” (Clarke, 1981, p.205).
2
Suppose that $ is a nonempty set of regions, and # a binary relation on % ; we let #&!('*)+"-,-$ !/#0"21 . If 3546$ , and !7,8$ , then ! is the fusion of 3 , if
.
#0!9';: #&" ?
(1.1)
Clarke assumes are two mereological axioms: A 1. # is reflexive and symmetric, A 2. If #0!9'@#&" , then !A'B" , and one axiom concerning the fusion: A 3. If 3C46$
is nonempty, then the fusion of 3
exists in $
..
If a region ! is not connected to every other region, then the complement DE! of ! is defined as the fusion of all regions F which are not connected to ! . In other words,
#HG DI!/JK'
(1.2)
: #VFW? L MONQPSRUT
Biacino and Gerla (1991) show that the domains satisfying A 1 – A 3 are exactly the complete orthocomplemented lattices in which
] !/#0"YXHZ5!\[ DE" ?
(1.3)
]
Here, is the lattice ordering; the fusion is just the lattice join. It may be worthy to point out that, although Clarke calls his operations “quasi–Boolean”, the models for his calculus are not necessarily (quasi–) Boolean algebras. Clarke (1985) adds another axiom, the purpose of which is to define a ‘point’ within his calculus. Unfortunately, the full system collapses to classical mereology, as Biacino and Gerla (1991) show, and they suggest a modification of Clarke’s system calculus: “The new system should still admit as models the class of the nonempty regular open sets of a topological space1 . . . But in these models the connection relation should be as follows: (1.4)
!S#&"HXHZ
!Y^ "7'` [ _ .”
Here, ! is the topological closure of ! . Such a system, the “region connection calculus” (RCC), was presented by Randell et al. (1992a,b), and has since received prominence in spatial reasoning (see Cohn et al., 1997, for an overview). The differences to Clarke’s system are that only the existence of the fusion of finite sets is postulated, and different notion of “complement”. Already in the early RCC presentation of Randell et al. (1992a), the importance of relational transitivity tables for qualitative reasoning about regions was recognized; recently, Bennett et al. (1997) 1
These are the models of classical mereology.
3
have raised several questions regarding the expressiveness of relational reasoning, in particular with respect to the RCC. Relational reasoning as algebraic manipulation of relations has a long-standing tradition, going back to A. De Morgan, C.S. Peirce, and E. Schröder (cf. Anellis and Houser, 1991). From the 1940s onwards, A. Tarski (who, incidentally, was Le´sniewski’s only doctoral student) and his colleagues have continued the work on the calculus of relations which eventually led to an algebraization of first order logic via cylindric algebras (Henkin et al., 1971, 1985), and its finite fragments, in particular, first order logic with three variables via relation algebras (cf. Tarski and Givant, 1987). In this paper we shall explore the relation – algebraic aspects of the RCC relations, and suggest some modifications. We also hope to answer some of the questions raised in Bennett et al. (1997). The paper is structured as follows: We first introduce the necessary machinery of relation algebras; based on these, we will then discuss some aspects of Bennett et al. (1997) from a relation – algebraic point of view. Section 4 introduces the RCC and lists some of its properties. We show that the algebraic part of the RCC leads to quasi – Boolean operations, and present a refined (weak) composition table which contains additional definable relations which do not appear in the original RCC. Finally, we investigate a reduced set of RCC relations (RCC5).
2 Relations and their algebras A relation algebra (RA)
acbedfgdihUd
awvrdxvrd+mndkjldkjldxvrd+mndkjpt is a structure of type
acb|dfgdihUd
1. 2.
D acb|dopdq}d+m s t
dkjld+mt
D
dkjld+mndopdqrd+m sut
which satisfies for all y
dxz dk{ ,
b ,
is a Boolean algebra (BA). is an involuted monoid, i.e.
acb|dopd+m s t ms (a) q~q d is aI o semigroup z q znqo with q identity , (b) y 'y Gcy J ' y .
3. The following conditions are equivalent: (2.1)
oz +h { jld qo{ hz jld {oEznq h j cG y J ' Gcy J ' G J H y ' ?
In the sequel, we will usually identify algebras with their base set.
a dd d d d dopdq d+m s t The full algebra of binary relations is a structured%Vd +G VJ on a set d , where ^ D _ ( %V+G VJ is the set of all binary relations on , ^ D areo the usual set theoretic operations, q _ are, respectively,q the empty and the universal relation, is relational composition, the a d t a d t ms b relational converse (i.e. ') ! " . " ! ,Y1 ), and is the identity relation on . A subset of %V G VJ which is closed under the distinguished operations of %V G VJ and contains the distinguished constants is called an algebra of binary relations (BRA) on . It is a subalgebra of %V+G VJ , a fact which we 4
b
]
a by t %V+G VJ . a If )%., t 194@%VG 0J , we denote the BRA generated by )%.,1 denote by % .Q, . If , % .&, , we say that is RA – definable by )% .&,W1 . If the set of generators is understood, we shall usually omit mentioning it, and just say that is RA definable. b A relation algebra is called representable if it is isomorphic to a subalgebra of a product of full algebras of binary relations. The logic of RAs is a fragment of first order logic, and the following fundamental result is due to A. Tarski (see Tarski and Givant, 1987):
d
d
a
d
d
t
d t relations on % is thea setd ofd all binary Proposition 2.1. If %& ?i?i? %&,%V+G VJ , then % ?i?i? which are definable in the (language of the) relational structure %& ?i?i? & % by first order formulas using at most three variables. b
ms
b
b
a d t b = An RA is called integral, if is an atom of . If ' % a is d a BRA, t then is integral if and only if no proper nonempty subset of is definable in the model % = by a formula with at
most three variables (see Andréka det al., 1995b). b b d d a d t ] and ! " F8, we usually write !Q0" if ! " , , and Suppose that %V G VJ . For , !QV" F means !Q0" and " F . With some abuse of notation, we let %&!5. ')+" ,¡ ¢.! %&"21 be the range of ! w.r.t. R. a d t a d t
b group of , and ¥,B£I¤ ; we will write ¥ Let £I¤ be the symmetric under ¥ is denoted by %V¨ , i.e. The image of %§,
! "
a d t a d t % ¨ ')¥ ! " . ! " ,©%g1~?
(2.2)
instead of ¥G¦!/J ¥G¦"J .
b
If %V¨©'ª% b , we call % invariant under ¥ . The permutation ¥ is called ab base automorphism b¬« of , if every %§, b« is invariant under ¥ . The set of b all base automorphisms b« of is denoted by ; itd is easy to see that is a subgroup of £ ¤ . We call b permutational, if is transitive, i.e. for all ! "8,\ b« there is some ¥, such that ¥G¦!/J' a " d . Ift is permutational, then no proper nonempty subset of is first order definable in the model % d =® . Conversely, if ¯ is a subgroup of £¬¤ and !
",- , we set d d ¯ Tn° < ')¥G¦! "WJE.r¥\,(¯|1 d and let ¯&± be the BRA on generated by )¯ Tn° < .r! "©, e1 . Observe that the sets ¯ Tn° < are just the b ¯&± , and orbits of the action of ¯ on &² , and hence a partition of ² . Indeed, each ¯ Tn° < is an atom of every atom of ¯&b¬ ± « has this b form. The assignments ³ and ´ form a Galois connection, and is called Galois closed if ±µ' (see Jónsson, 1984, Börner and Pöschel, 1991, Andréka et al., 1995a). b
b b first order closed if every first order definable binary relation in the language A BRA on is called of is an element of . The next result is most likely known, and its easy proof is left to the reader: b
Lemma 2.2. If
b«
is integral and first order closed, then
is transitive.
The following result gives a connection between Galois closure and first order closure (Jónsson, 1991, Andréka and Németi, 1991): 5
b Proposition 2.3. If
b
'
b «
± iff
b
is finite, then is closed under every permutation invariant operation on binary relations.
We shall need this in our discussion of RCC5 in Section 4.4. The concept of residuation will be of importance in our later considerations. It will turn out that many theorems of the mereological part of spatial relations are consequences of the residual operators, since the “part of” relation turns out to bedxthe residual of the “connected to” b z right b o relation. z o z
Suppose that is an RA, and that y , . Even though z the equations z¬¸ y !©' and ! y¶' do not always have a solution, there are always elements y|· and z y , called, respectively, the right and left residual of by y such that
o
y o !
z
]
zd
]
y& z XHZC! z¹¸ · ] ! y XHZC! y2? z The residuals can be expressed as RA terms in y and by z qo z d (2.3) y0¸ · 'ºDeGcy zKo D q J z¹ (2.4) yµ'ºDeG D y J? z If y' , we shall only speak of the right (left) residual of y . These residuals have the following properties: Lemma 2.4.
¸
1. y0·y and y
2. If y is reflexive, then y0·y
]
]
y are reflexive and transitive.
3. If y is symmetric, then y&·y
]
y . y if and only if GcyV·yWJ
qo
Gcy0·yWJ
]
y .o
Proof. A proof ofm s1.o can be found in Pratt (1990). For 2., o o the monotony of ] ] imply that DIyµ' DIy y D¬y . Thus, yV·yµ'§DeGcy DIylJ y . Suppose that y is symmetric. Then, one line implies the next:
and the reflexivity of y
] yV y o o ·y ] Gch yV·yWJ o DIy D y o jy I GGcyV·yWJ DI D Gcy h DIyWJ»' j qyWo J e GGcy0·yWJ cG yV·yWJJ DIyµ' ? q o q\h j ] qom suppose j Gcy©·gyWJ Gcy·Yo yWJ h~ym s , andj assume that Gcy©·gyWJ D y¼' [ . Then, I Conversely, that s h GGcyV·yWJ J DIy7' [ , and hence, GcyV·y DIyWJ ' [ . On the other hand, qo qo h j o h j ] cG y&·yWJ y0·y yµXHZ½GGcyV·yWJ y0·ylJ yY' XHZ;GcyV·y DIylJ yV · yH' ? (2.5) Reflexivity of y0·y implies that (2.6)
Gcy0·y
o
DIylJ
hpm s ]
GcyV·y
which proves our claim. 6
o
h
DIyWJ yH'
jld
If %
d
¡,%V+G VJ , then the residuals are given by the conditions d
! Gc%6¸ ·IJ"µXHZ¿GÀ FpJ GcF~%&!©Á5Fp¾"J d ¾ !¾G %eJ"µXHZ¿GÀ FpJ G¦"l%0FgÁC!SÂFpJ (see e.g. Jónsson, 1991). We also use the following conditions (Jipsen, 1992):
b
Lemma 2.5. Let
ms
be an RA and y
oz
,
{
z
b
j
·) 1 . Then, qo{
] ] , then y . (Integral lemma) z z o{ K z o {Ãq ] ] 2. If is an atom and y , then . (Atom lemma) y b bÄ b d d Suppose that is a complete and atomic RA with b atoms b bÄ relational b { G b JY'Å)yÆ ?i?idx?z y}ÇQ1 . { Then, ] composition can be interpreted as a mapping È7. oEz ÁÊÉ8b¬G Ä b J withbÈËÄ Gcy b JE'Ì) , G J. y 1 . Indeed, it is sufficient G J. b to look d at the restriction of È to d G Ï J 1. If
]
q»o
dxz dk{
y
y , and y
a list of all atoms below Theo composition table of is an G¦Í ÍÎJ –matrix where entry G¦ J contains b y ynÐ ; an example is shown in Table 1. A complete and atomic RA is completely determined by
o Table 1:Ñ A composition Ѿq Ò tableÓ Ò Ñ Ñ Ò Ó Ñ¾q ÑÔdcÑÂqd+m s DѾq Ò Ñ¾qdkÓ Ò Ò Ò Ò Ò DÒ Ó ÑÔdkÓ Ó Ò D bÄ b m s m s table of its b set of atoms G J . When writing such a table, we will omit the relational composition column and row
, if
is an atom of .
In our construction of RAs we have been aided by the RA Scratchpad, designed and written by Peter Jipsen (1992). For other properties of relations and their algebras see Jónsson (1982, 1991) and Andréka et al. (1998); we recommend Grätzer (1978) as a reference text for lattice theory, and Koppelberg (1989) for Boolean algebras.
3 Weak composition A more general view of composition is taken in Randell et al. (1992a) and Bennett et al. (1997); there, a compositiona table d t (CT) is just a daÖlmapping t ÈA.2ÕA©ÕÁCÉ8G Ö ÕËJ , where Õ is a set of relational symbols. AÖ model of Õ È is a pair , where is a set and .IÕ@Á %V G VJ is a mapping such that ) GcyWJ.~yA, ÕÎ1 is a partition of - . The model is called consistent if
{
(3.1)
,©ÈËGcy
dxz
Ö
oÖ z Ö { JXHZ½G GcyWJ G J JË^ G JV'` [ _r?
7
dxzdk{
for all y ,×Õ . In the sequel, we will call such a table a weak composition (table) to distinguish it from the usual relational composition. Bennett et al. (1997) call a weak composition table extensional, if
Ö
(3.2)
GcyWJ
oÖ z G J'
Ö { : G J? Ø = ÙMÛÚi° ÜR
Then, they write “One might . . . conjecture that by refining relations in a set Rels one can always arrive at a set Rels’ which is more expressive than Rels and whose CT can be interpreted extensionally.”
a d t a dÖlt Ö { { If Õ È has a model at all, Ö then, { in { case ) G J. ,§ÕÎ1 is closed under converse, and the identity is a union of elements of ) G J0. 6 , ÕÎ1 , its elements are the atoms of a relation algebra iff the table is extensional (see e.g. Jónsson, 1984). Given a partition Ý of @8 , the relations in Ý will always generate a relation algebra which has an extensional table in case it is atomic and closed under arbitrary unions. A weak composition table is called complete w.r.t. a theory Þ whose language contains Õ ßnà á if, “. . . whenever a set of (ground) facts involving only relations in Õ/ßnà á and constants is inconsistent, this can be detected by reference to the table.” (Bennett et al., 1997) Bennett et al. then conjecture that “. . . a CT is complete w.r.t. a theory Þ iff Þ extensional interpretation of composition.”
implies all formulae corresponding to the
ms
j s following is a simple b The counterexample: Consider the RA with the two atoms (identity) and Ï d Ï on a three element set. Let b Þ say that there are four (diversity), and let j s be itsd representation ]6â ¬' [ 1 . Then, Þ is not satisfiable in , but each triangle is. elements, e.g. Þ@')+!2 ! Ð .~
4 The region connection calculus The Region Connection Calculus (RCC) introduced in Randell et al. (1992a) is a system similar to the mereological and quasi–Boolean part of Clarke’s calculus of individuals, the difference being the definition of complementation. The base relation is a connection # , and further relations are obtained
8
from # by the relational operators as in Clarke’s system:
'@#6·E# m s V'q^-o D ã ' q ã ã ' ^-D|Gc o J ä V'V^ GcåY# åYe # J æ ä ä V'V^-D V ã åg#'@#6^-D ç #'ºD#
part of proper part of overlap partially overlap tangential proper part non–tangential proper part externally connected disconnected ?
4.1 RCC axioms
æ
d
æ
A model for the RCC consists of a base set è'é% , where % are disjoint,f a distinguished ê ,©% , a unary operation h D`.}%&¬Á5%& , where ê %&&.r'%6·¬) 1 , a binary operation .}§ % ©%Áë% , æ , and a binary relation # on % . another binary operation .~%§8%Á5% There are 8 axioms for the RCC: RCC 1. GÀ !(d ,©%eJ!S#&!
RCC 2. GÀ !
",%eJ ì !/#0"µZC"#&!2í RCC 3. GÀ !(,©%eJd !S# ê RCC 4. GÀ !(,© a % d "At ,©%&J ,
æ ä
D " t ,(#X¶Zëî! V0" (a) a ! d E ã (b) d ! d E D " , a d X¶Zë f ît ! 0"
RCC 5. GÀ ! d " d F¶,©%eJ ì a ! d " h Ft ,(#XHZï!/#0" or !S#0F RCC 6. GÀ ! d " ¶ F ,©%eJ ì h ! " F ,8#X¶Z½Gðrñº,©%eJ G¦ñ0" and ñVF and !/#0ñ&J í RCC 7. GÀ !
",%eJ ì !
"A,©%XHZ5!
ã
"pí
RCC 8. If ! 0" and "r0! , then !A'B" . We have added RCC 8, since this is what is intended, but it does not seem to follow from the other axioms. By the definition of , Lemma 2.4.1, and RCC 8, we see that is a partial order. The original RCC system contains another axiom:
æ ä
d
“. . . A rather deep theorem of the theory is given by the formula ÀQ!Sðr"/ì VòG¦" !SJ í which was demonstrated by informal argument in Randell, Cohn, and Cui (1992a). Because we have so far not been able to give a fully formal proof of this theorem we often regard the formula as an additional axiom of the theory” (Cohn et al., 1997). Below follows a simple proof of this property:
9
Lemma 4.1.
æ ä
GÀQ!7,©%eJ Gðr"A,©%eJ"
(4.1)
VV! æ ä
V0! ; By RCC 4a, this Proof. Assume that there is some !§,% such that for all "@,% , îÂ" o t is the largest relation Ä on R with implies that "W#ªD¡a ! d for t all ",% . Since ó'ôo #a ·V# d – i.e. ] ] # a¦Äd t # –, and ! DE! ,© [ , we obtain that # ) ! DE! 19[ # . Hence, there is some ,8% such DE! ,7 [ # , a contradiction. According to Randell et al. (1992a), the weak composition table has the form given in Tab. 2. Since there are eight base relations, the system is called RCC8. Table 2: RCC8 weak composition table C
õ ö
O
DR DC
PP EC
PO
DC EC
1 DR,PO,PP
PO TPP
DR,PO,PP DC
NTPP TPP
DC DR,PO,PP
÷
DC EC,PO,PP
÷
DR,PO,PP
÷
PO,PP
÷
NTPP
÷
DR,PO,PP 1’,DR,PO, TPP TPP DR,PO,PP DR
÷
÷
÷ ÷
NTPP
TPP
PP
÷
DR,PO,PP EC,PO,PP
DR,PO,PP PO,PP
DC DR
1 DR,PO,PP
PO,PP PP
PO,PP NTPP
DR,PO,PP PO,PP
NTPP 1’,PO, TPP,TPP PO,PP
NTPP PO,PP
DR,PO,PP 1’,DR,PO, TPP,TPP DR,PO,PP PP
PO,PP
÷
÷
÷
ø
O 1’
o ù
÷
DC DC
÷
÷
DR,PO,PP DR,PO, PP
÷
1 NTPP
÷
NTPP
÷ NTPP
DR,PO,PP DR,PO,PP
÷
÷
TPP
÷
÷
÷
÷ o ù
ç
NTPP
that the weak compositiono åg#ú' Recall is to be interpreted as minimalo inclusion, e.g. # ã ã ç ç ç ç % V , and # åY# intersects each relation on % V means that # åg#ª4 the right hand side. The table is, in a way, a minimal requirement: It follows from the RCC axioms that each RCC model must satisfy the condition given in the table. Michael Winter has pointed out, that Table 2 has an extensional interpretation, namely, the closed circle algebra introduced in Düntsch et al. (1999b). There, the domain of regions is the collection of closed circles in the Euclidean plane, and !/#0"YXHZ !µ^9"7'@ [ _.
4.2 RCC models are Boolean algebras As in Clarke’s system, the operations of the RCC axioms are called “quasi–Boolean”. In contrast to Clarke’s operations – which define the more general orthocomplemented lattices –, our next result shows that RCC operations indeed define a Boolean algebra, if we extend them and the relation j æ over the set ') 1 in a natural way2 . Proposition 4.2. Suppose that
2
æ
j
j
') 1 , and let %Vû '% ) 1 . Then, a dfgdihUd dkjld t ü ê ' % û D
The same result has been independently obtained by John Stell (1997).
10
is an atomless Boolean algebra with natural order .
j
jld j ê h jþ ý ÿ such a way that 0 in ! for all !f , %Vû ; furthermore, we set D ' D ' ê , Proof. We extend ã and also, ! " ' XHZ î¾! " . Finally, we extend in the obvious way. It is clear that all these
extensionsa can d be t reversed uniquely, so that we can always return to the original structure. Note that %
is atomless by Lemma 4.1. The claim follows now from the statements below:
1. GÀ !(,©%eJ! ê , and ê is the only element with this property. 2. ! 3. !
f
h
" is the supremum of ! and " w.r.t. . " is the infimum of ! and " w.r.t. .
4. DI! is the unique complement of ! .
a
5. The lattice % û
dfgdihUd
dkjld t ê is modular. D Ö
Ö Ö 1. The first part follows immediately from RCC 3 and theÖ definition of . Suppose that !Q for all ê ê ê !(,8% ; then, in particular, by RCC 8. a d f t . Since , we have ' a d f t " ,7 [ . Then, there is some F9,7% sucha that d f Fp#0! t and F ! " ,[ # . This 2. Assume that ! ! o F , and that ñ ! " ,¡# . By RCC 5, we can contradicts RCC 5. Now, suppose that ! VF and "re a f d that t ñ0#&! . Now, ñ0#&! VF , and # ô4§# implies ñ0#0F . The definition of now assume w.l.o.g. " F ,© . h gives us ! a d h t
d td¹a F d ! t " o,6# . By RCC 6, there a ish dsome t ñ¼,¡ a % h such d t that 3. Suppose w.l.o.g that ! "\,¡a % .d Let ñV! ñV" , andd Fp#&ñ d , and thus, F ! F " ,(# *46# .a This d h shows t ! " ! , , ! " " ,© . Now, let F¶,©% F~V! F~V" , and ñ0#0F ; we need to show that ñ ! " ,(# . This follows immediately from RCC h j a d 6. t ã
, then î¾!Q0! f by RCC 4, a contradiction. Thus, ! DI!' by RCC 7. Let If ! DE! , F,6% , and assume that î\Fp#HG¦! DI!/J . Then, q»byo RCC 5, îFr#&! , î\Fp#ºD! , and RCC a d 4t implies ã ã æ ä æ ä æ ä æ ä that F V0! and F @D! . Now, V V V*4 , and it follows that ! DI! , ,a contradiction. a d t
4.
Next, we show that D|G DE!/JK'B! which will be needed for the uniqueness proof. Since DeG DI!/J DE! , [ ã ] ] , we obtain DeG DI!/J ! . For the converse, assume that ![ DeG DI!/J . By definition of , there is æ ä some o ñ½, % such that ñ&#&! and q o î¾ñ&# DºG DI!/J , i.e. ñ V D! . Now, ! åg#¢DB! , and thus æ ä æ ä ç !QåY# V F ; however, åY# Vª4 # , contradicting !/#0ñ . f d h j
It remains to show that DE! is the unique complement of ! . Suppose that ! "µ' ê ! "H' . Then, ã ] ] î¾! " by RCC 7, and h RCC j 4b implies that " DI! . Assume that DE!B[ " ; then, again d by RCC 4b, ã æ ä æ ä DE! D(f" , i.e. DE! DE" . With (4.1) choose some ñº,© a % d such t that ñ `D7! ñ V VD7" . æ ä ê Since ! "µ' , let w.l.o.g ñ�! . By RCC 4a we have ñ DE! , [ V , a contradiction.
]
]
We also note that ! " implies DI" D ! : Assume not; then, there is someo F¶,8% such that Fp#D7" . E æ ä æ ä æ ä æ ä ] i.e. îÂF V0" , and îKFp#@D! , i.e. F 0! . Since ! e " and e ¢4 V , we obtain æ ä F V0" , a contradiction. 11
It follows that D also fulfills the De Morgan condition
]
“ ”: Since !
]
f !
]
f
" and " ]
a d! t " D G DE! e “ ”: We show ,[ % . contradicting ! DE! ©
D|G¦! f
!
"J»'ºDE!
]
"J
DE"9. f
DE! and D6G¦!
]
"J
DE" ?
] DI"WJ . Assume w.l.o.g. that !B[ e D G DE!
5. To show modularity, it is enough to prove
d
If F}0!
(4.2)
h
" , we have f
D|G¦! h
f
f
then F
h
G¦!
h
"WJK'!
f
G¦"
FpJ
h
DE"WJ ; then, !
ã
G DE!
h
DE"WJ ,
d
see Grätzer (1978), Lemma I.4.12.
]
“ ”: In any lattice we have
f F
(4.3)
]
see Grätzer (1978), Lemma I.4.9. F “ ”: We first show
Ä
(4.4)
Ä
If
!
h "
]
f
GcF
Ä
DIF and
]
f
!/J GcF f
! implies F
]
h
"J
d
!A'! , and the claim follows. f "
d F
Ä then
]
"Q?
ã
ã
Proof. Assume not;Ä then, D\" by RCC 4b. By (4.1) and theÄ definition of there is some ñÌ,-% æ ä æ ä ç ç such that Ä ñ V and ñ e D" . It Ä followsf that ñ&# , ñ " , and ñ F , the latter because æ äIä ] ] ñ DIF . On the other hand, since " F , we have #0" or n#0F for any with n# , a contradiction. Next, we need (4.5) Proof. Since !
h "
]
h
! and !
!
h
"
f
!
h
DE"H'B!Î? f h ] ] D " I ! , we have ! " ! DE" ! . h
Conversely,
! " # %$
&' %$( &) %$ &' %$
æ ä
and & %$+*
æ ä
f
Thus, if this is true, there some ñ such that ñ V0! and ñ V¶G DI! DE"J , and from (4.4) we ã ã ] obtain that ñ DE" ; it follows that ! D " . Similarly, we see that ! " , a contradiction.
h
f
]
Now, assume that ! G¦" FpJ0[ F is some ñ,©% such that
f
!
h
" , and w.l.o.g. !
12
h
G¦"
f
FpJ,©% . Then, by definition of , there
1. ñ0#HG¦!
h
2. î8ñ0#HGcF
G¦" f
f
FpJJ , and h ! "J . Ä
The first condition says with RCC 6 that there is a ,©% such that
Ä
]
(4.6)
! and
Ä
]
Ä " and & # ñe?
The second condition tells us with RCC 5, RCC 4a, and RCC 6 that ñ
æ äIä
DF , and
] ! and " imply î8ñ&# ií? Ä h Î j Ä d æ ä D F . If ' , then î #0ñ , since ñ ¬ DF , a contradiction. Otherwise, e,8% V Set ' ] " , the latter by (4.4). Now, and Ä ÄÂh f×ÄÎh by (4.5) ñ&# X¶Zïñ&#HÄÂG h F D¬ ľFph J by RCC 5 X¶Zïñ&# ÄÂh F or ñ&# DIF d æ ä since ñ X¶Zïñ&# DIF VDF X¶Zïñ&# GÀ,nJ ì-
(4.7)
]
]
! ,
which contradicts (4.7).
4.3 Refining the RCC table It is pointed out in Bennett et al. (1997) that the RCC axioms do not take into account that the largest Ï RCC m table to take care of this fact. Set @'`% , region ê isd definable. Our next task will be to refined the ] ê ê d Ï Î¶ m Ð for ¾¬') 1 ¹Æ¹'%(·¾) 1 , and ¾ Ð ' ; it is easy to check that for all base relations ] of the RCC (listed on p. 8), and ,
¾ Ð 4 or Î Ð ^\'`_r? Now,
ms
ms
o
¾m s ¬' m s ^-DeìUGcV d V . ' m s ^o D& Âo& m s d ÆÆ' . o ² o . d ¾Æ'  o ² o ÆÆ d Æ ' ÆÆ ²
q
ms d J ^ í S
which shows that all ¾ Ð and . are RA definable. The equation which tells us that ê is the largest element with respect to now is (4.8)
Æ 046E? 13
d Thus, in the sequel, we shall restrict the relations to %§·Y) ê 1 . In order to show that the defining ã ä equations on p. 8 and the axioms still hold, it suffices to prove it for V , and the axiom RCC 6, since all other definitions, respectively axioms, are universal, and thus carry over to substructures. This is straightforward, and is left to the reader; note that complementation of relations is restricted to %6·) ê 1|©%6·¬) ê 1 .
Let / be the incomparability relation, i.e. ª / 'ºDeGc åg# by
å # Y æ åY# and
ã
by
q
J . We extend the original RCC8 by replacing
o
q» qo d 'ºDeGcV V d V V|J ç 'åg#^7DIåg#
ç
q¹o o q d ã æ ' /@^-GcV q¹o V 0 |JË^-Gce V o J q ã ç '0/@^-GcV |JË^7D|GcV V V J ?
Then,
d
ç
Q! åY# "µXHZ5!A'ºDE" f d æ ê !QåY# µ " XHZ5!Qåg#0d " and " '[ f h ! jld d ã æ ê !Q " XHZ5!,/g" d ! h "7' [ jld ! f µ " '[ ã ç !Q " XHZ5!,/g" ! "7' [ µ ! " ' ê ? H This gives us 10 base relations, and we call the resulting system RCC10. The extended weak composition can be found in Table 3 on the following page. For cells containing ' , the RCC axioms together with general RA properties such as Lemma 2.4 or the equations (2.1) imply that strict composition (i.e. equality) holds; for cells containing ' [ , there is a model in which the composition is strictly smaller than the cell entry. For cell entries which can be shown to be below the weak composition we use the superscript 1 . In this way, we indicate in which cells the composition is extensional, and when it need not be. In computing the table, we have used the RA scratchpad, which in turn uses Lemma 2.5 and the equations (2.1); we are grateful to Michael Winter who spotted and corrected several inaccuracies. We have also used the following properties, which may be interesting in their own right.
æ
ä "HXHZC! V`D " . f ç ä FpJ . 2. If ! #VF , then ! VòG¦! f æ ä æ ä æ ä 3. ! VVF and " VVFgXHZ½G¦! "WJ VF . V h æ ä ä 4. If ! eVF , then DE! F VVF .
Lemma 4.3.
1. ! åg#
14
23 4
4
4
4 576
8
4
4
15
TPP, NTPP, PON, POD, ECN, ECD ( ), POD TPP, NTPP, PON, ECN, DC
4
ECN
4 4
4
4
4
4
4
4
4
4
4
8
8
4
8
TPP, NTPP,= NTPP, =
4
DC DC
8
POD TPP, NTPP, PON, POD, ECN, ECD, DC,=
576 4
TPP ,= TPP, NTPP, PON, ECN, DC, =
1’, TPP, TPP , PON, ECN, DC
1’,= NTPP, =
TPP, =
4
4
ECN,= DC =
4
4
TPP , NTPP ( ), PON, ECN, DC NTPP ,= 1’, TPP, TPP , NTPP, NTPP , PON, ECN, DC
NTPP ,= 4
TPP, NTPP, PON ( ), ECN, DC ( ), PON,= TPP, NTPP, PON, ECN, DC,=
4
TPP , NTPP ( )
TPP , NTPP
4
TPP, NTPP, PON, POD
1’, TPP, TPP , NTPP, NTPP , PON, POD,= TPP, NTPP ( )
4
TPP , NTPP , PON, POD, ECN, ECD, DC,= DC, =
PON, =
4
TPP , NTPP , PON, ECN, DC,
4
POD
TPP, NTPP, PON, POD, =
TPP , NTPP , PON, POD, ECN, ECD, DC, = TPP , NTPP , PON, ECN, DC, =
DC
TPP , NTPP , PON, ECN, DC
DC, =
DC
4
TPP , NTPP , PON, POD( ), ECN, ECD( ) ECN, DC
4
POD
4
1’, TPP, TPP , NTPP, NTPP , PON, POD, ECN, ECD, DC,= TPP , NTPP , PON, POD,=
4
POD
4
TPP , NTPP , PON, ECN, DC,=
POD
TPP , NTPP , PON, POD
DC, =
POD
ECN, =
ECD
4
8
ECD DC
576
TPP , NTPP , PON, ECN, DC
8
TPP , NTPP , PON, POD, =
4
4
TPP, NTPP( ), PON( ), POD
4 4
PON
4
1’, TPP, TPP , NTPP, NTPP , PON, POD, = TPP( ), NTPP, PON, POD( )
8
NTPP , =
576
TPP , NTPP , PON, POD
4
TPP, NTPP, PON, POD, ECN, ECD, DC,= POD
4
TPP , NTPP , PON, POD, ECN, ECD, DC,=
ECN, DC
ECN
4
NTPP
576
1’, TPP, TPP , NTPP, NTPP , PON, ECN, DC, = NTPP
8
NTPP
4
TPP, NTPP, PON, ECN, DC,
TPP, NTPP, PON, POD ( ), ECN, ECD ( ) POD
POD
8
NTPP, =
8
TPP, NTPP ( ), PON ( ), ECN, DC ( ) TPP , NTPP ( ), PON ( ), POD TPP, NTPP, PON, ECN, DC, =
PON
4
NTPP
TPP, NTPP, PON, POD
4
TPP , NTPP
8
TPP , NTPP ( ), PON, ECN, DC, NTPP , =
NTPP 8
1’, TPP, TPP , PON, POD
NTPP, =
NTPP 8
TPP
4
1’, TPP, TPP , PON, ECN, DC,
TPP
4
TPP, NTPP
TPP 4
TPP
Table 3: The RCC 10 weak composition table
4 4
4
576
8
4
4
8
4
4
4 8
576
8
4
4
8
8
f
"7' [ ê . Then, h lj d æ ! åg# "HXHZï! "µ' d !S#&" XHZï! 9@DE" d !/#0" æ ä XHZï! 9@DE" !¾G D |JÂD-" e ä XHZï! V@D " ?
Proof. 1: Suppose that !'º [ DE" , so that !
f f f # F ; then, F:9CDE! and !¾G D#|J F . Since 0 !/#*D6! , and DE!'óF DeG¦! FpJ , we have æ ä !/#@D6G¦! FpJ by RCC 5.d Thus, !¾G D VgJ G¦! FrJ by f RCC 4a. f æ ä æ ä æ ä 3: “ Z ”: Let ! VVF " VVF and assume îG¦! "J VVF . Then, by RCC 4a, G¦! "J#0F , æ ä VVF , a contradiction. and RCC 5 implies that w.l.o.g. !S#0F . RCC 4a now implies îÂ! f f æ ä æ ä “ X ”: Let G¦! "J eVF , and assume that î¾! VVF . Then, !/#VF , which, together with îEG¦! "J#0F contradicts RCC 5. h æ ä VgJ! and 3 by setting "µ'ºDI! F . 4: This follows from !¾G D 2: Let ! f
ç
There is no relation algebra which is a model of the RCC10 table. In the standard model of nonempty proper regular open sets of a regular connected topological space, there are at least 25 atoms Düntsch et al. (1999a).
4.4 A reduced set of RCC relations
m sd
d
ç
ã
d
d
q
% V V 1 of RCC relations has received some attention, and is usually The subset ) ã ä called RCC5. It arises from disregarding the split of # into and åg# , and e into V and ã æ ä V ; in other words, one adds the additional axiom #¢' . If one takes the weak composition induced by the RCC8 table, one arrives at Table 4. o ù DR PO PPq PP
Table 4: RCC5 weak composition table C=O DR PO PP 1 q DR, PO, PP DR q DR, PO, PP
DR, PO, PP 1 DR, PO, PP q PO, PP
DR, PO, PP PO, PP PP
D
ç
%
q PP DR q DR, PO, PP 1q PP
If we take into consideration that the largest region ê is RA definable and work within %6·¬) ê 1 , then, ã ã æ ã ç ç çòæ çòç similar to the RCC10 table, splits into and , and % splits into and , where ç9ç çòæ ç çòç is the complement, and . The composition induced by the RCC10 table is ' %`^D given in Table 5.
jld+m The next proposition shows that the RCC7 table – and thus RCC5 – has a very simple interpretation ü ü ü ·e) 1; (Düntsch et al., 1998). Suppose that is an atomless Boolean algebra, and that ¶' 16
Table 5: RCC7 table DR ;
DN DD PON POD PP PP @
also, let §' ü on :
O
DN
DD
ACB
