Algebras of approximating regions

Fundamenta Informaticae XX (2000) 1 12 1 IOS Press Algebras of approximating regions Ivo Düntsch School of Information and Software Engineering University of Ulster at Jordanstown Newtownabbey, BT 37 0QB, N.Ireland I.Duentsch@ulst.ac.uk Ewa Orłowska Institute of Telecommunications Szachowa 1 04 894, Warszawa, Poland orlowska@itl.waw.pl Hui Wang School of Information and Software Engineering University of Ulster at Jordanstown Newtownabbey, BT 37 0QB, N.Ireland H.Wang@ulst.ac.uk 1. Introduction It is rarely the case that spatial regions can be determined up to their true boundaries, if, indeed, they have such boundaries; in most cases, we can only observe regions up to a certain granularity. Often, this is a desirable feature, since too much detail can disturb the view, and we will not be able to see the forest for the trees, if our desire is to see the forest. Having as our basic assumption that regions can (or need to) be observed only approximately, we want to find an operationalisation of the domain of regions, which is broad enough to express the properties The authors gratefully acknowledge support by the KBN/British Council Grant No WAR/992/174

2 author / short title which we want to study, and, at the same time, has enough mathematical structure to serve as a reasoning mechanism without being overly restrictive to our intuition. We make three model assumptions: 1. The first assumption is that there is a partial order, called the part-of relation (or relation of part ), defined on the collection of regions. We say that two regions overlap, if they have a common part, in other words, if there is some such that and. 2. The second assumption is that there is a collection of crisp or definable regions, which forms a Boolean algebra with natural order, such that restricted to coincides with. The crisp regions delineate the bounds up to the granularity of which other regions can be observed. The power of observation is expressed by pairs of the form, where are definable regions. In other words, to each (unknown) region there is associated a lower bound and an upper bound!", both of which are crisp, up to which is discernible. If!#, then itself is definable. The pair $ % is called an approximating region. 3. The final assumption is that the bounds are best possible; in other words, if is a region approximated by, then (1.1) (1.2) No definable region & with (')& is a part of If &*', then overlaps with +,&. This implies that for each approximating region -. there is a collection /0 1 of regions each of which is approximated by, and for which (1.1) and (1.2) hold. Furthermore, if is an approximating region different from, then /0 243-/0 5!#6. The example below shows that these assumptions are fulfilled in an important area of application, namely, screen resolution: Consider the region 7 in the Euclidean plane, depicted in Figure 1. We suppose in our example that granularity in the plane is determined by an equivalence relation on the points, the classes of which are the atoms of the Boolean algebra of definable regions; these are drawn as squares. We can, for example, think of the squares as pixels on a computer screen. The region 7 can only be discerned up to the bounds given by its lower and upper approximation, each of which is a union of squares, i.e. (1.3) (1.4) 8709;:< -=?>A@2BC EDF7HGI 8709;:< -=?>A@2BC J3K7ML #65G is the lower, resp. upper approximation of 7. Here, B< N:< O@I BC PG is the equivalence class containing. It is obvious that our three model assumptions are fulfilled. This is related to the rough set approach to data analysis of [16]; similar paradigms have been put forward in the field of spatial reasoning by Lehmann & Cohn [12] aand Worboys [20]. This paper is organised as follows: Firstly, we will define the class of approximation algebras which is then shown to be equipollent to a class of well known algebraic structures. Secondly, we suggest a way to generalise contact of regions to the approximate case. Finally,we will give an outlook to future work.

author / short title 3 Figure 1. An approximated region 2. Approximation algebras In this Section, we will translate the intuitive assumptions of the previous Section into a formal system which will reflect the ontology that we have in mind. Even though we will appeal to the spatial intuition, we will, however, put no formal restrictions on the kind of object that we want to approximate. We will then proceed to show that this system is term equivalent to a well known equational class of algebras. We assume a basic knowledge of the theory of ordered structures, lattices, and Boolean algebras, and we invite the reader to consult [3] for terminology and concepts not explained here. If no confusion can arise, we shall identify algebras with their base set, and, with some abuse of language, we will describe algebras by the name of their classes. For example, BA will be the class of Boolean algebras, and we also call a Boolean algebra a BA. If Q is an ordered structure with smallest element R, we set Q!S,#Q-TU:<R5G. Throughout, O VW YXZ [+\ R] [^ will denote a Boolean algebra (BA), with the base set possibly indexed. We may think of as an algebra of definable (or crisp) objects within some domain as mentioned in the introduction. Since we intend to identify approximate objects with pairs of definable objects from below and above, we start by setting (2.1) W_ `$ab:i c@2dygie

4 author / short title We regard _ `$a as a sublattice of gfh, so that Lower and upper approximation are defined by We observe that Vi&C j59nkvf&c 9VFj5 XC&C j59nlx&c mx[j5e n$ %4o 5 n % oc e (2.2) 9p n q%4# n$ % 4 n %%4# n %e We can recover by identifying with :I 5,@]=0 rg. Thus, an approximating region is definable, if it is equal to its lower and upper approximation. The operators and are a co-normal multiplicative interior, respectively, a normal additive closure operator, i.e. for O=- _ `sa, (2.3) (2.4) (2.5) (2.6) (2.7) n^y9o^1 WXY I!# X[ 5 WtM u) 5 u " q9# " Co-normal Multiplicative Interior and (2.8) (2.9) (2.10) (2.11) (2.12) R2!#RP lv) I!# VF9 5 WtM u)9 5 - " 9p %!# e " Normal Additive Closure Furthermore, we see that for W=- _ `$a, (2.13) 9" 5 and!# 5 imply d# e This expresses the intuition that approximating regions are uniquely determined by their lower and upper bound. The algebra _ `$a may be too large for certain situations. It describes the situation that for each v with '# there are true regions which are approximated by ; however, this may not be always the case. Thus, a less restrictive notion is required, and we generalise _ `$a as follows:

author / short title 5 An approximating algebra (AA) Q VW [Xw R] ^2 %q % is a structure of type xp x] R] %R] ^2 ^[ such that for all (=Q, (2.14) (2.15) (2.16) (2.17) (2.18) (2.19) Q VW YXZ R] ^Y is a bounded distributive lattice. is a co-normal multiplicative interior operator on Q. is a normal additive closure operator on Q. 9 % #9 9 p % # p e # p I and 9!#9 5 imply d# e Each closed element has a complement. It is not hard to see that _ `$a is an AA, and one can show that each AA is a subalgebra of some _ `sa [4]. We will denote by ypqm or just by if no confusion can arise the set of closed elements of Q. By (2.17), is also the set of interior elements of Q. Our first aim is to show that does what we want it to do: Proposition 2.1. is a subalgebra of Q and a Boolean algebra. Proof: By (2.17), the functions and are the identity on, and thus, is closed under and. Furthermore, R] [^z=- since is normal and is co normal. If 9 5m=-, then, 9 VF9p I!# lv) I since is additive, and thus, is closed under V. Now, Thus, {X[9 5!#9 %XY Iq by (2.17) #9 X 5% # X[9p I%qe by (2.17) e since is multiplicative is a RP [^ sublattice of Q. Since Q, and thus, is distributive, the complement whose existence is guaranteed by (2.19) is unique, and it follows, that is a Boolean algebra. }~ 2.1. AA and regular double Stone algebras In this section we will show that the class AA is definitionally equivalent to the well known class of regular double Stone algebras (RDSA) which have been investigated, among others, in [11, 19]. Many other classes definitionally equivalent to RDSA are known, for example, the class of three valued Łukasiewicz algebras; for more details we invite the reader to consult [2, 15].

6 author / short title For each =-Q we define (2.20) (2.21) where + Eo+, p uo+,9p denotes the complement operation in. Observe that W 4 and 9p W % for all =-Q, and that for =- we have +, d# ". Proposition 2.2. Let K=-Q. (2.22) (2.23) (2.24) is the smallest Ö=Q for which WVF lo^. is the largest Ö=-Q for which OX[ \#R. dqmcb:< @2 =-QƒGkb:< @2 =-QuG. Proof: (2.22): Since p u, and 4Vi+, p!b^, we have \VF N^. Suppose that lvh lo^.since is additive, normal, and order preserving, we have 4VF ln^ which implies that ( "+,!#. (2.23): This is shown analogously. (2.24): This follows from # 4 and # % %, respectively. }~ The operation is called pseudocomplementation, and the operation is called dual pseudocomplementation. An algebra Q, VW [XZ R] [^Y of type xp x] Y^2 ^2 %R] %R2 is called a regular double Stone algebra (RDSA), if 1. Q, VW [XZ R] ^ is a bounded distributive lattice. 2. satisfies (2.23), and satisfies (2.24). 3. % distributes over V, and 4 distributes over X. 4. # and # imply d#. In the sequel, ^ is the identity relation on the set under consideration. The following result is now straightforward to prove: Proposition 2.3. Let AA be as above, and RDSA be the class of regular double Stone algebras, regarded as categories with the appropriate homomorphisms. Define mappings and ˆ in the following way: Let Q, VW [XZ % % %R] ^[ = z, and let and be defined as in (2.22) and (2.23). Set mqc9o$q, VW YXZ R] Y^[, and mš4coš for any AA morphism. Conversely, let Q, VW [XZ R] [^Yu=0 uœl 5, and set p uo 4 and 9p!# %. Define ˆ]Qc!oQ VW [Xw % % R5 [^, and ˆ]Š4!#Š for any RDSA morphism. Then, -@2 z ŽM ƒœl 5 and ˆ @I uœl 5 #Ž z are covariant functors, and ˆ,iˆ5 % yo^.

author / short title 7 We call dqm the centre of Q, and the set hqc!b:< =Q@2 % o^cg its dense set. We shall usually just write instead of EQc, if no confusion can arise. The name of derives from the property (2.25) i.e.!# % o^2 (2.26) OX[ ( R for all =- 9 d=q!s. Note that Q0# if and only if b:5^cg. We can regard as a crude measure of the granularity level: If g:i^cg, then Qo, and all regions under consideration are definable. If W is isomorphic to, then for all z=- with d there is a unique -=Q with!# and!# ; _ `sa is an example for this situation. 3. Contact relations and their approximations A contact relation š on a set (3.1) (3.2) (3.3) (of regions) has the properties š is reflexive. š is symmetric. If šœ!išœ 5 then (" e Here, for (=-, we let šœ 2!;:[ ž= @I Ÿšl %G. š is called proper, if it is not the identity. By the extensionality axiom (3.3), a region is determined by all regions to which it is in contact. It was shown by [8] that, in the presence of (3.1) and (3.2), the extensionality axiom (3.3) is equivalent to (3.4) +l8šy Y+ušJ is a partial order., and is relational composition. This partial order will usually be denoted by, and we call it the part of relation of š. It is well known that is the right residual of š, i.e. that is the largest relation on such that šy #DFš [10]. For later use, we also let \ [ %$>\ be the set of all binary relations on a set >. Here, the complement is taken with respect to the set of all binary relations on If H [, is an ordered structure, and š a contact relation on, we say that š is compatible with (or ) if for all O=-, (3.5) It is not hard to see that (3.5) is equivalent to (3.6) l Wt e - W Wt šœ mdfš 5e

8 author / short title It was shown in [7] that on each atomless BA there are at least two compatible proper contact relations. The next result shows that on an AA Q which is not a Boolean algebra, there is no proper compatible contact relation: Proposition 3.1. If Q is an AA, and š a proper compatible contact relation on Q!S, then QH#. Proof: Let =H. Then, by (2.26), XC E AR for all E= Q S. Then, dxc 2šW and šœp (X2 5\D#š show that ]šw.it follows that šœ!išœ ^[, and therefore, do^ by (3.3). }~ Thus, if Q"L #, we cannot define a proper compatible contact relation on Q. Since each AA is obtained from its Boolean algebra of definable elements via the approximation functions, we suppose that we have a contact relation š on, which we want to approximate in a similar way. It seems natural for (=-Q to say that Thus, given ( qšj, we let and are certainly connected Jt p nšw 5 and are possibly connected Jt 9p nšw9 5e (3.7) (3.8) ]šl W Wtª nšw 5 PšJ«C W Wtª nšw9 5e For formal reasons, however, we do not want to start with š on, but with the approximation relations š %š «on Q, whose restriction to is a contact relation š such that (3.7) and (3.8) hold. Recall that for a relation its conversew is defined as W,b:5 c@2 \ PGIe An approximate contact algebra (ACA) is a structure Q, % %š qš «such that Q % is an AA and (3.9) (3.10) (3.11) (3.12) (3.13) (3.14) š š 2 šw«kišw«] ^ D š 3dš «Pšl J Wt nšl Z I ]šw«c J Wt 9 nšj«y9 5 9p nš I! Wt 9 nš «9 5 š «p9 %md š «9p I%! Wt 9 u9p Ie

author / short title 9 Proposition 3.2. 1. Let $Q, q %š qš «be an ACA, and set šaiš «`. Then, š is a compatible contact relation on, and (3.7) and (3.8) hold. Furthermore, š «` iš `. 2. If Q, % is an AA and š a contact relation on, then, š and š «, defined by (3.7) and (3.8) satisfy (3.9) (3.14). Furthermore, šoiš «`. Proof: 1. The facts that š is symmetric and reflexive follow immediately from (3.9) and (3.10). Let 9 nšj 8 s! Wt 9 5nšW98 n, and assume *L o9 5. Then, by (3.14), there is some d=hq such that 9 nš «, but not 9 5nš «. It follows from the fact that is idempotent and from (3.12), that 9 nš «9p 2, and not 9 5nš «9p 2, contradicting our hypothesis. Condition (3.8) follows immediately from (3.12), and the definition of š. Now, «š ]š W Wt nš p I By (3.11) Wt 9 % š 9 5% By (2.17) Wt 9 % š 9p 5% By (3.13) Wt nšj«y 5 By (2.17) Wt nšj Ie By definition of The fact that š is compatible with the Boolean order follows from (3.14), and š «` š ` follows from (2.17). 2. The proofs are straightforward computations. By way of example, we show (3.13): ««š I! Wt qnšj9 5% By (3.7) Wt 9 nšj 5 By (2.17) Wt 9 qnš 9p9 5% By (3.8) Wt 9 nš Ie By (2.12) šaiš «` now follows from (3.8). }~ In the sequel, we let Q q qš %š «be a generic ACA, and š±²š «relation on. ` be the associated contact Let (3.15) (3.16) B W Wtª # p I B «l Wt 9!#9 5eZe Then, B is an Q, YXZ RP ^2 % congruence, B «is an Q, VW R] Y^2 % congruence, and B 3-B «v^ by (2.13). The following is easy to prove, observing that šv=- \ Y % l :

µ ` S µ ` ` 10 author / short title Figure 2. Approximate part-of relations ³ { Proposition 3.3. š #B qšy B and š «#B «qšy B «. The question arises, how the part-of relation belonging to š (in the sense of (3.4)) can be sensibly approximated, in other words, what does it mean to say that the approximating region is (approximately) part of the approximating region? We know that the part-of relation on generated by š is the Boolean order. Besides the lattice ordering on Q which extends, there are, on first glance, several possibilities to generalise to a relation on Q : (3.17) (3.18) (3.19) (3.20) S[ W WtM9p,) 5 W WtM9p,)9 5 W WtM p u I ³ W WtM p u Ie A sketch of the D relationships among these part-of relations is given in Figure 2. Note that { 3 Following our assumptions (1.1) and (1.2), we think of lower bound as certainty and upper bound as possibility, where both bounds are best possible. With this in mind, we see that an approximated part-of \¹ relation on an ACA must satisfy (3.21) (3.22) ¹ The restriction of to is equal to e ¹ Jt uº 5 and 9p u Ie While the first condition is clear, we should explain the second one: If is approximately a part of, and if is certainly part of, i.e. part of all regions which approximates, then should be certainly part of anything which is approximated by. If 9 WL DA I, then no region approximated by has a i.

~} author / short title 11 common part with 9!XI+, 5, but every region approximated by has. Thus, there can be no region in 9 which is part of some region in 9 5. With these observations, there can be only two suitable ordering relations, namely, the lattice ordering, and the ordering S, denoted by», which is defined by (3.23) -» Wtª 5e The lattice ordering can be interpreted as possible part-of, while»v says that anything which approximates is certainly a part of anything which approximates. Finally, we present the following compatibility Theorem: Proposition 3.4. For all (=-Q, (3.24) Jt šl % udfšl % 5 and š «udfš «5e Proof: Let in L; then, - Wtª u 5 and 9 u9 5 WtMš %cdfšœ 5% and š9p %udfš9p I% Wt¼w½ 1¾ p nšw 2 tm 5nšW 2sÀ and w½ 1¾ 9p nšw9 2 tm I šj9p 2$Às Wt¼w½ 1¾ ]š lt 2š ŸÀ and w½ 2¾Á ]š «\t 2š «ŸÀ WtMš udfš 5 and š «p udfš «p Ie 4. Summary and outlook In this paper we have provided an algebraic foundation for the intuitive notion of approximating region. In a second step, we have suggested a way to define approximately in contact and approximately partof on these algebras. For the construction, we have made three model assumptions which are fulfilled in the important example of screen resolution. Currently, we are developing a logic for AA and ACA using the relational semantics introduced in [13] which seems to us most appropriate for the spatial context. Many constraints which are not modally expressible receive an explicit representation in the form of a relational rule or a relational axiomatic sequence. Pertaining to our present context, it was shown in [6] that the extensionality axiom (3.2) of contact relations is not expressible in a classical modal logic, nor, as shown in [5], in its sufficiency counterpart defined in [9]. It is, however, expressible in a mixed logic with relational semantics. A further task is the comparison of approximate part-of with the notion of rough inclusion described in connection with rough mereology [17, 18].

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