Revisable Spatial Knowledge by means of a Spatial ...

Revisable Spatial Knowledge by means of a Spatial Modal Logic Robert JEANSOULIN, Christophe MATHIEU L.I.M URA CNRS 1787 - GDR 1041 CASSINI CMI, Technopole de Chateau Gombert 39, rue Joliot Curie 13452 Marseille CEDEX 13, FRANCE Phone: (33) 91 11 36 08 Fax: (33) 91 11 36 02 E-Mail: {mathieu,jeansoul}@gyptis.univ-mrs.fr

Abstract. Our aim is to define a formal system able to represent both spatial information and reasoning about these information. The representation must hold as a logical model as well as a data model. The logical model is especially tailored for two frequent form of reasoning in geography : the disaggregation of information inside a zone, and the propagation around a zone of known information. We also want this logical model to handle reasoning with incomplete knowledge. The underlaying data model - discussed in previous work (the so-called "goodies" model) - belongs to the "extended" relational family of spatial databases, for now popular in the GIS research community. In this paper, we propose to use a propositional multimodal logic as formalism of representation. The main reason of using such a formalism is its ability to express in one formal system several kinds of information. Indeed it provides a logical framework for a triple translation mechanism: the translation of facts using classical propositions, the translation of spatial relations between these facts using some appropriate modal operators (inside and around), and the translation of revisable information using "hypothesis" modal operators. We show here how spatial information and reasoning can be merged in a multimodal system. It appears that this process should open a new way of search for the representation of geographical reasoning in GIS (Geographic Information System). KEYWORD: spatial logic, topological relations, modal logic, hypothesis theory, revision in GIS, reasoning This paper has not already been accepted by and is not currently under review for a journal or another conference. Nor will it be submitted for such during IJCAI's review period.

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1 Introduction In the spatial domain, we may reason with notions of very different nature (topological, sociological, economical, …). Knowledge may be symbolic as well as numerical, may refer to "spatial objects" which can be "mapped" with precise borders, or may refer to spatial variables which can't. Geographers aim to construct models making easier the simultaneous use of these information. Thus GIS (Geographic Information System) were created to manage the various knowledge. At this moment, most of the GIS are computed through extended relational language or object oriented language. For instance, in our previous work [Jeansoulin 90] , a model including vector and raster and using an object oriented approach has been elaborated based on extended relational language. If GIS deal easily with geometrical data (points, lines, …), they can hardly, indeed even not at all, be used to represent more general data as uncertain or incomplete ones. Our purpose is to give a framework in which it's possible to formalize reasoning in a spatial context with information known as uncertain or incomplete ones. In this paper, a representation is proposed for the spatial reasoning, based on the propositional modal logic approach. Modal logic are used because modal operator can be chosen to formalize spatial relations. The semantical definition of modal logic needs a logical universe (in this case propositional calculus) and an accessibility structure. We investigate on which spatial relations one can build a structure for the set of parts of the 2D plane: the topological relations and their properties. Then, we propose a bimodal logic for spatial logic: one with the "inside" operator, one with the "around" operator. The basic axiomatic system shows analogies, but also differences, with the classical temporal logic system. The second aim is to make spatial reasoning based on hypothesis. Indeed, we have knowledge on a global area and we want to know if these knowledge remain verified inside this area (in the sub area) or around this area. We have to deal here with incomplete or uncertain data, as some conclusions can or can't be made, or may be or not may be still verified in another context. Hence, we introduce Hypothesis theory (a non monotonic modal formalism [Siegel, Schwind 93] ) to formalize the two corresponding forms of knowledge: "make the assumption inside and around". We associate the two spatial modal operators to those of modal hypothesis theory. The resulting system supplies a formal framework for the description of spatial propagation mechanisms or of hypothesis spatial disintegration, as used by geographer. Finally, we show how to compute the two spatial modal operators with the Goodies data model (our prototypal GIS). The deductive schema in spatial context allows us to recover previous works on the spatial database with extended relational language. The paper consists of two main parts. The first one presents multimodal logic (section 2), describes the underlying topology (section 3) and present how spatial notion can be translate into this multimodal formalism (section 4). The second

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one is reasoning oriented. We introduce a non monotonic formalism (hypothesis theory) based on modal logic and we present how spatial notion and this kind of reasoning can be merged into one multimodal system (section 5). Just before conclusion, a Goodies computation example of the spatial logic is given.

2 Modal Logic 2.1 Language Within our framework, the language of multimodal logic is given by a set of propositions Q labelled q1, q2,… that translate facts - such as "dense urban area", see an example in part 4 - by a set of modal operators (necessity operators written Lx and possible ones written Mx where x characterises the (mono)modal system) and by logical connectors as (or), (implies). The semantic is based on Kripke possible world semantics. The meaning of modal operators is given by the semantic. A complete presentation of multimodal logic is given into [Catach 89] . 2.2 Axiomatics Roughly speaking a multimodal system is a combination of one or more (mono)modal systems (the same ones or different ones) with possible linking axiom between these (mono)modal systems. Let us define the basic (mono)modal systems: A modal system is said to be normal if each modal operator L x verify all A, A B instances of tautologies of propositional calculus, Modus Ponens: B , the inference rule:

Lx

, and the K axiom: Lx (

)

(Lx

Lx

.

Adding axioms to the minimal normal system K, we can obtain different modal systems such as T adding L x f f, S4 adding L x f f and L x f LxLx f, and S5 adding Mx f LxM x f to S4. A multimodal system is said to be homogeneous if all its basic (mono)modal systems are the same one, and non homogeneous otherwise. It's said to be with interaction if it contains axioms that uses modal operators of different basic monomodal systems. According to the [Catach 89] notation, a multimodal system composed by a K system, a T one and an interaction axiom I (for example LKf Tf)between K and T is written: KxT+I{K,T}. 2.3 Semantics A Kripke multimodel is a triple =(W, ,V), where W is a set of state or possible-worlds, a set of binary relations R x of WxW (s is in relation with t is written sR xt) and V is a truth assignment to the propositions of Q (i.e. V: WxQ {true, false}). (W, ) is a Kripke structure. We define an inductive "satisfaction" relation between a model and a formula. =(W,R,V) s g means that the model satisfies the formula g in the world s. s q iff V(s,q)=true, q Q

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/sg h) iff: if s (g s g then sh (W,Rx,V) s Lxf iff sR xt implies (W,R x,V) t f for all world t of W. (W,Rx,V) s M x f iff there exists a world t of W such that sR x t and (W,Rx,V) t f s

We say that a formula g is valid in a model (W, ,V) or satisfiable and write (W, ,V) g iff (W, ,V) s g for all s W; we say that a formula g is valid in a structure (W, ) and write (W, ) g iff g is valid in all models (W, ,V); we say that a formula g is valid and write g iff g is valid in all the structures (W, ). 2.4 Link between Axiomatics and Semantics There is a consistency and completeness theorem saying: A is valid iff it's provable in K. Notation: A iff K A (A can be deduced by means of the axioms and inference rules of K). Each modal system (T, S4, S5, …) has a semantic counterpart. Let us call respectively Ref, Ref-trans and Equiv the classes of models where the accessibility relation is reflexive, reflexive and transitive, or an equivalence relation. Let us represent by P the fact that a formula is valid with respect to a class of models where the accessibility relation possesses the property P. The following completeness theorem can be shown: T A iff Ref A, S4 A iff Ref-trans A, S5 A iff Equiv A.

3 Topology Before defining the modal spatial logic, we are going to precise the underlying topology we want to formalize. Our universe is the set of parts of the 2D space W = 2RxR, and we call spatial objects the elements of W (noted s, t, u, …). One can define an accessibility structure between spatial objects, by means of binary relations on these elements, for example topological relations. This is a partial analogy with the classical choice of temporal modal logic in which an "current time" splits the temporal axis into past and future. We propose a topological splitting based on borders, between inside and outside the "current spatial object". 3.1 Topological relations The topological concept of neighbourhood let define "open sets", "exteriors" and "borders" (noted O , E , B ). The basic intersection operators classically implemented in most GIS are: point and line, point and polygon, line and line. That makes possible the computation of intersection between borders ( B1 B2) open set or their outside (O 1 O 2 or E 1 O 2). Several combinations of these intersections are possible, but to translate the useful cases of disintegration and propagation, we choose respectively: Ri (O 1 ,O 2 ) = O 1 includes O 2, (where Ri is the disintegration relation, i for inside) i.e.: O1 O2 and E 1 O2 = and O1 E 2 )

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Ra( O1,O2) = O1 and O2 are externally adjacent, (where R a is the propagation relation, a for around) i.e.: O1 O2 = and E 1 O2 and O1 E 2 and

B1 B2 3.2 Topological relations properties The following array illustrates some classical properties the binary relations R i and Ra verify: Property Name: Formula: s (sRs) Reflexive s t (sRt tRs) Symmetrical s t (sRt) Reproductive s t u (sRt tRu sRu) Transitive s t unique (sRt) Functional s t (sRt u (sRu uRt)) Weakly Dense (*) the R i relation is anti symmetric.

Ri yes no(*) yes yes no yes

Ra no yes yes no no yes

4 Spatial Logic: a modal approach We can define two structures, called topological accessibility structures, based on the W universe and the two previous accessibility relations Ri and Ra. To build a modal logic, we must choose modal systems (i.e. first modal operators, then axiomatics) that formalize the access between objects by means of the above structures. 4.1 Inside and Around modal operators The modal operators, noted L i and L a are directly linked to Ri and R a. Their meaning can be seen as: Li P: "P remains true inside", (everywhere inside) La P: "P remains true in the immediate vicinity", (everywhere around) These two modal operators are necessity kind of, their dual M i i Ma a Mi P: "P is true at least once inside ", (somewhere inside) Ma P: "P is true at least once in the immediate vicinity", (somewhere around) Li is called inside necessity modal operator and L a around necessity modal operator. Remark: The inside necessity operator translates the stability of propositions with respect to a splitting of the current spatial object. It's very useful to synthetically express that a proposition must be (or not) disintegrated on such object (see examples 1 and 2 below). On the other hand, one can represent the "reference sector" notion, that thematicians often use for restraining a large study domain to a smaller

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"prototype" sector. This approach is implemented in the pedological application of [Ledreux and al. 94]: the sector predicate remains true everywhere in the domain, where is representative. This can be written: P is true on Li P is true on . Examples: (1) Let P be the proposition "it's sunny" and W be the set of district of France. Li P means that the assertion P is true everywhere inside the current district, thus it's sunny everywhere inside this district. (2) On the other hand, with the proposition Q "the NO for the Maastricht vote is ", the valuation will not be necessary the same for the district and for each polling i Q) but somewhere (Mi Q: there is at least one polling station where Q is true). 4.2 Axiomatic System To construct our spatial logic (respectively inside and around logic) on the previous defined topological structures, we take the axioms of propositional calculus, to which we add: the K axiom: Li (P Q) ( Li P Li Q) means that the deduction scheme remains valid in all cutting of the associated geometrical place. the K axiom: La(P Q) (LaP LaQ) means that the deduction scheme remains valid in all vicinity of the associated geometrical place. , and Li La Furthermore, with respect to the underlying topology (see subsection 3.2), we add the axiom T ( Li P P) and 4 for L i (the R i relation is reflexive and transitive). The modal system associated to Li is S4. The axiomatic system associated to La is a K one, but not a T or S4 one because of the lack of reflexivity and transitivity properties. On the other hand, we can had the 5 axiom (translating the symmetrical property). Therefore, it is lacking many classical properties used in temporal logic. But, we can admit the reproductivity property induced by the topology (each place have an inside and a vicinity), and the weak density (a third part is always possible as intermediate inclusion or in the vicinity). the two necessity modal inference rules:

These two modal logics are made up by all the theorems that are valid formulas. According to the notation introduced before, we obtain the bimodal system without interaction axiom: KxS4. The spatial modal system of disintegration and propagation is: KxS4. 4.3 Semantic model Let us note the model (W,{R i,Ra},S) where S represent a spatialization mapping from the set of propositions to the set of parts of W, that relates each proposition P to its geometrical place: the subset S(P) of W where P is true. The following semantics (classical Kripke's one) stand: P is true in s: s S(P), P Q is true in s: s S(P) implies s S(Q),

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Li P is true in s: its dual MiP: LaP is true in s: its dual MaP:

t (sR it) implies t S(P), t (sR it) and t S(P), t (sR at) implies t S(P) t (sR at) and t S(P).

Example: Just recall the previous example. Let us consider the universe of French district. Let P and Q be the propositions P = "it's sunny" and Q = "it rains". Let s be the district on which we have the knowledge. Then, the meaning of LiP is "it's sunny everywhere inside the district" and those of M vP M vQ is "possibly it's sunny around the district and possibly it's raining around the district (somewhere)"

P

P

P P P P P P

? Q Q

L P i Ma P

Ma Q

Object "s" in bold line

5 Hypothesis Spatial Logic Within the logical framework previously established, we want to make reasoning about information contained in a spatial database, described within the spatial modal logic. These data are incomplete, then to reason required non classical formalisms as non monotonic ones. In the lack of information, we are obliged to make some hypothesises to follow up the inference of deductions. In order to handle such a reasoning, we propose to use the formalism of Hypothesis Theory [Siegel, Schwind 93] . Indeed, this theory is modal logic based (its integration in spatial logic can be considered), and captures the notion of Reiter's defaults [Reiter 80] . 5.1 Default Logic and hypothesis Theory Non monotonicity allows to express notions as "Generally, A are B" said to be common sense reasoning or default reasoning. Among all the existing formalisms defined to handle such kind of reasoning, the Reiter's approach [Reiter 80] allows A, B to formalize default reasoning using particular inference rules C that can be understood as: if A is known and B is consistent with the knowledge, then we can infer C. In the case of C=B, defaults are said to be normal. The Hypothesis Theory is a bimodal language that models three kinds of knowledge: valid facts (represented by propositional formulas), known facts (represented by the knowledge modal operator L c of the T system) and hypothesis (represented by the existential modal operator M h of the K system). M h is defined

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as MhP c negation is known. This axiom, an interaction axiom, translates semantically that the R h accessibility relation associated to M h is included into the one associated to Lc. In fact, the meaning of this inclusion is: a known fact can contradict an hypothesis, but not the opposite. Using multimodal notation, Hypothesis Theory is the multimodal system KxT+I{K,T}. [Siegel, Schwind 93] established a precise relationship between default logic and hypothesis theory. A normal default is translated in L cA M hB LcB. Then, it's possible to represent default rules using modal formulas. Moreover, a deduction method is given for this theory in [Mathieu 93] . 5.2 Hypothesis Spatial Logic Reasoning by hypothesis can be done with this formalism. The interest of presenting the above formalisms is to show the easy way to combine it with the inclusion and proximity modal logic. The idea is to join spatial logic and hypothesis theory in order to obtain a logic able to reason with four different knowledge: known information using L c, inside and around information with L i and L a, hypothesis with Mh. Multimodal logic can integrated the four modalities, elaborating a multimodal system based on four different modal systems. Syntactically, the combining is done by a simple concatenation: MhL i P for example. This formula express a different degrees of knowledge than the Mi P one. We can read the first one as: one can make the hypothesis that P remains true everywhere inside, expressing in this sense that without any contradictions we can admit that P is true inside everywhere. By this way, we introduce reasoning in the disintegration. In the second formula, only the possibility for P to be true at least once inside is expressed. Nothing can be deduce on the database. Formally, the resulting system, merging spatial logic and hypothesis theory, is the multimodal system K1xTxK 2xS4+I{K 1,T} where K1 and K2 are introduced to distinguish the K system of Hypothesis Theory (K 1) to the one of spatial logic (K 2). Example: Let be a first spatial cutting corresponding to districts. This allows to express for example the formula P: "district of more than 10,000 inhabitants". Let be a second spatial cutting giving the land-use, based on "IGN-BDCarto" nomenclature, that expresses facts as Q1 "dense urban area", Q2 "joint urban area", Q3 "farm area" or Q4 "fallow land". We can have a given model, true for s (valid facts for a district) and allowing to make an hypothesis on land-use areas which it includes. Thus, LcP MhLi (Q1 Q2) can be understood as: the knowledge P on s implies the Q1 and Q2 hypothesises on all area included in s.

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6 Merging Modal Formulas within the Goodies software The operators of extended relational algebra, developed in the Goodies model, allow to compute the four modal operators built in this paper. Recall: a map in the Goodies terminology is a relation among a Cartesian product (D1 xD2x … xDn xW) where each D i is a thematic domain and W, the required spatial domain to handle a map, is the chosen universe. The algebra defined on this Cartesian product allows to elaborate a kind of SQL language which mixes thematic and topological criteria. Let map(s) be the map that contains the reference object. Let map(t) be the one including the objects which we want to test the accessibility from s and on which P stands. For the existential operators , just test if the resulting selection is not empty: {Select * From map(s),map(t) Where P(t) Ri (s,t)} ? . For the universal operators, test the equality with the selection map without P: {Select * From map(s),map(t) Where P(t) Ri (s,t)} =? {Select * From map(s),map(t) Where Ri (s,t)}. Example: To illustrate both Goodies capacity (as model and software) and the hypothesis modal construction, we show below the result of processing cited in subsection 4.1. The selection of Spot image pixels on the selected area S(P) [more than 10,000 inhabitants] is shown at left, and at right are shown the pixels verifying the hypothesis M hLi (Q1 Q2) [dense or joint urban area].

2 images here

Figure : [left] : "ndvi" pixels belonging to S(P) and [right] : where hypothesis MhLi (Q1 Q2) is verified.

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Three "map relations" are involved in this example : mapspot = [ ndv-index, region ], where region is of "pixel type", this map is used only for visualisation purpose (ndvi = "normalised difference vegetation index", is computed from a Spot multi spectral image. Thanks to Spot-Image), mapland = [ landuse-code, region ], where region is of "zone type". mapadm = [ commune-name, commune-population, region ], where region is of "zone type". These two last maps are gracefully provided IGN-France. The figures are obtained with the requests : Select (mapspot.ndvi-index, mapspot.region) into left-figure From mapspot and mapadm Where mapadm.population > 10000 and mapspot.region inside mapadm.region. Select (mapspot.ndvi-index, mapspot.region) into right-figure From mapspot and current = Select (mapland.region) From mapland and mapadm Where mapadm.population > 10000 and mapland.landuse = dense or mapland.landuse = mixt urban and mapland.region intersects mapadm.region. Where mapspot.region = current.region. Actually, the "current" map represents Li (dense or mixt), and we can observe that the hypothesis is not fully verified because in our case we own the full information. Thus we can compute the percentage of correctness as the ratio : area(right-figure) / area(left-figure), where area(any-figure) is merely equal to the number of pixel for each picture.

7 Conclusion and perspectives We have presented in this paper, a general formalism able to represent both spatial and reasoning notions. We have seen that it was possible to build a multimodal system with modal operators translating spatial knowledge (propagation and disintegration), and with modal operators translating reasoning (Hypothesis Theory). The main interest of using multimodal logic lies in the possibility to elaborate a system joining different notions such as topology and default reasoning. One possible extension is to introduce temporal notion. Indeed, temporal logic are specific modal logic. It seems to be possible to merge temporality in the same way as spatial logic. Another extension lies in the fact that most of geographical information are numerical ones. Spatial databases contain a lot of numerical information. To take this into account, we investigate formalisms able to manage these information. A good candidate seems to be the graded multimodal logic [Chatalic, Froidevaux 93] . This formalism is a qualitative approach which allows for attaching partially

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ordered symbolic grades to logical formulas. The ordered set of grades is used to represent degrees of certainty. Uncertain information are expressed by means of parametized modal operators. Intuitively, [ ]p corresponds to a formula p which is known at least grade . As surface and distance measure occurs frequently in geography and are integrated in geographical reasoning, one can associated area(t ) measure to grade. We can define as area(s) i , on all ti such that (sR iti ) and ti S(P). This measure is a probability one. Further more, [Chatalic 94] proposed a multimodal graded approach for Hypothesis Theory. In conclusion, the work presented in this paper can be considered as a further step toward a better understanding of some kind of reasoning in GIS. Furthermore, new improvements in logics and non monotonicity could provide basics for conceptualize next generation GIS including hypothesis management and data revision.

References [Catach 89] L. Catach, Les logiques multimodales [Chatalic 94] P. Chatalic, Towards Graded Hypothesis Theories, Rapport de Recherche n [Chatalic, Froidevaux 93] P. Chatalic and C. Froidevaux, A Multimodal Approach to Graded Logic, Rapport de Recherche n 1994 [Egenhofer 89] M.J. Egenhofer, A formal definition of Binary Topological Relationships, in Proc. First Intl. Conf. Foundations of Data Organization, FODO, 1989. [Jeansoulin 90] R. Jeansoulin, Basic Tools for Spatial Logic, COGNITIVA'90, Madrid, nov. 1990. [Ledreux and al. 94] C. Ledreux, P. Lagacherie, R. Jeansoulin, , in Revue 1, 1994. [Mathieu 93] C. Mathieu, monotone Marseille, 1993 [Reiter 80] R. Reiter, A Logic for Default Reasoning, Artificial intelligence 13, p. 81-132, 1980 [Siegel, Schwind 93] P. Siegel and C. Schwind, A Modal Logic Based Theory for Nonmonotonic Reasoning, Journal of Applied Non Classical Logic. p.73-92,

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