Qualitative reasoning about space with hybrid logic

Transcription

1 Qualitative reasoning about space with hybrid logic Stanislovas Norgėla Julius Andrikonis Arūnas Stočkus Vilnius University The Faculty of Mathematics and Informatics Department of Computer Science Naugarduko 24, Vilnius 2012 July 11

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4 RCC with 5 predicates Any relation between two areas of the plane can be defined using following predicates: x y x y x y DC(x, y) EC(x, y) PO(x, y) x x y y TPP(x, y) NTPP(x, y)

5 RCC with 8 predicates However, usually eight predicates are used: x y x y x y x y DC(x, y) EC(x, y) PO(x, y) EQ(x, y) x x y y y y x x TPP(x, y) NTPP(x, y) TPPI(x, y) NTPPI(x, y) Definition Such theory is called RCC-8.

6 Spatial information of points, lines and regions The aim is to show how spatial information can be presented using Hybrid logic. But spatial information may consist of lines and points too. The definition of predicates DC, EC, PO, EQ, TPP, NTPP, TPPI and NTPPI can be extended to cover lines and points. However to cover all the possible relations, predicate Cr is needed: x y x y x y Cr(x, y) Cr(x, y) Cr(x, y)

7 The results The work demonstrates: How spatial information of regions, lines and points can be presented using multimodal hybrid logic. How spatial information of regions can be presented using monomodal hybrid logic. How spatial information of regions, lines and points can be presented using monomodal hybrid logic.

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9 Classical propositional logic Language consists of: p, q, r,... propositional variables; conjunction; disjunction; implication; (, ) parentheses; Interpretation is function ν : P {, }. P set of propositional variables.

10 Modal logic Additional modal operators: necessity; possibility. Kripke structure for modal logic W, R, ν : W set of worlds; R relation between elements of W; ν : W P {, } interpretation function. For some Kripke structure S = W, R, ν : F is true in world w W, iff F is true in every w W such that wrw ; F is true in world w W, iff F is true in some w W such that wrw.

11 Example of Kripke structure for modal logic w 2 p = w 1 w 4 p = p = w 3 p = Formula p is true in worlds w 1 and w 3 ; Formula p is true in worlds w 1 and w 2 ;

12 Multimodal logic n of each modal operator: i, i [1, n] necessity; i, i [1, n] possibility. Kripke structure for multimodal logic W, R 1, R 2,..., R n, ν : W set of worlds; R i relation between elements of W for i th modality; ν : W P {, } interpretation function. For some Kripke structure S = W, R 1, R 2,..., R n, ν : i F is true in world w W, iff F is true in every w W such that wr i w ; i F is true in world w W, iff F is true in some w W such that wr i w.

13 Hybrid logic ) A set of nominals N = {i, j,... } is used; Each nominal is true in exactly one world of Kripke structure; A nominal is a formula; Two new and ; i F is true, if F is true in world i; Formula x.f is true, if F is true in current state, which is denoted x in formula F.

14 Example of Kripke structure for hybrid logic w 2 p =, i 2 w 1 w 4 p =, i 1 p =, i 4 w 3 p =, i 3 Formula i1 p) is false in all the worlds; Formula i2 p) is true in worlds w 1 and w 3 ; Formula x. (p x) is true in worlds w 2 and w 4 ;

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17 Kripke structure is: W consists of regions, lines and points; Nine relations: R DC, R EC, R PO, R EQ, R TPP, R NTPP, R TPPI, R NTPPI and R Cr ; ν(w, r) =, iff w is a region; ν(w, l) =, iff w is a line; ν(w, p) =, iff w is a point; Set of nominals consists of object names; Other propositional variables, that describe the object, might be included: f forest, s see, c country, d district

18 Example r,c,a B C F A D E TPP(E,A) EC(B,A) TPP(C,A) TPP(D,A) r,d,e NTPP(F,A) r,d,c r,d,d EC(D,E) EC(B,C) EC(C,D) r,s,b PO(C,F) PO(D,F) r,f,f

19 Examples Is there a see, which is next to a forest? s EC f ; Is there any forest in Lithuania (c NTPP f ); Is there any country, without a land boarder? c EC c; Is there a forest, which is in more than one country? x.f PO ( c y.@x PO (c y) ) ;

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21 Districts It is known how predicates of RCC-8 can be expressed using predicate C and formulas of predicate logic. C(x, y) is reflexive and symmetric. Let s divide the set of regions into a finite number of closed areas, called districts, such that: each district is bounded by several closed lines, which are part of the outset of some regions; the district can not be divided further; two districts do not intersect.

22 Example of regions and districts A * C * F * D *

23 Kripke structure for RCC-8 W consists of regions, districts and two other worlds: complement of the union of all the worlds; world g; R is defined as follows: if C(w 1, w 2 ), then w 1 Rw 2 ; grw for every w g; in addition ν(w, r) =, iff w is a region ν(w, d) =, iff w is a district;

24 Additional predicates Several auxiliary predicates are used: PP(x, y) = TPP(x, y) NTPP(x, y); P(x, y) = PP(x, y) EQ(x, y); O(x, y) = P(x, y) PO(x, y) P(y, x).

25 Formulas of hybrid logic for RCC-8 DC(x, x y; P(x, x g ( x y); PP(x, y): P(x, y) P(y, x); EQ(x, y): P(x, y) P(y, x); O(x, g z. ( P(z, x) P(z, y) ) ; PO(x, y): O(x, y) P(x, y) P(y, x); EC(x, x y O(x, y); TPP(x, y): PP(x, g z. ( EC(z, x) EC(z, y) ) ; NTPP(x, y): PP(x, g z. ( EC(z, x) EC(z, y) ) ; TPPI(x, y): TPP(y, x); NTPPI(x, y): NTPP(y, x).

26 Corollary for RCC-8 It is known that the complexity of model checking algorithm in logic ) is O(l n), where l is the length of the formula (query) and n is the number of the nodes; Complexity of model checking algorithm, that decides if a Kripke structure of the spatial information is a model of the query, is O((2 + k + r) l), where r is the number of regions, k is the number of districts and l is the length of the formula (query).

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28 Segments Let s divide each line into segments which end at the point in the outline of some district or in the crossing of several lines; R 1 R 3 R 2

29 Kripke structure for regions, lines and points Let s introduce additional information into Kripke structure: all the segments and points are part of W; the inside of every region and district is part of W; w 1 Rw 2 if: w 1 is the inside of region or district w 2 ; w 1 is a segment or point on the outline of region or district w 2 ; w 1 is a segment or point inside region or district and w 2 is the inside of the region or district; w 1 and w 2 are two segments with common point; w 1 is a segment, which crosses point w 2 ; ν(w, l) =, if w is a segment; ν(w, p) =, if w is a point; ν(w, in) =, if w is an inside of region or district;

30 Formulas of hybrid logic for regions, lines and points Relation between regions are already covered; EQ(x, y) : (@ x y x y) (@ x y x y); Segment x is inside region x y x (in y); Segment x is on the outline of region x y x y; Segment x touches segment x y x y; Point x is on segment x y x y; Point x is inside region x y x (in y); Point x is on the outline of region x y x y;

31 Corollary for regions, lines and points There are 2r + 2k + s + p + 2 = m nodes, where r is the number of regions, k is the number of districts, s is the number of segments and p is the number of points. Complexity of model checking algorithm, that decides if a Kripke structure of the spatial information is a model of the query, is O(m n), where l is the length of the formula (query).