Holes, Knots and Shapes: A Spatial Ontology of a Puzzle

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1 Holes, Knots and Sapes: A Spatial Ontology of a Puzzle Paulo Santos Centro Universitário da FEI, São Paulo, psantos@fei.edu.br Pedro Cabalar Coruña University, Spain, cabalar@dc.fi.udc.es Abstract In tis paper we propose a spatial ontology for reasoning about oles, rigid objects and strings, taking a classical puzzle as a motivating example. In tis ontology te domain is composed of spatial regions wereby a teory about oles is defined over a mereological basis. We also assume primitives for representing sapes of objects (including te string). From tese primitives we propose a sufficient condition for object s penetrability troug oles. Additionally, a string is represented as a data structure defined upon a sequence of sections limited by points were te string crosses itself or points were it passes troug a ole. Introduction Real life situations were we must deal wit strings tying objects and passing troug oles appear from time to time in very different contexts. Examples range from tying soelaces, to andling ropes in a sailboat or organising te cable connections map inside an office, a building or a wole city. Altoug umans sow an amazing intuition for solving problems of tis nature, a formal representation for reasoning about oles and strings is still a relatively unexplored area. To understand te problem, note for instance tat using a fully detailed matematical model of te involved objects does not seem feasible for computational purposes, let alone wen we consider deformable objects like a string. Moreover, umans typically describe solutions to spatial reasoning problems in terms of qualitative descriptions instead. Tis is, in fact, te orientation followed by Qualitative Spatial Reasoning (QSR) (Stock 1997), a field tat attempts te logical formalisation of spatial knowledge based on primitive relations defined over elementary spatial entities. To obtain a suitable representation for strings and oles we ave adopted te following metodology. We begin from specific formalisations of particular scenarios, wic usually imply a more abstract and simplified description level, and advance tem towards more general representations to cover different domains, implying a more fine-grained ontology. As a starting point, puzzle-like examples constitute a good test bed, as tey offer a small number of objects wile keeping enoug complexity for a callenging problem of KR. Copyrigt c 2007, Association for te Advancement of Artificial Intelligence ( All rigts reserved. Following tis line, we take as a starting point te work developed in (Cabalar & Santos 2006), wic presented an automated solution to a classical string puzzle called Fiserman s Folly (see Figure 1). To solve te problem, te autors applied several strong assumptions like identifying a singleoled object wit its unique ole, assuming tat eac ole always as two entry boundaries or ignoring tat strings may form knots. Te treatment of knots, for instance, is unnecessary for te final solution, but it may be reasonably objected tat tere is no direct justification for discarding knots from te very beginning, or tat a sligt cange in te puzzle goal could easily require andling knots. In tis paper we go one step furter and remove tese assumptions to propose a more general ontology applicable to oter scenarios. On te one and, we describe a teory about oles defined over a mereological basis, proposing a sufficient condition for object s penetrability troug oles. In tis way, we can derive information of wic objects can pass troug a given ole, someting tat was taken as given in (Cabalar & Santos 2006). On te oter and, we also extend te representation of strings used in (Cabalar & Santos 2006), making a combination wit te approac in (Takamatsu et al. 2006) for dealing wit knots. Te goal of tis paper is solely to define te basic elements of an ontology about rigid, flexible and oled object, terefore we leave for future work te problem of representing actions and cange in tis domain as well as te investigation of te possible consequences of te resulting ontology. Te Fiserman s Folly Te elements of te puzzle are a oled post (P ) fixed to a wooden base (B), a string (Str), a ring (R), a pair of speres (S 1, S 2 ) and a pair of disks (D 1, D 2 ). Te speres can be moved along te string, wereas te disks are fixed at eac string endpoint. Te string passes troug te post s ole in a way tat one spere and one disk remain on eac side of te post. It is wort pointing out tat te speres are larger tan te post s ole, terefore te string cannot be separated from te post witout cutting eiter te post, or te string, or destroying one of te speres. Te disks and te ring, in contrast, can pass troug te post s ole. In tis work we assume tat neiter te lengt nor te tickness of te string constrain any solution to te puzzle, i.e. te string is infinitely extensible and one-dimensional. Relaxing tese

2 assumptions is a matter for future work. Initial Goal Figure 1: A spatial puzzle: te Fiserman s Folly. In te initial state (sown in Figure 1) te post is in te middle of te ring, wic in its turn is supported on te post s base. Te goal of tis puzzle is to find a sequence of transformations tat, wile maintaining te pysical integrity of te domain objects, allow us to free te ring from te rest of objects, regardless teir final configuration. Figure 1 sows one possible goal state. As we sall see, te representation for initial and (one possible) goal states is sown, respectively, on line 0 and line 5 of Figure 2. A crucial observation is tat te puzzle actually deals wit four oles: te post ole (P), te ring ole R and te two spere oles S 1 and S 2. Note tat in a natural language description we would probably identify oles wit teir ost objects, saying tat te string passes troug te spere (ole) or tat te post passes troug te ring (ole). Furtermore, we would talk about sliding te ring upwards te post, rater tan moving te post downwards te ring ole. In (Cabalar & Santos 2006) te puzzle entities were classified into tree different sorts: long objects, regular objects and oles, corresponding in te puzzle to te sets {P, Str}, {R, S 1, S 2, D 1, D 2, B} and {P, R, S 1, S 2 }, respectively. A distinguising feature of a long object x is tat we usually identify te two opposite extremities of its major axis. Tese extremities, denoted as x and x +, respectively receive te names of negative terminal and positive terminal of x. As a tumb rule, wen not stated, we assume in all figures tat rigtmost or topmost extremities are positive, wereas leftmost or bottom are negative (were te left-rigt dicotomy dominates te top-bottom one to solve any ambiguity). To put an example of tis notation, te rigt disk D 2 is linked to Str +, wile te post base B is linked to P. A second important observation is tat a long object may be simultaneously crossing several oles. In fact, altoug te post just crosses R, wen executing te puzzle s solution, te string may be simultaneously passing troug all te oles and, moreover, it may cross te same ole several times. Tus, we associate to eac long object x a list Cain(x) collecting te sequence of all ole crossings made by x following, for instance, te arbitrary ordering from x to x +. Furtermore, te direction in wic te string crosses te ole is also relevant in order to provide an unambiguous description of distinct puzzle states. To understand wy, just consider te partial configuration represented in Figure 2. If we represent tis situation using Cain(Str) = [..., P, P], ten it would not be possible to distinguis it from te state sown in Figure 2, wic clearly represents a substantially different situation: te disk D 2 is now to te rigt (or positive side) of te post ole P. Terefore, in an analogous way to long object terminals, we also denote two poles 1, and + per eac ole in te puzzle, considering tat in tis case oles ave only two entry boundaries. Tus, we can describe Cain(x) as a list of (outgoing) ole poles tose troug wic x exits wen going from x to x +. As a result, Cain(Str) = [..., P, P ] would represent Figure 2 wereas Cain(Str) = [..., P +, P + ] would correspond to te crossings in Figure 2. Str+ D 2 - P + Str - Str+ D 2 - P + Str- Figure 2: Two distinct puzzle states. Under tis setting, te solution deals wit two elementary actions: an action P assob(t, p) for passing an object terminal t towards a ole pole p, and an action PassH(, p) for passing te (object containing) ole towards te ole pole p. For brevity sake, we omit ere te detailed effects of tese actions (see (Cabalar & Santos 2006)), altoug tey can be guessed from te description of te puzzle s solution in Figure 3. In tat figure, eac state is identified by its sequence number plus te pair of lists Cain(P) and Cain(Str) in tis order. Te performed actions in eac transition are interlaced between eac state i and te next one i : [R + ], [S + 1, P+, S + 2 ] PassOb(Str +, P ) 1 : [R + ], [S + 1, P+, S + 2, P ] PassOb(P +, R ) & PassH(P, R ) 2 : [ ], [S + 1, R, P +, R +, S + 2, R, P, R + ] PassH(S 2, R ) 3 : [ ], [S + 1, R, P +, S + 2, P, R + ] PassH(R, P + ) 4 : [ ], [S + 1, P+, R, S + 2, R+, P ] PassH(S 2, R + ) 5 : [ ], [S + 1, P+, S + 2, P ] Figure 3: A formal solution for te Fiserman s puzzle. Note tat State 5 as actually reaced te goal since, at tis point, te ring ole R does not occur in any list, i.e., it is not crossed by any long object. Rater tan te particular mecanics of te puzzle, our main concern in tis work is to analyse in more detail te spatial knowledge representation 1 We follow te same tumb rule criterion of identifying rigt or top as positive.

3 used to obtain tis solution. For instance, it must be noticed tat (Cabalar & Santos 2006) used additional information to constrain te possible actions to be performed, including a predicate CannotP ass(x, ) to describe wen an object x cannot pass troug a ole. Te information for tis predicate was assumed as given and ad te form of an explicit list of ground atoms. We claim tat, wit a suitable representation for oles and objects, tis predicate sould be derived. Furtermore, it sould also account for groups of objects rater tan for a single object x. To understand wy, note tat if we just consider tat te spere S 2 can pass troug te ring ole R, te action PassOb(S 2, R ) (tat is, moving S 2 down te ring ole) could be performed in te initial situation. But tis movement is pysically impossible, since tere would be a moment in wic te post P and te spere S 2 would cross R, and bot objects altogeter cannot pass troug te ring. A teory about oles In tis section we follow te guidelines proposed in (Varzi 1996; Casati & Varzi 1999) and construct a basic ontology about oles using mereological relations. Te domain objects in tis work are identified wit teir occupancy regions. Holes are defined as te spatial region tat is part of te portion of an object s complement tat lies inside tat object s occupancy region. We name tis object te ost of a ole. Tere are at least tree distinct types of oles: cavities, i.e. oles tat are entirely idden inside teir osts; ollows, wic are superficial depressions on te ost; and, perforating oles (or tunnels), wic are oles tat ave at least two distinct entrance boundaries. In tis paper we sall deal only wit perforating oles, since only tese are relevant to te puzzle s solutions 2. In te formalisation described below, oles are assumed as open regions wose boundaries belong to teir ost objects. Te relationsip between oles and teir osts is formalised using te elementary relation: H(, x), meaning is a ole in te object x (conversely, x is te ost of ) (Casati & Varzi 1999). For example, it is a fact about te puzzle domain described above tat H(R, R). As we assumed tat te space is only populated by spatial regions, apart from te relation H/2, it is convenient to include in te basic teory about oles a set of mereological relations accounting for te degree of connectedness between regions. In tis work we assume RCC-8 ((Randell, Cui, & Con 1992)) wic is a first-order axiomatisation of spatial relations based on a dyadic primitive relation of connectivity (C/2) between two regions. Informally, assuming two regions x and y, te relation C(x, y), read as x is connected wit y, is true if and only if te closures of x and y ave a point in common. Assuming te C/2 relation as primitive, and tat x, y and z are variables for spatial regions, te following mereological relations can be defined: DC(x, y), wic stands for x is disconnected from y ; EQ(x, y), for x is equal to y ; PO(x, y), for x partially overlaps y ; 2 Terefore, in te remainder of tis paper we will use te words: tunnels, perforating oles and oles intercangeably. EC(x, y), for te closure of x and y are externally connected ; TPP(x, y), for x is a tangential proper part of y ; NTPP(x, y), for x is a non-tangential proper part of y ; and, TPPi/2 and NTPPi/2 are te inverse relations of TPP/2 and NTPP/2 respectively. Assuming RCC, te relation H(, x) can be constrained by te axioms (1) and (2) below. Axiom (1) guarantees tat te ost of a ole is not itself a ole; wereas Axioms (2) states tat te ole and it s ost object are externally connected. H(, x) H(x, y) (1) H(, x) EC(, x) (2) Moreover, Axiom 1 implies tat te relation H is irreflexive (meaning tat no ole osts itself) and anti-symmetric (i.e., te ost cannot be a ole of its ole). An essential caracteristic of oles is tat tey can be interpenetrated by oter objects. Terefore, te ole ontology as to include relations about te relative location of a ole wrt te penetrating object. In a world uniquely populated by spatial regions, relative location can be expressed by mereological relations. In order to define relative location wrt a ole, we need te concept of a ole entry boundary (EB) tat is defined in (Casati & Varzi 1999) by te relation EB( i,, x), read as i is te maximally connected part of te ole (fiat) boundary tat is nowere a boundary of te ost x. If a ole as n entry boundaries, we denote tem as i wit 1 i n (as we deal wit tunnels, n 2). We can now express te following relations wrt an object x and a ole : x is wolly outside (W Out(x, )) iff DC(x, ); x is just outside wrt te ole entry boundary i (JOut(x,, i )) iff y(h(, y) EB( i,, y)) EC(x, i ) TPP(x, ); x is partially outside wrt te EB i (POut(x,, i )) iff y(h(, y) EB( i,, y)) PO(x, ) PO(x, y); x is just inside wrt te EB i (JIn(x,, i )) iff y(h(, y) EB( i,, y)) EC(x, i ) TPP(x, ). x is wolly inside (W In(x, )) iff TPP(x, ) NTPP(x, )) i JIn(x,, i ). WOut, JOut, POut, WIn and JIn are scematised in Figure 4, were te ost object is te cuboid, te ole is te cylindrical figure inside te cuboid and te penetrating object is te v-saped figure. It is wort pointing out tat, in contrast to (Casati & Varzi 1999), encoding te relative location of an object wrt a ole using RCC relations allowed us to include bot JOut and JIn into te same formalism since RCC is defined over te closure of regions. Terefore, te concepts of just inside and just outside can coexist wit te initial assumption of oles as open regions. Anoter difference between te formalism presented above wrt tat proposed in (Casati & Varzi 1999)

4 WOut JOut (c) POut (d) JIn (e) WIn Figure 4: Relative location of an object v wrt a ole. is te inclusion of te ole entry boundary in te definitions of JOut, POut and JIn, in order to account for te action of an object passing troug a particular ole entry. Figure 4 can be understood as a sequence of continuous transitions from te relation wolly outside to wolly inside. In order to provide a formal solution to te Fiserman s Folly, owever, we need to be able to locate an object in space tat is WOut, wit respect to every ole, but tat is near a particular entry boundary of a tunnel. In effect, tunnels are important qualitative landmarks tat could be used as local reference frames. Tis idea is developed in te next section. Hole subspaces It is not unusual in te common language to caracterise sections of a road by te sections before and after a tunnel. In a domestic domain, we decide were to locate (non-wireless) electronic objects according to te nearby plugs (wic are, in fact, tunnel entry boundaries). Te issue of reasoning about tunnels becomes quite critical wen te problem is to locate buried infrastructure so tat repairs can be conducted on a particular network of pipes and cables underground. However, to te best of our knowledge, tere are no references tat account to te potential use of ole entry boundaries as local reference frames. Tis section describes an initial attempt to cope wit tis issue. Following te previously introduced notation, eac entry boundary is uniquely identified by a symbol referring to its ost ole plus a subscript number. If a global reference frame is assumed in te domain, te entry boundaries can be identified by te 3D Cartesian coordinates of teir respective centre points; tus, te local reference provided by te entry boundaries could be associated to a global reference frame. In tis work, owever, objects are only located wit respect to te near neigbouroods of ole entry boundaries. An object x is in te near neigbourood of a ole EB i iff x is just outside i or tere is eiter anoter object y connected to x or a ole x in x connected to y and y is just outside i or partially outside i. More formally: NN(x, i ) iff JOut(x,, i) y x (H(x, x) (C(x, y) C(x, y)) (JOut(y,, i ) POut(y,, i ))). Terefore, we can say tat (in te situation of Figure 1) te left spere, left disk and a part of te string are in te NN of P. It is wort noting also tat, for an object x and a ole, NN(x, ) and NN(x, + ) are not inconsistent as tere are feasible situations were a ole as two entry boundaries closer to eac oter. In effect tis is equivalent to describing te position of an object wrt various distinct local reference frames. We are now capable of expressing formally tat an object is near a tunnel (e.g., a car is parked outside te Eurotunnel entrance) or tat objects are related to a complex arrangement of objects and oles (wic is te case of te puzzle in question). However, in order to account for te main issues involved in te Fiserman s Folly, te teory as to include some basic ideas about object s sape so tat it is capable of expressing object s penetrability troug oles. Te next section discusses some insigts about tis issue. Te sapes of objects Representing and reasoning about objects sape are, at te same time, te most elusive and te most important issues in reasoning about te common sense space (Con & Hazarika 2001). In tis paper we cannot escape from taking into account objects sape, since te solution of puzzles suc as tat sown in Figure 1 involves passing an object of a particular sape and size, troug a ole entry boundary, also of a particular sape and size. Tis section presents some primitives to account for te sapes of rigid objects and strings. We assume in tis work ellipsoids and elliptic cylinders (Figures 5 and 5, respectively) as te basic primitives to describe te sapes of rigid objects. C F A B D E D E Figure 5: Base sape primitive. An ellipsoid (Figure 5) is a 3D figure wic every planar cross section is an ellipsis. Tis figure as tree symmetry axes: AB, CD and EF (cf. Figure 5) tat are called, respectively, major, mean and minor axes. Tus, te speres in our puzzle ave ellipsoid sapes wose tree symmetry axes are of te same lengt. Te post is ellipsoid saped, were its major axis is muc greater tan bot its mean and minor axes. In an analogous way, we use an elliptic cylinder (Figure 5) to account for te sapes of te puzzle objects not represented by te ellipsoid. An elliptic cylinder is a cylinder wose base is an ellipsis. Tis figure also as a major, mean and minor axes (respectively axes AB, CD and EF in Figure 5). Tus, te sape of te puzzle s disks and ring can be approximated to cylinders wose axis AB is muc smaller tan CD and EF, and te last two are of equal lengt. Te sape of te post base can also be approximated to a cylinder. Te string s sape as an extra complication wic is related to tis object s intrinsic flexibility. Consequently, every different sape resulting from non-destructive deformations A B F C

5 of a string is also a sape of te string. In order to cope wit tis issue we define te string s sape as tat of an elongated elliptical cylinder, were AB is muc greater tan CD and EF, or te sape resulting from te application of any sequence of te transformations depicted in Figure 6 on suc elongated elliptical cylinder. Dealing wit a string Similarly to te idea of cain described above, (Takamatsu et al. 2006) proposes a representation of te various sapes a string assumes by collecting te points were te string crosses itself, sweeping it from one of its terminals to te oter. Tis representation, called p-data projection (Figure 7), is based on a 2D projection of a 3D knot and facilitates an algebraic treatment of Reidemeister moves. (c) (d) Figure 6: Te Reidemeister moves and te cross move. Figure 6 sows te tree Reidemeister moves (Reidemeister 1983) (Figures 6, 6 and 6(c)) and te cross move (Takamatsu et al. 2006) (Figure 6(d)), tat can be described as follows: Reidemeister move I (Figure 6) adds or deletes a simple twist in te string; Reidemeister move II (Figure 6) allows te inclusion (or exclusion) of two crossings in te string; Reidemeister move III (Figure 6(c)) slides a strand of te string from one side of a crossing to te oter; te cross move (Figure 6(d)) is defined on simple open curves and adds or removes a string crossing by sliding an open end of it over a continuous part of te string. In te next section we use te sape primitives above to propose a sufficient condition for an object to pass troug an entry boundary. Passing an object troug a ole We first assume tat every object is conducted troug a ole in te direction of te largest semi-line connecting any two points of its boundary, tis semi-line we call conducting line. Tus, te post is always conducted troug te ring ole in te direction of its major axis; similarly, te disks are conducted troug te post ole via teir diameters (i.e., via teir mean or minor axes). Let s define te region of te ortogonal projection of an object o (taken troug te object s conducting line) as pl(o) and te region defined by te ortogonal projection of a ole entry boundary i as p( i ). Now we say tat te object can pass troug te ole if it is possible to superimpose pl(o) and p( i ) so tat TPP(pl(o), p( i )) NTPP(pl(o), p( i )) (3) EQ(pl(o), p( i )). Tis condition can be extended for te case of a group of objects passing troug a ole by simply considering, instead of te object o in (3), te convex ull of te group of objects. Now tat we ave a way of cecking weter a particular object can pass troug a determined ole, te next section defines a suitable representation for expressing te various states of te puzzle U U L L L U Figure 7: A knot and its p-data representation. Te p-data representation of a string is constructed as follows. We first consider te direction of te string as te direction in wic it is being swept. From one terminal of te string to te oter, eac point were te string crosses itself receives an ID number (starting from te number 1). Eac ID appears twice in te p-data structure, since a crossing is noted two times wen sweeping a string. Tis amounts to te first two lines of te structure sown in Figure 7. At eac crossing we ave to annotate also a sign of te crossing and weter te string crosses over or under itself (respectively sown in te tird and fourt lines in Figure 7). Te latter is called te vertical position of te crossing and is determined by verifying te position of te string at eac crossing found. Te former can be obtained from te sign of te expression: ( l upper l lower ) e z, were l upper and l lower are, respectively, te directions of te upper and lower parts of te string at te crossing and e z is te normal vector of te projection plane. Te sign and te vertical position of eac crossing is summarised in te fift line sown in Figure 7, were te symbol 1 represents an upper crossing wose sign is ; 2 represents a lower/ ; 3 an upper/+; and 4 a lower/+ crossing. Te p-data representation captures te essential caracteristics of crossings of a string on itself. Tis representation, owever, assumes tat tere are only single crossings of te string; i.e., te string cannot cross itself more tan one time at eac point. Now, a way of combining bot, cains (representing ole subspaces) and p-data projections, encoding string crossings is obtained by simply linking bot structures, incorporating information about te crossings witin cains and about te occurrence of objects (and oles) along te p-data. Tis combined structure we call p-cain and is represented in Figure 8. Figure 8 sows a state of te Fiserman s Folly puzzle wereby te string crosses itself. Te p-cain structure representing tis state is depicted in Figure 8 tat sows a modified structure for Cain(Str) and p-data. Te structure Cain(Str) was extended wit te numbers 1 and 2

6 S1 P S2 D1 2 D2 Cain(Str) = [1, S1+, P+, S2+, 2] p data = 1 * U L 1 2 * = [S1+, P+, S2+] te puzzle objects. Tis association of data structures provides a powerful way to compute problems involving oles and strings, wic may be useful for solving problems suc as planning maintenance work in a network of underground pipes an cables 3. Scaling te framework discussed in tis paper to solve suc problems is subject for future researc. Acknowledgements: Paulo Santos acknowleges support from FAPESP and CNPq, Brazil; Pedro Cabalar is partially supported by CICyT project TIC C02, Spain. Follow-up notes: ttp:// papers/notes/santos-and-cabalar/ Figure 8: A state and its p-cain representation. representing te string crossing. Similarly, te p-data structure for tis state was extended wit a symbol tat represents te cain tat occurs between te crossing 1 and 2. For clarity, tis example sows a particular case were te string crossing does not divide te cain structure. In te general case te p-data may be annotated wit symbols representing subsets of cains divided by string crossings. Care sould be taken for updating te states of bot cain and p-data structures wen actions are performed eiter on te string or on te objects connected to te string. For te purposes of tis work, we conjecture tat a set of actions tat would completely describe te canges in te puzzle sown in Figure 1 would be composed of te tree Reidemeister moves and te cross move (Figure 6), tat operate on te sape of te string; and te two actions for passing objects troug oles, as described above. We also conjecture tat te data structures described above follow logically from te completion of te ontology about oles, strings and rigid objects presented in tis paper. Te development of suc extended teories is te subject of furter investigations. Concluding remarks In tis work we investigated knowledge representation issues regarding te spatial aspects of a puzzle. Te puzzle cosen is called Fiserman s Folly and is constituted by an arrangement of rigid objects and non-trivial elements suc as oles and a string. Te goal of tis paper is to define te basic elements of an ontology about rigid, flexible and oled object, terefore we leave for future work te problem of representing actions and cange in tis domain as well as te investigation of te possible consequences of te resulting ontology. How elements of our framework can be used to plan a solution of Fiserman s Folly as being presented elsewere (Cabalar & Santos 2006). Te first representation issue tis paper tackled is te development of a mereological teory about oles, based on (Casati & Varzi 1999). Tis teory facilitates te definition of ole subspaces, wereby it is possible to localise objects wit respect to ole entry boundaries. We also proposed a combination of two novel data structures, cains (Cabalar & Santos 2006) and p-data projections (Takamatsu et al. 2006), for representing te arrangement of References Cabalar, P., and Santos, P Strings and oles: an exercise on spatial reasoning. In Proc. of SBIA-IBERAMIA, volume 4140 of LNAI, Casati, R., and Varzi, A. C Parts and Places: te structures of spatial representation. MIT Press. Con, A. G., and Hazarika, S. M Qualitative spatial representation and reasoning: An overview. Fundamenta Informaticae 46(1-2):1 29. Randell, D.; Cui, Z.; and Con, A A spatial logic based on regions and connection. In Proc. of KR, Reidemeister, K Knot Teory. BCS Associates. Stock, O., ed Spatial and Temporal Reasoning. Kluwer Academic Publisers. Takamatsu, J.; Morita, T.; Ogawara, K.; Kimura, H.; and Ikeuci, K Representation for knot-tying tasks. IEEE Transactions on Robotics and Automation 22(1): Varzi, A. C Reasoning about space: Te ole story. Logic and Logical Pilosopy 4: Ibid. footnote 2.