Querying Geometric Figures Using a Controlled Language, Ontological Graphs and Dependency Lattices

Transcription

1 Querying Geoetric Figure Uing a Controlled Language, Ontological Graph and Dependency Lattice Yanni Haralabou 1 and Pedro Quarea 2 1 Intitut Mine-Téléco, Téléco retagne Coputer Science Departent UMR CNRS 6285 Lab-STICC Technopôle ret Iroie CS 83818, ret Cedex 3, France 2 CISUC/Departaent of Matheatic, Univerity of Coibra P Coibra, Portugal btract. Dynaic geoetry yte (DGS) have becoe baic tool in any area of geoetry a, for exaple, in education. Geoetry utoated Theore Prover (GTP) are an active area of reearch and are conidered a being baic tool in future enhanced educational oftware a well a in a next generation of echanized atheatic aitant. Recently eerged Web repoitorie of geoetric knowledge, like TGTP and Intergeo, are an attept to ake the already vat data et of geoetric knowledge widely available. Conidering the large aount of geoetric inforation already available, we face the need of a query echani for decription of geoetric contruction. In thi paper we dicu two approache for decribing geoetric figure (declarative and procedural), and preent algorith for querying geoetric figure in declaratively and procedurally decribed corpora, by uing a DGS or a dedicated controlled natural language for querie. Introduction Dynaic geoetry yte (DGS) ditinguih theelve fro drawing progra in two ajor way. The firt i their knowledge of geoetry: fro a initial et of object drawn freely in the Carteian plane (or aybe, on another odel of geoetry), one can pecify/contruct a given geoetric figure uing relation between the object, e.g., the interection of two non-parallel line, a line perpendicular to a given line and containing a given point, etc. nother ajor feature of a DGS i it capability to introduce dynaic to a given geoetric contruction oving a (free) baic object alway preerving the geoetric propertie of the contruction [18]. That i, one ue a DGS by contructing a geoetric figure with geoetric object and geoetric relation between the, and not by placing point on pecific Carteian coordinate. Mot (if not all) DGS poe a foral language for the pecification of geoetric contruction. In oe yte thi foral The final publication i available at

2 language i explicit, in other it i hidden fro the uer by the graphical interface. The Intergeo project deigned a coon forat, called I2G, for thi foral language which i already accepted by oe DGS [16,17]. Geoetry autoated theore prover (GTP), being foral yte, need a foral language to decribe geoetric conjecture. GTP are nowaday ature tool capable of proving hundred of geoetric conjecture [2,8]. The I2GTP foral language i an extenion of the foral language ued by the DGS. The I2GTP project goal i to define a coon language, an extenion of the I2G language, to the DGS/GTP tool [12]. The deign of coon language, and the eergence of Web repoitorie of geoetric knowledge i an attept to ake the already vat data et of geoetric knowledge widely available. The Intergeo project [9], GeoTh [7] and TGTP [11] yte already eet oe of thee goal, having provided a large data et of geoetric inforation widely available. In thee yte the quetion of querying the geoetric contruction i not olved, that i, it i not yet poible to query the data et for a contruction iilar to oe other contruction, or to query for all contruction having oe coon geoetric propertie. The goal of our reearch i to develop a earch echani for geoetric contruction (done by a DGS or a GTP) uing the different way of geoetric contruction decription. 1 What You See and How to Get It: Declarative v. Procedural v. nalytic Figure Decription On Fig. 1 the reader can ee (a viual repreentation of) the centroid theore, a iple geoetric figure taken fro the TGTP corpu [11, Fig. 13]. There are any approache for decribing thi figure: the declarative one: we have point, E, on the ae line uch that E = E, point, D, C on the ae line uch that D = DC, point C, F, on the ae line uch that CF = F, and diplayed are C, C,, F, D, CE. The interection of D, CE, F i called G ; the procedural one: draw egent, C, C, take their idpoint E, D, F, draw F, D and CE, take the interection of D and CE and call it G ; the analytic one: point,, C have coordinate (35, 40), (10, 10), (40, 10); point D, E, F, C D F G E Fig. 1. Contruction 13 of the corpu. have coordinate (37.5, 25), (22.5, 25), (25, 10); egent have coordinate ((35, 40), (40, 10)), ((35, 40), (10, 10)), etc. In thi paper we will concentrate on the procedural and declarative decription of a figure.

3 The declarative decription i about what the part of the figure are and how they relate to each other, while the procedural decription i about how to contruct the figure. In the forer we can upply arbitrary (and potentially redundant) inforation about the figure; in the latter we provide only intruction that reult into the given figure. The firt proble we encountered when querying figure wa the fact that a given figure can often be contructed (and hence, procedurally decribed) in everal way. For exaple, Fig. 1 can be procedurally obtained in (at leat) the following two way (cf. [4] for the econd): tart with point,, C draw idpoint of, C, C call the D, E, F draw the egent tart with point D, E, F draw a line at F parallel to DE draw a line at E parallel to DF draw a line at D parallel to EF call the interection, and C reove the line draw the egent That i, we can tart with the triangle and find the idpoint, or we can tart with the idpoint and find the triangle. oth the DGS to be ued (GeoGebra [6]) and the controlled language we will define ( 3) are procedural, and hence decribe a figure by it contruction. there are any contruction reulting in the ae figure, we concluded that our earch yte hould better ue a declarative approach. For thi (ee alo [13]), we convert procedural decription into declarative one and repreent the inforation they contain by the ue of ontological graph. Thi operation i done both for the earch corpu and for the querie, o that a figure query becoe the earch of a graph pattern inide a corpu of ontological graph. The econd proble we encountered i that procedural decription are oetie lacunary, provided the correct viual reult i obtained. For exaple, in the procedural decription of Fig. 1, a it i included in our corpu, the creator of the figure ha defined G a being the interection of D and CE, without going any further. Since the goal wa to obtain the correct viual repreentation of the figure, it wa not neceary to tate that G i alo the interection of F and D a well a of F and CE. Thi ean that, after converion into the declarative repreentation, the inforation provided in it will lack thee fact. Inference can fill oe of the gap and ake a declarative decription ore coplete. For exaple it can detect paralleli or orthogonality relation that are not explicitly tated. 1 The way we propoe to olve thi proble i by going the other way around : intead of aking the corpu richer, we can weaken the query. Thi ethod i called query reduction and i ueful when the query contain too uch inforation and cannot be found in the corpu. 1 In a future developent we plan to ue the deductive databae ethod to find all the fix-point for a given contruction, finding in thi way the iing fact [3].

4 The proble then i, how do we reduce the query? Indeed, when in front of an ontological graph query where all ingredient of the query figure have becoe node, and their relation have becoe edge, how do we chooe the ot uitable node or edge to reove? It i the procedural decription of the query that provide u with an anwer 2 to thi quetion. Fro the procedural data, we build a dependency lattice of the query figure. The lattice tructure provide u with the node to reove, and the order in which to reove the. For thee reaon, we have developed, and will dicu in thi paper, both procedural and declarative decription of geoetric figure. Thank to their copleentarity we obtain en efficient geoetric figure earch yte. 2 Ontological Graph 2.1 Decribing a Geoetric Figure by an Ontological Graph In the following we will ue an ontology pecific to geoetric figure on the plane. Thi ontology contain concept: point: a point of the plane; egent: a egent, defined by two point. It ha an attribute length which induce a relation of ratio aong egent intance; line: a line, defined by two point or in oe other way (for exaple, by a point and a property like perpendicularity); conic: a conic defined in variou way, and, in particular, a circle, defined by it center point and another point; angle: the angle of two egent/line, it ha an attribute value which can have a nueric value or the odal value traight. The relation between intance will be 3 : belong to: a relation whoe doain are both point (belonging to egent, line, circle and angle), and egent (belonging to line); ha ratio: can be ued for length and angle value. It i a 3-ary reified relation, the eber of which are the noinator, denoinator and ratio value; i center of : connect a point with the circle of which it i the center; i parallel to: connect two parallel line (uing inference, we will find all parallel line); i perpendicular to: connect two perpendicular line or egent (uing inference, we will find all perpendicular line or egent, knowing that the perpendicular of a perpendicular i a parallel); i radiu of : connect a egent with the circle of which it i the radiu. 2 Well undertood, the anwer i not unique ince it trongly depend on the way the figure ha been contructed, which i not unique. 3 The lit i not exhautive.

5 Thee concept and relation have been inpired by the eleent type of DTD GeoCon.dtd [10] (the ontology doe not cover XML eleent toward, tranlation, rotation which are ueful for drawing but do not affect the ontological graph of the figure) and of GeoGebra XML chea ggb.xd [6]. Every figure becoe a graph of intance of concept and of relation. Not only thi approach i independent of the way the figure ha been contructed, but it i alo independent of intance nae and allow to focu on the network of relation between the ingredient of the figure. Our choice of concept and relation ake oe graphical contruction obtainable by a ingle relation, for exaple: C C i repreented by intance,, C (point), and C (egent) and C (angle), and the following graph of relation: C C C value= where olid arrow denote the belong to relation, and i the right angle value of the value attribute. Other contruction, although trivial, are ore difficult to encode. For exaple: i tangent at circle c at point c cannot be encoded by a ingle relation. We need to ue the radiu O and ay that c O O c and the graph of relation will be O value= O O c

6 where the dotted arrow repreent the relation i center of and the dahed one, the relation i radiu of. The ontological graph of a geoetrical figure can rapidly increae in ize. It generation i done in a two-tep proce: 1. every XML eleent of the figure decription i converted into a et of concept and relation; 2. then, inference i applied to generate additional relation: (a) we calculate the tranitive cloure of paralleli and orthogonality relation (a // b b // c a // c and a b b c a // c); (b) node having equal length or equal angle value by contruction 4 obtain a ha ratio relation with value attribute equal to 1; (c) if neceary, angle are intantiated for every pair of egent with a coon point. Our corpu of 134 figure, encoded a an XML file of 3,137 eleent reulted into graph of a total of 5,282 concept intance and 10,211 relation intance. 2.2 Exaple Take Fig. 1 repreenting Figure 13 of the corpu (illutrating the fact that edian interect at the barycenter of the triangle). The contruction, a given in the XML file, take arbitrary point,, C, define D (rep. E, F ) a the idpoint of C (rep., C), and G a the interection of D and CE. Furtherore, the egent F i drawn. The ontological graph will contain concept for point,, C, D, E, F, G, and egent, C, D, E, F, C, D, E, F, CD, CE, CF. The relation will all be of type belong to, except for oe 3-ary ha ratio relation repreenting equal length. In Fig. 2, unlabelled arrow denote the belong to relation. 2.3 Querying Ontological Graph To be able to earch in a corpu, we convert all figure of the corpu into ontological graph and we tore the in a graph databae (we ue a neo4j databae [15]). The uer query i a figure drawn by uing a DGS, or a query uing the controlled query language ( 3). Thi figure or CQL tateent i converted into an ontological graph on-the-fly, and then into a Cypher query (Cypher i the query language of the neo4j graph databae yte). The query i end to the databae and return graph intance containing the query a ub-graph. t thi tep, ranking i perfored to preent the reult to the uer in a pertinent way. Our ranking criterion (which we will try to iprove in the future) i the ratio between nuber of node and relation of the query and the nuber of node and relation of the atched graph. Uing thi criterion we obtain firt 4 We ephaize the fact that equality i explicitly given in the contruction and i not the reult of eaureent between object of the figure.

7 value ha ratio 1 noinator E denoin. E E D C CE G D C F noinator noinator D CD F CF F denoin. value denoin. ha ratio 1 C 1 ha ratio value Fig. 2. The ontological graph of Fig. 1. the allet figure poible figure containing the query ubgraph. We intend to ue graphical echani to highlight the atched pattern in the reulting graph by uing, for exaple, a different color. 3 The Controlled Query Language In oe cae the uer ay not wih to ue the DGS to build the query, either becaue it i cuberoe to ue or becaue it doe not provide the neceary abtraction. We propoe, a an alternative to the DGS, a controlled query language that allow the (procedural) decription of a figure in a way that i iple and cloe to natural language. 3.1 Decription of the controlled query language Here i the graar of the controlled query language: S -> query query -> ent drawvp PERIOD query -> ent PERIOD drawvp -> DRW ent ent -> ent SEMICOLON ent ent -> ent ent -> np vrb pp ent -> np vrb ent -> INST LEL ent -> LEL

8 vrb -> VER DJE vrb -> VER NOUN vrb -> VER pp -> ent pent pp -> pent pp -> ent pent -> pent pent pent -> pent pent -> PREP ent ent -> ent ND ent ent -> ent np -> ent NOUN -> /(idpoint foot ediatrix interection biector)[]?/ INST -> /(point egent line angle circle center point egent line angle circle center)/ VER -> /(i are interect[]? connect[]?)/ DJE -> /(perpendicular parallel defined right)/ SEMICOLON -> /;/ ND -> /(, and)/ DRW -> /draw/ PREP -> /(at of by to on)/ PERIOD -> /\./ LEL -> /[-Z]([_]?[0-9]+)?(-[-Z]([_]?[0-9]+)?)*/ and here are it rule 5 : 1. Every query i of the for: [lit of entence eparated by ;] draw [lit of intance eparated by,]. 2. n intance conit of a type and a nae (or jut a nae, if there i no abiguity). It i written in the for type [nae]. 3. priitive intance i an intance of a point, a line or a circle. 4. The nae of a priitive intance atche the regular expreion [-Z]([_]?[0-9]+)?. 5. Nae of non-priitive intance are copoite: they are written by joining nae of point uing the hyphen character (for exaple: egent -). 6. n intance type can be followed by ore than one intance, in that cae it i written in plural for and the intance are eparated by coa (for exaple: point,, C). 7. The firt part of a query define (and draw) new intance, the econd draw already known intance. 8. The following entence can be ued 6 : (a) line? interect line? at point? (b) point? i the idpoint of egent? (c) line? i perpendicular to line? at point? 5 We will ue an erif font for illutrating the controlled language. 6 ll type nae are optional except for center in (h).

9 (d) point? i the foot of point? on line? (e) line? i the ediatrix of egent? (f) line? i parallel to line? at point? (g) line? connect point?,? (h) circle? i defined by center? and point? (i) point?,? are the interection of circle?,? (j) point?,? are the interection of circle? and line? (k) line? i the biector of angle? (l) angle? i right 9. ll entence have plural for where arguent are ditributed at all poition and eparated by coa (for exaple: line L 1, L 2, L 3 connect point 1, 1, 2, 2, 3, 3, which ean that { i, i } L i ). 10. In all entence except (8e), the ter egent and line are ynonyou, with the yntactic difference that egent ut be followed by a copoite nae (for exaple - 1), while line ut be followed by a priitive nae, ince line i a priitive intance. 11. Soe variation i allowed, for exaple and i a ynony of the coa, article the in front of noun are optional. 12. Querie end by a period.. Here i, for exaple, a decription of Fig. 1 in the controlled query language: D, E, F are idpoint of -C, -, -C ; C-E interect -D at G ; draw -C, -F, -, -C, -D, C-E. The query language i copiled, producing a Cypher query, which i then ubitted to the graph databae exactly a when uing the DGS. The copiler ha been developed uing the Python PLY package [1]. 3.2 Future plan for the controlled language In future verion of the controlled language, we plan to introduce the poibility of extending the query ontology by introducing new concept and/or new relation. For exaple, it ay be intereting to define a type quare a Point,, C, D for a quare --C-D when we draw equal egent -, -C, C-D, D- where angle --C, -C-D, C-D-, D-- are right. Thi would allow querie of the for (which will draw the notoriou figure of the Pythagorean theore) ngle --C i right ; -C-C 1-2, , C-- 2-C 2 are quare. 4 Reduced Querie The algorith we decribe in thi paper can be quite ucceful in finding exact atche of querie in the corpu. ut what happen when the figure in the corpu atch only partially the query?

10 4.1 Ontological Graph For exaple, let u conider Fig. 2 anew. The ontological graph of the figure ha been build olely uing the XML data of Figure 13 of the corpu (cf. Fig. 1). What i not viible on Fig. 1 i the fact that G ha not been defined a lying on F, and hence the belong to edge between G and F i iing in the ontological graph. Thi i alo reflected in the CQL query exaple we gave in 3.1, where we requet that C-E interect -D at G but not that -F interect -D at G, probably becaue thi could be inferred fro the previou one, if we had the external Euclidean Geoetry knowledge of the fact that the three edian of a triangle have a coon interection. Neverthele, the uer eeking Fig. 1 i not necearily aware of thi ubtlety, and will earch for a triangle with three edian, which will reult in an ontological graph iilar to the one of Fig. 2 but containing an additional edge G F, and thi graph, of coure, will not atch Figure 13 of the corpu, ince it i not a ub-graph of it. To olve thi proble, a long a a query doe not return any reult, we retry with reduced querie, in the ene of the ae query graph with one or ore intance (or relation) reoved. ut how do we decide which node and edge to reove fro a query, and in what order? The anwer to thi quetion i provided by dependency lattice, decribed in the next ection. 4.2 Dependency Lattice Let u return to the procedural approach of decribing geoetric figure. How do we decribe a figure uing the operation that led to it contruction? Strictly peaking, uch a decription would require a erge-acyclic hypergraph [5, 3], where each operation would be a hyper-edge, connecting the input (the et of known node) and the output (the et of new node), for exaple, in the cae of the idpoint operation on egent C, the hyper-edge would connect {, C} (input) and {} (output). ut there i a ipler way. In fact, it uffice for our need to repreent dependencie a edge of a directed graph. For exaple, in the idpoint exaple, i dependent of and C, ince the latter have been ued to calculate the forer: C y adding a global ource node (located above all ource node) and a global ink node (underneath all final reult ), thi graph becoe a lattice, the partial order of which i the dependency relation. On Fig. 3, the uer can ee the dependency lattice of Fig. 1. We have ued only node that are ued in calculation, o that, for exaple, egent, E, etc. do not appear in the lattice. S and T are the global ource and global ink

11 S C D E F CE F D i G i T Fig. 3. The dependency lattice of Fig. 1, a it i procedurally decribed in the corpu. node, they are connected to node of the lattice by dahed arrow. Full arrow repreent operation and are labelled by their initial letter ( = idpoint, = egent drawing, i = interection). 4.3 Uing Dependency Lattice for Reduced Querie Let u now ee how the dependency lattice would be affected if the XML decription of Fig. 1 had an additional intruction, aying that G i (alo) the interection of D and F. On Fig. 4 one can copare the two graph, on the right ide one can ee the one with the additional intruction. S C D E F CE F D i G T i S C D E F CE F D i i G T i Fig. 4. The dependency lattice of Figure 13 of the corpu (left) and the dependency lattice of Figure 13 plu an additional intruction egent -F interect egent -D at point G (right).

12 Data: corpu of declaratively decribed figure, a query Reult: One or ore figure atching the query if uing controlled query language then write the query in controlled query language; ele draw the query in a DGS; end convert query into ontological graph; apply inference to ontological graph; convert ontological graph to Cypher; ubit to neo4j databae; if no reult returned then convert query into dependency lattice; while no reult returned do extract node or relation fro botto of dependency lattice; reove that node or that relation fro the ontological graph; convert ontological graph to Cypher; ubit to neo4j databae; end end lgorith 1: The Query lgorith for a Declaratively Decribed Corpu. Indeed, the new dependency lattice ha one additional edge F G. On the other hand, the edge F T diappear ince there i a path fro F to T, and F i not a ink anyore. dependencie have to be repected, if we reove a node fro the upper part of the lattice, we will have to reove all decendant of it. For thi reaon, the only reaonable query reduction trategy would be to reove node or edge fro the lower part of the lattice. If we reove, for exaple, the node G (and hence the edge CE G, F G and D G), then we obtain a triangle with three edian but where the barycenter ha not been explicitly drawn. (Interetingly, we till obtain a figure that i viually identical to Fig. 1.) If we go further and reove one of the relation aong CE G, F G and D G (for yetry reaon it doen t atter which relation we reove), then the query will ucceed, while the viual repreentation of the figure till ha not changed. lgorith 1 i a ynthei of the geoetric figure query algorith we propoe. 5 Evaluation We plan to evaluate the algorith decribed in thi paper, in the following way:

13 Querying a ub-figure in a corpu of declaratively decribed figure give a binary reult: either the figure i atching the ub-figure or it i not, o evaluation i iply counting the nuber of uccee. n intereting paraeter to oberve i the nuber and nature of query reduction that were neceary to obtain reult, correlated with the nuber of reult obtained. We will proceed a follow: after viually inpecting the corpu (and hence with no knowledge about the procedural and declarative decription of the figure) we will forulate 20 querie and anually annotate the figure we expect to find. fter uing the algorith, we will count the nuber of uccee and tudy the nuber of reult v. the paraeter of query reduction. 6 Future Work future work, beide extending the controlled natural language ( 3.2), we plan to integrate thi earch echani in repoitorie uch a TGTP and Intergeo, and in learning environent like the Web Geoetry Laboratory [14]. In a ore generic approach, we hould ue a coon forat and develop an application prograing interface that will allow to integrate the earching echani in any geoetric yte in need of it. 7 Concluion In thi paper we have preented algorith for querying geoetric figure in either declaratively or analytically decribed corpora, by uing either a DGS or a dedicated controlled query language. t the tie of ubiion of the article, evaluation wa not copleted, hence it i preented a a plan. Reference 1. eazley, D.: Python Lex-Yacc, 2. Chou, S.C., Gao, X.S.: utoated reaoning in geoetry. In: Handbook of utoated Reaoning. pp Elevier Science Publiher (2001) 3. Chou, S.C., Gao, X.S., Zhang, J.Z.: deductive databae approach to autoated geoetry theore proving and dicovering. Journal of utoated Reaoning 25, (2000) 4. DiffeoR: nwer to I it poible to recontruct a triangle fro the idpoint of it ide? (2014/2/20), 5. Fagin, R.: Degree of acyclicity for hypergraph and relational databae chee. Journal of the CM (JCM) 30(3), (Jul 1983) 6. Hohenwarter, M., Preiner, J.: Dynaic atheatic with GeoGebra. The Journal of Online Matheatic and It pplication 7 (2007), ID 1448

14 7. Janičić, P., Quarea, P.: Syte decription: GCLCprover + GeoTh. In: Furbach, U., Shankar, N. (ed.) utoated Reaoning. Lecture Note in Coputer Science, vol. 4130, pp Springer (2006) 8. Jiang, J., Zhang, J.: review and propect of readable achine proof for geoetry theore. Journal of Syte Science and Coplexity 25, (2012) 9. Kortenkap, U., Dohrann, C., Krei, Y., Dording, C., Libbrecht, P., Mercat, C.: Uing the Intergeo platfor for teaching and reearch. In: Proceeding of the 9th International Conference on Technology in Matheatic Teaching (ICTMT-9) (2009) 10. Quarea, P., Janičić, P., T.J., Vujošević-Janičić, M., Tošić, D.: Counicating Matheatic in The Digital Era, chap. XML-ae Forat for Decription of Geoetric Contruction and Proof, pp K. Peter, Ltd. (2008) 11. Quarea, P.: Thouand of Geoetric proble for geoetric Theore Prover (TGTP). In: Schreck, P., Narboux, J., Richter-Gebert, J. (ed.) utoated Deduction in Geoetry, Lecture Note in Coputer Science, vol. 6877, pp Springer (2011) 12. Quarea, P.: n XML-forat for conjecture in geoetry. pp No. 921 in CEUR Workhop Proceeding, achen (2012), Quarea, P., Haralabou, Y.: Geoetry contruction recognition by the ue of eantic graph. In: RECPD 2012: 18th Portuguee Conference on Pattern Recognition, Coibra: October pp (2012) 14. Quarea, P., Santo, V., ouallegue, S.: The Web Geoetry laboratory project. In: CICM 2013 Proceeding. Lecture Note in rtificial Intelligence, vol. 7961, pp Springer (2013) 15. Robinon, I., Webber, J., Eifre, E.: Graph Databae. O Reilly (2013) 16. Santiago, E., Hendrik, M., Krei, Y., Kortenkap, U., Marquè, D.: i2g Coon File Forat Final Verion. Tech. Rep. D3.10, The Intergeo Conortiu (2010), The Intergeo Conortiu: Intergeo ipleentation table. xwiki/bin/view/i2gforat/ipleentationtable (2012) 18. Wikipedia: Lit of interactive geoetry oftware. wiki/lit_of_interactive_geoetry_oftware (February 2014)