Spider Diagrams: a diagrammatic reasoning system

Transcription

1 Spider iagram: a diagrammaic reaoning yem John Howe, Fernando Molina, John Taylor School of Compuing and Mahemaical Science niveriy of righon, K {John.Howe, F.Molina, John.Taylor}@brighon.ac.uk Suar Ken Compuing Laboraory niveriy of Ken, Canerbury, K S.J.H.Ken@ukc.ac.uk Joeph (Yoi) Gil eparmen of Compuing Science Technion IIT, Haifa 32000, Irael yogi@c.echnion.ac.il 1

2 brac Spider diagram combine and exend Venn diagram and Euler circle o expre conrain on e and heir relaionhip wih oher e. Thee diagram can be ued in conjuncion wih objec-oriened modelling noaion uch a he nified Modeling Language. Thi paper ummarie he main ynax and emanic of pider diagram. I alo inroduce inference rule for reaoning wih pider diagram and a rule for combining pider diagram. Thi yem i hown o be ound bu no complee. ijuncive diagram are conidered a one way of enriching he yem o allow combinaion of diagram o ha no emanic informaion i lo. The relaionhip of hi yem of pider diagram o oher imilar yem, which are known o be ound and complee, i explored briefly. Keyword iagrammaic reaoning, viual formalim. 2

3 1. Inroducion iagrammaic noaion involving circle or cloed curve, which we will call conour, have been in ue ince a lea he Middle ge [11]. In he middle of he 18 h cenury, he Swi mahemaician Leonhard Euler inroduced he noaion we now call Euler circle (or Euler diagram) [2] for he repreenaion of claical yllogim. Thi noaion ue he opological properie of encloure, excluion and inerecion o repreen he e-heoreic noion of ube, dijoin e, and inerecion, repecively. The 19 h cenury logician John Venn [16] modified hi noaion o repreen logical propoiion. In Venn diagram all conour mu inerec. Moreover, for each non-empy ube of he conour, here mu be a conneced region of he diagram, uch ha he conour in hi ube inerec a exacly ha region. Shading i hen ued o how ha a paricular region repreen he empy e. Venn diagram are expreive a a viual noaion for wriing conrain on e and heir relaionhip wih oher e, bu complicaed o draw becaue all poible inerecion have o be drawn and hen ome region haded. rawing he Venn diagram of four or more e i quie challenging. More [12], in he lae 1950, developed an algorihm for adding a new conour o a Venn diagram. I i poible o add conour indefiniely, bu he conour quickly aume weird and wonderful hape, and he reuling diagram i very complicaed and difficul o follow. Indeed, i i rare o ee Venn diagram of four or more conour. On he oher hand, Euler circle are inuiive and eaier o draw, bu are no a expreive a Venn diagram becaue hey lack proviion for hading. n indicaion of he populariy and inuiivene of Venn and Euler diagram i he fac ha hey are ued in elemenary chool for eaching e heory a an inroducion o mahemaic. In fac, i i uually a hybrid of he wo noaion ha i ued for eaching purpoe; in view of heir relaive meri, i doe eem naural o combine he wo noaion, by relaxing he demand ha all curve in Venn diagram mu inerec or by inroducing hading ino Euler diagram. Thi combined noaion form he bai of pider diagram. In he 1890, Peirce modified Venn diagram by including X-equence o inroduce elemen and dijuncive informaion ino he yem [13]. Recenly, full formal emanic and inference rule have been developed for Venn- Peirce diagram [15] and Euler diagram [6]; ee alo [1, 5] for relaed work. Shin [15] prove oundne and compleene reul for wo yem of Venn-Peirce diagram. In objec-oriened ofware developmen, diagrammaic modelling noaion are ued o pecify yem. Recenly, he nified Modelling Language (ML) [14] ha become he Objec Managemen Group (OMG) andard for uch 3

4 noaion. In ML, conrain, uch a invarian, precondiion and pocondiion, are expreed uing he Objec Conrain Language (OCL) [17] which i eenially a ylied, exual form of fir-order predicae logic and i par of he ML andard. Conrain diagram [10, 4] provide a diagrammaic noaion for expreing conrain and can be ued in conjuncion wih ML and OCL. Librarie collecion Copie OnHold Publicaion publicaion onholdfor reerved Reervaion Figure 1.1 The conrain diagram in figure 1.1 expree (among oher conrain) an invarian on a model of a library yem: for any library objec, and any of ha library copie which i on hold, ha copy publicaion mu be he ame a ha aociaed wih he reervaion for which i i on hold. The noaion i baed on a mixure of Venn and Euler diagram. Spider diagram [3] emerged from work on conrain diagram. They combine and exend Venn diagram and Euler circle o expre conrain on e and heir relaionhip wih oher e. Thi paper ummarie he ynax and emanic of pider diagram and exend he diagrammaic inference rule for Venn-Peirce diagram o pider diagram. more deailed dicuion of pider diagram i conduced in ecion 2, where he main ynax and emanic of he noaion i inroduced. Secion 3 inroduce inference rule for reaoning wih pider diagram and a rule which govern he equivalence of Venn and Euler form of pider diagram, and dicue he rule for combining wo pider diagram. Secion 4 dicue oundne and compleene of he yem and indicae one poible way of enriching he yem in order o combine pider diagram o ha no emanic informaion i lo. Subyem of pider diagram which include dijuncive diagram are hown o be ound and complee. 4

5 2. Spider diagram Thi ecion inroduce he main ynax and emanic of pider diagram; ee [3] for more deail and example. Spider diagram are Euler circle augmened wih haded region and pider. Spider diagram alo include he concep of Schrödinger pider and projecion; hee are no neceary for hi paper and are omied from hi dicuion. In [3], he diincion i made beween given and exienial pider; a given pider denoe a given elemen of he correponding e (in he ame way ha conour repreen given e) wherea an exienial pider denoe exienial quanificaion over he correponding e. In hi paper, all pider are given (excep for he yem inroduced in ecion 4.3) Synacic elemen of pider diagram conour i a imple cloed plane curve. boundary recangle properly conain all oher conour. diric (or baic region) i he bounded area of he plane encloed by a conour. region i defined a follow: any diric i a region; if r 1 and r 2 are region, hen he union, inerecion, or difference, of r 1 and r 2 are region provided hee are nonempy. zone (or minimal region) i a region having no oher region conained wihin i. Conour and region denoe e. pider i a ree wih node (called fee) placed in differen zone; node are repreened by mall quare and he connecing edge (called leg) are raigh line. pider ouche a zone if one of i fee appear in ha region. pider may only ouch a zone once. pider i aid o inhabi he region which i he union of he zone i ouche. For any pider, he habia of, denoed η(), i he region inhabied by. The e of pider ouching region r i denoed by T(r). Spider are ued o denoe elemen; in hi paper, all pider repreen given elemen of he correponding e. Two diinc pider denoe diinc elemen, unle hey are joined by a ie or by a rand. ie i a double, raigh line (an equal ign) connecing wo fee, from differen pider, placed in he ame zone. Tie indicae equaliy of he correponding elemen. The ne of pider and, wrien τ(, ), i he union of hoe zone z having he propery ha he fee of and are conneced by a ie in z. Two pider which have a non-empy ne are referred o a mae. If boh he elemen denoed by pider and belong o he e denoed by he ame zone in he ne of and, hen and denoe he ame elemen. 5

6 rand i a wavy line connecing wo fee, from differen pider, placed in he ame zone. Srand indicae ha he correponding elemen may (bu no necearily mu) be equal. The web of pider and, wrien ζ(, ), i he union of zone z having he propery ha here i a equence of pider = 0, 1, 2,, n = uch ha, for i = 0,, n 1, i and i+1 are conneced by a ie or by a rand in z. So τ(, ) i a ubregion of ζ(, ). Two pider wih a non-empy web are referred o a friend. Two pider and may (bu no necearily mu) denoe he ame elemen if ha elemen i in he e denoed by he web of and. Clearly, if here i a ie beween fee, hen a rand beween hoe fee i redundan. Similarly, muliple rand or ie beween he ame pair of fee are redundan. Thu, on he ynacic level, we allow a mo one ie or rand beween any pair of fee. In laer ecion, we will need o compare region acro diagram. To faciliae hi, we exend he noaion and ue, for example, ζ(,, ) and τ(,, ) o denoe he web and ne repecively of pider and in he diagram. Every region i a union of zone. region i haded if each of i componen zone i haded. haded region conaining no pider denoe he empy e. Shading a region r which include pider ha he effec of placing an upper limi on he number of elemen in he e denoed by he region. n upper bound i T(r), bu hi migh no be a lea upper bound. pider diagram i a finie collecion of conour (exacly one of which mu be a boundary conour ), pider, rand, ie and haded region. For any pider diagram, we ue C = C(), R = R(), Z = Z(), Z* = Z*() and S = S() o denoe he e of conour, region, zone, haded zone and pider of, repecively. The Venn form of a pider diagram conain every poible inerecion of conour; oherwie, he diagram i in Euler form. pider diagram wih n (non-boundary) conour ha 2 n zone if and only if i i in Venn form. The pider diagram in figure 2.1 i in Venn form. I ha hree non-boundary conour,, C and wo pider and. The label refer o he whole pider and no ju o any paricular node. There i a ie beween and in ( C), a rand beween and in ( C) and no ynacic connecion beween and in ( ) C. Hence, if he elemen denoed by and boh belong o ( C) hen hey are equal; if hee elemen boh belong o ( C) hey may be equal or diinc; and if hee elemen boh belong o ( ) C hen hey are diinc. elow are ome properie of (he denoaion of) where, for impliciy, we ue he ame label for a conour and he e i denoe and we ue he ame label for a pider and he elemen i denoe. 6

7 ( C) =, ( C) 1 ( C) ( C ), ( C) ( C ),, C =,, C. C Figure Semanic of pider diagram The emanic of a pider diagram i given in erm of he emanic funcion Ψ : C Se, ψ : S where i a given univeral e of and Se denoe he power e of. Conour and region are inerpreed a ube of, and pider a elemen of. The boundary conour i inerpreed a. zone i uniquely defined by he conour conaining i and he conour no conaining i; i inerpreaion i he inerecion of he e denoed by he conour conaining i and he complemen of he e denoed by hoe conour no conaining i. We exend he domain of Ψ o inerpre region a ube of. Fir define Ψ: Z Se by Ψ() z = Ψ() c Ψ() c + c C () z c C () z 7

8 where C+(z) i he e of conour conaining he zone z, C (z) i he e of conour no conaining z and Ψ( c) = Ψ( c), he complemen of Ψ(c). Since any region i a union of zone, we may define Ψ: R Se by Ψ() r = Ψ() z z Z() r where, for any region r, Z(r) i he e of zone conained in r. The emanic of a diagram i he conjuncion of he following condiion. Plane Tiling Condiion: ll elemen fall wihin e denoed by zone: z Z Ψ( z) = Spider Condiion: The elemen denoed by a pider i in he e denoed by he habia of he pider: S ψ () Ψ ( η ()) Sranger Condiion: The elemen denoed by wo diinc pider are diinc unle hey fall wihin he e denoed by he pider web:, S ( ψ() = ψ() ψ(), ψ() Ψ( ζ(,)) ) Maing Condiion: If he elemen denoed by wo diinc pider fall wihin he e denoed by he ame zone in he pider ne, hen he elemen are equal:, S z Z( τ (, )) ( ψ(), ψ() Ψ () z ψ() = ψ() ) 8

9 Shading Condiion: The e denoed by a haded zone conain no elemen oher han hoe denoed by he pider: Ψ( z) { ψ ( )} z Z S We will require he following lemma which follow from he pider and hading condiion. Lemma 2.1. The e denoed by a haded zone no conaining he fee of any pider i empy. 3. Reaoning wih pider diagram In hi ecion we inroduce rule for manipulaing ingle diagram. Excep he la rule, each i an inference rule ha allow u o obain one diagram from a given diagram by adding or removing diagrammaic elemen. The la rule govern he equivalence of he Euler and Venn form of pider diagram. Throughou hi ecion we ue and repecively o denoe he diagram before and afer a ingle applicaion of one of he rule. To link he emanic of he before and afer diagram, we aume ha and denoe he ame (univeral) e, which we denoe, and ha any wo conour or pider agged wih he ame label in and denoe he ame e or elemen Rule of ranformaion We inroduce even rule for manipulaing ingle diagram. The fir ix are inference rule ha allow u o obain one diagram from a given diagram by removing, adding or modifying diagrammaic elemen. The la rule govern he equivalence of he Euler and Venn form of pider diagram. Rule 1: Inroducion of a rand. rand may be drawn beween he fee of any wo pider in he ame zone. Similarly, any ie may be replaced wih a rand. 9

10 Example 3.1 Inroducing a rand beween wo non-conneced fee in a zone weaken he informaion conained in he diagram. In figure 3.1, he pider and u in diagram repreen diinc elemen bu in hey may repreen he ame elemen of. Similarly, replacing a ie beween he fee of wo pider wih a rand alo weaken he emanic informaion given by he diagram. If he elemen denoed by lie in, hen, in, and are necearily equal wherea in hey need no be. u u ' Figure 3.1 Rule 2: Spreading he fee of a pider. If a diagram ha a pider, hen we may draw a node in any zone z which doe no conain a foo of and connec i o. If z conain he foo of anoher pider, hen we may join he fee of and wih a rand or a ie or leave he fee eparaed in z. Example 3.2 Rule 2 i illuraed by he diagram in figure 3.2. The inference from o require wo applicaion of rule 2, bu i clearly valid ince i ju repreen a weakening of informaion. From we know ha he elemen correponding o belong o. Having pread i fee in, we may only infer ha hi elemen belong o. In he zone correponding o, we have choen o keep he fee of and eparaed; in he zone correponding o, we have joined he fee of and wih a rand. 10

11 ' Figure 3.2 Rule 3: Eraure of a pider. We may erae a complee pider on any non-haded region and any rand or ie conneced o i. If removing a pider diconnec any componen of he rand-ie graph in a zone, hen he componen o formed hould be reconneced uing one or more rand o reore he original componen. Example 3.3 In figure 3.3a, eraing he pider u and i wo connecing rand diconnec pider and in he zone. However, he web of and i he region, and hi hould no change wih he deleion of u. Hence in he pider are explicily reconneced by joining hem wih a rand. In he diagram given in figure 3.3b, he elemen denoed by pider and need no be equal (unle he elemen denoed by u belong o again reconneced by a rand and no a ie in he diagram. ) which i why hey are u u ' ' Figure 3.3a Figure 3.3b Example 3.4 The requiremen ha he region from which a pider i removed hould be non-haded i a neceary one. Figure 3.4 illurae ha he removal of a pider from a haded zone may reul in an invalid inference (ee ecion 3.2). In diagram, he e correponding o region conain a ingle elemen, wherea in, he correponding e i empy. 11

12 ' Figure 3.4 Rule 4: Eraure of hading. We may erae he hading in an enire zone. Example 3.5 In he diagram given in figure 3.5, he e correponding o region conain a mo a ingle elemen, wherea in, he correponding e i no conrained. ' Figure 3.5 Rule 5: Eraure of a conour. We may erae a conour. When a conour i eraed: any hading remaining in only a par of a zone hould alo be eraed. if a pider ha fee in wo region which combine o form a ingle zone wih he eraure of he conour, hen hee fee are replaced wih a ingle foo conneced o he re of he pider and any ie connecing i in he new zone hould be replaced by rand. Example 3.6 Eraing a conour can caue boh ynacic and emanic difficulie. 12

13 ' Figure 3.6 Figure 3.6 illurae he ynacic difficulie. Simply eraing he conour in he diagram, he (new) zone become parially haded and he pider ha wo fee in he new zone. To enure ha he reuling diagram i wellformed, he parial hading mu be eraed and he fee of in hould be replaced wih a ingle foo. The la par of rule 5 concern emanic difficulie conneced wih eraing a conour and i a lile more uble. ' Figure 3.7 Conider he diagram hown in figure 3.7. The diagram ha a model in which he elemen correponding o pider and boh belong o he e bu are diinc; namely, he model where and. When he conour i removed, hee wo zone and combine o form he ingle zone in. Since i i poible for and o repreen diinc elemen of, he ie connecing hem mu be replaced wih a rand. Rule 6: Inroducion of a conour. new conour may be drawn inerior o he bounding recangle oberving he parialoverlapping rule: each zone pli ino wo zone wih he inroducion of he new conour. If he zone i haded, hen boh correponding new zone are haded. Each foo of a pider i replaced wih a conneced pair of fee, one in each new zone. Likewie, each rand or ie bifurcae and become a pair of rand or ie, one in each new zone. 13

14 Example 3.7 In figure 3.8, a new conour i inroduced aifying he parial overlapping rule. Each zone in become a pair of zone in and each foo of pider, and u bifurcae o become wo fee, one in each new zone. The rand and ie alo bifurcae. The zone in correponding o he haded zone in alo become haded. u u ' Figure 3.8 Rule 7: Equivalence of Venn and Euler form. We may replace a diagram in which ome region do no exi by a diagram V() in Venn form where hoe region are haded. ll oher diagrammaic elemen oher haded region, pider, rand and ie remain unchanged. Converely, we may replace a diagram in Venn form which ha a e of haded zone conaining no pider by a diagram E where (ome of) hoe region do no exi. gain, all oher diagrammaic elemen oher haded region, pider, rand and ie remain unchanged. The raniion from he Euler o he Venn form of a pider diagram i algorihmic. There are variou known algorihm for conrucing a Venn diagram wih n conour for example, ee [6]. Given a pider diagram in Euler form, fir conruc he underlying Venn diagram whoe e of conour i C(). Shade any zone which were no preen in he original Euler form. Finally add pider, rand and ie in order o replicae he rand-ie graph in each zone of. The reuling pider diagram i V(), he Venn form of. C r r C ' Figure

15 Example 3.8 Figure 3.9 illurae he equivalence beween he Euler and Venn form of a pider diagram. The Euler form doe no conain zone correponding o C or C. In he Venn form, he correponding region are haded, bu he rand-ie graph in every oher zone i he ame a he correponding graph in Comparing region Laer we will need o be able o idenify correponding region in differen diagram. For impliciy, we conider he cae where a diagram i obained from a diagram by adding conour, o ha C() C( ). There i a naural mapping α=: Z() R( ) which idenifie zone in wih heir correponding region in. The mapping may be defined inducively, wih he inducive ep a follow. Suppoe ha i obained from by adding a ingle conour. ccording o Rule 6, each zone z in bifurcae ino wo zone z in and z ou in ; z in i ha par of z encloed wihin he new conour and z ou i ha par of z lying ouide he new conour (ee figure 3.10). In hi cae, we define α(z) = z in z ou. Given any zone z in, here i a unique zone z in uch ha z α(z). The aociaion z z define a mapping β=: Z( ) Z() o ha β(z ) i he unique zone in ha denoe a upere of he e repreened by z. The mapping α and β are illuraed in figure

16 z ou z z in C ' α(z) = z in z ou, β(z in ) = z = β(z ou ) Figure 3.10 y aking union of zone, hee mapping exend o mapping α=: R() R( ), β=: R( ) R(). Thee mapping are relaed a follow. For all region r R(), βα(r) = r and for all region r R( ), r αβ(r ). The fir of hee aemen ay ha β i a lef invere for α and α i a righ invere for β. I follow ha α i injecive and β i urjecive. We ay ha a region r R() correpond o a region r R( ) if α(r) = r. We will need he following lemma. Lemma 3.1 (i) Le be he diagram formed from by adding a conour C aifying he parial overlapping rule. If he zone z in and z ou in are formed from he zone z in a decribed above, hen Ψ(z, ) = Ψ(z in, ) Ψ(z ou, ). (ii) Le be he diagram formed from by adding a conour aifying he parial overlapping rule. If a region r R() correpond o a region r R( ) hen hey denoe he ame e: Ψ(r) = Ψ(r ) Combining diagram Given wo diagram, 1 and 2, we wih o combine hem o produce a ingle diagram which reain a much of heir combined emanic informaion a poible. Of coure, hi i only meaningful if he pair 1, 2 i conien. In hi ecion we decribe he conrucion of uch a combined diagram. Even in imple cae, ome informaion conained in he pair 1, 2 will be lo in he combinaion. In he nex ecion, we will indicae one poible way of enriching he yem of pider diagram o overcome hi problem. 16

17 Suppoe wo diagram 1 and 2 are given which do no conain conflicing informaion. To implify he proce of combinaion, we fir conruc he equivalen Venn form of each diagram, V( 1 ) and V( 2 ) repecively. The combined diagram clearly mu conain any conour which appear in eiher 1 or 2, o he fir ep in combining he diagram i o conruc a Venn diagram whoe e of conour i C( 1 ) C( 2 ). From hi underlying Venn diagram, we add diagrammaic elemen hading, pider, rand and ie o produce he final combined diagram. Since i obained from each of he diagram V( 1 ) and V( 2 ) by adding conour, he correponding region mapping inroduced in he previou ecion are defined beween V( 1 ) and and beween V( 2 ) and. Thee are denoed, repecively, α 1, β 1 and α 2, β 2. ny haded zone in he Venn form V( 1 ) or V( 2 ) mu correpond o a haded region in. Hence a zone z of i haded if and only if β 1 (z) Z*(V( 1 )) or β 2 (z) Z*(V( 2 )). a conequence, we have: z = α1() z α2() z. z Z ( ) z Z ( V( 1)) z Z ( V( 2)) Thi ep i illuraed in figure 3.11 (where 1 = V( 1 ) and 2 = V( 2 )). C 1 2 C Figure

18 Nex, we add pider o. Since η() define he region o which belong, inuiion ugge ha, for each pider, i habia in hould be he inerecion of he correponding habia in V( 1 ) and V( 2 ). Thi i no quie correc, however, ince i doe no ake accoun of region which are known o be empy. 1 2 Figure 3.12 Thi i illuraed in figure The habia of he pider in he combined diagram mu exclude he region ince, from 2, hi correpond o an empy e. We define a region of a pider diagram o be empy if i i haded and i no ouched by any pider foo. We denoe by E(V()) he e of he empy zone of V(): E(V()) = Z*(V()) {z Z(V()) T(z) = }. For each pider S(V( 1 )) S(V( 2 )), we need o define i habia in. There are eenially wo cae. If belong o boh diagram 1 and 2 hen i habia in i he inerecion of i habia in each diagram: S(V( 1 )) S(V( 2 )) η(, ) = α 1 (η(, (V( 1 ))) α 2 (η(, V( 2 ))) If belong o exacly one of he diagram 1 and 2 hen i habia in i reduced by removing from i he empy zone in he oher diagram: S(V( 1 )) S(V( 2 )) η(, ) = α 1 (η(, (V( 1 ))) 7 α2 ( z). z E( V( 2 )) Wih hee definiion, he compoiion of he wo diagram in figure 3.12 i given in figure

19 Figure 3.13 Finally, we conider rand and ie. Suppoe wo pider are uch ha each ha a foo in a zone z of he combined diagram. Then z correpond o zone z 1 = β 1 (z) and z 2 = β 2 (z) in V( 1 ) and V( 2 ), repecively. gain here are everal cae o conider. If neiher diagram V( 1 ) nor V( 2 ) conain boh pider, hen hey hould be joined by a rand in z. In hi cae, one pider belong o V( 1 ) and he oher belong o V( 2 ), o we have no informaion concerning heir equaliy or oherwie if hey belong o z; hence he pider hould be conneced in he mo general way. If exacly one of he diagram, V( i ) ay, conain boh pider, hen hey hould be conneced in z in he ame manner a in z i. If boh diagram conain boh pider hen: hey are conneced by a ie in z if hey are joined by a ie in one of he region z 1, z 2 and a ie or rand in he oher region; hey are no conneced in z if hey are no conneced in one of he region z 1, z 2 and are eiher no conneced or conneced by a rand in he oher region; oherwie hey are conneced by a rand in z. Example 3.9. Conider he diagram given in figure Since C( 1 ) = C( 2 ), i follow ha each of he correpondence mapping α 1, β 1, α 2, β 2 defined above i he ideniy mapping. Since here are alo no haded region, i follow ha η(, ) = η(, 1 ) η(, 2 ). The habia of he pider i equal in all hree diagram. We need o conider eparaely each zone in which conain fee of boh pider. For, he pider are conneced by a ie in one diagram ( 2 ) and a rand in he oher; hence in he combined diagram, hey are conneced by a 19

20 ie. For, he pider are eparaed in one diagram ( 1 ) and joined by a rand in he oher; hence he pider hould be eparaed in. 1 2 Figure 3.14 Example Thi example illurae ha i i poible for wo pider, and, o be eparaed in z 1 and be joined by a ie in z 2. before, z 1 and z 2 denoe zone in 1 and 2, repecively, which correpond o he zone z in he combined diagram conaining fee of boh and. Conider he zone z = C in he compoie diagram hown in figure 3.15 below. Thi zone conain fee boh of and of. The zone z ogeher wih he correponding zone z 1 = β 1 (z) and z 2 = β 2 (z) are illuraed wih hickened border. Noe ha and are eparaed in z 1 bu are ied in z 2. lhough i i no poible for he elemen correponding o and boh o belong o C, hi informaion i no capured in. Thu i could be argued ha i i immaerial how and are conneced in z. We have choen o connec heir fee wih a rand o ha each pair of diagram, 1, and 2,, i conien. 20

21 C 1 2 C Figure Soundne and compleene In hi ecion, we ouline he proof of validiy of he inference rule inroduced in he previou ecion and dicu oundne and compleene of hi yem and relaed yem. model for a diagram i a riple m = (, Ψ, ψ) where Ψ: C Se and ψ: S are he emanic funcion defined in ecion 2.2. We ay a model m complie wih, denoed m, if i aifie he conjuncion of he emanic predicae inroduced in ecion 2.2. diagram i a conequence of a diagram, denoed, if every complian model for i alo a complian model for. We aume ha conour wih he ame label in and repreen he ame e. To ay ha a rule i valid, we mean ha whenever a diagram i obained from anoher diagram by a ingle applicaion of he rule, hen Validiy of he inference rule. Several of he rule amoun o hrowing away ome of he emanic informaion conained in a diagram, in he ene decribed in he following lemma. Noe ha we adop he convenion ha he conjuncion of an empy e of propoiion equae o rue. 21

22 Lemma 4.1 If diagram and have emanic of he form i I inference from. P i and i J P i repecively, where J I, hen i a valid Rule 1: Inroducing a rand. Suppoe wo pider and have fee which are eparaed (ha i, no joined by a rand or a ie) in a zone z belonging o diagram. Le be he diagram obained from by adding a rand beween he fee of and in z. Then ζ(,, ) = ζ(,, ) z. The Sranger Condiion i he only emanic condiion which involve he web of and ; for hee pider he condiion i ψ() = ψ() ψ(), ψ() Ψ( ζ(,, )). Since ζ(,, ) ζ(,, ), we can infer he correponding condiion for. ll he oher emanic condiion are idenical for and, o he fir par of rule 1 i valid. To juify he validiy of he econd par of he rule, uppoe and are a decribed above excep ha, in, he pider and are joined by a ie in z. In hi cae, he web of and i unchanged, bu heir ne change beween he diagram: τ(,, ) = τ(,, ) z. Thu i i only he Maing Condiion which change in. For and, he Maing Condiion i a conjuncion of erm of he form ( ) ψ() Ψ() z ψ() Ψ () z ψ() = ψ(), one erm for each zone z in he ne of and. y lemma 3.1, we may infer he Maing Condiion of from ha of. Rule 2: Spreading he fee of a pider. Suppoe i obained from by preading he fee of pider ino he zone z. We conider he emanic condiion ha are changed in paing from o. 22

23 Spider Condiion. Spreading he fee of exend i habia o ha η(, ) = η(, ) z. Since η(, ) η(, ), he pider condiion for follow from ha for. To complee he proof, we uppoe z conain a pider and conider he hree cae given in rule 2. (a) If and are joined by a rand in z hen ζ(,, ) = ζ(,, ) z o ζ(,, ) ζ(,, ). Hence he Sranger Condiion for follow from ha for. In hi cae he Maing Condiion i unchanged. (b) If and remain eparaed in m hen heir web and ne are unchanged and hence o are he Sranger and Maing Condiion. (c) If and are joined by a ie in z hen ζ(,, ) = ζ(,, ) z and τ(,, ) = τ(,, ) z. The Sranger Condiion for follow a in cae (a). To obain he Maing Condiion for, we add he conjunc ψ(), ψ() Ψ(z) ψ() = ψ() o he Maing Condiion for. However, from he Spider Condiion for, we know ψ() Ψ(z) ince z doe no form par of he habia of in. Therefore he addiional conjunc i rue and he Maing Condiion for follow. Rule 3: Eraure of hading. Eraing he hading in a zone only change he Shading Condiion by removing conjunc, o he validiy of he fir par of rule 2 follow by lemma 4.1. Rule 4: Eraure of a pider. The validiy of he rule for eraing a pider follow imilarly. However, in paing from he emanic of o ha of, one or more conjunc may be lo from he Spider, Sranger and Maing condiion. 23

24 Rule 5: Eraure of a conour. When we erae a conour C, here exi wo poibiliie: zone z in i correpond o a zone z in (e.g., zone z 3 and z 3 in figure 4.1). here exi zone z in and z ou in ha become a ingle zone z in (e.g., zone z 1, z 2 and z in figure 4.1). z 1 E z z 3 z 2 z 3 E C Figure 4.1 When a conour i eraed all he emanic condiion are changed. We will conider each condiion eparaely. The Plane Tiling Condiion. Since C( ) {C} = C(), every region in ha a correponding region in and herefore he Plane Tiling Condiion for follow from he Plane Tiling Condiion for. The Shading Condiion. Eraing any hading remaining in only a par of a zone (z 4 for example in figure 4.2) only change he Shading Condiion by removing conjunc. Therefore he validiy of he fir par of he rule follow by lemma 4.1. For any pair of haded zone z in, z ou in ha combine o form a ingle haded zone z in afer he eraure of conour C, a pair of conjunc in he Shading Condiion for become a conjunc in he Shading Condiion for (for example, in figure 4.2, z 1 and z 2 in combine o form z in ). 24

25 z 1 z 2 z 3 E z z 3 E z 4 C ' Figure 4.2 y he Shading Condiion for we have Ψ( zin) { ψ ( )} and Ψ( zou ) { ψ ( )} S S whence ( Ψ( zin ) Ψ( zou )) { ψ ( )} S which i equivalen, by lemma 3.1, o Ψ( z ) { ψ ( )} S he Shading Condiion for z in. The hird and final poibiliy i ha a haded zone z in (e.g., z 3 in figure 4.2) correpond o a unique zone z in. I conjunc in he Shading Condiion for ha an equivalen for he haded zone z in he Shading Condiion for. Spider Condiion. For any pider, he e denoed by i habia in i a ube of he e denoed by i habia in : η(, ) η(, ) Therefore, we may infer he Spider Condiion for from ha of. Maing Condiion. For any ie replaced by a rand connecing pider r and in a zone z he Maing Condiion change by removing he conjunc ψ(), r ψ() Ψ () z ψ() r = ψ(). 25

26 For any oher ie connecing pider r and in zone z and no having o be replaced by a rand, here exi no region k conaining fee of r or uch ha Ψ(z) Ψ(k) = Ψ (z ) for ome z in (oherwie, he ie hould be replaced by a rand) or here exi a correponding region n in. Therefore, heir ne change beween he diagram: τ(,, ) τ(,, ) and for any conjunc including pider r and in he Maing Condiion for, here i a correponding conjunc in he Maing Condiion for. Sranger Condiion. y lemma 3.1, i follow ha ζ(r,, ) ζ(r,, ) Therefore for any wo pider r and in a zone z in, ψ() r = ψ() ψ(), r ψ() Ψ( ζ(,)) The Sranger Condiion in follow: ψ( r) = ψ( ) ψ( r), ψ( ) Ψ ( ζ(, )). Rule 6: Inroducion of a conour. To idenify and keep rack of he zone in ha arie from pliing zone in wih he inroducion of a conour C. We denoe by z in and z ou he wo zone in which are formed by pliing a zone z in ; z in i ha par of z encloed wihin he new conour C and z ou i ha par of z lying ouide C (ee figure 4.3). z ou z z in C' ' Figure 4.3 In paing from o by adding conour C a decribed in he rule, a number of emanic condiion change. We conider each condiion in urn. 26

27 Plane Tiling Condiion. The zone in may be grouped in pair of he form z in, z ou for ome zone z in. Hence he Plane Tiling Condiion for follow from he correponding condiion for by lemma 3.1. Spider Condiion. Suppoe a pider ha a foo in he zone z of. In he foo bifurcae giving a foo in each of he zone z in and z ou in. Hence, by lemma 3.1, Ψ(η(, )) = Ψ(η(, )), o he Spider condiion for follow from ha of. Maing Condiion. Each ie connecing pider and bifurcae in, o lemma 3.1 enure ha he ne of and i unchanged: τ(,, ) = τ(,, ). Suppoe ha pider and are joined by a ie in a zone z in. The Maing Condiion for z in give ψ(), ψ() Ψ () z ψ() = ψ(). If ψ(), ψ() Ψ (z in ) Ψ(z) hen ψ() = ψ(); imilarly, if ψ(), ψ() Ψ (z ou ) Ψ(z) hen ψ() = ψ(). The Maing Condiion for follow. Sranger Condiion. Thi i imilar o he pider condiion. ifurcaing each ie, rand and node of pider and we enure ha e denoed by heir web in and are idenical ζ(,, ) = ζ(,, ). The Sranger Condiion for and in hen follow from he correponding condiion for. Shading Condiion. Suppoe ha z i a zone in. Then boh he zone z in and z ou in are alo haded. The Shading Condiion for z, i he following. Ψ( z) { ψ ( )} S 27

28 Since Ψ(z) = Ψ (z in ) Ψ (z ou ), hi give Ψ( zin) { ψ ( )} and Ψ( zou ) { ψ ( )} S S Thee are preciely he Shading Condiion for z in and z ou in. Rule 7: Equivalence of Venn and Euler form. Given ha he raniion beween he Venn and he Euler form of a pider diagram only affec he repreenaion of empy zone i i only he Plane Tiling Condiion and he Shading Condiion which change. Suppoe a diagram in Venn form ha a e ZE * () of haded zone ha are no conained in he Euler form. The Shading Condiion for may be eparaed ino erm whoe zone belong o Z * () ZE * () and erm whoe zone belong o ZE * (): 7 and * z Z ( ) ZE ( ) Ψ( z) { ψ ( )} S 7 Ψ( z) { ψ ( )} z ZE * ( ) S Noe ha, ince Z * () ZE * () = Z * ( ), he fir collecion of conjunc, 7 * z Z ( ) ZE ( ) Ψ( z) { ψ ( )} S i equivalen o he Shading Condiion for. I can alo be hown ha he econd collecion of conjunc, i equivalen o he Plane Tiling Condiion for 7 Ψ( z) { ψ ( )} z ZE * ( ) S z Z( ) Ψ ( z) = ince any empy zone in ZE * () doe no exi in and herefore i i no included in he Plane Tiling Condiion for. Noe ha he Plane Tiling Condiion for he diagram in Venn form i rue ince conain all poible region. 28

29 Combining diagram Suppoe wo diagram 1 and 2 are given which do no conain conflicing informaion. Their combined diagram i formed by adding ynacic elemen ino he Venn diagram whoe e of conour i C( 1 ) C( 2 ). y he rule of equivalence of Venn and Euler form and he rule of inroducing conour, we may aume, wihou lo of generaliy, ha 1 and 2 are in Venn form and have he ame e of conour. Thu C() = C( 1 ) = C( 2 ). In hi cae, 1 = V( 1 ), 2 = V( 2 ) and he correponding region mapping, α 1, β 1 and α 2, β 2, are ideniy mapping. We conider each of he emanic condiion for in urn. Plane Tiling Condiion. Since i in Venn form, all poible zone appear in he diagram and he Plane Tiling Condiion for follow. Spider Condiion. Le be a pider in. There are wo cae o conider. Suppoe ha belong o boh 1 and 2. Then i habia in i he inerecion of i habia in 1 and 2 : η(, ) = η(, 1 ) η(, 2 ). In hi cae we have: ψ() Ψ(η(, 1 )) ψ() Ψ(η(, 2 )) (from he Spider Condiion in 1 and 2 ) ψ() Ψ(η(, 1 )) Ψ(η(, 2 )) ψ() Ψ(η(, 1 ) =η(, 2 )) (ince correponding region denoe he ame e) ψ() Ψ(η(, )) Suppoe ha belong exacly one of he diagram; ay, belong o boh 1 bu no 2. Then i habia in i η (, ) = η(, 1 ) z. z E( 2 ) Now ψ() Ψ(η(, 1 )) from he Spider Condiion in 1 and = ψ () Ψ() z ince Ψ(z) = for any zone z E( ) z E( 2 ), by lemma 2.1. Hence ψ() Ψ(η(, )). Therefore he Spider Condiion i aified in. 2 29

30 Sranger Condiion. The rule for combining diagram implie ha, for all pider, in, ζ(,, 1 ) ζ(,, 2 ) ζ(,, ). Hence he Sranger Condiion for follow from he Sranger Condiion for 1 and 2. Maing Condiion. Le z be a zone in which form par of he ne of pider and ; ha i, z τ(,, ). Then z form par of he ne of and in a lea one of he diagram 1 and 2. Therefore he Maing Condiion for follow from he Maing Condiion for 1 and 2. Shading Condiion. Le z be a haded zone of. Then z i haded in a lea one of he diagram 1 and 2. Suppoe he pider ha a foo in z in he combined diagram ; ha i, z η(, ). Then in a lea one of he diagram 1 and 2, ha a foo in z and z i haded; ha i, z η(, i ) {z : z = Z ( i )} for i = 1 or 2. Hence if x Ψ(z) hen x = ψ() for ome S( i ) for i = 1 or 2, by he Shading Condiion for i. Since S() = S( 1 ) S( 2 ), he Shading Condiion for follow from he Shading Condiion for 1 and 2. Hence he rule of combining diagram i valid Soundne We wrie o denoe ha he diagram can be obained from he diagram by applying a finie equence of ranformaion. Similarly, we wrie { 1, 2,, n } if can be obained from he e of diagram { 1, 2,, n } by applying a finie equence of ranformaion, including he rule for (pairwie) combinaion of diagram. The emanic of a e of diagram i he conjuncion of he emanic of he individual diagram; he boundary recangle of all diagram are inerpreed a he ame e and conour wih he ame label in differen individual diagram are inerpreed a he ame e. The following oundne heorem for he pider diagram yem follow by inducion from he validiy of each of he ranformaion rule and he rule for combining diagram, eablihed in he previou ecion. Theorem 4.1 (Soundne Theorem) If { 1, 2,, n } hen { 1, 2,, n }. 30

31 The yem of pider diagram inroduced here i no complee, a he following example how. Example 4.1. In figure 4.4, diagram can be inferred from diagram,. However, when removing pider from, rule 2 would require a rand beween pider r and in he reuling diagram, a weaker reul. The rule of inference do no allow o be obained ynacically from. r r ' Figure Relaionhip o oher yem. In order o obain a yem ha i complee, we would require a rule for combining diagram in which no emanic informaion i lo. In hi ecion we decribe one poible oluion. We could give figure 4.5 a he combined diagram for he diagram 1 and 2 in example 3.10 (ee figure 3.15). C C Figure 4.5 Figure 4.5 i a compound (or dijuncive) pider diagram. Eiher he lef-hand componen hold or he righ-hand componen hold (ee [15] where compound Venn-Peirce diagram are ued). I i no poible for he elemen correponding o and boh o belong o C. In, eiher can be in C and no, a in he lef-hand 31

32 componen, or vice vera, a in he righ-hand componen. ll he emanic informaion of 1 and 2 i capured in dijuncive diagram. The emanic of a compound diagram i he dijuncion of he emanic of i componen uniary diagram; he boundary recangle of he componen uniary diagram are inerpreed a he ame e. Conour wih he ame label in differen componen uniary diagram of a compound diagram are inerpreed a he ame e. Thu he compound diagram in figure 4.5 aer ha: ( C ( C ) ( C) ) ( C ( C) ). We have no conidered in deail hi yem of pider diagram wih dijuncion. Inead, we have explored a variey of relaed yem wih lighly differen (ynax and) emanic. In [7, 8, 9] we conider diagram wih exienial pider which have fee denoed by mall dic (raher han quare) and which repreen he exience of an elemen in he correponding e. Tie play no par in hee yem ince x, y x X y X x = y i logically equivalen o x x X. Figure 4.6 i an example of uch a diagram; i aer ha: (( x, y x C y C) ( C ( ) = ) ) ( x, y x y ( C) ). C 1 2 C Figure 4.6 The diagram in figure 4.6 i an example of a yem of diagram, called imple pider diagram, conidered in [7]. The yem i baed on Venn (raher han Euler) diagram, include compound (dijuncive) diagram, doe no allow pider fee o ouch haded region and doe no conain rand. The ranformaion rule given in hi paper are adaped and exended in [7] and we include a definiion of combining diagram ha doe no loe emanic informaion. Thi yem i boh ound and complee. The baic raegy o prove compleene (i.e., if { 1, 2,, n } hen { 1, 2,, n } 32

33 ), i firly combine he individual diagram in { 1, 2,, n } ino a ingle diagram ; hen expand boh and ino compound diagram in a way imilar o dijuncive normal form in ymbolic logic; i hen follow ha for each uniary componen in he expanded form of here exi a componen in he expanded form of ha follow logically from i; finally from he diagrammaic condiion ha mu hold beween hee wo componen, i follow ha one can be ranformed ino he oher ynacically. Thi proof raegy exend, wih ome modificaion, o oher pider and conrain diagram yem. In [8] we increae he expreivene of he yem by allowing pider fee o ouch haded region (a we do for he yem conidered in hi paper). The ynacic elemen of he yem are furher exended in [9] where we allow Euler diagram and we include rand and Schrödinger pider. The exended yem in [9] alo allow he negaion of a diagram o be repreened in a raighforward way. oh hee exended yem are hown o be ound and complee [8, 9]. 5. Concluion In hi paper, we have given he main ynax and emanic of pider diagram. We have given inference rule, a rule governing he equivalence of he Venn and Euler form of pider diagram and a rule for combining pider diagram. Thee rule have been hown o be ound; bu, in ome cae, he rule do no give a rong an inference a poible and o he yem i no complee. There are relaed yem of pider diagram ha have exienial pider and include compound diagram. Thee yem are known o be boh ound and complee. The general aim of hi work i o provide he neceary mahemaical underpinning for he developmen of ofware ool o aid reaoning wih diagram. In paricular, we aim o prove imilar reul for conrain diagram and o develop he ool ha will enable hem o become par of he ofware developmen andard. 33

34 cknowledgemen We graefully acknowledge Richard oworh, Paul Courney, Richard Michell, an Simpon and Kee van eemper for commen on earlier verion of hi paper. uhor Howe and Ken acknowledge uppor of he K EPSRC gran number GR/M Reference 1. G. llwein & J. arwie (edior) (1996) Logical Reaoning wih iagram. Oxford niveriy Pre, New York. 2. L. Euler (1772) Lere a ne Princee d llemagne. Vol. 2, Sur diver ubjec de phyique e de philoophie, Leer No (Reprin of 1795 ediion: (1997) Thoemme Pre, riol) 3. Y Gil, J Howe & S Ken (1999) Formalizing Spider iagram. In: Proceeding of 1999 IEEE Sympoium on Viual Language (VL99), IEEE Compuer Sociey Pre, Lo lamio, pp Gil, Y., Howe, J., Ken, S. (1999) Conrain iagram: a ep beyond ML. In: Proceeding of Technology of Objec-Oriened Language and Syem (TOOLS 30), IEEE Compuer Sociey Pre, Lo lamio, pp Glagow, J, Narayanan, N, Chandraekaran, (edior) (1995) iagrammaic Reaoning, I Pre, Menlo Park Ca. 6. Hammer, E.M. (1995) Logic and Viual Informaion. CSLI Publicaion, Sanford. 7. J. Howe, F. Molina, & J. Taylor (2000) Sound and Complee iagrammaic Reaoning Syem. In: Proceeding of 3rd ISTE Inernaional Conference on rificial Inelligence and Sof Compuing (SC 2000), (o appear). 8. J. Howe, F. Molina, & J. Taylor (2000) S2: Sound and Complee iagrammaic Reaoning Syem. In: Proceeding of 2000 IEEE Sympoium on Viual Language (VL2000), IEEE Compuer Sociey Pre, Lo lamio, (o appear). 9. J. Howe, F. Molina, & J. Taylor (2000) On he Compleene and Expreivene of Spider iagram Syem. In: Proceeding of iagram 2000, lecure Noe in rificial Inelligence, Springer, (o appear). 34

35 10. S. Ken (1997) Conrain iagram: Viualiing Invarian in Objec Oriened Model. In: Proceeding of 1997 CM SIGPLN Conference on Objec-Oriened Programming Syem, Language and pplicaion (OOPSL 97), CM Pre, pp R. Lull (1517) r Magma. Lyon. 12. T. More (1959) On he conrucion of Venn diagram. Journal of Symbolic Logic, 24, C. Peirce (1933) Colleced Paper. Vol. 4. Ed. C Harhorne & P Wei, Harvard niveriy Pre, Cambridge Ma. 14. J. Rumbaugh, I. Jacobon & G. ooch (1999) nified Modeling Language Reference Manual. ddion-weley, Reading Ma. 15. S-J. Shin (1994) The Logical Sau of iagram. Cambridge niveriy Pre, Cambridge. 16. J. Venn (1880) On he iagrammaic and Mechanical Repreenaion of Propoiion and Reaoning. Phil. Mag.(5) 9, Warmer, J. and Kleppe,. (1998) The Objec Conrain Language: Precie Modeling wih ML. ddion-weley, Reading Ma. 35