Bounded Rationality for Data Reasoning based on Formal Concept Analysis

Bounded Ra ionali y o Da a Reasoning based on Fo mal Concep Analysis Gonzalo A. A anda-Co al Depa men o In o ma ion Technology Uni e sidad de Huel a Palos de La F on e a, Spain Email:[email p o ec ed] Joaqu´ ın Bo ego-D´ ıaz and Juan Gal´ an-P´ aez Depa men o Compu e Science and A i icial In elligence Uni e sidad de Se illa Se illa, Spain Email:jbo e[email p o ec ed], [email p o ec ed] Abs ac —Fo mal Concep Analysis (FCA) is a heo y whose goal is o disco e and ex ac Knowledge om quali a i e da a. I also p o ides ools o sound easoning (implica ion basis and associa ion ules). The aim o his pape is o apply FCA o a new model o bounded a ionali y based on he implica ional easoning o e con ex ual knowledge bases which a e ob ained om con ex ual selec ions. A con ex ual selec ion is a selec ion o e en s and a ibu es abou hem which induces pa ial con ex s om a global o mal con ex . In o de o a oid inconsis encies, associa ion ules a e selec ed as easoning engine. The model is applied o o ecas spo esul s. Keywo ds-Fo mal Concep Analysis, Bounded Ra ionali y, Con idence Reasoning I. INTRODUCTION Bounded Ra ionali y (BR) is in ima ely ela ed wi h he human capaci y o making in e ences unde limi ed ime and Knowledge [1]. F om he iewpoin o A i icial In- elligence (AI), BR comp ises easoning echniques ha acili a e, o example, con ex and empo al easoning. Psy- chological esea ch on speci ic heu is ics in human in e ence p ocessing e eals a complex amewo k whe e adi ional app oaches o classical logic is no sound o explaining he success o se e al o hem, as o example Recogni ion Heu is ic (RH) [2]. A numbe o expe imen s show ha cogni i e mechanisms capable o success ul pe o mance in he eal wo ld do no need o sa is y he classical no ms o a ional in e ence (c . [3]; see also [4]). In ac , an in iguing ques ion om ecological a ionali y analysis is: How could mo e knowledge be no be e —o wo se— han signi ican ly less knowledge? [2]. One o he key ea u es in BR is ha in e ence p ocess is concen a ed on a limi ed se o expe iences in which objec s, p ope ies and ac ions a e selec ed. In his pape we aim o model his ea u e wi h Fo mal Concep Analysis. Fo mal Concep Analysis (FCA) [5] is a ma hema ical heo y o da a analysis, using o mal con ex s and concep la ices as key ools. Domains can be o mally modelled acco ding o he ex en and he in en o each o mal concep . In FCA, he basic da a s uc u e is a o mal con ex (wi h a quali a i e na u e) which ep esen s a se o objec s and hei p ope ies. I is use ul bo h o de ec and o desc ibe egula i ies and he ela ionship s uc u es among concep s. I also p o ides a sound o malism o easoning wi h such s uc u es, mainly implica ion basis and associa ion ules. Roughly speaking, o mal con ex s ep esen weak s uc- u es easily buil om expe ience ha allow he ex ac ion o knowledge om hem. Despi e i s simple da a s uc u e, o - mal con ex s a e use ul s uc u es o knowledge ex ac ion (c . [5]) and easoning. Mo eo e , in BR i is well known ha in se e al cases simple s a is ical o ecas ing ules, which a e usually simpli ica ions o models, ha e been shown o make be e p edic ions han mo e complex ules, especially when he u u e alues o a c i e ion a e highly unce ain [6]. The hesis o he pape is ha associa ion ules associa ed o o mal con ex s can be an in e es ing sou ce o BR. The aim is o p esen a logical model o BR based on easoning on subcon ex s o a p ede e mined (global) con ex which plays he ole o global memo y/quali a i e da ase . The model is based on he exis ence o some selec i e p ocesses (named con ex ual selec ion he e) which induce speci ic con ex s, and implica ional basis a e ex ac ed om hem (namely S em Basis [7] and associa ion ules). The easoning wi h hese Knowledge Bases (KB) (called con- ex ual KB) is he model easoning p oposed in he pape . Logical combina ion o con ex ual KBs in o de o a oid inconsis encies wi h backg ound Knowledge can be made. The model has been used in [8] o desc ibe a con idence- based (and con ex ual) easoning sys em o o ecas ing spo s be ing. In his pape we analyse he soundness o he o mal model as one o bounded a ionali y, p esen ing he heo e ical amewo k. The s uc u e o he pape is as ollows. The nex sec ion e iews he main elemen s o FCA and i s logical ea u es. In sec ion 3 he ole o o mal con ex s as basic b icks o a model o bounded a ionali y is p esen ed. Sec ion 4 e iews an expe imen by using he model o o ecas ing in spo s be ing. Sec ion 5 is de o ed o desc ibe u u e wo k. II. BACKGROUND: FORMAL CONCEPT ANALYSIS Acco ding o R. Wille, FCA [5] ma hema izes he philo- sophical unde s anding o a concep as a uni o hough s composed o wo pa s: he ex en and he in en . The ex en co e s all objec s belonging o his concep , while he in en comp ises o all common a ibu es alid o all he Figu e 1. pa ial con ex om obse a ion and S em Basis objec s unde conside a ion. I also allows he compu a ion o concep hie a chies om da a ables. In his sec ion, we succinc ly p esen basic FCA elemen s (see [5] o de ails). A o mal con ex M= (O, A, I)consis s o wo se s, O(objec s) and A(a ibu es) and a ela ion I⊆O×A. Fini e con ex s can be ep esen ed by a 1-0- able (iden i ying Iwi h a Boolean unc ion on O×A). See Fig. 1 o an example o o mal con ex abou li e beings. The FCA main goal is he compu a ion o he concep la ice om he con ex . Fo X⊆Oand Y⊆Awe de ine X0:= {a∈A|oIa o all o∈X} Y0:= {o∈O|oIa o all a∈Y} A ( o mal) concep is a pai (X, Y )such ha X0=Yand Y0=X. In his pape i wo ks wi h logical ela ions on a ibu es which a e alid in he con ex and he s anda d implica ional logic in FCA (see, e.g., [5]), called implica ions be ween a ibu es. De ini ion. 1 An implica ion be ween a ibu es is a pai o se s o a ibu es, w i en as Y1→Y2, An implica ion is ue wi h espec o a o mal con ex M= (O, A, I)acco ding o he ollowing de ini ion. A subse T⊆A espec s Y1→Y2i Y16⊆ To Y2⊆T. I says ha Y1→Y2holds in M(M|=Y1→Y2) i o all o∈O, he se {o}0 espec s Y1→Y2. In ha case, i is said ha Y1→Y2is an implica ion o M. De ini ion. 2 Le Lbe a se o implica ions and Lbe an implica ion o M. 1) L ollows om L(L |=L) i each subse o A espec ing Lalso espec s L. 2) Lis comple e i e e y implica ion o he con ex ollows om L. 3) Lis non- edundan i o each L∈ L,L {L} 6|=L. 4) I Lis a (implica ion) basis o Mis comple e and non- edundan . A well-known me hod o compu ing speci ic implica- ional basis, called S em Basis (SB), exis s [7]. I is imple- men ed in o Conexp (h p://sou ce o ge.ne /p ojec s/conexp/) so wa e. A SB o li e beings’ o mal con ex is p o ided in Fig. 1. I is impo an o ema k ha SB is only an example o a basis o a o mal con ex . In his pape any speci ic p ope y o he SB can be used, so i can be eplaced by any implica ion basis. I is possible o ex end |=in ela ion o any p oposi ional o mula wi h p oposi ional a iables in A, by conside ing each objec o∈Mas a alua ion oon Ade ining o(A)=1⇐⇒ (o, A)∈I So M|=Fi and only i o|=F o any o∈O. By de ining `Aas he p oo ela ion induced by A m- s ong ules [9], R1 : X→XR2 : X→Y X∪Z→YR3 : X→Y, Y ∪Z→W X∪Z→W i holds ha he implica ional bases a e `A-comple e (a s aigh o wa d consequence o A ms ong’s esul [9]): Theo em 3 Le Lbe a basis o a o mal con ex M, and Lan implica ion. Then M|=Li and only i L `AL. In o de o wo k wi h o mal con ex s, s em basis and associa ion ules, he Conexp has been selec ed. I is used as a lib a y o build he module which p o ides he implica ions (and associa ion ules) o he easoning module o ou sys em. The easoning module is a p oduc ion sys em based on wha was designed o [10]. Ini ially i wo ks wi h S em Basis and en ailmen is based on he ollowing esul : Theo em 4 Le Lbe a basis o he con ex Mand {A1, . . . , An} ∪ Y⊆A. The ollowing condi ions a e equi alen : 1) S ∪ {A1,...An} `pY(`pis he en ailmen wi h he p oduc ion sys em). 2) S`AA1,...An→Y 3) M|={A1,...An} → Y. A. Associa ion ules o a a o mal con ex We can conside a S em Basis as an adequa e knowledge base o a p oduc ion sys em in o de o eason. Howe e , S em Basis is designed o en ailing ue implica ions only, wi hou any excep ions in he objec se no implica ions wi h a low numbe o coun e examples in he con ex . Ano he mo e impo an ques ion a ises when i wo ks on p edic ions. In his case we a e in e es ed in ob aining me hods o selec ing a esul among all ob ained esul s (e en i hey a e mu ually incohe en ). Theo em 4 does no p o ide such a me hod. The e o e, i is be e o conside as- socia ion ules (wi h con idence) ins ead o ue implica ions. Mo eo e , he ini ial p oduc ion sys em mus be e ised o wo king wi h con idence. In es iga ions on sound logical easoning me hods wi h associa ion ules is a ela i ely ecen esea ch line wi h p omising applica ions [11]. In FCA, associa ion ules a e implica ions among se s o a ibu es. Con idence and sup- po a e de ined as usual. Recall ha he suppo o X, Figu e 2. Model o easoning based on `∃ supp(X)o a se o a ibu es X is de ined as he p opo ion o objec s which sa is y e e y a ibu e o X, and he con idence o a associa ion ule is con (X→Y) = supp(X∪Y)/supp(X). Con idence can be in e p e ed as an es ima e o he p obabili y P(Y|X), he p obabili y o an objec sa is ying e e y a ibu e o Yunde he condi ion ha i also sa is ies e e y one o X. Conexp so wa e p o ides associa ion ules (and hei con idence) o o mal con ex s. III. FORMAL CONTEXTS AS KNOWLEDGE STRUCTURES Global memo y is composed o e en s (objec s) which ha e a numbe o p ope ies (a ibu es). They cons i u e a global o mal con ex M= (O,A,I)(which we call mons e con ex ollowing he adi ion in Model Theo y) om which subcon ex s a e ex ac ed. Once he speci ic subcon ex is conside ed, i is also possible o conside backg ound knowl- edge ∆which would be combined wi h he KB ex ac ed om o mal con ex (S em basis o associa ion ules). De ini ion. 5 Le Mbe a mons e con ex , and le O⊆O. 1) A con ex on Ois a con ex M= (O1, A, I)whe e O⊆O1⊆O,A⊆Aand I⊆I. 2) Acon ex ual selec ion on Oand Mis a map s:O→ P(O1)× P(A). 3) Acon ex ual KB o an objec o∈Ow. . . a selec ion swi h con idence γis a subse o associa ion ules wi h con idence g ea e o equal o γo he o mal con ex associa ed o s(o) = (s1(o), s2(o)), ha is, o he con ex M(s(o)) := (s1(o), s2(o), Is1(o)×s2(o))(no e ha when con i- dence is 1 he con ex ual KB is a implica ional basis). The easoning model on Mis a gumen a i e, whe e he a gumen is based on KBs ex ac ed om subcon ex s. Anagously o [12], he exis en ial a gumen s a e conside ed, bu eplacing he consis en se by subcon ex : De ini ion. 6 Le Lbe an implica ion and ∆a backg ound knowledge. I is said ha Lis a possible consequence o M unde he backg ound knowledge ∆,M|=∆ ∃L, i he e exis s Ma nonemp y subcon ex o Msuch ha M|= ∆ ∪ {L}. No e ha by heo em 4, when ∆is a se o implica ions, i holds ha |=∃is equi alen o `∃which is de ined by: M`∃Li he e exis s M|= ∆ a subcon ex o Msuch ha S`pL(whe e Sis a SB o M). In he example desc ibed in Sec . 4, he easoning model is based on `pon con ex ual KBs o an objec o∈Ow. . . a selec ion gi en by an expe . To compu e all consequences by `∆ ∃implies o conside he en i e model. Howe e we only need consequences en ailed by a submodel. See sec ion IV bellow. Gi en Mi= (Oi, Ai, Ii), i = 1,2 wo subcon ex s o M he in e sec ion o M1and M2,M1∩M2is (O1∩O2, A1∪A2, I1∩((O1∩O2)×A1)∪I2∩((O1∩O2)×A2)) In o de o s udy `∃unde backg ound knowledge, i is necessa y o s udy he ela ionship among a gumen s based on dis inc con ex s. Two compa ibili y no ions can be used. De ini ion. 7 Le Mi= (Oi, Ai, Ii), i = 1,2be wo subcon ex s o M, and le ∆be a backg ound p oposi ional knowledge on he language o A1∩A2. •I is said ha M1and M2a e compa ible w. . ∆i he e exis s a supe con ex Mo M1and M2such ha M|= ∆. •I is said ha M1and M2a e downwa d compa ible w. . ∆i M1∩M2|= ∆. Compa ible con ex s a e also downwa d compa ible. The e o e, i can join ly ex end downwa d compa ible con- ex s bu hey can no be es ic ed wi h logical eliabili y. Thus, o mal con ex s can be ex ended in o de o e ine esul s. I is also possible o wo k wi h any con ex whose objec s sa is y backg ound knowledge ∆ o ob ain `∃con- sequences. P oposi ion. 8 I wo con ex s a e compa ible hen hey a e downwa d compa ible P oo : Suppose ha M1and M2a e compa ible. Le M be he supe con ex o M1and M2. By conside ing each objec o∈Mas a alua ion oon Ade ined by o(A)=1⇐⇒ (o, A)∈I he objec s in M1∩M2a e models o ∆. Thus M1∩M2|= ∆ The ecip ocal is no ue: Conside he con ex M= (O, A, I)wi h O={o1, o2, o3}and A=a1, a2, a3and le I={(o1, a1),(o1, a3),(o2, a2),(o3, a1),(o3, a3)}. Le M1 be he subcon ex wi h O1={o1, o2}and A1={a1, a2} and le M2be he subcon ex wi h O2={o2, o3}and A2=a2, a3. The in e sec ion M3=M1∩M2has O3={o2}and A3=a1, a2, a3and I3={(o2, a2)}. Since M3|=a2, we ha e ha M1and M2a e downwa d compa ible w. . . {a2}(seen as a p oposi ional o mula), ye he e is no supe con ex o M1and M2sa is ying a2. The in e ence p ocess o `∆ ∃has ee s eps (Fig. 2): 1) A ques ion on whe he a new e en (objec ) has a p ope y (a ibu e) is aised. On he new objec some p ope ies a e known (a ibu e alues) {A1,...An}. Figu e 3. Con ex based easoning sys em 2) A con ex ual selec ion ou pu s a subcon ex o M. A con ex ual KB L( o some con idence h eshold) is compu ed o he subcon ex . Selec ion made by a use is composed by a small se o a ibu es. 3) The p oduc ion sys em is execu ed on L ∪ {A1,...An}. The esul s ob aineda e he a ibu es in e ed abou he e en . I has ha , i A0is in e ed by he p oduc ion sys em, hen M`∆ ∃{A1,...An}→{A0} No e ha i does no compu e all implica ions, only hose which a e en ailed om he a ibu es selec ed by he use . A. Incompa ible a ibu es Implica ion logics do no su e om inconsis ency issues. Howe e , in FCA i can be usual o conside incompa ible a ibu es. A pai A1, A2o incompa ible a ibu es e i ies ha M|=¬(A1∧A2). I such a o mula is included in backg ound knowledge ∆i is possible o deal wi h incon- sis ency issues, because `∃is an a gumen a i e en ailmen which wo ks on subcon ex s (see de . o `∃in [12]). Two op ions ha e been conside ed o sol ing his p ob- lem by FCA. The i s one is based on a oiding inconsis en- cies using conse a i e e ac ion [13]. The aim is o emo e one a ibu e om incompa ible pai s in con ex ual KB, nex o p esen ela ed a ibu e p oo . We ha e selec ed a second op ion: o use associa ion ules and do no conside any backg ound knowledge. In his way, in e ed a ibu e wi h maximum con idence is selec ed [8]. IV. EXAMPLE: DATA ON SOCCER LEAGUE MATCHES We ha e applied he model o socce be ing. The p ojec s a s wi h he hypo hesis ha pas da a hides ends o socce eams ha expe s use o o ecas ing. The mons e model is composed o da a (pas and cu en ) on ma ches ( he objec s, wi h empo al s amp). The expe imen ca ies ou se en s ages (see Fig. 3): 1) Selec ion o he se o ele an a ibu es o conside . I is made by he use . 2) Da a ex ac ion. Wi h his da a he sys em is capable o building any subcon ex needed. In he da a ime s amps a e impo an , because a numbe o a ibu es deal wi h pas ma ches. Da a has been ex ac ed om he RSSSF A chi e (h p://www. sss .com) om he pas en yea s. Objec s a e ma ches (wi h empo al s amp), and a ibu es a e compu ed o each objec . The ele an p ope ies (a ibu es) ha expe s se- lec ed was 17, se e al o hem a e pa ame ized ( o example, anking di e ence abo e a h eshold). An a ibu e wi h h eshold can p oduce a la ge numbe o bina y a ibu es by changing he h eshold. Thus he explici compu a ion o Mis no easible. I has h ee dis inguished a ibu es, co esponding o T eam1wins (1), Team2wins (2) and d aws (X). 3) Selec ion o a ( u u e) ma ch. 4) Con ex ual selec ion, based on he selec ion o h esh- olds o a ibu es. 5) Compu ing o a ibu e alues o he ma ch om da a ( o build he subcon ex ), excep ob iously he alue o dis inguished a ibu es. 6) Execu ion o he Sys em (associa ion ules as KB and a ibu es o he objec as ac s). Se e al modes o con idence compu ing, based on unce ain easoning echniques by Expe Sys ems a e conside ed [8]. 7) Resul : a iple <(1, c1),(X, cx),(2, c2)>o pai s (a ibu e, con idence), o he selec ed ma ch. V. EXPERIMENTS Two expe imen s we e launched o Spanish socce league, on 2009-10 and 2010-11 seasons. A ibu es we e selec ed acco ding o au ho s’ knowledge abou Spanish socce league (which a e no expe s). F om his con ex ual selec ion, `∃was compu ed o all ma ches and weeks. 2009-10 season: Expe imen s wi h he sys em show o e- cas s o abou 58.16% by a con ex ual selec ion based on he p e ious 38 ma ches. Such a pe cen age o hi s o a quali a i e easoning sys em may be conside ed as an accep able esul compa able wi h expec able esul s o expe s [14]. Expe imen s also shows an inc ease in he numbe o hi s by abou 7% in he second hal o he season. The eason is ha da a om he i s hal p o ides mo e ecen in o ma ion on eams and pas ma ches. 2010-11 season: A way o e alua e how good is his o ecas ing sis em is compa ing numbe o successes in ou pool wi h he mos popula be ing selec ions. This popula selec ions a e collec ed om he mos o ed esul s o each ma ch, published a s a e agency web ha con ols socce pools. In Fig. 4 bo h esul s a e compa ed. Ou hi s a e in blue and popula ones in g een and las se en een weeks om 2010-11 season a e ep esen ed. No e ha Spanish socce pools a e o e 15 ma ches. VI. CONCLUDING REMARKS AND FUTURE WORK The model p esen ed is conce ned wi h associa ion ule easoning and i does no use -in i s cu en o m- mo e sophis ica ed p obabili y ools (se e.g. [15]). As is s a ed in [16], he heo y o p obabilis ic men al models assumes Figu e 4. Co ec p edic ions on he las 17 weeks o he season 2010-11 compa ed wi h popula he mos popula be s ha in e ences abou unknown s a es o he wo ld a e based on p obabili y cues [17]. In some sense, associa ion ules’s con idence plays he ole o p obabili y cues in he model. The ela ionship o ou p oposal wi h RH [2] ( oughly speaking, i one o he possibili ies is ecognized and he o he is no , hen in e ha he ecognized objec has he highe alue wi h espec o he c i e ion) is no clea . We may asse ha ou model ecognises ends in con ex s. T ends ( ep esen ed as associa ion ules o implica ion basis) can be conside ed as a kind o ecognizing me hod, hough. I is wo h no ing ha i only uses `∃because he aim is o simula e bounded easoning. O he en ailmen ela ionships om a gumen a i e amewo k, as o example `∀, ha e no been conside ed in his pape , because i equi es o an exhaus i e explo a ion o M. Pa o ou ongoing wo k includes wo esea ch lines. The i s one is he analysis o conse a i e e ac ion me hod o wo king wi h incompa ible a ibu es. The second one is o simula e he a ibu e lea ning p ocess in ou model, applying non mono one easoning echniques. ACKNOWLEDGMENTS Suppo ed by TIN2009-09492 p ojec o Spanish Minis y o Science and Inno a ion, and Excellence p ojec TIC-6064 o Jun a de Andaluc´ ıa co inanced wi h FEDER ounds. REFERENCES [1] H. A. Simon, Models o bounded a ionali y. MIT P ess, 1982. [2] D. G. Golds ein and G. Gige enze , “Models o ecological a ionali y: he ecogni ion heu is ic.” Psychological Re iew, ol. 109, no. 1, pp. 75–90, 2002. [3] D. Kahneman, P. Slo ic, and A. T e sky, (eds.) Judgmen unde unce ain y: Heu is ics and biases. Camb idge Uni- e si y P ess, 1982. [4] T. Pachu and G. Biele, “Fo ecas ing om igno ance: he use and use ulness o ecogni ion in lay p edic ions o spo s e en s.” Ac a Psychologica, ol. 125, no. 1, pp. 99–116, 2007. [5] B. Gan e and R. Wille, Fo mal Concep Analysis: Ma he- ma ical Founda ions. Sp inge , 1999. [6] D. G. Golds ein and G. Gige enze , “Fas and ugal o ecas - ing,” In e na ional Jou nal o Fo ecas ing, ol. 25, no. 4, pp. 760–772, 2009. [7] J.-L. Guigues and V. Duquenne, “Familles minimales d’implica ions in o ma i es esul an d’un ableau de donnees binai es,” Ma hema iques e Sciences Humaines, ol. 95, pp. 5–18, 1986. [8] G. A. A anda-Co al, J. Bo ego-D´ ıaz, and J. Gal´ an-P´ aez, “Con idence-based easoning wi h local empo al o mal con- ex s,” in IWANN (2), 2011, pp. 461–468. [9] W. W. A ms ong, “Dependency s uc u es o da a base ela- ionships.” in IFIP Cong ess, 1974, pp. 580–583. [10] G. A. A anda-Co al and J. Bo ego-D´ ıaz, “Reconciling knowledge in social agging web se ices,” in HAIS (2), 2010, pp. 383–390. [11] J. L. Balc´ aza , “Redundancy, deduc ion schemes, and minimum-size bases o associa ion ules,” Logical Me hods in Compu e Science, ol. 6, no. 2, 2010. [12] A. Hun e , “Reasoning wi h inconsis ency in s uc u ed ex ,” Knowledge Enginee ing Re iew, ol. 15, pp. 200–0, 1999. [13] G. A. A anda-Co al, J. Bo ego-D´ ıaz, and M. M. Fe n´ andez- Leb ´ on, “Conse a i e e ac ions o p oposi ional logic he- o ies by means o boolean de i a i es: Theo e ical ounda- ions,” in Calculemus/MKM, 2009, pp. 45–58. [14] P. Ande sson, M. Ekman, and J. Edman, “Fo ecas ing he as and ugal way: A s udy o pe o mance and in o ma ion- p ocessing s a egies o expe s and non-expe s when p e- dic ing he wo ld cup 2002 in socce ,” S ockholm School o Economics, Tech. Rep. 2003:9, May 2003. [15] B. Min, J. Kim, C. Choe, H. Eom, and R. Bobmckay, “A compound amewo k o spo s esul s p edic ion: A oo ball case s udy,” Knowledge-Based Sys ems, ol. 21, no. 7, pp. 551–562, 2008. [16] G. Gige enze and D. G. Golds ein, “Reasoning he as and ugal way: models o bounded a ionali y.” Psychological Re iew, ol. 103, no. 4, pp. 650–669, 1996. [17] E. B unswik, “Rep esen a i e design and p obabilis ic heo y in a unc ional psychology.” Psychological Re iew, ol. 62, no. 3, pp. 193–217, 1955.