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.
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