Completing causal networks by meta-level abduction

37 Pages·2017·0.49 MB·English

Completing causal networks by meta-level abduction
Katsumi Inoue, Andrei Doncescu, Hidetomo Nabeshima
To cite this version:
Katsumi Inoue, Andrei Doncescu, Hidetomo Nabeshima. Completing causal networks by meta-level
abduction. Machine Learning, 2013, 91 (2), pp.239-277. ￿hal-00999312￿
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MachineLearningmanuscriptNo.
(willbeinsertedbytheeditor)
Completing causal networks by meta-level abduction
KatsumiInoue · AndreiDoncescu · Hidetomo
Nabeshima
Received:date/Accepted:date
Abstract Meta-levelabductionisamethodtoabducemissingrulesinexplainingobserva-
tions.Byrepresentingrulestructuresofaprobleminaformofcausalnetworks,meta-level
abductioninfersmissinglinksandunknownnodesfromincompletenetworkstocomplete
pathsforobservations.Weexamineapplicabilityofmeta-levelabductiononnetworkscon-
taining both positive and negative causal effects. Such networks appear in many domains
includingbiology,inwhichinhibitoryeffectsareimportantinseveralbiologicalpathways.
Reasoninginnetworkswithinhibitioninvolvesnonmonotonicinference,whichcanbere-
alizedbymakingdefaultassumptionsinabduction.Weshowthatmeta-levelabductioncan
consistentlyproducebothpositiveandnegativecausalrelationsaswellasinventednodes.
Case studies of meta-level abduction are presented in p53 signaling networks, in which
causalrelationsareabducedtosuppressatumorwithanewproteinandtostopDNAsynthe-
siswhendamagehasoccurred.Effectsofourmethodarealsoanalyzedthroughexperiments
ofcompletingnetworksrandomlygeneratedwithbothpositiveandnegativelinks.
Keywords abduction·theorycompletion·meta-reasoning·causalnetwork·inhibition·
predicateinvention·defaultreasoning·InductiveLogicProgramming·SystemsBiology
This research was supported in part by the 2008-2011 JSPS Grant-in-Aid for Scientific Research (A)
No.20240016.
K.Inoue
NationalInstituteofInformatics
2-1-2Hitotsubashi,Chiyoda-ku,Tokyo 101-8430, Japan
E-mail:[e-mail]
A.Doncescu
LAAS-CNRSUPR8001
AvenueduColonelRoche, 31007Toulouse, France
E-mail:[e-mail]
H.Nabeshima
DivisionofMedicineandEngineeringScience, UniversityofYamanashi
4-3-11Takeda,Kofu,Yamanashi 400-8511, Japan
E-mail:[e-mail]
2
1 Introduction
Abductionandinductionarebothampliativereasoning,andplayessentialrolesinknowl-
edge discovery and development in science and technology. Integration of abduction and
inductionhasbeendiscussedinsuchdiverseaspectsasimplementationofinductivelogic
programming(ILP)systemsusingabductivemethods[67,14,52,54,27,3]and“closingthe
loop” methodologies in scientific discovery [10,28,55]. The use of prior or background
knowledge in scientific applications has directed our attention to theory completion [40]
rather than classical learning tasks such as concept learning and classification. There, ab-
duction is mainly used to complete proofs of observations from incomplete background
knowledge,whileinductionreferstogeneralizationofabducedcases.
Inscientificdomains,backgroundknowledgeisoftenstructuredinaformofnetworks.
Inbiology,asequenceofsignalingsorbiochemicalreactionsconstitutesanetworkcalleda
pathway,whichspecifiesamechanismtoexplainhowgenesorcellscarryouttheirfunc-
tions.However,informationofbiologicalnetworksinpublic-domaindatabasesisgenerally
incompleteinthatsomedetailsofreactions,intermediarygenes/proteinsorkineticinforma-
tionareeitheromittedorundiscovered.Todealwithincompletenessofpathways,weneed
topredictthestatusofrelationswhichisconsistentwiththestatusofnodes[63,68,51,21,
55],orinsertmissingarcsbetweennodestoexplainobservations[69,28,65,1].Thesegoals
are characterized by abduction as theory completion, in which status of nodes or missing
arcsareaddedtoaccountforobservations.
If a network is represented in a logical theory, inference on the network can be real-
ized on a language in which each network element such as a node and a link is itself an
entity of the language. Such a theory (or program) refers a language of the network, and
is hence regarded as a meta-theory (or meta-program), while a theory (or program) rep-
resenting networks is referred to as an objective theory (or objective program). Then, to
performabductiononnetworks,weneedabductiononmeta-theories,whichisreferredto
asmeta-levelabduction.Notethatmeta-reasoninghasbeenintensivelyinvestigatedinlogic
programming,e.g.,[29,12,4],yetmaininferenceconsideredtherehasbeendeduction.
Meta-levelabductionwasintroducedin[18]asamethodtodiscoverunknownrelations
from incomplete networks. Given a network representing causal relations, called a causal
network,missinglinksandnodesareabducedinthenetworktoaccountforobservations.
Themainobjectivein[18]istoprovidealogicalfoundationandknowledgerepresentation
forabducingrules,whichisanimportantabductiveprobleminILP.Itisnotablethatother
abductive methods [49,3] need a predetermined set of candidate rules (called abducible
rules or abducibles) and then select consistent combinations of abducibles to form expla-
nations. In contrast, meta-level abduction does not need any such abducibles in advance.
Meta-level abduction is implemented in SOLAR [44,45], an automated deduction system
forconsequencefinding,usingafirst-orderrepresentationforalgebraicpropertiesofcausal-
ityandafull-clausalformofnetworkinformationandconstraints.Meta-levelabductionby
SOLARispowerfulenoughtoinfermissingrules,missingfacts,andunknowncausesin-
volvingpredicateinvention[41]intheformofexistentiallyquantifiedhypotheses.Notethat
predicateinventionhadbeenintensivelyinvestigatedintheinitialstageofILPresearchand
hasrecentlybeenrevisited[42],sinceitshouldplayanimportantroleindiscovery.
Meta-level abduction has been applied to discover physical skills in terms of hidden
rulestoexplaingivenempiricalrulesincelloplayingexamplesin[18],andathoroughex-
perimentalanalysiswithavarietyofprobleminstanceshasbeenpresentedin[45].However,
allthoseexamplesofmeta-levelabductionin[18,45]containonlyonekindofcausaleffects,
whicharepositive,anditwasleftopenhowtodealwithbothpositiveandnegativeeffects.
3
Then,weshallexamineapplicabilityofmeta-levelabductiontodealwithnetworksexpress-
ingbothpositiveandnegativecausaleffects.Suchnetworksareoftenusedinbiology,where
inhibitoryeffectsareessentialingeneregulatory,signalingandmetabolicnetworks.
Inthispaper,wepresentaxiomsformeta-levelabductiontoproducebothpositiveand
negative causal relations as well as newly invented nodes. We show two axiomatizations
forsuchmeta-levelabduction.Oneisasetofalternatingaxiomswhichdefinerelationsof
positiveandnegativecausaleffectsinadoubleinductivemanner.Thisaxiomsetreducesto
theaxiomsforordinarymeta-levelabductiondefinedin[18]whenthereisnonegativecausal
link. The other axiomatization is a variant of the alternating axioms, but prefers negative
links to positive ones if both are connected to the same node. In this case, reasoning in
causalnetworksbecomesnonmonotonic,andinvolvesdefaultassumptionsinabduction.
Then,applicationstop53signalnetworks[50,36]arepresentedascasestudiesofour
framework,inwhichmeta-levelabductionreproducestheoriesexplaininghowtumorsup-
pressorswork[65]andhowDNAsynthesisstops[62].Analysisofsuchabstractsignaling
networks, although simple, provides one of the most fundamental inference problems in
computer-aidedscientificresearchincludingSystemsBiology:Givenanincompletecausal
network, infer possible connections and functions of network entities to reach the target
entitiesfromthesources.Meta-levelabductioninthispaperiscrucialinthistaskforthefol-
lowingreasons.First,suggestionofpossibleadditionsinpriornetworksenablesscientiststo
conducthypothesis-drivenexperimentswiththosefocusedcases.Ifasuggestedhypothesis
isjustifiedthroughathroughsetofexperiments,thecorrespondingnewlinksand/ornodes
are considered to be discovered. In network completion, however, the larger the network
becomes,themoreabductiveinferencestepsarerequiredtogetahypothesisandthemore
candidatehypothesesareinferred.Then,itishardforhumanscientiststoconsiderallpossi-
bilitieswithoutlosinganyimportantones.Therefore,automationofhypothesisenumeration
isveryimportant.Second,abductioninsuchnetworkdomainsofteninvolvesagoalrather
thananobservation:Ahypothesisisinferredtoachievethegoalthathasnotbeenobserved
yet.Forexample,indrugdesignandpharmacologyaswellastherapeuticresearch,theef-
fectofintroductionofnewentitiesandlinkstoaknownnetworkisgoal-orientedandthe
samehypothesescannotbeappliedtoothergoalsingeneral.Thisfeatureofgoal-oriented
abductionalsoexistsincompletingcausalnetworksforimprovementofphysicaltechniques
inmusicalperformance[18],inwhichspecificskillsarerequiredforrequestedtasks.
Finally, scalability of meta-level abduction is analyzed through experiments of com-
pletingnetworksrandomlygeneratedwithbothpositiveandnegativelinks.Byvaryingthe
averagedegreeofnodestoedges,fromsparsetodensenetworksaregeneratedwithseveral
sizes.Wewillseethatitisnoteasytogeneratealargeanddensenetworkbykeepingthe
consistency, since there are more chances to become inconsistent in such networks. Then
thegrowthrateofhypothesesinthesizeofnetworksratherdecreasesindensenetworks.
Thisisanextendedversionofthepaper[17],andcontainsseveraltechnicaldetailssuch
asanabductiveprocedurebasedonconsequencefinding,theoreticalcorrectnessforthepro-
posedformalizationandtheirproofs,detailedanalysisofexperimentsonp53pathways,and
scalabilityissues.Therestofthispaperisorganizedasfollows.Section2offerstheessen-
tialofmeta-levelabductionanditsuseforruleabduction.Section3thenextendsmeta-level
abductiontoallowfortwotypesofcausaleffects,inwhichpositiveandnegativerulesare
called triggers and inhibitors, respectively, and investigate properties of two axiomatiza-
tions. Section 4 presents two case studies of meta-level abduction applied to completion
ofsub-networksinp53signalnetworks.Section5showsexperimentsoncausalnetworks,
andanalyzesscalabilityofourmethodincompletingnetworks.Section6discussesrelated
work,andSection7givesasummaryandfuturework.
4
2 Meta-levelAbduction
Thissectionrevisitstheframeworkformeta-levelabduction[18],andprovidesthecorrect-
nessofruleabduction.
2.1 CausalNetworks
Wesupposeabackgroundtheoryrepresentedinanetworkstructurecalledacausalgraph
oracausalnetwork.Acausalgraphisadirectedgraphrepresentingcausalrelations,which
consistsofasetofnodesandasetof(directed)arcs(orlinks).1Eachnodeinacausalgraph
representssomeevent,factorproposition.Adirectcausalrelationcorrespondstoadirected
arc,andacausalchainisrepresentedbythereachabilitybetweentwonodes.Theinterpre-
tationofa“cause”hereiskeptratherinformal,andjustrepresentstheconnectivity,which
mayrefertoamathematical,physical,chemical,conceptual,epidemiological,structural,or
statisticaldependency[48].Similarly,a“directcause”heresimplyrepresentstheadjacent
connectivity,whileitseffectisdirectonlyrelativetoacertainlevelofabstraction.
Wethenconsiderafirst-orderlanguagetoexpresscausalnetworks.Eachnodeisrep-
resentedasapropositionora(ground)atominthelanguage.Whenthereisadirectcausal
relation from a node s to a node g, we define that connected(g,s) is true as in (1). Note
thatconnected(g,s)onlyshowsthatsisoneofpossiblecausesofg,andthustheexistence
of another connected(g,t) (s (cid:2)= t) means that s and t are alternative causes for g. Here,
expressionofcausalrelationsisdoneatthemetalevelusingthemeta-predicateconnected,
whiletheobjectlevelreferstonodesinacausalnetwork.Anatomconnected(s,t)atthe
metalevelcorrespondstoarule(s←t)attheobjectlevel.Thefactthatadirectcausallink
cannotexistfromstogisrepresentedinan(integrity)constraintoftheform(2).
✛
g s connected(g,s) (1)
✛
g / s ¬connected(g,s) (2)
Adirectcausalrelationfromswhichhasnondeterministiceffectsg andh,writtenas(g∨
h ← s)attheobjectlevel,isrepresentedinadisjunctionoftheform(3)atthemetalevel.
Ontheotherhand,therelationthat“gisjointlycausedbysandt”,writtenas(g ← s∧t)
at the object level, is expressed in a disjunction of the form (4) at the meta level, viz.,
(g←s∧t)≡(g←s)∨(g←t).
g✐
✛
OR s connected(g,s)∨connected(h,s)
✮ (3)
h
s
✛ ✮
g ✐AND connected(g,s)∨connected(g,t)
(4)
t
There can be more than two atoms in a disjunction of the form (3) or (4). For example,
(g←s∧t∧u)attheobjectlevelcanbeexpressedasconnected(g,s)∨connected(g,t)∨
connected(g,u).Acomplexrelationoftheform(g∨h←s∧t)thathasmorethanonenode
inboththeleft-handandright-handsidesoftherulecanbedecomposedintotworelations,
1 Preciselyspeaking,ourcausalnetworksbringusmoreinformationthandirectedgraphs,sincenegation,
disjunctiveeffectsandjointcausesareallrepresentedinanetwork.
5
(s-t←s∧t)and(g∨h←s-t),wheres-trepresentstheintermediatecomplex.Then,any
directcausalrelationinacausalnetworkcanberepresentedbyatmosttwodisjunctionsof
atomsoftheforms(3)and(4)atthemetalevel,usingintermediatecomplexes.
Intheaboveexpression,(i)eachatomattheobjectlevelisrepresentedasatermatthe
metalevel,and(ii)eachruleattheobjectlevelisrepresentedasafact oradisjunctionof
factsatthemetalevel.Thepoint(ii)cannotonlyholdforrulesgivenintheaxioms,butcan
alsobeappliedtoexpressinferredrulesatthemetalevel.Now,toexpressinferredrules,we
introduceanothermeta-predicatecaused.Forobject-levelpropositionsg ands,wedefine
thatcaused(g,s)istrueifthereisacausalchainfromstog.Then,thecausalchainsare
generallydefinedtransitivelyintermsofconnectedastheaxiomswithvariables:
caused(X,Y)←connected(X,Y). (5)
caused(X,Y)←connected(X,Z)∧caused(Z,Y). (6)
Here, the caused/2 relation is recursively defined with the connected/2 facts. The first-
order expression with variables is thus used to represent that these axioms hold for all
instances of them. Other algebraic properties as well as some particular constraints (e.g.,
¬caused(a,b)) can also be defined if necessary. Variables in object-level expressions like
g(T) and s(T) can be allowed in the meta-level expression like connected(g(T),s(T)),
where the predicates g and s are treated as function symbols in the same way that Prolog
canallowhigher-orderexpressions.Here,anexpressiong(T)canrepresentasetofnodes
withthesamepropertygwithdifferentvaluesoftheargumentT suchastime.
2.2 RuleAbduction
Reasoningaboutcausalnetworksisrealizedbydeductionandabductionfromthemeta-level
expression of causal networks together with the axioms for causal relations including (5)
and(6).Inrulededuction,wewilllaterproveinProposition1that,ifameta-levelexpression
oftheformcaused(g,s)forsomefactsgandscanbederived,itmeansthattherule(g←s)
canbederivedattheobjectlevel.
Similarly, we can realize rule abduction in the meta-level representation as follows.
Supposethatafactgissomehowcausedbyafacts,whichcanalsoberegardedasaninput-
outputrelationthatanoutputgisobtainedgivenaninputs.Here,gandsarecalledagoal
(fact)(oratarget(fact))andasource(fact),respectively.SettinganobservationO asthe
causalchaincaused(g,s),wewanttoexplainwhyorhowOiscaused.Attheobjectlevel,
O correspondstotherule(g ← s),whichcanbegivenaseitherarealobservation(called
anempiricalrule)oravirtualgoaltobeachieved.Anabductivetaskisthentofindhidden
rulesthatestablishaconnectionfromthesourcestothegoalgbyfillingthegapsincausal
networks.AswewilllaterseeinTheorem1,suchanempiricalrulecanhavemorethanone
antecedent.Forexample,anobject-levelobservation(g ← s∧t)canbeexpressedasthe
meta-levelformula(caused(g,s)∨caused(g,t))inthesamewayas(4).
Logically speaking, a background theory B consists of the meta-level expression of a
causalnetworkNandtheaxiomsforcausalrelationsatthemetalevelcontaining(5)and(6).
WhenBisincomplete,theremaybenopathbetweengandsinB,thatis,caused(g,s)can-
notbederivedfromB.Then,abductioninfersanexplanation(orhypothesis)Hconsistingof
missingrelations(links)andmissingfacts(nodes).Thisisrealizedbysettingtheabducibles
Γ, the set of candidate literals to be assumed, as the atoms with the predicate connected:
Γ = {connected( , )}. It is sometimes declared that there cannot exist any direct causal
relationbetweenthesourceandthegoal,i.e.,¬connected(g,s).
6
Formally, given a set O of formulas, a set H of instances of elements of Γ is an (ab-
ductive)explanationofO (withrespecttoB)ifB∪H |= O andB∪H isconsistent.A
set of formulas can be interpreted as the conjunction of them. An explanation H of O is
minimalifitdoesnotimplyanyexplanationofOthatisnotlogicallyequivalenttoH.Min-
imalexplanationsinmeta-levelabductioncorrespondtominimaladditionsincausalgraphs,
andarereasonableaccordingtotheprincipleofOccam’srazor.Forexample,supposethe
observation O = caused(g,s)∧caused(h,s), that is, the multiple causal chains between
twogoalfactsg,handthesourcefacts.ExamplesofminimalexplanationsofOcontaining
twointermediatenodesareasfollows.
✛
g X✐
∃X∃Y(connected(g,X)∧connected(h,Y)
H1: s
✛ ✮ ∧connected(X,s)∧connected(Y,s))
h Y
✛
g X
♦ ∃X∃Y(connected(g,X)∧connected(X,Y)
H2: s
✙ ∧connected(h,Y)∧connected(Y,s))
✛
h Y
H1andH2representdifferentconnectivities,andwemaywanttoenumeratedifferenttypes
ofnetworkstructuresthataremissingintheoriginalcausalnetwork.Here,H1corresponds
tothefourrules{(g←χ), (χ←s), (h←ψ), (ψ←s)},henceruleabduction,i.e.,abduc-
tion of rules, is realized. Moreover, these hypotheses contain existentially quantified vari-
ables,whereχandψ arenewlyinventedhere.Thosenewtermscanberegardedaseither
some existing nodes or new unknown nodes. Since new formulas can be produced at the
objectlevel,predicateinvention[41]ispartiallyrealizedinmeta-levelabduction.2
Ahypothesiswithajointcauseoftheform(4)canbeobtainedbytakingadisjunction
ofexplanationsoftheformconnected(g, )orbyobtainingadisjunctiveanswer[53]foran
observationcontainingafreevariableX oftheformcaused(g,X).Alternatively,thiscan
berealizedbyaddingameta-levelaxiom:
connected(X,Y)∨connected(X,Z)←jointly connected(X,Y,Z). (7)
to the background theory B and the literals of the form jointly connected( , , ) to the
abduciblesΓ.Causesconsistingofmorethantwojointlinkscanberepresentedinasimilar
way.Atomsoftheformjointly connected( , , )togetherwithaxiom(7)canalsobeused
to represent conjunctive causes of the form (4) in a causal network. That is, to express
(g←s∧t)attheobjectlevel,theatomjointly connected(g,s,t)canbeusedinsteadofthe
formula(connected(g,s)∨connected(g,t))atthemetalevel.
Thesoundnessandcompletenessofruleabductioninmeta-levelabductioncanbede-
rivedasfollows.3 Foranymeta-leveltheoryN suchthatthepredicateofanyformulaap-
pearinginN isconnectedonly,letλ(N)betheobject-leveltheoryobtainedbyreplacing
everyconnected(t1,t2)(t1 andt2 areterms)appearinginN withtheformula(t1 ← t2).
Wefirstshowthecorrectnessofmeta-leveldeductioninthenextproposition.
Proposition1 Suppose a meta-level theory N, which consists of disjunctions of facts of
the form connected( , ). Let the background theory be B = N ∪{(5),(6)}. Then, B |=
(caused(g,s1)∨···∨caused(g,sn))ifandonlyifλ(N)|=(g←s1∧···∧sn).
2 Predicateinventioninthisformcanalsoberegardedasarealizationofhiddenobjectinventiondiscussed
byMuggletonin[7].
3 The properties presented here are generalizations of preliminary results in [18, Section 3.3], which
provedthecorrectnesswhentheobservationisafact,i.e.,asingleatom,oftheformcaused(g,s).
7
Proof Weprovethepropositionbyinductiononthedepthofprooftrees.4
Induction basis. It holds by the meaning of causal networks that caused(g,s) is de-
rivedinaproofhavingdepth1iffconnected(g,s)∈B(by(5))iff(g ←s)∈λ(N).Then,
(caused(g,s1)∨···∨caused(g,sn))isderivedinaproofhavingdepth1iff(connected(g,s1)∨
···∨connected(g,sn)) ∈ N iff((g ← si)∨···∨(g ← sn)) ∈ λ(N)iffλ(N) |= (g ←
s1∧···∧sn).
Inductionhypothesis.Supposethatthepropositionholdsforanyformula(caused(g,s1)∨
···∨caused(g,sn))derivedfromBinaprooftreehavingdepthdsuchthatd≤k.
Inductionstep.Aformula(caused(g,s1)∨···∨caused(g,sn))isderivedinaprooftree
havingdepthk+1iffB |=((connected(g,s1)∨∃s′1(connected(g,s′1)∧caused(s′1,s1)))∨
···∨(connected(g,sn)∨∃s′n(connected(g,s′n)∧caused(s′n,sn))))(by(5)and(6))such
thatcaused(s′j,sj)isderivedinaprooftreehavingdepthk forj = 1,...,niffλ(N) |=
(((g←s1)∨∃s′1((g←s′1)∧(s′1 ←s1)))∨···∨((g←sn)∨∃s′n((g←s′n)∧(s′n ←s1))))
(bytheinductionhypothesis)iffλ(N) |= ((g ← s1)∨···∨(g ← sn))iffλ(N) |= (g ←
s1∧···∧sn). ⊓⊔
NoteinProposition1thatwedonotneedtheλ-counterpartsofaxioms(5)and(6)in
theobjectlevel.Thislogicreflectstheassumptionthattransitivityofcauseby←holdsfor
achaininacausalnetwork.Nowwehavethecorrectnessofmeta-levelabduction.
Theorem1 LetN andBbethesametheoriesasinProposition1.Supposetheobservation
O = (caused(g,s1)∨···∨caused(g,sn)), and let Γ = {connected( , )}. Then, H is
anabductiveexplanationofcaused(g,s)withrespecttoB andΓ ifandonlyifλ(H)isa
hypothesissatisfyingthat
λ(N)∪λ(H)|=(g←s1∧···∧sn), and (8)
λ(N)∪λ(H)isconsistent. (9)
Proof The equivalence between the relation that B ∪H |= O and the abductive deriva-
tion(8)holdsbyProposition1.TheequivalencebetweentheconsistencyofB∪H andthe
consistency(9)isobvious:B∪H isconsistentbecauseitcontainsnointegrityconstraint,
andsoisλ(N)∪λ(H). ⊓⊔
2.3 AbductionofRulesandFacts
Besidestheuseinruleabduction,meta-levelabductioncanalsobeappliedtofactabduction
[18],whichhasbeenfocusedonalmostexclusivelyinresearchofabductioninAI.5Abduc-
tionoffactsattheobjectlevelcanbeformalizedasqueryansweringatthemetalevel.Given
agoaloftheformcaused(g,X),abductionofcausesiscomputedbyanswersubstitutions
tothevariableX.Tothisend,eachabducibleliteralaattheobjectlevelisassociatedwith
the fact caused(a,a) at the meta level. That is, an abducible can hold by assuming itself.
Equivalently,theaxiomforabduciblesisexpressedusingthemeta-predicateabdas:
caused(X,X)←abd(X).
4 Forexample,SOLtableaux[44,45](Section2.4)canbeusedasprooftreesindeducingtargetconse-
quences.SinceSOL(AR)iscompleteforconsequence-finding[13,45],foranyminimalclauseC derived
fromaconsistentaxiomset,thereisanSOLtableauproducingCwithacertaindepth.
5 Inaphilosophicalwork[60],factabductionandruleabductionareclassifiedasfactualabductionand
law-abduction,respectively.Ourmeta-levelabductionalsogivesarealizationof2ndorderexistentialabduc-
tion,whichismostimportanttoproducenewtheorieswithnewconcepts[60].
8
Then,eachabducibleaattheobjectlevelshouldbedeclaredasabd(a).Answerextraction
forthequery ←caused(g,X)canberealizedbygivingthemeta-levelformulaoftheform:
ans(X)←caused(g,X)∧abd(X). (10)
Here,ansistheanswerpredicate[23],andthevariableXisusedtocollectonlyabducibles
whichcauseg.Anintegrityconstraintthattwofactspandq cannotholdatthesametime
(←p∧q)canberepresentedas:
←caused(p,X)∧caused(q,Y)∧abd(X)∧abd(Y).
This makes any combination of abducibles that causes p and q incompatible. Such a set
of incompatible abducibles is called a nogood. Finally, by combining rule abduction and
factabductionintheformofconditionalqueryanswering[23],whichextractsanswersina
querywithadditionalabducedconditions,meta-levelabductionenablesustoabduceboth
rulesandfacts[18].
2.4 ComputationbyConsequenceFinding
Alltypesofmeta-levelinferencesinthissection,includinggenerationofexistentiallyquan-
tifiedhypothesesinmeta-levelabductionaswellasconditionalqueryansweringtoabduce
rulesandfacts,canberealizedbySOLAR[44,45].SOLARisaconsequence-findingsystem
basedonSOLresolution[13]andtheconnectiontableaux.
InSOLAR,thenotionofproductionfields[13]isusedtorepresentlanguagebiasesfor
hypotheses.Aclauseisadisjunctionofliterals.Aproductionfield P isapair(cid:13)L,Cond(cid:14),
whereLisasetofliteralsandCondisacertaincondition.IfCondistrue(empty),P is
denotedas(cid:13)L(cid:14).AclauseC belongstoP = (cid:13)L,Cond(cid:14)ifeveryliteralinC isaninstance
ofaliteralinLandC satisfiesCond.LetΣ beaclausaltheory.Thesetofconsequences
ofΣbelongingtoP isdenotedasThP(Σ).ThecharacteristicclausesofΣwithrespectto
P aredefinedasCarc(Σ,P) = µThP(Σ),whereµT denotesthesetofclausesinT that
areminimalwithrespecttosubsumption.ThenewcharacteristicclausesofaclauseCwith
respecttoΣandP aredefinedasNewcarc(Σ,C,P)=µ[ThP(Σ∪{C})\ThP(Σ)].
Let B be a clausal theory (background theory) and O a set of literals (observations).
Then,asetH ofliteralsisobtainedasanabductiveexplanationofObyinverseentailment
[13]:
B∪{¬O}|=¬H, (11)
whereboth¬O = (cid:2) ¬Land¬H = (cid:2) ¬Lareclauses(becauseOandH aresets
L∈O L∈H
ofliteralsandareinterpretedasconjunctionsofthem).Similarly,theconditionthatB∪H
is consistent is equivalent to B (cid:2)|= ¬H. Hence, for any hypothesis H, its negated form
¬H is deductively obtained as a “new” consequence of B ∪{¬O} which is not an “old”
consequenceofBalone.GiventheabduciblesΓ,anyliteralin¬Hisaninstanceofaliteral
inΓ ={¬L|L∈Γ}.Hence,thesetofminimalexplanationsofOwithrespecttoBandΓ
ischaracterizedas{H |¬H ∈Newcarc(B,¬O,(cid:13)Γ(cid:14))},whilethesetofminimalnogoods
withrespecttoBandΓ is{H |¬H ∈Carc(B,(cid:13)Γ(cid:14))}.
SOLAR is complete for finding (new) characteristic clauses with respect to a given
production field. SOLAR can thus be used to implement a complete abductive system for
finding and enumerating minimal explanations from full clausal theories containing non-
Horn clauses. A simple way to compute Newcarc(Σ,C,P) in SOLAR is: (1) enumerate
Carc(Σ,P),andthen(2)enumeratetheSOLtableaudeductionsfromΣ∪{C}withthetop
clauseC[45]byremovingeachproducedclausesubsumedbysomeclauseinCarc(Σ,P).
9
3 ReasoningaboutPositiveandNegativeCausalEffects
Sofar,meta-levelabductionhasbeendefinedforcausalnetworkswithoutexplicitlyargu-
ing the meaning of causes. Indeed, links in a causal network have been of one kind, and
connected(g,s) at the meta level, i.e., (g ← s) at the object level, just represents that g
directlydependsonssomehow.However,mixingdifferenttypesofcausalitiesinonetype
oflinksoftenmakesanalysisofactualcausescomplicated[48].Forexample,supposethat
increaseoftheamountofpdecreasestheamountofqandthatincreaseofqcausesincrease
of r. In this case, we cannot say that increase of p causes increase of r because q cannot
mediate between p and r. For this problem, it is not appropriate to represent the causali-
ties as (p → q → r) because transitivity does not hold. Instead, (inc(p) → dec(q)) and
(inc(q)→inc(r))wouldbemoreprecisebutweneedmoreentitiesandrelationsbetween
inc( )anddec( ).Inthissection,weconsideroneofthemostimportantproblemsofthis
kind:networkswithtwotypesofcausalities,i.e.,positiveandnegativecausaleffects.With
this regard, from now on we can understand that each arc of the form connected(g,s) in
Section2onlyrepresentspositiveeffects.
Weextendapplicabilityofmeta-levelabductiontodealwithnetworksexpressingboth
positiveandnegativecausaleffects.Suchnetworksareseeninbiologicaldomains,where
inhibitioneffectsnegativelyingeneregulatory,signalingandmetabolicpathways.Nowwe
considertwotypesofdirectcausalrelations:triggered andinhibited.Fortwonodesgand
t,therelationtriggered(g,t)representsapositivecausesuchthattisatriggerofg,written
asg ←− tinacausalnetwork.Ontheotherhand,therelationinhibited(g,s)representsa
negativecausesuchthatsisaninhibitorofg,writtenasg |—–sinacausalnetwork.6The
meaningoftheselinkswillbegivenintwowaysinSections3.1and3.2.
AsinSection2.1,negation,disjunctiveeffectsandconjunctivecausescanbedefinedfor
triggeredandinhibited,cf.,(2),(3)and(4),andcomplexcausalrelationscanberepresented
usingthosecombinationsandintermediatecomplexes.Forinstance,gisjointlytriggeredby
t1andt2canbeexpressedastriggered(g,t1)∨triggered(g,t2).
The notion of causal chains is also divided into two types: the positive one (written
promoted)andthenegativeone(writtensuppressed),respectivelycorrespondingtotriggered
andinhibited.Nowourtaskistodesigntheaxiomsforthesetwometa-predicates.
3.1 AlternatingAxiomsforCausality
Supposefirstthatthereisnoinhibitorinacausalnetwork,thatis,alllinksarepositive.In
thiscase,theaxiomsforpromoted shouldcoincidewith(5)and(6):
promoted(X,Y)←triggered(X,Y). (12)
promoted(X,Y)←triggered(X,Z)∧promoted(Z,Y). (13)
Next,letusinterpretthemeaningofaninhibitorasatoggleswitchofsignalsflowedinthe
inhibitor,justasaninverterinalogiccircuit[62].Then,inthepresenceofinhibitors,we
needonemoreaxiom(14),whichblocksanadjacentinhibitorforXinordertopromoteX:
promoted(X,Y)←inhibited(X,Z)∧suppressed(Z,Y). (14)
6 Atriggerandaninhibitorareoftencalledanactivatorandarepressor,respectively.