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￿
HAL Id: hal-00999312
https://hal.archives-ouvertes.fr/hal-00999312
Submitted on 4 Jun 2014
HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est
archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents
entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non,
lished or not. The documents may come from émanant des établissements d’enseignement et de
teaching and research institutions in France or recherche français ou étrangers, des laboratoires
abroad, or from public or private research centers. publics ou privés.
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:[email protected]
A.Doncescu
LAAS-CNRSUPR8001
AvenueduColonelRoche, 31007Toulouse, France
E-mail:[email protected]
H.Nabeshima
DivisionofMedicineandEngineeringScience, UniversityofYamanashi
4-3-11Takeda,Kofu,Yamanashi 400-8511, Japan
E-mail:[email protected]
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.