Deduction, Abduction, and Induction - International Center for

40 Pages·2012·0.81 MB·English

Deduction, Abduction, and Induction
SteffenHo¨lldobler
InternationalCenterforComputationalLogic
TechnischeUniversita¨tDresden
Germany
(cid:73) Introduction
(cid:73) Deduction
(cid:73) Sorts
(cid:73) Abduction
(cid:73) Induction
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Deduction,Abduction,andInduction 1
AnIntroductoryExample: Abduction
(cid:73) ConsiderK|=F,
whereKisasetofformulascalledknowledgebaseandF isaformula.
(cid:73) InthefollowingexampleIwillusethefollowingpropositionalatoms:
grassIsWet, wheelsAreWet, sprinklerIsRunning, raining.
(cid:73) LetK={g →w, s→w, r →g}.
(cid:46) DoesK|=w hold?
(cid:73) Idea FindanatomAsuchthatK∪{A}|=w andK∪{A}issatisfiable.
(cid:46) A=w
(cid:46) A=g
(cid:46) A=s or A=r
(cid:73) Thisprocessiscalledabduction.
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Deduction,Abduction,andInduction
(cid:73) Peirce1931 K ∪K |=G .
facts rules result
(cid:46) Deduction
isananalyticprocessbasedontheapplicationofgeneralrulestoparticular
facts,withtheinferenceasaresult.
(cid:46) Abduction
issyntheticreasoningwhichinfersafactfromtherulesandtheresult.
(cid:46) Induction
issyntheticreasoningwhichinfersarulefromthefactsandtheresult.
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Deduction
(cid:73) AllreasoningprocessesconsideredinthemoduleFoundationssofarare
deductions.
(cid:73) Thelogics(first-order,equational)areunsorted.
(cid:73) Theycanbeeasilyextendedtosortedlogics.
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Sorts
(cid:73) (∀X,Y)(number(X)∧number(Y)→plus(X,Y)≈plus(Y,X))
(cid:46) (∀X,Y:number)plus(X,Y)≈plus(Y,X).
(cid:73) Afirstorderlanguagewithsortsconsistsof
(cid:46) afirstorderlanguageL(R,F,V)and
(cid:46) afunctionsort:V →2RS,
whereR ⊆Risafinitesetofunarypredicatesymbolscalledbasesorts.
S
(cid:73) Elementsof2RS arecalledsorts;∅∈2RS iscalledtopsort.
(cid:73) WewriteX:sifsort(X)=s.
(cid:73) WeassumethatforeverysortstherearecountablymanyvariablesX:s∈V.
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Deduction,Abduction,andInduction 5
Sorts–Semantics
(cid:73) LetIbeaninterpretationwithdomainD,
I : s={p1,...,pn}(cid:55)→sI =D∩p1I ∩...∩pnI.
(cid:46) I : ∅(cid:55)→D.
(cid:73) AvariableassignmentZissortediffforallX:s∈VwefindXZ ∈sI.
(cid:73) Weassumethatallsortsarenon-empty.
(cid:73) FI,Z isdefinedasusualexceptfor
[(∃X:s)F]I,Z =(cid:62) iff thereexistsd ∈sI suchthatFI,{X(cid:55)→d}Z =(cid:62).
[(∀X:s)F]I,Z =(cid:62) iff foralld ∈sI wefindFI,{X(cid:55)→d}Z =(cid:62).
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Relativization
(cid:73) Sortedformulascanbemappedontounsortedonesbymeansofa
relativizationfunctionrel:
rel(p(t1,...,tn)) = p(t1,...,tn)
rel(¬F) = ¬rel(F)
rel(F ∧F ) = rel(F )∧rel(F )
1 2 1 2
rel(F ∨F ) = rel(F )∨rel(F )
1 2 1 2
rel(F →F ) = rel(F )→rel(F )
1 2 1 2
rel(F ↔F ) = rel(F )↔rel(F )
1 2 1 2
rel((∀X:s)F) = (∀Y)(p1(Y)∧...∧pn(Y)→rel(F{X (cid:55)→Y}))
ifsort(X)=s={p1,...,pn}andY isanewvariable
rel((∃X:s)F) = (∃Y)(p1(Y)∧...∧pn(Y)∧rel(F{X (cid:55)→Y}))
ifsort(X)=s={p1,...,pn}andY isanewvariable
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SortingFunctionandRelationSymbols
(cid:73) Eachatomoftheformp(t1,...,tn)canbeequivalentlyreplacedby
(∀X1...Xn)(p(X1,...,Xn)←X1 ≈t1∧...∧Xn ≈tn).
(cid:73) EachatomA(cid:100)f(t1,...,tn)(cid:101)canbeequivalentlyreplacedby
(∀X1...Xn)A(cid:100)f(t1,...,tn)/f(X1,...,Xn)(cid:101)←X1 ≈t1∧...∧Xn ≈tn.
(cid:73) EachformulaF canbetransformedintoanequivalentformulaF(cid:48),inwhich
(cid:46) allargumentsoffunctionandrelationsymbolsdifferentfrom≈
arevariablesand
(cid:46) allequationsareoftheformt1 ≈t2orf(X1,...,Xn)≈t,where
X1,...,Xnarevariablesandt, t1,andt2arevariablesorconstants.
(cid:73) SortingthevariablesoccurringinF(cid:48)effectivelysortsthefunctionandrelation
symbols.
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SortDeclaration
(cid:73) F(cid:48)isusuallyquitelengthyandcumbersometoread.
(cid:73) Ifsort(X)=sthenthesortdeclarationforthevariableX is
X:s.
(cid:73) Lets,1≤i ≤n,andsbesorts,f afunctionandparelationsymbol,both
i
witharityn.Then
f:s1×...×sn →s
and
p:s1×...×sn
aresortdeclarationsforf andp,respectively.
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Deduction,Abduction,andInduction 9
Abduction
(cid:73) Example Startingacar.
(cid:73) Applications
(cid:46) faultdiagnosis
(cid:46) medicaldiagnosis
(cid:46) highlevelvision
(cid:46) naturallanguageunderstanding
(cid:46) reasoningaboutstates,actions,andcausality
(cid:46) knowledgeassimilation
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