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ICM
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(Propositional) LP – Some Notations
Clauses: IF burglary and earthquake are true THEN alarm is true
Clause
burglary.
earthquake.
alarm :- burglary, earthquake.
marycalls :- alarm.
johncalls :- alarm.
Herbrand Base (HB) = all atoms in the program
burglary, earthquake, alarm, marycalls, johncalls
Program
atom
body
head
ICM
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Model Theoretic: Restrictions on Possible Worlds
burglary.
earthquake.
alarm :- burglary, earthquake.
marycalls :- alarm.
johncalls :- alarm.
• Herbrand Interpretation– Truth assigments to all elements of HB
• An interpretation is a model of a clause C If the body of C holds then the head holds, too.
burglary
earthquake
alarm
marycalls
johncalls
truefalse
truefalse
truefalse
truefalse
truefalse
ICM
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004Goal
Proof Theoretic (Entailment):Restrictions on Possible Derivations
burglary.
earthquake.
alarm :- burglary, earthquake.
marycalls :- alarm.
johncalls :- alarm.
:- johncalls.
:- alarm.
:- burglary, earthquake.
:- earthquake.
{}
• A set of clauses can be used to prove that atoms are entailed by the set of clauses.
ICM
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Bayesian Networks [Pearl 91]
Qualitative part:
Directed acyclic graph• Nodes - random vars. • Edges - direct influence
Compact representation of joint probability distributions
Quantitative part: Set of conditional probability distributions
0.9 0.1
e
b
e
0.2 0.8
0.01 0.99
0.9 0.1
be
b
b
e
BE P(A | B,E)Earthquake
JohnCalls
Alarm
MaryCalls
Burglary
P(E,B,A,M,J)
Together:Define a unique distribution in a compact, factored form
P(E,B,A,M,J)=P(E) * P(B) * P(A|E,B) * P(M|A) * P(J|A)
[illustration inspired by Kevin Murphy]
ICM
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Traditional Approaches
P(j) = P(j|a) * P(m|a) * P(a|e,b) * P(e) * P(b)
+ P(j|a) * P(m|a) * P(a|e,b) * P(e) * P(b)
0.9 0.1
e
b
e
0.2 0.8
0.01 0.99
0.9 0.1
be
b
b
e
BE P(A | B,E)Earthquake
JohnCalls
Alarm
MaryCalls
Burglary
Model Theoretic
...
+ P(j|a) * P(m|a) * P(a|e,b) * P(e) * P(b)
burglary.
earthquake.
alarm :- burglary, earthquake.
marycalls :- alarm.
johncalls :- alarm.
Bayesian Networks [Pearl 91]
ICM
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(Hidden) Markov Models[Rabiner 89]
coin2
coin1
0.5
0.5
0.50.5
0.3 : head
0.7 : tail
0.5 : head
0.5 : tailMoore
coin2
coin1
0.5*0.5 : head
0.5*0.5 : tail
0.5*0.3 : tail
0.5*0.7 : head
0.5*
0.5
: he
ad
0.5*
0.5
: ta
il
0.5*
0.7
: he
ad
0.5*
0.3
: ta
ilMealy
Observations: t,
Hidden States: c1, c2, c1,c2, ...
Statistical models for sequences, i.e.
observations over time T=0,1,2,3,...
h, t, t, ...
Not obse
rved
ICM
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(Hidden) Markov Models
coin2
coin1
0.5*0.5 : head
0.5*0.5 : tail
0.5*0.3 : tail
0.5*0.7 : head
0.5*
0.5
: he
ad
0.5*
0.5
: ta
il
0.5*
0.7
: he
ad
0.5*
0.3
: ta
il
coin2.coin10.5*0.3 : tail
[Rabiner 89]
coin2
coin1
coin2
coin1
tail
coin2
coin1
tail
coin2
coin1
head
Prio
r
...
= P
P1
+ P2
+ P3
P11 P12 P13* *
P10 =
P20 + P4
...
Proof Theoretic
*
ICM
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Stochastic Grammars
Weighted Rewrite Rules
SNP VP
VP PP
i saw
V NP P NP
Det N Det N
man with the telescopethe
1.0 : S NP, VP
1/3 : NP i 1/3 : NP Det, N 1/3 : NP NP, PP
1.0 : Det the
0.5 : N man 0.5 : N telescope
0.5 : VP V, NP 0.5 : VP VP, PP
1.0 : PP P, NP
1.0 : V saw
1.0 : P with1.0 * 1/3 * 0.5 * 0.5 * 1.0 * ...= 0.00231
Proof Theoretic
[Manning, Schütze 99]
Upgrade HMMs (regular
languages) to more complex
languages such as
context-free languages.
ICM
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Upgrading to First-Order Logic
The maternal information mc/2 depends on the maternal and paternal pc/2 information of the mother mother/2: mchrom(fred,a). mchrom(fred,b),...
or better mc(P,a) :- mother(M,P), pc(M,a), mc(M,a). mc(P,a) :- mother(M,P), pc(M,a), mc(M,b). mc(P,b) :- mother(M,P), pc(M,a), mc(M,b). ...
father(rex,fred). mother(ann,fred).
father(brian,doro). mother(utta, doro).
father(fred,henry). mother(doro,henry).
pc(rex,a). mc(rex,a).
pc(ann,a). mc(ann,b).
...
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Upgrading - continued
Propositional Clausal LogicExpressions can be true or false
Relational Clausal LogicConstants and variables refer to objects
Full Clausal LogicFunctors aggregate objects
alarm :- burglary, earthquake.
atom
clause
head body
Substitution: Maps variables to terms: {M / ann}:
mc(P,a) :- mother(ann,P),pc(ann,a),mc(ann,a).
Herbrand base: set of ground atoms (no variables):
{mc(fred,fred),mc(rex,fred),…}
atom
mc(P,a) :- mother(ann,P),pc(ann,a),mc(ann,a).clause
head body
variable (placeholder)constant
terms
nat(0).
nat(succ(X)) :- nat(X).
atom
clause
head body
variable
constant
functor
term
Interpretations can be infinite !
nat(0),nat(succ(0)),
nat(succ(succ(0))), ...
ICM
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Forward Chainingfather(rex,fred). mother(ann,fred). father(brian,doro). mother(utta, doro). father(fred,henry). mother(doro,henry).pc(rex,a). mc(rex,a). pc(ann,a). mc(ann,b)....
mc(P,a) :- mother(M,P), pc(M,a), mc(M,a).
mc(P,a) :- mother(M,P), pc(M,a), mc(M,b).
{M/ann, P/fred}
mc(P,a):- mother(M,P), pc(M,a), mc(M,b).
mc(fred,a)
...
mother(ann,fred). pc(ann,a) mc(ann,b) father(rex,fred). ......
...Set of derivable ground atoms = least Herbrand model
ICM
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Backward Chainingfather(rex,fred). mother(ann,fred). father(brian,doro). mother(utta, doro). father(fred,henry). mother(doro,henry).pc(rex,a). mc(rex,a). pc(ann,a). mc(ann,b)....
mc(P,a) :- mother(M,P), pc(M,a), mc(M,a).
mc(P,a) :- mother(M,P), pc(M,a), mc(M,b).
mother(ann,fred).
{M/ann}pc(ann,a),mc(ann,a)
mother(ann,fred).
{M/ann}pc(ann,a),mc(ann,b)
pc(ann,a).
mc(ann,a)
fail
pc(ann,a).
mc(ann,b)
success
mc(fred,a)
{P/fred}
mother(M,fred),pc(M,a),mc(M,a)
mc(P,a):- mother(M,P), pc(M,a), mc(M,a).
mother(M,fred),pc(M,a),mc(M,b)
mc(P,a):- mother(M,P), pc(M,a), mc(M,b).{P/fred}
