Biomedical Engineering Reference
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∀
x HeavilyDrinks
(
x
)
⇒
CarAccidents
(
x
)
∀
x
,
y Friends
(
x
,
y
)
⇒
(
HeavilyDrinks
(
x
)
⇔
HeavilyDrinks
(
x
))
Fig. 4.1
Example of a knowledge base in first-order logic
1.8
∀
x HeavilyDrinks
(
x
) ⇒
CarAccidents
(
x
)
0.7
∀
x
,
y Friends
(
x
,
y
) ⇒ (
HeavilyDrinks
(
x
) ⇔
HeavilyDrinks
(
x
))
Fig. 4.2
Example of a knowledge base in Markov logic
1.8
∀
x HeavilyDrinks
(
x
) ⇒
CarAccidents
(
x
)
0.7 ∀
x
,
y Friends
(
x
,
y
) ⇒ (
HeavilyDrinks
(
x
) ⇔
HeavilyDrinks
(
x
))
Constants: Paolo (P) and Cesare (C)
HeavilyDrinks
(P)
HeavilyDrinks
(C)
CarAccidents
(P)
CarAccidents
(C)
Fig. 4.3
Partial construction of the nodes of the ground Markov network
Since FOL statements, in practice are not always true, it is necessary to soften
these hard constraints. For example, in practice it is not always true that if someone
drinks heavily, he will have a car accident. In Fig.
4.2
, it is presented a KB in Markov
Logic. As it can be seen, formulas have weights attached and statements are not
always true any more. Their degree of truth depends on the weight attached. For
instance, the first formula expresses a stronger constraint than the second.
The simple KB in Fig.
4.2
together with a set of constants defines an MN. For
example, suppose we have two constants in the domain that represent two persons,
Paolo and Cesare. Then, the first step in the construction on the MN is given by
the grounding of each predicate in the domain according to the constants of the
domain. Partial grounding is shown in Fig.
4.3
where only groundings of
HDrinks
and
CarAcc
are considered. The complete nodes are shown in Fig.
4.4
where all the
groundings of the predicates represent nodes in the graph.
In the next step, any two nodes whose corresponding predicates appear to-
gether is some ground formula are connected. For example, in Fig.
4.5
, the nodes
HDrinks(P)
and
CarAcc(P)
are connected through an arc, because the two predi-
cates appear together in the grounding of the second formula. The complete graph
is presented in Fig.
4.6
.
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