Machine Learning - DISCo

Father (Shannon, Tom)
GrandChild(Bob,x) v
Father(x,Tom) I
GrandChild(Bob, Shannon)
FIGURE 10.3
A multistep inverse resolution. In each case, the boxed clause is the result of the inference step. For
each step, C is the clause at the bottom, C1 the clause to the left, and C2 the boxed clause to the
right. In both inference steps here, el is the empty substitution (1, and 0;' is the substitution shown
below C2. Note the final conclusion (the boxed clause at the top right) is the alternative form of the
Horn clause GrandChild(y, x) c Father(x, z) A Father(z, y).
and select the clause C1 = Father(Shannon, Tom) from the background information.
To apply the inverse resolution operator we have only one choice
for the literal L1, namely Father(Shannon, Tom). Suppose we choose the inverse
substitutions 9;' = {} and 9;' = {Shannon/x}. In this case, the resulting
clause C2 is the union of the clause (C- (C1 - {Ll})91)9;~ = (~91)9;'
= GrandChild(Bob, x), and the clause {-~~9~9,')
= -.Father(x, Tom). Hence
the result is the clause GrandChild(Bob, x) v -Father(x, Tom), or equivalently
(GrandChild(Bob, x) t Father(x, Tom)). Note this general rule, together with
C1 entails the training example GrandChild(Bob, Shannon).
In similar fashion, this inferred clause may now be used as the conclusion
C for a second inverse resolution step, as illustrated in Figure 10.3. At each such
step, note there are several possible outcomes, depending on the choices for the
substitutions. (See Exercise 10.7.) In the example of Figure 10.3, the particular set
of choices produces the intuitively satisfying final clause GrandChild(y, x) t
Father(x, 2) A Father(z, y).
10.7.3 Summary of Inverse Resolution
To summarize, inverse resolution provides a general approach to automatically
generating hypotheses h that satisfy the constraint (B A h A xi) t- f (xi). This is
accomplished by inverting the general resolution rule given by Equation (10.3).
Beginning with the resolution rule and solving for the clause C2, the inverse
resolution rule of Equation (10.4) is easily derived.
Given a set of beginning clauses, multiple hypotheses may be generated by
repeated application of this inverse resolution rule. Note the inverse resolution rule
has the advantage that it generates only hypotheses that satisfy (B ~h AX^) t- f (xi).