Machine Learning - DISCo

CHAPTER 10 LEARNING SETS OF RULES 2!)7
1. Find a literal L1 from clause C1, literal Lz from clause Cz, and substitution 0 such that LIB =
-L28.
2. Form the resolvent C by including all literals from CIB and C28, except for L1 B and -L2B. More
precisely, the set of literals occurring in the conclusion C is
c = (Cl- (L11)O lJ (C2- ILzI)@
TABLE 10.7
Resolution operator (first-order form).
10.7.2 Inverting Resolution: First-Order Case
We can derive the inverse resolution operator analytically, by algebraic manipulation
of Equation (10.3) which defines the resolution rule. First, note the unifying
substitution 8 in Equation (10.3) can be uniquely factored into 81 and 82, where
0 = Ole2, where contains all substitutions involving variables from clause C1,
and where O2 contains all substitutions involving variables from C2. This factorization
is possible because C1 and C2 will always begin with distinct variable
names (because they are distinct universally quantified statements). Using this
factorization of 8, we can restate Equation (10.3) as
Keep in mind that "-" here stands for set difference. Now if we restrict inverse
resolution to infer only clauses C2 that contain no literals in common with C1
(corresponding to a preference for shortest C2 clauses), then we can re-express
the above as
c - (Cl - {LlHel = (C2 - IL2W2
Finally we use the fact that by definition of the resolution rule L2 = -~1818;',
and solve for C2 to obtain
Inverse resolution:
cz = (c - (CI - { ~~~)e~)e,-l u { -~,e~e;'~ (10.4)
Equation (10.4) gives the inverse resolution rule for first-order logic. As in the
propositional case, this inverse entailment operator is nondeterministic. In particular,
in applying it we may in general find multiple choices for the clause Cr to
be resolved and for the unifying substitutions and 82. Each set of choices may
yield a different solution for C2.
Figure 10.3 illustrates a multistep application of this inverse resolution rule
for a simple example. In this figure, we wish to learn rules for the target predicate
GrandChild(y, x), given the training data D = GrandChild(Bob, Shannon) and
the background information B = {Father(Shannon, Tom), Father (Tom, Bob)).
Consider the bottommost step in the inverse resolution tree of Figure 10.3. Here,
we set the conclusion C to the training example GrandChild(Bob, Shannon)