Machine Learning - DISCo

It is easiest to introduce the resolution rule in propositional form, though it is
readily extended to first-order representations. Let L be an arbitrary propositional
literal, and let P and R be arbitrary propositional clauses. The resolution rule is
P V L
-L v R
P V R
which should be read as follows: Given the two clauses above the line, conclude
the clause below the line. Intuitively, the resolution rule is quite sensible. Given
the two assertions P v L and -L v R, it is obvious that either L or -L must be
false. Therefore, either P or R must be true. Thus, the conclusion P v R of the
resolution rule is intuitively satisfying.
The general form of the propositional resolution operator is described in
Table 10.5. Given two clauses C1 and C2, the resolution operator first identifies
a literal L that occurs as a positive literal in one of these two clauses and as
a negative literal in the other. It then draws the conclusion given by the above
formula. For example, consider the application of the resolution operator illustrated
on the left side of Figure 10.2. Given clauses C1 and C2, the first step of the
procedure identifies the literal L = -KnowMaterial, which is present in C1, and
whose negation -(-KnowMaterial) = KnowMaterial is present in C2. Thus the
conclusion is the clause formed by the union of the literals C1- (L} = Pass Exam
and C2 - (-L} = -Study. As another example, the result of applying the resolution
rule to the clauses C1 = A v B v C v -D and C2 = -B v E v F is the clause
AvCV-DvEvF.
It is easy to invert the resolution operator to form an inverse entailment
operator O(C, C1) that performs inductive inference. In general, the inverse entailment
operator must derive one of the initial clauses, C2, given the resolvent C
and the other initial clause C1. Consider an example in which we are given the
resolvent C = A v B and the initial clause C1 = B v D. How can we derive a
clause C2 such that C1 A C2 F C? First, note that by the definition of the resolution
operator, any literal that occurs in C but not in C1 must have been present in C2.
In our example, this indicates that C2 must contain the literal A. Second, the literal
1. Given initial clauses C1 and C2, find a literal L from clause C1 such that -L occurs in clause C2.
2. Form the resolvent C by including all literals from C1 and C2, except for L and -L. More
precisely, the set of literals occurring in the conclusion C is
where u denotes set union, and "-" denotes set difference.
TABLE 10.5
Resolution operator (propositional form). Given clauses C1 and C2, the resolution operator constructs
a clause C such that C1 A C2 k C.