Machine Learning - DISCo

C : KnowMaterial v -Study C: KnowMaterial V 7SNdy
C : Passh v ~KnawMaferial C : PIISS~ V 1KnowMafcrial
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FIGURE 10.2
On the left, an application of the (deductive) resolution rule inferring clause C from the given clauses
C1 and C2. On the right, an application of its (inductive) inverse, inferring Cz from C and C1.
that occurs in C1 but not in C must be the literal removed by the resolution rule,
and therefore its negation must occur in C2. In our example, this indicates that C2
must contain the literal -D. Hence, C:! = A v -D. The reader can easily verify
that applying the resolution rule to C1 and C2 does, in fact, produce the desired
resolvent C.
Notice there is a second possible solution for C2 in the above example. In
particular, C2 can also be the more specific clause A v -D v B. The difference
between this and our first solution is that we have now included in C2 a literal
that occurred in C1. The general point here is that inverse resolution is not
deterministic-in general there may be multiple clauses C2 such that C1 and C2
produce the resolvent C. One heuristic for choosing among the alternatives is to
prefer shorter clauses over longer clauses, or equivalently, to assume C2 shares no
literals in common with C1. If we incorporate this bias toward short clauses, the
general statement of this inverse resolution procedure is as shown in Table 10.6.
We can develop rule-learning algorithms based on inverse entailment operators
such as inverse resolution. In particular, the learning algorithm can use
inverse entailment to construct hypotheses that, together with the background
information, entail the training data. One strategy is to use a sequential covering
algorithm to iteratively learn a set of Horn clauses in this way. On each
iteration, the algorithm selects a training example (xi, f (xi)) that is not yet covered
by previously learned clauses. The inverse resolution rule is then applied to
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1. Given initial clauses C1 and C, find a literal L that occurs in clause C1, but not in clause C.
2. Form the second clause Cz by including the following literals
TABLE 10.6
Inverse resolution operator (propositional form). Given two clauses C and Cl. this computes a clause
C2 such that C1 A Cz I- C.