Machine Learning - DISCo

FIGURE 11.3
Computing the weakest preimage of SafeToStack(0 bj 1, Obj2) with respect to the explanation. The
target concept is regressed from the root (conclusion) of the explanation, down to the leaves. At each
step (indicated by the dashed lines) the current frontier set of literals (underlined in italics) is regressed
backward over one rule in the explanation. When this process is completed, the conjunction of
resulting literals constitutes the weakest preimage of the target concept with respect to the explanation.
This weakest preimage is shown by the italicized literals at the bottom of the figure.
resulting in a set of generalized literals for the leaf nodes of the tree. This final
set of literals, shown at the bottom of Figure 11.3, forms the body of the final
rule.
The heart of the regression procedure is the algorithm that at each step regresses
the current frontier of expressions through a single Horn clause from the
domain theory. This algorithm is described and illustrated in Table 11.3. The illustrated
example in this table corresponds to the bottommost single regression step
of Figure 11.3. As shown in the table, the REGRESS algorithm operates by finding
a substitution that unifies the head of the Horn clause rule with the corresponding
literal in the frontier, replacing this expression in the frontier by the rule body,
then applying a unifying substitution to the entire frontier.
The final Horn clause rule output by PROLOG-EBG is formulated as follows:
The clause body is defined to be the weakest preconditions calculated by the above
procedure. The clause head is the target concept itself, with each substitution from
each regression step (i.e., the substitution Oh[ in Table 11.3) applied to it. This
substitution is necessary in order to keep consistent variable names between the
head and body of the created clause, and to specialize the clause head when the