Suppose we describe patterns by means of predicates of a particular rank r∗. Show the Ugly Duckling Theorem (Theorem 9.2) applies to any single level r∗, and thus for all predicates up to an arbitrary maximum level.

Theorem 9.2

(Ugly Duckling) Given that we use a finite set of predicates that enables us to distinguish any two patterns under consideration, the number of predicates shared by any two such patterns is constant and independent of the choice of those patterns. Furthermore, if pattern similarity is based on the total number of predicates shared by two patterns, then any two patterns are “equally similar.” ∗In summary, then, the Ugly Duckling Theorem states something quite simple yetimportant: there is no problem-independent or privileged or “best” set of features or feature attributes. Moreover, while the above was derived using d-tuples of binary values, it also applies to a continuous feature spaces too, if such as space is discretized (at any resolution). The Theorem forces us to acknowledge that even the apparently simple notion of similarity between patterns is fundamentally based on implicit assumptions about the problem domain (Problem 12).