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		We also define .

		It is easy to verify that .

		We next show that Consider any scientist M that identifies Let  s  be such that , for some and WM( s ) = L1. (Note that there exists such a  s , since M identifies L1.)

		Let be such that , for some n > m, and . (Note that there exists such a  s ' since M identifies .)

		Let be such that and (Note that there exists such a  s '', since M' identifies .)

		We finally claim that M fails to be monotonic while identifying . To see this it is sufficient to observe that but .

		§5.5.3 Specialization Strategies

		Generalization strategies improve upon their successive conjectures by emitting indexes for larger and larger languages. Alternatively, a scientist can begin with an index for a language larger than the target language and then "cut down" her hypotheses to converge to an index for the target language. Such strategies are referred to as specialization strategies; they may be viewed as models for top-down or general-to-specific searches in artificial systems of empirical inquiry (see for example, Shapiro's MIS [171] and Quinlan's FOIL [157]).

		Three specialization strategies can be defined as duals to the three generalization strategies. We consider only the dual of weak-monotonic scientists. Duals of strong-monotonic scientists and monotonic scientists are considered in the exercises.

		5.59 Definition (Kapur [103])

		(a) M is dual-weak-monotonic on L just in case, for each with and each with , we have .

		(b) M is dual-weak-monotonic on just in case M is dual-weak-monotonic on each .

		(c) M is dual-weak-monotonic just in case M is dual-weak-monotonic on each .

		(d) .