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(a) M is said to be strong-monotonic on L just in case, for each  with  and each  , we have  . |
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(b) M is said to be strong monotonic on  just in case M is strong-monotonic on each  . |
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(c) M is strong monotonic just in case M is strong monotonic on each  . |
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So, a strong-monotonic scientist, upon being fed a text for a language, outputs a chain of hypotheses such that if index j is hypothesized after index i, then  . A consequence of this requirement is that if a scientist incorrectly hypothesizes that a particular element belongs to the target language, then it cannot revise this assumption by emitting a hypothesis that excludes this element. |
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(a)  |
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(b)  . |
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We thus consider only the global version in the sequel. |
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Wiehagen [200] suggested that the requirement of strong monotonicity is too stringent. He proposed the weaker notion of monotonic strategy, which simply requires that a scientist's successive hypotheses be more general only with respect to the target language. More precisely: |
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5.50 Definition (Wiehagen [200]) |
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(a) M is said to be monotonic on L just in case, for each  with  and each  , we have  . |
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(b) M is said to be monotonic on  just in case M is monotonic on each  . |
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Thus a monotonic strategy is allowed to correct its mistaken assumption that certain nonelements of L belong to L, but once it has correctly concluded that an element of L belongs to L it is not allowed to output a hypothesis that contradicts such a conclusion. The reader should note that we deliberately omitted the third clause in the above definition because the global version of monotonicity is equivalent to the requirement of strong |
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