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By Ulf Grenander

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When we proceed to groups that are locally compact but not commutative, our knowledge is less complete. See, however, the monumental work of Heyer (1977), which contains a wealth of information. The basic normalization problem has not been settled: How should we normalize in (31) in order to arrive at nontrivial probabilistic limit theorems? How should be transformed by a one-to-one mapping into some other space? In a few special cases we know how to do it, but not in general. In view of this, it may appear too early to turn to groups that are not even locally compact.

The learning takes place in two phases: listening and speaking. During the listening phase the learner “hears” one more sk. During the speaking phase the learner produces a sentence and is told whether it is grammatically correct or not. We include no semantics in this set up; the problem of learning semantics in a formal sense is being studied at this time and will be reported elsewhere. We want to construct an algorithm such that we can prove that it converges to the true grammar G as n + x .

To Step 4 if x = y, otherwise go to Step 5. Step 4 . Select another terminal y from the (current) class containing y. Replacex by 2 in the sentence. Treat 2 asx is treated in Steps 6-8. Go to Step 2. Step 5. Test the grammaticality of the sentence withx replaced by y. If it is not grammatical go to Step 7.

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