Learning Correction Grammars Lorenzo Carlucci1 , John Case2 , and Sanjay Jain 2
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1 Via Elea 8, 00183, Roma, Italy.
[email protected] Department of Computer and Information Sciences, University of Delaware, Newark, DE 19716-2586, U.S.A.
[email protected] 3 Department of Computer Science, National University of Singapore, Singapore 117543, Republic of Singapore.
[email protected]
Abstract. We investigate a new paradigm in the context of learning in the limit, namely, learning correction grammars for classes of r.e. languages. Knowing a language may feature a representation of the target language in terms of two sets of rules (two grammars). The second grammar is used to make corrections to the first grammar. Such a pair of grammars can be seen as a single description of (or grammar for) the language. We call such grammars correction grammars. Correction grammars capture the observable fact that people do correct their linguistic utterances during their usual linguistic activities. Is the need for self-corrections implied by using correction grammars instead of normal grammars compensated by a learning advantage? We show that learning correction grammars for classes of r.e. languages in the TxtEx-model (i.e., converging to a single correct correction grammar in the limit) is sometimes more powerful than learning ordinary grammars even in the TxtBc-model (where the learner is allowed to converge to infinitely many syntactically distinct but correct conjectures in the limit). For each n ≥ 0, there is a similar learning advantage, again in learning correction grammars for classes of r.e. languages, but where we compare learning correction grammars that make n + 1 corrections to those that make n corrections. The concept of a correction grammar can be extended into the constructive transfinite, using the idea of counting-down from notations for transfinite constructive ordinals. This transfinite extension can also be conceptualized as being about learning Ershov-descriptions for r.e. langauges. For u a notation in Kleene’s general system (O,
