Abstract
We investigate a new paradigm in the context of learning in the limit: 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.
We show that learning correction grammars for classes of r.e. languages in the -model (i.e., converging to a single correct correction grammar in the limit) is sometimes more powerful than learning ordinary grammars even in the -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, 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. For u a notation in Kleene’s general system (O, < o ) of ordinal notations, we introduce the concept of an u-correction grammar, where u is used to bound the number of corrections that the grammar is allowed to make. We prove a general hierarchy result: if u and v are notations for constructive ordinals such that u < o v, then there are classes of r.e. languages that can be -learned by conjecturing v-correction grammars but not by conjecturing u-correction grammars.
Surprisingly, we show that — above “ω-many” corrections — it is not possible to strengthen the hierarchy: -learning u-correction grammars of classes of r.e. languages, where u is a notation in O for any ordinal, can be simulated by -learning w-correction grammars, where w is any notation for the smallest infinite ordinal ω.
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References
Ambainis, A., Case, J., Jain, S., Suraj, M.: Parsimony Hierarchies for Inductive Inference. Journal of Symbolic Logic 69(1), 287–327 (2004)
Ash, J., Knight, J.F.: Recursive Structures and Ershov’s Hierarchy. Mathematical Logic Quarterly 42, 461–468 (1996)
Blum, L., Blum, M.: Toward a mathematical theory of inductive inference. Information and Control 28, 125–155 (1975)
Blum, M.: A machine-independent theory of the complexity of recursive functions. Journal of the ACM 14, 322–336 (1967)
Burgin, M.: Grammars with prohibition and human-computer interaction. In: Proceedings of the 2005 Business and Industry Symposium and the 2005 Military, Government, and Aerospace Simulation Symposium, pp. 143–147. Society for Modeling and Simulation (2005)
Carlucci, L., Case, J., Jain, S.: Learning correction grammars. TR12/06, National University of Singapore (December 2006)
Case, J.: The power of vacillation in language learning. SIAM Journal on Computing 28(6), 1941–1969 (1999)
Case, J., Jain, S., Sharma, A.: On learning limiting programs. International Journal of Foundations of Computer Science 3(1), 93–115 (1992)
Case, J., Lynes, C.: Machine inductive inference and language identification. In: Nielsen, M., Schmidt, E.M. (eds.) Automata, Languages, and Programming. LNCS, vol. 140, pp. 107–115. Springer, Heidelberg (1982)
Case, J., Royer, J.: Program size complexity of correction grammars. Preprint (2006)
Case, J., Smith, C.: Comparison of identification criteria for machine inductive inference. Theoretical Computer Science 25, 193–220 (1983)
Chen, K.: Tradeoffs in inductive inference of nearly minimal sized programs. Information and Control 52, 68–86 (1982)
Ershov, Y.L.: A hierarchy of sets I. Algebra and Logic 7, 23–43 (1968)
Ershov, Y.L.: A hierarchy of sets II. Algebra and Logic 7, 212–232 (1968)
Ershov, Y.L.: A hierarchy of sets III. Algebra and Logic 9, 20–31 (1970)
Freivalds, R.: Minimal Gödel numbers and their identification in the limit. LNCS, vol. 32, pp. 219–225. Springer, Heidelberg (1975)
Freivalds, R.: Inductive inference of minimal programs. In: Fulk, M., Case, J. (eds.) Proceedings of the Third Annual Workshop on Computational Learning Theory, pp. 3–20. Morgan Kaufmann Publishers, Inc. San Francisco (1990)
Freivalds, R., Smith, C.: On the role of procrastination in machine learning. Information and Computation 107(2), 237–271 (1993)
Gold, E.M.: Language identification in the limit. Information and Control 10, 447–474 (1967)
Hopcroft, J., Ullman, J.: Introduction to Automata Theory, Languages and Computation. Addison-Wesley, London (1979)
Jain, S., Osherson, D., Royer, J., Sharma, A.: Systems that Learn: An Introduction to Learning Theory, 2nd edn. MIT Press, Cambridge (1999)
Jain, S., Sharma, A.: Program Size Restrictions in Computational Learning. Theoretical Computer Science 127, 351–386 (1994)
Kinber, E.: On the synthesis in the limit of almost minimal Gödel numbers. Theory Of Algorithms and Programs, LSU, Riga. 1, 221–223 (1974)
Kleene, S.C.: On notation for ordinal numbers. Journal of Symbolic Logic 3, 150–155 (1938)
Kleene, S.C.: On the forms of predicates in the theory of constructive ordinals. American Journal of Mathematics 66, 41–58 (1944)
Kleene, S.C.: On the forms of predicates in the theory of constructive ordinals (second paper). American Journal of Mathematics 77, 405–428 (1955)
Osherson, D., Weinstein, S.: Criteria for language learning. Information and Control 52, 123–138 (1982)
Rogers, H.: Theory of Recursive Functions and Effective Computability. McGraw-Hill, 1967. Reprinted by MIT Press in (1987)
Royer, J., Case, J.: Subrecursive Programming Systems: Complexity & Succinctness. Birkhäuser (1994)
Schaefer, M.: A guided tour of minimal indices and shortest descriptions. Archive for Mathematical Logic 18, 521–548 (1998)
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Carlucci, L., Case, J., Jain, S. (2007). Learning Correction Grammars . In: Bshouty, N.H., Gentile, C. (eds) Learning Theory. COLT 2007. Lecture Notes in Computer Science(), vol 4539. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-72927-3_16
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DOI: https://doi.org/10.1007/978-3-540-72927-3_16
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