Skip to main content Accessibility help
×
×
Home

Learning correction grammars

  • Lorenzo Carlucci (a1) (a2), John Case (a3) and Sanjay Jain (a4)

Abstract

We investigate a new paradigm in the context of learning in the limit, namely, learning correction grammars for classes of computably enumerable (c.e.) languages. Knowing a language may feature a representation of it in terms of 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 c.e. languages in the TxtEx-mode (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 c.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 c.e. languages. For u a notation in Kleene's general system (O, <o) of ordinal notations for constructive ordinals, 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 <ov. then there are classes of c.e. languages that can be TxtEx-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: TxtEx-learning u-correction grammars of classes of c.e. languages, where u is a notation in O for any ordinal, can be simulated by TxtBc-learning w-correction grammars, where w is any notation for the smallest infinite ordinal ω.

Copyright

References

Hide All
[1]Ambainis, A., Case, J., Jain, S., and Suraj, M., Parsimony hierarchies for inductive inference, this Journal, vol. 69 (2004), no. 1, pp. 287327.
[2]Angluin, D., Inductive inference of formal languages from positive data. Information and Control, vol. 45 (1980), pp. 117135.
[3]Ash, J. and Knight, J. F., Recursive structures and Ershov's hierarchy, Mathematical Logic Quarterly, vol. 42 (1996), pp. 461468.
[4]Baliga, G., Case, J., Jain, S., and Suraj, M., Machine learning of higher order programs, this Journal, vol. 59 (1994), no. 2, pp. 486500.
[5]Barzdiņš, J. M., Inductive inference of automata, functions and programs, Proceedings of the 20th International Congress of Mathematicians (James, R. D., editor), 1974, pp. 455460.
[6]Blum, L. and Blum, M., Toward a mathematical theory of inductive inference. Information and Control, vol. 28 (1975), pp. 125155.
[7]Blum, M., A machine-independent theory of the complexity of recursive functions, Journal of the ACM, vol. 14 (1967), pp. 322336.
[8]Buchholz, W., Proof-theoretic analysis of termination proofs, Annals of Pure and Applied Logic, vol. 75 (1995), pp. 5765.
[9]Burgin, M., Grammars with prohibition and human-computer interaction. Proceedings of the Business and Industry Symposium and the Military, Government, and Aerospace Simulation Symposium (Hill, J., editor), 2005, pp. 143147.
[10]Carlucci, L., Case, J., and Jain, S., Learning correction grammars, Proceedings of the 20th Annual Conference on Learning Theory (Bshouty, N. and Gentile, C., editors), Lecture Notes in Artificial Intelligence, vol. 4539, Spring Verlag, 2007, pp. 203217.
[11]Case, J., The power of vacillation in language learning, SIAM Journal on Computing, vol. 28 (1999), no. 6. pp. 19411969.
[12]Case, J., Jain, S., and Sharma, A., On learning limiting programs. International Journal of Foundations of Computer Science, vol. 3 (1992), no. 1, pp. 93115.
[13]Case, J. and Lynes, C., Machine inductive inference and language identification, Proceedings of the 9th International Colloquium on Automata, Languages and Programming, Lecture Notes in Computer Science, vol. 140, Springer Verlag, 1982, pp. 107115.
[14]Case, J. and Royer, J., Program size complexity of correction grammars, Working paper, 2008.
[15]Case, J. and Smith, C., Comparison of identification criteria for machine inductive inference. Theoretical Computer Science, vol. 25 (1983), pp. 193220.
[16]Chen, K., Tradeoffs in inductive inference of nearly minimal sized programs, Information and Control, vol. 52 (1982), pp. 6886.
[17]Epstein, R. L., Haas, R., and Kramer, R., Hierarchies of sets and degrees below 0′, Logic Year 1979-1980 (Lerman, M., Schmerl, J. H., and Soare, R. I., editors), Lecture Notes in Mathematics, vol. 859, Springer Verlag, 1981, pp. 3248.
[18]Ershov, Y. L., A hierarchy of sets I, Algebra and Logic, vol. 7 (1968), pp. 2343.
[19]Ershov, Y. L., A hierarchy of sets II, Algebra and Logic, vol. 7 (1968), pp. 212232.
[20]Ershov, Y. L., A hierarchy of sets III, Algebra and Logic, vol. 9 (1970), pp. 2031.
[21]Freivalds, R., Minimal Gödel numbers and their identification in the limit, Proceedings of the 4th Symposium on Mathematical Foundations of Computer Science (Becvar, Jiri, editor), Lecture Notes in Computer Science, vol. 32, 1975, pp. 219225.
[22]Freivalds, R., Inductive inference of minimal programs, Proceedings of the 3rd Annual Workshop on Computational Learning Theory (Fulk, M. and Case, J., editors), Morgan Kaufmann Publishers, 1990, pp. 320.
[23]Freivalds, R. and Smith, C., On the role of procrastination in machine learning, Information and Computation, vol. 107 (1993), no. 2, pp. 237271.
[24]Gold, E. M., Language identification in the limit. Information and Control, vol. 10 (1967), pp. 447474.
[25]Hopcroft, J. and Ullman, J., Introduction to Automata Theory, Languages and Computation, Addison-Wesley, 1979.
[26]Jain, S., Osherson, D.. Royer, J., and Sharma, A., Systems that Learn: An Introduction to Learning Theory, 2nd ed., MIT Press, 1999.
[27]Jain, S. and Sharma, A.. Program size restrictions in computational learning, Theoretical Computer Science, vol. 127 (1994), pp. 351386.
[28]Kinber, E. B., On the synthesis in the limit of almost minimal Godel numbers, Theory of Algorithms and Programs (Bārzdiñš, J. M., editor), vol. 1, Latvian State University, 1974, pp. 221223.
[29]Kleene, S. C., On notation for ordinal numbers, this Journal, vol. 3 (1938), pp. 150155.
[30]Kleene, S. C.. On the forms of predicates in the theory of constructive ordinals, American Journal of Mathematics, vol. 66 (1944). pp. 4158.
[31]Kleene, S. C., On the forms of predicates in the theory of constructive ordinals (second paper), American Journal of Mathematics, vol. 77 (1955), pp. 405428.
[32]Machtey, M. and Young, P., An Introduction to the General Theory of Algorithms, North Holland, New York, 1978.
[33]Osherson, D., Stob, M., and Weinstein, S., Systems that Learn: An Introduction to Learning Theory for Cognitive and Computer Scientists, MIT Press, 1986.
[34]Osherson, D. and Weinstein, S., Criteria for language learning, Information and Control, vol. 52 (1982), pp. 123138.
[35]Osherson, D. and Weinstein, S., A note on formal learning theory, Cognition, vol. 11 (1982), pp. 7788.
[36]Osherson, D. and Weinstein, S., Learning theory and natural language, Cognition, vol. 17 (1984), pp. 128.
[37]Peter, R., Recursive Functions, Academic Press, New York, 1967.
[38]Pinker, S., Formal models of language learning, Cognition, vol. 7 (1979), pp. 217283.
[39]Rathjen, M., The realm of ordinal analysis, Sets and Proofs (Cooper, S. and Truss, J., editors), Cambridge University Press, 1999, pp. 219279.
[40]Rogers, H., Theory of Recursive Functions and Effective Computability, McGraw-Hill, 1967, Reprinted by MIT Press in 1987.
[41]Royer, J. and Case, J., Subrecursive Programming Systems: Complexity & Succinctness, Birkhäuser, 1994.
[42]Schaefer, M., A guided tour of minimal indices andsliortest descriptions, Archive for Mathematical Logic, vol. 18 (1998), pp. 521548.
[43]Takeuti, G., Proof Theory, 2nd ed., North Holland, 1987.
[44]Weiermann, A., Proving termination for term rewriting systems, Proceedings of the 5th Workshop on Computer Science Logic (Börger, E., Jäger, G., Büning, H. K., and Richter, M. M., editors), Lecture Notes in Computer Science, vol. 626, 1992, pp. 419428.
[45]Wexler, K., On extensional learnability, Cognition, vol. 11 (1982), pp. 8995.
[46]Wexler, K. and Culicover, P., Formal Principles of Language Acquisition, MIT Press, 1980.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed