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A Note on Inadequacy of the Model for Learning from Queries
Ryuichi NAKANISHI Hiroyuki SEKI Tadao KASAMI
Publication
IEICE TRANSACTIONS on Information and Systems
Vol.E77D
No.8
pp.861868 Publication Date: 1994/08/25
Online ISSN:
DOI:
Print ISSN: 09168532 Type of Manuscript: PAPER Category: Automata, Languages and Theory of Computing Keyword: computational learning theory, learning correctly from queries, formal language, extracting long counter example,
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Summary:
Learning correctly from queries" is a formal learning model proposed by Angluin. In this model, for a class Γ of language representations, a learner asks queries to a teacher of an unknown language L_{q} which can be represented by some G_{q}Γ, and eventually outputs a language representation GΓ which represents L_{q} and halts. An algorithm (leaner) A is said to learn a class of languages represented by Γ in the weak definition if the time complexity of A is some polynomial of n and m, where n is the minimum size of the lagunage representations in Γ which represent L_{q}, and m is the maximum length of the counterexamples returned in an execution. On the other band, A is said to learn represented by Γ in the strong definition if at any point τ of the execution, the time consumed up to τ is some polynomial of n and m, where n is the same as above, and m is the maximum length of the counterexamples returned up to τ. In this paper, adequacy of the model is examined, and it is shown that both in the weak and strong definitions, there exist learners which extract a long counterexample, and identify L_{q} by using equivalence queries exhaustively. For example, there exists a learner which learns the class CFL of contextfree languages represented by the class CFG of contextfree grammars in the weak definition using only equivalence queries. Next, two restrictions concerning with learnability criteria are introduced. Proper termination condition is that when a teacher replies with yes" to an equivalence query, then the learner must halt immediately. The other condition, called LBCcondition, is that in the weak/strong definition, the time complexity must be some polynomial of n and log m. In this paper, it is shown that under these conditions, there still exist learners which execute exhaustive search. For instance, there exists a learner which learns CFL represented by CFG in the weak definition using membership queries and equivalence queries under the proper termination condition, and there also exists a learner that learns CFL represented by CFG in the strong definition using subset queries and superset queries under LBCcondition. These results suggest that the weak definition is not an adequate learning model even if the proper termination condition is assumed. Also, the model becomes inadequate in the strong definition if some combination of queries, such as subset queries and superset queries, is used instead of equivalence queries. Many classes of languages become learnable by our extracting long counterexample" technique. However, it is still open whether or not CFL represented by CFG is learnable in the strong definition from membership queries and equivalence queries, although the answer is known to be negative if at least one of (1) quadratic residues modulo a composite, (2) inverting RSA encryption, or (3) factoring Blum integers, is intractable.

