The complexity of learning SUBSEQ(A)
Stephen Fenner, William Gasarch, and Brian Postow
Source: J. Symbolic Logic Volume 74, Issue 3 (2009), 939-975.
Abstract
Higman essentially showed that if A is any language then SUBSEQ(A) is regular, where SUBSEQ(A) is the language of all subsequences of strings in A. Let s₁,s₂,s₃,… be the standard lexicographic enumeration of all strings over some finite alphabet. We consider the following inductive inference problem: given A(s₁), A(s₂), A(s₃), …, learn, in the limit, a DFA for SUBSEQ(A). We consider this model of learning and the variants of it that are usually studied in Inductive Inference: anomalies, mind-changes, teams, and combinations thereof.
This paper is a significant revision and expansion of an earlier conference version [10].
Keywords: machine learning; inductive inference; automata; computability; subsequence
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Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.jsl/1245158093
Digital Object Identifier: doi:10.2178/jsl/1245158093
Zentralblatt MATH identifier:
05321129
References
A. Ambainis, S. Jain, and A. Sharma, Ordinal mind change complexity of language identification, Theoretical Computer Science, vol. 220 (1999), no. 2, pp. 323--343.
Mathematical Reviews (MathSciNet):
MR1693792
Digital Object Identifier: doi:10.1016/S0304-3975(99)00005-5
D. Angluin, Inductive inference of formal languages from positive data, Information and Control, vol. 45 (1980), no. 2, pp. 117--135.
Mathematical Reviews (MathSciNet):
MR584828
Digital Object Identifier: doi:10.1016/S0019-9958(80)90285-5
G. Baliga and J. Case, Learning with higher order additional information, Proceedings of the 5th international workshop on Algorithmic Learning Theory, 1994, pp. 64--75.
J. M. Barzdin, Two theorems on the limiting synthesis of functions, Theory of algorithms and programs (J. Barzdin, editor), Latvian State University, Riga, U.S.S.R., 1974, pp. 82--88.
Mathematical Reviews (MathSciNet):
MR505358
L. Blum and M. Blum, Towards a mathematical theory of inductive inference, Information and Control, vol. 28 (1975), pp. 125--155.
Mathematical Reviews (MathSciNet):
MR395312
J. Case, S. Jain, and S. N. Manguelle, Refinements of inductive inference by Popperian and reliable machines, Kybernetika, vol. 30--1 (1994), pp. 23--52.
Mathematical Reviews (MathSciNet):
MR1267470
J. Case and C. H. Smith, Comparison of identification criteria for machine inductive inference, Theoretical Computer Science, vol. 25 (1983), pp. 193--220.
Mathematical Reviews (MathSciNet):
MR693078
Digital Object Identifier: doi:10.1016/0304-3975(83)90061-0
R. Daley, B. Kalyanasundaram, and M. Velauthapillai, Breaking the probability 1/2 barrier in FIN-type learning, Journal of Computer and System Sciences, vol. 50 (1995), pp. 574--599.
Mathematical Reviews (MathSciNet):
MR1339564
Digital Object Identifier: doi:10.1006/jcss.1995.1045
Y. L. Ershov, S. S. Goncharov, A. Nerode, and J. B. Remmel (editors), Handbook of recursive mathematics, Elsevier North-Holland, New York, 1998, Volume 1: Recursive Model Theory; Volume 2: Recursive Algebra, Analysis, and Combinatorics.
Mathematical Reviews (MathSciNet):
MR1673582
S. Fenner and W. Gasarch, The complexity of learning SUBSEQ$(A)$, Proceedings of the 17th international conference on algorithmic learning theory, Lecture Notes in Artificial Intelligence, vol. 4264, Springer-Verlag, 2006, pp. 109--123.
Mathematical Reviews (MathSciNet):
MR2324117
Digital Object Identifier: doi:10.1007/11894841_12
S. Fenner, W. Gasarch, and B. Postow, The complexity of finding SUBSEQ$(A)$, Theory of Computing Systems, (2008), to appear.
L. Fortnow, S. Jain, W. Gasarch, E. Kinber, M. Kummer, S. Kurtz, M. Pleszkoch, T. Slaman, F. Stephan, and R. Solovay, Extremes in the degrees of inferability, Annals of Pure and Applied Logic, vol. 66 (1994), no. 3, pp. 231--276.
Mathematical Reviews (MathSciNet):
MR1270964
Digital Object Identifier: doi:10.1016/0168-0072(94)90035-3
R. Freivalds and C. H. Smith, On the role of procrastination for machine learning, Information and Computation, vol. 107 (1993), no. 2, pp. 237--271.
Mathematical Reviews (MathSciNet):
MR1251620
Digital Object Identifier: doi:10.1006/inco.1993.1068
R. Freivalds, C. H. Smith, and M. Velauthapillai, Trade-off among parameters affecting inductive inference, Information and Computation, vol. 82 (1989), no. 3, pp. 323--349.
Mathematical Reviews (MathSciNet):
MR1016686
Digital Object Identifier: doi:10.1016/0890-5401(89)90005-9
W. Gasarch, E. Kinber, M. Pleszkoch, C. H. Smith, and T. Zeugmann, Learning via queries, teams, and anomalies, Fundamenta Informaticae, vol. 23 (1995), pp. 67--89, prior version in Computational Learning Theory (COLT), 1990.
Mathematical Reviews (MathSciNet):
MR1351576
W. Gasarch and A. Lee, Inferring answers to queries, Proceedings of the 10th annual ACM conference on Computational Learning Theory, 1997, journal version to appear in Journal of Computer and System Sciences in 2008, pp. 275--284.
Mathematical Reviews (MathSciNet):
MR2416785
Digital Object Identifier: doi:10.1016/j.jcss.2007.06.009
W. Gasarch, M. Pleszkoch, and R. Solovay, Learning via queries to $[+,<]$, Journal of Symbolic Logic, vol. 57 (1992), no. 1, pp. 53--81.
Mathematical Reviews (MathSciNet):
MR1150925
Digital Object Identifier: doi:10.2307/2275176
JSTOR: links.jstor.org
W. Gasarch, M. Pleszkoch, F. Stephan, and M. Velauthapillai, Classification using information, Annals of Mathematics and Artificial Intelligence, vol. 23 (1998), pp. 147--168, earlier version in Proceedings of the 5th international workshop on algorithmic learning theory, 1994, pp. 290--300.
Mathematical Reviews (MathSciNet):
MR1661596
Digital Object Identifier: doi:10.1023/A:1018916324775
W. Gasarch and C. H. Smith, Learning via queries, Journal of the ACM, vol. 39 (1992), no. 3, pp. 649--675, prior version in Proceedings of the 29th IEEE Symposium on Foundations of Computer Science (FOCS), 1988.
Mathematical Reviews (MathSciNet):
MR1177958
Digital Object Identifier: doi:10.1145/146637.146670
E. M. Gold, Language identification in the limit, Information and Control, vol. 10 (1967), no. 10, pp. 447--474.
J. Hartmanis, Context-free languages and Turing machine computations. Mathematical aspects of computer science, Proceedings of symposia in Applied Mathematics (J. T. Schwartz, editor), vol. 19, American Mathematical Society, Providence, RI, 1967, pp. 42--51.
Mathematical Reviews (MathSciNet):
MR235938
A. G. Higman, Ordering by divisibility in abstract algebras, Proceedings of the London Mathematical Society, no. 1, 1952, s3--2, pp. 326--336.
Mathematical Reviews (MathSciNet):
MR49867
Digital Object Identifier: doi:10.1112/plms/s3-2.1.326
S. Jain personal communication, 2008.,
S. Jain and A. Sharma, Computational limits on team identification of languages, Information and Computation, vol. 130 (1996), no. 1, pp. 19--60.
Mathematical Reviews (MathSciNet):
MR1423480
Digital Object Identifier: doi:10.1006/inco.1996.0081
S. Jain, A. Sharma, and M. Velauthapillai, Finite identification of functions by teams with success ratio 1/2 and above, Information and Computation, vol. 121 (1995), no. 2, pp. 201--213.
Mathematical Reviews (MathSciNet):
MR1348527
Digital Object Identifier: doi:10.1006/inco.1995.1133
M. Kummer and F. Stephan, On the structure of the degrees of inferability, Journal of Computer and System Sciences, vol. 52 (1996), no. 2, pp. 214--238, prior version in Sixth annual conference on Computational Learning Theory (COLT), 1993.
Mathematical Reviews (MathSciNet):
MR1393991
Digital Object Identifier: doi:10.1006/jcss.1996.0018
G. Metakides and A. Nerode, Effective content of field theory, Annals of Mathematical Logic, vol. 17 (1979), pp. 289--320.
Mathematical Reviews (MathSciNet):
MR556895
Digital Object Identifier: doi:10.1016/0003-4843(79)90011-1
L. Pitt, Probabilistic inductive inference, Journal of the ACM, vol. 36 (1989), no. 2, pp. 383--433.
Mathematical Reviews (MathSciNet):
MR1072427
Digital Object Identifier: doi:10.1145/62044.62053
L. Pitt and C. H. Smith, Probability and plurality for aggregations of learning machines, Information and Computation, vol. 77 (1988), pp. 77--92.
Mathematical Reviews (MathSciNet):
MR937376
Digital Object Identifier: doi:10.1016/0890-5401(88)90046-6
H. Rogers, Theory of recursive functions and effective computability, McGraw-Hill, 1967, reprinted by MIT Press, 1987.
Mathematical Reviews (MathSciNet):
MR886890
G. E. Sacks, Higher recursion theory, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1990.
Mathematical Reviews (MathSciNet):
MR1080970
C. H. Smith, The power of pluralism for automatic program synthesis, Journal of the ACM, vol. 29 (1982), pp. 1144--1165, prior version in Proceedings of the 22nd IEEE symposium on Foundations of Computer Science (FOCS), 1981.
Mathematical Reviews (MathSciNet):
MR674261
Digital Object Identifier: doi:10.1145/322344.322356
R. I. Soare, Recursively enumerable sets and degrees, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1987.
Mathematical Reviews (MathSciNet):
MR882921
J. van Leeuwen, Effective constructions in well-partially-ordered free monoids, Discrete Mathematics, vol. 21 (1978), pp. 237--252.
Mathematical Reviews (MathSciNet):
MR522898
Digital Object Identifier: doi:10.1016/0012-365X(78)90156-5
