John Case, Susanne Kaufmann, E m Kinber and Martin Kummer. Learning recursive functions from approximations. Journal of Computer and System Sciences, 55:183-196, 1997.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....to theory revision. The study of drifting concepts in learning theory [12] may perhaps be viewed as a dynamic version of theory revision. Much has been written about learning from partial information. A few examples include SOAR [16] in systems building machine learning, the work of Case et al. [8] and Jain and Sharma [13] in inductive inference, and Rivest and Sloan [24] in PAC learning. Of course, theory revision can be viewed as simply learning from a lot of partial information. And all machine learning is then the special case of learning from partial information where the partial ....

J. Case, S. Kaufmann, E. Kinber, and M. Kummer. Learning recursive functions from approximations. J. Comput. Syst. Sci., 55:183--196, 1997.

....for S on f . If N converges on f , then H converges on f to a finite number and thus f is recursive. Furthermore, also M must converge to a program e and since M is reliable and f is recursive, pad(e; c) is a program for f . Thus N is totally reliable. Case, Kaufmann, Kinber and Kummer [4] showed that there is a family of binary recursive trees of width 2 whose infinite branches are not Ex learnable. This yields a class S which is totally reliably BC learnable but not reliably Ex learnable. For bounded classes, the concept of totally reliable BC learning is also a proper ....

....all functions which are infinite branch of some bounded primitive recursive tree of rank up to k. Given f , the learning algorithm first finds (in the limit) a tree T such that f is an infinite branch of T . Having found this tree T , one uses the algorithm of Case, Kaufmann, Kinber and Kummer [4] who showed that knowing an index of the tree and having a primitive recursive function majorizing all infinite branches, one can learn the function by a team of k 1 Ex learners or k BC learners, respectively. The team size is also optimal. The class S k is closed, that is, S k ] S k . So, it ....

John Case, Susanne Kaufmann, Efim Kinber and Martin Kummer. Learning recursive functions from approximations. Journal of Computer and System Sciences, 55:183--196, 1997.

....(m; n) admissible vector set V is not (h; k) admissible then one can, given some e , not always find some e 0 looks a bit artificial since it is principally unsolvable and seems to be repairable with some finite case distinction and knowledge about A. But Case, Kaufmann, Kinber and Kummer [13] showed that it is impossible to close the gap. They used for this negative result some notion from learning theory which might also be formalized as follows: there is no limiting recursive process which has access to A as an oracle and translates a program which (m; n) computes A into one which ....

....learning theory which might also be formalized as follows: there is no limiting recursive process which has access to A as an oracle and translates a program which (m; n) computes A into one which (h; k) computes A. Putting all these things together one obtains the following theorem. Theorem 7. 4 [13] The following statements are equivalent: a) Every (m; n) admissible vector set is (h; k) admissible. 24 Gasarch and Stephan (b) Every (m; n) admissible vector set V f0; 1g k is (h; k) admissible. c) For each e there is an e 0 such that every set which is (m; n) computable via e is ....

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J. Case, S. Kaufmann, E. Kinber and M. Kummer. Learning recursive functions from approximations. Journal of Computer and System Sciences, 55:183--196, 1997.

....recursive branches of recursive trees is of independent interest in recursion theory [7, 24] In [11] it is studied to which extend (in the sense of so called k selectors) infinite recursive branches of trees can be computed uniformly. This approach was combined with inductive inference in [5]. Here the learner receives input output examples of f and as additional information an index of a tree T such that f is a branch of T . 2 Notation and Definitions The natural numbers are denoted by . We identify sets A with their characteristic function. #A denotes the cardinality of A ....

J. Case, S. Kaufmann, E. Kinber, and M. Kummer. Learning recursive functions from approximations. In EuroCOLT'95, volume 904 of LNCS, pages 140--153. Springer-Verlag, 1995.

....a large and varied collection of domains [41, 89] In the practical context of robot planning, McDermott [60] says, Learning makes the most sense when it is thought of as filling in the details in an algorithm that is 1966 JOHN CASE already nearly right. In the context of function learning, [27] provides several models of learning from examples together with approximately correct programs. Included are models in which the maximal probability of learning all the computable functions is proportional to how tightly the approximately correct programs envelope the data. Unexplored, but very ....

J. Case, S. Kaufmann, E. Kinber, and M. Kummer, Learning recursive functions from approximations, J. Comput. System Sci., 55 (1997), pp. 183--196.

....Due to space limitations all proofs have been omitted, except, for illustration, a proof sketch of the upper bound of Theorem 13. The results of Sections 4, 5 originally appeared in [8] where additional material can be found. Very recently our results have been applied in inductive inference [4]. Notation and Definitions The notation is standard (see e.g. the textbooks [14, 16, 17] f0; 1; 2; g. i is the i th partial recursive function in the standard enumeration, W i is the i th r.e. set in the standard enumeration (W i = dom( i ) For a given r.e. set A, let fA s g ....

J. Case, S. Kaufmann, E. Kinber, M. Kummer. Learning recursive functions from approximations. In: Proceedings of EuroCOLT'95. LNCS 904, 140--153, SpringerVerlag, 1995.

....the upper bounds require refinements of the proofs of the original theorems. The lower bounds are shown by suitable diagonalizations. The results of Sections 4, 5 originally appeared in [8] where additional material can be found. Very recently our results have been applied in inductive inference [4]. Notation and Definitions The notation follows the books of Odifreddi and Soare [14, 17] f0; 1; 2; g. i is the i th partial recursive function in the standard enumeration, W i is the i th r.e. set in the standard enumeration (W i = dom( i ) For a given r.e. set A, let fA s g ....

J. Case, S. Kaufmann, E. Kinber, M. Kummer. Learning recursive functions from approximations. To appear in: Proceedings of EuroCOLT'95. Springer-Verlag, Berlin, 1995. S. Kaufmann and M. Kummer / Quantitative Notion of Uniformity 20

....implies that even our most powerful notion of master learning, namely, learning an arbitrary branch from several selected masters, is not strong enough to learn a branch for every tree: Corollary 10. For all m 1: Tree 62 SelectMa m . Theorem 9 can be proven using a team learning result from [4]. However, interestingly, the separation of SelectMa 2 and SelectMa 1 is witnessed by natural classes of trees, in particular, by the class TreeInf of all trees which contain infinitely many infinite computable branches. Since such separation results, which are witnessed by natural classes, ....

....which are witnessed by natural classes, are particularly interesting for inductive inference, we present here the proof of this result instead of proving the general statement from Theorem 9. Besides TreeInf , also the following natural classes witness SelectMa 2 6 SelectMa 1 (see, e.g. [4] for definitions) the binary trees which contain only computable infinite branches, the binary trees of bounded width, the binary trees of bounded rank, and the binary trees of bounded variation. Theorem 11. The class TreeInf of all trees, which contain infinitely many infinite ....

J. Case, S. Kaufmann, E. Kinber, and M. Kummer. Learning recursive functions from approximations. Journal of Computer and System Sciences, 55:183--196, 1997.

....and Degtev [8] until the early eighties. Recently an upsurge of interest occurred in connection with the investigation of bounded query computations in complexity theory [1, 2, 3, 6] and computability theory [4, 5, 9] Very recently frequency computations were also studied in inductive inference [7, 13, 17]. In the present paper we characterize the fundamental inclusion problem of frequency computation by a combinatorial property. This yields a short proof of McNicholl s new result [18] that the inclusion problem is decidable. Similar characterizations are provided for the restriction of the ....

J. Case, S. Kaufmann, E. Kinber, M. Kummer. Learning recursive functions from approximations. In Proceedings EuroCOLT'95, Lecture Notes in Computer Science 904, pp. 140--153, 1995. Springer-Verlag

....cases, increases the difficulty of learning, sometimes in interesting ways. It would be good to assuage the difficulty of learning from noisy data, in the future, by finding natural forms of innate knowledge or additional information (as, for example, was done for noise free function learning in [11]) 7 Appendix The Appendix contains the proofs for three theorems. Theorem 3.8 (a) NoisyInfFex n 2 6 NoisyInfFex n 1 . Proof: The noninclusion is witnessed by L = fL = fx 1 x 2 . g : jLj n 2 (9i n 2) L = W x i ]g: The following M NoisyInfOex n 2 infers L: On input oe, M ....

Case, J., Kaufmann, S., Kinber, E., and Kummer, M. (1995), Learning Recursive Functions From Approximations, in, "Proceedings of the European Conference on Computational Learning Theory (EuroCOLT)", Springer-Verlag, 140-153.

....trees. Infinite branches of recursive trees are of general interest in recursion theory [15] In [9] it is studied to which extend (in the sense of so called k selectors) infinite recursive branches of trees can be computed uniformly. This approach was combined with inductive inference in [4]. Here the learner receives input output examples of f and as additional information an index of a tree T such that f is a branch of T . In an abstract view T can be seen as a problem specification and the infinite branches of T as the solutions. We ask whether it is possible to learn solutions ....

J. Case, S. Kaufmann, E. Kinber, and M. Kummer. Learning recursive functions from approximations. In EuroCOLT'95, volume 904 of LNCS, pages 140--153. Springer-Verlag, 1995.

....TreeInf is a natural example witnessing that SelectMa 2 6 SelectMa 1 . For each m 1, actually, as in the case of arbitrary masters, one can show that watching m 1 selected masters is also strictly more powerful than watching m. In order to prove this we use a team learning result from [5]. Basically, a team of m machines is successful, if at least one of them is (and we may not know which one) 17 In [5] the identification of computable functions is investigated when a tree is given as additional input to the learner, which contains the function as an infinite computable branch. ....

....of arbitrary masters, one can show that watching m 1 selected masters is also strictly more powerful than watching m. In order to prove this we use a team learning result from [5] Basically, a team of m machines is successful, if at least one of them is (and we may not know which one) 17 In [5] the identification of computable functions is investigated when a tree is given as additional input to the learner, which contains the function as an infinite computable branch. 5] studied the team size which is necessary to identify all infinite computable branches of some naturally defined ....

[Article contains additional citation context not shown here]

J. Case, S. Kaufmann, E. Kinber, and M. Kummer. Learning recursive functions from approximations. Journal of Computer and System Sciences, 25:59--78, 1997.

....cases, increases the difficulty of learning, sometimes in interesting ways. It would be good to assuage the difficulty of learning from noisy data, in the future, by finding natural forms of innate knowledge or additional information (as, for example, was done for noise free function learning in [CKKK95]) Acknowledgements Frank Stephan was supported by the Deutsche Forschungsgemeinschaft (DFG) Grant No. Am 60 9 1. We are grateful to Tom Nordahl for his information on schizophrenia. ....

J. Case, S. Kaufmann, E. Kinber, and M. Kummer. Learning recursive functions from approximations. In Paul Vitanyi, editor, Computational Learning Theory, Second European Conference, EuroCOLT'95, Barcelona, Spain, pages 140--153. Springer-Verlag, March 1995. Lecture Notes in Artificial Intelligence 904.

....cases, increases the difficulty of learning, sometimes in interesting ways. It would be good to assuage the difficulty of learning from noisy data, in the future, by finding natural forms of innate knowledge or additional information (as, for example, was done for noise free function learning in [13]) ....

Case, J., Kaufmann, S., Kinber, E., and Kummer, M. (1995), Learning Recursive Functions From Approximations, in, "Proceedings of the European Conference on Computational Learning Theory (EuroCOLT)", Springer-Verlag, 140-153.

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John Case, Susanne Kaufmann, E m Kinber and Martin Kummer. Learning recursive functions from approximations. Journal of Computer and System Sciences, 55:183-196, 1997.

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