Investigated is algorithmic learning, in the limit, of correct programs for recursive functions f from both input/output examples of f and several interesting varieties of approximate additional (algorithmic) information about f . Specifically considered, as such approximate additional information about f , are Rose's frequency computations for f and several natural generalizations from the literature, each generalization involving programs for restricted trees of recursive functions which have f as a branch. Considered as the types of trees are those with bounded variation, bounded width, and bounded rank. For the case of learning final correct programs for recursive functions, EX- learning, where the additional information involves frequency computations, an insightful and interestingly complex combinatorial characterization of learning power is presented as a function of the frequency parameters. For EX- learning (as well as for BC-learning, where a final sequence of cor...

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ABSTRACT. We continue the study of effective Hausdorff dimension as it was initiated by LUTZ. Whereas he uses a generalization of martingales on the Cantor space to introduce this notion we give a characterization in terms of effective s-dimensional Hausdorff measures, similar to the effectivization of Lebesgue measure by MARTIN-LÖF. It turns out that effective Hausdorff dimension allows to classify sequences according to their ‘degree ’ of algorithmic randomness, i.e., their algorithmic density of information. Earlier the works of STAIGER and RYABKO showed a deep connection between Kolmogorov complexity and Hausdorff dimension. We further develop this relationship and use it to give effective versions of some important properties of (classical) Hausdorff dimension. Finally, we determine the effective dimension of some objects arising in the context of computability theory, such as degrees and spans. 1.

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. The following theorems on the structure inside nonrecursive truthtable degrees are established: Degtev's result that the number of bounded truth-table degrees inside a truth-table degree is at least two is improved by showing that this number is infinite. There are even infinite chains and antichains of bounded truth-table degrees inside the truth-table degrees which implies an affirmative answer to a question of Jockusch whether every truthtable degree contains an infinite antichain of many-one degrees. Some but not all truth-table degrees have a least bounded truth-table degree. The technique to construct such a degree is used to solve an open problem of Beigel, Gasarch and Owings: there are Turing degrees (constructed as hyperimmunefree truth-table degrees) which consist only of 2-subjective sets and do therefore not contain any objective set. Furthermore a truth-table degree consisting of three positive degrees is constructed where one positive degree consists of enum...

y1) f(xk , yk )) communicating as few bits as possible. The Direct Sum Conjecture (henceforth DSC) of Karchmer, Raz, and Wigderson, states that the obvious way to compute it (computing f(x1 , y1 ), then f(x2 , y2 ), etc.) is, roughly speaking, the best. This conjecture arose in the study of circuits since a variant of it implies NC 1 #= NC 2 . We consider two related problems. Enumeration: Alice and Bob output e # 2 k - 1 elements of {0, 1} k , one of which is f(x1 , y1) f(xk , yk ). Elimination: Alice and Bob output # b such that # b #= f(x1 , y1 ) f(xk , yk ). Selection: (k = 2) Al

. The notion of an (m; n){computation was already introduced in 1960 by Rose and further investigated by Trakhtenbrot in 1963. It has been extended to nite automata by Kinber in 1976 and he has shown an analogue of Trakhtenbrot's result: The class of languages (m; n)- recognizable by deterministic nite automata is equal to the class of regular languages if and only if 2m > n. Furthermore, for a unary alphabet, the class of (m; n)-recognizable languages coincides with the class of regular languages for all m and n. In this paper, we will present the rst structural property of (m; n)- recognizable languages which is valid for all 1 m n and for all alphabets. Kinber's result for unary alphabets becomes a corollary. This property is also used to show that certain non-unary languages are not (m; n)-regular and that the class of all (m; n)-recognizable languages is not closed under the reversal operation. However, this class forms a Boolean algebra. 1 Introduction The ...

We study whether sets inside NP can be reduced to sets with low information content but possibly still high computational complexity. Examples of sets with low information content are tally sets, sparse sets, P-selective sets and membership comparable sets. For the graph automorphism...

Linial et al. raised the question of how diffcult the computation of the Vapnik-Cervonenkis dimension of a concept class over a finite universe is. Papadimitriou and Yannakakis obtained a first answer using matrix representations of concept classes. However, this approach does not capture classes having exponential size, like monomials, which are encountered in learning theory. We choose a more natural representation, which leads us to redefine the vc dimension problem. We establish that vc dimension is Σ^p_3-complete, thereby giving a rare natural example of a Σ^p_3-complete problem.

Abstract. A partial information algorithm for a language A computes, for some fixed m, for input words x1,..., xm a set of bitstrings containing χA(x1,..., xm). E.g., p-selective, approximable, and easily countable languages are defined by the existence of polynomial-time partial information algorithms of specific type. Self-reducible languages, for different types of self-reductions, form subclasses of PSPACE. For a self-reducible language A, the existence of a partial information algorithm sometimes helps to place A into some subclass of PSPACE. The most prominent known result in this respect is: P-selective languages which are self-reducible are in P [9]. Closely related is the fact that the existence of a partial information algorithm for A simplifies the type of reductions or self-reductions to A. The most prominent known result in this respect is: Turing reductions to easily countable languages simplify to truth-table reductions [8]. We prove new results of this type. We show: 1. Self-reducible languages which are easily 2-countable are in P. This partially confirms a conjecture of [8]. 2. Self-reducible languages which are (2m − 1, m)-verbose are truthtable self-reducible. This generalizes the result of [9] for p-selective languages, which are (m + 1, m)-verbose. 3. Self-reducible languages, where the language and its complement are strongly 2-membership comparable, are in P. This generalizes the corresponding result for p-selective languages of [9]. 4. Disjunctively truth-table self-reducible languages which are 2-membership comparable are in UP. Topic: Structural complexity 1