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12
On the Structure of Degrees of Inferability
 Journal of Computer and System Sciences
, 1993
"... Degrees of inferability have been introduced to measure the learning power of inductive inference machines which have access to an oracle. The classical concept of degrees of unsolvability measures the computing power of oracles. In this paper we determine the relationship between both notions. ..."
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Cited by 32 (19 self)
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Degrees of inferability have been introduced to measure the learning power of inductive inference machines which have access to an oracle. The classical concept of degrees of unsolvability measures the computing power of oracles. In this paper we determine the relationship between both notions. 1 Introduction We consider learning of classes of recursive functions within the framework of inductive inference [21]. A recent theme is the study of inductive inference machines with oracles ([8, 10, 11, 17, 24] and tangentially [12]; cf. [10] for a comprehensive introduction and a collection of all previous results.) The basic question is how the information content of the oracle (technically: its Turing degree) relates with its learning power (technically: its inference degreedepending on the underlying inference criterion). In this paper a definitive answer is obtained for the case of recursively enumerable oracles and the case when only finitely many queries to the oracle are allo...
Quantifying the Amount of Verboseness
, 1995
"... We study the fine structure of the classification of sets of natural numbers A according to the number of queries which are needed to compute the nfold characteristic function of A. A complete characterization is obtained, relating the question to finite combinatorics. In order to obtain an explic ..."
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Cited by 16 (6 self)
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We study the fine structure of the classification of sets of natural numbers A according to the number of queries which are needed to compute the nfold characteristic function of A. A complete characterization is obtained, relating the question to finite combinatorics. In order to obtain an explicit description we consider several interesting combinatorial problems. 1 Introduction In the theory of bounded queries, we measure the complexity of a function by the number of queries to an oracle which are needed to compute it. The field has developed in various directions, both in complexity theory and in recursion theory; see Gasarch [21] for a recent survey. One of the original concerns is the classification of sets A of natural numbers by their "query complexity," i.e., according to the number of oracle queries that are needed to compute the nfold characteristic function F A n = x 1 ; : : : ; x n : (ØA (x 1 ); : : : ; ØA (x n )). In [3, 8] a set A is called verbose iff F A n is com...
On the complexity of random strings (Extended Abstract)
 IN STACS 96
, 1996
"... We show that the set R of Kolmogorov random strings is truthtable complete. This improves the previously known Turing completeness of R and shows how the halting problem can be encoded into the distribution of random strings rather than using the time complexity of nonrandom strings. As an applic ..."
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Cited by 8 (1 self)
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We show that the set R of Kolmogorov random strings is truthtable complete. This improves the previously known Turing completeness of R and shows how the halting problem can be encoded into the distribution of random strings rather than using the time complexity of nonrandom strings. As an application we obtain that Post's simple set is truthtable complete in every Kolmogorov numbering. We also show that the truthtable completeness of R cannot be generalized to sizecomplexity with respect to arbitrary acceptable numberings. In addition we note that R is not frequency computable.
A Guided Tour of Minimal Indices and Shortest Descriptions
 Archives for Mathematical Logic
, 1997
"... The set of minimal indices of a G#del numbering ' is deøned as MIN' = fe : (8i ! e)[' i 6= 'e ]g. It has been known since 1972 that MIN' jT ; 00 , but beyond this MIN' has remained mostly uninvestigated. This thesis collects the scarce results on MIN' from the literature and adds some new observa ..."
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Cited by 8 (2 self)
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The set of minimal indices of a G#del numbering ' is deøned as MIN' = fe : (8i ! e)[' i 6= 'e ]g. It has been known since 1972 that MIN' jT ; 00 , but beyond this MIN' has remained mostly uninvestigated. This thesis collects the scarce results on MIN' from the literature and adds some new observations including that MIN' is autoreducible, but neither regressive nor (1; 2) computable. We also study several variants of MIN' that have been deøned in the literature like sizeminimal indices, shortest descriptions, and minimal indices of decision tables. Some challenging open problems are left for the adventurous reader. 1 Introduction How long is the shortest program that solves your problem? There are at least two ways to interpret this question depending on the type of problem involved. If the program's task is to output one speciøc object, we are looking for a shortest description of that object. This interpretation is closely related to Kolmogorov complexity. Although we have sev...
On the Structures Inside TruthTable Degrees
 Forschungsberichte Mathematische Logik 29 / 1997, Mathematisches Institut, Universitat
, 1997
"... . The following theorems on the structure inside nonrecursive truthtable degrees are established: Degtev's result that the number of bounded truthtable degrees inside a truthtable degree is at least two is improved by showing that this number is infinite. There are even infinite chains and anti ..."
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Cited by 4 (2 self)
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. The following theorems on the structure inside nonrecursive truthtable degrees are established: Degtev's result that the number of bounded truthtable degrees inside a truthtable degree is at least two is improved by showing that this number is infinite. There are even infinite chains and antichains of bounded truthtable degrees inside the truthtable degrees which implies an affirmative answer to a question of Jockusch whether every truthtable degree contains an infinite antichain of manyone degrees. Some but not all truthtable degrees have a least bounded truthtable 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 truthtable degrees) which consist only of 2subjective sets and do therefore not contain any objective set. Furthermore a truthtable degree consisting of three positive degrees is constructed where one positive degree consists of enum...
A Short History of Minimal Indices
, 1996
"... ing from concrete machine models the question translates into minimal indices with respect to a numbering of the computable, partial functions. The first part of the paper tells the history of this problem collecting the known results. The second part offers some new observations, and the last part ..."
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Cited by 2 (2 self)
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ing from concrete machine models the question translates into minimal indices with respect to a numbering of the computable, partial functions. The first part of the paper tells the history of this problem collecting the known results. The second part offers some new observations, and the last part concludes with a list of open problems. We will only consider Godel numberings. A Godel numbering is an effective numbering ' of all computable partial functions such that for every effective numbering / a 'index can be computed from a /index. We will also use Kolmogorov numberings. A Godel numbering is a Kolmogorov numbering, if there is a linearly bounded computable function that transforms /indices into 'indices. It is well known that Kolmogorov numberings exist. Definition 1.1 Let ' be a Godel numbering. Define MIN' := fe : (8i ! e)[' i 6= ' e ]g; the set of minimal indices of '. What would happen if instead of Godel numberings arbitrary numberings of the computable, partial fun...
A Structural Property of Regular Frequency Computations
 Theoretical Comput. Sci
, 2000
"... . 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 determinist ..."
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Cited by 2 (1 self)
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. 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 nonunary 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 ...
The Power of Frequency Computation (Extended Abstract)
 In: Proceedings FCT'95, Lecture Notes in Computer Science
, 1995
"... ) Martin Kummer and Frank Stephan ? Universitat Karlsruhe, Institut fur Logik, Komplexitat und Deduktionssysteme, D76128 Karlsruhe, Germany. fkummer; fstephang@ira.uka.de Abstract. The notion of frequency computation concerns approximative computations of n distinct parallel queries to a set A. A ..."
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Cited by 1 (1 self)
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) Martin Kummer and Frank Stephan ? Universitat Karlsruhe, Institut fur Logik, Komplexitat und Deduktionssysteme, D76128 Karlsruhe, Germany. fkummer; fstephang@ira.uka.de Abstract. The notion of frequency computation concerns approximative computations of n distinct parallel queries to a set A. A is called (m; n)recursive if there is an algorithm which answers any n distinct parallel queries to A such that at least m answers are correct. This paper gives natural combinatorial characterizations of the fundamental inclusion problem, namely the question for which choices of the parameters m; n; m 0 ; n 0 , every (m;n)recursive set is (m 0 ; n 0 )recursive. We also characterize the inclusion problem restricted to recursively enumerable sets and the inclusion problem for the polynomialtime bounded version of frequency computation. Furthermore, using these characterizations we obtain many explicit inclusions and noninclusions. 1 Introduction Frequency computation is a classic...
Max and Min Limiters
"... If A ⊆ ω, n ≥ 2, and the function max({x1,..., xn} ∩ A) is partial recursive, it is easily seen that A is recursive. In this paper, we weaken this hypothesis in various ways (and similarly for “min ” in place of “max”) and investigate what effect this has on the complexity of A. We discover a sharp ..."
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If A ⊆ ω, n ≥ 2, and the function max({x1,..., xn} ∩ A) is partial recursive, it is easily seen that A is recursive. In this paper, we weaken this hypothesis in various ways (and similarly for “min ” in place of “max”) and investigate what effect this has on the complexity of A. We discover a sharp contrast between retraceable and coretraceable sets, and we characterize sets which are the union of a recursive set and a cor.e., retraceable set. Most of our proofs are noneffective. Several open questions are raised. 2 1