Characterization problems in the theory of inductive inference

Wiehagen, Rolf

doi:10.1007/3-540-08860-1_37

Cited by 32 publications

(20 citation statements)

References 7 publications

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“…Theorem 2 is a consequence of Theorem 9 from Wiehagen (1978). Note that for an arbitrary Gödel numbering, U -consistency is undecidable for any non-empty class U ⊆ R.…”

Section: The General Inconsistency Phenomenonmentioning

confidence: 95%

“…However, an intuitively satisfying answer is provided by a characterization of identification in the limit in terms of computable numberings (cf. Wiehagen (1978), Theorem 8). This theorem actually states that a class U of recursive functions can be learned in the limit iff there are a space of hypotheses containing for each function at least one program, and a computable "discrimination" function d such that for any two programs i and j the value d(i, j) is an upper bound for an argument x on which program i behaves differently than program j does.…”

Section: Introductionmentioning

confidence: 98%

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Learning and consistency

Wiehagen

Zeugmann

1995

Algorithmic Learning for Knowledge-Based Systems

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In designing learning algorithms it seems quite reasonable to construct them in such a way that all data the algorithm already has obtained are correctly and completely reflected in the hypothesis the algorithm outputs on these data. However, this approach may totally fail. It may lead to the unsolvability of the learning problem, or it may exclude any efficient solution of it. Therefore we study several types of consistent learning in recursion-theoretic inductive inference. We show that these types are not of universal power. We give "lower bounds" on this power. We characterize these types by some versions of decidability of consistency with respect to suitable "non-standard" spaces of hypotheses. Then we investigate the problem of learning consistently in polynomial time. In particular, we present a natural learning problem and prove that it can be solved in polynomial time if and only if the algorithm is allowed to work inconsistently.

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“…Theorem 2 is a consequence of Theorem 9 from Wiehagen (1978). Note that for an arbitrary Gödel numbering, U -consistency is undecidable for any non-empty class U ⊆ R.…”

Section: The General Inconsistency Phenomenonmentioning

confidence: 95%

Section: Introductionmentioning

confidence: 98%

Learning and consistency

Wiehagen

Zeugmann

1995

Algorithmic Learning for Knowledge-Based Systems

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“…There are connections between numberings and Gold-style learning theory [3,4,7]. In particular, Wiehagen and Jung [18,19] gave characterizations of the basic notions of learning theory in terms of numberings. Kummer [8] characterized in terms of numberings those enumerable classes of total functions which are co-learnable.…”

Section: Connections To Learning Theorymentioning

confidence: 99%

Topological aspects of numberings

Menzel¹,

Stephan²

2003

Mathematical Logic Qtrly

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We investigate connections between the syntactic and semantic distance of programs on an abstract, recursion theoretic level. For a certain rather restrictive notion of interdependency of the two kinds of distances, there remain only few and “unnatural” numberings allowing such close relationship. Weakening the requirements leads to the discovery of universal metrics such that for an arbitrary recursively enumerable family of functions a numbering compatible with such a metric can uniformly be constructed. We conclude our considerations with some implications on learning theory.

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“…Moreover, the RefEx-characterization of Theorem 12 is incremental to a characterization of Ex, since the existence of a numbering with condition (1) above is necessary and sufficient for Ex-learning the class C (cf. [36]). Finally, the refutation space could be "economized" in the same manner as the learning space by making it one-to-one.…”

Section: Definition 17 ([32])mentioning

confidence: 99%

Learning Recursive Functions Refutably

Jain

Kinber²,

Wiehagen

et al. 2001

Lecture Notes in Computer Science

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Abstract. Learning of recursive functions refutably means that for every recursive function, the learning machine has either to learn this function or to refute it, i.e., to signal that it is not able to learn it. Three modi of making precise the notion of refuting are considered. We show that the corresponding types of learning refutably are of strictly increasing power, where already the most stringent of them turns out to be of remarkable topological and algorithmical richness. All these types are closed under union, though in different strengths. Also, these types are shown to be different with respect to their intrinsic complexity; two of them do not contain function classes that are "most difficult" to learn, while the third one does. Moreover, we present characterizations for these types of learning refutably. Some of these characterizations make clear where the refuting ability of the corresponding learning machines comes from and how it can be realized, in general. For learning with anomalies refutably, we show that several results from standard learning without refutation stand refutably. Then we derive hierarchies for refutable learning. Finally, we show that stricter refutability constraints cannot be traded for more liberal learning criteria.

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Characterization problems in the theory of inductive inference

Cited by 32 publications

References 7 publications

Learning and consistency

Learning and consistency

Topological aspects of numberings

Learning Recursive Functions Refutably

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