## On the Structure of Degrees of Inferability (1993)

Venue: | Journal of Computer and System Sciences |

Citations: | 32 - 19 self |

### BibTeX

@INPROCEEDINGS{Kummer93onthe,

author = {Martin Kummer and Frank Stephan},

title = {On the Structure of Degrees of Inferability},

booktitle = {Journal of Computer and System Sciences},

year = {1993},

pages = {117--126}

}

### Years of Citing Articles

### OpenURL

### Abstract

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 degree---depending 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...

### Citations

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Language identification in the limit
- Gold
- 1967
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Citation Context ...r EX-degree and this degree is called the omniscient EX-degree. ; defines the trivial EX-degree. In the same way it is possible to define BC-degrees, EX -degrees and BC - degrees, etc. Definition 2.4 =-=[13]-=- E denotes the class of all r.e. subsets of !, a set L 2 E is called a language. A text for L is any mapping T from ! to ! [ f?g such that L is the set of natural numbers in the range of T . T is call... |

484 |
Recursively enumerable sets and degrees
- Soare
- 1987
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Citation Context ...A. Turing reducibility is denoted bysT . If A is a set then A 0 is the halting problem relative to A, that is fe j ' A e (e) #g. A is high if K 0sT A 0 . (This differs slightly from the definition in =-=[27]-=- since we do not require that AsK.) A is low if A 0sT K. A total function f is increasing iff f(x)sf(x + 1) for all x. A total function g is said to dominate a set S of total functions iff for every f... |

265 |
Toward a mathematical theory of inductive inference
- Blum, Blum
- 1975
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Citation Context ...EX denote the set of all S ` REC such that S is EX-inferred by a reliable total inference machine. REX[A] is defined as usual. If A is high then REC 2 REX[A]. By a well-known result of Minicozzi (see =-=[5]-=-), REX[A] is closed under union. Since S 1 2 REX it follows that: S 0 2 REX[A] ) S 0 [ S 1 2 REX[A]. Hence we get from Lemma 8.5: Corollary 8.6 EX ` REX[A] ) A is high. An inference machine M consiste... |

163 |
Comparison of identification criteria for machine inductive inference, Theoretical Computer Science
- Case, Smith
- 1983
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Citation Context ...each r.e. set W ` f0; 1g there is a string oesA such that either oe 2 W ors= 2 W for allsoe. Degrees of Inferability 3 Remaining recursion theoretic notation is from Soare's book [27]. Definition 2.2 =-=[7]-=- An inductive inference machine (IIM) M is a total Turing machine that is trying to learn recursive functions f from their initial segments oesf . M BCinfers f , if for almost all oesf , M(oe) is an i... |

121 |
Systems that Learn
- Jain, Osherson, et al.
- 1999
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Citation Context ...g 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 result... |

74 | Degrees of unsolvability - Lerman - 1983 |

56 |
The power of pluralism for automatic program synthesis
- Smith
- 1982
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Citation Context ... Let S ` REC. Then S 2 EX (BC) if there exists an IIM M such that for all f 2 S, M EX-infers f (BC-infers f ). Let a; bs1 be such that asb. A set of recursive functions S is in [a; b]EX (concept from =-=[26]-=-, notation from [22]) if there exist b IIMs M 1 , M 2 ,: : :, M b such that for every f 2 S, there are a machines M i 1 ; M i 2 ; : : : ; M i a , which all EX-infer f , with 1si 1 ! \Delta \Delta \Del... |

43 |
Inductive Inference and Unsolvability
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- 1991
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Citation Context ...ults from [8] in an unexpected way. The technique can also be used to give a simplified proof of a result of Slaman and Solovay that characterizes the trivial inference degrees [24]. Adleman and Blum =-=[1]-=- characterized the oracles A that allow one to EX-infer all recursive functions. This is the case iff A has high information content (i.e., K 0sT A 0 ). We extend this result by showing that it alread... |

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- 1987
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Citation Context ... 1-generic sets below K which join up to K; they exist by [25, Theorem 2.1].) In all of our proofs for EX[A] 6` EX[B] this noninclusion was witnessed by a set S 2 EX[A] \Gamma EX[B] which belonged to =-=[1; 2]-=-EX. Is this true for all noninclusions? Another intriguing question is whether the EX-degrees and the Team-EX degrees coincide, i.e., whether EX[A] 6` EX[B] implies that [1; n]EX[A] 6` [1; n]EX[B]. Fo... |

40 |
Probability and plurality for aggregations of learning machines
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- 1988
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Citation Context ... 2 EX (BC) if there exists an IIM M such that for all f 2 S, M EX-infers f (BC-infers f ). Let a; bs1 be such that asb. A set of recursive functions S is in [a; b]EX (concept from [26], notation from =-=[22]-=-) if there exist b IIMs M 1 , M 2 ,: : :, M b such that for every f 2 S, there are a machines M i 1 ; M i 2 ; : : : ; M i a , which all EX-infer f , with 1si 1 ! \Delta \Delta \Delta ! i asb. If a = 1... |

35 | Learning via queries
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Citation Context ...ning 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... |

29 | Terse, superterse, and verbose sets
- Beigel, Gasarch, et al.
- 1993
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Citation Context ...[A]. The following theorem summarizes the results of the first part of this section: Theorem 8.4 If A has not omniscient EX-degree, i.e., if A is not high, then: EX[A] = [1; 1]EX[A] ae [1; 2]EX[A] ae =-=[1; 3]-=-EX[A] ae : : :, EX[A] ae EX 1 [A] ae EX 2 [A] ae : : : ae EX [A] ae BC[A]. If A has not omniscient EX -degree, i.e., if A \Phi K 6 T K 0 , then: EX[A] = [1; 1]EX[A] ae [1; 2]EX[A] ae [1; 3]EX[A] ae : ... |

27 | Degrees of generic sets - Jockusch - 1981 |

25 | Degrees joining to 0 - Posner, Robinson - 1981 |

22 |
Hierarchies of sets and degrees below 0
- Epstein, Haas, et al.
- 1981
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Citation Context ...s : g(x; s) 6= g(x; s + 1)gjsx for all x. Let !-REC denote the set of all !-r.e. functions. A set A is called !-r.e. iff A 2 !-REC. A Turing degree is called !-r.e. iff it contains an !-r.e. set (see =-=[9]-=- for more information). The following result is the analog of Theorem 9.4 for InfEx degrees. We just need to replace REC by !-REC in the definition of RA . Theorem 11.10 Let AsT K and let c A (x) = mi... |

19 |
Learning via queries to an oracle
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- 1989
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Citation Context ...not infer ~ S n (j) which contradicts the hypothesis. Inference with finitely many queries to A is equivalent to inference with finitely many queries to A \Phi K (a relativization of EX[K ] = EX, cf. =-=[11]-=-, [10, Note 4.10, Lemma 5.6]). Hence for a characterization of the -degrees it suffices to consider oracles A with KsT A. Theorem 5.5 The following statements are equivalent for all sets A; BsT K and ... |

19 | Definability in the Turing degrees - Slaman, Woodin - 1986 |

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Degrees of inferability
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- 1992
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The degrees below a 1-generic degree < 0
- Haught
- 1986
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Citation Context ...red). They use a rather complicated finite injury argument to show that if AsT KsEX[A] = EX then there is a 1-generic set GsT K such that AsT G. Then they conclude that A 2 G using a result of Haught =-=[14]-=- who proved that the nonrecursive predecessors of a 1-generic degree ! 0 0 are all 1-generic. Haught's proof is also a rather complicated finite injury argument. In this section we provide a simplifie... |

2 |
Extremes in the degrees of inferability. To appear in: Annals of Pure and Applied Logic. Degrees of Inferability 51
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