Lenny Adleman and Manuel Blum. Inductive inference and unsolvability. Journal of Symbolic Logic, 56:891-900, 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....computable learner for this collection. There exist classes which cannot be learned by a computable learner. For example the class REC of all computable functions cannot be learned by a computable machine which receives as input the sequence f(0)f(1) of the values of f . Adleman and Blum [1] as well as Gasarch and Pleszkoch [9] considered nonrecursive learners and measured their complexity in terms of Turing degrees. Adleman and Blum showed, for example, that a learner for REC exists in a Turing degree a if and only if a is high (a ) A learner which can learn every object in ....

....learners and measured their complexity in terms of Turing degrees. Adleman and Blum showed, for example, that a learner for REC exists in a Turing degree a if and only if a is high (a ) A learner which can learn every object in the target concept class is called omniscient. Adleman and Blum [1] constructed nonrecursive omniscient learners for function learning. But such omniscient learners do not exist for the model of learning from text: The class flNg [ fD lN : D is niteg cannot be learned from text relative to any oracle [10] Let denote the collection of all classes L of ....

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Lenny Adleman and Manuel Blum. Inductive inference and unsolvability. Journal of Symbolic Logic, 56:891-900, 1991.

....such that (Y [ D) U = and (Y [ D [ U) X. Note that D is not empty i (Y [ U) X. In the following theorem, we consider the class L of recursively enumerable submatroids of a given matroid. Furthermore, general learners which do not have to be computable are considered. Adleman and Blum [1] proposed to measure the complexity of such learners in terms of their Turing degrees. Together with Theorem 2.2, one has that, whenever a general SwEx learner exists, this learner can be chosen so that it is computable relative to the halting problem K. Theorem 5.4. Given a matroid (X; let L ....

Lenny Adleman and Manuel Blum. Inductive inference and unsolvability. The Journal of Symbolic Logic, 56(3):891-900, 1991.

....7.12 If A is a set of natural numbers, then G(A) means that one of the following holds. 1. A is recursive. 2. A T K, and A is in a 1 generic degree. In Table 7.1, U 1 ; U 5 are open questions. U 4 and U 5 may depend on the parameters a; b; n. The results for EX being omniscient are in [AB91, KS96] The results for EX being trivial are in [SS91, KS96] All the rest are in [FJG 94] We know more than what is in the table [FJG 94] 1. If A T K or A is in a hyperimmune free degree then U 1 (A) holds. 2. There are low sets A such that U 2 (A) and U 5 (A) hold. 3. For all sets ....

Lenny Adleman and Manuel Blum. Inductive inference and unsolvability. Journal of Symbolic Logic, 56(3):891--900, September 1991.

....(as e.g. EX , BC , or Team inference) This extends (and partially corrects) previous results 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 0 T A 0 ) We extend this result by showing that it already requires a high oracle to EX infer all Institut fur Logik, Komplexitat und ....

....and Solovay [24] We shall later present a simplified proof. If infinitely many queries are allowed then the omniscient degrees are no longer independent of the inference criterion. All of the following results, except the first, are from [10] ffl REC 2 EX[A] iff A is high (Adleman and Blum [1]) ffl REC 2 [a; b]EX[A] iff A is high. ffl There is a low set A with REC 2 BC[A] ffl For A r.e. REC 2 BC[A] iff A is high. Fact: There is a low set A with REC 2 LIMEX[A] Proof: By the Low Basis Theorem (see [27, VI.5.13, XII.3.9] there is a low set A 2 such that for all i; x: i ....

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L. Adleman, M. Blum. Inductive inference and unsolvability. Journal of Symbolic Logic, 56:891--900, 1991.

....We write AM(i 1 ; i n ) instead of the (formally correct) AM(fi 1 ; i n g) 3 Technical Summary We examine when a degree of inferability can be trivial, and when it can be omniscient. We then extend these questions to other notions of inferability. This paper, together with [AB91, JS, KS93b] describes all that is known for these questions. All results listed are in this paper unless otherwise noted. A more comprehensive summary is in Section 7. Notation 3.1 G(A) stands for the condition that either A is recursive, or A T K and is in a 1 generic Turing degree. 1) When ....

....of the proof of a similar result for EX which is in [SS91] or directly in [KS93b] This supersedes item a) but a) has an easy proof. c) BC[A] BC iff A T K. d) 8m) BC[A[m] BC iff A T K] 3) When are EX degrees (and variations) omniscient a) REC 2 EX[A] iff A is high (proven in [AB91] Also see [KS93b] b) REC 2 EX[A] iff ; 00 T A Phi K. c) 8m) 8A) REC = 2 EX[A[m] 4) When are BC degrees (and variations) omniscient a) If A is r.e. then REC 2 BC[A] iff A is high. b) There exists a low set A such that REC 2 BC[A] c) For all X there exists A such that X T A 00 ....

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Lenny Adleman and Manuel Blum. Inductive inference and unsolvability. Journal of Symbolic Logic, 56(3):891--900, September 1991.

....2.1 [15] A has trivial EX degree iff either A is recursive or A has the same Turing degree as a 1 generic set B T K. 3 Obviously if an IIM EX[A] infers the whole class REC then A belongs to the greatest inference degree. This degree is called the omniscient inference degree. Adleman and Blum [1] showed that for the criterion EX there exists an omniscient inference degree and that it has the following easy characterization: Fact 2.2 [1] A has omniscient EX degree iff A is high. A query language is a language that has the usual logical symbols (equality, constants, variables, quantifiers ....

....EX[A] infers the whole class REC then A belongs to the greatest inference degree. This degree is called the omniscient inference degree. Adleman and Blum [1] showed that for the criterion EX there exists an omniscient inference degree and that it has the following easy characterization: Fact 2. 2 [1] A has omniscient EX degree iff A is high. A query language is a language that has the usual logical symbols (equality, constants, variables, quantifiers and logical operations to link the atom) and a special symbol f . Query languages may contain further symbols, in particular Succ, and ....

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L. Adleman, M. Blum. Inductive inference and unsolvability. Journal of Symbolic Logic, 56:891--900, 1991.

....hard as the halting problem. This insight deserves some attention. Gold (1967) showed that no IIM can learn the class R of all recursive functions in the limit. On the other hand, the degree of the algorithmic unsolvability of R 2 LIM is strictly less than the degree of the halting problem (cf. Adleman and Blum (1991)) This puts the constraint to learn monotonically with respect to a particular hypothesis space into a new perspective. An algorithmically solvable learning problem (e.g. L csd 2 CSMON d 0TXT ) may become algorithmically unsolvable, if an at first glance natural demand is added (e.g. to learn ....

Adleman, L.M., and Blum, M. (1991), Inductive inference and unsolvability, Journal of Symbolic Logic 56, 891 -- 900.

....theorem shows the problem to finitely multi classify two sets that are themselves finitely classifiable might become at least as hard as the halting problem. This nicely contrasts the result that the degree of unsolvability for R 2 EX is strictly less than the degree of the halting problem (cf. Adleman and Blum (1991)) Theorem 12. There are nonempty sets S 0 ; S 1 R such that (1) S 0 ; S 1 ) 2 FCL n Multi FCL, 2) The degree of (S 0 ; S 1 ) 2 Multi FCL is equivalent to the degree of the halting problem. Proof. Let S 0 = ff f 2 R; f(0) 2 Kg, and let S 1 = ff f 2 R; f(1) 2 Kg. By definition, for any f 2 ....

Adleman, L., and Blum, M. (1991), Inductive inference and unsolvability, Journal of Symbolic Logic 56, 891 - 900.

....to the ability to solve learning problems of the type under consideration. A notational remark: For an identification type ID, ID[A] contains all sets of total recursive functions, which can be learned w.r.t. the identification criterion ID by a strategy equipped with an oracle A. Proposition 8 ([AB91]) For any oracle A, I R l.r.e. A] R2LIM[A] So for LIM, a validator, which claims to validate all learning strategies must be able to solve the learning task for arbitrary recursive functions, and vice versa. Theorem 9 For any oracle A, LIM2EVAL[A] R2LIM[A] The techniques used for ....

L. M. Adleman and M. Blum. Inductive inference and unsolvability. The Journal of Symbolic Logic, 56(3):891--900, Sept. 1991.

....that e is an infinite recursive branch of T . A class of trees C is EX [A] branch learnable (C 2 BranchEx [A] if there is a machine M A which EX [A] branch learns every T 2 C. The following results can be obtained by modifying the proofs from the corresponding results in inductive inference [1, 9, 14]. As in [9] we write G(A) if A T G T K for some 1 generic set G, i.e. if A is either recursive or has the same degree as a 1 generic Turing degree below K. Fact 6.2. 1. A T K = BranchFin[A] ae BranchEx . 2. For all A: BranchEx 1 6 BranchFin[A] where BranchEx 1 means EX branch learnable ....

L. Adleman and M. Blum. Inductive inference and unsolvability. Journal of Symbolic Logic, 56(3):891--900, 1991.

....Fact 2.1 [15] A has trivial EX degree iff either A is recursive or A has the Turing degree of a 1 generic set B T K. Obviously if an IIM EX[A] infers the whole class REC then A belongs to the greatest inference degree. This degree is called the omniscient inference degree. Adleman and Blum [1] showed that for the criterion EX there exists an omniscient inference degree and that it has the following easy characterization: Fact 2.2 [1] A has omniscient EX degree iff A is high. A query language is a language that has the usual logical symbols (equality, constants, variables, quantifiers ....

....EX[A] infers the whole class REC then A belongs to the greatest inference degree. This degree is called the omniscient inference degree. Adleman and Blum [1] showed that for the criterion EX there exists an omniscient inference degree and that it has the following easy characterization: Fact 2. 2 [1] A has omniscient EX degree iff A is high. A query language is a language that has the usual logical symbols (equality, constants, variables, quantifiers and logical operations to link the atom) and a special symbol f . Query languages may contain further symbols, in particular Succ, and ....

L. Adleman, M. Blum. Inductive inference and unsolvability. Journal of Symbolic Logic, 56:891--900, 1991.

....because the identification types are substantially different. They form a proper hierarchy as mentioned above. Experts have to exploit their skills for systems validation in different ways, respectively. Limiting recursive decidability is a concept adopted from computability theory (cf. (Adleman Blum 1991), e.g. that describes an expert s opinion which might change over time, but finally leads to the right decision. An even weaker variant is called limiting recursive enumerability. For determining whether or not some program computes a total recursive function, both concepts coincide (cf. ....

Adleman, L.M., and Blum, M. 1991. Inductive inference and unsolvability. The Journal of Symbolic Logic 56(3):891--900.

....the claim, and the theorem follows. Proposition 23 characterizes validation expertise. Next, we attack the general problem to relate validation expertise to domain expertise, i.e. the ability to solve learning problems in the required sense. The following result is due to Adlemann and Blum (cf. [1]) Proposition24. Let A be any oracle. Then, I R l.r.e. A ( R2LIM] A . Putting the last two results together, we arrive at the following insight. Theorem25. Let A be any oracle. Then, LIM2EVAL] A ( R2LIM] A . Consequently, an expert who has the ability to LIM evaluate all ....

Adleman, L. M. and Blum, M. (1991) Inductive inference and unsolvability. The Journal of Symbolic Logic, 56:891--900.

....the derived results on validating inductive inference devices. In all the results listed, the upper index A which occurs in terms like [ A denotes some expertise which, in formal terms, means relativized computability with respect to the power of deciding membership for the set A (cf. AB91] Klaus P. Jantke Domain Expertise vs. Validation Expertise: Essentials : 13 Towards an Interpretation: The necessary expertise for validation is so strong, that it is crushing the differences of originally distinguished learning problems. In other words, those being qualified to validate ....

L. M. Adleman and M. Blum. Inductive inference and unsolvability. The Journal of Symbolic Logic, 56:891--900, 1991.

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Adleman, L.M., and Blum, M. (1991), Inductive inference and unsolvability, Journal of Symbolic Logic 56, 891 - 900.

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Adleman, L.M., and Blum,M. (1991), Inductive inference and unsolvability, Journal of Symbolic Logic 56, 891 - 900.

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