## The intrinsic complexity of language identification (1996)

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Venue: | Journal of Computer and System Sciences |

Citations: | 17 - 7 self |

### BibTeX

@ARTICLE{Jain96theintrinsic,

author = {Sanjay Jain and Arun Sharma},

title = {The intrinsic complexity of language identification},

journal = {Journal of Computer and System Sciences},

year = {1996},

volume = {52},

pages = {278--286}

}

### Years of Citing Articles

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### Abstract

A new investigation of the complexity of language identification is undertaken using the notion of reduction from recursion theory and complexity theory. The approach, referred to as the intrinsic complexity of language identification, employs notions of ‘weak ’ and ‘strong ’ reduction between learnable classes of languages. The intrinsic complexity of several classes is considered and the results agree with the intuitive difficulty of learning these classes. Several complete classes are shown for both the reductions and it is also established that the weak and strong reductions are distinct. An interesting result is that the self referential class of Wiehagen in which the minimal element of every language is a grammar for the language and the class of pattern languages introduced by Angluin are equivalent in the strong sense. This study has been influenced by a similar treatment of function identification by Freivalds, Kinber, and Smith. 1

### Citations

4099 | Introduction to Automata Theory, Languages and Computation, 2nd ed - Hopcroft, Motwani, et al. - 2000 |

941 |
Language identification in the limit
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- 1967
(Show Context)
Citation Context ...e text, outputs an infinite sequence of grammars. Several criteria for the learning machine to be successful on a 2stext have been proposed. In the present paper we will concern ourselves with Gold’s =-=[11]-=- criterion of identification in the limit (referred to as TxtEx-identification). A sequence of grammars, G = g0, g1, . . ., is said to converge to g just in case, for all but finitely many n, gn = g. ... |

874 |
Theory of Recursive Functions and Effective Computability
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(Show Context)
Citation Context ...ies from language learning theory. In Section 3, we introduce our reducibilities. Results are presented in Section 4. 2 Notation and Preliminaries Any unexplained recursion theoretic notation is from =-=[18]-=-. The symbol N denotes the set of natural numbers, {0, 1, 2, 3, . . .}. Unless otherwise specified, e, g, i, j, k, l, m, n, q, r, s, t, w, x, y, with or without decorations 1 , range over N. Symbols ∅... |

304 |
Inductive inference of formal languages from positive data
- Angluin
- 1980
(Show Context)
Citation Context ...et code denote a 1-1 onto mapping from strings in C ∗ to N. The language associated with the pattern w is defined as L(w) = {code(f(w)) | f ∈ PatMap}. Then, PATTERN = {L(w) | w is a pattern}. Angluin =-=[2]-=- showed that PATTERN ∈ TxtEx. Our first result about PATTERN is that it is not ≤ TxtEx weak -complete. Corollary 2 FIN �≤ TxtEx weak PATTERN. The above Corollary follows directly from Theorem 3, since... |

224 |
Systems that learn, An introduction to learning theory
- Osherson, Stob, et al.
- 1986
(Show Context)
Citation Context ...mplete with respect to strong reduction. Suppose M0, M1, . . . is an enumeration of the learning machines such that, (∀L ∈ TxtEx)(∃i)[L ⊆ TxtEx(Mi)] (there exists such an enumeration, see for example =-=[16]-=-). For j ∈ N and L ∈ E, let S j L = {〈x, j〉 | x ∈ L}. Then, let LTxtEx = {S j L | L ∈ E ∧ j ∈ N ∧ L ∈ TxtEx(Mj)}. It is easy to see that LTxtEx ∈ TxtEx. Theorem 15 LTxtEx is ≤ TxtEx strong Proof. Let ... |

221 |
Finding patterns common to a set of strings
- Angluin
- 1980
(Show Context)
Citation Context ...merable language. We show that this class is not complete and is in fact equivalent to COINIT under the strong reduction. The second class is the collection of pattern languages introduced by Angluin =-=[1]-=-. Pattern languages have been studied extensively in the computational learning theory literature since their introduction as a nontrivial class of languages that could be learned in the limit from on... |

103 |
An introduction to the general theory of algorithms
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(Show Context)
Citation Context ...n be 1 Decorations are subscripts, superscripts and the like. 4sextended to n-tuples in a natural way. By ϕ we denote a fixed acceptable programming system for the partial computable functions: N → N =-=[18, 15]-=-. By ϕi we denote the partial computable function computed by the program with number i in the ϕ-system. The letter, p, in some contexts, with or without decorations, ranges over programs; in other co... |

92 |
Machine inductive inference and language identification
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(Show Context)
Citation Context ... L ∈ TxtEx(M)) just in case M TxtEx-identifies each text for L. (c) TxtEx = {L ⊆ E | (∃M)[L ⊆ TxtEx(M)]}. Other criteria of success are finite identification [11], behaviorally correct identification =-=[8, 17, 7]-=-, and vacillatory identification [17, 5]. In the present paper, we only discuss results about TxtEx-identification. 3 Weak and Strong Reductions We first present some technical machinery. We write “σ ... |

76 |
A machine independent theory of the complexity of recursive functions
- Blum
(Show Context)
Citation Context ...ith or without decorations, ranges over programs; in other contexts p ranges over total functions with its range being construed as programs. By Φ we denote an arbitrary fixed Blum complexity measure =-=[3, 12]-=- for the ϕ-system. By Wi we denote domain(ϕi). Wi is, then, the r.e. set/language (⊆ N) accepted (or equivalently, generated) by the ϕ-program i. We also say that i is a grammar for Wi. Symbol E will ... |

51 |
Periodicity in generations of automata
- Case
- 1974
(Show Context)
Citation Context ...N, g ′ n = min({n} ∪ Wgn,n). It is easy to see that Θ and Ψ witness WIEHAGEN ≤ TxtEx strong COINIT; we omit the details. Theorem 5 COINIT ≤ TxtEx strong WIEHAGEN. Proof. By operator recursion theorem =-=[4]-=- there exists a recursive 1–1 increasing function p such that for all i, Wp(i) = {x | x ≥ p(i)}. Let Θ be such that Θ(L) = {x | (∃i)[i ∈ L ∧ x ≥ p(i)]}. Note that such a Θ can be easily constructed. L... |

44 | The power of vacillation in language learning - Case - 1992 |

37 | Theory of Recursive Functions and E ective Computability - Rogers - 1967 |

35 |
Criteria of language learning
- Osherson, Weinstein
- 1982
(Show Context)
Citation Context ... L ∈ TxtEx(M)) just in case M TxtEx-identifies each text for L. (c) TxtEx = {L ⊆ E | (∃M)[L ⊆ TxtEx(M)]}. Other criteria of success are finite identification [11], behaviorally correct identification =-=[8, 17, 7]-=-, and vacillatory identification [17, 5]. In the present paper, we only discuss results about TxtEx-identification. 3 Weak and Strong Reductions We first present some technical machinery. We write “σ ... |

32 |
Some decidability results on grammatical inference and complexity
- Feldman
- 1972
(Show Context)
Citation Context ... L ∈ TxtEx(M)) just in case M TxtEx-identifies each text for L. (c) TxtEx = {L ⊆ E | (∃M)[L ⊆ TxtEx(M)]}. Other criteria of success are finite identification [11], behaviorally correct identification =-=[8, 17, 7]-=-, and vacillatory identification [17, 5]. In the present paper, we only discuss results about TxtEx-identification. 3 Weak and Strong Reductions We first present some technical machinery. We write “σ ... |

30 |
The power of vacillation
- Case
- 1988
(Show Context)
Citation Context ...ies each text for L. (c) TxtEx = {L ⊆ E | (∃M)[L ⊆ TxtEx(M)]}. Other criteria of success are finite identification [11], behaviorally correct identification [8, 17, 7], and vacillatory identification =-=[17, 5]-=-. In the present paper, we only discuss results about TxtEx-identification. 3 Weak and Strong Reductions We first present some technical machinery. We write “σ ⊆ τ” if σ is an initial segment of τ, an... |

25 | On the intrinsic complexity of learning
- Freivalds, Kinber, et al.
- 1994
(Show Context)
Citation Context ...es an insight into why certain classes are more easily learned than others. Our model adopts a similar study in the context of learning functions by Freivalds [9], and by Freivalds, Kinber, and Smith =-=[10]-=-. The main idea of the approach is to introduce 1sreductions between learnable classes of languages. If a collection of languages, L1, can be reduced to another collection of languages, L2, then the l... |

23 | Language identi cation in the limit - Gold - 1967 |

16 |
Identification of formal languages
- Wiehagen
- 1977
(Show Context)
Citation Context ...ong reduction. We now discuss two interesting collections of languages that are shown not to be complete with respect to either reduction. The first one is a class of languages introduced by Wiehagen =-=[19]-=- which contains all those languages L such that the minimum element in L is a grammar for L; we refer to this collection of languages as WIEHAGEN. This self-referential class, which can be TxtEx-ident... |

15 |
Inductive inference of recursive functions: Qualitative theory
- Freivalds
- 1991
(Show Context)
Citation Context ...sent paper describes a model which provides an insight into why certain classes are more easily learned than others. Our model adopts a similar study in the context of learning functions by Freivalds =-=[9]-=-, and by Freivalds, Kinber, and Smith [10]. The main idea of the approach is to introduce 1sreductions between learnable classes of languages. If a collection of languages, L1, can be reduced to anoth... |

15 | The structure of intrinsic complexity of learning
- Jain, Sharma
- 1995
(Show Context)
Citation Context ...hat any collection of languages that can be finitely identified (i.e., identified with 0 mind changes) from informants is ≤ InfEx strong SINGLE. 16sstructure of these reductions has also been studied =-=[14]-=-. However, it is felt that for these reductions to have an impact on the study of feasibility issues in language identification, their fidelity has to be improved. Acknowledgements Our study has clear... |

2 | Identi cation of formal languages - Wiehagen - 1977 |

1 |
On the intrinsic complexity of learning. unpublished
- Freivalds, Kinber, et al.
(Show Context)
Citation Context ...s a model which provides an insight into why certain classes are more easily learned than others. Our model adopts a similar study in the context of learning functions by Freivalds, Kinber, and Smith =-=[8]-=-. The main idea of the approach is to introduce reductions between collections of languages. If a collection L 1 can be reduced to a collection L 2 , then the learnability of L 1 is no more difficult ... |