## On Aggregating Teams of Learning Machines (1994)

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Venue: | Theoretical Computer Science A |

Citations: | 7 - 4 self |

### BibTeX

@ARTICLE{Jain94onaggregating,

author = {Sanjay Jain and Arun Sharma},

title = {On Aggregating Teams of Learning Machines},

journal = {Theoretical Computer Science A},

year = {1994},

volume = {137},

pages = {150--163}

}

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### Abstract

The present paper studies the problem of when a team of learning machines can be aggregated into a single learning machine without any loss in learning power. The main results concern aggregation ratios for vacillatory identification of languages from texts. For a positiveinteger n,amachine is said to TxtFex n -identify a language L just in case the machine converges to up to n grammars for L on any text for L.For such identification criteria, the aggregation ratio is derived for the n = 2 case. It is shown that the collection of languages that can be TxtFex 2 identified by teams with success ratio greater than 5=6 are the same as those collections of languages that can be TxtFex 2 - identified by a single machine. It is also established that 5=6 is indeed the cut-off point by showing that there are collections of languages that can be TxtFex 2 -identified bya team employing 6 machines, at least 5 of which are required to be successful, but cannot be TxtFex 2 -identified byany single machine. Additionally, aggregation ratios are also derived for finite identification of languages from positive data and for numerous criteria involving language learning from both positive and negative data.

### Citations

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Citation Context ... said to TxtEx identify a language L just in case M, fed any text for L, converges to a correct grammar for L. This is essentially the seminal notion of identification in the limit introduced by Gold =-=[13]-=- (see also Case and Lynes [7] and Osherson and Weinstein [21]). 1sA learning machine M is said to TxtBc-identify L just in case M, fed any text for L, outputs an infinite sequence of grammars such tha... |

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Citation Context ...eliminary notions and definitions from inductive inference literature. Our results are presented in Section 3. 2 Preliminaries 2.1 Notation Any unexplained recursion theoretic notation is from Rogers =-=[25]-=-. The symbol N denotes the set of natural numbers, {0, 1, 2, 3, . . .}. The symbol N + denotes the set of positive natural numbers, {1, 2, 3, . . .}. Unless otherwise specified, i, j, k, l, m, n, q, r... |

163 |
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(Show Context)
Citation Context ...that in the context of function learning, vacillatory identification turns out to be the same as identification in the limit. This was first shown by Barzdin and Podnieks [2] (see also Case and Smith =-=[8]-=-). Let n be a positive integer. A learning machine M is said to TxtFex n -identify a language L just in case M, fed any text for L, converges in the limit to a finite set, with cardinalitysn, of gramm... |

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An Introduction to the General Theory of Algorithms
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Citation Context ... ·〉 for encoding multiple tuples of natural numbers onto N. By ϕ we denote a fixed acceptable programming system for the partial computable functions: N → N (see Rogers [24, 25] and Machtey and Young =-=[19]-=-). By ϕi we denote the partial computable function computed by program i in the ϕ-system. The letter, p, in some contexts, with or without decorations, ranges over programs; in other contexts p ranges... |

92 |
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(Show Context)
Citation Context ...guage L just in case M, fed any text for L, converges to a correct grammar for L. This is essentially the seminal notion of identi cation in the limit introduced by Gold [13] (see also Case and Lynes =-=[7]-=- and Osherson and Weinstein [22]). A learning machine M is said to TxtBc-identify L just in case M, fedany text for L, outputs an in nite sequence of grammars such that after a nite number of incorrec... |

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Citation Context ...s, with or without decorations, ranges over programs� in other contexts p ranges over total functions with its range being construed as programs. By we denote an arbitrary xed Blum complexity measur=-=e [3, 14]-=- for the '-system. By Wi we denote domain('i). Wi is, then, the r.e. set/language ( N) accepted (or equivalently, generated) by the'-program i. Symbol E will denote the set of all r.e. languages. Symb... |

56 |
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(Show Context)
Citation Context ... of learning machines is essentially a multiset of learning machines. Definition 12 introduces team learning of functions and Definition 13 introduces team learning of languages. Definition 12 (Smith =-=[26]-=-, Osherson, Stob, and Weinstein [20]) Let I ∈ {Fin, Ex, Bc} and let m, n ∈ N + . (a) A team of n machines, M1, M2, . . . , Mn, is said to Team m n I-identify f (written: f ∈ Team m n I(M1, M2, . . . ,... |

50 |
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(Show Context)
Citation Context ...arly, we can define 〈·, . . . , ·〉 for encoding multiple tuples of natural numbers onto N. By ϕ we denote a fixed acceptable programming system for the partial computable functions: N → N (see Rogers =-=[24, 25]-=- and Machtey and Young [19]). By ϕi we denote the partial computable function computed by program i in the ϕ-system. The letter, p, in some contexts, with or without decorations, ranges over programs;... |

49 |
Periodicity in generations of automata
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(Show Context)
Citation Context ...valds [12] about probabilistic nite identi cation. Theorem 1(b) shows that 2=3 is the cut-o point for aggregation of Fin-identi cation� a diagonalization argument using the operator recursion theore=-=m [4] s-=-u ces to establish this latter result. Theorem 1 [28, 15] (a) (8m� n 2 N + j m=n > 2=3)[Team m n Fin = Fin]. (b) Fin Team 2 3Fin. Pitt and Smith [24] settled the question for function identi cation ... |

44 | The power of vacillation in language learning - Case - 1999 |

40 |
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- 1988
(Show Context)
Citation Context ...hree popularly investigated criteria of success, namely, Fin (finite identification), Ex (identification in the limit) and Bc (behaviorally correct identification). For both Ex and Bc, Pitt and Smith =-=[23]-=- showed the aggregation ratio to be 1/2. For finite function identification, Fin, it was reported in Jain and Sharma [15] that the aggregation ratio is 2/3 (this result can also be argued from a resul... |

35 |
Criteria of language learning
- Osherson, Weinstein
- 1982
(Show Context)
Citation Context ...text for L, converges to a correct grammar for L. This is essentially the seminal notion of identification in the limit introduced by Gold [13] (see also Case and Lynes [7] and Osherson and Weinstein =-=[21]-=-). 1sA learning machine M is said to TxtBc-identify L just in case M, fed any text for L, outputs an infinite sequence of grammars such that after a finite number of incorrect guesses, M outputs only ... |

30 |
The power of vacillation
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- 1969
(Show Context)
Citation Context ... next consider vacillatory identi cation of languages from texts in whichamachine is required to converge to a nite set of grammars. This notion was studied by Osherson and Weinstein [22] and by Case =-=[5]-=-. It should be noted that in the context of function learning, vacillatory identi cation turns out to be the same as identi cation in the limit. This was rst shown by Barzdin and Podnieks [2] (see als... |

27 |
Two Theorems on the Limiting Synthesis of Functions, in "Theory of Algorithms and Programs
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- 1974
(Show Context)
Citation Context ...2 Bc(M)) () ( 1 8 n)['M(f [n]) = f]. We de ne the class Bc = fS R j (9M)[S Bc(M)]g. The following proposition summarizes the relationship between the various function learning criteria. Proposition 1 =-=[8, 1]-=- Fin Ex Bc. 2.4 Language Learning A text T for a language L is a mapping from N into (N [f#g) such thatL is the set of natural numbers in the range of T . The content of a text T , denoted content(T )... |

24 |
Synthesizing inductive expertise
- Osherson, Stob, et al.
- 1988
(Show Context)
Citation Context ...uts only grammars for L. This criterion was first studied by Case and Lynes [7] and Osherson and Weinstein [21], and is also referred to as “extensional” identification. Osherson, Stob, and Weinstein =-=[20]-=- first observed that for TxtEx-identification, a team can be aggregated if its success ratio is greater than 2/3. Hence, in matters of aggregation, identification in the limit of languages from positi... |

18 | Relations between probabilistic and team one-shot learners - Daley, Pitt, et al. - 1991 |

17 | Computational Limits on Team Identification of Languages
- Jain, Sharma
- 1993
(Show Context)
Citation Context ...e result is known for TxtEx-identification and TxtBcidentification. It was shown by Osherson, Stob, and Weinstein [20] that aggregation ratio for TxtEx-identification is 2/3 (see also Jain and Sharma =-=[16, 18, 17]-=- for extension of this result to anomalies in the final grammar). 8sTheorem 17 (Osherson, Stob, and Weinstein [20]) (a) (∀m, n ∈ N + | m/n > 2/3)[Team m n TxtEx = TxtEx] (b) TxtEx ⊂ Team 2 3TxtEx. The... |

16 | Breaking the probability 1 2 barrier in FIN-type learning - Daley, Kalyanasundaram, et al. - 1995 |

15 |
The theory of inductive inference
- Bārzdiņˇs, Podnieks
- 1973
(Show Context)
Citation Context ... by Case [5]. It should be noted that in the context of function learning, vacillatory identi cation turns out to be the same as identi cation in the limit. This was rst shown by Barzdin and Podnieks =-=[2]-=- (see also Case and Smith [8]). Let n be a positive integer. A learning machine M is said to TxtFexn-identify a language L just in case M, fedany text for L, converges in the limit to a nite set, with... |

10 |
Finite learning by a team
- Jain, Sharma
- 1990
(Show Context)
Citation Context ...d Bc (behaviorally correct identification). For both Ex and Bc, Pitt and Smith [23] showed the aggregation ratio to be 1/2. For finite function identification, Fin, it was reported in Jain and Sharma =-=[15]-=- that the aggregation ratio is 2/3 (this result can also be argued from a result of Freivalds [12] about probabilistic finite function identification). The present paper describes aggregation results ... |

10 |
A Characterization of Probabilistic Inference
- PITT
(Show Context)
Citation Context ...han 2/3. Hence, in matters of aggregation, identification in the limit of languages from positive data turns out to be similar to finite function identification. On the other hand, a result from Pitt =-=[22]-=- can easily be used to show that for TxtBc-identification the aggregation ratio is 1/2. Thus, TxtEx and TxtBc exhibit different behavior with respect to aggregation. We now present two more criteria o... |

10 |
Inductive inference with bounded number of mind changes
- Velauthapillai
- 1989
(Show Context)
Citation Context ...eivalds [12] about probabilistic finite identification. Theorem 15(b) can be established via a diagonalization argument employing the operator recursion theorem (Case [4]). Theorem 15 (Velauthapillai =-=[27]-=-, Jain and Sharma [15]) (a) (∀m, n ∈ N + | m/n > 2/3)[Team m n Fin = Fin]. (b) Fin ⊂ Team 2 3Fin. Pitt and Smith [23] settled the question for function identification in the limit and behaviorally cor... |

6 |
Comparison of identi cation criteria for machine inductive inference, Theoretical Computer Science 25
- Case, Smith
- 1983
(Show Context)
Citation Context ...ted that in the context of function learning, vacillatory identi cation turns out to be the same as identi cation in the limit. This was rst shown by Barzdin and Podnieks [2] (see also Case and Smith =-=[8]-=-). Let n be a positive integer. A learning machine M is said to TxtFexn-identify a language L just in case M, fedany text for L, converges in the limit to a nite set, with cardinality n, of grammars f... |

5 |
Functions computable in the limit by probabilistic machines
- Freivalds
- 1975
(Show Context)
Citation Context ...regation ratio to be 1/2. For finite function identification, Fin, it was reported in Jain and Sharma [15] that the aggregation ratio is 2/3 (this result can also be argued from a result of Freivalds =-=[12]-=- about probabilistic finite function identification). The present paper describes aggregation results about language identification from positive data. The main results are in the context of vacillato... |

5 |
Probability is more powerful than team for language identification
- Jain, Sharma
- 1993
(Show Context)
Citation Context ...e result is known for TxtEx-identification and TxtBcidentification. It was shown by Osherson, Stob, and Weinstein [20] that aggregation ratio for TxtEx-identification is 2/3 (see also Jain and Sharma =-=[16, 18, 17]-=- for extension of this result to anomalies in the final grammar). 8sTheorem 17 (Osherson, Stob, and Weinstein [20]) (a) (∀m, n ∈ N + | m/n > 2/3)[Team m n TxtEx = TxtEx] (b) TxtEx ⊂ Team 2 3TxtEx. The... |

4 | Inductive inference hierarchies: Probabilistic vs pluralistic - Daley - 1985 |