## Generalized Notions of Mind Change Complexity (1997)

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Venue: | In Proceedings of the Tenth Annual Conference on Computational Learning Theory |

Citations: | 11 - 5 self |

### BibTeX

@INPROCEEDINGS{Sharma97generalizednotions,

author = {Arun Sharma and Frank Stephan and Yuri Ventsov},

title = {Generalized Notions of Mind Change Complexity},

booktitle = {In Proceedings of the Tenth Annual Conference on Computational Learning Theory},

year = {1997},

pages = {96--108},

publisher = {ACM Press}

}

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### Abstract

Speed of convergence in Gold's identification in the limit model can be measured by deriving bounds on the number of mind changes made by a learner before the onset of convergence. Two approaches to date are bounds given by constants (referred here as Type 1) and bounds expressed as constructive ordinals (referred as Type 2). The use of ordinals has recently been successfully employed to measure the mind change complexity of learning rich concept classes such as unions of pattern languages, elementary formal systems and logic programs. Motivated by these applications, the present work introduces two more general approaches to bounding mind changes. These are based on counting by going down in a linearly ordered set (Type 3) and on counting by going down in a partially ordered set (Type 4). In both cases the set must not contain infinite, descending, and computable sequences. These four types of mind changes yield a hierarchy and there are identifiable classes that cannot be learned wit...

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Citation Context ...lustrated by characterizing them in terms of branches of uniformly recursive families of binary trees. 1 Introduction The learning model considered in this paper is Gold's identification in the limit =-=[4, 7]-=- usually referred to as Ex-identification. In this model, a learner M --- a computable device --- receives increasing segments of the values, f(0); f(1); : : :, of a computable function f one element ... |

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Citation Context ...the learner make no more than a given fixed number of mind changes before it converges. This approach was first considered by Barzdins and Freivalds [3] and investigated extensively by Case and Smith =-=[5]. However,-=- this notion turns out to be too restrictive as it places the same constant bound on each concept in the class being learned. It does not allow the possibility that a more "complex" concept ... |

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Citation Context ...lustrated by characterizing them in terms of branches of uniformly recursive families of binary trees. 1 Introduction The learning model considered in this paper is Gold's identification in the limit =-=[4, 7]-=- usually referred to as Ex-identification. In this model, a learner M --- a computable device --- receives increasing segments of the values, f(0); f(1); : : :, of a computable function f one element ... |

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Citation Context ...n enumerable set and S is uniformly recursive iff f(x; e) : x 2 L e g is a computable set. Uniformly recursive families have been widely studied, in particular in context of monotonicity requirements =-=[2, 8, 12, 13, 15, 16, 17, 24]-=-. Uniformly enumerable families are just a natural generalization. Angluin [2] found a nice characterization which states when a uniformly recursive family can be learned from text; de Jongh and Kanaz... |

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Citation Context ... proof of Theorem 2.2. So only the part (b ) c) must be shown. Let M be a machine which converges on every computable text. N is now constructed according to the locking-sequence hunting construction =-=[21]-=-. Let oe 0 ; oe 1 ; : : : be an enumeration of all strings in lN and let a 0 a 1 : : : be some text of some language L. Now N on input a 0 a 1 : : : an is defined as follows: N (a 0 a 1 : : : an ) = M... |

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Citation Context ...n enumerable set and S is uniformly recursive iff f(x; e) : x 2 L e g is a computable set. Uniformly recursive families have been widely studied, in particular in context of monotonicity requirements =-=[2, 8, 12, 13, 15, 16, 17, 24]-=-. Uniformly enumerable families are just a natural generalization. Angluin [2] found a nice characterization which states when a uniformly recursive family can be learned from text; de Jongh and Kanaz... |

5 | Trees and learning
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Citation Context ...vide characterizations of the second and fourth types of mind change bound, for both Ex and BC, in terms of branches of uniformly recursive families of binary trees in the style of Merkle and Stephan =-=[19]-=-. Language Learning and Mind Changes: We also consider the generalized mind change bound for language learning from positive data (texts). We show that the four types of bounds can be adapted to this ... |

4 |
Sanjay Jain and Arun Sharma: Ordinal Mind Change Complexity of Language Identification
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Citation Context ...conjectured upper bound on the number of mind changes, and so on. The use of ordinals to model mind change complexity has recently been applied by Jain and Sharma [9] and by Ambainis, Jain and Sharma =-=[1]-=- to measure the complexity of learning pattern languages, unions of pattern languages and elementary formal systems (a logic programming system on strings). More recently, these techniques have direct... |

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Citation Context ...he number of times it may revise its conjectured upper bound on the number of mind changes, and so on. The use of ordinals to model mind change complexity has recently been applied by Jain and Sharma =-=[9]-=- and by Ambainis, Jain and Sharma [1] to measure the complexity of learning pattern languages, unions of pattern languages and elementary formal systems (a logic programming system on strings). More r... |

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Citation Context ...nding the number of mind changes is requiring that the learner make no more than a given fixed number of mind changes before it converges. This approach was first considered by Barzdins and Freivalds =-=[3]-=- and investigated extensively by Case and Smith [5]. However, this notion turns out to be too restrictive as it places the same constant bound on each concept in the class being learned. It does not a... |

3 |
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Citation Context ..."complex" concept may require a larger bound on the number of mind changes than a "simpler" concept. Motivated by such limitations, Freivalds and Smith [6] introduced the use of co=-=nstructive ordinals [23]-=- to bound the number of mind changes. This notion turns out to be more natural as it allows a learner to decide on a mind change bound based on the data received. This approach also allows the possibi... |

2 |
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Citation Context ...n enumerable set and S is uniformly recursive iff f(x; e) : x 2 L e g is a computable set. Uniformly recursive families have been widely studied, in particular in context of monotonicity requirements =-=[2, 8, 12, 13, 15, 16, 17, 24]-=-. Uniformly enumerable families are just a natural generalization. Angluin [2] found a nice characterization which states when a uniformly recursive family can be learned from text; de Jongh and Kanaz... |

2 |
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Citation Context ...Instead of directly attempting to adapt mind change complexity to BC learning, we incorporate mind change bounds in the model of "finite error next value prediction " NV 00 --- a model which=-= Podnieks [22]-=- proved to be equivalent to BC (see also Case and Smith [5]). In this model a learner is allowed to be partial recursive. We say that learner M is successful on a class of functionssS just in case for... |

2 |
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Citation Context |

1 |
and Arun Sharma: Mind Change Complexity of Learning Logic Programs. Under preparation
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Citation Context ... on strings). More recently, these techniques have directly been applied to measure the mind change complexity of learning logic programs from positive facts and from both positive and negative facts =-=[10]. Motivate-=-d by the above applications, this paper asks the question: "Are there more general ways of bounding the number of mind changes than ordinals?" We give an affirmative answer to this question ... |

1 |
de Jongh and Makoto Kanazawa: Angluin's Theorem for Indexed Families of r.e. Sets and Applications
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(Show Context)
Citation Context ...iformly enumerable families are just a natural generalization. Angluin [2] found a nice characterization which states when a uniformly recursive family can be learned from text; de Jongh and Kanazawa =-=[11]-=- generalized this criterion to uniformly enumerable families. Beside looking at the fact how mind change bounds generalize to language learning from sets, it is an interesting question which kind of A... |

1 |
Language learning under various types of constraint combinations
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(Show Context)
Citation Context |

1 |
Thomas Zeugmann: Types of Monotonic Language Learning and Their Characterization
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Citation Context |

1 |
and Thomas Zeugmann: Monotonic versus Non-Monotonic Language Learning
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(Show Context)
Citation Context |

1 |
Thomas Zeugmann: Language Learning in Dependence on the Space of Hypotheses
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Citation Context ... might be more difficult than to produce any programs. Therefore it can be expected that learning w.r.t. a given enumeration is more difficult than w.r.t. an arbitrary enumeration. Lange and Zeugmann =-=[17]-=- therefore considered three settings: (1) exact learning: here the learner has to use a given hypothesis space, that means the task is given as a set S plus an enumeration ff 0 ; f 1 ; : : :g of S. Le... |

1 |
Thomas Zeugmann: Language learning with bounded number of mind changes
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(Show Context)
Citation Context ...e work will explore Angluin [2] style characterization for learnability of uniformly enumerable collections of concepts in the context of mind change bounds of Type 2, 3 and 4 (see Lange and Zeugmann =-=[18]-=- for Type 1 characterization) . Refinements of results in this paper about the use of more expressive hypothesis spaces to reduce the mind change complexity are likely to yield insights into the desig... |