LNAI 2842 On Ordinal VC Dimension and Some
Notions of Complexity 1st Edition by Eric
Martin, Arun Sharma, Frank Stephan ISBN
9783540200574 354020057X pdf download
https://ebookball.com/product/lnai-2842-on-ordinal-vc-dimension-
and-some-notions-of-complexity-1st-edition-by-eric-martin-arun-
sharma-frank-stephan-isbn-9783540200574-354020057x-10630/
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LNAI 2842 Signal Extraction and Knowledge Discovery Based on
Statistical Modeling 1st Edition by Genshiro Kitagawa ISBN
9783540200574 354020057X
https://ebookball.com/product/lnai-2842-signal-extraction-and-
knowledge-discovery-based-on-statistical-modeling-1st-edition-by-
genshiro-kitagawa-isbn-9783540200574-354020057x-10042/
LNAI 2842 Transductive Confidence Machine Is Universal 1st Edition by
Ilia Nouretdinov, Vladimir Vyugin, Alex Gammerman ISBN 9783540200574
354020057X
https://ebookball.com/product/lnai-2842-transductive-confidence-
machine-is-universal-1st-edition-by-ilia-nouretdinov-vladimir-
vyugin-alex-gammerman-isbn-9783540200574-354020057x-9092/
LNAI 2842 Identification with Probability One of Stochastic
Deterministic Linear Languages 1st Edition by Colin de la Higuera,
Jose Oncina ISBN 9783540200574 354020057X
https://ebookball.com/product/lnai-2842-identification-with-
probability-one-of-stochastic-deterministic-linear-languages-1st-
edition-by-colin-de-la-higuera-jose-oncina-
isbn-9783540200574-354020057x-11750/
LNAI 2801 Culture and the Baldwin Effect 1st Edition by Diego Federici
ISBN 9783540200574 354020057X
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effect-1st-edition-by-diego-federici-
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,LNAI 2801 Developmental Neural Networks for Agents 1st Edition by Andy
Balaam ISBN 9783540200574 354020057X
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LNAI 2903 A Defeasible Logic of Policy Based Intention 1st Edition by
Guido Governatori, Vineet Padmanabhan ISBN 9783540200574 354020057X
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LNAI 2903 Design and Implementation of an Intelligent Information
Infrastructure 1st Edition by Henry Lau, Andrew Ning, Peggy Fung ISBN
9783540200574 354020057X
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edition-by-henry-lau-andrew-ning-peggy-fung-
isbn-9783540200574-354020057x-9392/
LNAI 2801 Distributed Genetic Algorithm Learning by Direct Exchange of
Chromosomes 1st Edition by AleÅ¡ KubÃ-k ISBN 9783540200574 35402005
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algorithm-learning-by-direct-exchange-of-chromosomes-1st-edition-
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LNAI 2801 An Imitation Game for Emerging Action Categories 1st Edition
by Bart Jansen ISBN 9783540200574 354020057X
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, On Ordinal VC-Dimension and Some Notions of
Complexity
Eric Martin1 , Arun Sharma2 , and Frank Stephan2
1
School of Computer Science and Engineering, UNSW Sydney, NSW 2052, Australia
2
National ICT Australia, UNSW Sydney, NSW 2052, Australia
{Arun.Sharma,Frank.Stephan}@nicta.com.au
Abstract. We generalize the classical notion of VC-dimension to ordi-
nal VC-dimension, in the context of logical learning paradigms. Logical
learning paradigms encompass the numerical learning paradigms com-
monly studied in Inductive inference. A logical learning paradigm is de-
fined as a set W of structures over some vocabulary, and a set D of
first-order formulas that represent data. The sets of models of ϕ in W,
where ϕ varies over D, generate a natural topology W over W.
We show that if D is closed under boolean operators, then the notion of
ordinal VC-dimension offers a perfect characterization for the problem
of predicting the truth of the members of D in a member of W, with an
ordinal bound on the number of mistakes. This shows that the notion
of VC-dimension has a natural interpretation in Inductive Inference,
when cast into a logical setting. We also study the relationships between
predictive complexity, selective complexity—a variation on predictive
complexity—and mind change complexity. The assumptions that D is
closed under boolean operators and that W is compact often play a
crucial role to establish connections between these concepts.
Keywords: Inductive inference, logical paradigms, VC-dimension, pre-
dictive complexity, mind change bounds.
1 Introduction
The notion of VC-dimension is a key concept in PAC-learning [6,12,13]. The
notion of finite telltale is a key concept in Inductive inference [3,8]. It can be
claimed that VC-dimension is to PAC-learning what finite telltales are to Induc-
tive inference. Both provide a characterization of learnability, for fundamental
classes of learning paradigms, in the respective settings. Both take the form of
a condition where finiteness is a key requirement, in frameworks that deal with
infinite objects. In logical learning paradigms of identification in the limit, it
has been shown that the finite telltale condition can be seen as a generalization
of the compactness property, the latter being the hallmark of, equivalently, fi-
nite learning or deductive inference [10]. The finite telltale condition can even be
generalized and be interpreted as a property of β-weak compactness, that charac-
terizes classification with less than β mind changes [10]. There are extensions of
R. Gavaldà et al. (Eds.): ALT 2003, LNAI 2842, pp. 54–68, 2003.
c Springer-Verlag Berlin Heidelberg 2003
Notions of Complexity 1st Edition by Eric
Martin, Arun Sharma, Frank Stephan ISBN
9783540200574 354020057X pdf download
https://ebookball.com/product/lnai-2842-on-ordinal-vc-dimension-
and-some-notions-of-complexity-1st-edition-by-eric-martin-arun-
sharma-frank-stephan-isbn-9783540200574-354020057x-10630/
Explore and download more ebooks or textbooks
at ebookball.com
, Get Your Digital Files Instantly: PDF, ePub, MOBI and More
Quick Digital Downloads: PDF, ePub, MOBI and Other Formats
LNAI 2842 Signal Extraction and Knowledge Discovery Based on
Statistical Modeling 1st Edition by Genshiro Kitagawa ISBN
9783540200574 354020057X
https://ebookball.com/product/lnai-2842-signal-extraction-and-
knowledge-discovery-based-on-statistical-modeling-1st-edition-by-
genshiro-kitagawa-isbn-9783540200574-354020057x-10042/
LNAI 2842 Transductive Confidence Machine Is Universal 1st Edition by
Ilia Nouretdinov, Vladimir Vyugin, Alex Gammerman ISBN 9783540200574
354020057X
https://ebookball.com/product/lnai-2842-transductive-confidence-
machine-is-universal-1st-edition-by-ilia-nouretdinov-vladimir-
vyugin-alex-gammerman-isbn-9783540200574-354020057x-9092/
LNAI 2842 Identification with Probability One of Stochastic
Deterministic Linear Languages 1st Edition by Colin de la Higuera,
Jose Oncina ISBN 9783540200574 354020057X
https://ebookball.com/product/lnai-2842-identification-with-
probability-one-of-stochastic-deterministic-linear-languages-1st-
edition-by-colin-de-la-higuera-jose-oncina-
isbn-9783540200574-354020057x-11750/
LNAI 2801 Culture and the Baldwin Effect 1st Edition by Diego Federici
ISBN 9783540200574 354020057X
https://ebookball.com/product/lnai-2801-culture-and-the-baldwin-
effect-1st-edition-by-diego-federici-
isbn-9783540200574-354020057x-11122/
,LNAI 2801 Developmental Neural Networks for Agents 1st Edition by Andy
Balaam ISBN 9783540200574 354020057X
https://ebookball.com/product/lnai-2801-developmental-neural-
networks-for-agents-1st-edition-by-andy-balaam-
isbn-9783540200574-354020057x-13726/
LNAI 2903 A Defeasible Logic of Policy Based Intention 1st Edition by
Guido Governatori, Vineet Padmanabhan ISBN 9783540200574 354020057X
https://ebookball.com/product/lnai-2903-a-defeasible-logic-of-
policy-based-intention-1st-edition-by-guido-governatori-vineet-
padmanabhan-isbn-9783540200574-354020057x-9684/
LNAI 2903 Design and Implementation of an Intelligent Information
Infrastructure 1st Edition by Henry Lau, Andrew Ning, Peggy Fung ISBN
9783540200574 354020057X
https://ebookball.com/product/lnai-2903-design-and-
implementation-of-an-intelligent-information-infrastructure-1st-
edition-by-henry-lau-andrew-ning-peggy-fung-
isbn-9783540200574-354020057x-9392/
LNAI 2801 Distributed Genetic Algorithm Learning by Direct Exchange of
Chromosomes 1st Edition by AleÅ¡ KubÃ-k ISBN 9783540200574 35402005
https://ebookball.com/product/lnai-2801-distributed-genetic-
algorithm-learning-by-direct-exchange-of-chromosomes-1st-edition-
by-alea-kubak-isbn-9783540200574-354020057x-14086/
LNAI 2801 An Imitation Game for Emerging Action Categories 1st Edition
by Bart Jansen ISBN 9783540200574 354020057X
https://ebookball.com/product/lnai-2801-an-imitation-game-for-
emerging-action-categories-1st-edition-by-bart-jansen-
isbn-9783540200574-354020057x-13286/
, On Ordinal VC-Dimension and Some Notions of
Complexity
Eric Martin1 , Arun Sharma2 , and Frank Stephan2
1
School of Computer Science and Engineering, UNSW Sydney, NSW 2052, Australia
2
National ICT Australia, UNSW Sydney, NSW 2052, Australia
{Arun.Sharma,Frank.Stephan}@nicta.com.au
Abstract. We generalize the classical notion of VC-dimension to ordi-
nal VC-dimension, in the context of logical learning paradigms. Logical
learning paradigms encompass the numerical learning paradigms com-
monly studied in Inductive inference. A logical learning paradigm is de-
fined as a set W of structures over some vocabulary, and a set D of
first-order formulas that represent data. The sets of models of ϕ in W,
where ϕ varies over D, generate a natural topology W over W.
We show that if D is closed under boolean operators, then the notion of
ordinal VC-dimension offers a perfect characterization for the problem
of predicting the truth of the members of D in a member of W, with an
ordinal bound on the number of mistakes. This shows that the notion
of VC-dimension has a natural interpretation in Inductive Inference,
when cast into a logical setting. We also study the relationships between
predictive complexity, selective complexity—a variation on predictive
complexity—and mind change complexity. The assumptions that D is
closed under boolean operators and that W is compact often play a
crucial role to establish connections between these concepts.
Keywords: Inductive inference, logical paradigms, VC-dimension, pre-
dictive complexity, mind change bounds.
1 Introduction
The notion of VC-dimension is a key concept in PAC-learning [6,12,13]. The
notion of finite telltale is a key concept in Inductive inference [3,8]. It can be
claimed that VC-dimension is to PAC-learning what finite telltales are to Induc-
tive inference. Both provide a characterization of learnability, for fundamental
classes of learning paradigms, in the respective settings. Both take the form of
a condition where finiteness is a key requirement, in frameworks that deal with
infinite objects. In logical learning paradigms of identification in the limit, it
has been shown that the finite telltale condition can be seen as a generalization
of the compactness property, the latter being the hallmark of, equivalently, fi-
nite learning or deductive inference [10]. The finite telltale condition can even be
generalized and be interpreted as a property of β-weak compactness, that charac-
terizes classification with less than β mind changes [10]. There are extensions of
R. Gavaldà et al. (Eds.): ALT 2003, LNAI 2842, pp. 54–68, 2003.
c Springer-Verlag Berlin Heidelberg 2003
