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In [[model theory]], a branch of [[mathematical logic]], a complete theory ''T'' is said to satisfy '''NIP''' (or "not the independence property") if none of its formulae satisfy the '''independence property''', that is if none of its formulae can pick out any given subset of an arbitrarily large finite set. | |||
==Definition== | |||
Let ''T'' be a [[Complete theory|complete]] ''L''-theory. An ''L''-formula φ('''''x''''','''''y''''') is said to have the independence property (with respect to '''''x''''', '''''y''''') if in every model ''M'' of ''T'' there is, for each ''n'' = {0,1,…''n'' − 1} < ω, a family of [[tuple]]s '''''b'''''<sub>0</sub>,…,'''''b'''''<sub>''n''−1</sub> such that for each of the 2<sup>''n''</sup> subsets ''X'' of ''n'' there is a tuple '''''a''''' in ''M'' for which | |||
:<math>M\models\varphi(\boldsymbol{a},\boldsymbol{b}_i)\quad\Leftrightarrow\quad i\in X.</math> | |||
The theory ''T'' is said to have the independence property if some formula has the independence property. If no ''L''-formula has the independence property then ''T'' is called dependent, or said to satisfy NIP. An ''L''-structure is said to have the independence property (respectively, NIP) if its theory has the independence theory (respectively, NIP). The terminology comes from the notion of independence in the sense of [[Boolean algebra (structure)|boolean algebras]]. | |||
In the nomenclature of [[Vapnik–Chervonenkis theory]], we may say that a collection '''''S''''' of subsets of ''X'' ''shatters'' a set ''B'' ⊆ ''X'' if every subset of ''B'' is of the form ''B'' ∩ ''S'' for some ''S'' ∈ '''''S'''''. Then ''T'' has the independence property if in some model ''M'' of ''T'' there is a definable family (''S''<sub>''a''</sub> | ''a''∈''M''<sup>''n''</sup>) ⊆ ''M''<sup>''k''</sup> that shatters arbitrarily large finite subsets of ''M''<sup>''k''</sup>. In other words, (''S''<sub>''a''</sub> | ''a''∈''M''<sup>''n''</sup>) has infinite [[VC dimension|Vapnik–Chervonenkis dimension]]. | |||
==Examples== | |||
Any complete theory ''T'' that has the independence property is [[Stable theory|unstable]].<ref>See Hodges.</ref> | |||
In arithmetic, i.e. the structure ('''N''',+,·), the formula "''y'' divides ''x''" has the independence property.<ref>See Poizat, page 249.</ref> This formula is just | |||
:<math>(\exists k)(y\cdot k=x).</math> | |||
So, for any finite ''n'' we take the ''n'' 1-tuples ''b''<sub>''i''</sub> to be the first ''n'' [[prime number]]s, and then for any subset ''X'' of {0,1,…''n'' − 1} we let ''a'' be the product of those ''b''<sub>''i''</sub> such that ''i'' is in ''X''. Then ''b''<sub>''i''</sub> divides ''a'' if and only if ''i'' ∈ ''X''. | |||
Every [[o-minimal theory]] satisfies NIP.<ref>Pillay and Steinhorn, corollary 3.10 and Knight, Pillay, and Steinhorn, theorem 0.2.</ref> This fact has had unexpected applications to neural network learning.<ref>See Anthony and Bartlett for details.</ref> | |||
==Notes== | |||
{{reflist}} | |||
==References== | |||
* {{ cite book | last=Anthony | first=Martin | coauthors=Bartlett, Peter L. | publisher=[[Cambridge University Press]] | title=Neural network learning: theoretical foundations | year=1999 | isbn=0-521-57353-X }} | |||
* {{ cite book | last=Hodges | first=Wilfrid | authorlink=Wilfrid Hodges | publisher=[[Cambridge University Press]] | title=Model theory | year=1993 | isbn=0-521-30442-3 }} | |||
* {{ cite journal | last=Knight | first=Julia | authorlink = Julia F. Knight | coauthors=Pillay, Anand; Steinhorn, Charles | title=Definable sets in ordered structures II | journal=[[Transactions of the American Mathematical Society]] | year=1986 | volume=295 | issue=2 | pages=593–605 | doi=10.2307/2000053}} | |||
* {{ cite journal | last=Pillay | first=Anand | coauthors=Steinhorn, Charles | title=Definable sets in ordered structures I | journal=Transactions of the American Mathematical Society | year=1986 | volume=295 | issue=2 | pages=565–592 }} | |||
* {{ cite book | last=Poizat | first=Bruno | publisher=Springer | title=A Course in Model Theory | year=2000 | isbn=0-387-98655-3 }} | |||
[[Category:Model theory]] | |||
Revision as of 15:47, 6 October 2013
In model theory, a branch of mathematical logic, a complete theory T is said to satisfy NIP (or "not the independence property") if none of its formulae satisfy the independence property, that is if none of its formulae can pick out any given subset of an arbitrarily large finite set.
Definition
Let T be a complete L-theory. An L-formula φ(x,y) is said to have the independence property (with respect to x, y) if in every model M of T there is, for each n = {0,1,…n − 1} < ω, a family of tuples b0,…,bn−1 such that for each of the 2n subsets X of n there is a tuple a in M for which
The theory T is said to have the independence property if some formula has the independence property. If no L-formula has the independence property then T is called dependent, or said to satisfy NIP. An L-structure is said to have the independence property (respectively, NIP) if its theory has the independence theory (respectively, NIP). The terminology comes from the notion of independence in the sense of boolean algebras.
In the nomenclature of Vapnik–Chervonenkis theory, we may say that a collection S of subsets of X shatters a set B ⊆ X if every subset of B is of the form B ∩ S for some S ∈ S. Then T has the independence property if in some model M of T there is a definable family (Sa | a∈Mn) ⊆ Mk that shatters arbitrarily large finite subsets of Mk. In other words, (Sa | a∈Mn) has infinite Vapnik–Chervonenkis dimension.
Examples
Any complete theory T that has the independence property is unstable.[1]
In arithmetic, i.e. the structure (N,+,·), the formula "y divides x" has the independence property.[2] This formula is just
So, for any finite n we take the n 1-tuples bi to be the first n prime numbers, and then for any subset X of {0,1,…n − 1} we let a be the product of those bi such that i is in X. Then bi divides a if and only if i ∈ X.
Every o-minimal theory satisfies NIP.[3] This fact has had unexpected applications to neural network learning.[4]
Notes
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A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
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