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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.
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==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''&nbsp;=&nbsp;{0,1,&hellip;''n''&nbsp;&minus;&nbsp;1}&nbsp;<&nbsp;ω, a family of [[tuple]]s '''''b'''''<sub>0</sub>,&hellip;,'''''b'''''<sub>''n''&minus;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''&nbsp;⊆&nbsp;''X'' if every subset of ''B'' is of the form ''B''&nbsp;∩&nbsp;''S'' for some ''S''&nbsp;∈&nbsp;'''''S'''''.  Then ''T'' has the independence property if in some model ''M'' of ''T'' there is a definable family (''S''<sub>''a''</sub>&nbsp;|&nbsp;''a''∈''M''<sup>''n''</sup>)&nbsp;⊆&nbsp;''M''<sup>''k''</sup> that shatters arbitrarily large finite subsets of ''M''<sup>''k''</sup>. In other words, (''S''<sub>''a''</sub>&nbsp;|&nbsp;''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,&hellip;''n''&nbsp;&minus;&nbsp;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''&nbsp;∈&nbsp;''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&ndash;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&ndash;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]]

Latest revision as of 21:41, 5 May 2014



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