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In [[formal language theory]], in particular in [[algorithmic learning theory]], a class ''C'' of [[formal language#Definition|languages]] has '''finite thickness''' if every string is contained in at most finitely many languages in ''C''. This condition was introduced by [[Dana Angluin]] as a [[sufficient condition]] for ''C'' being [[language identification in the limit#Finite thickness|identifiable in the limit]]. | |||
<ref>{{cite journal| author=Dana Angluin| title=Inductive Inference of Formal Languages from Positive Data| journal=Information and Control| year=1980| volume=45| pages=117–135| url=http://citeseer.ist.psu.edu/context/14508/0|url=http://www-personal.umich.edu/~yinw/papers/Angluin80.pdf}}; here: Condition 3, p.123 mid. Angluin's original requirement (every non-empty string ''set'' be contained in at most finitely many languages) is equivalent.</ref> | |||
==The related notion of M-finite thickness== | |||
Given a language ''L'' and an indexed class ''C'' = { ''L''<sub>1</sub>, ''L''<sub>2</sub>, ''L''<sub>3</sub>, ... } of languages, a member language ''L''<sub>''j''</sub> ∈ ''C'' is called a '''minimal concept''' of ''L'' within ''C'' if ''L'' ⊆ ''L''<sub>''j''</sub>, but not ''L'' ⊊ ''L''<sub>''i''</sub> ⊆ ''L''<sub>''j''</sub> for any ''L''<sub>''i''</sub> ∈ ''C''.<ref>{{cite book| author=Andris Ambainis, Sanjay Jain, Arun Sharma| chapter=Ordinal mind change complexity of language identification| title=Computational Learning Theory| year=1997| volume=1208| pages=301-315| publisher=Springer| series=LNCS| url=http://www.comp.nus.edu.sg/~sanjay/paps/efs2.pdf}}; here: Definition 25</ref> | |||
The class ''C'' is said to satisfy the '''MEF-condition''' if every finite subset ''D'' of a member language ''L''<sub>''i''</sub> ∈ ''C'' has a minimal concept ''L''<sub>''j''</sub> ⊆ ''L''<sub>''i''</sub>. Symmetrically, ''C'' is said to satisfy the '''MFF-condition''' if every nonempty finite set ''D'' has at most finitely many minimal concepts in ''C''. Finally, ''C'' is said to have '''M-finite thickness''' if it satisfies both the MEF- and the MFF-condition. | |||
<ref>Ambainis et. al. 1997, Definition 26</ref> | |||
<!---commented-out former, unsourced definitions:--- | |||
We say that <math>\mathcal L</math> satisfies the '''MEF-condition''' if for each string s and each consistent language L in the class, there is a minimal consistent language in <math>\mathcal L</math>, which is a sublanguage of L. Symmetrically, we say that <math>\mathcal L</math> satisfies the '''MFF-condition''' if for every string s there are only finitely many minimal consistent languages in <math>\mathcal L</math>. | |||
---> | |||
Finite thickness implies M-finite thickness.<ref>Ambainis et. al. 1997, Corollary 29</ref> However, there are classes that are of M-finite thickness but not of finite thickness (for example, any class of languages ''C'' = { ''L''<sub>1</sub>, ''L''<sub>2</sub>, ''L''<sub>3</sub>, ... } such that ''L''<sub>1</sub> ⊆ ''L''<sub>2</sub> ⊆ ''L''<sub>3</sub> ⊆ ...). | |||
{{reflist}} | |||
{{comp-sci-theory-stub}} | |||
[[Category:Formal languages]] | |||
Revision as of 23:53, 29 January 2014
In formal language theory, in particular in algorithmic learning theory, a class C of languages has finite thickness if every string is contained in at most finitely many languages in C. This condition was introduced by Dana Angluin as a sufficient condition for C being identifiable in the limit. [1]
Given a language L and an indexed class C = { L1, L2, L3, ... } of languages, a member language Lj ∈ C is called a minimal concept of L within C if L ⊆ Lj, but not L ⊊ Li ⊆ Lj for any Li ∈ C.[2] The class C is said to satisfy the MEF-condition if every finite subset D of a member language Li ∈ C has a minimal concept Lj ⊆ Li. Symmetrically, C is said to satisfy the MFF-condition if every nonempty finite set D has at most finitely many minimal concepts in C. Finally, C is said to have M-finite thickness if it satisfies both the MEF- and the MFF-condition. [3]
Finite thickness implies M-finite thickness.[4] However, there are classes that are of M-finite thickness but not of finite thickness (for example, any class of languages C = { L1, L2, L3, ... } such that L1 ⊆ L2 ⊆ L3 ⊆ ...).
43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro. Template:Comp-sci-theory-stub
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There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang; here: Condition 3, p.123 mid. Angluin's original requirement (every non-empty string set be contained in at most finitely many languages) is equivalent. - ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534; here: Definition 25 - ↑ Ambainis et. al. 1997, Definition 26
- ↑ Ambainis et. al. 1997, Corollary 29