No free lunch in search and optimization

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Language identification in the limit is a formal model for inductive inference. It was introduced by E. Mark Gold in his paper with the same title [1] In this model, a learner is provided with presentation (i.e. strings) of some formal language. The learning is seen as an infinite process. Each time an element of the presentation is read the learner should provide a representation (e.g. a formal grammar) for the language. It is said that a learner can identify in the limit a class of languages if given any presentation of any language in the class the learner will produce only a finite number of wrong representations, and therefore converge on the correct representation in a finite number of steps, without however necessarily being able to announce its correctness since a counterexample to that representation could appear as an element arbitrarily long after.

Gold defined two types of presentations:

  • Text (positive information): an enumeration of all strings the language consists of.
  • Complete presentation (positive and negative information): an enumeration of all possible strings, each with a label indicating if the string belongs to the language or not.

Learnability

This model is an early attempt to formally capture the notion of learnability. Gold's paper[2] introduces for contrast the stronger models

  • Finite identification (where the learner has to announce correctness after a finite number of steps), and
  • Fixed-time identification (where correctness has to be reached after an apriori-specified number of steps).

A weaker formal model of learnability is the Probably approximately correct learning (PAC) model, introduced by Leslie Valiant in 1984.

Examples

Example 4
Teacher Learner's
Guess Query
0. abab
1. 23 yr old Mining Engineer (Excluding Oil ) Roman Kull from Markham, enjoys to spend time metal detecting, property developers in singapore property index and history. In recent time took some time to check out Macquarie Island. abab baba
2. 23 yr old Mining Engineer (Excluding Oil ) Roman Kull from Markham, enjoys to spend time metal detecting, property developers in singapore property index and history. In recent time took some time to check out Macquarie Island. a*(ba)*b* aa
3. 23 yr old Mining Engineer (Excluding Oil ) Roman Kull from Markham, enjoys to spend time metal detecting, property developers in singapore property index and history. In recent time took some time to check out Macquarie Island. (ab)*(ba)*(ab)*(ba)* bababa
4. 23 yr old Mining Engineer (Excluding Oil ) Roman Kull from Markham, enjoys to spend time metal detecting, property developers in singapore property index and history. In recent time took some time to check out Macquarie Island. (ab+ba)* babb
5. 23 yr old Mining Engineer (Excluding Oil ) Roman Kull from Markham, enjoys to spend time metal detecting, property developers in singapore property index and history. In recent time took some time to check out Macquarie Island. (ab+ba)* baaa
... ...
Example 3
Teacher Learner
1. 23 yr old Mining Engineer (Excluding Oil ) Roman Kull from Markham, enjoys to spend time metal detecting, property developers in singapore property index and history. In recent time took some time to check out Macquarie Island. abab
2. 23 yr old Mining Engineer (Excluding Oil ) Roman Kull from Markham, enjoys to spend time metal detecting, property developers in singapore property index and history. In recent time took some time to check out Macquarie Island. a*(ba)*b*
3. 23 yr old Mining Engineer (Excluding Oil ) Roman Kull from Markham, enjoys to spend time metal detecting, property developers in singapore property index and history. In recent time took some time to check out Macquarie Island. (ab)*(ba)*(ab)*(ba)*
4. 23 yr old Mining Engineer (Excluding Oil ) Roman Kull from Markham, enjoys to spend time metal detecting, property developers in singapore property index and history. In recent time took some time to check out Macquarie Island. (ab+ba)*
5. 23 yr old Mining Engineer (Excluding Oil ) Roman Kull from Markham, enjoys to spend time metal detecting, property developers in singapore property index and history. In recent time took some time to check out Macquarie Island. (ab+ba)*
6. 23 yr old Mining Engineer (Excluding Oil ) Roman Kull from Markham, enjoys to spend time metal detecting, property developers in singapore property index and history. In recent time took some time to check out Macquarie Island. (ab+ba)*
7. 23 yr old Mining Engineer (Excluding Oil ) Roman Kull from Markham, enjoys to spend time metal detecting, property developers in singapore property index and history. In recent time took some time to check out Macquarie Island. (ab+ba)*
... ...
Example 2
Teacher Learner
1. abab abab
2. ba abab+ba
3. baba abab+ba+baba
4. ba abab+ba+baba
5. baba abab+ba+baba
6. abab abab+ba+baba
7. ε abab+ba+baba+ε
... ...
Example 1
Teacher Learner
1. abab abab
2. baba abab+baba
3. baabab (b+ε)(ab)*
4. baabab (b+ε)(ab)*+baabab
5. abbaabba (ab)*(ba)*(ab)*(ba)*
6. baabbaab (ab+ba)*
7. bababa (ab+ba)*
... ...

It is instructive to look at concrete examples of learning sessions the definition of identification in the limit speaks about.

  • Example 1: fictitious session to learn a regular language L over the alphabet {a,b} from text presentation. In each step, the teacher gives a string belonging to L, and the learner answers a guess for L, encoded as a regular expression. In step 3, the learner's guess is not consistent with the strings seen so far; in step 4, the teacher gives a string repeatedly. After step 6, the learner sticks to the regular expression (ab+ba)*. If this happens to be a description of the language L the teacher has in mind, it is said that the learner has learned that language. If a computer program for the learner's role would exist that was able to successfully learn each regular language, that class of languages would be identifiable in the limit. Gold has shown that this is not the case.[3]
  • Example 2: a particular learning algorithm always guessing L to be just the union of all strings seen so far. If L is a finite language, the learner will eventually guess it correctly, however, without being able to tell when. Although the guess didn't change during step 3 to 6, the learner couldn't be sure to be correct. Gold has shown that the class of finite languages is identifibale in the limit.;[4] however, this class is neither finitely nor fixed-time identifiable.
  • Example 3: learning from complete presentation by telling. In each step, the teacher gives a string and tells whether it belongs to L (23 yr old Mining Engineer (Excluding Oil ) Roman Kull from Markham, enjoys to spend time metal detecting, property developers in singapore property index and history. In recent time took some time to check out Macquarie Island.) or not (23 yr old Mining Engineer (Excluding Oil ) Roman Kull from Markham, enjoys to spend time metal detecting, property developers in singapore property index and history. In recent time took some time to check out Macquarie Island.). Each possible string is eventually classified in this way by the teacher.
  • Example 4: learning from complete presentation by request. The learner gives a query string, the teacher tell whether it belongs to L (23 yr old Mining Engineer (Excluding Oil ) Roman Kull from Markham, enjoys to spend time metal detecting, property developers in singapore property index and history. In recent time took some time to check out Macquarie Island.) or not (23 yr old Mining Engineer (Excluding Oil ) Roman Kull from Markham, enjoys to spend time metal detecting, property developers in singapore property index and history. In recent time took some time to check out Macquarie Island.); the learner then gives a guess for L, followed by the next query string. In this example, the learner happens to query in each step just the same string as given by the teacher in example 3. In general, Gold has shown that each language class identifiable in the request-presentation setting is also identifiable in the telling-presentation setting,[5] since the learner, instead of querying a string, just needs to wait until it is eventually given by the teacher.

Learnability characterization

Dana Angluin gave the characterizations of learnability from text (positive information) in a 1980 paper. [6] If a learner is required to be effective, then an indexed class of recursive languages is learnable in the limit if there is an effective procedure that uniformly enumerates tell-tales for each language in the class (Condition 1).[7] It is not hard to see that if we allow an ideal learner (i.e., an arbitrary function), then an indexed class of languages is learnable in the limit if each language in the class has a tell-tale (Condition 2).[8]

Language classes learnable in the limit

Dividing lines between identifiable and nonidentifiable language classes[9]
Learnability model Class of languages
Anomalous text presentation[note 1]
Recursively enumerable
Recursive
Complete presentation
Primitive recursive[note 2]
Context-sensitive
Context-free
Regular
Superfinite[note 3]
Normal text presentation[note 4]
Finite
Singleton[note 5]

The table shows which language classes are identifiable in the limit in which learning model. On the right hand side, each language class is a super class of all lower classes. Each learning model (i.e. type of presentation) can identify in the limit all classes below it. In particular, the class of finite languages is identifiable in the limit by text presentation (cf. Example 2 above), while the class of regular languages is not.

Pattern Languages, introduced by Dana Angluin in another 1980 paper, [10] are also identifiable by normal text presentation; they are omitted in the table, since they are above the singleton and below the primitive recursive language class, but incomparable to the classes in between.[note 6]Template:Clarify

Sufficient conditions for learnability

Condition 1 in Angluin's paper[7] is not always easy to verify. Therefore, people come up with various sufficient conditions for the learnability of a language class. See also Induction of regular languages for learnable subclasses of regular languages.

Finite thickness

A class of languages has finite thickness if every non-empty set of strings is contained in at most finitely many languages of the class. This is exactly Condition 3 in Angluin's paper.[11] Angluin showed that if a class of recursive languages has finite thickness, then it is learnable in the limit.[12]

A class with finite thickness certainly satisfies MEF-condition and MFF-condition; in other words, finite thickness implies M-finite thickness.[13]

Finite elasticity

A class of languages is said to have finite elasticity if for every infinite sequence of strings and every infinite sequence of languages in the class , there exists a finite number n such that implies is inconsistent with . [1]

It is shown that a class of recursively enumerable languages is learnable in the limit if it has finite elasticity.

Mind change bound

Template:Empty section

Other concepts

Infinite cross property

A language L has infinite cross property within a class of languages if there is an infinite sequence of distinct languages in and a sequence of finite subset such that:

Note that L is not necessarily a member of the class of language.

It is not hard to see that if there is a language with infinite cross property within a class of languages, then that class of languages has infinite elasticity.

Relations between concepts

  • Finite thickness implies finite elasticity;[13][14] the converse is not true.
  • Finite elasticity and conservatively learnable implies the existence of a mind change bound. [2]
  • Finite elasticity and M-finite thickness implies the existence of a mind change bound. However, M-finite thickness alone does not imply the existence of a mind change bound; neither does the existence of a mind change bound imply M-finite thickness. [3]
  • Existence of a mind change bound implies learnability; the converse is not true.
  • If we allow for noncomputable learners, then finite elasticity implies the existence of a mind change bound; the converse is not true.
  • If there is no accumulation order for a class of languages, then there is a language (not necessarily in the class) that has infinite cross property within the class, which in turn implies infinite elasticity of the class.

Open questions

  • If a countable class of recursive languages has a mind change bound for noncomputable learners, does the class also have a mind change bound for computable learners, or is the class unlearnable by a computable learner?

Notes

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References

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.

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    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  2. ↑ p.457
  3. ↑ Theorem I.8,I.9, p.470-471
  4. ↑ Theorem I.6, p.469
  5. ↑ Theorem I.3, p.467
  6. ↑ One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    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

    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
  7. ↑ 7.0 7.1 p.121 top
  8. ↑ p.123 top
  9. ↑ Table 1, p.452, in (Gold 1967)
  10. ↑ One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    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

    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
  11. ↑ p.123 mid
  12. ↑ p.123 bot, Corollary 2
  13. ↑ 13.0 13.1 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: Proof of Corollary 29
  14. ↑ Wright, Keith (1989) "Identification of Unions of Languages Drawn from an Identifiable Class". Proc. 2nd Workwhop on Computational Learning Theory, 328-333; with correction in: Motoki, Shinohara, and Wright (1991) "The correct definition of finite elasiticity: corrigendum to identification of unions", Proc. 4th Workshop on Computational Learing Theory, 375-375


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