Tubular neighborhood

The pumping lemma for context-free languages, also known as the Bar-Hillel lemma, is a lemma that gives a property shared by all context-free languages.

Formal statement

If a language L is context-free, then there exists some integer p ≥ 1 such that every string s in L with |s| ≥ p (where p is a "pumping length") can be written as

s = uvxyz

with substrings u, v, x, y and z, such that

1. |vxy| ≤ p,

2. |vy| ≥ 1, and

3. uv ⁿxy ⁿz is in L for all n ≥ 0.

Informal statement and explanation

The pumping lemma for context-free languages (called just "the pumping lemma" for the rest of this article) describes a property that all context-free languages are guaranteed to have.

The property is a property of all strings in the language that are of length at least p, where p is a constant—called the pumping length—that varies between context-free languages.

Say s is a string of length at least p that is in the language.

The pumping lemma states that s can be split into five substrings, $s=uvxyz$, where vy is non-empty and the length of vxy is at most p, such that repeating v and y any (and the same) number of times in s produces a string that is still in the language (it's possible and often useful to repeat zero times, which removes v and y from the string). This process of "pumping up" additional copies of v and y is what gives the pumping lemma its name.

Note that finite languages (which are regular and hence context-free) obey the pumping lemma trivially by having p equal to the maximum string length in L plus one. As there are no strings of this length the pumping lemma is not violated.

The pumping lemma is often used to prove that a given language is non-context-free by showing that for each p, we can find some string s of length at least p in the language that does not have the properties outlined above, i.e. that it cannot be "pumped" without producing some strings that are not in the language.

Usage of the lemma

The pumping lemma for context-free languages can be used to show that certain languages are not context-free.

For example, we can show that language $L=\left\{a^i{b}^i{c}^i\right|i>0\}$ is not context-free by using the pumping lemma in a proof by contradiction. First, assume that $L$ is context free. By the pumping lemma, there exists an integer $p$ which is the pumping length of language $L$. Consider the string $s=a^p{b}^p{c}^p$ in $L$. The pumping lemma tells us that $s$ can be written in the form $s=uvxyz$, where $u,v,x,y$, and $z$ are substrings, such that $\left|vxy\right|\le p$, $\left|vy\right|\ge 1$, and $u{v}^{i}x{y}^{i}z$ is in $L$ for every integer $i\ge 0$. By our choice of $s$ and the fact that $\left|vxy\right|\le p$, it is easily seen that the substring $vxy$ can contain no more than two distinct letters. That is, we have one of five possibilities for $vxy$:

1. $vxy=a^j$ for some $j\le p$.
2. $vxy=a^j{b}^k$ for some $j$ and $k$ with $j+k\le p$.
3. $vxy=b^j$ for some $j\le p$.
4. $vxy=b^j{c}^k$ for some $j$ and $k$ with $j+k\le p$.
5. $vxy=c^j$ for some $j\le p$.

For each case, it is easily verified that $u{v}^{i}x{y}^{i}z$ does not contain equal numbers of each letter for any $i\ne 1$. Thus, $u{v}^{2}x{y}^{2}z$ does not have the form $a^i{b}^i{c}^i$. This contradicts the definition of $L$. Therefore, our initial assumption that $L$ is context free must be false.

While the pumping lemma is often a useful tool to prove that a given language is not context-free, it does not give a complete characterization of the context-free languages. If a language does not satisfy the condition given by the pumping lemma, we have established that it is not context-free. On the other hand, there are languages that are not context-free, but still satisfy the condition given by the pumping lemma. There are more powerful proof techniques available, such as Ogden's lemma, but also these techniques do not give a complete characterization of the context-free languages.

See also

• Pumping lemma for regular languages
• Formal languages

References

• 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
• 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 Section 1.4: Nonregular Languages, pp. 77–83. Section 2.3: Non-context-free Languages, pp. 115–119.