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The '''star height problem''' in [[formal language theory]] is the question whether all [[regular language]]s can be expressed using [[Regular_expression#Formal_language_theory|regular expression]]s of limited [[star height]], i.e. with a limited nesting depth of [[Kleene star]]s. Specifically, is a nesting depth of one always sufficient? If not, is there an [[algorithm]] to determine how many are required? The problem was raised by {{harvtxt|Eggan|1963}}.
 
==Families of regular languages with unbounded star height==
The first question was answered in the negative when in 1963, Eggan gave examples of regular languages of [[star height]] ''n'' for every ''n''. Here, the star height ''h''(''L'') of a regular language ''L'' is defined as the minimum star height among all regular expressions representing ''L''. The first few languages found by {{harvtxt|Eggan|1963}} are described in the following, by means of giving a regular expression for each language:
 
:<math>\begin{alignat}{2}
e_1 &= a_1^* \\
e_2 &= \left(a_1^*a_2^*a_3\right)^*\\
e_3 &= \left(\left(a_1^*a_2^*a_3\right)^*\left(a_4^*a_5^*a_6\right)^*a_7\right)^*\\
e_4 &= \left(
\left(\left(a_1^*a_2^*a_3\right)^*\left(a_4^*a_5^*a_6\right)^*a_7\right)^*
\left(\left(a_8^*a_9^*a_{10}\right)^*\left(a_{11}^*a_{12}^*a_{13}\right)^*a_{14}\right)^*
a_{15}\right)^*
\end{alignat}
</math>
 
The construction principle for these expressions is that expression <math>e_{n+1}</math> is obtained by concatenating two copies of <math>e_n</math>, appropriately renaming the letters of the second copy using fresh alphabet symbols, concatenating the result with another fresh alphabet symbol, and then by surrounding the resulting expression with a Kleene star. The remaining, more difficult part, is to prove that for <math>e_n</math> there is no equivalent regular expression of star height less than ''n''; a proof is given in {{harv|Eggan|1963}}.
 
However, Eggan's examples use a large [[Alphabet (computer science)|alphabet]], of size 2<sup>''n''</sup>-1 for the language with star height ''n''. He thus asked whether we can also find examples over binary alphabets. This was proved to be true shortly afterwards by {{harvtxt|Dejean|Schützenberger|1966}}.
Their examples can be described by an [[inductive definition|inductively defined]] family of regular expressions over the binary alphabet <math>\{a,b\}</math> as follows&ndash;cf. {{harvtxt|Salomaa|1981}}:
:<math>\begin{alignat}{2}
e_1 & = (ab)^* \\
e_2 & = \left(aa(ab)^*bb(ab)^*\right)^* \\
e_3 & = \left(aaaa \left(aa(ab)^*bb(ab)^*\right)^* bbbb \left(aa(ab)^*bb(ab)^*\right)^*\right)^* \\
\, & \cdots \\
e_{n+1} & = (\,\underbrace{a\cdots a}_{2^n}\, \cdot \, e_n\, \cdot\, \underbrace{b\cdots b}_{2^n}\, \cdot\, e_n \,)^*
\end{alignat}
</math>
 
Again, a rigorous proof is needed for the fact that <math>e_n</math> does not admit an equivalent regular expression of lower star height. Proofs are given by {{harv|Dejean|Schützenberger|1966}} and by {{harv|Salomaa|1981}}.
 
==Computing the star height of regular languages==
In contrast, the second question turned out to be much more difficult, and the question became a famous open problem in formal language theory for over two decades {{harv|Brzozowski|1980}}. For years, there was only little progress. The [[pure-group language]]s were the first interesting family of regular languages for which the star height problem was proved to be [[decidable]] {{harv|McNaughton|1967}}. But the general problem remained open for more than 25 years until it was settled by [[Kosaburo Hashiguchi|Hashiguchi]], who in 1988 published an algorithm to determine the [[star height]] of any regular language. The algorithm wasn't at all practical, being of non-[[ELEMENTARY|elementary]] complexity. To illustrate the immense resource consumptions of that algorithm, Lombardy and Sakarovitch (2002) give some actual numbers:
 
{{cquote|
[The procedure described by Hashiguchi] leads to computations that are by far impossible, even for very small examples. For instance, if ''L'' is accepted by a 4 state automaton of loop complexity 3 (and with a small 10 element transition monoid), then a ''very low minorant'' of the number of languages to be tested with ''L'' for equality is:
 
<math>\left(10^{10^{10}}\right)^{\left(10^{10^{10}}\right)^{\left(10^{10^{10}}\right)}}.</math>
|4=S. Lombardy and J. Sakarovitch
|5=''Star Height of Reversible Languages and Universal Automata'', LATIN 2002
}}
Notice that alone the number <math>10^{10^{10}}</math> has 10 billion zeros when written down in [[decimal notation]], and is already ''by far'' larger than the [[Observable_universe#Matter_content|number of atoms in the observable universe]].
 
A much more efficient algorithm than Hashiguchi's procedure was devised by Kirsten in 2005. This algorithm runs, for a given [[nondeterministic finite automaton]] as input, within double-[[EXPSPACE|exponential space]]. Yet the resource requirements of this algorithm still greatly exceed the margins of what is considered practically feasible.
 
==See also==
*[[Generalized star height problem]]
 
==References==
*{{cite journal |first=Lawrence C. |last=Eggan |title=Transition graphs and the star-height of regular events | journal=[[Michigan Mathematical Journal]] | volume=10 | issue=4 | pages=385–397 | year=1963 | doi=10.1307/mmj/1028998975 | zbl=0173.01504 }}
*{{cite journal |first=Françoise |last=Dejean |authorlink2=Marcel-Paul Schützenberger |first2=Marcel-Paul |last2=Schützenberger |title=On a Question of Eggan |journal=[[Information and Control]] |volume=9 |issue=1 |pages=23–25 |year=1966 |doi=10.1016/S0019-9958(66)90083-0 }}
* {{cite journal
| title = The Loop Complexity of Pure-Group Events
| year = 1967
| last = McNaughton |first=Robert
| journal = Information and Control
| pages = 167–176
| volume = 11
| issue = 1–2
| doi=10.1016/S0019-9958(67)90481-0
}}
*{{cite book |authorlink=Janusz Brzozowski (computer scientist) |first=Janusz A. |last=Brzozowski |chapter=Open problems about regular languages |editor-first=Ronald V. |editor-last=Book |title=Formal language theory—Perspectives and open problems |pages=23–47 |publisher=Academic Press |location=New York |year=1980 |isbn=0-12-115350-9 }} [https://www.cs.uwaterloo.ca/research/tr/1980/CS-80-03.pdf (technical report version)]
* {{cite book |title=Jewels of Formal Language Theory |last= Salomaa |first= Arto |authorlink=Arto Salomaa |year=1981 |publisher=Pitman Publishing |location=Melbourne |isbn=0-273-08522-0 |zbl=0487.68063 }}
*{{cite journal |first=Kosaburo |last=Hashiguchi |title=Regular languages of star height one |journal=Information and Control |volume=53 |issue=2 |pages=199–210 |year=1982 |doi=10.1016/S0019-9958(82)91028-2 }}
*{{cite journal |first=Kosaburo |last=Hashiguchi |title=Algorithms for Determining Relative Star Height and Star Height |journal=Information and Computation |volume=78 |issue=2 |pages=124–169 |year=1988 |doi=10.1016/0890-5401(88)90033-8 }}
*{{cite paper |first=Sylvain |last=Lombardy |first2=Jacques |last2=Sakarovitch |title=Star Height of Reversible Languages and Universal Automata |work=5th Latin American Symposium on Theoretical Informatics (LATIN) 2002, vol. 2286 of LNCS |publisher=Springer |url=http://www-igm.univ-mlv.fr/~lombardy/publi/LATIN.pdf |year=2002 }}
*{{cite journal |first=Daniel |last=Kirsten |title=Distance Desert Automata and the Star Height Problem |journal=RAIRO - Informatique Théorique et Applications |volume=39 |issue=3 |pages=455–509 |year=2005 |doi=10.1051/ita:2005027 }}
* {{cite book | last=Sakarovitch | first=Jacques | title=Elements of automata theory | others=Translated from the French by Reuben Thomas | location=Cambridge | publisher=[[Cambridge University Press]] | year=2009 | isbn=978-0-521-84425-3 | zbl=1188.68177 }}
 
[[Category:Automata theory]]
[[Category:Formal languages]]
[[Category:Theorems in discrete mathematics]]

Revision as of 22:39, 10 December 2013

The star height problem in formal language theory is the question whether all regular languages can be expressed using regular expressions of limited star height, i.e. with a limited nesting depth of Kleene stars. Specifically, is a nesting depth of one always sufficient? If not, is there an algorithm to determine how many are required? The problem was raised by Template:Harvtxt.

Families of regular languages with unbounded star height

The first question was answered in the negative when in 1963, Eggan gave examples of regular languages of star height n for every n. Here, the star height h(L) of a regular language L is defined as the minimum star height among all regular expressions representing L. The first few languages found by Template:Harvtxt are described in the following, by means of giving a regular expression for each language:

The construction principle for these expressions is that expression is obtained by concatenating two copies of , appropriately renaming the letters of the second copy using fresh alphabet symbols, concatenating the result with another fresh alphabet symbol, and then by surrounding the resulting expression with a Kleene star. The remaining, more difficult part, is to prove that for there is no equivalent regular expression of star height less than n; a proof is given in Template:Harv.

However, Eggan's examples use a large alphabet, of size 2n-1 for the language with star height n. He thus asked whether we can also find examples over binary alphabets. This was proved to be true shortly afterwards by Template:Harvtxt. Their examples can be described by an inductively defined family of regular expressions over the binary alphabet as follows–cf. Template:Harvtxt:

Again, a rigorous proof is needed for the fact that does not admit an equivalent regular expression of lower star height. Proofs are given by Template:Harv and by Template:Harv.

Computing the star height of regular languages

In contrast, the second question turned out to be much more difficult, and the question became a famous open problem in formal language theory for over two decades Template:Harv. For years, there was only little progress. The pure-group languages were the first interesting family of regular languages for which the star height problem was proved to be decidable Template:Harv. But the general problem remained open for more than 25 years until it was settled by Hashiguchi, who in 1988 published an algorithm to determine the star height of any regular language. The algorithm wasn't at all practical, being of non-elementary complexity. To illustrate the immense resource consumptions of that algorithm, Lombardy and Sakarovitch (2002) give some actual numbers:

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A much more efficient algorithm than Hashiguchi's procedure was devised by Kirsten in 2005. This algorithm runs, for a given nondeterministic finite automaton as input, within double-exponential space. Yet the resource requirements of this algorithm still greatly exceed the margins of what is considered practically feasible.

See also

References

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    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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  • 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

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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

    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
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  • 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.

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  • 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
  • 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
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  • 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