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{{Programming paradigms}} | |||
'''Inductive logic programming''' ('''ILP''') is a subfield of [[machine learning]] which uses [[logic programming]] as a uniform representation for examples, background knowledge and hypotheses. Given an encoding of the known background knowledge and a set of examples represented as a logical database of facts, an ILP system will derive a hypothesised logic program which entails all the positive and none of the negative examples. | |||
Schema: ''positive examples'' + ''negative examples'' + ''background knowledge'' => ''hypothesis''. | |||
Inductive logic programming is particularly useful in [[bioinformatics]] and [[natural language processing]]. | |||
The term ''Inductive Logic Programming'' was first introduced<ref>Luc De Raedt. A Perspective on Inductive Logic Programming. The Workshop on Current and Future Trends in Logic Programming, Shakertown, to appear in Springer LNCS, 1999. [[CiteSeerX]]: {{url|1=http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.56.1790|2=10.1.1.56.1790}}</ref> in a paper by [[Stephen Muggleton]] in 1991.<ref>{{cite doi|10.1007/BF03037089}}</ref> The term "''inductive''" here refers to [[Inductive reasoning|philosophical]] (i.e. suggesting a theory to explain observed facts) rather than [[mathematical induction|mathematical]] (i.e. proving a property for all members of a well-ordered set) induction. | |||
==Formal definition== | |||
The ''background knowledge'' is given as a logical proposition <math>B</math>, commonly in the form of [[Horn clauses]] used in [[logic programming]]. | |||
The ''positive'' and ''negative'' examples are given as a conjunction <math>F^+</math> and <math>F^-</math> of unnegated and negated [[ground expression|ground]] [[Literal (mathematical logic)|literals]], respectively. | |||
A hypothesis <math>h</math> is a logical proposition satisfying the following requirements.<ref>{{cite journal|first1=Stephen|last1=Muggleton|title=Inductive Logic Programming: Issues, Results and the Challenge of Learning Language in Logic|journal=Artificial Intelligence|volume=114|pages=283–296|year=1999}}; here: Sect.2.1</ref> | |||
{| | |||
|- | |||
| Necessity: | |||
||<math>B</math> | |||
|| <math> \not\models</math> | |||
|| <math> F^+</math> | |||
|- | |||
| Sufficiency: | |||
|| <math>B \land h</math> | |||
|| [[Entailment#Semantic consequence|<math> \models</math>]] | |||
|| <math> F^+</math> | |||
|- | |||
| Weak consistency: | |||
|| <math>B \land h</math> | |||
|| <math> \not\models</math> | |||
|| <math> \textit{false}</math> | |||
|- | |||
| Strong consistency: | |||
|| <math>B \land h \land F^-</math> | |||
|| <math> \not\models</math> | |||
|| <math> \textit{false}</math> | |||
|} | |||
"''Necessity''" does not impose a restriction on <math>h</math>, but forbids any generation of a hypothesis as long as the positive facts are explainable without it. | |||
"''Sufficiency''" requires any generated hypothesis <math>h</math> to explain all positive examples <math>F^+</math>. | |||
"''Weak consistency''" forbids generation of any hypothesis <math>h</math> that contradicts the background knowledge <math>B</math>. | |||
"''Strong consistency''" also forbids generation of any hypothesis <math>h</math> that is inconsistent with the negative examples <math>F^-</math>, given the background knowledge <math>B</math>; it implies "''Weak consistency''"; if no negative examples are given, both requirements coincide. Džeroski <ref>{{citation|first1=Sašo|last1=Džeroski|contribution=Inductive Logic Programming and Knowledge Discovery in Databases|pages=117–152|editor1-first=U.M.|editor1-last=Fayyad|editor2-first=G.|editor2-last=Piatetsky-Shapiro|editor3-first=P.|editor3-last=Smith|editor4-first=R.|editor4-last=Uthurusamy|title=Advances in Knowledge Discovery and Data Mining|publisher=MIT Press|year=1996}}; here: Sect.5.2.4</ref> requires only "''Sufficiency''" (called "Completeness" there) and "''Strong consistency''". | |||
==Example== | |||
[[File:Family relations example for inductive logic programming article.gif|thumb|Assumed family relations in section "Example"]] | |||
The following well-known example about learning definitions of family relations uses the abbreviations <math>\textit{par}: \textit{parent}</math>, <math>\textit{fem}: \textit{female}</math>, <math>\textit{dau}: \textit{daughter}</math>, <math>g:\textit{George}</math>, <math>h:\textit{Helen}</math>, <math>m:\textit{Mary}</math>, <math>t:\textit{Tom}</math>, <math>n:\textit{Nancy}</math>, and <math>e:\textit{Eve}</math>. It starts from the background knowledge (cf. picture) | |||
:<math>\textit{par}(h,m) \land \textit{par}(h,t) \land \textit{par}(g,m) \land \textit{par}(t,e) \land \textit{par}(n,e) \land \textit{fem}(h) \land \textit{fem}(m) \land \textit{fem}(n) \land \textit{fem}(e)</math>, | |||
the positive examples | |||
:<math>\textit{dau}(m,h) \land \textit{dau}(e,t)</math>, | |||
and the trivial proposition | |||
<math>\textit{true}</math> | |||
to denote the absence of negative examples. | |||
Plotkin's <ref>{{cite journal|first1=Gordon D.|last1=Plotkin|title=A Note on Inductive Generalization|editor1-first=B.|editor1-last=Meltzer|editor2-first=D.|editor2-last=Michie|publisher=Edinburgh University Press|journal=Machine Intelligence|volume=5|pages=153–163|year=1970}}</ref><ref>{{cite journal|first1=Gordon D.|last1=Plotkin|title=A Further Note on Inductive Generalization|editor1-first=B.|editor1-last=Meltzer|editor2-first=D.|editor2-last=Michie|publisher=Edinburgh University Press|journal=Machine Intelligence|volume=6|pages=101–124|year=1971}}</ref> "''relative least general generalization (rlgg)''" approach to ''inductive logic programming'' shall be used to obtain a suggestion about how to formally define the daughter relation <math>\textit{dau}</math>. | |||
This approach uses the following steps. | |||
* Relativize each positive example literal with the complete background knowledge: | |||
**<math>\textit{dau}(m,h) \leftarrow \textit{par}(h,m) \land \textit{par}(h,t) \land \textit{par}(g,m) \land \textit{par}(t,e) \land \textit{par}(n,e) \land \textit{fem}(h) \land \textit{fem}(m) \land \textit{fem}(n) \land \textit{fem}(e)</math> | |||
**<math>\textit{dau}(e,t) \leftarrow \textit{par}(h,m) \land \textit{par}(h,t) \land \textit{par}(g,m) \land \textit{par}(t,e) \land \textit{par}(n,e) \land \textit{fem}(h) \land \textit{fem}(m) \land \textit{fem}(n) \land \textit{fem}(e)</math>, | |||
* Convert into [[clause normal form]]: | |||
**<math>\textit{dau}(m,h) \lor \lnot \textit{par}(h,m) \lor \lnot \textit{par}(h,t) \lor \lnot \textit{par}(g,m) \lor \lnot \textit{par}(t,e) \lor \lnot \textit{par}(n,e) \lor \lnot \textit{fem}(h) \lor \lnot \textit{fem}(m) \lor \lnot \textit{fem}(n) \lor \lnot \textit{fem}(e)</math> | |||
**<math>\textit{dau}(e,t) \lor \lnot \textit{par}(h,m) \lor \lnot \textit{par}(h,t) \lor \lnot \textit{par}(g,m) \lor \lnot \textit{par}(t,e) \lor \lnot \textit{par}(n,e) \lor \lnot \textit{fem}(h) \lor \lnot \textit{fem}(m) \lor \lnot \textit{fem}(n) \lor \lnot \textit{fem}(e)</math>, | |||
* [[anti-unification (computer science)|Anti-unify]] each compatible <ref>i.e. sharing the same predicate symbol and negated/unnegated status</ref> pair <ref>in general: <math>n</math>-tuple when <math>n</math> positive example literals are given</ref> of literals: | |||
**<math>\textit{dau}(x_{me},x_{ht})</math> from <math>\textit{dau}(m,h)</math> and <math>\textit{dau}(e,t)</math>, | |||
**<math>\lnot \textit{par}(x_{ht},x_{me})</math> from <math>\lnot \textit{par}(h,m)</math> and <math>\lnot \textit{par}(t,e)</math>, | |||
**<math>\lnot \textit{fem}(x_{me})</math> from <math>\lnot \textit{fem}(m)</math> and <math>\lnot \textit{fem}(e)</math>, | |||
**<math>\lnot \textit{par}(g,m)</math> from <math>\lnot \textit{par}(g,m)</math> and <math>\lnot \textit{par}(g,m)</math>, similar for all other background-knowledge literals | |||
**<math>\lnot \textit{par}(x_{gt},x_{me})</math> from <math>\lnot \textit{par}(g,m)</math> and <math>\lnot \textit{par}(t,e)</math>, and many more negated literals | |||
* Delete all negated literals containing variables that don't occur in a positive literal: | |||
**after deleting all negated literals containing other variables than <math>x_{me},x_{ht}</math>, only <math>\textit{dau}(x_{me},x_{ht}) \lor \lnot \textit{par}(x_{ht},x_{me}) \lor \lnot \textit{fem}(x_{me})</math> remains, together with all ground literals from the background knowledge | |||
* Convert clauses back to Horn form: | |||
** <math>\textit{dau}(x_{me},x_{ht}) \leftarrow \textit{par}(x_{ht},x_{me}) \land \textit{fem}(x_{me}) \land (\text{all background knowledge facts})</math> | |||
The resulting Horn clause is the hypothesis <math>h</math> obtained by the rlgg approach. Ignoring the background knowledge facts, the clause informally reads "''<math>x_{me}</math> is called a daughter of <math>x_{ht}</math> if <math>x_{ht}</math> is the parent of <math>x_{me}</math> and <math>x_{me}</math> is female''", which is a commonly accepted definition. | |||
Concerning the [[#Formal definition|above]] requirements, "''Necessity''" was satisfied because the predicate <math>\textit{dau}</math> doesn't appear in the background knowledge, which hence cannot imply any property containing this predicate, such as the positive examples are. | |||
"''Sufficiency''" is satisfied by the computed hypothesis <math>h</math>, since it, together with <math>\textit{par}(h,m) \land \textit{fem}(m)</math> from the background knowledge, implies the first positive example <math>\textit{dau}(m,h)</math>, and similarly <math>h</math> and <math>\textit{par}(t,e) \land \textit{fem}(e)</math> from the background knowledge implies the second positive example <math>\textit{dau}(e,t)</math>. "''Weak consistency''" is satisfied by <math>h</math>, since <math>h</math> holds in the (finite) [[Herbrand structure]] described by the background knowledge; similar for "''Strong consistency''". | |||
The common definition of the grandmother relation, viz. <math>\textit{gra}(x,z) \leftarrow \textit{fem}(x) \land \textit{par}(x,y) \land \textit{par}(y,z)</math>, cannot be learned using the above approach, since the variable <math>y</math> occurs in the clause body only; the corresponding literals would have been deleted in the 4th step of the approach. To overcome this flaw, that step has to be modified such that it can be parametrized with different ''literal post-selection heuristics''. Historically, the GOLEM implementation is based on the rlgg approach. | |||
== Implementations == | |||
* ACE (A Combined Engine) (http://dtai.cs.kuleuven.be/ACE/) | |||
* Aleph (http://web.comlab.ox.ac.uk/oucl/research/areas/machlearn/Aleph/) | |||
* Claudien (http://dtai.cs.kuleuven.be/claudien/) | |||
* DL-Learner (http://dl-learner.org) | |||
* DMax (http://dtai.cs.kuleuven.be/dmax/) | |||
* [[First Order Inductive Learner|FOIL]] (ftp://ftp.cs.su.oz.au/pub/foil6.sh) | |||
* [[Golem (ILP)]] (http://www.doc.ic.ac.uk/~shm/Software/golem) | |||
* Inthelex (INcremental THEory Learner from EXamples) (http://lacam.di.uniba.it:8000/systems/inthelex/) | |||
* Lime (http://cs.anu.edu.au/people/Eric.McCreath/lime.html) | |||
* Mio (http://libra.msra.cn/Publication/3392493/mio-user-s-manual) | |||
* MIS (Model Inference System) by Ehud Shapiro | |||
* [[PROGOL]] (http://www.doc.ic.ac.uk/~shm/Software/progol5.0) | |||
* RSD (http://labe.felk.cvut.cz/~zelezny/rsd/) | |||
* Warmr (now included in ACE) | |||
==See also== | |||
* [[Inductive reasoning]] | |||
* [[Inductive programming]] | |||
== References == | |||
{{reflist}} | |||
==Further reading== | |||
{{Refbegin}} | |||
* {{cite doi|10.1016/0743-1066(94)90035-3}} | |||
* {{cite book | first1 = N. | last1 = Lavrac | first2 = S. | last2 = Dzeroski | title = Inductive Logic Programming: Techniques and Applications | publisher = Ellis Horwood | location = New York | year = 1994 | isbn = 0-13-457870-8 | url = http://www-ai.ijs.si/SasoDzeroski/ILPBook/ }} | |||
{{Refend}} | |||
[[Category:Inductive logic programming|*]] |
Revision as of 22:39, 17 January 2014
Template:Programming paradigms
Inductive logic programming (ILP) is a subfield of machine learning which uses logic programming as a uniform representation for examples, background knowledge and hypotheses. Given an encoding of the known background knowledge and a set of examples represented as a logical database of facts, an ILP system will derive a hypothesised logic program which entails all the positive and none of the negative examples.
Schema: positive examples + negative examples + background knowledge => hypothesis.
Inductive logic programming is particularly useful in bioinformatics and natural language processing. The term Inductive Logic Programming was first introduced[1] in a paper by Stephen Muggleton in 1991.[2] The term "inductive" here refers to philosophical (i.e. suggesting a theory to explain observed facts) rather than mathematical (i.e. proving a property for all members of a well-ordered set) induction.
Formal definition
The background knowledge is given as a logical proposition , commonly in the form of Horn clauses used in logic programming. The positive and negative examples are given as a conjunction and of unnegated and negated ground literals, respectively. A hypothesis is a logical proposition satisfying the following requirements.[3]
Necessity: | |||
Sufficiency: | |||
Weak consistency: | |||
Strong consistency: |
"Necessity" does not impose a restriction on , but forbids any generation of a hypothesis as long as the positive facts are explainable without it. "Sufficiency" requires any generated hypothesis to explain all positive examples . "Weak consistency" forbids generation of any hypothesis that contradicts the background knowledge . "Strong consistency" also forbids generation of any hypothesis that is inconsistent with the negative examples , given the background knowledge ; it implies "Weak consistency"; if no negative examples are given, both requirements coincide. Džeroski [4] requires only "Sufficiency" (called "Completeness" there) and "Strong consistency".
Example
![](https://upload.wikimedia.org/wikipedia/commons/0/0a/Family_relations_example_for_inductive_logic_programming_article.gif)
The following well-known example about learning definitions of family relations uses the abbreviations , , , , , , , , and . It starts from the background knowledge (cf. picture)
the positive examples
and the trivial proposition to denote the absence of negative examples.
Plotkin's [5][6] "relative least general generalization (rlgg)" approach to inductive logic programming shall be used to obtain a suggestion about how to formally define the daughter relation .
This approach uses the following steps.
- Relativize each positive example literal with the complete background knowledge:
- Convert into clause normal form:
- Anti-unify each compatible [7] pair [8] of literals:
- Delete all negated literals containing variables that don't occur in a positive literal:
- Convert clauses back to Horn form:
The resulting Horn clause is the hypothesis obtained by the rlgg approach. Ignoring the background knowledge facts, the clause informally reads " is called a daughter of if is the parent of and is female", which is a commonly accepted definition.
Concerning the above requirements, "Necessity" was satisfied because the predicate doesn't appear in the background knowledge, which hence cannot imply any property containing this predicate, such as the positive examples are. "Sufficiency" is satisfied by the computed hypothesis , since it, together with from the background knowledge, implies the first positive example , and similarly and from the background knowledge implies the second positive example . "Weak consistency" is satisfied by , since holds in the (finite) Herbrand structure described by the background knowledge; similar for "Strong consistency".
The common definition of the grandmother relation, viz. , cannot be learned using the above approach, since the variable occurs in the clause body only; the corresponding literals would have been deleted in the 4th step of the approach. To overcome this flaw, that step has to be modified such that it can be parametrized with different literal post-selection heuristics. Historically, the GOLEM implementation is based on the rlgg approach.
Implementations
- ACE (A Combined Engine) (http://dtai.cs.kuleuven.be/ACE/)
- Aleph (http://web.comlab.ox.ac.uk/oucl/research/areas/machlearn/Aleph/)
- Claudien (http://dtai.cs.kuleuven.be/claudien/)
- DL-Learner (http://dl-learner.org)
- DMax (http://dtai.cs.kuleuven.be/dmax/)
- FOIL (ftp://ftp.cs.su.oz.au/pub/foil6.sh)
- Golem (ILP) (http://www.doc.ic.ac.uk/~shm/Software/golem)
- Inthelex (INcremental THEory Learner from EXamples) (http://lacam.di.uniba.it:8000/systems/inthelex/)
- Lime (http://cs.anu.edu.au/people/Eric.McCreath/lime.html)
- Mio (http://libra.msra.cn/Publication/3392493/mio-user-s-manual)
- MIS (Model Inference System) by Ehud Shapiro
- PROGOL (http://www.doc.ic.ac.uk/~shm/Software/progol5.0)
- RSD (http://labe.felk.cvut.cz/~zelezny/rsd/)
- Warmr (now included in ACE)
See also
References
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Further reading
- Template:Cite doi
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My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
- ↑ Luc De Raedt. A Perspective on Inductive Logic Programming. The Workshop on Current and Future Trends in Logic Programming, Shakertown, to appear in Springer LNCS, 1999. CiteSeerX: Template:Url
- ↑ Template:Cite doi
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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 - ↑ i.e. sharing the same predicate symbol and negated/unnegated status
- ↑ in general: -tuple when positive example literals are given