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In [[type theory]], '''containers''' are abstractions which permit various "collection types", such as [[list (computing)|lists]] and [[tree (data structure)|trees]], to be represented in a uniform way. A ([[unary operation|unary]]) container is defined by a type of ''shapes'' S and a type family of ''positions'' P, indexed by S. The ''extension'' of a container is a family of dependent pairs consisting of a shape (of type S) and a function from positions of that shape to the element type. Containers can be seen as [[canonical form]]s for collection types.<ref name='containers'>{{cite journal |author=Michael Abbott, Thorsten Altenkirch and Neil Ghani|year=2005 |title=Containers: Constructing strictly positive types |journal=Theoretical Computer Science |volume=342 |issue=1 |pages=3–27 |doi=10.1016/j.tcs.2005.06.002}}</ref> | |||
For lists, the shape type is the [[natural number]]s (including zero). The corresponding position types are the types of natural numbers less than the shape, for each shape. | |||
For trees, the shape type is the type of trees of units (that is, trees with no information in them, just structure). The corresponding position types are isomorphic to the types of valid paths from the root to particular nodes on the shape, for each shape. | |||
Note that the natural numbers are isomorphic to lists of units. In general the shape type will always be isomorphic to the original non-generic container type family (list, tree etc.) applied to unit. | |||
One of the main motivations for introducing the notion of containers is to support [[generic programming]] in a [[dependent type|dependently-typed]] setting.<ref name='containers'/> | |||
== Categorical aspects == | |||
{{Expand section|date=October 2008}} | |||
The extension of a container is an [[endofunctor]]. It takes a function <code>g</code> to | |||
<math>\lambda\left(s,f\right).\left(s,g\circ f\right)</math> | |||
This is equivalent to the familiar <code>map g</code> in the case of lists, and does something similar for other containers. | |||
== Indexed containers == | |||
{{Expand section|date=October 2008}} | |||
'''Indexed containers''' (also known as '''dependent polynomial functors''') are a generalisation of containers, which can represent a wider class of types, such as vectors (sized lists).<ref name='indexed-containers'>{{cite journal | |||
| author = Thorsten Altenkirch, Neil Ghani, Peter Hancock, [[Conor McBride]], and Peter Morris | |||
| title = Indexed Containers | |||
| version = Unpublished manuscript | |||
| url = http://strictlypositive.org/indexed-containers.pdf | |||
| format = PDF | |||
| accessdate = 2008-10-30 | |||
}} | |||
</ref> | |||
The element type (called the ''input type'') is indexed by shape and position, so it can vary by shape and position, and the extension (called the ''output type'') is also indexed by shape. | |||
== See also == | |||
* [[Container (abstract data type)]] | |||
* [[Functor (type theory)]] | |||
== References == | |||
{{reflist}} | |||
[[Category:Type theory]] | |||
{{type-theory-stub}} | |||
Revision as of 06:08, 20 February 2013
In type theory, containers are abstractions which permit various "collection types", such as lists and trees, to be represented in a uniform way. A (unary) container is defined by a type of shapes S and a type family of positions P, indexed by S. The extension of a container is a family of dependent pairs consisting of a shape (of type S) and a function from positions of that shape to the element type. Containers can be seen as canonical forms for collection types.[1]
For lists, the shape type is the natural numbers (including zero). The corresponding position types are the types of natural numbers less than the shape, for each shape.
For trees, the shape type is the type of trees of units (that is, trees with no information in them, just structure). The corresponding position types are isomorphic to the types of valid paths from the root to particular nodes on the shape, for each shape.
Note that the natural numbers are isomorphic to lists of units. In general the shape type will always be isomorphic to the original non-generic container type family (list, tree etc.) applied to unit.
One of the main motivations for introducing the notion of containers is to support generic programming in a dependently-typed setting.[1]
Categorical aspects
Template:Expand section
The extension of a container is an endofunctor. It takes a function g to
This is equivalent to the familiar map g in the case of lists, and does something similar for other containers.
Indexed containers
Template:Expand section Indexed containers (also known as dependent polynomial functors) are a generalisation of containers, which can represent a wider class of types, such as vectors (sized lists).[2]
The element type (called the input type) is indexed by shape and position, so it can vary by shape and position, and the extension (called the output type) is also indexed by shape.
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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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