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Relational algebra

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m (Just a thought: extensible record and more generally, type arithmetic)
(Other links: The λ Abroad. A Functional Approach To Software Components by Daniel Johannes Pieter Leijen)
 
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{{Foundations infobox}}
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==Definition==
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Relational algebra is an offshoot of first-order logic and is a set of relations closed under operators.
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== Pointfree ==
 
== Pointfree ==
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== Practice ==
 
== Practice ==
   
Thus, in contrast to direct SQL text manipulation, [[Libraries and tools/Database interfaces |database managemant]] systems can be approached also in declarative, type safe ways. More specifically, they may be implemented as domain specific embedded languages -- using e.g. Haskell for their host language. See the examples of
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Thus, in contrast to direct SQL text manipulation, [[Libraries and tools/Database interfaces|database management]] systems can be approached also in declarative, type safe ways. More specifically, they may be implemented as domain specific embedded languages -- using e.g. Haskell for their host language. See the examples of
 
* [[Libraries and tools/Database interfaces/HaskellDB|HaskellDB]]
 
* [[Libraries and tools/Database interfaces/HaskellDB|HaskellDB]]
 
* [[Libraries and tools/Database interfaces/CoddFish|CoddFish]]
 
* [[Libraries and tools/Database interfaces/CoddFish|CoddFish]]
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==Other links==
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* [http://en.wikipedia.org/wiki/Relational_algebra Wikipedia entry for relational algebra]
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* [http://research.microsoft.com/users/daan/download/papers/phd-thesis.pdf The λ Abroad. A Functional Approach To Software Components] by Daniel Johannes Pieter Leijen
   
 
[[Category:Theoretical foundations]]
 
[[Category:Theoretical foundations]]

Latest revision as of 21:00, 7 August 2007


Haskell theoretical foundations

General:
Mathematics - Category theory
Research - Curry/Howard/Lambek

Lambda calculus:
Alpha conversion - Beta reduction
Eta conversion - Lambda abstraction

Other:
Recursion - Combinatory logic
Chaitin's construction - Turing machine
Relational algebra

[edit] 1 Definition

Relational algebra is an offshoot of first-order logic and is a set of relations closed under operators.


[edit] 2 Pointfree

José Nuno Oliveira: First Steps in Pointfree Functional Dependency Theory. A concise and deep approach, it is pointfree. See also the author's homepage and also his many other papers -- many materials related to in this topic can be found.

[edit] 3 Just a thought

An early, immature thought of mine to represent relational algebra expressions:

data Query :: * -> * -> * where
        Identity :: Scheme a => Query a a
        Restrict :: (Scheme a, Scheme b) => Expr b Bool -> Query a b -> Query a b
        Project :: (Scheme a, Scheme b, Scheme b', Sub b' b) => b' -> Query a b -> Query a b'
        Rename :: (Scheme a, Scheme b, Scheme b', Iso b b') => Query a b -> Query a b'
        Product :: (Scheme a, Scheme b1, Scheme b2, Scheme b, Sum b1 b2 b) =>
                   Query a b1 -> Query a b2 -> Query a b
        Union :: (Scheme a, Scheme b, Id b) => Query a b -> Query a b -> Query a b
        Difference :: (Scheme a, Scheme b, Id b) => Query a b -> Query a b -> Query a b

... using the concepts / ideas of

The case of
Restrict
uses
Expr
. I think, the concept of
Expr
is an inside approach (making the relational algebra -- regarded as an embedded language -- richer, more autonome from the host language, but also more restricted):
data Expr :: * -> * -> * where
        Constant :: (Scheme sch, Literal a) => a -> Expr sch a
        Attribute :: (Scheme sch, Match attr a, Context attr sch) => attr -> Expr sch a
        Not :: Scheme sch => Expr sch Bool -> Expr sch Bool
        And :: Scheme sch => Expr sch Bool -> Expr sch Bool -> Expr sch Bool
        Or :: Scheme sch => Expr sch Bool -> Expr sch Bool -> Expr sch Bool
        Equal :: (Scheme sch, Eq a) => Expr sch a -> Expr sch a -> Expr sch Bool
        Less :: (Scheme sch, Ord a) => Expr sch a -> Expr sch a -> Expr sch Bool

Maybe an outside approach (exploiting the host language more, thus enjoying more generality) would be also appropriate:

data Query :: * -> * -> * where
        ...
        Restrict :: (Scheme a, Scheme b, Record br, On b br) => (br -> Bool) -> Query a b -> Query a b
        ...
        Rename :: (Scheme a, Scheme b, Scheme b', Iso b b') => (b -> b') -> Query a b -> Query a b'

[edit] 4 Practice

Thus, in contrast to direct SQL text manipulation, database management systems can be approached also in declarative, type safe ways. More specifically, they may be implemented as domain specific embedded languages -- using e.g. Haskell for their host language. See the examples of

[edit] 5 Other links