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Relational algebra

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Contents


1 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.

2 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'

3 Practice

Database managemant systems can be approached also in declarative, type safe ways. See the examples of

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Category: Theoretical foundations