# 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

- generalised algebraic datatype
- a sort of differential approach (I think I took it from Zipper). Can be regarded as an arrow, and if so, is it worth of doing so?Query

Restrict

Expr

Expr

*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