Relational algebra

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

Definition

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

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.

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 Query be regarded as an arrow, and if so, is it worth of doing so?
• extensible record and more generally, type arithmetic

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'

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

• HaskellDB
• CoddFish

Other links

• Wikipedia entry for relational algebra
• The λ Abroad. A Functional Approach To Software Components by Daniel Johannes Pieter Leijen