Quotients and Transitive Closure

Computing the transitive closure of a relation and computing the quotient of a set by an equivalence relation are two common data integration operations that are unwieldly in SQL but are easy in AQL, as we demonstrate below (built-in to the IDE with example name Quotient).

The example defines a source schema about people and who likes whom and a target schema with a single entity representing groups connected by liking. There is a single schema mapping from the source to the target, and AQL's sigma operation along this mapping computes the connected groups.

We start by defining a source schema for people and who likes whom. For brevity, we do not define any types (String, Integer, etc) and use the empty typeside:

typeside Ty = empty schema Likes = literal : Ty { entities Like Person foreign_keys likee : Like -> Person liker : Like -> Person }

Here is some sample data, taken from a popular TV show:

instance SimpsonsLikes = literal : Likes { generators Ned Maud Rodd Todd MrBurns Smithers : Person l1 l2 l3 l4 : Like equations l1.liker = Ned l1.likee = Maud l2.liker = Maud l2.likee = Rodd l3.liker = Rodd l3.likee = Todd l4.liker = Smithers l4.likee = MrBurns }


Our goal is to group together all the people who are transitively connected by liking. The target schema contains a single entity called Connection:

schema Connections = literal : Ty { entities Connection }

Next, we define a schema mapping (in fact, the only mapping possible) from Likes to Connections:

mapping FindConnections = literal : Likes -> Connections { entities Person -> Connection Like -> Connection foreign_keys likee -> Connection liker -> Connection }

To find the connected groups, we use AQL's sigma operation:

instance SimpsonsConnections = sigma FindConnections SimpsonsLikes

The IDE reports that there are two groups, each of which is named by a particular representative of the group:


To determine which group a person is in, we use AQL's unit operation:

transform whichConnection = unit FindConnections SimpsonsLikes


A screen shot of the entire development is shown below: