Reachability

Problem description

Reachability is one of the best studied problems in computer science. Instances of the reachability problem occurr, directly or indirectly, in a lot of relevant real world applications, ranging from databases to product configurations and networks.

Given a directed graph G=(V,E), a couple of nodes of V belongs to the Reachability relation (also called the transitive closure relation) if there is a -empty path from a to b through a non-empty sequence of edges in E. The input is provided by a relation edge(X,Y) where a fact edge(i,j) states that node j is directly reachable by an edge in E from node i, and by one tuple (fact) of the form query(a,b) .

In database terms, determining all pairs of reachable nodes in G amounts to computing the transitive closure of the relation storing the edges.

Input format

The input graph is represented by the set of its edges. The predicate used to define edges is a binary predicate "edge" where edge(a,b) means that there is a directed edge going from a to b:

edge(1,3). edge(3,4). edge(4,2). edge(3,5). edge(2,5).

The query is encoded by a single fact for the binary predicate "query", where query(a,b) asks whether node 'b' is reachable starting from node 'a'.

Output format

If the answer to the input query, say, query(a,b), is positive (i.e., b is reachable from a), then the output should be given by the predicate reaches which is a subset of the reachability relation of the edge relation such that for some path (a n1 ... nk b), the output includes at least all atoms reaches(a,n1), reaches(a,n2),..,reaches(a,nk), reaches(a,b).

Example

Input:

edge(1,3). edge(3,4). edge(3,5). edge(4,2). edge(2,5). query(1,2).

Possible Output:

reaches(1,3). reaches(1,4). reaches(1,2).

Author: Giorgio Terracina