= Reachability =

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== Problem Description ==

Reachability is one of the best studied problems in computer science. Instances of the reachability problem occur{, directly or indirectly, in a lot of relevant real world applications, ranging from databases to product configurations and networks.
In database terms, determining all pairs of reachable nodes in a graph G amounts to computing the transitive closure of the relation storing the edges. The version of the problem for this competition comes in terms of queries to the transitive closure of G. 

Given a directed graph {{{G=(V,E)}}} and a couple {{{<a,b>}}} of nodes of {{{{G}}}, the truth of the atom {{{reaches(a,b)}}} determines whether node {{{b}}} is reachable from node {{{a}}} through a 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 query of the form {{{reaches(a,b)?}}}.


== Predicates ==

 * '''Input''': {{{edge/2}}}

 * '''Query''': {{{reaches/2}}}

 * '''Output''': {{{reaches/2}}}

== 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 expressed as a ground atom of the binary predicate {{{reaches}}}: e.g. {{{reaches(a,b)?}}} asks whether node 'b' is reachable from node 'a'. Only ground queries are provided.

== Output format ==

If the answer to the input query, say, {{{reaches(a,b)}}}, is positive (i.e., b is reachable from a), then the output should be a single fact representing the query itself;
if the query is false, the output is nothing (thus resulting in an empty row), according the Competition Input and Output specification.

== Notes and differences with previous versions of this problem ==

With respect to the Second ASP Competition, the goal query is herein made explicit by means of the {{{reaches(a,b)?}}} which shall be found in input instances.

== Example(s) ==

Input:
{{{edge(1,3). edge(3,4). edge(3,5). edge(4,2). edge(2,5). reaches(1,2)?}}}

Possible Output:
{{{reaches(1,2).}}}

== Problem Peculiarities ==

'''Type''': Polynomial/Query 
'''Competition''': both Model & Solve and System competition

Reachability can be considered a classic polynomial problem, not to be considered too trivial. The instance graphs will be indeed quite large: rather than modelling abilities (we expect all competitors will adopt a pretty standard encoding in the M&S track also), this problem will stress memory usage and the ability of tailoring the search space at hand to what is needed for answering the goal query only.


== Author(s) ==
 * Author: Giorgio Terracina
   * Affiliation: University of Calabria, Italy