Find the shortest path in DLV - path

Question

I am trying to find all the paths in a graph with minimum distance using DLV. Say I have the following graph:
I am expecting to obtain the predicates (I hope I don't skip any):
path(a, b, 1), path(a, d, 1), path(a, e, 1), path(a, c, 2)
path(b, a, 1), path(b, c, 1), path(d, d, 2), path(b, e, 2)
path(c, b, 1), path(c, e, 1), path(c, a, 2), path(c, d, 3)
path(d, a, 1), path(d, b, 2), path(d, e, 2), path(d, c, 3)
path(e, a, 1), path(e, c, 1), path(e, d, 2), path(e, b, 2)
I assume that you can travel an arch both left or right. So, I tried the following:
path(X, Y, 1) :- arc(X, Y).
path(Y, X, 1) :- arc(X, Y).
path(X, Z, L) :- path(X, Y, M), path(Y, Z, N),
X!=Z,
L = M + N,
not path(X, Z, V), V < L, #int(V)
The idea of the third rule was to add 2 existing paths if they are not going back (X!=Z) and there is not already a path connecting the same edges with a shorter distance (not path(X, Z, V), V < L, #int(V)). I had to add #int(V) because otherwise the rule was not safe. I don't know if there is a better way of resolving this safety issue with an integer value.
When I run this code (with the flag -N=5 to set #maxint=5) I get paths that should not be there, for example, path(d,a,5). I don't know if the problem is with the #int(V) or something else but I wouldn't expect these paths to appear since I already have a path(d,a,1). Probably it is because of #int(V) but I can't figure out how to do this right.
Can anyone help me solve this? Thanks in advance.

Answer 1

Solution to the problem using lists to keep track of the path:
path(X, Y, [X, Y], 1) :- arc(X, Y).
path(Y, X, [Y, X], 1) :- arc(X, Y).
path(X, Z, P, D) :- path(X, Y, P1, D1),
path(Y, Z, P2, 1),
#insLast(P1, Z, P),
D = D1 + 1,
not #member(Z, P1).
shortest_path(X, Y, D) :- node(X), node(Y),
#min{L: path(X, Y, P, L)} = D.
Solution without the need of lists (with the help of CapelliC)
path(X, Y, 1) :- arc(X,Y).
path(Y, X, 1) :- arc(X,Y).
path(X, Y, D) :- path(X,Z,D0), arc(Z,Y),
#count{A: node(A)} = Max,
D0<Max, X != Y,
D = D0+1.
shorter_paths(X, Y, D) :- node(X), node(Y),
#min{L: path(X, Y, L)} = D.
Note that we need to define all nodes with a predicate node() and that the predicate arc() assumes that the edge of the graph is bidirectional.

Answer 2

examples/spaths.dl from DES distribution. See the commented code below... -
%
% Shortest Paths in a Graph
%
% Datalog Formulation
%
% Program: Shortest paths in a graph
% Author : Fernando Sáenz-Pérez
% Date : September, 2009
edge(a,b).
edge(a,c).
edge(b,a).
edge(b,d).
path(X,Y,1) :-
edge(X,Y).
path(X,Y,L) :-
path(X,Z,L0),
edge(Z,Y),
count(edge(A,B),Max),
L0<Max,
L is L0+1.
spaths(X,Y,L) :-
min(path(X,Y,Z),Z,L).
% Note that the following is not stratifiable in DES
%sp(X,Y,1) :-
% edge(X,Y).
%sp(X,Y,L) :-
% sp(X,Z,L0),
% not(shorter(X,Z,L0)),
% edge(Z,Y),
% L is L0+1.
%shorter(X,Y,L) :-
% sp(X,Y,L0),
% L0<L.