Prolog ways to compare variables - reference

Question

I was trying to implement some graph algorithms in Prolog. I came up with an idea to use unification to build a tree from the graph structure:
The graph would be defined as follows:
A list of Vertex-Variable pairs where Vertex is a constant representing the vertex and Variable is a corresponding variable, that would be used as a "reference" to the vertex. e.g.:
[a-A, b-B, c-C, d-D]
A list of VertexVar-NeighboursList pairs, where the VertexVar and the individual neighbours in the NeighboursList are the "reference variables". e.g.:
[A-[B, C, D], B-[A, C], C-[A, B], D-[A]] meaning b, c, d are neighbours of a etc.
Then before some graph algorithm (like searching for components, or simple DFS/BFS etc.) that could use some kind of tree built from the original graph, one could use some predicate like unify_neighbours that unifies the VertexVar-NeighbourList pairs as VertexVar = NeighboursList. After that, the vertex variables may be interpreted as lists of its neighbours, where each neighbour is again a list of its neighbours.
So this would result in a good performance when traversing the graph, as there is no need in linear search for some vertex and its neighbours for every vertex in the graph.
But my problem is: How to compare those vertex variables? (To check if they're the same.) I tried to use A == B, but there are some conflicts. For the example above, (with the unify_neighbours predicate) Prolog interprets the graph internally as:
[a-[S_1, S_2, S_3], b-S_1, c-S_2, d-S_3]
where:
S_1 = [[S_1, S_2, S_3], S_2]
S_2 = [[S_1, S_2, S_3], S_1]
S_3 = [[S_1, S_2, S_3]]
The problem is with S_1 and S_2 (aka b and c) as X = [something, Y], Y = [something, X], X == Y is true. The same problem would be with vertices, that share the same neighbours. e.g. U-[A, B] and V-[A, B].
So my question is: Is there any other way to compare variables, that could help me with this? Something that compares "the variables themselves", not the content, like comparing addresses in procedural programming languages? Or would that be too procedural and break the declarative idea of Prolog?
Example
graph_component(Vertices, Neighbours, C) :-
% Vertices and Neighbours as explained above.
% C is some component found in the graph.
vertices_refs(Vertices, Refs),
% Refs are only the variables from the pairs.
unify_neighbours(Neighbours), % As explained above.
rec_(Vertices, Refs, [], C).
rec_(Vertices, Refs, Found, RFound) :-
% Vertices as before.
% Refs is a stack of the vertex variables to search.
% Found are the vertices found so far.
% RFound is the resulting component found.
[Ref|RRest] = Refs,
vertices_pair(Vertices, Vertex-Ref),
% Vertex is the corresponding Vertex for the Ref variable
not(member(Vertex, Found)),
% Go deep:
rec_(Vertices, Ref, [Vertex|Found], DFound),
list_revpush_result([Vertex|Found], DFound, Found1),
% Go wide:
rec_(Vertices, RRest, Found1, RFound).
rec_(Vertices, Refs, Found, []) :-
% End of reccursion.
[Ref|_] = Refs,
vertices_pair(Vertices, Vertex-Ref),
member(Vertex, Found).
This example doesn't really work, but it's the idea. (Also, checking whether the vertices were found is done linearly, so the performance is still not good, but it's just for demonstration.) Now the predicate, that finds the corresponding vertex for the variable is implemented as:
vertices_pair([Vertex-Ref|_], Vertex-Ref).
vertices_pair([_-OtherRef|Rest], Vertex-Ref) :-
Ref \== OtherRef,
vertices_pair(Rest, Vertex-Ref).
where the \== operator is not really what I want and it creates those conflicts.

Answer 1

It is an intrinsic feature of Prolog that, once you have bound a variable to a term, it becomes indistinguishable from the term itself. In other words, if you bind two variables to the same term, you have two identical things, and there is no way to tell them apart.
Applied to your example: once you have unified every vertex-variable with the corresponding neighbours-list, all the variables are gone: you are left simply with a nested (and most likely circular) data structure, consisting of a list of lists of lists...
But as you suggest, the nested structure is an attractive idea because it gives you direct access to adjacent nodes. And although Prolog system vary somewhat in how well they support circular data structures, this need not stop you from exploiting this idea.
The only problem with your design is that a node is identified purely by the (potentially deeply nested and circular) data structure that describes the sub-graph that is reachable from it. This has the consequence that
two nodes that have the same descendants are indistinguishable
it can be very expensive to check whether two "similar looking" sub-graphs are identical or not
A simple way around that is to include a unique node identifier (such as a name or number) in your data structure. To use your example (slightly modified to make it more interesting):
make_graph(Graph) :-
Graph = [A,B,C,D],
A = node(a, [C,D]),
B = node(b, [A,C]),
C = node(c, [A,B]),
D = node(d, [A]).
You can then use that identifier to check for matching nodes, e.g. in a depth-first traversal:
dfs_visit_nodes([], Seen, Seen).
dfs_visit_nodes([node(Id,Children)|Nodes], Seen1, Seen) :-
( member(Id, Seen1) ->
Seen2 = Seen1
;
writeln(visiting(Id)),
dfs_visit_nodes(Children, [Id|Seen1], Seen2)
),
dfs_visit_nodes(Nodes, Seen2, Seen).
Sample run:
?- make_graph(G), dfs_visit_nodes(G, [], Seen).
visiting(a)
visiting(c)
visiting(b)
visiting(d)
G = [...]
Seen = [d, b, c, a]
Yes (0.00s cpu)

Answer 2

Thanks, #jschimpf, for the answer. It clarified a lot of things for me. I just got back to some graph problems with Prolog and thought I'd give this recursive data structure another try and came up with the following predicates to construct this data structure from a list of edges:
The "manual" creation of the data structure, as proposed by #jschimpf:
my_graph(Nodes) :-
Vars = [A, B, C, D, E],
Nodes = [
node(a, [edgeTo(1, B), edgeTo(5, D)]),
node(b, [edgeTo(1, A), edgeTo(4, E), edgeTo(2, C)]),
node(c, [edgeTo(2, B), edgeTo(6, F)]),
node(d, [edgeTo(5, A), edgeTo(3, E)]),
node(e, [edgeTo(3, D), edgeTo(4, B), edgeTo(1, F)]),
node(e, [edgeTo(1, E), edgeTo(6, C)])
],
Vars = Nodes.
Where edgeTo(Weight, VertexVar) represents an edge to some vertex with a weight assosiated with it. The weight is just to show that this can be customized for any additional information. node(Vertex, [edgeTo(Weight, VertexVar), ...]) represents a vertex with its neighbours.
A more "user-friendly" input format:
[edge(Weight, FromVertex, ToVertex), ...]
With optional list of vertices:
[Vertex, ...]
For the example above:
[edge(1, a, b), edge(5, a, d), edge(2, b, c), edge(4, b, e), edge(6, c, f), edge(3, d, e), edge(1, e, f)]
This list can be converted to the recursive data structure with the following predicates:
% make_directed_graph(+Edges, -Nodes)
make_directed_graph(Edges, Nodes) :-
vertices(Edges, Vertices),
vars(Vertices, Vars),
pairs(Vertices, Vars, Pairs),
nodes(Pairs, Edges, Nodes),
Vars = Nodes.
% make_graph(+Edges, -Nodes)
make_graph(Edges, Nodes) :-
vertices(Edges, Vertices),
vars(Vertices, Vars),
pairs(Vertices, Vars, Pairs),
directed(Edges, DiretedEdges),
nodes(Pairs, DiretedEdges, Nodes),
Vars = Nodes.
% make_graph(+Edges, -Nodes)
make_graph(Edges, Nodes) :-
vertices(Edges, Vertices),
vars(Vertices, Vars),
pairs(Vertices, Vars, Pairs),
directed(Edges, DiretedEdges),
nodes(Pairs, DiretedEdges, Nodes),
Vars = Nodes.
% make_directed_graph(+Vertices, +Edges, -Nodes)
make_directed_graph(Vertices, Edges, Nodes) :-
vars(Vertices, Vars),
pairs(Vertices, Vars, Pairs),
nodes(Pairs, Edges, Nodes),
Vars = Nodes.
The binary versions of these predicates assume, that every vertex can be obtained from the list of edges only - There are no "edge-less" vertices in the graph. The ternary versions take an additional list of vertices for exactly these cases.
make_directed_graph assumes the input edges to be directed, make_graph assumes them to be undirected, so it creates additional directed edges in the opposite direction:
% directed(+UndirectedEdges, -DiretedEdges)
directed([], []).
directed([edge(W, A, B)|UndirectedRest], [edge(W, A, B), edge(W, B, A)|DirectedRest]) :-
directed(UndirectedRest, DirectedRest).
To get all the vertices from the list of edges:
% vertices(+Edges, -Vertices)
vertices([], []).
vertices([edge(_, A, B)|EdgesRest], [A, B|VerticesRest]) :-
vertices(EdgesRest, VerticesRest),
\+ member(A, VerticesRest),
\+ member(B, VerticesRest).
vertices([edge(_, A, B)|EdgesRest], [A|VerticesRest]) :-
vertices(EdgesRest, VerticesRest),
\+ member(A, VerticesRest),
member(B, VerticesRest).
vertices([edge(_, A, B)|EdgesRest], [B|VerticesRest]) :-
vertices(EdgesRest, VerticesRest),
member(A, VerticesRest),
\+ member(B, VerticesRest).
vertices([edge(_, A, B)|EdgesRest], VerticesRest) :-
vertices(EdgesRest, VerticesRest),
member(A, VerticesRest),
member(B, VerticesRest).
To construct uninitialized variables for every vertex:
% vars(+List, -Vars)
vars([], []).
vars([_|ListRest], [_|VarsRest]) :-
vars(ListRest, VarsRest).
To pair up verticies and vertex variables:
% pairs(+ListA, +ListB, -Pairs)
pairs([], [], []).
pairs([AFirst|ARest], [BFirst|BRest], [AFirst-BFirst|PairsRest]) :-
pairs(ARest, BRest, PairsRest).
To construct the recursive nodes:
% nodes(+Pairs, +Edges, -Nodes)
nodes(Pairs, [], Nodes) :-
init_nodes(Pairs, Nodes).
nodes(Pairs, [EdgesFirst|EdgesRest], Nodes) :-
nodes(Pairs, EdgesRest, Nodes0),
insert_edge(Pairs, EdgesFirst, Nodes0, Nodes).
First, a list of empty nodes for every vertex is initialized:
% init_nodes(+Pairs, -EmptyNodes)
init_nodes([], []).
init_nodes([Vertex-_|PairsRest], [node(Vertex, [])|NodesRest]) :-
init_nodes(PairsRest, NodesRest).
Then the edges are inserted one by one:
% insert_edge(+Pairs, +Edge, +Nodes, -ResultingNodes)
insert_edge(Pairs, edge(W, A, B), [], [node(A, [edgeTo(W, BVar)])]) :-
vertex_var(Pairs, B, BVar).
insert_edge(Pairs, edge(W, A, B), [node(A, EdgesTo)|NodesRest], [node(A, [edgeTo(W, BVar)|EdgesTo])|NodesRest]) :-
vertex_var(Pairs, B, BVar).
insert_edge(Pairs, edge(W, A, B), [node(X, EdgesTo)|NodesRest], [node(X, EdgesTo)|ResultingNodes]) :-
A \= X,
insert_edge(Pairs, edge(W, A, B), NodesRest, ResultingNodes).
To get a vertex variable for a given vertex: (This actually works in both directions.)
% vertex_var(+Pairs, +Vertex, -Var)
vertex_var(Pairs, Vertex, Var) :-
member(Vertex-Var, Pairs).
```Prolog
This, of course, brings additional time overhead, but you can do this once and then just copy this data structure every time you need to perform some graph algorithm on it and access neighbours in constant time.
You can also add additional information to the `node` predicate. For example:
```Prolog
node(Vertex, Neighbours, OrderingVar)
Where the uninitialized variable OrderingVar can be "assigned" (initialized) in constant time with information about the vertex' position in a partial ordering of the graph, for example. So this may be used as output. (As sometimes denoted by +- in Prolog comments - an uninitialized variable as a part of an input term, that is yet to be initialized by the used predicate and provides output.)