Defining Prolog DCG with extra arguments - prolog

Question

I am trying to define a Prolog DCG for the set of strings 0^N 1^M 2^N+M of length 2N + 2M for N, M >= 0 using extra arguments. An example of a correct string would be "011222" but not "012".
I have used the following code to create this DCG.
s --> a(N), b(M), c(N), c(M).
a(0) --> [].
a(succ(X)) --> [0], a(X).
b(0) --> [].
b(succ(X)) --> [1], b(X).
c(0) --> [].
c(succ(X)) --> [2], c(X).
When I run the query
s([0,1,1,2,2,2], []).
Prolog returns true as expected.
However when I run
s(X, []).
Prolog returns the following:
X = []
X = [1,2]
X = [1,1,2,2]
X = [1,1,1,2,2,2]
These are not valid strings. I think this may be because N and M are being decremented by the c predicate before prolog runs the a and b predicates. Is this the case? How could this be resolved?
Edit:
I've tried modifying the s production to this:
s --> a(N), b(M), c(NplusM), {NplusM is N + M}.
but that gives an error when running queries.

Answer 1

IMO the answers you are getting are correct!
I renamed your grammar from s to aN_bM_cNM and added two additional arguments, one for N, the other for M. Also, I renamed succ to s:
aN_bM_cNM(N, M) --> n_reps(N, 0), n_reps(M, 1), n_reps(N, 2), n_reps(M, 2).
n_reps( 0 , _) --> [].
n_reps(s(N), E) --> [E], n_reps(N, E).
Now let's run the query that #CapelliC gave. The goal length(Xs, _) ensures fair enumeration of the infinite solution set of aN_bM_cNM//2:
?- length(Xs, _), phrase(aN_bM_cNM(N,M), Xs).
( Xs = [] , N = 0 , M = 0
; Xs = [1,2] , N = 0 , M = s(0)
; Xs = [0,2] , N = s(0) , M = 0
; Xs = [1,1,2,2] , N = 0 , M = s(s(0))
; Xs = [0,1,2,2] , N = s(0) , M = s(0)
; Xs = [0,0,2,2] , N = s(s(0)) , M = 0
; Xs = [1,1,1,2,2,2] , N = 0 , M = s(s(s(0)))
; Xs = [0,1,1,2,2,2] , N = s(0) , M = s(s(0))
; Xs = [0,0,1,2,2,2] , N = s(s(0)) , M = s(0)
; Xs = [0,0,0,2,2,2] , N = s(s(s(0))), M = 0
; Xs = [1,1,1,1,2,2,2,2], N = 0 , M = s(s(s(s(0))))
...
To raise the lower bound of N or M, just state an additional goal of the form X = s(s(_)) (for a minimum value of 2).
In the following query both N and M are to be greater than 0:
?- N = s(_) , M = s(_) , length(Xs, _), phrase(aN_bM_cNM(N,M), Xs).
( N = s(0) , M = s(0) , Xs = [0,1,2,2]
; N = s(0) , M = s(s(0)) , Xs = [0,1,1,2,2,2]
; N = s(s(0)), M = s(0) , Xs = [0,0,1,2,2,2]
; N = s(0) , M = s(s(s(0))), Xs = [0,1,1,1,2,2,2,2]
...

Answer 2

You're misusing succ/2, maybe because you expect Prolog evaluates functions in head patterns. It doesn't. Then, try to replace your rules with
a(0) --> [].
a(Y) --> {succ(X,Y)}, [0], a(X).
etc etc
edit since succ/2 needs at least one argument instantiated to an integer, we could supply N,M to the DCG entry, or, using CLP(FD):
:- use_module(library(clpfd)).
s --> a(N), b(M), c(N), c(M).
a(0) --> [].
a(Y) --> {Y #= X-1}, [0], a(X).
b(0) --> [].
b(Y) --> {Y #= X-1}, [1], b(X).
c(0) --> [].
c(Y) --> {Y #= X-1}, [2], c(X).
but still, list' length must be provided. For example
?- length(L,_),phrase(s,L).
L = [] ;
L = [1, 2] ;
L = [0, 2] ;
L = [1, 1, 2, 2] ;
L = [0, 1, 2, 2] ;
L = [0, 0, 2, 2] ;
...