Runtime Complexity of the query pattern
q(g)
w.r.t. the given Prolog program could be shown to be BOUNDS(O(1), O(1)).

(0) Obligation:

Clauses:

q(X) :- ','(grammar(G), ','(-(terminal, string(X)), produces(G, X))).
grammar(S) :- ','(equal(S, or(N, and(a, and(S, b)))), equal(N, or(c, and(c, N)))).
equal(X, X).
-(terminal, string(nil)).
-(terminal, string(and(A, X))) :- ','(terminal(A), -(terminal, string(X))).
terminal(a).
terminal(b).
terminal(c).
produces(G, X) :- gives(G, X, nil).
gives(or(P, Q), X, Y) :- gives(P, X, Y).
gives(or(P, Q), X, Y) :- gives(Q, X, Y).
gives(and(P, Q), X, Z) :- ','(gives(P, X, Y), gives(Q, Y, Z)).
gives(A, and(A, X), X) :- terminal(A).

Query: q(g)

(1) LPReorderTransformerProof (EQUIVALENT transformation)

Reordered facts before rules in definite LP [PROLOG].

(2) Obligation:

Clauses:

equal(X, X).
-(terminal, string(nil)).
terminal(a).
terminal(b).
terminal(c).
q(X) :- ','(grammar(G), ','(-(terminal, string(X)), produces(G, X))).
grammar(S) :- ','(equal(S, or(N, and(a, and(S, b)))), equal(N, or(c, and(c, N)))).
-(terminal, string(and(A, X))) :- ','(terminal(A), -(terminal, string(X))).
produces(G, X) :- gives(G, X, nil).
gives(or(P, Q), X, Y) :- gives(P, X, Y).
gives(or(P, Q), X, Y) :- gives(Q, X, Y).
gives(and(P, Q), X, Z) :- ','(gives(P, X, Y), gives(Q, Y, Z)).
gives(A, and(A, X), X) :- terminal(A).

Query: q(g)

(3) PrologToCdtProblemTransformerProof (UPPER BOUND (ID) transformation)

Built complexity over-approximating cdt problems from derivation graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
2 [label="2: q(T1)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
3 [label="3: q(T1), SCOPE: 1, CLAUSE: 5\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(grammar(X5), ','(-(terminal, string(T4)), produces(X5, T4)))\n\nT4 is ground\nX5 is free", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: ','(grammar(X5), ','(-(terminal, string(T4)), produces(X5, T4))), SCOPE: 2, CLAUSE: 6\n\nT4 is ground\nX5 is free", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ','(','(equal(X13, or(X12, and(a, and(X13, b)))), equal(X12, or(c, and(c, X12)))), ','(-(terminal, string(T4)), produces(X13, T4)))\n\nT4 is ground\nX13 is free\nX12 is free", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(','(equal(X13, or(X12, and(a, and(X13, b)))), equal(X12, or(c, and(c, X12)))), ','(-(terminal, string(T4)), produces(X13, T4))), SCOPE: 3, CLAUSE: 0\n\nT4 is ground\nX13 is free\nX12 is free", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 15 [arrowhead = none ];
15 -> 3;

16 [label="ONLY EVAL with clause\nq(X4) :- ','(grammar(X5), ','(-(terminal, string(X4)), produces(X5, X4))).\nand substitutionT1 -> T4,\nX4 -> T4", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 16 [arrowhead = none ];
16 -> 6;

17 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
6 -> 17 [arrowhead = none ];
17 -> 8;

18 [label="ONLY EVAL with clause\ngrammar(X11) :- ','(equal(X11, or(X12, and(a, and(X11, b)))), equal(X12, or(c, and(c, X12)))).\nand substitutionX5 -> X13,\nX11 -> X13", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 18 [arrowhead = none ];
18 -> 10;

19 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
10 -> 19 [arrowhead = none ];
19 -> 13;

20 [label="BACKTRACK\nfor clause: equal(X, X)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 20 [arrowhead = none ];
20 -> 14;

}

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:none

Compound Symbols:none

(5) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(6) BOUNDS(O(1), O(1))

(7) PrologToCdtProblemTransformerProof (UPPER BOUND (ID) transformation)

Built complexity over-approximating cdt problems from derivation graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: q(T1)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: q(T1), SCOPE: 1, CLAUSE: 5\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(grammar(X3), ','(-(terminal, string(T3)), produces(X3, T3)))\n\nT3 is ground\nX3 is free", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(grammar(X3), ','(-(terminal, string(T3)), produces(X3, T3))), SCOPE: 2, CLAUSE: 6\n\nT3 is ground\nX3 is free", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: ','(','(equal(X11, or(X10, and(a, and(X11, b)))), equal(X10, or(c, and(c, X10)))), ','(-(terminal, string(T3)), produces(X11, T3)))\n\nT3 is ground\nX11 is free\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(','(equal(X11, or(X10, and(a, and(X11, b)))), equal(X10, or(c, and(c, X10)))), ','(-(terminal, string(T3)), produces(X11, T3))), SCOPE: 3, CLAUSE: 0\n\nT3 is ground\nX11 is free\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 13 [arrowhead = none ];
13 -> 4;

14 [label="ONLY EVAL with clause\nq(X2) :- ','(grammar(X3), ','(-(terminal, string(X2)), produces(X3, X2))).\nand substitutionT1 -> T3,\nX2 -> T3", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 14 [arrowhead = none ];
14 -> 5;

15 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
5 -> 15 [arrowhead = none ];
15 -> 7;

16 [label="ONLY EVAL with clause\ngrammar(X9) :- ','(equal(X9, or(X10, and(a, and(X9, b)))), equal(X10, or(c, and(c, X10)))).\nand substitutionX3 -> X11,\nX9 -> X11", fontsize=14, style = filled, fillcolor = lightblue];
7 -> 16 [arrowhead = none ];
16 -> 9;

17 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
9 -> 17 [arrowhead = none ];
17 -> 11;

18 [label="BACKTRACK\nfor clause: equal(X, X)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 18 [arrowhead = none ];
18 -> 12;

}

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:none

Compound Symbols:none