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

(0) Obligation:

Clauses:

dyck(X) :- ','(grammar(G), ','(-(terminal, string(X)), produces(G, X))).
grammar(S) :- =(S, or(e, or(and(S, S), and(o, and(S, c))))).
equal(X, X).
-(terminal, string(nil)).
-(terminal, string(and(A, X))) :- ','(terminal(A), -(terminal, string(X))).
terminal(o).
terminal(c).
terminal(e).
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: dyck(g)

(1) 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: dyck(T1)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
3 [label="3: dyck(T1), SCOPE: 1, CLAUSE: 0\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: 1\n\nT4 is ground\nX5 is free", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(=(X11, or(e, or(and(X11, X11), and(o, and(X11, c))))), ','(-(terminal, string(T4)), produces(X11, T4)))\n\nT4 is ground\nX11 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];
2 -> 13 [arrowhead = none ];
13 -> 3;

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

15 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
6 -> 15 [arrowhead = none ];
15 -> 8;

16 [label="ONLY EVAL with clause\ngrammar(X10) :- =(X10, or(e, or(and(X10, X10), and(o, and(X10, c))))).\nand substitutionX5 -> X11,\nX10 -> X11", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 16 [arrowhead = none ];
16 -> 11;

17 [label="UNIFY-FAIL\nbecause of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 17 [arrowhead = none ];
17 -> 12;

}

(2) 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

(3) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(4) BOUNDS(O(1), O(1))

(5) 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: dyck(T1)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: dyck(T1), SCOPE: 1, CLAUSE: 0\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: 1\n\nT3 is ground\nX3 is free", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: ','(=(X9, or(e, or(and(X9, X9), and(o, and(X9, c))))), ','(-(terminal, string(T3)), produces(X9, T3)))\n\nT3 is ground\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 11 [arrowhead = none ];
11 -> 4;

12 [label="ONLY EVAL with clause\ndyck(X2) :- ','(grammar(X3), ','(-(terminal, string(X2)), produces(X3, X2))).\nand substitutionT1 -> T3,\nX2 -> T3", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 12 [arrowhead = none ];
12 -> 5;

13 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
5 -> 13 [arrowhead = none ];
13 -> 7;

14 [label="ONLY EVAL with clause\ngrammar(X8) :- =(X8, or(e, or(and(X8, X8), and(o, and(X8, c))))).\nand substitutionX3 -> X9,\nX8 -> X9", fontsize=14, style = filled, fillcolor = lightblue];
7 -> 14 [arrowhead = none ];
14 -> 9;

15 [label="UNIFY-FAIL\nbecause of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 15 [arrowhead = none ];
15 -> 10;

}

(6) 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