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

(0) Obligation:

Clauses:

q :- p(X, Y).
p(a, b) :- !.
p(a, a) :- p(a, a).

Query: q()

(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: q\n\n", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
3 [label="3: q, SCOPE: 1, CLAUSE: 0\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: p(X3, X4)\n\nX3 is free\nX4 is free", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: p(X3, X4), SCOPE: 2, CLAUSE: 1 | p(X3, X4), SCOPE: 2, CLAUSE: 2\n\nX3 is free\nX4 is free", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: !_2 | p(X3, X4), SCOPE: 2, CLAUSE: 2\n\nX3 is free\nX4 is free", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: true\n\n", 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 :- p(X3, X4).\nand substitution", 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 -> 11;

18 [label="ONLY EVAL with clause\np(a, b) :- !_2.\nand substitutionX3 -> a,\nX4 -> b", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 18 [arrowhead = none ];
18 -> 12;

19 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
12 -> 19 [arrowhead = none ];
19 -> 13;

20 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 20 [arrowhead = none ];
20 -> 14;

}

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in → f2_out1

Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:

f2_in

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: q\n\n", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: q, SCOPE: 1, CLAUSE: 0\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: p(X1, X2)\n\nX1 is free\nX2 is free", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: p(X1, X2), SCOPE: 2, CLAUSE: 1 | p(X1, X2), SCOPE: 2, CLAUSE: 2\n\nX1 is free\nX2 is free", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: !_2 | p(X1, X2), SCOPE: 2, CLAUSE: 2\n\nX1 is free\nX2 is free", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: true\n\n", 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\nq :- p(X1, X2).\nand substitution", 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\np(a, b) :- !_2.\nand substitutionX1 -> a,\nX2 -> b", fontsize=14, style = filled, fillcolor = lightblue];
7 -> 14 [arrowhead = none ];
14 -> 8;

15 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 15 [arrowhead = none ];
15 -> 9;

16 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 16 [arrowhead = none ];
16 -> 10;

}

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in → f1_out1

Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:

f1_in

Defined Pair Symbols:none

Compound Symbols:none