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

(0) Obligation:

Clauses:

p :- ','(q, ','(r, !)).
q.
q :- q.
r.

Query: p()

(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: p\n\n", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: p, SCOPE: 1, CLAUSE: 0\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(q, ','(r, !_1))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: ','(q, ','(r, !_1)), SCOPE: 2, CLAUSE: 1 | ','(q, ','(r, !_1)), SCOPE: 2, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ','(r, !_1) | ','(q, ','(r, !_1)), SCOPE: 2, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: ','(r, !_1), SCOPE: 3, CLAUSE: 3 | ?_3 | ','(q, ','(r, !_1)), SCOPE: 2, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: !_1 | ?_3 | ','(q, ','(r, !_1)), SCOPE: 2, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 19 [arrowhead = none ];
19 -> 4;

20 [label="ONLY EVAL with clause\np :- ','(q, ','(r, !_1)).\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 20 [arrowhead = none ];
20 -> 6;

21 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
6 -> 21 [arrowhead = none ];
21 -> 8;

22 [label="ONLY EVAL with clause\nq.\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 22 [arrowhead = none ];
22 -> 10;

23 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
10 -> 23 [arrowhead = none ];
23 -> 12;

24 [label="ONLY EVAL with clause\nr.\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 24 [arrowhead = none ];
24 -> 14;

25 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
14 -> 25 [arrowhead = none ];
25 -> 16;

26 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
16 -> 26 [arrowhead = none ];
26 -> 18;

}

(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: p\n\n", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
3 [label="3: p, SCOPE: 1, CLAUSE: 0\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(q, ','(r, !_1))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(q, ','(r, !_1)), SCOPE: 2, CLAUSE: 1 | ','(q, ','(r, !_1)), SCOPE: 2, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: ','(r, !_1) | ','(q, ','(r, !_1)), SCOPE: 2, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(r, !_1), SCOPE: 3, CLAUSE: 3 | ?_3 | ','(q, ','(r, !_1)), SCOPE: 2, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: !_1 | ?_3 | ','(q, ','(r, !_1)), SCOPE: 2, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 18 [arrowhead = none ];
18 -> 3;

19 [label="ONLY EVAL with clause\np :- ','(q, ','(r, !_1)).\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 19 [arrowhead = none ];
19 -> 5;

20 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
5 -> 20 [arrowhead = none ];
20 -> 7;

21 [label="ONLY EVAL with clause\nq.\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
7 -> 21 [arrowhead = none ];
21 -> 9;

22 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
9 -> 22 [arrowhead = none ];
22 -> 11;

23 [label="ONLY EVAL with clause\nr.\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 23 [arrowhead = none ];
23 -> 13;

24 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 24 [arrowhead = none ];
24 -> 15;

25 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
15 -> 25 [arrowhead = none ];
25 -> 17;

}

(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