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

(0) Obligation:

Clauses:

a :- b.
b :- c.
c :- d.
d :- ','(e, !).
e.
e :- a.

Query: a()

(1) 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: a\n\n", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: a, SCOPE: 1, CLAUSE: 0\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: b\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: b, SCOPE: 2, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: c\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: c, SCOPE: 3, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: d\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: d, SCOPE: 4, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: ','(e, !_4)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ','(e, !_4), SCOPE: 5, CLAUSE: 4 | ','(e, !_4), SCOPE: 5, CLAUSE: 5\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: !_4 | ','(e, !_4), SCOPE: 5, CLAUSE: 5\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 26 [arrowhead = none ];
26 -> 4;

27 [label="ONLY EVAL with clause\na :- b.\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 27 [arrowhead = none ];
27 -> 6;

28 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
6 -> 28 [arrowhead = none ];
28 -> 10;

29 [label="ONLY EVAL with clause\nb :- c.\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 29 [arrowhead = none ];
29 -> 11;

30 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
11 -> 30 [arrowhead = none ];
30 -> 13;

31 [label="ONLY EVAL with clause\nc :- d.\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 31 [arrowhead = none ];
31 -> 15;

32 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
15 -> 32 [arrowhead = none ];
32 -> 16;

33 [label="ONLY EVAL with clause\nd :- ','(e, !_4).\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
16 -> 33 [arrowhead = none ];
33 -> 19;

34 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
19 -> 34 [arrowhead = none ];
34 -> 21;

35 [label="ONLY EVAL with clause\ne.\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 35 [arrowhead = none ];
35 -> 23;

36 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
23 -> 36 [arrowhead = none ];
36 -> 24;

37 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
24 -> 37 [arrowhead = none ];
37 -> 25;

}

(2) 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

(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];
2 [label="2: a\n\n", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
3 [label="3: a, SCOPE: 1, CLAUSE: 0\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: b\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: b, SCOPE: 2, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: c\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: c, SCOPE: 3, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: d\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: d, SCOPE: 4, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: ','(e, !_4)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: ','(e, !_4), SCOPE: 5, CLAUSE: 4 | ','(e, !_4), SCOPE: 5, CLAUSE: 5\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: !_4 | ','(e, !_4), SCOPE: 5, CLAUSE: 5\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 27 [arrowhead = none ];
27 -> 3;

28 [label="ONLY EVAL with clause\na :- b.\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 28 [arrowhead = none ];
28 -> 5;

29 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
5 -> 29 [arrowhead = none ];
29 -> 7;

30 [label="ONLY EVAL with clause\nb :- c.\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
7 -> 30 [arrowhead = none ];
30 -> 8;

31 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
8 -> 31 [arrowhead = none ];
31 -> 9;

32 [label="ONLY EVAL with clause\nc :- d.\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 32 [arrowhead = none ];
32 -> 12;

33 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
12 -> 33 [arrowhead = none ];
33 -> 14;

34 [label="ONLY EVAL with clause\nd :- ','(e, !_4).\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 34 [arrowhead = none ];
34 -> 17;

35 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
17 -> 35 [arrowhead = none ];
35 -> 18;

36 [label="ONLY EVAL with clause\ne.\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
18 -> 36 [arrowhead = none ];
36 -> 20;

37 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
20 -> 37 [arrowhead = none ];
37 -> 22;

38 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
22 -> 38 [arrowhead = none ];
38 -> 26;

}

(6) 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