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

(0) Obligation:

Clauses:

f(0, 1, X) :- f(X, X, X).

Query: f(a,a,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: f(T1, T2, T3)\n\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: f(T1, T2, T3), SCOPE: 1, CLAUSE: 0\n\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: f(T6, T6, T6)\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: f(T6, T6, T6), SCOPE: 2, CLAUSE: 0\n\nT6 is ground", 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 -> 4;

14 [label="EVAL with clause\nf(0, 1, X3) :- f(X3, X3, X3).\nand substitutionT1 -> 0,\nT2 -> 1,\nT3 -> T6,\nX3 -> T6", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 14 [arrowhead = none ];
14 -> 6;

15 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 15 [arrowhead = none ];
15 -> 7;

16 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
6 -> 16 [arrowhead = none ];
16 -> 11;

17 [label="BACKTRACK\nfor clause: f(0, 1, X) :- f(X, X, X)because 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: f(T1, T2, T3)\n\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
3 [label="3: f(T1, T2, T3), SCOPE: 1, CLAUSE: 0\n\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: f(T5, T5, T5)\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: f(T5, T5, T5), SCOPE: 2, CLAUSE: 0\n\nT5 is ground", 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 -> 3;

12 [label="EVAL with clause\nf(0, 1, X2) :- f(X2, X2, X2).\nand substitutionT1 -> 0,\nT2 -> 1,\nT3 -> T5,\nX2 -> T5", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 12 [arrowhead = none ];
12 -> 5;

13 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 13 [arrowhead = none ];
13 -> 8;

14 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
5 -> 14 [arrowhead = none ];
14 -> 9;

15 [label="BACKTRACK\nfor clause: f(0, 1, X) :- f(X, X, X)because 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