Runtime Complexity proof of /tmp/tmpIjt

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

0 Prolog

↳1 LPReorderTransformerProof (⇔, 0 ms)

↳2 Prolog

↳3 PrologToCdtProblemTransformerProof (UPPER BOUND (ID), 0 ms)

↳4 CdtProblem

    ↳5 SIsEmptyProof (⇔, 3 ms)

    ↳6 BOUNDS(O(1), O(1))

↳7 PrologToCdtProblemTransformerProof (UPPER BOUND (ID), 0 ms)

↳8 CdtProblem

(0) Obligation:

Clauses:

p(.(A, [])) :- l(.(A, [])).
r(1).
l([]).
l(.(H, T)) :- ','(r(H), l(T)).

Query: p(a)

(1) LPReorderTransformerProof (EQUIVALENT transformation)

Reordered facts before rules in definite LP [PROLOG].

(2) Obligation:

Clauses:

r(1).
l([]).
p(.(A, [])) :- l(.(A, [])).
l(.(H, T)) :- ','(r(H), l(T)).

Query: p(a)

(3) 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(T1)\n\n", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
3 [label="3: p(T1), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: l(.(T5, []))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: l(.(T5, [])), SCOPE: 2, CLAUSE: 1 | l(.(T5, [])), SCOPE: 2, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: l(.(T5, [])), SCOPE: 2, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ','(r(T10), l([]))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: ','(r(T10), l([])), SCOPE: 3, CLAUSE: 0\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: l([])\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: l([]), SCOPE: 4, CLAUSE: 1 | l([]), SCOPE: 4, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: true | l([]), SCOPE: 4, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: l([]), SCOPE: 4, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 29 [arrowhead = none ];
29 -> 3;

30 [label="EVAL with clause\np(.(X3, [])) :- l(.(X3, [])).\nand substitutionX3 -> T5,\nT1 -> .(T5, []),\nT4 -> T5", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 30 [arrowhead = none ];
30 -> 6;

31 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 31 [arrowhead = none ];
31 -> 8;

32 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
6 -> 32 [arrowhead = none ];
32 -> 16;

33 [label="BACKTRACK\nfor clause: l([])because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
16 -> 33 [arrowhead = none ];
33 -> 17;

34 [label="ONLY EVAL with clause\nl(.(X8, X9)) :- ','(r(X8), l(X9)).\nand substitutionT5 -> T10,\nX8 -> T10,\nX9 -> [],\nT9 -> T10", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 34 [arrowhead = none ];
34 -> 21;

35 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
21 -> 35 [arrowhead = none ];
35 -> 22;

36 [label="EVAL with clause\nr(1).\nand substitutionT10 -> 1", fontsize=14, style = filled, fillcolor = lightblue];
22 -> 36 [arrowhead = none ];
36 -> 23;

37 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
22 -> 37 [arrowhead = none ];
37 -> 24;

38 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
23 -> 38 [arrowhead = none ];
38 -> 25;

39 [label="ONLY EVAL with clause\nl([]).\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
25 -> 39 [arrowhead = none ];
39 -> 26;

40 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
26 -> 40 [arrowhead = none ];
40 -> 27;

41 [label="BACKTRACK\nfor clause: l(.(H, T)) :- ','(r(H), l(T))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
27 -> 41 [arrowhead = none ];
41 -> 28;

}

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in → f2_out1(.(1, []))

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

f2_in

Defined Pair Symbols:none

Compound Symbols:none

(5) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(6) BOUNDS(O(1), O(1))

(7) 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(T1)\n\n", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: p(T1), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: l(.(T4, []))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: l(.(T4, [])), SCOPE: 2, CLAUSE: 1 | l(.(T4, [])), SCOPE: 2, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: l(.(T4, [])), SCOPE: 2, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(r(T9), l([]))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: ','(r(T9), l([])), SCOPE: 3, CLAUSE: 0\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: l([])\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: l([]), SCOPE: 4, CLAUSE: 1 | l([]), SCOPE: 4, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: true | l([]), SCOPE: 4, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: l([]), SCOPE: 4, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 21 [arrowhead = none ];
21 -> 4;

22 [label="EVAL with clause\np(.(X2, [])) :- l(.(X2, [])).\nand substitutionX2 -> T4,\nT1 -> .(T4, []),\nT3 -> T4", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 22 [arrowhead = none ];
22 -> 5;

23 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 23 [arrowhead = none ];
23 -> 7;

24 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
5 -> 24 [arrowhead = none ];
24 -> 9;

25 [label="BACKTRACK\nfor clause: l([])because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 25 [arrowhead = none ];
25 -> 10;

26 [label="ONLY EVAL with clause\nl(.(X7, X8)) :- ','(r(X7), l(X8)).\nand substitutionT4 -> T9,\nX7 -> T9,\nX8 -> [],\nT8 -> T9", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 26 [arrowhead = none ];
26 -> 11;

27 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
11 -> 27 [arrowhead = none ];
27 -> 12;

28 [label="EVAL with clause\nr(1).\nand substitutionT9 -> 1", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 28 [arrowhead = none ];
28 -> 13;

29 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
12 -> 29 [arrowhead = none ];
29 -> 14;

30 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
13 -> 30 [arrowhead = none ];
30 -> 15;

31 [label="ONLY EVAL with clause\nl([]).\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 31 [arrowhead = none ];
31 -> 18;

32 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
18 -> 32 [arrowhead = none ];
32 -> 19;

33 [label="BACKTRACK\nfor clause: l(.(H, T)) :- ','(r(H), l(T))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
19 -> 33 [arrowhead = none ];
33 -> 20;

}

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in → f1_out1(.(1, []))

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

f1_in

Defined Pair Symbols:none

Compound Symbols:none