Runtime Complexity proof of /tmp/tmpEDj7CE/list2.pl

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

0 Prolog

↳1 PrologToCdtProblemTransformerProof (UPPER BOUND (ID), 61 ms)

↳2 CdtProblem

    ↳3 SIsEmptyProof (⇔, 0 ms)

    ↳4 BOUNDS(O(1), O(1))

↳5 PrologToCdtProblemTransformerProof (UPPER BOUND (ID), 66 ms)

↳6 CdtProblem

(0) Obligation:

Clauses:

list([]).
list(X) :- ','(no(empty(X)), ','(tail(X, T), list(T))).
tail([], []).
tail(.(X, Xs), Xs).
empty(X).
no(X) :- ','(X, ','(!, failure(a))).
no(X1).
failure(b).

Query: list(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: list(T1)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
3 [label="3: list(T1), SCOPE: 1, CLAUSE: 0 | list(T1), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: true | list([]), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: list(T1), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nlist(T1) !=? list([])", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: list([]), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: ','(no(empty([])), ','(tail([], X6), list(X6)))\n\nX6 is free", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: ','(no(empty([])), ','(tail([], X6), list(X6))), SCOPE: 2, CLAUSE: 5 | ','(no(empty([])), ','(tail([], X6), list(X6))), SCOPE: 2, CLAUSE: 6\n\nX6 is free", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: ','(','(call(empty([])), ','(!_2, failure(a))), ','(tail([], X6), list(X6))) | ','(no(empty([])), ','(tail([], X6), list(X6))), SCOPE: 2, CLAUSE: 6\n\nX6 is free", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: ','(empty([]), ','(','(!_2, failure(a)), ','(tail([], X6), list(X6)))) | ?_3 | ','(no(empty([])), ','(tail([], X6), list(X6))), SCOPE: 2, CLAUSE: 6\n\nX6 is free", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: ','(empty([]), ','(','(!_2, failure(a)), ','(tail([], X6), list(X6)))), SCOPE: 4, CLAUSE: 4 | ?_4 | ?_3 | ','(no(empty([])), ','(tail([], X6), list(X6))), SCOPE: 2, CLAUSE: 6\n\nX6 is free", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: ','(','(!_2, failure(a)), ','(tail([], X6), list(X6))) | ?_4 | ?_3 | ','(no(empty([])), ','(tail([], X6), list(X6))), SCOPE: 2, CLAUSE: 6\n\nX6 is free", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: ','(failure(a), ','(tail([], X6), list(X6)))\n\nX6 is free", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: ','(failure(a), ','(tail([], X6), list(X6))), SCOPE: 5, CLAUSE: 7\n\nX6 is free", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: ','(no(empty(T4)), ','(tail(T4, X17), list(X17)))\n\nT4 is ground\nX17 is free\nlist(T4) !=? list([])", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: ','(no(empty(T4)), ','(tail(T4, X17), list(X17))), SCOPE: 6, CLAUSE: 5 | ','(no(empty(T4)), ','(tail(T4, X17), list(X17))), SCOPE: 6, CLAUSE: 6\n\nT4 is ground\nX17 is free\nlist(T4) !=? list([])", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: ','(','(call(empty(T7)), ','(!_6, failure(a))), ','(tail(T7, X17), list(X17))) | ','(no(empty(T7)), ','(tail(T7, X17), list(X17))), SCOPE: 6, CLAUSE: 6\n\nT7 is ground\nX17 is free\nlist(T7) !=? list([])", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: ','(empty(T7), ','(','(!_6, failure(a)), ','(tail(T7, X17), list(X17)))) | ?_7 | ','(no(empty(T7)), ','(tail(T7, X17), list(X17))), SCOPE: 6, CLAUSE: 6\n\nT7 is ground\nX17 is free\nlist(T7) !=? list([])", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: ','(empty(T7), ','(','(!_6, failure(a)), ','(tail(T7, X17), list(X17)))), SCOPE: 8, CLAUSE: 4 | ?_8 | ?_7 | ','(no(empty(T7)), ','(tail(T7, X17), list(X17))), SCOPE: 6, CLAUSE: 6\n\nT7 is ground\nX17 is free\nlist(T7) !=? list([])", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: ','(','(!_6, failure(a)), ','(tail(T10, X17), list(X17))) | ?_8 | ?_7 | ','(no(empty(T10)), ','(tail(T10, X17), list(X17))), SCOPE: 6, CLAUSE: 6\n\nT10 is ground\nX17 is free\nlist(T10) !=? list([])", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: ','(failure(a), ','(tail(T10, X17), list(X17)))\n\nT10 is ground\nX17 is free\nlist(T10) !=? list([])", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: ','(failure(a), ','(tail(T10, X17), list(X17))), SCOPE: 9, CLAUSE: 7\n\nT10 is ground\nX17 is free\nlist(T10) !=? list([])", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 45 [arrowhead = none ];
45 -> 3;

46 [label="EVAL with clause\nlist([]).\nand substitutionT1 -> []", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 46 [arrowhead = none ];
46 -> 5;

47 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 47 [arrowhead = none ];
47 -> 9;

48 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 48 [arrowhead = none ];
48 -> 10;

49 [label="ONLY EVAL with clause\nlist(X16) :- ','(no(empty(X16)), ','(tail(X16, X17), list(X17))).\nand substitutionT1 -> T4,\nX16 -> T4", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 49 [arrowhead = none ];
49 -> 31;

50 [label="ONLY EVAL with clause\nlist(X5) :- ','(no(empty(X5)), ','(tail(X5, X6), list(X6))).\nand substitutionX5 -> []", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 50 [arrowhead = none ];
50 -> 12;

51 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
12 -> 51 [arrowhead = none ];
51 -> 14;

52 [label="ONLY EVAL with clause\nno(X9) :- ','(call(X9), ','(!_2, failure(a))).\nand substitutionX9 -> empty([])", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 52 [arrowhead = none ];
52 -> 16;

53 [label="CALL", fontsize=14, style = filled, fillcolor = lightcyan];
16 -> 53 [arrowhead = none ];
53 -> 17;

54 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
17 -> 54 [arrowhead = none ];
54 -> 18;

55 [label="ONLY EVAL with clause\nempty(X12).\nand substitutionX12 -> []", fontsize=14, style = filled, fillcolor = lightblue];
18 -> 55 [arrowhead = none ];
55 -> 20;

56 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
20 -> 56 [arrowhead = none ];
56 -> 22;

57 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
22 -> 57 [arrowhead = none ];
57 -> 25;

58 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
25 -> 58 [arrowhead = none ];
58 -> 26;

59 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
31 -> 59 [arrowhead = none ];
59 -> 32;

60 [label="ONLY EVAL with clause\nno(X20) :- ','(call(X20), ','(!_6, failure(a))).\nand substitutionT4 -> T7,\nX20 -> empty(T7)", fontsize=14, style = filled, fillcolor = lightblue];
32 -> 60 [arrowhead = none ];
60 -> 33;

61 [label="CALL", fontsize=14, style = filled, fillcolor = lightcyan];
33 -> 61 [arrowhead = none ];
61 -> 35;

62 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
35 -> 62 [arrowhead = none ];
62 -> 37;

63 [label="ONLY EVAL with clause\nempty(X23).\nand substitutionT7 -> T10,\nX23 -> T10", fontsize=14, style = filled, fillcolor = lightblue];
37 -> 63 [arrowhead = none ];
63 -> 39;

64 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
39 -> 64 [arrowhead = none ];
64 -> 40;

65 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
40 -> 65 [arrowhead = none ];
65 -> 43;

66 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
43 -> 66 [arrowhead = none ];
66 -> 44;

}

(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: list(T1)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: list(T1), SCOPE: 1, CLAUSE: 0 | list(T1), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: true | list([]), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: list(T1), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nlist(T1) !=? list([])", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: list([]), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(no(empty([])), ','(tail([], X4), list(X4)))\n\nX4 is free", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(no(empty([])), ','(tail([], X4), list(X4))), SCOPE: 2, CLAUSE: 5 | ','(no(empty([])), ','(tail([], X4), list(X4))), SCOPE: 2, CLAUSE: 6\n\nX4 is free", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(','(call(empty([])), ','(!_2, failure(a))), ','(tail([], X4), list(X4))) | ','(no(empty([])), ','(tail([], X4), list(X4))), SCOPE: 2, CLAUSE: 6\n\nX4 is free", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: ','(empty([]), ','(','(!_2, failure(a)), ','(tail([], X4), list(X4)))) | ?_3 | ','(no(empty([])), ','(tail([], X4), list(X4))), SCOPE: 2, CLAUSE: 6\n\nX4 is free", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ','(empty([]), ','(','(!_2, failure(a)), ','(tail([], X4), list(X4)))), SCOPE: 4, CLAUSE: 4 | ?_4 | ?_3 | ','(no(empty([])), ','(tail([], X4), list(X4))), SCOPE: 2, CLAUSE: 6\n\nX4 is free", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: ','(','(!_2, failure(a)), ','(tail([], X4), list(X4))) | ?_4 | ?_3 | ','(no(empty([])), ','(tail([], X4), list(X4))), SCOPE: 2, CLAUSE: 6\n\nX4 is free", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: ','(failure(a), ','(tail([], X4), list(X4)))\n\nX4 is free", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: ','(failure(a), ','(tail([], X4), list(X4))), SCOPE: 5, CLAUSE: 7\n\nX4 is free", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: ','(no(empty(T3)), ','(tail(T3, X13), list(X13)))\n\nT3 is ground\nX13 is free\nlist(T3) !=? list([])", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: ','(no(empty(T3)), ','(tail(T3, X13), list(X13))), SCOPE: 6, CLAUSE: 5 | ','(no(empty(T3)), ','(tail(T3, X13), list(X13))), SCOPE: 6, CLAUSE: 6\n\nT3 is ground\nX13 is free\nlist(T3) !=? list([])", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: ','(','(call(empty(T6)), ','(!_6, failure(a))), ','(tail(T6, X13), list(X13))) | ','(no(empty(T6)), ','(tail(T6, X13), list(X13))), SCOPE: 6, CLAUSE: 6\n\nT6 is ground\nX13 is free\nlist(T6) !=? list([])", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: ','(empty(T6), ','(','(!_6, failure(a)), ','(tail(T6, X13), list(X13)))) | ?_7 | ','(no(empty(T6)), ','(tail(T6, X13), list(X13))), SCOPE: 6, CLAUSE: 6\n\nT6 is ground\nX13 is free\nlist(T6) !=? list([])", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: ','(empty(T6), ','(','(!_6, failure(a)), ','(tail(T6, X13), list(X13)))), SCOPE: 8, CLAUSE: 4 | ?_8 | ?_7 | ','(no(empty(T6)), ','(tail(T6, X13), list(X13))), SCOPE: 6, CLAUSE: 6\n\nT6 is ground\nX13 is free\nlist(T6) !=? list([])", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: ','(','(!_6, failure(a)), ','(tail(T9, X13), list(X13))) | ?_8 | ?_7 | ','(no(empty(T9)), ','(tail(T9, X13), list(X13))), SCOPE: 6, CLAUSE: 6\n\nT9 is ground\nX13 is free\nlist(T9) !=? list([])", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: ','(failure(a), ','(tail(T9, X13), list(X13)))\n\nT9 is ground\nX13 is free\nlist(T9) !=? list([])", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: ','(failure(a), ','(tail(T9, X13), list(X13))), SCOPE: 9, CLAUSE: 7\n\nT9 is ground\nX13 is free\nlist(T9) !=? list([])", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 47 [arrowhead = none ];
47 -> 4;

48 [label="EVAL with clause\nlist([]).\nand substitutionT1 -> []", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 48 [arrowhead = none ];
48 -> 6;

49 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 49 [arrowhead = none ];
49 -> 7;

50 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 50 [arrowhead = none ];
50 -> 8;

51 [label="ONLY EVAL with clause\nlist(X12) :- ','(no(empty(X12)), ','(tail(X12, X13), list(X13))).\nand substitutionT1 -> T3,\nX12 -> T3", fontsize=14, style = filled, fillcolor = lightblue];
7 -> 51 [arrowhead = none ];
51 -> 29;

52 [label="ONLY EVAL with clause\nlist(X3) :- ','(no(empty(X3)), ','(tail(X3, X4), list(X4))).\nand substitutionX3 -> []", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 52 [arrowhead = none ];
52 -> 11;

53 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
11 -> 53 [arrowhead = none ];
53 -> 13;

54 [label="ONLY EVAL with clause\nno(X7) :- ','(call(X7), ','(!_2, failure(a))).\nand substitutionX7 -> empty([])", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 54 [arrowhead = none ];
54 -> 15;

55 [label="CALL", fontsize=14, style = filled, fillcolor = lightcyan];
15 -> 55 [arrowhead = none ];
55 -> 19;

56 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
19 -> 56 [arrowhead = none ];
56 -> 21;

57 [label="ONLY EVAL with clause\nempty(X10).\nand substitutionX10 -> []", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 57 [arrowhead = none ];
57 -> 23;

58 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
23 -> 58 [arrowhead = none ];
58 -> 24;

59 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
24 -> 59 [arrowhead = none ];
59 -> 27;

60 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
27 -> 60 [arrowhead = none ];
60 -> 28;

61 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
29 -> 61 [arrowhead = none ];
61 -> 30;

62 [label="ONLY EVAL with clause\nno(X16) :- ','(call(X16), ','(!_6, failure(a))).\nand substitutionT3 -> T6,\nX16 -> empty(T6)", fontsize=14, style = filled, fillcolor = lightblue];
30 -> 62 [arrowhead = none ];
62 -> 34;

63 [label="CALL", fontsize=14, style = filled, fillcolor = lightcyan];
34 -> 63 [arrowhead = none ];
63 -> 36;

64 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
36 -> 64 [arrowhead = none ];
64 -> 38;

65 [label="ONLY EVAL with clause\nempty(X19).\nand substitutionT6 -> T9,\nX19 -> T9", fontsize=14, style = filled, fillcolor = lightblue];
38 -> 65 [arrowhead = none ];
65 -> 41;

66 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
41 -> 66 [arrowhead = none ];
66 -> 42;

67 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
42 -> 67 [arrowhead = none ];
67 -> 45;

68 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
45 -> 68 [arrowhead = none ];
68 -> 46;

}

(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