Runtime Complexity proof of /tmp/tmpsTd9q9/ts03.pl

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 PrologToCdtProblemTransformerProof (UPPER BOUND (ID), 56 ms)

↳2 CdtProblem

    ↳3 SIsEmptyProof (⇔, 0 ms)

    ↳4 BOUNDS(O(1), O(1))

↳5 PrologToCdtProblemTransformerProof (UPPER BOUND (ID), 77 ms)

↳6 CdtProblem

(0) Obligation:

Clauses:

p(X) :- ','(no(letter(X)), q(X)).
q(x) :- p(X1).
letter(x).
no(X) :- ','(X, ','(!, failure(a))).
no(X2).
failure(b).

Query: p(a)

(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(T1)\n\n", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
3 [label="3: p(T1), SCOPE: 1, CLAUSE: 0\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(no(letter(T6)), q(T6))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: ','(no(letter(T6)), q(T6)), SCOPE: 2, CLAUSE: 3 | ','(no(letter(T6)), q(T6)), SCOPE: 2, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ','(','(call(letter(T11)), ','(!_2, failure(a))), q(T11)) | ','(no(letter(T6)), q(T6)), SCOPE: 2, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: ','(letter(T11), ','(','(!_2, failure(a)), q(T11))) | ?_3 | ','(no(letter(T6)), q(T6)), SCOPE: 2, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: ','(letter(T11), ','(','(!_2, failure(a)), q(T11))), SCOPE: 4, CLAUSE: 2 | ?_4 | ?_3 | ','(no(letter(T6)), q(T6)), SCOPE: 2, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: ','(','(!_2, failure(a)), q(x)) | ?_4 | ?_3 | ','(no(letter(T6)), q(T6)), SCOPE: 2, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: ?_4 | ?_3 | ','(no(letter(T6)), q(T6)), SCOPE: 2, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: ','(failure(a), q(x))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: ','(failure(a), q(x)), SCOPE: 5, CLAUSE: 5\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: ?_3 | ','(no(letter(T6)), q(T6)), SCOPE: 2, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: ','(no(letter(T6)), q(T6)), SCOPE: 2, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: q(T16)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: q(T16), SCOPE: 6, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: p(X18)\n\nX18 is free", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: p(X18), SCOPE: 7, CLAUSE: 0\n\nX18 is free", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: ','(no(letter(X27)), q(X27))\n\nX27 is free", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: ','(no(letter(X27)), q(X27)), SCOPE: 8, CLAUSE: 3 | ','(no(letter(X27)), q(X27)), SCOPE: 8, CLAUSE: 4\n\nX27 is free", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: ','(','(call(letter(X34)), ','(!_8, failure(a))), q(X34)) | ','(no(letter(X27)), q(X27)), SCOPE: 8, CLAUSE: 4\n\nX27 is free\nX34 is free", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: ','(letter(X34), ','(','(!_8, failure(a)), q(X34))) | ?_9 | ','(no(letter(X27)), q(X27)), SCOPE: 8, CLAUSE: 4\n\nX27 is free\nX34 is free", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: ','(letter(X34), ','(','(!_8, failure(a)), q(X34))), SCOPE: 10, CLAUSE: 2 | ?_10 | ?_9 | ','(no(letter(X27)), q(X27)), SCOPE: 8, CLAUSE: 4\n\nX27 is free\nX34 is free", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: ','(','(!_8, failure(a)), q(x)) | ?_10 | ?_9 | ','(no(letter(X27)), q(X27)), SCOPE: 8, CLAUSE: 4\n\nX27 is free", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: ','(failure(a), q(x))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: ','(failure(a), q(x)), SCOPE: 11, CLAUSE: 5\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 57 [arrowhead = none ];
57 -> 3;

58 [label="ONLY EVAL with clause\np(X6) :- ','(no(letter(X6)), q(X6)).\nand substitutionT1 -> T6,\nX6 -> T6,\nT5 -> T6", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 58 [arrowhead = none ];
58 -> 6;

59 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
6 -> 59 [arrowhead = none ];
59 -> 8;

60 [label="ONLY EVAL with clause\nno(X10) :- ','(call(X10), ','(!_2, failure(a))).\nand substitutionT6 -> T11,\nX10 -> letter(T11),\nT10 -> T11", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 60 [arrowhead = none ];
60 -> 10;

61 [label="CALL", fontsize=14, style = filled, fillcolor = lightcyan];
10 -> 61 [arrowhead = none ];
61 -> 12;

62 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
12 -> 62 [arrowhead = none ];
62 -> 14;

63 [label="EVAL with clause\nletter(x).\nand substitutionT11 -> x", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 63 [arrowhead = none ];
63 -> 16;

64 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
14 -> 64 [arrowhead = none ];
64 -> 18;

65 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
16 -> 65 [arrowhead = none ];
65 -> 19;

66 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
18 -> 66 [arrowhead = none ];
66 -> 26;

67 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
19 -> 67 [arrowhead = none ];
67 -> 22;

68 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
22 -> 68 [arrowhead = none ];
68 -> 24;

69 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
26 -> 69 [arrowhead = none ];
69 -> 28;

70 [label="ONLY EVAL with clause\nno(X15).\nand substitutionT6 -> T16,\nX15 -> letter(T16),\nT15 -> T16", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 70 [arrowhead = none ];
70 -> 30;

71 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
30 -> 71 [arrowhead = none ];
71 -> 32;

72 [label="EVAL with clause\nq(x) :- p(X18).\nand substitutionT16 -> x", fontsize=14, style = filled, fillcolor = lightblue];
32 -> 72 [arrowhead = none ];
72 -> 34;

73 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
32 -> 73 [arrowhead = none ];
73 -> 36;

74 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
34 -> 74 [arrowhead = none ];
74 -> 38;

75 [label="ONLY EVAL with clause\np(X26) :- ','(no(letter(X26)), q(X26)).\nand substitutionX18 -> X27,\nX26 -> X27", fontsize=14, style = filled, fillcolor = lightblue];
38 -> 75 [arrowhead = none ];
75 -> 40;

76 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
40 -> 76 [arrowhead = none ];
76 -> 42;

77 [label="ONLY EVAL with clause\nno(X33) :- ','(call(X33), ','(!_8, failure(a))).\nand substitutionX27 -> X34,\nX33 -> letter(X34)", fontsize=14, style = filled, fillcolor = lightblue];
42 -> 77 [arrowhead = none ];
77 -> 44;

78 [label="CALL", fontsize=14, style = filled, fillcolor = lightcyan];
44 -> 78 [arrowhead = none ];
78 -> 46;

79 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
46 -> 79 [arrowhead = none ];
79 -> 48;

80 [label="ONLY EVAL with clause\nletter(x).\nand substitutionX34 -> x", fontsize=14, style = filled, fillcolor = lightblue];
48 -> 80 [arrowhead = none ];
80 -> 51;

81 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
51 -> 81 [arrowhead = none ];
81 -> 52;

82 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
52 -> 82 [arrowhead = none ];
82 -> 55;

83 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
55 -> 83 [arrowhead = none ];
83 -> 56;

}

(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: p(T1)\n\n", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: p(T1), SCOPE: 1, CLAUSE: 0\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(no(letter(T4)), q(T4))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(no(letter(T4)), q(T4)), SCOPE: 2, CLAUSE: 3 | ','(no(letter(T4)), q(T4)), SCOPE: 2, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: ','(','(call(letter(T9)), ','(!_2, failure(a))), q(T9)) | ','(no(letter(T4)), q(T4)), SCOPE: 2, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(letter(T9), ','(','(!_2, failure(a)), q(T9))) | ?_3 | ','(no(letter(T4)), q(T4)), SCOPE: 2, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(letter(T9), ','(','(!_2, failure(a)), q(T9))), SCOPE: 4, CLAUSE: 2 | ?_4 | ?_3 | ','(no(letter(T4)), q(T4)), SCOPE: 2, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(','(!_2, failure(a)), q(x)) | ?_4 | ?_3 | ','(no(letter(T4)), q(T4)), SCOPE: 2, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: ?_4 | ?_3 | ','(no(letter(T4)), q(T4)), SCOPE: 2, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: ','(failure(a), q(x))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ','(failure(a), q(x)), SCOPE: 5, CLAUSE: 5\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: ?_3 | ','(no(letter(T4)), q(T4)), SCOPE: 2, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: ','(no(letter(T4)), q(T4)), SCOPE: 2, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: q(T14)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: q(T14), SCOPE: 6, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: p(X15)\n\nX15 is free", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: p(X15), SCOPE: 7, CLAUSE: 0\n\nX15 is free", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: ','(no(letter(X24)), q(X24))\n\nX24 is free", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: ','(no(letter(X24)), q(X24)), SCOPE: 8, CLAUSE: 3 | ','(no(letter(X24)), q(X24)), SCOPE: 8, CLAUSE: 4\n\nX24 is free", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: ','(','(call(letter(X31)), ','(!_8, failure(a))), q(X31)) | ','(no(letter(X24)), q(X24)), SCOPE: 8, CLAUSE: 4\n\nX24 is free\nX31 is free", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: ','(letter(X31), ','(','(!_8, failure(a)), q(X31))) | ?_9 | ','(no(letter(X24)), q(X24)), SCOPE: 8, CLAUSE: 4\n\nX24 is free\nX31 is free", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: ','(letter(X31), ','(','(!_8, failure(a)), q(X31))), SCOPE: 10, CLAUSE: 2 | ?_10 | ?_9 | ','(no(letter(X24)), q(X24)), SCOPE: 8, CLAUSE: 4\n\nX24 is free\nX31 is free", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: ','(','(!_8, failure(a)), q(x)) | ?_10 | ?_9 | ','(no(letter(X24)), q(X24)), SCOPE: 8, CLAUSE: 4\n\nX24 is free", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: ','(failure(a), q(x))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: ','(failure(a), q(x)), SCOPE: 11, CLAUSE: 5\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 55 [arrowhead = none ];
55 -> 4;

56 [label="ONLY EVAL with clause\np(X4) :- ','(no(letter(X4)), q(X4)).\nand substitutionT1 -> T4,\nX4 -> T4,\nT3 -> T4", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 56 [arrowhead = none ];
56 -> 5;

57 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
5 -> 57 [arrowhead = none ];
57 -> 7;

58 [label="ONLY EVAL with clause\nno(X7) :- ','(call(X7), ','(!_2, failure(a))).\nand substitutionT4 -> T9,\nX7 -> letter(T9),\nT8 -> T9", fontsize=14, style = filled, fillcolor = lightblue];
7 -> 58 [arrowhead = none ];
58 -> 9;

59 [label="CALL", fontsize=14, style = filled, fillcolor = lightcyan];
9 -> 59 [arrowhead = none ];
59 -> 11;

60 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
11 -> 60 [arrowhead = none ];
60 -> 13;

61 [label="EVAL with clause\nletter(x).\nand substitutionT9 -> x", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 61 [arrowhead = none ];
61 -> 15;

62 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 62 [arrowhead = none ];
62 -> 17;

63 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
15 -> 63 [arrowhead = none ];
63 -> 20;

64 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 64 [arrowhead = none ];
64 -> 25;

65 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
20 -> 65 [arrowhead = none ];
65 -> 21;

66 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 66 [arrowhead = none ];
66 -> 23;

67 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
25 -> 67 [arrowhead = none ];
67 -> 27;

68 [label="ONLY EVAL with clause\nno(X12).\nand substitutionT4 -> T14,\nX12 -> letter(T14),\nT13 -> T14", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 68 [arrowhead = none ];
68 -> 29;

69 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
29 -> 69 [arrowhead = none ];
69 -> 31;

70 [label="EVAL with clause\nq(x) :- p(X15).\nand substitutionT14 -> x", fontsize=14, style = filled, fillcolor = lightblue];
31 -> 70 [arrowhead = none ];
70 -> 33;

71 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
31 -> 71 [arrowhead = none ];
71 -> 35;

72 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
33 -> 72 [arrowhead = none ];
72 -> 37;

73 [label="ONLY EVAL with clause\np(X23) :- ','(no(letter(X23)), q(X23)).\nand substitutionX15 -> X24,\nX23 -> X24", fontsize=14, style = filled, fillcolor = lightblue];
37 -> 73 [arrowhead = none ];
73 -> 39;

74 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
39 -> 74 [arrowhead = none ];
74 -> 41;

75 [label="ONLY EVAL with clause\nno(X30) :- ','(call(X30), ','(!_8, failure(a))).\nand substitutionX24 -> X31,\nX30 -> letter(X31)", fontsize=14, style = filled, fillcolor = lightblue];
41 -> 75 [arrowhead = none ];
75 -> 43;

76 [label="CALL", fontsize=14, style = filled, fillcolor = lightcyan];
43 -> 76 [arrowhead = none ];
76 -> 45;

77 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
45 -> 77 [arrowhead = none ];
77 -> 47;

78 [label="ONLY EVAL with clause\nletter(x).\nand substitutionX31 -> x", fontsize=14, style = filled, fillcolor = lightblue];
47 -> 78 [arrowhead = none ];
78 -> 49;

79 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
49 -> 79 [arrowhead = none ];
79 -> 50;

80 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
50 -> 80 [arrowhead = none ];
80 -> 53;

81 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
53 -> 81 [arrowhead = none ];
81 -> 54;

}

(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