Runtime Complexity proof of /tmp/tmp7kCdzr/plumer453b.pl

Runtime Complexity of the query pattern
f(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), 75 ms)

↳2 CdtProblem

↳3 PrologToCdtProblemTransformerProof (UPPER BOUND (ID), 55 ms)

↳4 CdtProblem

    ↳5 CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID), 0 ms)

    ↳6 CdtProblem

    ↳7 CdtGraphRemoveTrailingTuplepartsProof (BOTH BOUNDS(ID, ID), 0 ms)

    ↳8 CdtProblem

    ↳9 CdtKnowledgeProof (⇔, 0 ms)

    ↳10 BOUNDS(O(1), O(1))

(0) Obligation:

Clauses:

f(X) :- ','(p(X), q(X)).
p(a).
p(X) :- ','(p(a), !).
q(b).

Query: f(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: f(T1)\n\n", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: f(T1), SCOPE: 1, CLAUSE: 0\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(p(T4), q(T4))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(p(T4), q(T4)), SCOPE: 2, CLAUSE: 1 | ','(p(T4), q(T4)), SCOPE: 2, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: q(a) | ','(p(T4), q(T4)), SCOPE: 2, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: ','(p(T4), q(T4)), SCOPE: 2, CLAUSE: 2\n\np(T4) !=? p(a)", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: q(a), SCOPE: 3, CLAUSE: 3 | ?_3 | ','(p(T4), q(T4)), SCOPE: 2, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ?_3 | ','(p(T4), q(T4)), SCOPE: 2, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: ','(p(T4), q(T4)), SCOPE: 2, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(','(p(a), !_2), q(T9))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: ','(','(p(a), !_2), q(T9)), SCOPE: 4, CLAUSE: 1 | ','(','(p(a), !_2), q(T9)), SCOPE: 4, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(!_2, q(T9)) | ','(','(p(a), !_2), q(T9)), SCOPE: 4, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: q(T9)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: q(T9), SCOPE: 5, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ','(','(p(a), !_2), q(T14))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: ','(','(p(a), !_2), q(T14)), SCOPE: 6, CLAUSE: 1 | ','(','(p(a), !_2), q(T14)), SCOPE: 6, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: ','(!_2, q(T14)) | ','(','(p(a), !_2), q(T14)), SCOPE: 6, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: q(T14)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: q(T14), SCOPE: 7, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 31 [arrowhead = none ];
31 -> 4;

32 [label="ONLY EVAL with clause\nf(X2) :- ','(p(X2), q(X2)).\nand substitutionT1 -> T4,\nX2 -> T4,\nT3 -> T4", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 32 [arrowhead = none ];
32 -> 5;

33 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
5 -> 33 [arrowhead = none ];
33 -> 7;

34 [label="EVAL with clause\np(a).\nand substitutionT4 -> a", fontsize=14, style = filled, fillcolor = lightblue];
7 -> 34 [arrowhead = none ];
34 -> 8;

35 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 35 [arrowhead = none ];
35 -> 9;

36 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
8 -> 36 [arrowhead = none ];
36 -> 10;

37 [label="ONLY EVAL with clause\np(X8) :- ','(p(a), !_2).\nand substitutionT4 -> T14,\nX8 -> T14,\nT13 -> T14", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 37 [arrowhead = none ];
37 -> 21;

38 [label="BACKTRACK\nfor clause: q(b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 38 [arrowhead = none ];
38 -> 11;

39 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 39 [arrowhead = none ];
39 -> 12;

40 [label="ONLY EVAL with clause\np(X5) :- ','(p(a), !_2).\nand substitutionT4 -> T9,\nX5 -> T9,\nT8 -> T9", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 40 [arrowhead = none ];
40 -> 13;

41 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
13 -> 41 [arrowhead = none ];
41 -> 14;

42 [label="ONLY EVAL with clause\np(a).\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 42 [arrowhead = none ];
42 -> 15;

43 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
15 -> 43 [arrowhead = none ];
43 -> 16;

44 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
16 -> 44 [arrowhead = none ];
44 -> 17;

45 [label="EVAL with clause\nq(b).\nand substitutionT9 -> b", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 45 [arrowhead = none ];
45 -> 18;

46 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 46 [arrowhead = none ];
46 -> 19;

47 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
18 -> 47 [arrowhead = none ];
47 -> 20;

48 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
21 -> 48 [arrowhead = none ];
48 -> 22;

49 [label="ONLY EVAL with clause\np(a).\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
22 -> 49 [arrowhead = none ];
49 -> 23;

50 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
23 -> 50 [arrowhead = none ];
50 -> 26;

51 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
26 -> 51 [arrowhead = none ];
51 -> 27;

52 [label="EVAL with clause\nq(b).\nand substitutionT14 -> b", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 52 [arrowhead = none ];
52 -> 28;

53 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
27 -> 53 [arrowhead = none ];
53 -> 29;

54 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
28 -> 54 [arrowhead = none ];
54 -> 30;

}

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in → f2_out1(b)

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

f2_in

Defined Pair Symbols:none

Compound Symbols:none

(3) 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)\n\n", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
3 [label="3: f(T1), SCOPE: 1, CLAUSE: 0\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(p(T6), q(T6))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: p(T6)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: q(T7)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: p(T6), SCOPE: 2, CLAUSE: 1 | p(T6), SCOPE: 2, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: true | p(T6), SCOPE: 2, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: p(T6), SCOPE: 2, CLAUSE: 2\n\np(T6) !=? p(a)", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: p(T6), SCOPE: 2, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: ','(p(a), !_2)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: ','(p(a), !_2), SCOPE: 3, CLAUSE: 1 | ','(p(a), !_2), SCOPE: 3, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: !_2 | ','(p(a), !_2), SCOPE: 3, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: ','(p(a), !_2)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: ','(p(a), !_2), SCOPE: 4, CLAUSE: 1 | ','(p(a), !_2), SCOPE: 4, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: !_2 | ','(p(a), !_2), SCOPE: 4, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: q(T7), SCOPE: 5, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 49 [arrowhead = none ];
49 -> 3;

50 [label="ONLY EVAL with clause\nf(X3) :- ','(p(X3), q(X3)).\nand substitutionT1 -> T6,\nX3 -> T6,\nT5 -> T6", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 50 [arrowhead = none ];
50 -> 6;

51 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
6 -> 51 [arrowhead = none ];
51 -> 24;

52 [label="SPLIT 2\nreplacements:T6 -> T7", fontsize=14, style = filled, fillcolor = violet];
6 -> 52 [arrowhead = none ];
52 -> 25;

53 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
24 -> 53 [arrowhead = none ];
53 -> 31;

54 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
25 -> 54 [arrowhead = none ];
54 -> 45;

55 [label="EVAL with clause\np(a).\nand substitutionT6 -> a", fontsize=14, style = filled, fillcolor = lightblue];
31 -> 55 [arrowhead = none ];
55 -> 32;

56 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
31 -> 56 [arrowhead = none ];
56 -> 33;

57 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
32 -> 57 [arrowhead = none ];
57 -> 34;

58 [label="ONLY EVAL with clause\np(X9) :- ','(p(a), !_2).\nand substitutionT6 -> T13,\nX9 -> T13", fontsize=14, style = filled, fillcolor = lightblue];
33 -> 58 [arrowhead = none ];
58 -> 40;

59 [label="ONLY EVAL with clause\np(X6) :- ','(p(a), !_2).\nand substitutionT6 -> T10,\nX6 -> T10", fontsize=14, style = filled, fillcolor = lightblue];
34 -> 59 [arrowhead = none ];
59 -> 35;

60 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
35 -> 60 [arrowhead = none ];
60 -> 36;

61 [label="ONLY EVAL with clause\np(a).\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
36 -> 61 [arrowhead = none ];
61 -> 37;

62 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
37 -> 62 [arrowhead = none ];
62 -> 38;

63 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
38 -> 63 [arrowhead = none ];
63 -> 39;

64 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
40 -> 64 [arrowhead = none ];
64 -> 41;

65 [label="ONLY EVAL with clause\np(a).\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
41 -> 65 [arrowhead = none ];
65 -> 42;

66 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
42 -> 66 [arrowhead = none ];
66 -> 43;

67 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
43 -> 67 [arrowhead = none ];
67 -> 44;

68 [label="EVAL with clause\nq(b).\nand substitutionT7 -> b", fontsize=14, style = filled, fillcolor = lightblue];
45 -> 68 [arrowhead = none ];
68 -> 46;

69 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
45 -> 69 [arrowhead = none ];
69 -> 47;

70 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
46 -> 70 [arrowhead = none ];
70 -> 48;

}

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in → U1(f6_in)
U1(f6_out1(z0)) → f1_out1(z0)
f24_in → f24_out1
f25_in → f25_out1(b)
f6_in → U2(f24_in)
U2(f24_out1) → U3(f25_in)
U3(f25_out1(z0)) → f6_out1(z0)

Tuples:

F1_IN → c(U1'(f6_in), F6_IN)
F6_IN → c4(U2'(f24_in), F24_IN)
U2'(f24_out1) → c5(U3'(f25_in), F25_IN)

S tuples:

F1_IN → c(U1'(f6_in), F6_IN)
F6_IN → c4(U2'(f24_in), F24_IN)
U2'(f24_out1) → c5(U3'(f25_in), F25_IN)

K tuples:none
Defined Rule Symbols:

f1_in, U1, f24_in, f25_in, f6_in, U2, U3

Defined Pair Symbols:

F1_IN, F6_IN, U2'

Compound Symbols:

c, c4, c5

(5) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID) transformation)

Split RHS of tuples not part of any SCC

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in → U1(f6_in)
U1(f6_out1(z0)) → f1_out1(z0)
f24_in → f24_out1
f25_in → f25_out1(b)
f6_in → U2(f24_in)
U2(f24_out1) → U3(f25_in)
U3(f25_out1(z0)) → f6_out1(z0)

Tuples:

F1_IN → c1(U1'(f6_in))
F1_IN → c1(F6_IN)
F6_IN → c1(U2'(f24_in))
F6_IN → c1(F24_IN)
U2'(f24_out1) → c1(U3'(f25_in))
U2'(f24_out1) → c1(F25_IN)

S tuples:

F1_IN → c1(U1'(f6_in))
F1_IN → c1(F6_IN)
F6_IN → c1(U2'(f24_in))
F6_IN → c1(F24_IN)
U2'(f24_out1) → c1(U3'(f25_in))
U2'(f24_out1) → c1(F25_IN)

K tuples:none
Defined Rule Symbols:

f1_in, U1, f24_in, f25_in, f6_in, U2, U3

Defined Pair Symbols:

F1_IN, F6_IN, U2'

Compound Symbols:

c1

(7) CdtGraphRemoveTrailingTuplepartsProof (BOTH BOUNDS(ID, ID) transformation)

Removed 4 trailing tuple parts

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in → U1(f6_in)
U1(f6_out1(z0)) → f1_out1(z0)
f24_in → f24_out1
f25_in → f25_out1(b)
f6_in → U2(f24_in)
U2(f24_out1) → U3(f25_in)
U3(f25_out1(z0)) → f6_out1(z0)

Tuples:

F1_IN → c1(F6_IN)
F6_IN → c1(U2'(f24_in))
F1_IN → c1
F6_IN → c1
U2'(f24_out1) → c1

S tuples:

F1_IN → c1(F6_IN)
F6_IN → c1(U2'(f24_in))
F1_IN → c1
F6_IN → c1
U2'(f24_out1) → c1

K tuples:none
Defined Rule Symbols:

f1_in, U1, f24_in, f25_in, f6_in, U2, U3

Defined Pair Symbols:

F1_IN, F6_IN, U2'

Compound Symbols:

c1, c1

(9) CdtKnowledgeProof (EQUIVALENT transformation)

The following tuples could be moved from S to K by knowledge propagation:

F1_IN → c1(F6_IN)
F6_IN → c1(U2'(f24_in))
F1_IN → c1
F6_IN → c1
U2'(f24_out1) → c1
U2'(f24_out1) → c1
F6_IN → c1(U2'(f24_in))
F6_IN → c1
U2'(f24_out1) → c1
U2'(f24_out1) → c1

Now S is empty

(0) Obligation:

(1) PrologToCdtProblemTransformerProof (UPPER BOUND (ID) transformation)

(2) Obligation:

(3) PrologToCdtProblemTransformerProof (UPPER BOUND (ID) transformation)

(4) Obligation:

(5) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID) transformation)

(6) Obligation:

(7) CdtGraphRemoveTrailingTuplepartsProof (BOTH BOUNDS(ID, ID) transformation)

(8) Obligation:

(9) CdtKnowledgeProof (EQUIVALENT transformation)

(10) BOUNDS(O(1), O(1))