Runtime Complexity proof of /tmp/tmpT

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), 58 ms)

↳2 CdtProblem

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

    ↳4 CdtProblem

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

    ↳6 CdtProblem

    ↳7 CdtKnowledgeProof (⇔, 0 ms)

    ↳8 BOUNDS(O(1), O(1))

↳9 PrologToCdtProblemTransformerProof (UPPER BOUND (ID), 78 ms)

↳10 CdtProblem

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

    ↳12 CdtProblem

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

    ↳14 CdtProblem

(0) Obligation:

Clauses:

f(X) :- ','(p(X), q(X)).
p(a) :- !.
p(X) :- p(Y).
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];
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];
15 [label="15: p(T6)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: q(T7)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: p(T6), SCOPE: 2, CLAUSE: 1 | p(T6), SCOPE: 2, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: !_2 | p(T6), SCOPE: 2, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: p(T6), SCOPE: 2, CLAUSE: 2\n\np(T6) !=? p(a)", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: p(X8)\n\nX8 is free", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: p(X8), SCOPE: 3, CLAUSE: 1 | p(X8), SCOPE: 3, CLAUSE: 2\n\nX8 is free", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: !_3 | p(X8), SCOPE: 3, CLAUSE: 2\n\nX8 is free", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: q(T7), SCOPE: 4, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 40 [arrowhead = none ];
40 -> 3;

41 [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 -> 41 [arrowhead = none ];
41 -> 6;

42 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
6 -> 42 [arrowhead = none ];
42 -> 15;

43 [label="SPLIT 2\nreplacements:T6 -> T7", fontsize=14, style = filled, fillcolor = violet];
6 -> 43 [arrowhead = none ];
43 -> 17;

44 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
15 -> 44 [arrowhead = none ];
44 -> 18;

45 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
17 -> 45 [arrowhead = none ];
45 -> 36;

46 [label="EVAL with clause\np(a) :- !_2.\nand substitutionT6 -> a", fontsize=14, style = filled, fillcolor = lightblue];
18 -> 46 [arrowhead = none ];
46 -> 20;

47 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
18 -> 47 [arrowhead = none ];
47 -> 22;

48 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
20 -> 48 [arrowhead = none ];
48 -> 25;

49 [label="ONLY EVAL with clause\np(X7) :- p(X8).\nand substitutionT6 -> T10,\nX7 -> T10", fontsize=14, style = filled, fillcolor = lightblue];
22 -> 49 [arrowhead = none ];
49 -> 31;

50 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
25 -> 50 [arrowhead = none ];
50 -> 26;

51 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
31 -> 51 [arrowhead = none ];
51 -> 32;

52 [label="ONLY EVAL with clause\np(a) :- !_3.\nand substitutionX8 -> a", fontsize=14, style = filled, fillcolor = lightblue];
32 -> 52 [arrowhead = none ];
52 -> 33;

53 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
33 -> 53 [arrowhead = none ];
53 -> 34;

54 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
34 -> 54 [arrowhead = none ];
54 -> 35;

55 [label="EVAL with clause\nq(b).\nand substitutionT7 -> b", fontsize=14, style = filled, fillcolor = lightblue];
36 -> 55 [arrowhead = none ];
55 -> 37;

56 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
36 -> 56 [arrowhead = none ];
56 -> 38;

57 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
37 -> 57 [arrowhead = none ];
57 -> 39;

}

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in → U1(f6_in)
U1(f6_out1(z0)) → f2_out1(z0)
f15_in → f15_out1
f17_in → f17_out1(b)
f6_in → U2(f15_in)
U2(f15_out1) → U3(f17_in)
U3(f17_out1(z0)) → f6_out1(z0)

Tuples:

F2_IN → c(U1'(f6_in), F6_IN)
F6_IN → c4(U2'(f15_in), F15_IN)
U2'(f15_out1) → c5(U3'(f17_in), F17_IN)

S tuples:

F2_IN → c(U1'(f6_in), F6_IN)
F6_IN → c4(U2'(f15_in), F15_IN)
U2'(f15_out1) → c5(U3'(f17_in), F17_IN)

K tuples:none
Defined Rule Symbols:

f2_in, U1, f15_in, f17_in, f6_in, U2, U3

Defined Pair Symbols:

F2_IN, F6_IN, U2'

Compound Symbols:

c, c4, c5

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

Split RHS of tuples not part of any SCC

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in → U1(f6_in)
U1(f6_out1(z0)) → f2_out1(z0)
f15_in → f15_out1
f17_in → f17_out1(b)
f6_in → U2(f15_in)
U2(f15_out1) → U3(f17_in)
U3(f17_out1(z0)) → f6_out1(z0)

Tuples:

F2_IN → c1(U1'(f6_in))
F2_IN → c1(F6_IN)
F6_IN → c1(U2'(f15_in))
F6_IN → c1(F15_IN)
U2'(f15_out1) → c1(U3'(f17_in))
U2'(f15_out1) → c1(F17_IN)

S tuples:

F2_IN → c1(U1'(f6_in))
F2_IN → c1(F6_IN)
F6_IN → c1(U2'(f15_in))
F6_IN → c1(F15_IN)
U2'(f15_out1) → c1(U3'(f17_in))
U2'(f15_out1) → c1(F17_IN)

K tuples:none
Defined Rule Symbols:

f2_in, U1, f15_in, f17_in, f6_in, U2, U3

Defined Pair Symbols:

F2_IN, F6_IN, U2'

Compound Symbols:

c1

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

Removed 4 trailing tuple parts

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in → U1(f6_in)
U1(f6_out1(z0)) → f2_out1(z0)
f15_in → f15_out1
f17_in → f17_out1(b)
f6_in → U2(f15_in)
U2(f15_out1) → U3(f17_in)
U3(f17_out1(z0)) → f6_out1(z0)

Tuples:

F2_IN → c1(F6_IN)
F6_IN → c1(U2'(f15_in))
F2_IN → c1
F6_IN → c1
U2'(f15_out1) → c1

S tuples:

F2_IN → c1(F6_IN)
F6_IN → c1(U2'(f15_in))
F2_IN → c1
F6_IN → c1
U2'(f15_out1) → c1

K tuples:none
Defined Rule Symbols:

f2_in, U1, f15_in, f17_in, f6_in, U2, U3

Defined Pair Symbols:

F2_IN, F6_IN, U2'

Compound Symbols:

c1, c1

(7) CdtKnowledgeProof (EQUIVALENT transformation)

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

F2_IN → c1(F6_IN)
F6_IN → c1(U2'(f15_in))
F2_IN → c1
F6_IN → c1
U2'(f15_out1) → c1
U2'(f15_out1) → c1
F6_IN → c1(U2'(f15_in))
F6_IN → c1
U2'(f15_out1) → c1
U2'(f15_out1) → c1

Now S is empty

(8) BOUNDS(O(1), O(1))

(9) 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];
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: ','(!_2, 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)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: q(a), SCOPE: 3, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(p(X7), q(T9))\n\nX7 is free", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: p(X7)\n\nX7 is free", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: q(T9)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: p(X7), SCOPE: 4, CLAUSE: 1 | p(X7), SCOPE: 4, CLAUSE: 2\n\nX7 is free", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: !_4 | p(X7), SCOPE: 4, CLAUSE: 2\n\nX7 is free", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: q(T9), SCOPE: 5, 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];
1 -> 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) :- !_2.\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="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 36 [arrowhead = none ];
36 -> 10;

37 [label="ONLY EVAL with clause\np(X6) :- p(X7).\nand substitutionT4 -> T9,\nX6 -> T9,\nT8 -> T9", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 37 [arrowhead = none ];
37 -> 13;

38 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
10 -> 38 [arrowhead = none ];
38 -> 11;

39 [label="BACKTRACK\nfor clause: q(b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 39 [arrowhead = none ];
39 -> 12;

40 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
13 -> 40 [arrowhead = none ];
40 -> 14;

41 [label="SPLIT 2\nreplacements:X7 -> T10", fontsize=14, style = filled, fillcolor = violet];
13 -> 41 [arrowhead = none ];
41 -> 16;

42 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
14 -> 42 [arrowhead = none ];
42 -> 19;

43 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
16 -> 43 [arrowhead = none ];
43 -> 27;

44 [label="ONLY EVAL with clause\np(a) :- !_4.\nand substitutionX7 -> a", fontsize=14, style = filled, fillcolor = lightblue];
19 -> 44 [arrowhead = none ];
44 -> 21;

45 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 45 [arrowhead = none ];
45 -> 23;

46 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
23 -> 46 [arrowhead = none ];
46 -> 24;

47 [label="EVAL with clause\nq(b).\nand substitutionT9 -> b", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 47 [arrowhead = none ];
47 -> 28;

48 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
27 -> 48 [arrowhead = none ];
48 -> 29;

49 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
28 -> 49 [arrowhead = none ];
49 -> 30;

}

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in → U1(f13_in)
U1(f13_out1(z0)) → f1_out1(z0)
f14_in → f14_out1
f16_in → f16_out1(b)
f13_in → U2(f14_in)
U2(f14_out1) → U3(f16_in)
U3(f16_out1(z0)) → f13_out1(z0)

Tuples:

F1_IN → c(U1'(f13_in), F13_IN)
F13_IN → c4(U2'(f14_in), F14_IN)
U2'(f14_out1) → c5(U3'(f16_in), F16_IN)

S tuples:

F1_IN → c(U1'(f13_in), F13_IN)
F13_IN → c4(U2'(f14_in), F14_IN)
U2'(f14_out1) → c5(U3'(f16_in), F16_IN)

K tuples:none
Defined Rule Symbols:

f1_in, U1, f14_in, f16_in, f13_in, U2, U3

Defined Pair Symbols:

F1_IN, F13_IN, U2'

Compound Symbols:

c, c4, c5

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

Split RHS of tuples not part of any SCC

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in → U1(f13_in)
U1(f13_out1(z0)) → f1_out1(z0)
f14_in → f14_out1
f16_in → f16_out1(b)
f13_in → U2(f14_in)
U2(f14_out1) → U3(f16_in)
U3(f16_out1(z0)) → f13_out1(z0)

Tuples:

F1_IN → c1(U1'(f13_in))
F1_IN → c1(F13_IN)
F13_IN → c1(U2'(f14_in))
F13_IN → c1(F14_IN)
U2'(f14_out1) → c1(U3'(f16_in))
U2'(f14_out1) → c1(F16_IN)

S tuples:

F1_IN → c1(U1'(f13_in))
F1_IN → c1(F13_IN)
F13_IN → c1(U2'(f14_in))
F13_IN → c1(F14_IN)
U2'(f14_out1) → c1(U3'(f16_in))
U2'(f14_out1) → c1(F16_IN)

K tuples:none
Defined Rule Symbols:

f1_in, U1, f14_in, f16_in, f13_in, U2, U3

Defined Pair Symbols:

F1_IN, F13_IN, U2'

Compound Symbols:

c1

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

Removed 4 trailing tuple parts

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in → U1(f13_in)
U1(f13_out1(z0)) → f1_out1(z0)
f14_in → f14_out1
f16_in → f16_out1(b)
f13_in → U2(f14_in)
U2(f14_out1) → U3(f16_in)
U3(f16_out1(z0)) → f13_out1(z0)

Tuples:

F1_IN → c1(F13_IN)
F13_IN → c1(U2'(f14_in))
F1_IN → c1
F13_IN → c1
U2'(f14_out1) → c1

S tuples:

F1_IN → c1(F13_IN)
F13_IN → c1(U2'(f14_in))
F1_IN → c1
F13_IN → c1
U2'(f14_out1) → c1

K tuples:none
Defined Rule Symbols:

f1_in, U1, f14_in, f16_in, f13_in, U2, U3

Defined Pair Symbols:

F1_IN, F13_IN, U2'

Compound Symbols:

c1, c1