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

(0) Obligation:

Clauses:

q :- p(s(s(0))).
p(s(X)) :- p(X).
p(0).
p(s(s(s(0)))) :- p(s(s(s(0)))).

Query: q()

(1) LPReorderTransformerProof (EQUIVALENT transformation)

Reordered facts before rules in definite LP [PROLOG].

(2) Obligation:

Clauses:

p(0).
q :- p(s(s(0))).
p(s(X)) :- p(X).
p(s(s(s(0)))) :- p(s(s(s(0)))).

Query: q()

(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: q\n\n", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
3 [label="3: q, SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: p(s(s(0)))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: p(s(s(0))), SCOPE: 2, CLAUSE: 0 | p(s(s(0))), SCOPE: 2, CLAUSE: 2 | p(s(s(0))), SCOPE: 2, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: p(s(s(0))), SCOPE: 2, CLAUSE: 2 | p(s(s(0))), SCOPE: 2, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: p(s(s(0))), SCOPE: 2, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: p(s(s(0))), SCOPE: 2, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: p(s(0))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: p(s(0)), SCOPE: 3, CLAUSE: 0 | p(s(0)), SCOPE: 3, CLAUSE: 2 | p(s(0)), SCOPE: 3, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: p(s(0)), SCOPE: 3, CLAUSE: 2 | p(s(0)), SCOPE: 3, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: p(s(0)), SCOPE: 3, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: p(s(0)), SCOPE: 3, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: p(0)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: p(0), SCOPE: 4, CLAUSE: 0 | p(0), SCOPE: 4, CLAUSE: 2 | p(0), SCOPE: 4, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: true | p(0), SCOPE: 4, CLAUSE: 2 | p(0), SCOPE: 4, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: p(0), SCOPE: 4, CLAUSE: 2 | p(0), SCOPE: 4, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: p(0), SCOPE: 4, CLAUSE: 3\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="40: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 41 [arrowhead = none ];
41 -> 3;

42 [label="ONLY EVAL with clause\nq :- p(s(s(0))).\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 42 [arrowhead = none ];
42 -> 6;

43 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
6 -> 43 [arrowhead = none ];
43 -> 13;

44 [label="BACKTRACK\nfor clause: p(0)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 44 [arrowhead = none ];
44 -> 15;

45 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
15 -> 45 [arrowhead = none ];
45 -> 18;

46 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
15 -> 46 [arrowhead = none ];
46 -> 19;

47 [label="ONLY EVAL with clause\np(s(X5)) :- p(X5).\nand substitutionX5 -> s(0)", fontsize=14, style = filled, fillcolor = lightblue];
18 -> 47 [arrowhead = none ];
47 -> 21;

48 [label="BACKTRACK\nfor clause: p(s(s(s(0)))) :- p(s(s(s(0))))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
19 -> 48 [arrowhead = none ];
48 -> 40;

49 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
21 -> 49 [arrowhead = none ];
49 -> 22;

50 [label="BACKTRACK\nfor clause: p(0)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
22 -> 50 [arrowhead = none ];
50 -> 23;

51 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
23 -> 51 [arrowhead = none ];
51 -> 25;

52 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
23 -> 52 [arrowhead = none ];
52 -> 27;

53 [label="ONLY EVAL with clause\np(s(X10)) :- p(X10).\nand substitutionX10 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
25 -> 53 [arrowhead = none ];
53 -> 29;

54 [label="BACKTRACK\nfor clause: p(s(s(s(0)))) :- p(s(s(s(0))))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
27 -> 54 [arrowhead = none ];
54 -> 39;

55 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
29 -> 55 [arrowhead = none ];
55 -> 34;

56 [label="ONLY EVAL with clause\np(0).\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
34 -> 56 [arrowhead = none ];
56 -> 35;

57 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
35 -> 57 [arrowhead = none ];
57 -> 36;

58 [label="BACKTRACK\nfor clause: p(s(X)) :- p(X)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
36 -> 58 [arrowhead = none ];
58 -> 37;

59 [label="BACKTRACK\nfor clause: p(s(s(s(0)))) :- p(s(s(s(0))))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
37 -> 59 [arrowhead = none ];
59 -> 38;

}

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in → U1(f15_in)
U1(f15_out1) → f2_out1
U1(f15_out2) → f2_out1
f18_in → U2(f23_in)
U2(f23_out1) → f18_out1
U2(f23_out2) → f18_out1
f25_in → f25_out1
f15_in → U3(f18_in, f19_in)
U3(f18_out1, z0) → f15_out1
U3(z0, f19_out1) → f15_out2
f23_in → U4(f25_in, f27_in)
U4(f25_out1, z0) → f23_out1
U4(z0, f27_out1) → f23_out2

Tuples:

F2_IN → c(U1'(f15_in), F15_IN)
F18_IN → c3(U2'(f23_in), F23_IN)
F15_IN → c7(U3'(f18_in, f19_in), F18_IN)
F23_IN → c10(U4'(f25_in, f27_in), F25_IN)

S tuples:

F2_IN → c(U1'(f15_in), F15_IN)
F18_IN → c3(U2'(f23_in), F23_IN)
F15_IN → c7(U3'(f18_in, f19_in), F18_IN)
F23_IN → c10(U4'(f25_in, f27_in), F25_IN)

K tuples:none
Defined Rule Symbols:

f2_in, U1, f18_in, U2, f25_in, f15_in, U3, f23_in, U4

Defined Pair Symbols:

F2_IN, F18_IN, F15_IN, F23_IN

Compound Symbols:

c, c3, c7, c10

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

Split RHS of tuples not part of any SCC

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in → U1(f15_in)
U1(f15_out1) → f2_out1
U1(f15_out2) → f2_out1
f18_in → U2(f23_in)
U2(f23_out1) → f18_out1
U2(f23_out2) → f18_out1
f25_in → f25_out1
f15_in → U3(f18_in, f19_in)
U3(f18_out1, z0) → f15_out1
U3(z0, f19_out1) → f15_out2
f23_in → U4(f25_in, f27_in)
U4(f25_out1, z0) → f23_out1
U4(z0, f27_out1) → f23_out2

Tuples:

F2_IN → c1(U1'(f15_in))
F2_IN → c1(F15_IN)
F18_IN → c1(U2'(f23_in))
F18_IN → c1(F23_IN)
F15_IN → c1(U3'(f18_in, f19_in))
F15_IN → c1(F18_IN)
F23_IN → c1(U4'(f25_in, f27_in))
F23_IN → c1(F25_IN)

S tuples:

F2_IN → c1(U1'(f15_in))
F2_IN → c1(F15_IN)
F18_IN → c1(U2'(f23_in))
F18_IN → c1(F23_IN)
F15_IN → c1(U3'(f18_in, f19_in))
F15_IN → c1(F18_IN)
F23_IN → c1(U4'(f25_in, f27_in))
F23_IN → c1(F25_IN)

K tuples:none
Defined Rule Symbols:

f2_in, U1, f18_in, U2, f25_in, f15_in, U3, f23_in, U4

Defined Pair Symbols:

F2_IN, F18_IN, F15_IN, F23_IN

Compound Symbols:

c1

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

Removed 5 trailing tuple parts

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in → U1(f15_in)
U1(f15_out1) → f2_out1
U1(f15_out2) → f2_out1
f18_in → U2(f23_in)
U2(f23_out1) → f18_out1
U2(f23_out2) → f18_out1
f25_in → f25_out1
f15_in → U3(f18_in, f19_in)
U3(f18_out1, z0) → f15_out1
U3(z0, f19_out1) → f15_out2
f23_in → U4(f25_in, f27_in)
U4(f25_out1, z0) → f23_out1
U4(z0, f27_out1) → f23_out2

Tuples:

F2_IN → c1(F15_IN)
F18_IN → c1(F23_IN)
F15_IN → c1(F18_IN)
F2_IN → c1
F18_IN → c1
F15_IN → c1
F23_IN → c1

S tuples:

F2_IN → c1(F15_IN)
F18_IN → c1(F23_IN)
F15_IN → c1(F18_IN)
F2_IN → c1
F18_IN → c1
F15_IN → c1
F23_IN → c1

K tuples:none
Defined Rule Symbols:

f2_in, U1, f18_in, U2, f25_in, f15_in, U3, f23_in, U4

Defined Pair Symbols:

F2_IN, F18_IN, F15_IN, F23_IN

Compound Symbols:

c1, c1

(9) CdtKnowledgeProof (EQUIVALENT transformation)

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

F2_IN → c1(F15_IN)
F15_IN → c1(F18_IN)
F2_IN → c1
F18_IN → c1
F15_IN → c1
F15_IN → c1(F18_IN)
F15_IN → c1
F18_IN → c1(F23_IN)
F18_IN → c1
F23_IN → c1
F23_IN → c1

Now S is empty

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

(11) 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: q\n\n", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: q, SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: p(s(s(0)))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: p(s(s(0))), SCOPE: 2, CLAUSE: 0 | p(s(s(0))), SCOPE: 2, CLAUSE: 2 | p(s(s(0))), SCOPE: 2, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: p(s(s(0))), SCOPE: 2, CLAUSE: 2 | p(s(s(0))), SCOPE: 2, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: p(s(s(0))), SCOPE: 2, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: p(s(s(0))), SCOPE: 2, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: p(s(0))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: p(s(0)), SCOPE: 3, CLAUSE: 0 | p(s(0)), SCOPE: 3, CLAUSE: 2 | p(s(0)), SCOPE: 3, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: p(s(0)), SCOPE: 3, CLAUSE: 2 | p(s(0)), SCOPE: 3, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: p(s(0)), SCOPE: 3, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: p(s(0)), SCOPE: 3, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: p(0)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: p(0), SCOPE: 4, CLAUSE: 0 | p(0), SCOPE: 4, CLAUSE: 2 | p(0), SCOPE: 4, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: true | p(0), SCOPE: 4, CLAUSE: 2 | p(0), SCOPE: 4, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: p(0), SCOPE: 4, CLAUSE: 2 | p(0), SCOPE: 4, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: p(0), SCOPE: 4, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 34 [arrowhead = none ];
34 -> 4;

35 [label="ONLY EVAL with clause\nq :- p(s(s(0))).\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 35 [arrowhead = none ];
35 -> 5;

36 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
5 -> 36 [arrowhead = none ];
36 -> 7;

37 [label="BACKTRACK\nfor clause: p(0)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 37 [arrowhead = none ];
37 -> 8;

38 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
8 -> 38 [arrowhead = none ];
38 -> 9;

39 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
8 -> 39 [arrowhead = none ];
39 -> 10;

40 [label="ONLY EVAL with clause\np(s(X5)) :- p(X5).\nand substitutionX5 -> s(0)", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 40 [arrowhead = none ];
40 -> 11;

41 [label="BACKTRACK\nfor clause: p(s(s(s(0)))) :- p(s(s(s(0))))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 41 [arrowhead = none ];
41 -> 33;

42 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
11 -> 42 [arrowhead = none ];
42 -> 12;

43 [label="BACKTRACK\nfor clause: p(0)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
12 -> 43 [arrowhead = none ];
43 -> 14;

44 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
14 -> 44 [arrowhead = none ];
44 -> 16;

45 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
14 -> 45 [arrowhead = none ];
45 -> 17;

46 [label="ONLY EVAL with clause\np(s(X10)) :- p(X10).\nand substitutionX10 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
16 -> 46 [arrowhead = none ];
46 -> 20;

47 [label="BACKTRACK\nfor clause: p(s(s(s(0)))) :- p(s(s(s(0))))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 47 [arrowhead = none ];
47 -> 32;

48 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
20 -> 48 [arrowhead = none ];
48 -> 24;

49 [label="ONLY EVAL with clause\np(0).\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
24 -> 49 [arrowhead = none ];
49 -> 26;

50 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
26 -> 50 [arrowhead = none ];
50 -> 28;

51 [label="BACKTRACK\nfor clause: p(s(X)) :- p(X)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
28 -> 51 [arrowhead = none ];
51 -> 30;

52 [label="BACKTRACK\nfor clause: p(s(s(s(0)))) :- p(s(s(s(0))))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
30 -> 52 [arrowhead = none ];
52 -> 31;

}

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in → U1(f8_in)
U1(f8_out1) → f1_out1
U1(f8_out2) → f1_out1
f9_in → U2(f14_in)
U2(f14_out1) → f9_out1
U2(f14_out2) → f9_out1
f16_in → f16_out1
f8_in → U3(f9_in, f10_in)
U3(f9_out1, z0) → f8_out1
U3(z0, f10_out1) → f8_out2
f14_in → U4(f16_in, f17_in)
U4(f16_out1, z0) → f14_out1
U4(z0, f17_out1) → f14_out2

Tuples:

F1_IN → c(U1'(f8_in), F8_IN)
F9_IN → c3(U2'(f14_in), F14_IN)
F8_IN → c7(U3'(f9_in, f10_in), F9_IN)
F14_IN → c10(U4'(f16_in, f17_in), F16_IN)

S tuples:

F1_IN → c(U1'(f8_in), F8_IN)
F9_IN → c3(U2'(f14_in), F14_IN)
F8_IN → c7(U3'(f9_in, f10_in), F9_IN)
F14_IN → c10(U4'(f16_in, f17_in), F16_IN)

K tuples:none
Defined Rule Symbols:

f1_in, U1, f9_in, U2, f16_in, f8_in, U3, f14_in, U4

Defined Pair Symbols:

F1_IN, F9_IN, F8_IN, F14_IN

Compound Symbols:

c, c3, c7, c10

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

Split RHS of tuples not part of any SCC

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in → U1(f8_in)
U1(f8_out1) → f1_out1
U1(f8_out2) → f1_out1
f9_in → U2(f14_in)
U2(f14_out1) → f9_out1
U2(f14_out2) → f9_out1
f16_in → f16_out1
f8_in → U3(f9_in, f10_in)
U3(f9_out1, z0) → f8_out1
U3(z0, f10_out1) → f8_out2
f14_in → U4(f16_in, f17_in)
U4(f16_out1, z0) → f14_out1
U4(z0, f17_out1) → f14_out2

Tuples:

F1_IN → c1(U1'(f8_in))
F1_IN → c1(F8_IN)
F9_IN → c1(U2'(f14_in))
F9_IN → c1(F14_IN)
F8_IN → c1(U3'(f9_in, f10_in))
F8_IN → c1(F9_IN)
F14_IN → c1(U4'(f16_in, f17_in))
F14_IN → c1(F16_IN)

S tuples:

F1_IN → c1(U1'(f8_in))
F1_IN → c1(F8_IN)
F9_IN → c1(U2'(f14_in))
F9_IN → c1(F14_IN)
F8_IN → c1(U3'(f9_in, f10_in))
F8_IN → c1(F9_IN)
F14_IN → c1(U4'(f16_in, f17_in))
F14_IN → c1(F16_IN)

K tuples:none
Defined Rule Symbols:

f1_in, U1, f9_in, U2, f16_in, f8_in, U3, f14_in, U4

Defined Pair Symbols:

F1_IN, F9_IN, F8_IN, F14_IN

Compound Symbols:

c1

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

Removed 5 trailing tuple parts

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in → U1(f8_in)
U1(f8_out1) → f1_out1
U1(f8_out2) → f1_out1
f9_in → U2(f14_in)
U2(f14_out1) → f9_out1
U2(f14_out2) → f9_out1
f16_in → f16_out1
f8_in → U3(f9_in, f10_in)
U3(f9_out1, z0) → f8_out1
U3(z0, f10_out1) → f8_out2
f14_in → U4(f16_in, f17_in)
U4(f16_out1, z0) → f14_out1
U4(z0, f17_out1) → f14_out2

Tuples:

F1_IN → c1(F8_IN)
F9_IN → c1(F14_IN)
F8_IN → c1(F9_IN)
F1_IN → c1
F9_IN → c1
F8_IN → c1
F14_IN → c1

S tuples:

F1_IN → c1(F8_IN)
F9_IN → c1(F14_IN)
F8_IN → c1(F9_IN)
F1_IN → c1
F9_IN → c1
F8_IN → c1
F14_IN → c1

K tuples:none
Defined Rule Symbols:

f1_in, U1, f9_in, U2, f16_in, f8_in, U3, f14_in, U4

Defined Pair Symbols:

F1_IN, F9_IN, F8_IN, F14_IN

Compound Symbols:

c1, c1