Runtime Complexity proof of /tmp/tmp1p3NOJ/evenodd1.pl

Runtime Complexity of the query pattern
even(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), 79 ms)

↳2 CdtProblem

    ↳3 SIsEmptyProof (⇔, 0 ms)

    ↳4 BOUNDS(O(1), O(1))

↳5 PrologToCdtProblemTransformerProof (UPPER BOUND (ID), 59 ms)

↳6 CdtProblem

(0) Obligation:

Clauses:

even(0) :- !.
even(N) :- ','(p(N, P), odd(P)).
odd(s(0)) :- !.
odd(s(N)) :- even(P).
p(0, 0).
p(s(X), X).

Query: even(g)

(1) 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: even(T1)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
3 [label="3: even(T1), SCOPE: 1, CLAUSE: 0 | even(T1), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: !_1 | even(0), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: even(T1), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\neven(T1) !=? even(0)", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: ','(p(T4, X5), odd(X5))\n\nT4 is ground\nX5 is free\neven(T4) !=? even(0)", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: ','(p(T4, X5), odd(X5)), SCOPE: 2, CLAUSE: 4 | ','(p(T4, X5), odd(X5)), SCOPE: 2, CLAUSE: 5\n\nT4 is ground\nX5 is free\neven(T4) !=? even(0)", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: ','(p(T4, X5), odd(X5)), SCOPE: 2, CLAUSE: 5\n\nT4 is ground\nX5 is free\neven(T4) !=? even(0)", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: odd(T7)\n\nT7 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: odd(T7), SCOPE: 3, CLAUSE: 2 | odd(T7), SCOPE: 3, CLAUSE: 3\n\nT7 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: !_3 | odd(s(0)), SCOPE: 3, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: odd(T7), SCOPE: 3, CLAUSE: 3\n\nT7 is ground\nodd(T7) !=? odd(s(0))", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: even(X13)\n\nX13 is free", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: even(X13), SCOPE: 4, CLAUSE: 0 | even(X13), SCOPE: 4, CLAUSE: 1\n\nX13 is free", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: !_4 | even(X13), SCOPE: 4, CLAUSE: 1\n\nX13 is free", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 43 [arrowhead = none ];
43 -> 3;

44 [label="EVAL with clause\neven(0) :- !_1.\nand substitutionT1 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 44 [arrowhead = none ];
44 -> 5;

45 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 45 [arrowhead = none ];
45 -> 7;

46 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 46 [arrowhead = none ];
46 -> 8;

47 [label="ONLY EVAL with clause\neven(X4) :- ','(p(X4, X5), odd(X5)).\nand substitutionT1 -> T4,\nX4 -> T4", fontsize=14, style = filled, fillcolor = lightblue];
7 -> 47 [arrowhead = none ];
47 -> 14;

48 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 48 [arrowhead = none ];
48 -> 11;

49 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
14 -> 49 [arrowhead = none ];
49 -> 16;

50 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (even(T4), even(0))", fontsize=14, style = filled, fillcolor = lightgreen];
16 -> 50 [arrowhead = none ];
50 -> 18;

51 [label="EVAL with clause\np(s(X8), X8).\nand substitutionX8 -> T7,\nT4 -> s(T7),\nX5 -> T7", fontsize=14, style = filled, fillcolor = lightblue];
18 -> 51 [arrowhead = none ];
51 -> 19;

52 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
18 -> 52 [arrowhead = none ];
52 -> 21;

53 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
19 -> 53 [arrowhead = none ];
53 -> 24;

54 [label="EVAL with clause\nodd(s(0)) :- !_3.\nand substitutionT7 -> s(0)", fontsize=14, style = filled, fillcolor = lightblue];
24 -> 54 [arrowhead = none ];
54 -> 25;

55 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
24 -> 55 [arrowhead = none ];
55 -> 27;

56 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
25 -> 56 [arrowhead = none ];
56 -> 29;

57 [label="EVAL with clause\nodd(s(X12)) :- even(X13).\nand substitutionX12 -> T10,\nT7 -> s(T10)", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 57 [arrowhead = none ];
57 -> 33;

58 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
27 -> 58 [arrowhead = none ];
58 -> 35;

59 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
29 -> 59 [arrowhead = none ];
59 -> 31;

60 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
33 -> 60 [arrowhead = none ];
60 -> 37;

61 [label="ONLY EVAL with clause\neven(0) :- !_4.\nand substitutionX13 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
37 -> 61 [arrowhead = none ];
61 -> 39;

62 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
39 -> 62 [arrowhead = none ];
62 -> 41;

63 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
41 -> 63 [arrowhead = none ];
63 -> 42;

}

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(0) → f1_out1
f1_in(s(s(0))) → f1_out1
f1_in(s(s(z0))) → f1_out1

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

f1_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];
2 [label="2: even(T1)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: even(T1), SCOPE: 1, CLAUSE: 0 | even(T1), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: !_1 | even(0), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: even(T1), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\neven(T1) !=? even(0)", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(p(T3, X3), odd(X3))\n\nT3 is ground\nX3 is free\neven(T3) !=? even(0)", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(p(T3, X3), odd(X3)), SCOPE: 2, CLAUSE: 4 | ','(p(T3, X3), odd(X3)), SCOPE: 2, CLAUSE: 5\n\nT3 is ground\nX3 is free\neven(T3) !=? even(0)", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: ','(p(T3, X3), odd(X3)), SCOPE: 2, CLAUSE: 5\n\nT3 is ground\nX3 is free\neven(T3) !=? even(0)", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: odd(T6)\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: odd(T6), SCOPE: 3, CLAUSE: 2 | odd(T6), SCOPE: 3, CLAUSE: 3\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: !_3 | odd(s(0)), SCOPE: 3, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: odd(T6), SCOPE: 3, CLAUSE: 3\n\nT6 is ground\nodd(T6) !=? odd(s(0))", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: even(X11)\n\nX11 is free", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: even(X11), SCOPE: 4, CLAUSE: 0 | even(X11), SCOPE: 4, CLAUSE: 1\n\nX11 is free", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: !_4 | even(X11), SCOPE: 4, CLAUSE: 1\n\nX11 is free", 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="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 45 [arrowhead = none ];
45 -> 4;

46 [label="EVAL with clause\neven(0) :- !_1.\nand substitutionT1 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 46 [arrowhead = none ];
46 -> 6;

47 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 47 [arrowhead = none ];
47 -> 9;

48 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 48 [arrowhead = none ];
48 -> 10;

49 [label="ONLY EVAL with clause\neven(X2) :- ','(p(X2, X3), odd(X3)).\nand substitutionT1 -> T3,\nX2 -> T3", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 49 [arrowhead = none ];
49 -> 13;

50 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 50 [arrowhead = none ];
50 -> 12;

51 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
13 -> 51 [arrowhead = none ];
51 -> 15;

52 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (even(T3), even(0))", fontsize=14, style = filled, fillcolor = lightgreen];
15 -> 52 [arrowhead = none ];
52 -> 17;

53 [label="EVAL with clause\np(s(X6), X6).\nand substitutionX6 -> T6,\nT3 -> s(T6),\nX3 -> T6", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 53 [arrowhead = none ];
53 -> 20;

54 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 54 [arrowhead = none ];
54 -> 22;

55 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
20 -> 55 [arrowhead = none ];
55 -> 23;

56 [label="EVAL with clause\nodd(s(0)) :- !_3.\nand substitutionT6 -> s(0)", fontsize=14, style = filled, fillcolor = lightblue];
23 -> 56 [arrowhead = none ];
56 -> 26;

57 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
23 -> 57 [arrowhead = none ];
57 -> 28;

58 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
26 -> 58 [arrowhead = none ];
58 -> 30;

59 [label="EVAL with clause\nodd(s(X10)) :- even(X11).\nand substitutionX10 -> T9,\nT6 -> s(T9)", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 59 [arrowhead = none ];
59 -> 34;

60 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
28 -> 60 [arrowhead = none ];
60 -> 36;

61 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
30 -> 61 [arrowhead = none ];
61 -> 32;

62 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
34 -> 62 [arrowhead = none ];
62 -> 38;

63 [label="ONLY EVAL with clause\neven(0) :- !_4.\nand substitutionX11 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
38 -> 63 [arrowhead = none ];
63 -> 40;

64 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
40 -> 64 [arrowhead = none ];
64 -> 43;

65 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
43 -> 65 [arrowhead = none ];
65 -> 44;

}

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(0) → f2_out1
f2_in(s(s(0))) → f2_out1
f2_in(s(s(z0))) → f2_out1

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

f2_in

Defined Pair Symbols:none

Compound Symbols:none