Runtime Complexity proof of /tmp/tmpglYs8E/example1.pl

Runtime Complexity of the query pattern
p(g,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), 59 ms)

↳2 CdtProblem

    ↳3 SIsEmptyProof (⇔, 0 ms)

    ↳4 BOUNDS(O(1), O(1))

↳5 PrologToCdtProblemTransformerProof (UPPER BOUND (ID), 76 ms)

↳6 CdtProblem

(0) Obligation:

Clauses:

p(X, g(X)).
p(X, f(Y)) :- p(X, g(Y)).

Query: p(g,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, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: p(T1, T2), SCOPE: 1, CLAUSE: 0 | p(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: true | p(T4, T2), SCOPE: 1, CLAUSE: 1\n\nT4 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: p(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nX2 is free\np(T1, T2) !=? p(X2, g(X2))", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: p(T4, T2), SCOPE: 1, CLAUSE: 1\n\nT4 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: p(T7, g(T9))\n\nT7 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: p(T7, g(T9)), SCOPE: 2, CLAUSE: 0 | p(T7, g(T9)), SCOPE: 2, CLAUSE: 1\n\nT7 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: true | p(T12, g(T9)), SCOPE: 2, CLAUSE: 1\n\nT12 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: p(T7, g(T9)), SCOPE: 2, CLAUSE: 1\n\nT7 is ground\nX9 is free\np(T7, g(T9)) !=? p(X9, g(X9))", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: p(T12, g(T9)), SCOPE: 2, CLAUSE: 1\n\nT12 is ground", 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];
23 [label="23: p(T17, g(T19))\n\nT17 is ground\nX2 is free\np(T17, T2) !=? p(X2, g(X2))", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: p(T17, g(T19)), SCOPE: 3, CLAUSE: 0 | p(T17, g(T19)), SCOPE: 3, CLAUSE: 1\n\nT17 is ground\nX2 is free\np(T17, T2) !=? p(X2, g(X2))", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: true | p(T22, g(T19)), SCOPE: 3, CLAUSE: 1\n\nT22 is ground\nX2 is free\np(T22, T2) !=? p(X2, g(X2))", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: p(T17, g(T19)), SCOPE: 3, CLAUSE: 1\n\nT17 is ground\nX2 is free\nX20 is free\np(T17, T2) !=? p(X2, g(X2))\np(T17, g(T19)) !=? p(X20, g(X20))", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: p(T22, g(T19)), SCOPE: 3, CLAUSE: 1\n\nT22 is ground\nX2 is free\np(T22, T2) !=? p(X2, g(X2))", 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="EVAL with clause\np(X2, g(X2)).\nand substitutionT1 -> T4,\nX2 -> T4,\nT2 -> g(T4)", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 32 [arrowhead = none ];
32 -> 6;

33 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 33 [arrowhead = none ];
33 -> 8;

34 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 34 [arrowhead = none ];
34 -> 9;

35 [label="EVAL with clause\np(X16, f(X17)) :- p(X16, g(X17)).\nand substitutionT1 -> T17,\nX16 -> T17,\nX17 -> T19,\nT2 -> f(T19),\nT18 -> T19", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 35 [arrowhead = none ];
35 -> 23;

36 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 36 [arrowhead = none ];
36 -> 24;

37 [label="EVAL with clause\np(X5, f(X6)) :- p(X5, g(X6)).\nand substitutionT4 -> T7,\nX5 -> T7,\nX6 -> T9,\nT2 -> f(T9),\nT8 -> T9", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 37 [arrowhead = none ];
37 -> 13;

38 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 38 [arrowhead = none ];
38 -> 14;

39 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
13 -> 39 [arrowhead = none ];
39 -> 15;

40 [label="EVAL with clause\np(X9, g(X9)).\nand substitutionT7 -> T12,\nX9 -> T12,\nT9 -> T12", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 40 [arrowhead = none ];
40 -> 16;

41 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
15 -> 41 [arrowhead = none ];
41 -> 17;

42 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
16 -> 42 [arrowhead = none ];
42 -> 18;

43 [label="BACKTRACK\nfor clause: p(X, f(Y)) :- p(X, g(Y))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 43 [arrowhead = none ];
43 -> 20;

44 [label="BACKTRACK\nfor clause: p(X, f(Y)) :- p(X, g(Y))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
18 -> 44 [arrowhead = none ];
44 -> 19;

45 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
23 -> 45 [arrowhead = none ];
45 -> 25;

46 [label="EVAL with clause\np(X20, g(X20)).\nand substitutionT17 -> T22,\nX20 -> T22,\nT19 -> T22", fontsize=14, style = filled, fillcolor = lightblue];
25 -> 46 [arrowhead = none ];
46 -> 26;

47 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
25 -> 47 [arrowhead = none ];
47 -> 27;

48 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
26 -> 48 [arrowhead = none ];
48 -> 28;

49 [label="BACKTRACK\nfor clause: p(X, f(Y)) :- p(X, g(Y))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
27 -> 49 [arrowhead = none ];
49 -> 30;

50 [label="BACKTRACK\nfor clause: p(X, f(Y)) :- p(X, g(Y))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
28 -> 50 [arrowhead = none ];
50 -> 29;

}

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0) → f2_out1(g(z0))
f2_in(z0) → f2_out1(f(z0))

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

f2_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];
1 [label="1: p(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
3 [label="3: p(T1, T2), SCOPE: 1, CLAUSE: 0 | p(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: true | p(T5, T2), SCOPE: 1, CLAUSE: 1\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: p(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nX3 is free\np(T1, T2) !=? p(X3, g(X3))", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: p(T5, T2), SCOPE: 1, CLAUSE: 1\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: p(T10, g(T12))\n\nT10 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: p(T19, g(T21))\n\nT19 is ground\nX3 is free\np(T19, T2) !=? p(X3, g(X3))", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 23 [arrowhead = none ];
23 -> 3;

24 [label="EVAL with clause\np(X3, g(X3)).\nand substitutionT1 -> T5,\nX3 -> T5,\nT2 -> g(T5)", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 24 [arrowhead = none ];
24 -> 5;

25 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 25 [arrowhead = none ];
25 -> 7;

26 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 26 [arrowhead = none ];
26 -> 10;

27 [label="EVAL with clause\np(X16, f(X17)) :- p(X16, g(X17)).\nand substitutionT1 -> T19,\nX16 -> T19,\nX17 -> T21,\nT2 -> f(T21),\nT20 -> T21", fontsize=14, style = filled, fillcolor = lightblue];
7 -> 27 [arrowhead = none ];
27 -> 21;

28 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 28 [arrowhead = none ];
28 -> 22;

29 [label="EVAL with clause\np(X8, f(X9)) :- p(X8, g(X9)).\nand substitutionT5 -> T10,\nX8 -> T10,\nX9 -> T12,\nT2 -> f(T12),\nT11 -> T12", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 29 [arrowhead = none ];
29 -> 11;

30 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 30 [arrowhead = none ];
30 -> 12;

31 [label="INSTANCE with matching:\nT1 -> T10\nT2 -> g(T12)", fontsize=14, style = filled, fillcolor = lightgrey];
11 -> 31 [arrowhead = none , style=dashed];
31 -> 1[style=dashed];

32 [label="INSTANCE with matching:\nT1 -> T19\nT2 -> g(T21)", fontsize=14, style = filled, fillcolor = lightgrey];
21 -> 32 [arrowhead = none , style=dashed];
32 -> 1[style=dashed];

}

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0) → f1_out1(g(z0))
f1_in(z0) → U1(f1_in(z0), z0)
f1_in(z0) → U2(f1_in(z0), z0)
U1(f1_out1(g(z0)), z1) → f1_out1(f(z0))
U2(f1_out1(g(z0)), z1) → f1_out1(f(z0))

Tuples:

F1_IN(z0) → c1(U1'(f1_in(z0), z0), F1_IN(z0))
F1_IN(z0) → c2(U2'(f1_in(z0), z0), F1_IN(z0))

S tuples:

F1_IN(z0) → c1(U1'(f1_in(z0), z0), F1_IN(z0))
F1_IN(z0) → c2(U2'(f1_in(z0), z0), F1_IN(z0))

K tuples:none
Defined Rule Symbols:

f1_in, U1, U2

Defined Pair Symbols:

F1_IN

Compound Symbols:

c1, c2