Runtime Complexity proof of /tmp/tmp5qZu

Runtime Complexity of the query pattern
p(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), 55 ms)

↳2 CdtProblem

    ↳3 SIsEmptyProof (⇔, 0 ms)

    ↳4 BOUNDS(O(1), O(1))

↳5 PrologToCdtProblemTransformerProof (UPPER BOUND (ID), 70 ms)

↳6 CdtProblem

(0) Obligation:

Clauses:

p(X) :- q(X).
p(X) :- ','(no(q(X)), p(X)).
q(X).
no(X) :- ','(X, ','(!, failure(a))).
no(X1).
failure(b).

Query: p(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)\n\n", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
3 [label="3: p(T1), SCOPE: 1, CLAUSE: 0 | p(T1), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: q(T6) | p(T1), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: q(T6), SCOPE: 2, CLAUSE: 2 | ?_2 | p(T1), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: true | ?_2 | p(T1), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: ?_2 | p(T1), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: p(T1), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: ','(no(q(T15)), p(T15))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: ','(no(q(T15)), p(T15)), SCOPE: 3, CLAUSE: 3 | ','(no(q(T15)), p(T15)), SCOPE: 3, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ','(','(call(q(T21)), ','(!_3, failure(a))), p(T21)) | ','(no(q(T15)), p(T15)), SCOPE: 3, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: ','(q(T21), ','(','(!_3, failure(a)), p(T21))) | ?_4 | ','(no(q(T15)), p(T15)), SCOPE: 3, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: ','(q(T21), ','(','(!_3, failure(a)), p(T21))), SCOPE: 5, CLAUSE: 2 | ?_5 | ?_4 | ','(no(q(T15)), p(T15)), SCOPE: 3, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: ','(','(!_3, failure(a)), p(T26)) | ?_5 | ?_4 | ','(no(q(T15)), p(T15)), SCOPE: 3, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: ','(failure(a), p(T26))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: ','(failure(a), p(T26)), SCOPE: 6, CLAUSE: 5\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 32 [arrowhead = none ];
32 -> 3;

33 [label="ONLY EVAL with clause\np(X4) :- q(X4).\nand substitutionT1 -> T6,\nX4 -> T6,\nT5 -> T6", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 33 [arrowhead = none ];
33 -> 6;

34 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
6 -> 34 [arrowhead = none ];
34 -> 8;

35 [label="ONLY EVAL with clause\nq(X7).\nand substitutionT6 -> T9,\nX7 -> T9", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 35 [arrowhead = none ];
35 -> 10;

36 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 36 [arrowhead = none ];
36 -> 12;

37 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
12 -> 37 [arrowhead = none ];
37 -> 14;

38 [label="ONLY EVAL with clause\np(X11) :- ','(no(q(X11)), p(X11)).\nand substitutionT1 -> T15,\nX11 -> T15,\nT14 -> T15", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 38 [arrowhead = none ];
38 -> 16;

39 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
16 -> 39 [arrowhead = none ];
39 -> 18;

40 [label="ONLY EVAL with clause\nno(X15) :- ','(call(X15), ','(!_3, failure(a))).\nand substitutionT15 -> T21,\nX15 -> q(T21),\nT20 -> T21", fontsize=14, style = filled, fillcolor = lightblue];
18 -> 40 [arrowhead = none ];
40 -> 21;

41 [label="CALL", fontsize=14, style = filled, fillcolor = lightcyan];
21 -> 41 [arrowhead = none ];
41 -> 23;

42 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
23 -> 42 [arrowhead = none ];
42 -> 24;

43 [label="ONLY EVAL with clause\nq(X18).\nand substitutionT21 -> T26,\nX18 -> T26,\nT25 -> T26", fontsize=14, style = filled, fillcolor = lightblue];
24 -> 43 [arrowhead = none ];
43 -> 27;

44 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
27 -> 44 [arrowhead = none ];
44 -> 28;

45 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
28 -> 45 [arrowhead = none ];
45 -> 29;

46 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
29 -> 46 [arrowhead = none ];
46 -> 31;

}

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in → f2_out1

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)\n\n", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: p(T1), SCOPE: 1, CLAUSE: 0 | p(T1), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: q(T4) | p(T1), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: q(T4), SCOPE: 2, CLAUSE: 2 | ?_2 | p(T1), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: true | ?_2 | p(T1), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ?_2 | p(T1), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: p(T1), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(no(q(T13)), p(T13))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: ','(no(q(T13)), p(T13)), SCOPE: 3, CLAUSE: 3 | ','(no(q(T13)), p(T13)), SCOPE: 3, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: ','(','(call(q(T19)), ','(!_3, failure(a))), p(T19)) | ','(no(q(T13)), p(T13)), SCOPE: 3, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: ','(q(T19), ','(','(!_3, failure(a)), p(T19))) | ?_4 | ','(no(q(T13)), p(T13)), SCOPE: 3, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: ','(q(T19), ','(','(!_3, failure(a)), p(T19))), SCOPE: 5, CLAUSE: 2 | ?_5 | ?_4 | ','(no(q(T13)), p(T13)), SCOPE: 3, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: ','(','(!_3, failure(a)), p(T24)) | ?_5 | ?_4 | ','(no(q(T13)), p(T13)), SCOPE: 3, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: ','(failure(a), p(T24))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: ','(failure(a), p(T24)), SCOPE: 6, CLAUSE: 5\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 33 [arrowhead = none ];
33 -> 4;

34 [label="ONLY EVAL with clause\np(X3) :- q(X3).\nand substitutionT1 -> T4,\nX3 -> T4,\nT3 -> T4", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 34 [arrowhead = none ];
34 -> 5;

35 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
5 -> 35 [arrowhead = none ];
35 -> 7;

36 [label="ONLY EVAL with clause\nq(X6).\nand substitutionT4 -> T7,\nX6 -> T7", fontsize=14, style = filled, fillcolor = lightblue];
7 -> 36 [arrowhead = none ];
36 -> 9;

37 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 37 [arrowhead = none ];
37 -> 11;

38 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 38 [arrowhead = none ];
38 -> 13;

39 [label="ONLY EVAL with clause\np(X10) :- ','(no(q(X10)), p(X10)).\nand substitutionT1 -> T13,\nX10 -> T13,\nT12 -> T13", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 39 [arrowhead = none ];
39 -> 15;

40 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
15 -> 40 [arrowhead = none ];
40 -> 17;

41 [label="ONLY EVAL with clause\nno(X14) :- ','(call(X14), ','(!_3, failure(a))).\nand substitutionT13 -> T19,\nX14 -> q(T19),\nT18 -> T19", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 41 [arrowhead = none ];
41 -> 19;

42 [label="CALL", fontsize=14, style = filled, fillcolor = lightcyan];
19 -> 42 [arrowhead = none ];
42 -> 20;

43 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
20 -> 43 [arrowhead = none ];
43 -> 22;

44 [label="ONLY EVAL with clause\nq(X17).\nand substitutionT19 -> T24,\nX17 -> T24,\nT23 -> T24", fontsize=14, style = filled, fillcolor = lightblue];
22 -> 44 [arrowhead = none ];
44 -> 25;

45 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
25 -> 45 [arrowhead = none ];
45 -> 26;

46 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
26 -> 46 [arrowhead = none ];
46 -> 30;

47 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
30 -> 47 [arrowhead = none ];
47 -> 32;

}

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in → f1_out1

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

f1_in

Defined Pair Symbols:none

Compound Symbols:none