Runtime Complexity proof of /tmp/tmpikeOGr/lateinstance.pl

Runtime Complexity of the query pattern
p(a,a,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), 60 ms)

↳2 CdtProblem

    ↳3 SIsEmptyProof (⇔, 0 ms)

    ↳4 BOUNDS(O(1), O(1))

↳5 PrologToCdtProblemTransformerProof (UPPER BOUND (ID), 73 ms)

↳6 CdtProblem

(0) Obligation:

Clauses:

p(X, X, a) :- !.
p(X, Y, Z) :- ','(=(Z, a), ','(=(X, Y), p(X, Y, Z))).

Query: p(a,a,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, T3)\n\n", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
3 [label="3: p(T1, T2, T3), SCOPE: 1, CLAUSE: 0 | p(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: !_1 | p(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: p(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\nX3 is free\np(T1, T2, T3) !=? p(X3, X3, a)", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: ','(=(T19, a), ','(=(T20, T21), p(T20, T21, T19)))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ','(=(T22, T23), p(T22, T23, a))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: p(T26, T26, a)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: p(T26, T26, a), SCOPE: 2, CLAUSE: 0 | p(T26, T26, a), SCOPE: 2, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: !_2 | p(T26, T26, a), SCOPE: 2, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: true\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 -> 3;

32 [label="EVAL with clause\np(X3, X3, a) :- !_1.\nand substitutionT1 -> T6,\nX3 -> T6,\nT2 -> T6,\nT3 -> a", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 32 [arrowhead = none ];
32 -> 6;

33 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 33 [arrowhead = none ];
33 -> 8;

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

35 [label="ONLY EVAL with clause\np(X10, X11, X12) :- ','(=(X12, a), ','(=(X10, X11), p(X10, X11, X12))).\nand substitutionT1 -> T20,\nX10 -> T20,\nT2 -> T21,\nX11 -> T21,\nT3 -> T19,\nX12 -> T19,\nT18 -> T19,\nT16 -> T20,\nT17 -> T21", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 35 [arrowhead = none ];
35 -> 19;

36 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 36 [arrowhead = none ];
36 -> 12;

37 [label="UNIFY CASE with substitutionT19 -> a,\nT20 -> T22,\nT21 -> T23", fontsize=14, style = filled, fillcolor = lightblue];
19 -> 37 [arrowhead = none ];
37 -> 21;

38 [label="UNIFY-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
19 -> 38 [arrowhead = none ];
38 -> 23;

39 [label="UNIFY CASE with substitutionT22 -> T26,\nT23 -> T26,\nT25 -> T26", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 39 [arrowhead = none ];
39 -> 25;

40 [label="UNIFY-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 40 [arrowhead = none ];
40 -> 26;

41 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
25 -> 41 [arrowhead = none ];
41 -> 27;

42 [label="ONLY EVAL with clause\np(X16, X16, a) :- !_2.\nand substitutionT26 -> T30,\nX16 -> T30", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 42 [arrowhead = none ];
42 -> 28;

43 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
28 -> 43 [arrowhead = none ];
43 -> 29;

44 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
29 -> 44 [arrowhead = none ];
44 -> 30;

}

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in → f2_out1(a)

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, T3)\n\n", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: p(T1, T2, T3), SCOPE: 1, CLAUSE: 0 | p(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: !_1 | p(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: p(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\nX2 is free\np(T1, T2, T3) !=? p(X2, X2, a)", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(=(T12, a), ','(=(T13, T14), p(T13, T14, T12)))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: ','(=(T15, T16), p(T15, T16, a))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: p(T19, T19, a)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: p(T19, T19, a), SCOPE: 2, CLAUSE: 0 | p(T19, T19, a), SCOPE: 2, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: !_2 | p(T19, T19, a), SCOPE: 2, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 25 [arrowhead = none ];
25 -> 4;

26 [label="EVAL with clause\np(X2, X2, a) :- !_1.\nand substitutionT1 -> T5,\nX2 -> T5,\nT2 -> T5,\nT3 -> a", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 26 [arrowhead = none ];
26 -> 5;

27 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 27 [arrowhead = none ];
27 -> 7;

28 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 28 [arrowhead = none ];
28 -> 10;

29 [label="ONLY EVAL with clause\np(X6, X7, X8) :- ','(=(X8, a), ','(=(X6, X7), p(X6, X7, X8))).\nand substitutionT1 -> T13,\nX6 -> T13,\nT2 -> T14,\nX7 -> T14,\nT3 -> T12,\nX8 -> T12,\nT11 -> T12,\nT9 -> T13,\nT10 -> T14", fontsize=14, style = filled, fillcolor = lightblue];
7 -> 29 [arrowhead = none ];
29 -> 13;

30 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 30 [arrowhead = none ];
30 -> 11;

31 [label="UNIFY CASE with substitutionT12 -> a,\nT13 -> T15,\nT14 -> T16", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 31 [arrowhead = none ];
31 -> 14;

32 [label="UNIFY-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 32 [arrowhead = none ];
32 -> 15;

33 [label="UNIFY CASE with substitutionT15 -> T19,\nT16 -> T19,\nT18 -> T19", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 33 [arrowhead = none ];
33 -> 16;

34 [label="UNIFY-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
14 -> 34 [arrowhead = none ];
34 -> 17;

35 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
16 -> 35 [arrowhead = none ];
35 -> 18;

36 [label="ONLY EVAL with clause\np(X11, X11, a) :- !_2.\nand substitutionT19 -> T22,\nX11 -> T22", fontsize=14, style = filled, fillcolor = lightblue];
18 -> 36 [arrowhead = none ];
36 -> 20;

37 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
20 -> 37 [arrowhead = none ];
37 -> 22;

38 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
22 -> 38 [arrowhead = none ];
38 -> 24;

}

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in → f1_out1(a)

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

f1_in

Defined Pair Symbols:none

Compound Symbols:none