Runtime Complexity proof of /tmp/tmpsN8NBb/lateinst.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), 65 ms)

↳2 CdtProblem

    ↳3 SIsEmptyProof (⇔, 0 ms)

    ↳4 BOUNDS(O(1), O(1))

↳5 PrologToCdtProblemTransformerProof (UPPER BOUND (ID), 60 ms)

↳6 CdtProblem

(0) Obligation:

Clauses:

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

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];
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\nX3 is free\np(T1, T2, T3) !=? p(X3, X3, 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];
20 [label="20: ','(eq(T20, a), ','(eq(T21, T22), p(T21, T22, T20)))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ','(eq(T20, a), ','(eq(T21, T22), p(T21, T22, T20))), SCOPE: 2, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: ','(eq(T27, T28), p(T27, T28, a))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: ','(eq(T27, T28), p(T27, T28, a)), SCOPE: 3, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: p(T33, T33, a)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: p(T33, T33, a), SCOPE: 4, CLAUSE: 0 | p(T33, T33, a), SCOPE: 4, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: !_4 | p(T33, T33, a), SCOPE: 4, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 35 [arrowhead = none ];
35 -> 4;

36 [label="EVAL with clause\np(X3, X3, a) :- !_1.\nand substitutionT1 -> T6,\nX3 -> T6,\nT2 -> T6,\nT3 -> a", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 36 [arrowhead = none ];
36 -> 5;

37 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 37 [arrowhead = none ];
37 -> 7;

38 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 38 [arrowhead = none ];
38 -> 10;

39 [label="ONLY EVAL with clause\np(X11, X12, X13) :- ','(eq(X13, a), ','(eq(X11, X12), p(X11, X12, X13))).\nand substitutionT1 -> T21,\nX11 -> T21,\nT2 -> T22,\nX12 -> T22,\nT3 -> T20,\nX13 -> T20,\nT19 -> T20,\nT17 -> T21,\nT18 -> T22", fontsize=14, style = filled, fillcolor = lightblue];
7 -> 39 [arrowhead = none ];
39 -> 20;

40 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 40 [arrowhead = none ];
40 -> 11;

41 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
20 -> 41 [arrowhead = none ];
41 -> 21;

42 [label="EVAL with clause\neq(X17, X17).\nand substitutionT20 -> a,\nX17 -> a,\nT26 -> a,\nT21 -> T27,\nT22 -> T28", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 42 [arrowhead = none ];
42 -> 23;

43 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 43 [arrowhead = none ];
43 -> 24;

44 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
23 -> 44 [arrowhead = none ];
44 -> 25;

45 [label="EVAL with clause\neq(X21, X21).\nand substitutionT27 -> T33,\nX21 -> T33,\nT28 -> T33,\nT32 -> T33", fontsize=14, style = filled, fillcolor = lightblue];
25 -> 45 [arrowhead = none ];
45 -> 29;

46 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
25 -> 46 [arrowhead = none ];
46 -> 30;

47 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
29 -> 47 [arrowhead = none ];
47 -> 31;

48 [label="ONLY EVAL with clause\np(X25, X25, a) :- !_4.\nand substitutionT33 -> T37,\nX25 -> T37", fontsize=14, style = filled, fillcolor = lightblue];
31 -> 48 [arrowhead = none ];
48 -> 32;

49 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
32 -> 49 [arrowhead = none ];
49 -> 33;

50 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
33 -> 50 [arrowhead = none ];
50 -> 34;

}

(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];
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\nX2 is free\np(T1, T2, T3) !=? p(X2, X2, 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];
13 [label="13: ','(eq(T12, a), ','(eq(T13, T14), p(T13, T14, T12)))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: ','(eq(T12, a), ','(eq(T13, T14), p(T13, T14, T12))), SCOPE: 2, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(eq(T18, T19), p(T18, T19, a))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: ','(eq(T18, T19), p(T18, T19, a)), SCOPE: 3, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: p(T24, T24, a)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: p(T24, T24, a), SCOPE: 4, CLAUSE: 0 | p(T24, T24, a), SCOPE: 4, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: !_4 | p(T24, T24, a), SCOPE: 4, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 29 [arrowhead = none ];
29 -> 3;

30 [label="EVAL with clause\np(X2, X2, a) :- !_1.\nand substitutionT1 -> T5,\nX2 -> T5,\nT2 -> T5,\nT3 -> a", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 30 [arrowhead = none ];
30 -> 6;

31 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 31 [arrowhead = none ];
31 -> 8;

32 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 32 [arrowhead = none ];
32 -> 9;

33 [label="ONLY EVAL with clause\np(X6, X7, X8) :- ','(eq(X8, a), ','(eq(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];
8 -> 33 [arrowhead = none ];
33 -> 13;

34 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 34 [arrowhead = none ];
34 -> 12;

35 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
13 -> 35 [arrowhead = none ];
35 -> 14;

36 [label="EVAL with clause\neq(X11, X11).\nand substitutionT12 -> a,\nX11 -> a,\nT17 -> a,\nT13 -> T18,\nT14 -> T19", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 36 [arrowhead = none ];
36 -> 15;

37 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
14 -> 37 [arrowhead = none ];
37 -> 16;

38 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
15 -> 38 [arrowhead = none ];
38 -> 17;

39 [label="EVAL with clause\neq(X15, X15).\nand substitutionT18 -> T24,\nX15 -> T24,\nT19 -> T24,\nT23 -> T24", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 39 [arrowhead = none ];
39 -> 18;

40 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 40 [arrowhead = none ];
40 -> 19;

41 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
18 -> 41 [arrowhead = none ];
41 -> 22;

42 [label="ONLY EVAL with clause\np(X19, X19, a) :- !_4.\nand substitutionT24 -> T28,\nX19 -> T28", fontsize=14, style = filled, fillcolor = lightblue];
22 -> 42 [arrowhead = none ];
42 -> 26;

43 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
26 -> 43 [arrowhead = none ];
43 -> 27;

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

}

(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