Runtime Complexity proof of /tmp/tmp1co62k/minimum-bf.pl

Runtime Complexity of the query pattern
minimum(g,a)
w.r.t. the given Prolog program could be shown to be BOUNDS(O(1), O(n^1)).

0 Prolog

↳1 PrologToCdtProblemTransformerProof (UPPER BOUND (ID), 80 ms)

↳2 CdtProblem

    ↳3 CdtGraphRemoveTrailingTuplepartsProof (BOTH BOUNDS(ID, ID), 0 ms)

    ↳4 CdtProblem

    ↳5 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 122 ms)

    ↳6 CdtProblem

    ↳7 SIsEmptyProof (⇔, 0 ms)

    ↳8 BOUNDS(O(1), O(1))

↳9 PrologToCdtProblemTransformerProof (UPPER BOUND (ID), 67 ms)

↳10 CdtProblem

    ↳11 CdtGraphRemoveTrailingTuplepartsProof (BOTH BOUNDS(ID, ID), 0 ms)

    ↳12 CdtProblem

    ↳13 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 105 ms)

    ↳14 CdtProblem

(0) Obligation:

Clauses:

minimum(tree(X, void, X1), X).
minimum(tree(X2, Left, X3), X) :- minimum(Left, X).

Query: minimum(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: minimum(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
3 [label="3: minimum(T1, T2), SCOPE: 1, CLAUSE: 0 | minimum(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: true | minimum(tree(T7, void, T8), T2), SCOPE: 1, CLAUSE: 1\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: minimum(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nX8 is free\nX9 is free\nminimum(T1, T2) !=? minimum(tree(X8, void, X9), X8)", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: minimum(tree(T7, void, T8), T2), SCOPE: 1, CLAUSE: 1\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: minimum(void, T19)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: minimum(void, T19), SCOPE: 2, CLAUSE: 0 | minimum(void, T19), SCOPE: 2, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: minimum(void, T19), SCOPE: 2, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: minimum(T29, T32)\n\nT28 is ground\nT29 is ground\nT30 is ground\nX8 is free\nX9 is free\nminimum(tree(T28, T29, T30), T2) !=? minimum(tree(X8, void, X9), X8)", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 27 [arrowhead = none ];
27 -> 3;

28 [label="EVAL with clause\nminimum(tree(X8, void, X9), X8).\nand substitutionX8 -> T7,\nX9 -> T8,\nT1 -> tree(T7, void, T8),\nT2 -> T7", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 28 [arrowhead = none ];
28 -> 5;

29 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 29 [arrowhead = none ];
29 -> 7;

30 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 30 [arrowhead = none ];
30 -> 10;

31 [label="EVAL with clause\nminimum(tree(X44, X45, X46), X47) :- minimum(X45, X47).\nand substitutionX44 -> T28,\nX45 -> T29,\nX46 -> T30,\nT1 -> tree(T28, T29, T30),\nT2 -> T32,\nX47 -> T32,\nT31 -> T32", fontsize=14, style = filled, fillcolor = lightblue];
7 -> 31 [arrowhead = none ];
31 -> 25;

32 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 32 [arrowhead = none ];
32 -> 26;

33 [label="ONLY EVAL with clause\nminimum(tree(X22, X23, X24), X25) :- minimum(X23, X25).\nand substitutionT7 -> T16,\nX22 -> T16,\nX23 -> void,\nT8 -> T17,\nX24 -> T17,\nT2 -> T19,\nX25 -> T19,\nT18 -> T19", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 33 [arrowhead = none ];
33 -> 14;

34 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
14 -> 34 [arrowhead = none ];
34 -> 19;

35 [label="BACKTRACK\nfor clause: minimum(tree(X, void, X1), X)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
19 -> 35 [arrowhead = none ];
35 -> 20;

36 [label="BACKTRACK\nfor clause: minimum(tree(X2, Left, X3), X) :- minimum(Left, X)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
20 -> 36 [arrowhead = none ];
36 -> 21;

37 [label="INSTANCE with matching:\nT1 -> T29\nT2 -> T32", fontsize=14, style = filled, fillcolor = lightgrey];
25 -> 37 [arrowhead = none , style=dashed];
37 -> 2[style=dashed];

}

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(tree(z0, void, z1)) → f2_out1(z0)
f2_in(tree(z0, z1, z2)) → U1(f2_in(z1), tree(z0, z1, z2))
U1(f2_out1(z0), tree(z1, z2, z3)) → f2_out1(z0)

Tuples:

F2_IN(tree(z0, z1, z2)) → c1(U1'(f2_in(z1), tree(z0, z1, z2)), F2_IN(z1))

S tuples:

F2_IN(tree(z0, z1, z2)) → c1(U1'(f2_in(z1), tree(z0, z1, z2)), F2_IN(z1))

K tuples:none
Defined Rule Symbols:

f2_in, U1

Defined Pair Symbols:

F2_IN

Compound Symbols:

c1

(3) CdtGraphRemoveTrailingTuplepartsProof (BOTH BOUNDS(ID, ID) transformation)

Removed 1 trailing tuple parts

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(tree(z0, void, z1)) → f2_out1(z0)
f2_in(tree(z0, z1, z2)) → U1(f2_in(z1), tree(z0, z1, z2))
U1(f2_out1(z0), tree(z1, z2, z3)) → f2_out1(z0)

Tuples:

F2_IN(tree(z0, z1, z2)) → c1(F2_IN(z1))

S tuples:

F2_IN(tree(z0, z1, z2)) → c1(F2_IN(z1))

K tuples:none
Defined Rule Symbols:

f2_in, U1

Defined Pair Symbols:

F2_IN

Compound Symbols:

c1

(5) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

F2_IN(tree(z0, z1, z2)) → c1(F2_IN(z1))

We considered the (Usable) Rules:none
And the Tuples:

F2_IN(tree(z0, z1, z2)) → c1(F2_IN(z1))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(F2_IN(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(tree(x₁, x₂, x₃)) = [1] + x₂

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(tree(z0, void, z1)) → f2_out1(z0)
f2_in(tree(z0, z1, z2)) → U1(f2_in(z1), tree(z0, z1, z2))
U1(f2_out1(z0), tree(z1, z2, z3)) → f2_out1(z0)

Tuples:

F2_IN(tree(z0, z1, z2)) → c1(F2_IN(z1))

S tuples:none
K tuples:

F2_IN(tree(z0, z1, z2)) → c1(F2_IN(z1))

Defined Rule Symbols:

f2_in, U1

Defined Pair Symbols:

F2_IN

Compound Symbols:

c1

(7) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(8) BOUNDS(O(1), O(1))

(9) 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: minimum(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: minimum(T1, T2), SCOPE: 1, CLAUSE: 0 | minimum(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: true | minimum(tree(T5, void, T6), T2), SCOPE: 1, CLAUSE: 1\n\nT5 is ground\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: minimum(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nX6 is free\nX7 is free\nminimum(T1, T2) !=? minimum(tree(X6, void, X7), X6)", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: minimum(tree(T5, void, T6), T2), SCOPE: 1, CLAUSE: 1\n\nT5 is ground\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: minimum(void, T13)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: minimum(void, T13), SCOPE: 2, CLAUSE: 0 | minimum(void, T13), SCOPE: 2, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: minimum(void, T13), SCOPE: 2, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: minimum(T19, T22)\n\nT18 is ground\nT19 is ground\nT20 is ground\nX6 is free\nX7 is free\nminimum(tree(T18, T19, T20), T2) !=? minimum(tree(X6, void, X7), X6)", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: minimum(T19, T22), SCOPE: 3, CLAUSE: 0 | minimum(T19, T22), SCOPE: 3, CLAUSE: 1\n\nT18 is ground\nT19 is ground\nT20 is ground\nX6 is free\nX7 is free\nminimum(tree(T18, T19, T20), T2) !=? minimum(tree(X6, void, X7), X6)", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: true | minimum(tree(T27, void, T28), T22), SCOPE: 3, CLAUSE: 1\n\nT27 is ground\nT28 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: minimum(T19, T22), SCOPE: 3, CLAUSE: 1\n\nT18 is ground\nT19 is ground\nT20 is ground\nX6 is free\nX7 is free\nX34 is free\nX35 is free\nminimum(tree(T18, T19, T20), T2) !=? minimum(tree(X6, void, X7), X6)\nminimum(T19, T22) !=? minimum(tree(X34, void, X35), X34)", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: minimum(tree(T27, void, T28), T22), SCOPE: 3, CLAUSE: 1\n\nT27 is ground\nT28 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: minimum(void, T39)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: minimum(T49, T52)\n\nT48 is ground\nT49 is ground\nT50 is ground\nX34 is free\nX35 is free\nminimum(tree(T48, T49, T50), T22) !=? minimum(tree(X34, void, X35), X34)", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 30 [arrowhead = none ];
30 -> 4;

31 [label="EVAL with clause\nminimum(tree(X6, void, X7), X6).\nand substitutionX6 -> T5,\nX7 -> T6,\nT1 -> tree(T5, void, T6),\nT2 -> T5", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 31 [arrowhead = none ];
31 -> 6;

32 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 32 [arrowhead = none ];
32 -> 8;

33 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 33 [arrowhead = none ];
33 -> 9;

34 [label="EVAL with clause\nminimum(tree(X26, X27, X28), X29) :- minimum(X27, X29).\nand substitutionX26 -> T18,\nX27 -> T19,\nX28 -> T20,\nT1 -> tree(T18, T19, T20),\nT2 -> T22,\nX29 -> T22,\nT21 -> T22", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 34 [arrowhead = none ];
34 -> 16;

35 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 35 [arrowhead = none ];
35 -> 17;

36 [label="ONLY EVAL with clause\nminimum(tree(X12, X13, X14), X15) :- minimum(X13, X15).\nand substitutionT5 -> T10,\nX12 -> T10,\nX13 -> void,\nT6 -> T11,\nX14 -> T11,\nT2 -> T13,\nX15 -> T13,\nT12 -> T13", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 36 [arrowhead = none ];
36 -> 11;

37 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
11 -> 37 [arrowhead = none ];
37 -> 12;

38 [label="BACKTRACK\nfor clause: minimum(tree(X, void, X1), X)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
12 -> 38 [arrowhead = none ];
38 -> 13;

39 [label="BACKTRACK\nfor clause: minimum(tree(X2, Left, X3), X) :- minimum(Left, X)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 39 [arrowhead = none ];
39 -> 15;

40 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
16 -> 40 [arrowhead = none ];
40 -> 18;

41 [label="EVAL with clause\nminimum(tree(X34, void, X35), X34).\nand substitutionX34 -> T27,\nX35 -> T28,\nT19 -> tree(T27, void, T28),\nT22 -> T27", fontsize=14, style = filled, fillcolor = lightblue];
18 -> 41 [arrowhead = none ];
41 -> 22;

42 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
18 -> 42 [arrowhead = none ];
42 -> 23;

43 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
22 -> 43 [arrowhead = none ];
43 -> 24;

44 [label="EVAL with clause\nminimum(tree(X64, X65, X66), X67) :- minimum(X65, X67).\nand substitutionX64 -> T48,\nX65 -> T49,\nX66 -> T50,\nT19 -> tree(T48, T49, T50),\nT22 -> T52,\nX67 -> T52,\nT51 -> T52", fontsize=14, style = filled, fillcolor = lightblue];
23 -> 44 [arrowhead = none ];
44 -> 28;

45 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
23 -> 45 [arrowhead = none ];
45 -> 29;

46 [label="ONLY EVAL with clause\nminimum(tree(X48, X49, X50), X51) :- minimum(X49, X51).\nand substitutionT27 -> T36,\nX48 -> T36,\nX49 -> void,\nT28 -> T37,\nX50 -> T37,\nT22 -> T39,\nX51 -> T39,\nT38 -> T39", fontsize=14, style = filled, fillcolor = lightblue];
24 -> 46 [arrowhead = none ];
46 -> 27;

47 [label="INSTANCE with matching:\nT13 -> T39", fontsize=14, style = filled, fillcolor = lightgrey];
27 -> 47 [arrowhead = none , style=dashed];
47 -> 11[style=dashed];

48 [label="INSTANCE with matching:\nT1 -> T49\nT2 -> T52", fontsize=14, style = filled, fillcolor = lightgrey];
28 -> 48 [arrowhead = none , style=dashed];
48 -> 1[style=dashed];

}

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(tree(z0, void, z1)) → f1_out1(z0)
f1_in(tree(z0, void, z1)) → U1(f11_in, tree(z0, void, z1))
f1_in(tree(z0, tree(z1, void, z2), z3)) → f1_out1(z1)
f1_in(tree(z0, tree(z1, void, z2), z3)) → U2(f11_in, tree(z0, tree(z1, void, z2), z3))
f1_in(tree(z0, tree(z1, z2, z3), z4)) → U3(f1_in(z2), tree(z0, tree(z1, z2, z3), z4))
U1(f11_out1(z0), tree(z1, void, z2)) → f1_out1(z0)
U2(f11_out1(z0), tree(z1, tree(z2, void, z3), z4)) → f1_out1(z0)
U3(f1_out1(z0), tree(z1, tree(z2, z3, z4), z5)) → f1_out1(z0)

Tuples:

F1_IN(tree(z0, void, z1)) → c1(U1'(f11_in, tree(z0, void, z1)))
F1_IN(tree(z0, tree(z1, void, z2), z3)) → c3(U2'(f11_in, tree(z0, tree(z1, void, z2), z3)))
F1_IN(tree(z0, tree(z1, z2, z3), z4)) → c4(U3'(f1_in(z2), tree(z0, tree(z1, z2, z3), z4)), F1_IN(z2))

S tuples:

F1_IN(tree(z0, void, z1)) → c1(U1'(f11_in, tree(z0, void, z1)))
F1_IN(tree(z0, tree(z1, void, z2), z3)) → c3(U2'(f11_in, tree(z0, tree(z1, void, z2), z3)))
F1_IN(tree(z0, tree(z1, z2, z3), z4)) → c4(U3'(f1_in(z2), tree(z0, tree(z1, z2, z3), z4)), F1_IN(z2))

K tuples:none
Defined Rule Symbols:

f1_in, U1, U2, U3

Defined Pair Symbols:

F1_IN

Compound Symbols:

c1, c3, c4

(11) CdtGraphRemoveTrailingTuplepartsProof (BOTH BOUNDS(ID, ID) transformation)

Removed 3 trailing tuple parts

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(tree(z0, void, z1)) → f1_out1(z0)
f1_in(tree(z0, void, z1)) → U1(f11_in, tree(z0, void, z1))
f1_in(tree(z0, tree(z1, void, z2), z3)) → f1_out1(z1)
f1_in(tree(z0, tree(z1, void, z2), z3)) → U2(f11_in, tree(z0, tree(z1, void, z2), z3))
f1_in(tree(z0, tree(z1, z2, z3), z4)) → U3(f1_in(z2), tree(z0, tree(z1, z2, z3), z4))
U1(f11_out1(z0), tree(z1, void, z2)) → f1_out1(z0)
U2(f11_out1(z0), tree(z1, tree(z2, void, z3), z4)) → f1_out1(z0)
U3(f1_out1(z0), tree(z1, tree(z2, z3, z4), z5)) → f1_out1(z0)

Tuples:

F1_IN(tree(z0, void, z1)) → c1
F1_IN(tree(z0, tree(z1, void, z2), z3)) → c3
F1_IN(tree(z0, tree(z1, z2, z3), z4)) → c4(F1_IN(z2))

S tuples:

F1_IN(tree(z0, void, z1)) → c1
F1_IN(tree(z0, tree(z1, void, z2), z3)) → c3
F1_IN(tree(z0, tree(z1, z2, z3), z4)) → c4(F1_IN(z2))

K tuples:none
Defined Rule Symbols:

f1_in, U1, U2, U3

Defined Pair Symbols:

F1_IN

Compound Symbols:

c1, c3, c4

(13) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

F1_IN(tree(z0, void, z1)) → c1
F1_IN(tree(z0, tree(z1, void, z2), z3)) → c3

We considered the (Usable) Rules:none
And the Tuples:

F1_IN(tree(z0, void, z1)) → c1
F1_IN(tree(z0, tree(z1, void, z2), z3)) → c3
F1_IN(tree(z0, tree(z1, z2, z3), z4)) → c4(F1_IN(z2))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(F1_IN(x₁)) = [2]
POL(c1) = 0
POL(c3) = 0
POL(c4(x₁)) = x₁
POL(tree(x₁, x₂, x₃)) = 0
POL(void) = 0

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(tree(z0, void, z1)) → f1_out1(z0)
f1_in(tree(z0, void, z1)) → U1(f11_in, tree(z0, void, z1))
f1_in(tree(z0, tree(z1, void, z2), z3)) → f1_out1(z1)
f1_in(tree(z0, tree(z1, void, z2), z3)) → U2(f11_in, tree(z0, tree(z1, void, z2), z3))
f1_in(tree(z0, tree(z1, z2, z3), z4)) → U3(f1_in(z2), tree(z0, tree(z1, z2, z3), z4))
U1(f11_out1(z0), tree(z1, void, z2)) → f1_out1(z0)
U2(f11_out1(z0), tree(z1, tree(z2, void, z3), z4)) → f1_out1(z0)
U3(f1_out1(z0), tree(z1, tree(z2, z3, z4), z5)) → f1_out1(z0)

Tuples:

F1_IN(tree(z0, void, z1)) → c1
F1_IN(tree(z0, tree(z1, void, z2), z3)) → c3
F1_IN(tree(z0, tree(z1, z2, z3), z4)) → c4(F1_IN(z2))

S tuples:

F1_IN(tree(z0, tree(z1, z2, z3), z4)) → c4(F1_IN(z2))

K tuples:

F1_IN(tree(z0, void, z1)) → c1
F1_IN(tree(z0, tree(z1, void, z2), z3)) → c3

Defined Rule Symbols:

f1_in, U1, U2, U3

Defined Pair Symbols:

F1_IN

Compound Symbols:

c1, c3, c4