Runtime Complexity proof of /tmp/tmpYgCVe0/list.pl

Runtime Complexity of the query pattern
list(g)
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), 56 ms)

↳2 CdtProblem

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

    ↳4 CdtProblem

    ↳5 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 115 ms)

    ↳6 CdtProblem

    ↳7 SIsEmptyProof (⇔, 0 ms)

    ↳8 BOUNDS(O(1), O(1))

↳9 PrologToCdtProblemTransformerProof (UPPER BOUND (ID), 74 ms)

↳10 CdtProblem

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

    ↳12 CdtProblem

    ↳13 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 115 ms)

    ↳14 CdtProblem

(0) Obligation:

Clauses:

list([]) :- !.
list(X) :- ','(tail(X, T), list(T)).
tail([], []).
tail(.(X, Xs), Xs).

Query: list(g)

(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: list(T1)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
3 [label="3: list(T1), SCOPE: 1, CLAUSE: 0 | list(T1), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: !_1 | list([]), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: list(T1), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nlist(T1) !=? list([])", 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];
14 [label="14: ','(tail(T4, X5), list(X5))\n\nT4 is ground\nX5 is free\nlist(T4) !=? list([])", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: ','(tail(T4, X5), list(X5)), SCOPE: 2, CLAUSE: 2 | ','(tail(T4, X5), list(X5)), SCOPE: 2, CLAUSE: 3\n\nT4 is ground\nX5 is free\nlist(T4) !=? list([])", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: ','(tail(T4, X5), list(X5)), SCOPE: 2, CLAUSE: 3\n\nT4 is ground\nX5 is free\nlist(T4) !=? list([])", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: list(T10)\n\nT10 is ground", 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];
2 -> 23 [arrowhead = none ];
23 -> 3;

24 [label="EVAL with clause\nlist([]) :- !_1.\nand substitutionT1 -> []", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 24 [arrowhead = none ];
24 -> 6;

25 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 25 [arrowhead = none ];
25 -> 8;

26 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 26 [arrowhead = none ];
26 -> 9;

27 [label="ONLY EVAL with clause\nlist(X4) :- ','(tail(X4, X5), list(X5)).\nand substitutionT1 -> T4,\nX4 -> T4", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 27 [arrowhead = none ];
27 -> 14;

28 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 28 [arrowhead = none ];
28 -> 12;

29 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
14 -> 29 [arrowhead = none ];
29 -> 16;

30 [label="BACKTRACK\nfor clause: tail([], [])\nwith clash: (list(T4), list([]))", fontsize=14, style = filled, fillcolor = lightgreen];
16 -> 30 [arrowhead = none ];
30 -> 18;

31 [label="EVAL with clause\ntail(.(X10, X11), X11).\nand substitutionX10 -> T9,\nX11 -> T10,\nT4 -> .(T9, T10),\nX5 -> T10", fontsize=14, style = filled, fillcolor = lightblue];
18 -> 31 [arrowhead = none ];
31 -> 21;

32 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
18 -> 32 [arrowhead = none ];
32 -> 22;

33 [label="INSTANCE with matching:\nT1 -> T10", fontsize=14, style = filled, fillcolor = lightgrey];
21 -> 33 [arrowhead = none , style=dashed];
33 -> 2[style=dashed];

}

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in([]) → f2_out1
f2_in(.(z0, z1)) → U1(f2_in(z1), .(z0, z1))
U1(f2_out1, .(z0, z1)) → f2_out1

Tuples:

F2_IN(.(z0, z1)) → c1(U1'(f2_in(z1), .(z0, z1)), F2_IN(z1))

S tuples:

F2_IN(.(z0, z1)) → c1(U1'(f2_in(z1), .(z0, z1)), 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([]) → f2_out1
f2_in(.(z0, z1)) → U1(f2_in(z1), .(z0, z1))
U1(f2_out1, .(z0, z1)) → f2_out1

Tuples:

F2_IN(.(z0, z1)) → c1(F2_IN(z1))

S tuples:

F2_IN(.(z0, z1)) → 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(.(z0, z1)) → c1(F2_IN(z1))

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

F2_IN(.(z0, z1)) → c1(F2_IN(z1))

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

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

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in([]) → f2_out1
f2_in(.(z0, z1)) → U1(f2_in(z1), .(z0, z1))
U1(f2_out1, .(z0, z1)) → f2_out1

Tuples:

F2_IN(.(z0, z1)) → c1(F2_IN(z1))

S tuples:none
K tuples:

F2_IN(.(z0, z1)) → 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: list(T1)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: list(T1), SCOPE: 1, CLAUSE: 0 | list(T1), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: !_1 | list([]), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: list(T1), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nlist(T1) !=? list([])", 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: ','(tail(T3, X3), list(X3))\n\nT3 is ground\nX3 is free\nlist(T3) !=? list([])", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(tail(T3, X3), list(X3)), SCOPE: 2, CLAUSE: 2 | ','(tail(T3, X3), list(X3)), SCOPE: 2, CLAUSE: 3\n\nT3 is ground\nX3 is free\nlist(T3) !=? list([])", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: ','(tail(T3, X3), list(X3)), SCOPE: 2, CLAUSE: 3\n\nT3 is ground\nX3 is free\nlist(T3) !=? list([])", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: list(T9)\n\nT9 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 21 [arrowhead = none ];
21 -> 4;

22 [label="EVAL with clause\nlist([]) :- !_1.\nand substitutionT1 -> []", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 22 [arrowhead = none ];
22 -> 5;

23 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 23 [arrowhead = none ];
23 -> 7;

24 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 24 [arrowhead = none ];
24 -> 10;

25 [label="ONLY EVAL with clause\nlist(X2) :- ','(tail(X2, X3), list(X3)).\nand substitutionT1 -> T3,\nX2 -> T3", fontsize=14, style = filled, fillcolor = lightblue];
7 -> 25 [arrowhead = none ];
25 -> 13;

26 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 26 [arrowhead = none ];
26 -> 11;

27 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
13 -> 27 [arrowhead = none ];
27 -> 15;

28 [label="BACKTRACK\nfor clause: tail([], [])\nwith clash: (list(T3), list([]))", fontsize=14, style = filled, fillcolor = lightgreen];
15 -> 28 [arrowhead = none ];
28 -> 17;

29 [label="EVAL with clause\ntail(.(X8, X9), X9).\nand substitutionX8 -> T8,\nX9 -> T9,\nT3 -> .(T8, T9),\nX3 -> T9", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 29 [arrowhead = none ];
29 -> 19;

30 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 30 [arrowhead = none ];
30 -> 20;

31 [label="INSTANCE with matching:\nT1 -> T9", fontsize=14, style = filled, fillcolor = lightgrey];
19 -> 31 [arrowhead = none , style=dashed];
31 -> 1[style=dashed];

}

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in([]) → f1_out1
f1_in(.(z0, z1)) → U1(f1_in(z1), .(z0, z1))
U1(f1_out1, .(z0, z1)) → f1_out1

Tuples:

F1_IN(.(z0, z1)) → c1(U1'(f1_in(z1), .(z0, z1)), F1_IN(z1))

S tuples:

F1_IN(.(z0, z1)) → c1(U1'(f1_in(z1), .(z0, z1)), F1_IN(z1))

K tuples:none
Defined Rule Symbols:

f1_in, U1

Defined Pair Symbols:

F1_IN

Compound Symbols:

c1

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

Removed 1 trailing tuple parts

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in([]) → f1_out1
f1_in(.(z0, z1)) → U1(f1_in(z1), .(z0, z1))
U1(f1_out1, .(z0, z1)) → f1_out1

Tuples:

F1_IN(.(z0, z1)) → c1(F1_IN(z1))

S tuples:

F1_IN(.(z0, z1)) → c1(F1_IN(z1))

K tuples:none
Defined Rule Symbols:

f1_in, U1

Defined Pair Symbols:

F1_IN

Compound Symbols:

c1

(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(.(z0, z1)) → c1(F1_IN(z1))

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

F1_IN(.(z0, z1)) → c1(F1_IN(z1))

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

POL(.(x₁, x₂)) = [1] + x₂
POL(F1_IN(x₁)) = x₁
POL(c1(x₁)) = x₁

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in([]) → f1_out1
f1_in(.(z0, z1)) → U1(f1_in(z1), .(z0, z1))
U1(f1_out1, .(z0, z1)) → f1_out1

Tuples:

F1_IN(.(z0, z1)) → c1(F1_IN(z1))

S tuples:none
K tuples:

F1_IN(.(z0, z1)) → c1(F1_IN(z1))

Defined Rule Symbols:

f1_in, U1

Defined Pair Symbols:

F1_IN

Compound Symbols:

c1