Runtime Complexity proof of /tmp/tmpwPj4WU/app1.pl

Runtime Complexity of the query pattern
app(g,a,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), 55 ms)

↳2 CdtProblem

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

    ↳4 CdtProblem

    ↳5 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 124 ms)

    ↳6 CdtProblem

    ↳7 SIsEmptyProof (⇔, 0 ms)

    ↳8 BOUNDS(O(1), O(1))

↳9 PrologToCdtProblemTransformerProof (UPPER BOUND (ID), 76 ms)

↳10 CdtProblem

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

    ↳12 CdtProblem

    ↳13 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 123 ms)

    ↳14 CdtProblem

(0) Obligation:

Clauses:

app([], Y, Z) :- ','(!, eq(Y, Z)).
app(X, Y, .(H, Z)) :- ','(head(X, H), ','(tail(X, T), app(T, Y, Z))).
head([], X1).
head(.(X, X2), X).
tail([], []).
tail(.(X3, Xs), Xs).
eq(X, X).

Query: app(g,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: app(T1, T2, T3)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: app(T1, T2, T3), SCOPE: 1, CLAUSE: 0 | app(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(!_1, eq(T10, T11)) | app([], T2, T3), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: app(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nX8 is free\nX9 is free\napp(T1, T2, T3) !=? app([], X8, X9)", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: eq(T10, T11)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: eq(T10, T11), SCOPE: 2, CLAUSE: 6\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: ','(head(T23, T27), ','(tail(T23, X26), app(X26, T28, T29)))\n\nT23 is ground\nX8 is free\nX9 is free\nX26 is free\napp(T23, T2, T3) !=? app([], X8, X9)", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: ','(head(T23, T27), ','(tail(T23, X26), app(X26, T28, T29))), SCOPE: 3, CLAUSE: 2 | ','(head(T23, T27), ','(tail(T23, X26), app(X26, T28, T29))), SCOPE: 3, CLAUSE: 3\n\nT23 is ground\nX8 is free\nX9 is free\nX26 is free\napp(T23, T2, T3) !=? app([], X8, X9)", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: ','(head(T23, T27), ','(tail(T23, X26), app(X26, T28, T29))), SCOPE: 3, CLAUSE: 3\n\nT23 is ground\nX8 is free\nX9 is free\nX26 is free\napp(T23, T2, T3) !=? app([], X8, X9)", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: ','(tail(.(T37, T38), X26), app(X26, T39, T40))\n\nT37 is ground\nT38 is ground\nX26 is free", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: ','(tail(.(T37, T38), X26), app(X26, T39, T40)), SCOPE: 4, CLAUSE: 4 | ','(tail(.(T37, T38), X26), app(X26, T39, T40)), SCOPE: 4, CLAUSE: 5\n\nT37 is ground\nT38 is ground\nX26 is free", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: ','(tail(.(T37, T38), X26), app(X26, T39, T40)), SCOPE: 4, CLAUSE: 5\n\nT37 is ground\nT38 is ground\nX26 is free", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: app(T50, T39, T40)\n\nT50 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 37 [arrowhead = none ];
37 -> 4;

38 [label="EVAL with clause\napp([], X8, X9) :- ','(!_1, eq(X8, X9)).\nand substitutionT1 -> [],\nT2 -> T10,\nX8 -> T10,\nT3 -> T11,\nX9 -> T11,\nT8 -> T10,\nT9 -> T11", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 38 [arrowhead = none ];
38 -> 5;

39 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 39 [arrowhead = none ];
39 -> 7;

40 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 40 [arrowhead = none ];
40 -> 10;

41 [label="EVAL with clause\napp(X22, X23, .(X24, X25)) :- ','(head(X22, X24), ','(tail(X22, X26), app(X26, X23, X25))).\nand substitutionT1 -> T23,\nX22 -> T23,\nT2 -> T28,\nX23 -> T28,\nX24 -> T27,\nX25 -> T29,\nT3 -> .(T27, T29),\nT25 -> T27,\nT24 -> T28,\nT26 -> T29", fontsize=14, style = filled, fillcolor = lightblue];
7 -> 41 [arrowhead = none ];
41 -> 25;

42 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 42 [arrowhead = none ];
42 -> 26;

43 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
10 -> 43 [arrowhead = none ];
43 -> 12;

44 [label="EVAL with clause\neq(X12, X12).\nand substitutionT10 -> T14,\nX12 -> T14,\nT11 -> T14", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 44 [arrowhead = none ];
44 -> 14;

45 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
12 -> 45 [arrowhead = none ];
45 -> 20;

46 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
14 -> 46 [arrowhead = none ];
46 -> 21;

47 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
25 -> 47 [arrowhead = none ];
47 -> 29;

48 [label="BACKTRACK\nfor clause: head([], X1)\nwith clash: (app(T23, T2, T3), app([], X8, X9))", fontsize=14, style = filled, fillcolor = lightgreen];
29 -> 48 [arrowhead = none ];
48 -> 31;

49 [label="EVAL with clause\nhead(.(X34, X35), X34).\nand substitutionX34 -> T37,\nX35 -> T38,\nT23 -> .(T37, T38),\nT27 -> T37,\nT28 -> T39,\nT29 -> T40", fontsize=14, style = filled, fillcolor = lightblue];
31 -> 49 [arrowhead = none ];
49 -> 32;

50 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
31 -> 50 [arrowhead = none ];
50 -> 33;

51 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
32 -> 51 [arrowhead = none ];
51 -> 34;

52 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
34 -> 52 [arrowhead = none ];
52 -> 35;

53 [label="ONLY EVAL with clause\ntail(.(X44, X45), X45).\nand substitutionT37 -> T49,\nX44 -> T49,\nT38 -> T50,\nX45 -> T50,\nX26 -> T50", fontsize=14, style = filled, fillcolor = lightblue];
35 -> 53 [arrowhead = none ];
53 -> 36;

54 [label="INSTANCE with matching:\nT1 -> T50\nT2 -> T39\nT3 -> T40", fontsize=14, style = filled, fillcolor = lightgrey];
36 -> 54 [arrowhead = none , style=dashed];
54 -> 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: app(T1, T2, T3)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
3 [label="3: app(T1, T2, T3), SCOPE: 1, CLAUSE: 0 | app(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(!_1, eq(T8, T9)) | app([], T2, T3), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: app(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nX6 is free\nX7 is free\napp(T1, T2, T3) !=? app([], X6, X7)", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: eq(T8, T9)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: eq(T8, T9), SCOPE: 2, CLAUSE: 6\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: ','(head(T17, T21), ','(tail(T17, X19), app(X19, T22, T23)))\n\nT17 is ground\nX6 is free\nX7 is free\nX19 is free\napp(T17, T2, T3) !=? app([], X6, X7)", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: ','(head(T17, T21), ','(tail(T17, X19), app(X19, T22, T23))), SCOPE: 3, CLAUSE: 2 | ','(head(T17, T21), ','(tail(T17, X19), app(X19, T22, T23))), SCOPE: 3, CLAUSE: 3\n\nT17 is ground\nX6 is free\nX7 is free\nX19 is free\napp(T17, T2, T3) !=? app([], X6, X7)", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: ','(head(T17, T21), ','(tail(T17, X19), app(X19, T22, T23))), SCOPE: 3, CLAUSE: 3\n\nT17 is ground\nX6 is free\nX7 is free\nX19 is free\napp(T17, T2, T3) !=? app([], X6, X7)", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: ','(tail(.(T29, T30), X19), app(X19, T31, T32))\n\nT29 is ground\nT30 is ground\nX19 is free", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: ','(tail(.(T29, T30), X19), app(X19, T31, T32)), SCOPE: 4, CLAUSE: 4 | ','(tail(.(T29, T30), X19), app(X19, T31, T32)), SCOPE: 4, CLAUSE: 5\n\nT29 is ground\nT30 is ground\nX19 is free", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: ','(tail(.(T29, T30), X19), app(X19, T31, T32)), SCOPE: 4, CLAUSE: 5\n\nT29 is ground\nT30 is ground\nX19 is free", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: app(T42, T31, T32)\n\nT42 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 31 [arrowhead = none ];
31 -> 3;

32 [label="EVAL with clause\napp([], X6, X7) :- ','(!_1, eq(X6, X7)).\nand substitutionT1 -> [],\nT2 -> T8,\nX6 -> T8,\nT3 -> T9,\nX7 -> T9,\nT6 -> T8,\nT7 -> T9", 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="EVAL with clause\napp(X15, X16, .(X17, X18)) :- ','(head(X15, X17), ','(tail(X15, X19), app(X19, X16, X18))).\nand substitutionT1 -> T17,\nX15 -> T17,\nT2 -> T22,\nX16 -> T22,\nX17 -> T21,\nX18 -> T23,\nT3 -> .(T21, T23),\nT19 -> T21,\nT18 -> T22,\nT20 -> T23", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 35 [arrowhead = none ];
35 -> 17;

36 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 36 [arrowhead = none ];
36 -> 18;

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

38 [label="EVAL with clause\neq(X10, X10).\nand substitutionT8 -> T12,\nX10 -> T12,\nT9 -> T12", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 38 [arrowhead = none ];
38 -> 13;

39 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 39 [arrowhead = none ];
39 -> 15;

40 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 40 [arrowhead = none ];
40 -> 16;

41 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
17 -> 41 [arrowhead = none ];
41 -> 19;

42 [label="BACKTRACK\nfor clause: head([], X1)\nwith clash: (app(T17, T2, T3), app([], X6, X7))", fontsize=14, style = filled, fillcolor = lightgreen];
19 -> 42 [arrowhead = none ];
42 -> 22;

43 [label="EVAL with clause\nhead(.(X25, X26), X25).\nand substitutionX25 -> T29,\nX26 -> T30,\nT17 -> .(T29, T30),\nT21 -> T29,\nT22 -> T31,\nT23 -> T32", fontsize=14, style = filled, fillcolor = lightblue];
22 -> 43 [arrowhead = none ];
43 -> 23;

44 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
22 -> 44 [arrowhead = none ];
44 -> 24;

45 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
23 -> 45 [arrowhead = none ];
45 -> 27;

46 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
27 -> 46 [arrowhead = none ];
46 -> 28;

47 [label="ONLY EVAL with clause\ntail(.(X35, X36), X36).\nand substitutionT29 -> T41,\nX35 -> T41,\nT30 -> T42,\nX36 -> T42,\nX19 -> T42", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 47 [arrowhead = none ];
47 -> 30;

48 [label="INSTANCE with matching:\nT1 -> T42\nT2 -> T31\nT3 -> T32", fontsize=14, style = filled, fillcolor = lightgrey];
30 -> 48 [arrowhead = none , style=dashed];
48 -> 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