Runtime Complexity proof of /tmp/tmp22TmLx/map.pl

Runtime Complexity of the query pattern
map(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), 74 ms)

↳2 CdtProblem

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

    ↳4 CdtProblem

    ↳5 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 98 ms)

    ↳6 CdtProblem

    ↳7 SIsEmptyProof (⇔, 0 ms)

    ↳8 BOUNDS(O(1), O(1))

↳9 PrologToCdtProblemTransformerProof (UPPER BOUND (ID), 63 ms)

↳10 CdtProblem

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

    ↳12 CdtProblem

    ↳13 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 94 ms)

    ↳14 CdtProblem

(0) Obligation:

Clauses:

map([], L) :- ','(!, eq(L, [])).
map(X, .(Y, Ys)) :- ','(head(X, H), ','(tail(X, T), ','(p(H, Y), map(T, Ys)))).
head([], X1).
head(.(H, X2), H).
tail([], []).
tail(.(X3, T), T).
p(X, Y).
eq(X, X).

Query: map(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];
1 [label="1: map(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
3 [label="3: map(T1, T2), SCOPE: 1, CLAUSE: 0 | map(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(!_1, eq(T6, [])) | map([], T2), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: map(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nX6 is free\nmap(T1, T2) !=? map([], X6)", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: eq(T6, [])\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: eq(T6, []), SCOPE: 2, CLAUSE: 7\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: ','(head(T16, X21), ','(tail(T16, X22), ','(p(X21, T19), map(X22, T20))))\n\nT16 is ground\nX6 is free\nX21 is free\nX22 is free\nmap(T16, T2) !=? map([], X6)", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: ','(head(T16, X21), ','(tail(T16, X22), ','(p(X21, T19), map(X22, T20)))), SCOPE: 3, CLAUSE: 2 | ','(head(T16, X21), ','(tail(T16, X22), ','(p(X21, T19), map(X22, T20)))), SCOPE: 3, CLAUSE: 3\n\nT16 is ground\nX6 is free\nX21 is free\nX22 is free\nmap(T16, T2) !=? map([], X6)", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: ','(head(T16, X21), ','(tail(T16, X22), ','(p(X21, T19), map(X22, T20)))), SCOPE: 3, CLAUSE: 3\n\nT16 is ground\nX6 is free\nX21 is free\nX22 is free\nmap(T16, T2) !=? map([], X6)", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: ','(tail(.(T26, T27), X22), ','(p(T26, T19), map(X22, T20)))\n\nT26 is ground\nT27 is ground\nX22 is free", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: ','(tail(.(T26, T27), X22), ','(p(T26, T19), map(X22, T20))), SCOPE: 4, CLAUSE: 4 | ','(tail(.(T26, T27), X22), ','(p(T26, T19), map(X22, T20))), SCOPE: 4, CLAUSE: 5\n\nT26 is ground\nT27 is ground\nX22 is free", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: ','(tail(.(T26, T27), X22), ','(p(T26, T19), map(X22, T20))), SCOPE: 4, CLAUSE: 5\n\nT26 is ground\nT27 is ground\nX22 is free", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: ','(p(T34, T19), map(T35, T20))\n\nT34 is ground\nT35 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: ','(p(T34, T19), map(T35, T20)), SCOPE: 5, CLAUSE: 6\n\nT34 is ground\nT35 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: map(T35, T45)\n\nT35 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 41 [arrowhead = none ];
41 -> 3;

42 [label="EVAL with clause\nmap([], X6) :- ','(!_1, eq(X6, [])).\nand substitutionT1 -> [],\nT2 -> T6,\nX6 -> T6,\nT5 -> T6", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 42 [arrowhead = none ];
42 -> 5;

43 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 43 [arrowhead = none ];
43 -> 8;

44 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 44 [arrowhead = none ];
44 -> 9;

45 [label="EVAL with clause\nmap(X18, .(X19, X20)) :- ','(head(X18, X21), ','(tail(X18, X22), ','(p(X21, X19), map(X22, X20)))).\nand substitutionT1 -> T16,\nX18 -> T16,\nX19 -> T19,\nX20 -> T20,\nT2 -> .(T19, T20),\nT17 -> T19,\nT18 -> T20", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 45 [arrowhead = none ];
45 -> 23;

46 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 46 [arrowhead = none ];
46 -> 24;

47 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
9 -> 47 [arrowhead = none ];
47 -> 12;

48 [label="EVAL with clause\neq(X9, X9).\nand substitutionT6 -> [],\nX9 -> [],\nT9 -> []", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 48 [arrowhead = none ];
48 -> 14;

49 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
12 -> 49 [arrowhead = none ];
49 -> 17;

50 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
14 -> 50 [arrowhead = none ];
50 -> 18;

51 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
23 -> 51 [arrowhead = none ];
51 -> 25;

52 [label="BACKTRACK\nfor clause: head([], X1)\nwith clash: (map(T16, T2), map([], X6))", fontsize=14, style = filled, fillcolor = lightgreen];
25 -> 52 [arrowhead = none ];
52 -> 28;

53 [label="EVAL with clause\nhead(.(X30, X31), X30).\nand substitutionX30 -> T26,\nX31 -> T27,\nT16 -> .(T26, T27),\nX21 -> T26", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 53 [arrowhead = none ];
53 -> 31;

54 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
28 -> 54 [arrowhead = none ];
54 -> 32;

55 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
31 -> 55 [arrowhead = none ];
55 -> 33;

56 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
33 -> 56 [arrowhead = none ];
56 -> 34;

57 [label="ONLY EVAL with clause\ntail(.(X38, X39), X39).\nand substitutionT26 -> T34,\nX38 -> T34,\nT27 -> T35,\nX39 -> T35,\nX22 -> T35", fontsize=14, style = filled, fillcolor = lightblue];
34 -> 57 [arrowhead = none ];
57 -> 37;

58 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
37 -> 58 [arrowhead = none ];
58 -> 38;

59 [label="ONLY EVAL with clause\np(X46, X47).\nand substitutionT34 -> T43,\nX46 -> T43,\nT19 -> T44,\nX47 -> T44,\nT20 -> T45", fontsize=14, style = filled, fillcolor = lightblue];
38 -> 59 [arrowhead = none ];
59 -> 40;

60 [label="INSTANCE with matching:\nT1 -> T35\nT2 -> T45", fontsize=14, style = filled, fillcolor = lightgrey];
40 -> 60 [arrowhead = none , style=dashed];
60 -> 1[style=dashed];

}

(2) 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

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

Removed 1 trailing tuple parts

(4) 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

(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.

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

(6) 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

(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];
2 [label="2: map(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: map(T1, T2), SCOPE: 1, CLAUSE: 0 | map(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(!_1, eq(T5, [])) | map([], T2), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: map(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nX5 is free\nmap(T1, T2) !=? map([], X5)", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: eq(T5, [])\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: eq(T5, []), SCOPE: 2, CLAUSE: 7\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];
19 [label="19: ','(head(T12, X15), ','(tail(T12, X16), ','(p(X15, T15), map(X16, T16))))\n\nT12 is ground\nX5 is free\nX15 is free\nX16 is free\nmap(T12, T2) !=? map([], X5)", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ','(head(T12, X15), ','(tail(T12, X16), ','(p(X15, T15), map(X16, T16)))), SCOPE: 3, CLAUSE: 2 | ','(head(T12, X15), ','(tail(T12, X16), ','(p(X15, T15), map(X16, T16)))), SCOPE: 3, CLAUSE: 3\n\nT12 is ground\nX5 is free\nX15 is free\nX16 is free\nmap(T12, T2) !=? map([], X5)", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: ','(head(T12, X15), ','(tail(T12, X16), ','(p(X15, T15), map(X16, T16)))), SCOPE: 3, CLAUSE: 3\n\nT12 is ground\nX5 is free\nX15 is free\nX16 is free\nmap(T12, T2) !=? map([], X5)", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: ','(tail(.(T21, T22), X16), ','(p(T21, T15), map(X16, T16)))\n\nT21 is ground\nT22 is ground\nX16 is free", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: ','(tail(.(T21, T22), X16), ','(p(T21, T15), map(X16, T16))), SCOPE: 4, CLAUSE: 4 | ','(tail(.(T21, T22), X16), ','(p(T21, T15), map(X16, T16))), SCOPE: 4, CLAUSE: 5\n\nT21 is ground\nT22 is ground\nX16 is free", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: ','(tail(.(T21, T22), X16), ','(p(T21, T15), map(X16, T16))), SCOPE: 4, CLAUSE: 5\n\nT21 is ground\nT22 is ground\nX16 is free", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: ','(p(T29, T15), map(T30, T16))\n\nT29 is ground\nT30 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: ','(p(T29, T15), map(T30, T16)), SCOPE: 5, CLAUSE: 6\n\nT29 is ground\nT30 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: map(T30, T40)\n\nT30 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 40 [arrowhead = none ];
40 -> 4;

41 [label="EVAL with clause\nmap([], X5) :- ','(!_1, eq(X5, [])).\nand substitutionT1 -> [],\nT2 -> T5,\nX5 -> T5,\nT4 -> T5", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 41 [arrowhead = none ];
41 -> 6;

42 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 42 [arrowhead = none ];
42 -> 7;

43 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 43 [arrowhead = none ];
43 -> 10;

44 [label="EVAL with clause\nmap(X12, .(X13, X14)) :- ','(head(X12, X15), ','(tail(X12, X16), ','(p(X15, X13), map(X16, X14)))).\nand substitutionT1 -> T12,\nX12 -> T12,\nX13 -> T15,\nX14 -> T16,\nT2 -> .(T15, T16),\nT13 -> T15,\nT14 -> T16", fontsize=14, style = filled, fillcolor = lightblue];
7 -> 44 [arrowhead = none ];
44 -> 19;

45 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 45 [arrowhead = none ];
45 -> 20;

46 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
10 -> 46 [arrowhead = none ];
46 -> 11;

47 [label="EVAL with clause\neq(X8, X8).\nand substitutionT5 -> [],\nX8 -> [],\nT8 -> []", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 47 [arrowhead = none ];
47 -> 13;

48 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 48 [arrowhead = none ];
48 -> 15;

49 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 49 [arrowhead = none ];
49 -> 16;

50 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
19 -> 50 [arrowhead = none ];
50 -> 21;

51 [label="BACKTRACK\nfor clause: head([], X1)\nwith clash: (map(T12, T2), map([], X5))", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 51 [arrowhead = none ];
51 -> 22;

52 [label="EVAL with clause\nhead(.(X23, X24), X23).\nand substitutionX23 -> T21,\nX24 -> T22,\nT12 -> .(T21, T22),\nX15 -> T21", fontsize=14, style = filled, fillcolor = lightblue];
22 -> 52 [arrowhead = none ];
52 -> 26;

53 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
22 -> 53 [arrowhead = none ];
53 -> 27;

54 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
26 -> 54 [arrowhead = none ];
54 -> 29;

55 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
29 -> 55 [arrowhead = none ];
55 -> 30;

56 [label="ONLY EVAL with clause\ntail(.(X31, X32), X32).\nand substitutionT21 -> T29,\nX31 -> T29,\nT22 -> T30,\nX32 -> T30,\nX16 -> T30", fontsize=14, style = filled, fillcolor = lightblue];
30 -> 56 [arrowhead = none ];
56 -> 35;

57 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
35 -> 57 [arrowhead = none ];
57 -> 36;

58 [label="ONLY EVAL with clause\np(X39, X40).\nand substitutionT29 -> T38,\nX39 -> T38,\nT15 -> T39,\nX40 -> T39,\nT16 -> T40", fontsize=14, style = filled, fillcolor = lightblue];
36 -> 58 [arrowhead = none ];
58 -> 39;

59 [label="INSTANCE with matching:\nT1 -> T30\nT2 -> T40", fontsize=14, style = filled, fillcolor = lightgrey];
39 -> 59 [arrowhead = none , style=dashed];
59 -> 2[style=dashed];

}

(10) 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

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

Removed 1 trailing tuple parts

(12) 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

(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.

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

(14) 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