Runtime Complexity proof of /tmp/tmpytvDNY/add1.pl

Runtime Complexity of the query pattern
add(a,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), 76 ms)

↳2 CdtProblem

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

    ↳4 CdtProblem

    ↳5 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 93 ms)

    ↳6 CdtProblem

    ↳7 SIsEmptyProof (⇔, 0 ms)

    ↳8 BOUNDS(O(1), O(1))

↳9 PrologToCdtProblemTransformerProof (UPPER BOUND (ID), 57 ms)

↳10 CdtProblem

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

    ↳12 CdtProblem

    ↳13 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 114 ms)

    ↳14 CdtProblem

(0) Obligation:

Clauses:

add(X, 0, Y) :- ','(!, eq(X, Y)).
add(X, Y, s(Z)) :- ','(p(Y, P), add(X, P, Z)).
p(0, 0).
p(s(X), X).
eq(X, X).

Query: add(a,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: add(T1, T2, T3)\n\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
3 [label="3: add(T1, T2, T3), SCOPE: 1, CLAUSE: 0 | add(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(!_1, eq(T10, T11)) | add(T1, 0, T3), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: add(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT2 is ground\nX5 is free\nX6 is free\nadd(T1, T2, T3) !=? add(X5, 0, X6)", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: eq(T10, T11)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: eq(T10, T11), SCOPE: 2, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ','(p(T22, X20), add(T24, X20, T25))\n\nT22 is ground\nX5 is free\nX6 is free\nX20 is free\nadd(T1, T22, T3) !=? add(X5, 0, X6)", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: ','(p(T22, X20), add(T24, X20, T25)), SCOPE: 3, CLAUSE: 2 | ','(p(T22, X20), add(T24, X20, T25)), SCOPE: 3, CLAUSE: 3\n\nT22 is ground\nX5 is free\nX6 is free\nX20 is free\nadd(T1, T22, T3) !=? add(X5, 0, X6)", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: ','(p(T22, X20), add(T24, X20, T25)), SCOPE: 3, CLAUSE: 3\n\nT22 is ground\nX5 is free\nX6 is free\nX20 is free\nadd(T1, T22, T3) !=? add(X5, 0, X6)", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: add(T24, T30, T25)\n\nT30 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: \n\n", 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\nadd(X5, 0, X6) :- ','(!_1, eq(X5, X6)).\nand substitutionT1 -> T10,\nX5 -> T10,\nT2 -> 0,\nT3 -> T11,\nX6 -> T11,\nT8 -> T10,\nT9 -> T11", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 32 [arrowhead = none ];
32 -> 5;

33 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 33 [arrowhead = none ];
33 -> 7;

34 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 34 [arrowhead = none ];
34 -> 9;

35 [label="EVAL with clause\nadd(X17, X18, s(X19)) :- ','(p(X18, X20), add(X17, X20, X19)).\nand substitutionT1 -> T24,\nX17 -> T24,\nT2 -> T22,\nX18 -> T22,\nX19 -> T25,\nT3 -> s(T25),\nT21 -> T24,\nT23 -> T25", fontsize=14, style = filled, fillcolor = lightblue];
7 -> 35 [arrowhead = none ];
35 -> 21;

36 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 36 [arrowhead = none ];
36 -> 22;

37 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
9 -> 37 [arrowhead = none ];
37 -> 12;

38 [label="EVAL with clause\neq(X9, X9).\nand substitutionT10 -> T14,\nX9 -> T14,\nT11 -> T14", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 38 [arrowhead = none ];
38 -> 14;

39 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
12 -> 39 [arrowhead = none ];
39 -> 16;

40 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
14 -> 40 [arrowhead = none ];
40 -> 18;

41 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
21 -> 41 [arrowhead = none ];
41 -> 25;

42 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (add(T1, T22, T3), add(X5, 0, X6))", fontsize=14, style = filled, fillcolor = lightgreen];
25 -> 42 [arrowhead = none ];
42 -> 26;

43 [label="EVAL with clause\np(s(X25), X25).\nand substitutionX25 -> T30,\nT22 -> s(T30),\nX20 -> T30", fontsize=14, style = filled, fillcolor = lightblue];
26 -> 43 [arrowhead = none ];
43 -> 28;

44 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
26 -> 44 [arrowhead = none ];
44 -> 30;

45 [label="INSTANCE with matching:\nT1 -> T24\nT2 -> T30\nT3 -> T25", fontsize=14, style = filled, fillcolor = lightgrey];
28 -> 45 [arrowhead = none , style=dashed];
45 -> 1[style=dashed];

}

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(0) → f1_out1
f1_in(s(z0)) → U1(f1_in(z0), s(z0))
U1(f1_out1, s(z0)) → f1_out1

Tuples:

F1_IN(s(z0)) → c1(U1'(f1_in(z0), s(z0)), F1_IN(z0))

S tuples:

F1_IN(s(z0)) → c1(U1'(f1_in(z0), s(z0)), F1_IN(z0))

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(0) → f1_out1
f1_in(s(z0)) → U1(f1_in(z0), s(z0))
U1(f1_out1, s(z0)) → f1_out1

Tuples:

F1_IN(s(z0)) → c1(F1_IN(z0))

S tuples:

F1_IN(s(z0)) → c1(F1_IN(z0))

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(s(z0)) → c1(F1_IN(z0))

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

F1_IN(s(z0)) → c1(F1_IN(z0))

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

POL(F1_IN(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(s(x₁)) = [1] + x₁

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(0) → f1_out1
f1_in(s(z0)) → U1(f1_in(z0), s(z0))
U1(f1_out1, s(z0)) → f1_out1

Tuples:

F1_IN(s(z0)) → c1(F1_IN(z0))

S tuples:none
K tuples:

F1_IN(s(z0)) → c1(F1_IN(z0))

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: add(T1, T2, T3)\n\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: add(T1, T2, T3), SCOPE: 1, CLAUSE: 0 | add(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(!_1, eq(T8, T9)) | add(T1, 0, T3), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: add(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT2 is ground\nX3 is free\nX4 is free\nadd(T1, T2, T3) !=? add(X3, 0, X4)", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: eq(T8, T9)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: eq(T8, T9), SCOPE: 2, CLAUSE: 4\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];
17 [label="17: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: ','(p(T17, X14), add(T19, X14, T20))\n\nT17 is ground\nX3 is free\nX4 is free\nX14 is free\nadd(T1, T17, T3) !=? add(X3, 0, X4)", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: ','(p(T17, X14), add(T19, X14, T20)), SCOPE: 3, CLAUSE: 2 | ','(p(T17, X14), add(T19, X14, T20)), SCOPE: 3, CLAUSE: 3\n\nT17 is ground\nX3 is free\nX4 is free\nX14 is free\nadd(T1, T17, T3) !=? add(X3, 0, X4)", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: ','(p(T17, X14), add(T19, X14, T20)), SCOPE: 3, CLAUSE: 3\n\nT17 is ground\nX3 is free\nX4 is free\nX14 is free\nadd(T1, T17, T3) !=? add(X3, 0, X4)", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: add(T19, T23, T20)\n\nT23 is ground", 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];
2 -> 30 [arrowhead = none ];
30 -> 4;

31 [label="EVAL with clause\nadd(X3, 0, X4) :- ','(!_1, eq(X3, X4)).\nand substitutionT1 -> T8,\nX3 -> T8,\nT2 -> 0,\nT3 -> T9,\nX4 -> T9,\nT6 -> T8,\nT7 -> T9", 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="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 33 [arrowhead = none ];
33 -> 10;

34 [label="EVAL with clause\nadd(X11, X12, s(X13)) :- ','(p(X12, X14), add(X11, X14, X13)).\nand substitutionT1 -> T19,\nX11 -> T19,\nT2 -> T17,\nX12 -> T17,\nX13 -> T20,\nT3 -> s(T20),\nT16 -> T19,\nT18 -> T20", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 34 [arrowhead = none ];
34 -> 19;

35 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 35 [arrowhead = none ];
35 -> 20;

36 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
10 -> 36 [arrowhead = none ];
36 -> 11;

37 [label="EVAL with clause\neq(X7, X7).\nand substitutionT8 -> T12,\nX7 -> T12,\nT9 -> T12", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 37 [arrowhead = none ];
37 -> 13;

38 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 38 [arrowhead = none ];
38 -> 15;

39 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 39 [arrowhead = none ];
39 -> 17;

40 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
19 -> 40 [arrowhead = none ];
40 -> 23;

41 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (add(T1, T17, T3), add(X3, 0, X4))", fontsize=14, style = filled, fillcolor = lightgreen];
23 -> 41 [arrowhead = none ];
41 -> 24;

42 [label="EVAL with clause\np(s(X17), X17).\nand substitutionX17 -> T23,\nT17 -> s(T23),\nX14 -> T23", fontsize=14, style = filled, fillcolor = lightblue];
24 -> 42 [arrowhead = none ];
42 -> 27;

43 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
24 -> 43 [arrowhead = none ];
43 -> 29;

44 [label="INSTANCE with matching:\nT1 -> T19\nT2 -> T23\nT3 -> T20", fontsize=14, style = filled, fillcolor = lightgrey];
27 -> 44 [arrowhead = none , style=dashed];
44 -> 2[style=dashed];

}

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(0) → f2_out1
f2_in(s(z0)) → U1(f2_in(z0), s(z0))
U1(f2_out1, s(z0)) → f2_out1

Tuples:

F2_IN(s(z0)) → c1(U1'(f2_in(z0), s(z0)), F2_IN(z0))

S tuples:

F2_IN(s(z0)) → c1(U1'(f2_in(z0), s(z0)), F2_IN(z0))

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(0) → f2_out1
f2_in(s(z0)) → U1(f2_in(z0), s(z0))
U1(f2_out1, s(z0)) → f2_out1

Tuples:

F2_IN(s(z0)) → c1(F2_IN(z0))

S tuples:

F2_IN(s(z0)) → c1(F2_IN(z0))

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(s(z0)) → c1(F2_IN(z0))

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

F2_IN(s(z0)) → c1(F2_IN(z0))

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

POL(F2_IN(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(s(x₁)) = [1] + x₁

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(0) → f2_out1
f2_in(s(z0)) → U1(f2_in(z0), s(z0))
U1(f2_out1, s(z0)) → f2_out1

Tuples:

F2_IN(s(z0)) → c1(F2_IN(z0))

S tuples:none
K tuples:

F2_IN(s(z0)) → c1(F2_IN(z0))

Defined Rule Symbols:

f2_in, U1

Defined Pair Symbols:

F2_IN

Compound Symbols:

c1