Tool CaT
| Execution Time | Unknown |
|---|
| Answer | YES(?,O(n^2)) |
|---|
| Input | SK90 4.25 |
|---|
stdout:
YES(?,O(n^2))
Problem:
rev(a()) -> a()
rev(b()) -> b()
rev(++(x,y)) -> ++(rev(y),rev(x))
rev(++(x,x)) -> rev(x)
Proof:
Complexity Transformation Processor:
strict:
rev(a()) -> a()
rev(b()) -> b()
rev(++(x,y)) -> ++(rev(y),rev(x))
rev(++(x,x)) -> rev(x)
weak:
Matrix Interpretation Processor:
dimension: 3
max_matrix:
[1 0 0]
[0 0 3]
[0 0 0]
interpretation:
[1 0 0] [1 0 0] [1]
[++](x0, x1) = [0 0 0]x0 + [0 0 3]x1 + [1]
[0 0 0] [0 0 0] [1],
[4]
[b] = [4]
[1],
[1 0 0] [0]
[rev](x0) = [0 0 0]x0 + [4]
[0 0 0] [1],
[0]
[a] = [4]
[1]
orientation:
[0] [0]
rev(a()) = [4] >= [4] = a()
[1] [1]
[4] [4]
rev(b()) = [4] >= [4] = b()
[1] [1]
[1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [1]
rev(++(x,y)) = [0 0 0]x + [0 0 0]y + [4] >= [0 0 0]x + [0 0 0]y + [4] = ++(rev(y),rev(x))
[0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1]
[2 0 0] [1] [1 0 0] [0]
rev(++(x,x)) = [0 0 0]x + [4] >= [0 0 0]x + [4] = rev(x)
[0 0 0] [1] [0 0 0] [1]
problem:
strict:
rev(a()) -> a()
rev(b()) -> b()
rev(++(x,y)) -> ++(rev(y),rev(x))
weak:
rev(++(x,x)) -> rev(x)
Matrix Interpretation Processor:
dimension: 1
max_matrix:
1
interpretation:
[++](x0, x1) = x0 + x1,
[b] = 98,
[rev](x0) = x0 + 5,
[a] = 123
orientation:
rev(a()) = 128 >= 123 = a()
rev(b()) = 103 >= 98 = b()
rev(++(x,y)) = x + y + 5 >= x + y + 10 = ++(rev(y),rev(x))
rev(++(x,x)) = 2x + 5 >= x + 5 = rev(x)
problem:
strict:
rev(++(x,y)) -> ++(rev(y),rev(x))
weak:
rev(a()) -> a()
rev(b()) -> b()
rev(++(x,x)) -> rev(x)
Matrix Interpretation Processor:
dimension: 4
max_matrix:
[1 1 0 1]
[0 0 0 0]
[0 0 0 0]
[0 0 0 1]
interpretation:
[1 0 0 0] [1 1 0 0] [1]
[0 0 0 0] [0 0 0 0] [0]
[++](x0, x1) = [0 0 0 0]x0 + [0 0 0 0]x1 + [0]
[0 0 0 1] [0 0 0 1] [1],
[0]
[0]
[b] = [0]
[0],
[1 0 0 1]
[0 0 0 0]
[rev](x0) = [0 0 0 0]x0
[0 0 0 1] ,
[0]
[0]
[a] = [0]
[0]
orientation:
[1 0 0 1] [1 1 0 1] [2] [1 0 0 1] [1 0 0 1] [1]
[0 0 0 0] [0 0 0 0] [0] [0 0 0 0] [0 0 0 0] [0]
rev(++(x,y)) = [0 0 0 0]x + [0 0 0 0]y + [0] >= [0 0 0 0]x + [0 0 0 0]y + [0] = ++(rev(y),rev(x))
[0 0 0 1] [0 0 0 1] [1] [0 0 0 1] [0 0 0 1] [1]
[0] [0]
[0] [0]
rev(a()) = [0] >= [0] = a()
[0] [0]
[0] [0]
[0] [0]
rev(b()) = [0] >= [0] = b()
[0] [0]
[2 1 0 2] [2] [1 0 0 1]
[0 0 0 0] [0] [0 0 0 0]
rev(++(x,x)) = [0 0 0 0]x + [0] >= [0 0 0 0]x = rev(x)
[0 0 0 2] [1] [0 0 0 1]
problem:
strict:
weak:
rev(++(x,y)) -> ++(rev(y),rev(x))
rev(a()) -> a()
rev(b()) -> b()
rev(++(x,x)) -> rev(x)
Qed
Tool IRC1
| Execution Time | Unknown |
|---|
| Answer | YES(?,O(n^1)) |
|---|
| Input | SK90 4.25 |
|---|
stdout:
YES(?,O(n^1))
Tool IRC2
| Execution Time | Unknown |
|---|
| Answer | YES(?,O(n^1)) |
|---|
| Input | SK90 4.25 |
|---|
stdout:
YES(?,O(n^1))
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ rev(a()) -> a()
, rev(b()) -> b()
, rev(++(x, y)) -> ++(rev(y), rev(x))
, rev(++(x, x)) -> rev(x)}
Proof Output:
'Bounds with minimal-enrichment and initial automaton 'match'' proved the best result:
Details:
--------
'Bounds with minimal-enrichment and initial automaton 'match'' succeeded with the following output:
'Bounds with minimal-enrichment and initial automaton 'match''
--------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ rev(a()) -> a()
, rev(b()) -> b()
, rev(++(x, y)) -> ++(rev(y), rev(x))
, rev(++(x, x)) -> rev(x)}
Proof Output:
The problem is match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ rev_0(2) -> 1
, rev_1(2) -> 1
, rev_1(2) -> 3
, rev_1(2) -> 4
, a_0() -> 2
, a_1() -> 1
, a_1() -> 3
, a_1() -> 4
, b_0() -> 2
, b_1() -> 1
, b_1() -> 3
, b_1() -> 4
, ++_0(2, 2) -> 2
, ++_1(3, 4) -> 1
, ++_1(4, 4) -> 3
, ++_1(4, 4) -> 4}Tool RC1
| Execution Time | Unknown |
|---|
| Answer | YES(?,O(n^1)) |
|---|
| Input | SK90 4.25 |
|---|
stdout:
YES(?,O(n^1))
Tool RC2
| Execution Time | Unknown |
|---|
| Answer | YES(?,O(n^1)) |
|---|
| Input | SK90 4.25 |
|---|
stdout:
YES(?,O(n^1))
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: YES(?,O(n^1))
Input Problem: runtime-complexity with respect to
Rules:
{ rev(a()) -> a()
, rev(b()) -> b()
, rev(++(x, y)) -> ++(rev(y), rev(x))
, rev(++(x, x)) -> rev(x)}
Proof Output:
'Bounds with minimal-enrichment and initial automaton 'match'' proved the best result:
Details:
--------
'Bounds with minimal-enrichment and initial automaton 'match'' succeeded with the following output:
'Bounds with minimal-enrichment and initial automaton 'match''
--------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: runtime-complexity with respect to
Rules:
{ rev(a()) -> a()
, rev(b()) -> b()
, rev(++(x, y)) -> ++(rev(y), rev(x))
, rev(++(x, x)) -> rev(x)}
Proof Output:
The problem is match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ rev_0(2) -> 1
, rev_1(2) -> 1
, rev_1(2) -> 3
, rev_1(2) -> 4
, a_0() -> 2
, a_1() -> 1
, a_1() -> 3
, a_1() -> 4
, b_0() -> 2
, b_1() -> 1
, b_1() -> 3
, b_1() -> 4
, ++_0(2, 2) -> 2
, ++_1(3, 4) -> 1
, ++_1(4, 4) -> 3
, ++_1(4, 4) -> 4}Tool pair1rc
| Execution Time | Unknown |
|---|
| Answer | YES(?,O(n^1)) |
|---|
| Input | SK90 4.25 |
|---|
stdout:
YES(?,O(n^1))
We consider the following Problem:
Strict Trs:
{ rev(a()) -> a()
, rev(b()) -> b()
, rev(++(x, y)) -> ++(rev(y), rev(x))
, rev(++(x, x)) -> rev(x)}
StartTerms: basic terms
Strategy: none
Certificate: YES(?,O(n^1))
Application of 'pair1 (timeout of 60.0 seconds)':
-------------------------------------------------
The processor is not applicable, reason is:
Input problem is not restricted to innermost rewriting
We abort the transformation and continue with the subprocessor on the problem
Strict Trs:
{ rev(a()) -> a()
, rev(b()) -> b()
, rev(++(x, y)) -> ++(rev(y), rev(x))
, rev(++(x, x)) -> rev(x)}
StartTerms: basic terms
Strategy: none
1) 'Fastest' proved the goal fastest:
'Fastest' proved the goal fastest:
'Bounds with minimal-enrichment and initial automaton 'match'' proved the goal fastest:
The problem is match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ rev_0(2) -> 1
, rev_1(2) -> 1
, rev_1(2) -> 3
, rev_1(2) -> 4
, a_0() -> 2
, a_1() -> 1
, a_1() -> 3
, a_1() -> 4
, b_0() -> 2
, b_1() -> 1
, b_1() -> 3
, b_1() -> 4
, ++_0(2, 2) -> 2
, ++_1(3, 4) -> 1
, ++_1(4, 4) -> 3
, ++_1(4, 4) -> 4}
Hurray, we answered YES(?,O(n^1))Tool pair2rc
| Execution Time | Unknown |
|---|
| Answer | YES(?,O(n^1)) |
|---|
| Input | SK90 4.25 |
|---|
stdout:
YES(?,O(n^1))
We consider the following Problem:
Strict Trs:
{ rev(a()) -> a()
, rev(b()) -> b()
, rev(++(x, y)) -> ++(rev(y), rev(x))
, rev(++(x, x)) -> rev(x)}
StartTerms: basic terms
Strategy: none
Certificate: YES(?,O(n^1))
Application of 'pair2 (timeout of 60.0 seconds)':
-------------------------------------------------
The processor is not applicable, reason is:
Input problem is not restricted to innermost rewriting
We abort the transformation and continue with the subprocessor on the problem
Strict Trs:
{ rev(a()) -> a()
, rev(b()) -> b()
, rev(++(x, y)) -> ++(rev(y), rev(x))
, rev(++(x, x)) -> rev(x)}
StartTerms: basic terms
Strategy: none
1) 'Fastest' proved the goal fastest:
'Fastest' proved the goal fastest:
'Bounds with minimal-enrichment and initial automaton 'match'' proved the goal fastest:
The problem is match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ rev_0(2) -> 1
, rev_1(2) -> 1
, rev_1(2) -> 3
, rev_1(2) -> 4
, a_0() -> 2
, a_1() -> 1
, a_1() -> 3
, a_1() -> 4
, b_0() -> 2
, b_1() -> 1
, b_1() -> 3
, b_1() -> 4
, ++_0(2, 2) -> 2
, ++_1(3, 4) -> 1
, ++_1(4, 4) -> 3
, ++_1(4, 4) -> 4}
Hurray, we answered YES(?,O(n^1))Tool pair3irc
| Execution Time | Unknown |
|---|
| Answer | YES(?,O(n^1)) |
|---|
| Input | SK90 4.25 |
|---|
stdout:
YES(?,O(n^1))
We consider the following Problem:
Strict Trs:
{ rev(a()) -> a()
, rev(b()) -> b()
, rev(++(x, y)) -> ++(rev(y), rev(x))
, rev(++(x, x)) -> rev(x)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Application of 'pair3 (timeout of 60.0 seconds)':
-------------------------------------------------
The input problem contains no overlaps that give rise to inapplicable rules.
We abort the transformation and continue with the subprocessor on the problem
Strict Trs:
{ rev(a()) -> a()
, rev(b()) -> b()
, rev(++(x, y)) -> ++(rev(y), rev(x))
, rev(++(x, x)) -> rev(x)}
StartTerms: basic terms
Strategy: innermost
1) 'Fastest' proved the goal fastest:
'Fastest' proved the goal fastest:
'Bounds with minimal-enrichment and initial automaton 'match'' proved the goal fastest:
The problem is match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ rev_0(2) -> 1
, rev_1(2) -> 1
, rev_1(2) -> 3
, rev_1(2) -> 4
, a_0() -> 2
, a_1() -> 1
, a_1() -> 3
, a_1() -> 4
, b_0() -> 2
, b_1() -> 1
, b_1() -> 3
, b_1() -> 4
, ++_0(2, 2) -> 2
, ++_1(3, 4) -> 1
, ++_1(4, 4) -> 3
, ++_1(4, 4) -> 4}
Hurray, we answered YES(?,O(n^1))Tool pair3rc
| Execution Time | Unknown |
|---|
| Answer | YES(?,O(n^1)) |
|---|
| Input | SK90 4.25 |
|---|
stdout:
YES(?,O(n^1))
We consider the following Problem:
Strict Trs:
{ rev(a()) -> a()
, rev(b()) -> b()
, rev(++(x, y)) -> ++(rev(y), rev(x))
, rev(++(x, x)) -> rev(x)}
StartTerms: basic terms
Strategy: none
Certificate: YES(?,O(n^1))
Application of 'pair3 (timeout of 60.0 seconds)':
-------------------------------------------------
The processor is not applicable, reason is:
Input problem is not restricted to innermost rewriting
We abort the transformation and continue with the subprocessor on the problem
Strict Trs:
{ rev(a()) -> a()
, rev(b()) -> b()
, rev(++(x, y)) -> ++(rev(y), rev(x))
, rev(++(x, x)) -> rev(x)}
StartTerms: basic terms
Strategy: none
1) 'Fastest' proved the goal fastest:
'Fastest' proved the goal fastest:
'Bounds with minimal-enrichment and initial automaton 'match'' proved the goal fastest:
The problem is match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ rev_0(2) -> 1
, rev_1(2) -> 1
, rev_1(2) -> 3
, rev_1(2) -> 4
, a_0() -> 2
, a_1() -> 1
, a_1() -> 3
, a_1() -> 4
, b_0() -> 2
, b_1() -> 1
, b_1() -> 3
, b_1() -> 4
, ++_0(2, 2) -> 2
, ++_1(3, 4) -> 1
, ++_1(4, 4) -> 3
, ++_1(4, 4) -> 4}
Hurray, we answered YES(?,O(n^1))Tool rc
| Execution Time | Unknown |
|---|
| Answer | YES(?,O(n^1)) |
|---|
| Input | SK90 4.25 |
|---|
stdout:
YES(?,O(n^1))
We consider the following Problem:
Strict Trs:
{ rev(a()) -> a()
, rev(b()) -> b()
, rev(++(x, y)) -> ++(rev(y), rev(x))
, rev(++(x, x)) -> rev(x)}
StartTerms: basic terms
Strategy: none
Certificate: YES(?,O(n^1))
Application of 'rc (timeout of 60.0 seconds)':
----------------------------------------------
'Fastest' proved the goal fastest:
'Fastest' proved the goal fastest:
'Bounds with minimal-enrichment and initial automaton 'match' (timeout of 100.0 seconds)' proved the goal fastest:
The problem is match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ rev_0(2) -> 1
, rev_1(2) -> 1
, rev_1(2) -> 3
, rev_1(2) -> 4
, a_0() -> 2
, a_1() -> 1
, a_1() -> 3
, a_1() -> 4
, b_0() -> 2
, b_1() -> 1
, b_1() -> 3
, b_1() -> 4
, ++_0(2, 2) -> 2
, ++_1(3, 4) -> 1
, ++_1(4, 4) -> 3
, ++_1(4, 4) -> 4}
Hurray, we answered YES(?,O(n^1))Tool tup3irc
| Execution Time | 8.9154005e-2ms |
|---|
| Answer | YES(?,O(n^1)) |
|---|
| Input | SK90 4.25 |
|---|
stdout:
YES(?,O(n^1))
We consider the following Problem:
Strict Trs:
{ rev(a()) -> a()
, rev(b()) -> b()
, rev(++(x, y)) -> ++(rev(y), rev(x))
, rev(++(x, x)) -> rev(x)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Application of 'tup3 (timeout of 60.0 seconds)':
------------------------------------------------
The input problem contains no overlaps that give rise to inapplicable rules.
We abort the transformation and continue with the subprocessor on the problem
Strict Trs:
{ rev(a()) -> a()
, rev(b()) -> b()
, rev(++(x, y)) -> ++(rev(y), rev(x))
, rev(++(x, x)) -> rev(x)}
StartTerms: basic terms
Strategy: innermost
1) 'Fastest' proved the goal fastest:
'Fastest' proved the goal fastest:
'Bounds with minimal-enrichment and initial automaton 'match'' proved the goal fastest:
The problem is match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ rev_0(2) -> 1
, rev_1(2) -> 1
, rev_1(2) -> 3
, rev_1(2) -> 4
, a_0() -> 2
, a_1() -> 1
, a_1() -> 3
, a_1() -> 4
, b_0() -> 2
, b_1() -> 1
, b_1() -> 3
, b_1() -> 4
, ++_0(2, 2) -> 2
, ++_1(3, 4) -> 1
, ++_1(4, 4) -> 3
, ++_1(4, 4) -> 4}
Hurray, we answered YES(?,O(n^1))