Tool CaT
stdout:
MAYBE
Problem:
f(g(a(),a())) -> f(s(g(b(),b())))
f(f(x)) -> b()
g(x,x) -> f(g(x,x))
Proof:
Complexity Transformation Processor:
strict:
f(g(a(),a())) -> f(s(g(b(),b())))
f(f(x)) -> b()
g(x,x) -> f(g(x,x))
weak:
Matrix Interpretation Processor:
dimension: 4
max_matrix:
[1 2 1 0]
[0 0 0 1]
[0 0 0 1]
[0 0 0 0]
interpretation:
[1 2 0 0]
[0 0 0 0]
[s](x0) = [0 0 0 0]x0
[0 0 0 0] ,
[0]
[0]
[b] = [0]
[0],
[1 0 0 0] [0]
[0 0 0 0] [1]
[f](x0) = [0 0 0 0]x0 + [0]
[0 0 0 0] [0],
[1 0 1 0] [1 1 0 0] [0]
[0 0 0 1] [0 0 0 0] [1]
[g](x0, x1) = [0 0 0 1]x0 + [0 0 0 0]x1 + [0]
[0 0 0 0] [0 0 0 0] [0],
[0]
[1]
[a] = [2]
[0]
orientation:
[3] [2]
[1] [1]
f(g(a(),a())) = [0] >= [0] = f(s(g(b(),b())))
[0] [0]
[1 0 0 0] [0] [0]
[0 0 0 0] [1] [0]
f(f(x)) = [0 0 0 0]x + [0] >= [0] = b()
[0 0 0 0] [0] [0]
[2 1 1 0] [0] [2 1 1 0] [0]
[0 0 0 1] [1] [0 0 0 0] [1]
g(x,x) = [0 0 0 1]x + [0] >= [0 0 0 0]x + [0] = f(g(x,x))
[0 0 0 0] [0] [0 0 0 0] [0]
problem:
strict:
f(f(x)) -> b()
g(x,x) -> f(g(x,x))
weak:
f(g(a(),a())) -> f(s(g(b(),b())))
Matrix Interpretation Processor:
dimension: 1
max_matrix:
1
interpretation:
[s](x0) = x0 + 106,
[b] = 0,
[f](x0) = x0 + 18,
[g](x0, x1) = x0 + x1,
[a] = 65
orientation:
f(f(x)) = x + 36 >= 0 = b()
g(x,x) = 2x >= 2x + 18 = f(g(x,x))
f(g(a(),a())) = 148 >= 124 = f(s(g(b(),b())))
problem:
strict:
g(x,x) -> f(g(x,x))
weak:
f(f(x)) -> b()
f(g(a(),a())) -> f(s(g(b(),b())))
Open
Tool IRC1
stdout:
MAYBE
Warning when parsing problem:
Unsupported strategy 'OUTERMOST'Tool IRC2
stdout:
MAYBE
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ f(g(a(), a())) -> f(s(g(b(), b())))
, f(f(x)) -> b()
, g(x, x) -> f(g(x, x))}
Proof Output:
None of the processors succeeded.
Details of failed attempt(s):
-----------------------------
1) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: f^#(g(a(), a())) -> c_0(f^#(s(g(b(), b()))))
, 2: f^#(f(x)) -> c_1()
, 3: g^#(x, x) -> c_2(f^#(g(x, x)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{3} [ inherited ]
|
|->{1} [ NA ]
|
`->{2} [ MAYBE ]
Sub-problems:
-------------
* Path {3}: inherited
-------------------
This path is subsumed by the proof of path {3}->{1}.
* Path {3}->{1}: NA
-----------------
The usable rules for this path are:
{ g(x, x) -> f(g(x, x))
, f(g(a(), a())) -> f(s(g(b(), b())))
, f(f(x)) -> b()}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {3}->{2}: MAYBE
--------------------
The usable rules for this path are:
{ g(x, x) -> f(g(x, x))
, f(g(a(), a())) -> f(s(g(b(), b())))
, f(f(x)) -> b()}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ g^#(x, x) -> c_2(f^#(g(x, x)))
, f^#(f(x)) -> c_1()
, g(x, x) -> f(g(x, x))
, f(g(a(), a())) -> f(s(g(b(), b())))
, f(f(x)) -> b()}
Proof Output:
The input cannot be shown compatible
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: f^#(g(a(), a())) -> c_0(f^#(s(g(b(), b()))))
, 2: f^#(f(x)) -> c_1()
, 3: g^#(x, x) -> c_2(f^#(g(x, x)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{3} [ inherited ]
|
|->{1} [ NA ]
|
`->{2} [ MAYBE ]
Sub-problems:
-------------
* Path {3}: inherited
-------------------
This path is subsumed by the proof of path {3}->{1}.
* Path {3}->{1}: NA
-----------------
The usable rules for this path are:
{ g(x, x) -> f(g(x, x))
, f(g(a(), a())) -> f(s(g(b(), b())))
, f(f(x)) -> b()}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {3}->{2}: MAYBE
--------------------
The usable rules for this path are:
{ g(x, x) -> f(g(x, x))
, f(g(a(), a())) -> f(s(g(b(), b())))
, f(f(x)) -> b()}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ g^#(x, x) -> c_2(f^#(g(x, x)))
, f^#(f(x)) -> c_1()
, g(x, x) -> f(g(x, x))
, f(g(a(), a())) -> f(s(g(b(), b())))
, f(f(x)) -> b()}
Proof Output:
The input cannot be shown compatible
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: f^#(g(a(), a())) -> c_0(f^#(s(g(b(), b()))))
, 2: f^#(f(x)) -> c_1()
, 3: g^#(x, x) -> c_2(f^#(g(x, x)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{3} [ inherited ]
|
|->{1} [ NA ]
|
`->{2} [ MAYBE ]
Sub-problems:
-------------
* Path {3}: inherited
-------------------
This path is subsumed by the proof of path {3}->{1}.
* Path {3}->{1}: NA
-----------------
The usable rules for this path are:
{ g(x, x) -> f(g(x, x))
, f(g(a(), a())) -> f(s(g(b(), b())))
, f(f(x)) -> b()}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {3}->{2}: MAYBE
--------------------
The usable rules for this path are:
{ g(x, x) -> f(g(x, x))
, f(g(a(), a())) -> f(s(g(b(), b())))
, f(f(x)) -> b()}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ g^#(x, x) -> c_2(f^#(g(x, x)))
, f^#(f(x)) -> c_1()
, g(x, x) -> f(g(x, x))
, f(g(a(), a())) -> f(s(g(b(), b())))
, f(f(x)) -> b()}
Proof Output:
The input cannot be shown compatible
4) 'matrix-interpretation of dimension 1' failed due to the following reason:
The input cannot be shown compatible
5) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
6) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
Tool RC1
stdout:
MAYBE
Warning when parsing problem:
Unsupported strategy 'OUTERMOST'Tool RC2
stdout:
MAYBE
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ f(g(a(), a())) -> f(s(g(b(), b())))
, f(f(x)) -> b()
, g(x, x) -> f(g(x, x))}
Proof Output:
None of the processors succeeded.
Details of failed attempt(s):
-----------------------------
1) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: f^#(g(a(), a())) -> c_0(f^#(s(g(b(), b()))))
, 2: f^#(f(x)) -> c_1()
, 3: g^#(x, x) -> c_2(f^#(g(x, x)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{3} [ inherited ]
|
|->{1} [ NA ]
|
`->{2} [ MAYBE ]
Sub-problems:
-------------
* Path {3}: inherited
-------------------
This path is subsumed by the proof of path {3}->{1}.
* Path {3}->{1}: NA
-----------------
The usable rules for this path are:
{ g(x, x) -> f(g(x, x))
, f(g(a(), a())) -> f(s(g(b(), b())))
, f(f(x)) -> b()}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {3}->{2}: MAYBE
--------------------
The usable rules for this path are:
{ g(x, x) -> f(g(x, x))
, f(g(a(), a())) -> f(s(g(b(), b())))
, f(f(x)) -> b()}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ g^#(x, x) -> c_2(f^#(g(x, x)))
, f^#(f(x)) -> c_1()
, g(x, x) -> f(g(x, x))
, f(g(a(), a())) -> f(s(g(b(), b())))
, f(f(x)) -> b()}
Proof Output:
The input cannot be shown compatible
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: f^#(g(a(), a())) -> c_0(f^#(s(g(b(), b()))))
, 2: f^#(f(x)) -> c_1()
, 3: g^#(x, x) -> c_2(f^#(g(x, x)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{3} [ inherited ]
|
|->{1} [ NA ]
|
`->{2} [ MAYBE ]
Sub-problems:
-------------
* Path {3}: inherited
-------------------
This path is subsumed by the proof of path {3}->{1}.
* Path {3}->{1}: NA
-----------------
The usable rules for this path are:
{ g(x, x) -> f(g(x, x))
, f(g(a(), a())) -> f(s(g(b(), b())))
, f(f(x)) -> b()}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {3}->{2}: MAYBE
--------------------
The usable rules for this path are:
{ g(x, x) -> f(g(x, x))
, f(g(a(), a())) -> f(s(g(b(), b())))
, f(f(x)) -> b()}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ g^#(x, x) -> c_2(f^#(g(x, x)))
, f^#(f(x)) -> c_1()
, g(x, x) -> f(g(x, x))
, f(g(a(), a())) -> f(s(g(b(), b())))
, f(f(x)) -> b()}
Proof Output:
The input cannot be shown compatible
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: f^#(g(a(), a())) -> c_0(f^#(s(g(b(), b()))))
, 2: f^#(f(x)) -> c_1()
, 3: g^#(x, x) -> c_2(f^#(g(x, x)))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{3} [ inherited ]
|
|->{1} [ NA ]
|
`->{2} [ MAYBE ]
Sub-problems:
-------------
* Path {3}: inherited
-------------------
This path is subsumed by the proof of path {3}->{1}.
* Path {3}->{1}: NA
-----------------
The usable rules for this path are:
{ g(x, x) -> f(g(x, x))
, f(g(a(), a())) -> f(s(g(b(), b())))
, f(f(x)) -> b()}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {3}->{2}: MAYBE
--------------------
The usable rules for this path are:
{ g(x, x) -> f(g(x, x))
, f(g(a(), a())) -> f(s(g(b(), b())))
, f(f(x)) -> b()}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ g^#(x, x) -> c_2(f^#(g(x, x)))
, f^#(f(x)) -> c_1()
, g(x, x) -> f(g(x, x))
, f(g(a(), a())) -> f(s(g(b(), b())))
, f(f(x)) -> b()}
Proof Output:
The input cannot be shown compatible
4) 'matrix-interpretation of dimension 1' failed due to the following reason:
The input cannot be shown compatible
5) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
6) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.