Tool CaT
stdout:
MAYBE
Problem:
ack(0(),y) -> s(y)
ack(s(x),0()) -> ack(x,s(0()))
ack(s(x),s(y)) -> ack(x,ack(s(x),y))
f(s(x),y) -> f(x,s(x))
f(x,s(y)) -> f(y,x)
f(x,y) -> ack(x,y)
ack(s(x),y) -> f(x,x)
Proof:
OpenTool IRC1
stdout:
MAYBE
Tool IRC2
stdout:
MAYBE
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ ack(0(), y) -> s(y)
, ack(s(x), 0()) -> ack(x, s(0()))
, ack(s(x), s(y)) -> ack(x, ack(s(x), y))
, f(s(x), y) -> f(x, s(x))
, f(x, s(y)) -> f(y, x)
, f(x, y) -> ack(x, y)
, ack(s(x), y) -> f(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: ack^#(0(), y) -> c_0()
, 2: ack^#(s(x), 0()) -> c_1(ack^#(x, s(0())))
, 3: ack^#(s(x), s(y)) -> c_2(ack^#(x, ack(s(x), y)))
, 4: f^#(s(x), y) -> c_3(f^#(x, s(x)))
, 5: f^#(x, s(y)) -> c_4(f^#(y, x))
, 6: f^#(x, y) -> c_5(ack^#(x, y))
, 7: ack^#(s(x), y) -> c_6(f^#(x, x))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{2,6,7,3,5,4} [ inherited ]
|
`->{1} [ NA ]
Sub-problems:
-------------
* Path {2,6,7,3,5,4}: inherited
-----------------------------
This path is subsumed by the proof of path {2,6,7,3,5,4}->{1}.
* Path {2,6,7,3,5,4}->{1}: NA
---------------------------
The usable rules for this path are:
{ ack(0(), y) -> s(y)
, ack(s(x), 0()) -> ack(x, s(0()))
, ack(s(x), s(y)) -> ack(x, ack(s(x), y))
, ack(s(x), y) -> f(x, x)
, f(s(x), y) -> f(x, s(x))
, f(x, s(y)) -> f(y, x)
, f(x, y) -> ack(x, y)}
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.
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: ack^#(0(), y) -> c_0()
, 2: ack^#(s(x), 0()) -> c_1(ack^#(x, s(0())))
, 3: ack^#(s(x), s(y)) -> c_2(ack^#(x, ack(s(x), y)))
, 4: f^#(s(x), y) -> c_3(f^#(x, s(x)))
, 5: f^#(x, s(y)) -> c_4(f^#(y, x))
, 6: f^#(x, y) -> c_5(ack^#(x, y))
, 7: ack^#(s(x), y) -> c_6(f^#(x, x))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{2,6,7,3,5,4} [ inherited ]
|
`->{1} [ MAYBE ]
Sub-problems:
-------------
* Path {2,6,7,3,5,4}: inherited
-----------------------------
This path is subsumed by the proof of path {2,6,7,3,5,4}->{1}.
* Path {2,6,7,3,5,4}->{1}: MAYBE
------------------------------
The usable rules for this path are:
{ ack(0(), y) -> s(y)
, ack(s(x), 0()) -> ack(x, s(0()))
, ack(s(x), s(y)) -> ack(x, ack(s(x), y))
, ack(s(x), y) -> f(x, x)
, f(s(x), y) -> f(x, s(x))
, f(x, s(y)) -> f(y, x)
, f(x, y) -> ack(x, y)}
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:
{ ack^#(s(x), 0()) -> c_1(ack^#(x, s(0())))
, f^#(x, y) -> c_5(ack^#(x, y))
, ack^#(s(x), y) -> c_6(f^#(x, x))
, ack^#(s(x), s(y)) -> c_2(ack^#(x, ack(s(x), y)))
, f^#(x, s(y)) -> c_4(f^#(y, x))
, f^#(s(x), y) -> c_3(f^#(x, s(x)))
, ack^#(0(), y) -> c_0()
, ack(0(), y) -> s(y)
, ack(s(x), 0()) -> ack(x, s(0()))
, ack(s(x), s(y)) -> ack(x, ack(s(x), y))
, ack(s(x), y) -> f(x, x)
, f(s(x), y) -> f(x, s(x))
, f(x, s(y)) -> f(y, x)
, f(x, y) -> ack(x, y)}
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: ack^#(0(), y) -> c_0()
, 2: ack^#(s(x), 0()) -> c_1(ack^#(x, s(0())))
, 3: ack^#(s(x), s(y)) -> c_2(ack^#(x, ack(s(x), y)))
, 4: f^#(s(x), y) -> c_3(f^#(x, s(x)))
, 5: f^#(x, s(y)) -> c_4(f^#(y, x))
, 6: f^#(x, y) -> c_5(ack^#(x, y))
, 7: ack^#(s(x), y) -> c_6(f^#(x, x))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{2,6,7,3,5,4} [ inherited ]
|
`->{1} [ MAYBE ]
Sub-problems:
-------------
* Path {2,6,7,3,5,4}: inherited
-----------------------------
This path is subsumed by the proof of path {2,6,7,3,5,4}->{1}.
* Path {2,6,7,3,5,4}->{1}: MAYBE
------------------------------
The usable rules for this path are:
{ ack(0(), y) -> s(y)
, ack(s(x), 0()) -> ack(x, s(0()))
, ack(s(x), s(y)) -> ack(x, ack(s(x), y))
, ack(s(x), y) -> f(x, x)
, f(s(x), y) -> f(x, s(x))
, f(x, s(y)) -> f(y, x)
, f(x, y) -> ack(x, y)}
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:
{ ack^#(s(x), 0()) -> c_1(ack^#(x, s(0())))
, f^#(x, y) -> c_5(ack^#(x, y))
, ack^#(s(x), y) -> c_6(f^#(x, x))
, ack^#(s(x), s(y)) -> c_2(ack^#(x, ack(s(x), y)))
, f^#(x, s(y)) -> c_4(f^#(y, x))
, f^#(s(x), y) -> c_3(f^#(x, s(x)))
, ack^#(0(), y) -> c_0()
, ack(0(), y) -> s(y)
, ack(s(x), 0()) -> ack(x, s(0()))
, ack(s(x), s(y)) -> ack(x, ack(s(x), y))
, ack(s(x), y) -> f(x, x)
, f(s(x), y) -> f(x, s(x))
, f(x, s(y)) -> f(y, x)
, f(x, y) -> ack(x, y)}
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
Tool RC2
stdout:
MAYBE
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ ack(0(), y) -> s(y)
, ack(s(x), 0()) -> ack(x, s(0()))
, ack(s(x), s(y)) -> ack(x, ack(s(x), y))
, f(s(x), y) -> f(x, s(x))
, f(x, s(y)) -> f(y, x)
, f(x, y) -> ack(x, y)
, ack(s(x), y) -> f(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: ack^#(0(), y) -> c_0(y)
, 2: ack^#(s(x), 0()) -> c_1(ack^#(x, s(0())))
, 3: ack^#(s(x), s(y)) -> c_2(ack^#(x, ack(s(x), y)))
, 4: f^#(s(x), y) -> c_3(f^#(x, s(x)))
, 5: f^#(x, s(y)) -> c_4(f^#(y, x))
, 6: f^#(x, y) -> c_5(ack^#(x, y))
, 7: ack^#(s(x), y) -> c_6(f^#(x, x))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{2,6,7,3,5,4} [ inherited ]
|
`->{1} [ NA ]
Sub-problems:
-------------
* Path {2,6,7,3,5,4}: inherited
-----------------------------
This path is subsumed by the proof of path {2,6,7,3,5,4}->{1}.
* Path {2,6,7,3,5,4}->{1}: NA
---------------------------
The usable rules for this path are:
{ ack(0(), y) -> s(y)
, ack(s(x), 0()) -> ack(x, s(0()))
, ack(s(x), s(y)) -> ack(x, ack(s(x), y))
, ack(s(x), y) -> f(x, x)
, f(s(x), y) -> f(x, s(x))
, f(x, s(y)) -> f(y, x)
, f(x, y) -> ack(x, y)}
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.
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: ack^#(0(), y) -> c_0(y)
, 2: ack^#(s(x), 0()) -> c_1(ack^#(x, s(0())))
, 3: ack^#(s(x), s(y)) -> c_2(ack^#(x, ack(s(x), y)))
, 4: f^#(s(x), y) -> c_3(f^#(x, s(x)))
, 5: f^#(x, s(y)) -> c_4(f^#(y, x))
, 6: f^#(x, y) -> c_5(ack^#(x, y))
, 7: ack^#(s(x), y) -> c_6(f^#(x, x))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{2,6,7,3,5,4} [ inherited ]
|
`->{1} [ MAYBE ]
Sub-problems:
-------------
* Path {2,6,7,3,5,4}: inherited
-----------------------------
This path is subsumed by the proof of path {2,6,7,3,5,4}->{1}.
* Path {2,6,7,3,5,4}->{1}: MAYBE
------------------------------
The usable rules for this path are:
{ ack(0(), y) -> s(y)
, ack(s(x), 0()) -> ack(x, s(0()))
, ack(s(x), s(y)) -> ack(x, ack(s(x), y))
, ack(s(x), y) -> f(x, x)
, f(s(x), y) -> f(x, s(x))
, f(x, s(y)) -> f(y, x)
, f(x, y) -> ack(x, y)}
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:
{ ack^#(s(x), 0()) -> c_1(ack^#(x, s(0())))
, f^#(x, y) -> c_5(ack^#(x, y))
, ack^#(s(x), y) -> c_6(f^#(x, x))
, ack^#(s(x), s(y)) -> c_2(ack^#(x, ack(s(x), y)))
, f^#(x, s(y)) -> c_4(f^#(y, x))
, f^#(s(x), y) -> c_3(f^#(x, s(x)))
, ack^#(0(), y) -> c_0(y)
, ack(0(), y) -> s(y)
, ack(s(x), 0()) -> ack(x, s(0()))
, ack(s(x), s(y)) -> ack(x, ack(s(x), y))
, ack(s(x), y) -> f(x, x)
, f(s(x), y) -> f(x, s(x))
, f(x, s(y)) -> f(y, x)
, f(x, y) -> ack(x, y)}
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: ack^#(0(), y) -> c_0(y)
, 2: ack^#(s(x), 0()) -> c_1(ack^#(x, s(0())))
, 3: ack^#(s(x), s(y)) -> c_2(ack^#(x, ack(s(x), y)))
, 4: f^#(s(x), y) -> c_3(f^#(x, s(x)))
, 5: f^#(x, s(y)) -> c_4(f^#(y, x))
, 6: f^#(x, y) -> c_5(ack^#(x, y))
, 7: ack^#(s(x), y) -> c_6(f^#(x, x))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{2,6,7,3,5,4} [ inherited ]
|
`->{1} [ MAYBE ]
Sub-problems:
-------------
* Path {2,6,7,3,5,4}: inherited
-----------------------------
This path is subsumed by the proof of path {2,6,7,3,5,4}->{1}.
* Path {2,6,7,3,5,4}->{1}: MAYBE
------------------------------
The usable rules for this path are:
{ ack(0(), y) -> s(y)
, ack(s(x), 0()) -> ack(x, s(0()))
, ack(s(x), s(y)) -> ack(x, ack(s(x), y))
, ack(s(x), y) -> f(x, x)
, f(s(x), y) -> f(x, s(x))
, f(x, s(y)) -> f(y, x)
, f(x, y) -> ack(x, y)}
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:
{ ack^#(s(x), 0()) -> c_1(ack^#(x, s(0())))
, f^#(x, y) -> c_5(ack^#(x, y))
, ack^#(s(x), y) -> c_6(f^#(x, x))
, ack^#(s(x), s(y)) -> c_2(ack^#(x, ack(s(x), y)))
, f^#(x, s(y)) -> c_4(f^#(y, x))
, f^#(s(x), y) -> c_3(f^#(x, s(x)))
, ack^#(0(), y) -> c_0(y)
, ack(0(), y) -> s(y)
, ack(s(x), 0()) -> ack(x, s(0()))
, ack(s(x), s(y)) -> ack(x, ack(s(x), y))
, ack(s(x), y) -> f(x, x)
, f(s(x), y) -> f(x, s(x))
, f(x, s(y)) -> f(y, x)
, f(x, y) -> ack(x, y)}
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.