Tool CaT
stdout:
MAYBE
Problem:
f(0(),1(),x) -> f(g(x),g(x),x)
f(g(x),y,z) -> g(f(x,y,z))
f(x,g(y),z) -> g(f(x,y,z))
f(x,y,g(z)) -> g(f(x,y,z))
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:
{ f(0(), 1(), x) -> f(g(x), g(x), x)
, f(g(x), y, z) -> g(f(x, y, z))
, f(x, g(y), z) -> g(f(x, y, z))
, f(x, y, g(z)) -> g(f(x, y, z))}
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^#(0(), 1(), x) -> c_0(f^#(g(x), g(x), x))
, 2: f^#(g(x), y, z) -> c_1(f^#(x, y, z))
, 3: f^#(x, g(y), z) -> c_2(f^#(x, y, z))
, 4: f^#(x, y, g(z)) -> c_3(f^#(x, y, z))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{1,4,3,2} [ MAYBE ]
Sub-problems:
-------------
* Path {1,4,3,2}: MAYBE
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {1},
Uargs(c_1) = {1}, Uargs(c_2) = {1}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
0() = [0]
[0]
[0]
1() = [0]
[0]
[0]
g(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
f^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[3 3 3] [3 3 3] [3 3 3] [0]
[3 3 3] [3 3 3] [3 3 3] [0]
c_0(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
c_1(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
c_2(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
c_3(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: MAYBE
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules:
{ f^#(0(), 1(), x) -> c_0(f^#(g(x), g(x), x))
, f^#(x, y, g(z)) -> c_3(f^#(x, y, z))
, f^#(x, g(y), z) -> c_2(f^#(x, y, z))
, f^#(g(x), y, z) -> c_1(f^#(x, y, z))}
Weak Rules: {}
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^#(0(), 1(), x) -> c_0(f^#(g(x), g(x), x))
, 2: f^#(g(x), y, z) -> c_1(f^#(x, y, z))
, 3: f^#(x, g(y), z) -> c_2(f^#(x, y, z))
, 4: f^#(x, y, g(z)) -> c_3(f^#(x, y, z))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{1,4,3,2} [ MAYBE ]
Sub-problems:
-------------
* Path {1,4,3,2}: MAYBE
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {1},
Uargs(c_1) = {1}, Uargs(c_2) = {1}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
0() = [0]
[0]
1() = [0]
[0]
g(x1) = [1 0] x1 + [0]
[0 0] [0]
f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[3 3] [3 3] [3 3] [0]
c_0(x1) = [1 0] x1 + [0]
[0 1] [0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
c_2(x1) = [1 0] x1 + [0]
[0 1] [0]
c_3(x1) = [1 0] x1 + [0]
[0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: MAYBE
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules:
{ f^#(0(), 1(), x) -> c_0(f^#(g(x), g(x), x))
, f^#(x, y, g(z)) -> c_3(f^#(x, y, z))
, f^#(x, g(y), z) -> c_2(f^#(x, y, z))
, f^#(g(x), y, z) -> c_1(f^#(x, y, z))}
Weak Rules: {}
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^#(0(), 1(), x) -> c_0(f^#(g(x), g(x), x))
, 2: f^#(g(x), y, z) -> c_1(f^#(x, y, z))
, 3: f^#(x, g(y), z) -> c_2(f^#(x, y, z))
, 4: f^#(x, y, g(z)) -> c_3(f^#(x, y, z))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{1,4,3,2} [ MAYBE ]
Sub-problems:
-------------
* Path {1,4,3,2}: MAYBE
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {1},
Uargs(c_1) = {1}, Uargs(c_2) = {1}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
0() = [0]
1() = [0]
g(x1) = [1] x1 + [0]
f^#(x1, x2, x3) = [0] x1 + [0] x2 + [3] x3 + [0]
c_0(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: MAYBE
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules:
{ f^#(0(), 1(), x) -> c_0(f^#(g(x), g(x), x))
, f^#(x, y, g(z)) -> c_3(f^#(x, y, z))
, f^#(x, g(y), z) -> c_2(f^#(x, y, z))
, f^#(g(x), y, z) -> c_1(f^#(x, y, z))}
Weak Rules: {}
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:
{ f(0(), 1(), x) -> f(g(x), g(x), x)
, f(g(x), y, z) -> g(f(x, y, z))
, f(x, g(y), z) -> g(f(x, y, z))
, f(x, y, g(z)) -> g(f(x, y, z))}
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^#(0(), 1(), x) -> c_0(f^#(g(x), g(x), x))
, 2: f^#(g(x), y, z) -> c_1(f^#(x, y, z))
, 3: f^#(x, g(y), z) -> c_2(f^#(x, y, z))
, 4: f^#(x, y, g(z)) -> c_3(f^#(x, y, z))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{1,4,3,2} [ MAYBE ]
Sub-problems:
-------------
* Path {1,4,3,2}: MAYBE
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {1},
Uargs(c_1) = {1}, Uargs(c_2) = {1}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0 0 0] [0]
0() = [0]
[0]
[0]
1() = [0]
[0]
[0]
g(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
f^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
[3 3 3] [3 3 3] [3 3 3] [0]
[3 3 3] [3 3 3] [3 3 3] [0]
c_0(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
c_1(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
c_2(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
c_3(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: MAYBE
Input Problem: DP runtime-complexity with respect to
Strict Rules:
{ f^#(0(), 1(), x) -> c_0(f^#(g(x), g(x), x))
, f^#(x, y, g(z)) -> c_3(f^#(x, y, z))
, f^#(x, g(y), z) -> c_2(f^#(x, y, z))
, f^#(g(x), y, z) -> c_1(f^#(x, y, z))}
Weak Rules: {}
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^#(0(), 1(), x) -> c_0(f^#(g(x), g(x), x))
, 2: f^#(g(x), y, z) -> c_1(f^#(x, y, z))
, 3: f^#(x, g(y), z) -> c_2(f^#(x, y, z))
, 4: f^#(x, y, g(z)) -> c_3(f^#(x, y, z))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{1,4,3,2} [ MAYBE ]
Sub-problems:
-------------
* Path {1,4,3,2}: MAYBE
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {1},
Uargs(c_1) = {1}, Uargs(c_2) = {1}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
0() = [0]
[0]
1() = [0]
[0]
g(x1) = [1 0] x1 + [0]
[0 0] [0]
f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[3 3] [3 3] [3 3] [0]
c_0(x1) = [1 0] x1 + [0]
[0 1] [0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
c_2(x1) = [1 0] x1 + [0]
[0 1] [0]
c_3(x1) = [1 0] x1 + [0]
[0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: MAYBE
Input Problem: DP runtime-complexity with respect to
Strict Rules:
{ f^#(0(), 1(), x) -> c_0(f^#(g(x), g(x), x))
, f^#(x, y, g(z)) -> c_3(f^#(x, y, z))
, f^#(x, g(y), z) -> c_2(f^#(x, y, z))
, f^#(g(x), y, z) -> c_1(f^#(x, y, z))}
Weak Rules: {}
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^#(0(), 1(), x) -> c_0(f^#(g(x), g(x), x))
, 2: f^#(g(x), y, z) -> c_1(f^#(x, y, z))
, 3: f^#(x, g(y), z) -> c_2(f^#(x, y, z))
, 4: f^#(x, y, g(z)) -> c_3(f^#(x, y, z))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{1,4,3,2} [ MAYBE ]
Sub-problems:
-------------
* Path {1,4,3,2}: MAYBE
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(g) = {}, Uargs(f^#) = {}, Uargs(c_0) = {1},
Uargs(c_1) = {1}, Uargs(c_2) = {1}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
0() = [0]
1() = [0]
g(x1) = [1] x1 + [0]
f^#(x1, x2, x3) = [0] x1 + [0] x2 + [3] x3 + [0]
c_0(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: MAYBE
Input Problem: DP runtime-complexity with respect to
Strict Rules:
{ f^#(0(), 1(), x) -> c_0(f^#(g(x), g(x), x))
, f^#(x, y, g(z)) -> c_3(f^#(x, y, z))
, f^#(x, g(y), z) -> c_2(f^#(x, y, z))
, f^#(g(x), y, z) -> c_1(f^#(x, y, z))}
Weak Rules: {}
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.