Tool CaT
stdout:
MAYBE
Problem:
f(c(a(),z,x)) -> b(a(),z)
b(x,b(z,y)) -> f(b(f(f(z)),c(x,z,y)))
b(y,z) -> z
Proof:
OpenTool IRC1
stdout:
MAYBE
Tool IRC2
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:
{ f(c(a(), z, x)) -> b(a(), z)
, b(x, b(z, y)) -> f(b(f(f(z)), c(x, z, y)))
, b(y, z) -> z}
Proof Output:
'wdg' proved the best result:
Details:
--------
'wdg' succeeded with the following output:
'wdg'
-----
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ f(c(a(), z, x)) -> b(a(), z)
, b(x, b(z, y)) -> f(b(f(f(z)), c(x, z, y)))
, b(y, z) -> z}
Proof Output:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: f^#(c(a(), z, x)) -> c_0(b^#(a(), z))
, 2: b^#(x, b(z, y)) -> c_1(f^#(b(f(f(z)), c(x, z, y))))
, 3: b^#(y, z) -> c_2()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{1,2} [ YES(?,O(n^1)) ]
|
`->{3} [ YES(?,O(n^1)) ]
Sub-problems:
-------------
* Path {1,2}: YES(?,O(n^1))
-------------------------
The usable rules for this path are:
{ f(c(a(), z, x)) -> b(a(), z)
, b(x, b(z, y)) -> f(b(f(f(z)), c(x, z, y)))
, b(y, z) -> z}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {1}, Uargs(c) = {}, Uargs(b) = {1}, Uargs(f^#) = {1},
Uargs(c_0) = {1}, Uargs(b^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1) = [2 0] x1 + [0]
[2 0] [0]
c(x1, x2, x3) = [0 0] x1 + [1 2] x2 + [0 0] x3 + [2]
[0 0] [0 0] [0 0] [0]
a() = [0]
[0]
b(x1, x2) = [1 0] x1 + [1 3] x2 + [2]
[3 3] [2 3] [3]
f^#(x1) = [1 0] x1 + [0]
[3 3] [0]
c_0(x1) = [1 0] x1 + [0]
[0 1] [0]
b^#(x1, x2) = [3 3] x1 + [0 2] x2 + [0]
[0 0] [3 3] [0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
c_2() = [0]
[0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules:
{ f^#(c(a(), z, x)) -> c_0(b^#(a(), z))
, b^#(x, b(z, y)) -> c_1(f^#(b(f(f(z)), c(x, z, y))))}
Weak Rules:
{ f(c(a(), z, x)) -> b(a(), z)
, b(x, b(z, y)) -> f(b(f(f(z)), c(x, z, y)))
, b(y, z) -> z}
Proof Output:
The following argument positions are usable:
Uargs(f) = {}, Uargs(c) = {}, Uargs(b) = {}, Uargs(f^#) = {},
Uargs(c_0) = {1}, Uargs(b^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1) = [2 0] x1 + [0]
[1 0] [4]
c(x1, x2, x3) = [0 0] x1 + [1 1] x2 + [0 0] x3 + [1]
[0 0] [0 0] [0 0] [0]
a() = [0]
[0]
b(x1, x2) = [0 0] x1 + [1 2] x2 + [0]
[4 4] [0 1] [5]
f^#(x1) = [1 0] x1 + [0]
[0 0] [4]
c_0(x1) = [1 0] x1 + [0]
[0 0] [3]
b^#(x1, x2) = [4 0] x1 + [0 1] x2 + [0]
[4 4] [6 1] [4]
c_1(x1) = [4 0] x1 + [0]
[0 0] [7]
* Path {1,2}->{3}: YES(?,O(n^1))
------------------------------
The usable rules for this path are:
{ f(c(a(), z, x)) -> b(a(), z)
, b(x, b(z, y)) -> f(b(f(f(z)), c(x, z, y)))
, b(y, z) -> z}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {1}, Uargs(c) = {}, Uargs(b) = {1}, Uargs(f^#) = {1},
Uargs(c_0) = {1}, Uargs(b^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1) = [2 0] x1 + [0]
[2 0] [0]
c(x1, x2, x3) = [0 0] x1 + [1 2] x2 + [0 0] x3 + [1]
[0 0] [0 0] [0 0] [0]
a() = [0]
[0]
b(x1, x2) = [1 0] x1 + [1 3] x2 + [1]
[3 3] [2 3] [1]
f^#(x1) = [3 0] x1 + [0]
[0 0] [0]
c_0(x1) = [1 0] x1 + [0]
[0 1] [0]
b^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
c_2() = [0]
[0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {b^#(y, z) -> c_2()}
Weak Rules:
{ f^#(c(a(), z, x)) -> c_0(b^#(a(), z))
, b^#(x, b(z, y)) -> c_1(f^#(b(f(f(z)), c(x, z, y))))
, f(c(a(), z, x)) -> b(a(), z)
, b(x, b(z, y)) -> f(b(f(f(z)), c(x, z, y)))
, b(y, z) -> z}
Proof Output:
The following argument positions are usable:
Uargs(f) = {}, Uargs(c) = {}, Uargs(b) = {}, Uargs(f^#) = {},
Uargs(c_0) = {1}, Uargs(b^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1) = [2 0] x1 + [0]
[2 0] [4]
c(x1, x2, x3) = [0 0] x1 + [1 1] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
a() = [0]
[0]
b(x1, x2) = [0 0] x1 + [2 1] x2 + [0]
[4 4] [0 2] [2]
f^#(x1) = [2 0] x1 + [1]
[0 0] [4]
c_0(x1) = [1 0] x1 + [0]
[0 2] [3]
b^#(x1, x2) = [0 0] x1 + [0 2] x2 + [1]
[2 0] [0 0] [0]
c_1(x1) = [2 0] x1 + [1]
[0 0] [0]
c_2() = [0]
[0]Tool RC1
stdout:
MAYBE
Tool RC2
stdout:
YES(?,O(n^1))
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: YES(?,O(n^1))
Input Problem: runtime-complexity with respect to
Rules:
{ f(c(a(), z, x)) -> b(a(), z)
, b(x, b(z, y)) -> f(b(f(f(z)), c(x, z, y)))
, b(y, z) -> z}
Proof Output:
'wdg' proved the best result:
Details:
--------
'wdg' succeeded with the following output:
'wdg'
-----
Answer: YES(?,O(n^1))
Input Problem: runtime-complexity with respect to
Rules:
{ f(c(a(), z, x)) -> b(a(), z)
, b(x, b(z, y)) -> f(b(f(f(z)), c(x, z, y)))
, b(y, z) -> z}
Proof Output:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: f^#(c(a(), z, x)) -> c_0(b^#(a(), z))
, 2: b^#(x, b(z, y)) -> c_1(f^#(b(f(f(z)), c(x, z, y))))
, 3: b^#(y, z) -> c_2(z)}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{1,2} [ YES(?,O(n^1)) ]
|
`->{3} [ YES(?,O(n^1)) ]
Sub-problems:
-------------
* Path {1,2}: YES(?,O(n^1))
-------------------------
The usable rules for this path are:
{ f(c(a(), z, x)) -> b(a(), z)
, b(x, b(z, y)) -> f(b(f(f(z)), c(x, z, y)))
, b(y, z) -> z}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {1}, Uargs(c) = {1, 2, 3}, Uargs(b) = {1, 2},
Uargs(f^#) = {1}, Uargs(c_0) = {1}, Uargs(b^#) = {2},
Uargs(c_1) = {1}, Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1) = [1 0] x1 + [0]
[2 0] [1]
c(x1, x2, x3) = [1 0] x1 + [1 1] x2 + [1 0] x3 + [2]
[0 0] [0 0] [0 0] [0]
a() = [2]
[0]
b(x1, x2) = [1 0] x1 + [1 1] x2 + [1]
[3 2] [1 1] [3]
f^#(x1) = [1 0] x1 + [0]
[3 3] [0]
c_0(x1) = [1 0] x1 + [0]
[0 1] [0]
b^#(x1, x2) = [3 3] x1 + [1 1] x2 + [0]
[0 0] [3 3] [0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules:
{ f^#(c(a(), z, x)) -> c_0(b^#(a(), z))
, b^#(x, b(z, y)) -> c_1(f^#(b(f(f(z)), c(x, z, y))))}
Weak Rules:
{ f(c(a(), z, x)) -> b(a(), z)
, b(x, b(z, y)) -> f(b(f(f(z)), c(x, z, y)))
, b(y, z) -> z}
Proof Output:
The following argument positions are usable:
Uargs(f) = {}, Uargs(c) = {}, Uargs(b) = {}, Uargs(f^#) = {},
Uargs(c_0) = {1}, Uargs(b^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1) = [2 0] x1 + [0]
[1 0] [4]
c(x1, x2, x3) = [0 0] x1 + [1 1] x2 + [0 0] x3 + [1]
[0 0] [0 0] [0 0] [0]
a() = [0]
[0]
b(x1, x2) = [0 0] x1 + [1 2] x2 + [0]
[4 4] [0 1] [5]
f^#(x1) = [1 0] x1 + [0]
[0 0] [4]
c_0(x1) = [1 0] x1 + [0]
[0 0] [3]
b^#(x1, x2) = [4 0] x1 + [0 1] x2 + [0]
[4 4] [6 1] [4]
c_1(x1) = [4 0] x1 + [0]
[0 0] [7]
* Path {1,2}->{3}: YES(?,O(n^1))
------------------------------
The usable rules for this path are:
{ f(c(a(), z, x)) -> b(a(), z)
, b(x, b(z, y)) -> f(b(f(f(z)), c(x, z, y)))
, b(y, z) -> z}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {1}, Uargs(c) = {1, 2, 3}, Uargs(b) = {1, 2},
Uargs(f^#) = {1}, Uargs(c_0) = {1}, Uargs(b^#) = {2},
Uargs(c_1) = {1}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1) = [1 0] x1 + [0]
[1 0] [0]
c(x1, x2, x3) = [1 0] x1 + [1 1] x2 + [1 0] x3 + [2]
[0 0] [0 0] [0 0] [0]
a() = [0]
[0]
b(x1, x2) = [1 1] x1 + [1 1] x2 + [1]
[2 0] [1 1] [2]
f^#(x1) = [3 0] x1 + [0]
[0 0] [0]
c_0(x1) = [1 0] x1 + [0]
[0 1] [0]
b^#(x1, x2) = [0 0] x1 + [3 3] x2 + [0]
[0 0] [0 0] [0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
c_2(x1) = [1 0] x1 + [0]
[0 1] [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {b^#(y, z) -> c_2(z)}
Weak Rules:
{ f^#(c(a(), z, x)) -> c_0(b^#(a(), z))
, b^#(x, b(z, y)) -> c_1(f^#(b(f(f(z)), c(x, z, y))))
, f(c(a(), z, x)) -> b(a(), z)
, b(x, b(z, y)) -> f(b(f(f(z)), c(x, z, y)))
, b(y, z) -> z}
Proof Output:
The following argument positions are usable:
Uargs(f) = {}, Uargs(c) = {}, Uargs(b) = {}, Uargs(f^#) = {},
Uargs(c_0) = {1}, Uargs(b^#) = {}, Uargs(c_1) = {1},
Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1) = [2 0] x1 + [0]
[1 0] [4]
c(x1, x2, x3) = [0 0] x1 + [1 2] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [2]
a() = [0]
[0]
b(x1, x2) = [0 0] x1 + [1 1] x2 + [0]
[4 4] [0 2] [4]
f^#(x1) = [1 0] x1 + [2]
[4 0] [4]
c_0(x1) = [1 0] x1 + [0]
[2 0] [0]
b^#(x1, x2) = [2 4] x1 + [1 1] x2 + [2]
[4 4] [2 1] [4]
c_1(x1) = [1 0] x1 + [2]
[2 0] [0]
c_2(x1) = [1 1] x1 + [1]
[0 1] [0]