Tool CaT
stdout:
MAYBE
Problem:
g(tt(),x,y) -> g(f(x,y),s(x),s(y))
f(s(x),y) -> f(x,y)
f(x,s(y)) -> f(x,y)
f(0(),0()) -> tt()
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:
{ g(tt(), x, y) -> g(f(x, y), s(x), s(y))
, f(s(x), y) -> f(x, y)
, f(x, s(y)) -> f(x, y)
, f(0(), 0()) -> tt()}
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: g^#(tt(), x, y) -> c_0(g^#(f(x, y), s(x), s(y)))
, 2: f^#(s(x), y) -> c_1(f^#(x, y))
, 3: f^#(x, s(y)) -> c_2(f^#(x, y))
, 4: f^#(0(), 0()) -> c_3()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{2,3} [ YES(?,O(n^3)) ]
|
`->{4} [ YES(?,O(n^3)) ]
->{1} [ MAYBE ]
Sub-problems:
-------------
* Path {1}: MAYBE
---------------
The usable rules for this path are:
{ f(s(x), y) -> f(x, y)
, f(x, s(y)) -> f(x, y)
, f(0(), 0()) -> tt()}
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^#(tt(), x, y) -> c_0(g^#(f(x, y), s(x), s(y)))
, f(s(x), y) -> f(x, y)
, f(x, s(y)) -> f(x, y)
, f(0(), 0()) -> tt()}
Proof Output:
The input cannot be shown compatible
* Path {2,3}: YES(?,O(n^3))
-------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(g) = {}, Uargs(f) = {}, Uargs(s) = {}, Uargs(g^#) = {},
Uargs(c_0) = {}, Uargs(f^#) = {}, Uargs(c_1) = {1},
Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
g(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]
tt() = [0]
[0]
[0]
f(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
s(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
g^#(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]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
f^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[3 3 3] [3 3 3] [0]
[3 3 3] [3 3 3] [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() = [0]
[0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^3))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules:
{ f^#(s(x), y) -> c_1(f^#(x, y))
, f^#(x, s(y)) -> c_2(f^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(f^#) = {}, Uargs(c_1) = {1}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 4 2] x1 + [0]
[0 1 2] [4]
[0 0 1] [0]
f^#(x1, x2) = [0 2 0] x1 + [2 1 6] x2 + [0]
[0 0 0] [1 0 2] [0]
[0 2 4] [2 2 0] [0]
c_1(x1) = [1 0 0] x1 + [7]
[0 0 0] [0]
[0 0 0] [6]
c_2(x1) = [1 0 0] x1 + [3]
[0 0 0] [0]
[0 2 0] [7]
* Path {2,3}->{4}: YES(?,O(n^3))
------------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(g) = {}, Uargs(f) = {}, Uargs(s) = {}, Uargs(g^#) = {},
Uargs(c_0) = {}, Uargs(f^#) = {}, Uargs(c_1) = {1},
Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
g(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]
tt() = [0]
[0]
[0]
f(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
g^#(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]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
f^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [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() = [0]
[0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^3))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {f^#(0(), 0()) -> c_3()}
Weak Rules:
{ f^#(s(x), y) -> c_1(f^#(x, y))
, f^#(x, s(y)) -> c_2(f^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(f^#) = {}, Uargs(c_1) = {1}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 1 0] x1 + [0]
[0 1 3] [0]
[0 0 1] [2]
0() = [2]
[0]
[2]
f^#(x1, x2) = [1 3 2] x1 + [2 0 0] x2 + [0]
[0 1 1] [0 0 0] [0]
[0 3 0] [1 3 0] [4]
c_1(x1) = [1 0 0] x1 + [3]
[0 0 0] [2]
[0 0 0] [4]
c_2(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [3]
c_3() = [1]
[0]
[0]
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: g^#(tt(), x, y) -> c_0(g^#(f(x, y), s(x), s(y)))
, 2: f^#(s(x), y) -> c_1(f^#(x, y))
, 3: f^#(x, s(y)) -> c_2(f^#(x, y))
, 4: f^#(0(), 0()) -> c_3()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{2,3} [ YES(?,O(n^2)) ]
|
`->{4} [ YES(?,O(n^1)) ]
->{1} [ MAYBE ]
Sub-problems:
-------------
* Path {1}: MAYBE
---------------
The usable rules for this path are:
{ f(s(x), y) -> f(x, y)
, f(x, s(y)) -> f(x, y)
, f(0(), 0()) -> tt()}
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^#(tt(), x, y) -> c_0(g^#(f(x, y), s(x), s(y)))
, f(s(x), y) -> f(x, y)
, f(x, s(y)) -> f(x, y)
, f(0(), 0()) -> tt()}
Proof Output:
The input cannot be shown compatible
* Path {2,3}: YES(?,O(n^2))
-------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(g) = {}, Uargs(f) = {}, Uargs(s) = {}, Uargs(g^#) = {},
Uargs(c_0) = {}, Uargs(f^#) = {}, Uargs(c_1) = {1},
Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
g(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
tt() = [0]
[0]
f(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
0() = [0]
[0]
g^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
f^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
[3 3] [3 3] [0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
c_2(x1) = [1 0] x1 + [0]
[0 1] [0]
c_3() = [0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules:
{ f^#(s(x), y) -> c_1(f^#(x, y))
, f^#(x, s(y)) -> c_2(f^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(f^#) = {}, Uargs(c_1) = {1}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 1] x1 + [0]
[0 1] [4]
f^#(x1, x2) = [0 1] x1 + [0 2] x2 + [0]
[1 0] [0 0] [0]
c_1(x1) = [1 0] x1 + [3]
[0 0] [0]
c_2(x1) = [1 0] x1 + [7]
[0 0] [0]
* Path {2,3}->{4}: YES(?,O(n^1))
------------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(g) = {}, Uargs(f) = {}, Uargs(s) = {}, Uargs(g^#) = {},
Uargs(c_0) = {}, Uargs(f^#) = {}, Uargs(c_1) = {1},
Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
g(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
tt() = [0]
[0]
f(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
g^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
f^#(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(x1) = [1 0] x1 + [0]
[0 1] [0]
c_3() = [0]
[0]
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^#(0(), 0()) -> c_3()}
Weak Rules:
{ f^#(s(x), y) -> c_1(f^#(x, y))
, f^#(x, s(y)) -> c_2(f^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(f^#) = {}, Uargs(c_1) = {1}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 4] x1 + [2]
[0 0] [0]
0() = [2]
[2]
f^#(x1, x2) = [2 2] x1 + [2 0] x2 + [0]
[0 0] [0 2] [0]
c_1(x1) = [1 0] x1 + [3]
[0 0] [0]
c_2(x1) = [1 2] x1 + [3]
[0 0] [0]
c_3() = [1]
[0]
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: g^#(tt(), x, y) -> c_0(g^#(f(x, y), s(x), s(y)))
, 2: f^#(s(x), y) -> c_1(f^#(x, y))
, 3: f^#(x, s(y)) -> c_2(f^#(x, y))
, 4: f^#(0(), 0()) -> c_3()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{2,3} [ YES(?,O(n^1)) ]
|
`->{4} [ YES(?,O(n^1)) ]
->{1} [ MAYBE ]
Sub-problems:
-------------
* Path {1}: MAYBE
---------------
The usable rules for this path are:
{ f(s(x), y) -> f(x, y)
, f(x, s(y)) -> f(x, y)
, f(0(), 0()) -> tt()}
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^#(tt(), x, y) -> c_0(g^#(f(x, y), s(x), s(y)))
, f(s(x), y) -> f(x, y)
, f(x, s(y)) -> f(x, y)
, f(0(), 0()) -> tt()}
Proof Output:
The input cannot be shown compatible
* Path {2,3}: YES(?,O(n^1))
-------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(g) = {}, Uargs(f) = {}, Uargs(s) = {}, Uargs(g^#) = {},
Uargs(c_0) = {}, Uargs(f^#) = {}, Uargs(c_1) = {1},
Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
g(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
tt() = [0]
f(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [1] x1 + [0]
0() = [0]
g^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [0] x1 + [0]
f^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_1(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [0]
c_3() = [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules:
{ f^#(s(x), y) -> c_1(f^#(x, y))
, f^#(x, s(y)) -> c_2(f^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(f^#) = {}, Uargs(c_1) = {1}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
f^#(x1, x2) = [2] x1 + [6] x2 + [0]
c_1(x1) = [1] x1 + [3]
c_2(x1) = [1] x1 + [7]
* Path {2,3}->{4}: YES(?,O(n^1))
------------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(g) = {}, Uargs(f) = {}, Uargs(s) = {}, Uargs(g^#) = {},
Uargs(c_0) = {}, Uargs(f^#) = {}, Uargs(c_1) = {1},
Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
g(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
tt() = [0]
f(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
g^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [0] x1 + [0]
f^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [0]
c_3() = [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {f^#(0(), 0()) -> c_3()}
Weak Rules:
{ f^#(s(x), y) -> c_1(f^#(x, y))
, f^#(x, s(y)) -> c_2(f^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(f^#) = {}, Uargs(c_1) = {1}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [0]
0() = [2]
f^#(x1, x2) = [0] x1 + [2] x2 + [0]
c_1(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [0]
c_3() = [1]
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:
{ g(tt(), x, y) -> g(f(x, y), s(x), s(y))
, f(s(x), y) -> f(x, y)
, f(x, s(y)) -> f(x, y)
, f(0(), 0()) -> tt()}
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: g^#(tt(), x, y) -> c_0(g^#(f(x, y), s(x), s(y)))
, 2: f^#(s(x), y) -> c_1(f^#(x, y))
, 3: f^#(x, s(y)) -> c_2(f^#(x, y))
, 4: f^#(0(), 0()) -> c_3()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{2,3} [ YES(?,O(n^3)) ]
|
`->{4} [ YES(?,O(n^3)) ]
->{1} [ MAYBE ]
Sub-problems:
-------------
* Path {1}: MAYBE
---------------
The usable rules for this path are:
{ f(s(x), y) -> f(x, y)
, f(x, s(y)) -> f(x, y)
, f(0(), 0()) -> tt()}
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^#(tt(), x, y) -> c_0(g^#(f(x, y), s(x), s(y)))
, f(s(x), y) -> f(x, y)
, f(x, s(y)) -> f(x, y)
, f(0(), 0()) -> tt()}
Proof Output:
The input cannot be shown compatible
* Path {2,3}: YES(?,O(n^3))
-------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(g) = {}, Uargs(f) = {}, Uargs(s) = {}, Uargs(g^#) = {},
Uargs(c_0) = {}, Uargs(f^#) = {}, Uargs(c_1) = {1},
Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
g(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]
tt() = [0]
[0]
[0]
f(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
s(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
g^#(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]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
f^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[3 3 3] [3 3 3] [0]
[3 3 3] [3 3 3] [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() = [0]
[0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^3))
Input Problem: DP runtime-complexity with respect to
Strict Rules:
{ f^#(s(x), y) -> c_1(f^#(x, y))
, f^#(x, s(y)) -> c_2(f^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(f^#) = {}, Uargs(c_1) = {1}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 4 2] x1 + [0]
[0 1 2] [4]
[0 0 1] [0]
f^#(x1, x2) = [0 2 0] x1 + [2 1 6] x2 + [0]
[0 0 0] [1 0 2] [0]
[0 2 4] [2 2 0] [0]
c_1(x1) = [1 0 0] x1 + [7]
[0 0 0] [0]
[0 0 0] [6]
c_2(x1) = [1 0 0] x1 + [3]
[0 0 0] [0]
[0 2 0] [7]
* Path {2,3}->{4}: YES(?,O(n^3))
------------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(g) = {}, Uargs(f) = {}, Uargs(s) = {}, Uargs(g^#) = {},
Uargs(c_0) = {}, Uargs(f^#) = {}, Uargs(c_1) = {1},
Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
g(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]
tt() = [0]
[0]
[0]
f(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
g^#(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]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
f^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [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() = [0]
[0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^3))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {f^#(0(), 0()) -> c_3()}
Weak Rules:
{ f^#(s(x), y) -> c_1(f^#(x, y))
, f^#(x, s(y)) -> c_2(f^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(f^#) = {}, Uargs(c_1) = {1}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 1 0] x1 + [0]
[0 1 3] [0]
[0 0 1] [2]
0() = [2]
[0]
[2]
f^#(x1, x2) = [1 3 2] x1 + [2 0 0] x2 + [0]
[0 1 1] [0 0 0] [0]
[0 3 0] [1 3 0] [4]
c_1(x1) = [1 0 0] x1 + [3]
[0 0 0] [2]
[0 0 0] [4]
c_2(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [3]
c_3() = [1]
[0]
[0]
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: g^#(tt(), x, y) -> c_0(g^#(f(x, y), s(x), s(y)))
, 2: f^#(s(x), y) -> c_1(f^#(x, y))
, 3: f^#(x, s(y)) -> c_2(f^#(x, y))
, 4: f^#(0(), 0()) -> c_3()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{2,3} [ YES(?,O(n^2)) ]
|
`->{4} [ YES(?,O(n^1)) ]
->{1} [ MAYBE ]
Sub-problems:
-------------
* Path {1}: MAYBE
---------------
The usable rules for this path are:
{ f(s(x), y) -> f(x, y)
, f(x, s(y)) -> f(x, y)
, f(0(), 0()) -> tt()}
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^#(tt(), x, y) -> c_0(g^#(f(x, y), s(x), s(y)))
, f(s(x), y) -> f(x, y)
, f(x, s(y)) -> f(x, y)
, f(0(), 0()) -> tt()}
Proof Output:
The input cannot be shown compatible
* Path {2,3}: YES(?,O(n^2))
-------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(g) = {}, Uargs(f) = {}, Uargs(s) = {}, Uargs(g^#) = {},
Uargs(c_0) = {}, Uargs(f^#) = {}, Uargs(c_1) = {1},
Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
g(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
tt() = [0]
[0]
f(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
0() = [0]
[0]
g^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
f^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
[3 3] [3 3] [0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
c_2(x1) = [1 0] x1 + [0]
[0 1] [0]
c_3() = [0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: DP runtime-complexity with respect to
Strict Rules:
{ f^#(s(x), y) -> c_1(f^#(x, y))
, f^#(x, s(y)) -> c_2(f^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(f^#) = {}, Uargs(c_1) = {1}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 1] x1 + [0]
[0 1] [4]
f^#(x1, x2) = [0 1] x1 + [0 2] x2 + [0]
[1 0] [0 0] [0]
c_1(x1) = [1 0] x1 + [3]
[0 0] [0]
c_2(x1) = [1 0] x1 + [7]
[0 0] [0]
* Path {2,3}->{4}: YES(?,O(n^1))
------------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(g) = {}, Uargs(f) = {}, Uargs(s) = {}, Uargs(g^#) = {},
Uargs(c_0) = {}, Uargs(f^#) = {}, Uargs(c_1) = {1},
Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
g(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
tt() = [0]
[0]
f(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
g^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
f^#(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(x1) = [1 0] x1 + [0]
[0 1] [0]
c_3() = [0]
[0]
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^#(0(), 0()) -> c_3()}
Weak Rules:
{ f^#(s(x), y) -> c_1(f^#(x, y))
, f^#(x, s(y)) -> c_2(f^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(f^#) = {}, Uargs(c_1) = {1}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 4] x1 + [2]
[0 0] [0]
0() = [2]
[2]
f^#(x1, x2) = [2 2] x1 + [2 0] x2 + [0]
[0 0] [0 2] [0]
c_1(x1) = [1 0] x1 + [3]
[0 0] [0]
c_2(x1) = [1 2] x1 + [3]
[0 0] [0]
c_3() = [1]
[0]
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: g^#(tt(), x, y) -> c_0(g^#(f(x, y), s(x), s(y)))
, 2: f^#(s(x), y) -> c_1(f^#(x, y))
, 3: f^#(x, s(y)) -> c_2(f^#(x, y))
, 4: f^#(0(), 0()) -> c_3()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{2,3} [ YES(?,O(n^1)) ]
|
`->{4} [ YES(?,O(n^1)) ]
->{1} [ MAYBE ]
Sub-problems:
-------------
* Path {1}: MAYBE
---------------
The usable rules for this path are:
{ f(s(x), y) -> f(x, y)
, f(x, s(y)) -> f(x, y)
, f(0(), 0()) -> tt()}
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^#(tt(), x, y) -> c_0(g^#(f(x, y), s(x), s(y)))
, f(s(x), y) -> f(x, y)
, f(x, s(y)) -> f(x, y)
, f(0(), 0()) -> tt()}
Proof Output:
The input cannot be shown compatible
* Path {2,3}: YES(?,O(n^1))
-------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(g) = {}, Uargs(f) = {}, Uargs(s) = {}, Uargs(g^#) = {},
Uargs(c_0) = {}, Uargs(f^#) = {}, Uargs(c_1) = {1},
Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
g(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
tt() = [0]
f(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [1] x1 + [0]
0() = [0]
g^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [0] x1 + [0]
f^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_1(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [0]
c_3() = [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules:
{ f^#(s(x), y) -> c_1(f^#(x, y))
, f^#(x, s(y)) -> c_2(f^#(x, y))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(f^#) = {}, Uargs(c_1) = {1}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
f^#(x1, x2) = [2] x1 + [6] x2 + [0]
c_1(x1) = [1] x1 + [3]
c_2(x1) = [1] x1 + [7]
* Path {2,3}->{4}: YES(?,O(n^1))
------------------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(g) = {}, Uargs(f) = {}, Uargs(s) = {}, Uargs(g^#) = {},
Uargs(c_0) = {}, Uargs(f^#) = {}, Uargs(c_1) = {1},
Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
g(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
tt() = [0]
f(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
g^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [0] x1 + [0]
f^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [0]
c_3() = [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {f^#(0(), 0()) -> c_3()}
Weak Rules:
{ f^#(s(x), y) -> c_1(f^#(x, y))
, f^#(x, s(y)) -> c_2(f^#(x, y))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(f^#) = {}, Uargs(c_1) = {1}, Uargs(c_2) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [0]
0() = [2]
f^#(x1, x2) = [0] x1 + [2] x2 + [0]
c_1(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [0]
c_3() = [1]
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.