Tool CaT
stdout:
MAYBE
Problem:
f(x,x) -> f(i(x),g(g(x)))
f(x,y) -> x
g(x) -> i(x)
f(x,i(x)) -> f(x,x)
f(i(x),i(g(x))) -> a()
Proof:
Complexity Transformation Processor:
strict:
f(x,x) -> f(i(x),g(g(x)))
f(x,y) -> x
g(x) -> i(x)
f(x,i(x)) -> f(x,x)
f(i(x),i(g(x))) -> a()
weak:
Matrix Interpretation Processor:
dimension: 1
max_matrix:
1
interpretation:
[a] = 0,
[g](x0) = x0 + 1,
[i](x0) = x0,
[f](x0, x1) = x0 + x1 + 12
orientation:
f(x,x) = 2x + 12 >= 2x + 14 = f(i(x),g(g(x)))
f(x,y) = x + y + 12 >= x = x
g(x) = x + 1 >= x = i(x)
f(x,i(x)) = 2x + 12 >= 2x + 12 = f(x,x)
f(i(x),i(g(x))) = 2x + 13 >= 0 = a()
problem:
strict:
f(x,x) -> f(i(x),g(g(x)))
f(x,i(x)) -> f(x,x)
weak:
f(x,y) -> x
g(x) -> i(x)
f(i(x),i(g(x))) -> a()
Matrix Interpretation Processor:
dimension: 1
max_matrix:
1
interpretation:
[a] = 132,
[g](x0) = x0 + 44,
[i](x0) = x0 + 44,
[f](x0, x1) = x0 + x1
orientation:
f(x,x) = 2x >= 2x + 132 = f(i(x),g(g(x)))
f(x,i(x)) = 2x + 44 >= 2x = f(x,x)
f(x,y) = x + y >= x = x
g(x) = x + 44 >= x + 44 = i(x)
f(i(x),i(g(x))) = 2x + 132 >= 132 = a()
problem:
strict:
f(x,x) -> f(i(x),g(g(x)))
weak:
f(x,i(x)) -> f(x,x)
f(x,y) -> x
g(x) -> i(x)
f(i(x),i(g(x))) -> a()
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(x, x) -> f(i(x), g(g(x)))
, f(x, y) -> x
, g(x) -> i(x)
, f(x, i(x)) -> f(x, x)
, f(i(x), i(g(x))) -> a()}
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^#(x, x) -> c_0(f^#(i(x), g(g(x))))
, 2: f^#(x, y) -> c_1()
, 3: g^#(x) -> c_2()
, 4: f^#(x, i(x)) -> c_3(f^#(x, x))
, 5: f^#(i(x), i(g(x))) -> c_4()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{3} [ YES(?,O(1)) ]
->{1,4} [ NA ]
|
|->{2} [ NA ]
|
`->{5} [ NA ]
Sub-problems:
-------------
* Path {1,4}: NA
--------------
The usable rules for this path are:
{g(x) -> i(x)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(i) = {}, Uargs(g) = {1}, Uargs(f^#) = {2},
Uargs(c_0) = {1}, Uargs(g^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
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]
i(x1) = [1 0 0] x1 + [0]
[0 0 0] [1]
[0 0 0] [1]
g(x1) = [1 0 0] x1 + [3]
[0 0 0] [3]
[0 0 0] [3]
a() = [0]
[0]
[0]
f^#(x1, x2) = [0 0 0] x1 + [1 0 0] x2 + [0]
[3 3 3] [3 3 3] [0]
[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() = [0]
[0]
[0]
g^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2() = [0]
[0]
[0]
c_3(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
c_4() = [0]
[0]
[0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {1,4}->{2}: NA
-------------------
The usable rules for this path are:
{g(x) -> i(x)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(i) = {}, Uargs(g) = {1}, Uargs(f^#) = {2},
Uargs(c_0) = {1}, Uargs(g^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
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]
i(x1) = [1 1 1] x1 + [0]
[0 1 1] [1]
[0 0 1] [1]
g(x1) = [3 3 3] x1 + [3]
[0 3 3] [3]
[0 0 3] [3]
a() = [0]
[0]
[0]
f^#(x1, x2) = [0 0 0] x1 + [3 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
c_1() = [0]
[0]
[0]
g^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2() = [0]
[0]
[0]
c_3(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
c_4() = [0]
[0]
[0]
Complexity induced by the adequate RMI: YES(?,O(n^3))
We have not generated a proof for the resulting sub-problem.
* Path {1,4}->{5}: NA
-------------------
The usable rules for this path are:
{g(x) -> i(x)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(i) = {}, Uargs(g) = {1}, Uargs(f^#) = {2},
Uargs(c_0) = {1}, Uargs(g^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
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]
i(x1) = [1 1 1] x1 + [0]
[0 1 1] [1]
[0 0 1] [1]
g(x1) = [3 3 3] x1 + [3]
[0 3 3] [3]
[0 0 3] [3]
a() = [0]
[0]
[0]
f^#(x1, x2) = [0 0 0] x1 + [3 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
c_1() = [0]
[0]
[0]
g^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2() = [0]
[0]
[0]
c_3(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
c_4() = [0]
[0]
[0]
Complexity induced by the adequate RMI: YES(?,O(n^3))
We have not generated a proof for the resulting sub-problem.
* Path {3}: YES(?,O(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(f) = {}, Uargs(i) = {}, Uargs(g) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(g^#) = {}, Uargs(c_3) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
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]
i(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
g(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
a() = [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_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1() = [0]
[0]
[0]
g^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2() = [0]
[0]
[0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4() = [0]
[0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {g^#(x) -> c_2()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(g^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
g^#(x1) = [0 0 0] x1 + [7]
[0 0 0] [7]
[0 0 0] [7]
c_2() = [0]
[3]
[3]
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: f^#(x, x) -> c_0(f^#(i(x), g(g(x))))
, 2: f^#(x, y) -> c_1()
, 3: g^#(x) -> c_2()
, 4: f^#(x, i(x)) -> c_3(f^#(x, x))
, 5: f^#(i(x), i(g(x))) -> c_4()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{3} [ YES(?,O(1)) ]
->{1,4} [ MAYBE ]
|
|->{2} [ NA ]
|
`->{5} [ NA ]
Sub-problems:
-------------
* Path {1,4}: MAYBE
-----------------
The usable rules for this path are:
{g(x) -> i(x)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(i) = {}, Uargs(g) = {1}, Uargs(f^#) = {2},
Uargs(c_0) = {1}, Uargs(g^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
i(x1) = [1 0] x1 + [0]
[0 0] [1]
g(x1) = [1 0] x1 + [3]
[0 0] [3]
a() = [0]
[0]
f^#(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[3 3] [3 3] [0]
c_0(x1) = [1 0] x1 + [0]
[0 1] [0]
c_1() = [0]
[0]
g^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [1 0] x1 + [0]
[0 1] [0]
c_4() = [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: MAYBE
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules:
{ f^#(x, x) -> c_0(f^#(i(x), g(g(x))))
, f^#(x, i(x)) -> c_3(f^#(x, x))}
Weak Rules: {g(x) -> i(x)}
Proof Output:
The input cannot be shown compatible
* Path {1,4}->{2}: NA
-------------------
The usable rules for this path are:
{g(x) -> i(x)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(i) = {}, Uargs(g) = {1}, Uargs(f^#) = {2},
Uargs(c_0) = {1}, Uargs(g^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
i(x1) = [1 1] x1 + [0]
[0 1] [1]
g(x1) = [3 3] x1 + [3]
[0 3] [3]
a() = [0]
[0]
f^#(x1, x2) = [0 0] x1 + [3 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [1 0] x1 + [0]
[0 1] [0]
c_1() = [0]
[0]
g^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [1 0] x1 + [0]
[0 1] [0]
c_4() = [0]
[0]
Complexity induced by the adequate RMI: YES(?,O(n^2))
We have not generated a proof for the resulting sub-problem.
* Path {1,4}->{5}: NA
-------------------
The usable rules for this path are:
{g(x) -> i(x)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(i) = {}, Uargs(g) = {1}, Uargs(f^#) = {2},
Uargs(c_0) = {1}, Uargs(g^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
i(x1) = [1 1] x1 + [0]
[0 1] [1]
g(x1) = [3 3] x1 + [3]
[0 3] [3]
a() = [0]
[0]
f^#(x1, x2) = [0 0] x1 + [3 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [1 0] x1 + [0]
[0 1] [0]
c_1() = [0]
[0]
g^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [1 0] x1 + [0]
[0 1] [0]
c_4() = [0]
[0]
Complexity induced by the adequate RMI: YES(?,O(n^2))
We have not generated a proof for the resulting sub-problem.
* Path {3}: YES(?,O(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(f) = {}, Uargs(i) = {}, Uargs(g) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(g^#) = {}, Uargs(c_3) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
i(x1) = [0 0] x1 + [0]
[0 0] [0]
g(x1) = [0 0] x1 + [0]
[0 0] [0]
a() = [0]
[0]
f^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1() = [0]
[0]
g^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4() = [0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {g^#(x) -> c_2()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(g^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
g^#(x1) = [0 0] x1 + [7]
[0 0] [7]
c_2() = [0]
[1]
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: f^#(x, x) -> c_0(f^#(i(x), g(g(x))))
, 2: f^#(x, y) -> c_1()
, 3: g^#(x) -> c_2()
, 4: f^#(x, i(x)) -> c_3(f^#(x, x))
, 5: f^#(i(x), i(g(x))) -> c_4()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{3} [ YES(?,O(1)) ]
->{1,4} [ MAYBE ]
|
|->{2} [ NA ]
|
`->{5} [ NA ]
Sub-problems:
-------------
* Path {1,4}: MAYBE
-----------------
The usable rules for this path are:
{g(x) -> i(x)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(i) = {}, Uargs(g) = {1}, Uargs(f^#) = {2},
Uargs(c_0) = {1}, Uargs(g^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0] x1 + [0] x2 + [0]
i(x1) = [1] x1 + [1]
g(x1) = [1] x1 + [3]
a() = [0]
f^#(x1, x2) = [0] x1 + [1] x2 + [0]
c_0(x1) = [1] x1 + [0]
c_1() = [0]
g^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [1] x1 + [0]
c_4() = [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 1'
--------------------------------------
Answer: MAYBE
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules:
{ f^#(x, x) -> c_0(f^#(i(x), g(g(x))))
, f^#(x, i(x)) -> c_3(f^#(x, x))}
Weak Rules: {g(x) -> i(x)}
Proof Output:
The input cannot be shown compatible
* Path {1,4}->{2}: NA
-------------------
The usable rules for this path are:
{g(x) -> i(x)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(i) = {}, Uargs(g) = {1}, Uargs(f^#) = {2},
Uargs(c_0) = {1}, Uargs(g^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0] x1 + [0] x2 + [0]
i(x1) = [1] x1 + [0]
g(x1) = [3] x1 + [3]
a() = [0]
f^#(x1, x2) = [0] x1 + [3] x2 + [0]
c_0(x1) = [1] x1 + [0]
c_1() = [0]
g^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [1] x1 + [0]
c_4() = [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {1,4}->{5}: NA
-------------------
The usable rules for this path are:
{g(x) -> i(x)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(i) = {}, Uargs(g) = {1}, Uargs(f^#) = {2},
Uargs(c_0) = {1}, Uargs(g^#) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0] x1 + [0] x2 + [0]
i(x1) = [1] x1 + [0]
g(x1) = [3] x1 + [3]
a() = [0]
f^#(x1, x2) = [0] x1 + [3] x2 + [0]
c_0(x1) = [1] x1 + [0]
c_1() = [0]
g^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [1] x1 + [0]
c_4() = [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {3}: YES(?,O(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(f) = {}, Uargs(i) = {}, Uargs(g) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(g^#) = {}, Uargs(c_3) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0] x1 + [0] x2 + [0]
i(x1) = [0] x1 + [0]
g(x1) = [0] x1 + [0]
a() = [0]
f^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
c_1() = [0]
g^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {g^#(x) -> c_2()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(g^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
g^#(x1) = [0] x1 + [7]
c_2() = [0]
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(x, x) -> f(i(x), g(g(x)))
, f(x, y) -> x
, g(x) -> i(x)
, f(x, i(x)) -> f(x, x)
, f(i(x), i(g(x))) -> a()}
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^#(x, x) -> c_0(f^#(i(x), g(g(x))))
, 2: f^#(x, y) -> c_1(x)
, 3: g^#(x) -> c_2(x)
, 4: f^#(x, i(x)) -> c_3(f^#(x, x))
, 5: f^#(i(x), i(g(x))) -> c_4()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{3} [ YES(?,O(1)) ]
->{1,4} [ NA ]
|
|->{2} [ NA ]
|
`->{5} [ NA ]
Sub-problems:
-------------
* Path {1,4}: NA
--------------
The usable rules for this path are:
{g(x) -> i(x)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(i) = {1}, Uargs(g) = {1}, Uargs(f^#) = {2},
Uargs(c_0) = {1}, Uargs(c_1) = {}, Uargs(g^#) = {},
Uargs(c_2) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
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]
i(x1) = [1 0 0] x1 + [0]
[0 0 0] [1]
[0 0 0] [1]
g(x1) = [1 0 0] x1 + [3]
[0 0 0] [3]
[0 0 0] [3]
a() = [0]
[0]
[0]
f^#(x1, x2) = [0 0 0] x1 + [1 0 0] x2 + [0]
[3 3 3] [3 3 3] [0]
[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) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
g^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
c_4() = [0]
[0]
[0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {1,4}->{2}: NA
-------------------
The usable rules for this path are:
{g(x) -> i(x)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(i) = {1}, Uargs(g) = {1}, Uargs(f^#) = {2},
Uargs(c_0) = {1}, Uargs(c_1) = {}, Uargs(g^#) = {},
Uargs(c_2) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
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]
i(x1) = [1 1 1] x1 + [0]
[0 1 1] [1]
[0 0 1] [1]
g(x1) = [3 3 3] x1 + [3]
[0 3 3] [3]
[0 0 3] [3]
a() = [0]
[0]
[0]
f^#(x1, x2) = [3 3 3] x1 + [3 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
c_1(x1) = [1 3 1] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
g^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
c_4() = [0]
[0]
[0]
Complexity induced by the adequate RMI: YES(?,O(n^3))
We have not generated a proof for the resulting sub-problem.
* Path {1,4}->{5}: NA
-------------------
The usable rules for this path are:
{g(x) -> i(x)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(i) = {1}, Uargs(g) = {1}, Uargs(f^#) = {2},
Uargs(c_0) = {1}, Uargs(c_1) = {}, Uargs(g^#) = {},
Uargs(c_2) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
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]
i(x1) = [1 1 1] x1 + [0]
[0 1 1] [1]
[0 0 1] [1]
g(x1) = [3 3 3] x1 + [3]
[0 3 3] [3]
[0 0 3] [3]
a() = [0]
[0]
[0]
f^#(x1, x2) = [0 0 0] x1 + [3 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
g^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
c_4() = [0]
[0]
[0]
Complexity induced by the adequate RMI: YES(?,O(n^3))
We have not generated a proof for the resulting sub-problem.
* Path {3}: YES(?,O(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(f) = {}, Uargs(i) = {}, Uargs(g) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(g^#) = {}, Uargs(c_2) = {},
Uargs(c_3) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
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]
i(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
g(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
a() = [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_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
g^#(x1) = [3 3 3] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2(x1) = [1 1 1] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_4() = [0]
[0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {g^#(x) -> c_2(x)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(g^#) = {}, Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
g^#(x1) = [7 7 7] x1 + [7]
[7 7 7] [7]
[7 7 7] [7]
c_2(x1) = [3 3 3] x1 + [0]
[3 1 3] [1]
[1 1 1] [1]
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: f^#(x, x) -> c_0(f^#(i(x), g(g(x))))
, 2: f^#(x, y) -> c_1(x)
, 3: g^#(x) -> c_2(x)
, 4: f^#(x, i(x)) -> c_3(f^#(x, x))
, 5: f^#(i(x), i(g(x))) -> c_4()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{3} [ YES(?,O(1)) ]
->{1,4} [ MAYBE ]
|
|->{2} [ NA ]
|
`->{5} [ NA ]
Sub-problems:
-------------
* Path {1,4}: MAYBE
-----------------
The usable rules for this path are:
{g(x) -> i(x)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(i) = {1}, Uargs(g) = {1}, Uargs(f^#) = {2},
Uargs(c_0) = {1}, Uargs(c_1) = {}, Uargs(g^#) = {},
Uargs(c_2) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
i(x1) = [1 0] x1 + [0]
[0 0] [1]
g(x1) = [1 0] x1 + [3]
[0 0] [3]
a() = [0]
[0]
f^#(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[3 3] [3 3] [0]
c_0(x1) = [1 0] x1 + [0]
[0 1] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
g^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1) = [1 0] x1 + [0]
[0 1] [0]
c_4() = [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: MAYBE
Input Problem: DP runtime-complexity with respect to
Strict Rules:
{ f^#(x, x) -> c_0(f^#(i(x), g(g(x))))
, f^#(x, i(x)) -> c_3(f^#(x, x))}
Weak Rules: {g(x) -> i(x)}
Proof Output:
The input cannot be shown compatible
* Path {1,4}->{2}: NA
-------------------
The usable rules for this path are:
{g(x) -> i(x)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(i) = {1}, Uargs(g) = {1}, Uargs(f^#) = {2},
Uargs(c_0) = {1}, Uargs(c_1) = {}, Uargs(g^#) = {},
Uargs(c_2) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
i(x1) = [1 1] x1 + [0]
[0 1] [1]
g(x1) = [3 3] x1 + [3]
[0 3] [3]
a() = [0]
[0]
f^#(x1, x2) = [3 3] x1 + [3 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [1 0] x1 + [0]
[0 1] [0]
c_1(x1) = [1 3] x1 + [0]
[0 0] [0]
g^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1) = [1 0] x1 + [0]
[0 1] [0]
c_4() = [0]
[0]
Complexity induced by the adequate RMI: YES(?,O(n^2))
We have not generated a proof for the resulting sub-problem.
* Path {1,4}->{5}: NA
-------------------
The usable rules for this path are:
{g(x) -> i(x)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(i) = {1}, Uargs(g) = {1}, Uargs(f^#) = {2},
Uargs(c_0) = {1}, Uargs(c_1) = {}, Uargs(g^#) = {},
Uargs(c_2) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
i(x1) = [1 1] x1 + [0]
[0 1] [1]
g(x1) = [3 3] x1 + [3]
[0 3] [3]
a() = [0]
[0]
f^#(x1, x2) = [0 0] x1 + [3 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [1 0] x1 + [0]
[0 1] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
g^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
c_3(x1) = [1 0] x1 + [0]
[0 1] [0]
c_4() = [0]
[0]
Complexity induced by the adequate RMI: YES(?,O(n^2))
We have not generated a proof for the resulting sub-problem.
* Path {3}: YES(?,O(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(f) = {}, Uargs(i) = {}, Uargs(g) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(g^#) = {}, Uargs(c_2) = {},
Uargs(c_3) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
i(x1) = [0 0] x1 + [0]
[0 0] [0]
g(x1) = [0 0] x1 + [0]
[0 0] [0]
a() = [0]
[0]
f^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [0 0] x1 + [0]
[0 0] [0]
g^#(x1) = [3 3] x1 + [0]
[0 0] [0]
c_2(x1) = [1 1] x1 + [0]
[0 0] [0]
c_3(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4() = [0]
[0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {g^#(x) -> c_2(x)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(g^#) = {}, Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
g^#(x1) = [7 7] x1 + [7]
[7 7] [7]
c_2(x1) = [1 3] x1 + [0]
[3 1] [3]
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: f^#(x, x) -> c_0(f^#(i(x), g(g(x))))
, 2: f^#(x, y) -> c_1(x)
, 3: g^#(x) -> c_2(x)
, 4: f^#(x, i(x)) -> c_3(f^#(x, x))
, 5: f^#(i(x), i(g(x))) -> c_4()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{3} [ YES(?,O(1)) ]
->{1,4} [ MAYBE ]
|
|->{2} [ NA ]
|
`->{5} [ NA ]
Sub-problems:
-------------
* Path {1,4}: MAYBE
-----------------
The usable rules for this path are:
{g(x) -> i(x)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(i) = {1}, Uargs(g) = {1}, Uargs(f^#) = {2},
Uargs(c_0) = {1}, Uargs(c_1) = {}, Uargs(g^#) = {},
Uargs(c_2) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0] x1 + [0] x2 + [0]
i(x1) = [1] x1 + [1]
g(x1) = [1] x1 + [3]
a() = [0]
f^#(x1, x2) = [0] x1 + [1] x2 + [0]
c_0(x1) = [1] x1 + [0]
c_1(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [0]
c_4() = [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 1'
--------------------------------------
Answer: MAYBE
Input Problem: DP runtime-complexity with respect to
Strict Rules:
{ f^#(x, x) -> c_0(f^#(i(x), g(g(x))))
, f^#(x, i(x)) -> c_3(f^#(x, x))}
Weak Rules: {g(x) -> i(x)}
Proof Output:
The input cannot be shown compatible
* Path {1,4}->{2}: NA
-------------------
The usable rules for this path are:
{g(x) -> i(x)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(i) = {1}, Uargs(g) = {1}, Uargs(f^#) = {2},
Uargs(c_0) = {1}, Uargs(c_1) = {}, Uargs(g^#) = {},
Uargs(c_2) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0] x1 + [0] x2 + [0]
i(x1) = [1] x1 + [2]
g(x1) = [3] x1 + [3]
a() = [0]
f^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_0(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [0]
c_4() = [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {1,4}->{5}: NA
-------------------
The usable rules for this path are:
{g(x) -> i(x)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(i) = {1}, Uargs(g) = {1}, Uargs(f^#) = {2},
Uargs(c_0) = {1}, Uargs(c_1) = {}, Uargs(g^#) = {},
Uargs(c_2) = {}, Uargs(c_3) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0] x1 + [0] x2 + [0]
i(x1) = [1] x1 + [0]
g(x1) = [3] x1 + [3]
a() = [0]
f^#(x1, x2) = [0] x1 + [3] x2 + [0]
c_0(x1) = [1] x1 + [0]
c_1(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [0]
c_4() = [0]
Complexity induced by the adequate RMI: YES(?,O(n^1))
We have not generated a proof for the resulting sub-problem.
* Path {3}: YES(?,O(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(f) = {}, Uargs(i) = {}, Uargs(g) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(g^#) = {}, Uargs(c_2) = {},
Uargs(c_3) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0] x1 + [0] x2 + [0]
i(x1) = [0] x1 + [0]
g(x1) = [0] x1 + [0]
a() = [0]
f^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
g^#(x1) = [3] x1 + [0]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {g^#(x) -> c_2(x)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(g^#) = {}, Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
g^#(x1) = [7] x1 + [7]
c_2(x1) = [1] x1 + [0]
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.