Tool CaT
stdout:
MAYBE
Problem:
f(x,x) -> f(g(x),x)
g(x) -> s(x)
Proof:
Complexity Transformation Processor:
strict:
f(x,x) -> f(g(x),x)
g(x) -> s(x)
weak:
Matrix Interpretation Processor:
dimension: 1
max_matrix:
1
interpretation:
[s](x0) = x0,
[g](x0) = x0 + 225,
[f](x0, x1) = x0 + x1 + 79
orientation:
f(x,x) = 2x + 79 >= 2x + 304 = f(g(x),x)
g(x) = x + 225 >= x = s(x)
problem:
strict:
f(x,x) -> f(g(x),x)
weak:
g(x) -> s(x)
Open
Tool 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(x, x) -> f(g(x), x)
, g(x) -> s(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: f^#(x, x) -> c_0(f^#(g(x), x))
, 2: g^#(x) -> c_1()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{2} [ YES(?,O(1)) ]
->{1} [ MAYBE ]
Sub-problems:
-------------
* Path {1}: MAYBE
---------------
The usable rules for this path are:
{g(x) -> s(x)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(g) = {}, Uargs(s) = {}, Uargs(f^#) = {1},
Uargs(c_0) = {1}, Uargs(g^#) = {}
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]
g(x1) = [0 0 0] x1 + [3]
[3 3 3] [3]
[3 3 3] [3]
s(x1) = [0 0 0] x1 + [1]
[0 1 1] [1]
[0 0 1] [1]
f^#(x1, x2) = [2 0 0] x1 + [0 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]
g^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1() = [0]
[0]
[0]
Complexity induced by the adequate RMI: YES(?,O(n^2))
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^#(x, x) -> c_0(f^#(g(x), x))}
Weak Rules: {g(x) -> s(x)}
Proof Output:
The input cannot be shown compatible
* Path {2}: 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(g) = {}, Uargs(s) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(g^#) = {}
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]
g(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
s(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_0(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_1() = [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_1()}
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_1() = [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^#(g(x), x))
, 2: g^#(x) -> c_1()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{2} [ YES(?,O(1)) ]
->{1} [ MAYBE ]
Sub-problems:
-------------
* Path {1}: MAYBE
---------------
The usable rules for this path are:
{g(x) -> s(x)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(g) = {}, Uargs(s) = {}, Uargs(f^#) = {1},
Uargs(c_0) = {1}, Uargs(g^#) = {}
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]
g(x1) = [0 0] x1 + [3]
[3 3] [3]
s(x1) = [0 0] x1 + [2]
[0 1] [1]
f^#(x1, x2) = [2 0] x1 + [0 0] x2 + [0]
[3 3] [3 3] [0]
c_0(x1) = [1 0] x1 + [0]
[0 1] [0]
g^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1() = [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^#(g(x), x))}
Weak Rules: {g(x) -> s(x)}
Proof Output:
The input cannot be shown compatible
* Path {2}: 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(g) = {}, Uargs(s) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(g^#) = {}
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]
g(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 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]
g^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1() = [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_1()}
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_1() = [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^#(g(x), x))
, 2: g^#(x) -> c_1()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{2} [ YES(?,O(1)) ]
->{1} [ MAYBE ]
Sub-problems:
-------------
* Path {1}: MAYBE
---------------
The usable rules for this path are:
{g(x) -> s(x)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(g) = {}, Uargs(s) = {}, Uargs(f^#) = {1},
Uargs(c_0) = {1}, Uargs(g^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0] x1 + [0] x2 + [0]
g(x1) = [1] x1 + [3]
s(x1) = [1] x1 + [1]
f^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_0(x1) = [1] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_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 1'
--------------------------------------
Answer: MAYBE
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {f^#(x, x) -> c_0(f^#(g(x), x))}
Weak Rules: {g(x) -> s(x)}
Proof Output:
The input cannot be shown compatible
* Path {2}: 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(g) = {}, Uargs(s) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(g^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0] x1 + [0] x2 + [0]
g(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
f^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_1() = [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_1()}
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_1() = [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
Tool RC2
stdout:
MAYBE
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ f(x, x) -> f(g(x), x)
, g(x) -> s(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: f^#(x, x) -> c_0(f^#(g(x), x))
, 2: g^#(x) -> c_1(x)}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{2} [ YES(?,O(1)) ]
->{1} [ MAYBE ]
Sub-problems:
-------------
* Path {1}: MAYBE
---------------
The usable rules for this path are:
{g(x) -> s(x)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(g) = {}, Uargs(s) = {}, Uargs(f^#) = {1},
Uargs(c_0) = {1}, Uargs(g^#) = {}, Uargs(c_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]
g(x1) = [0 0 0] x1 + [3]
[3 3 3] [3]
[3 3 3] [3]
s(x1) = [0 0 0] x1 + [1]
[0 1 1] [1]
[0 0 1] [1]
f^#(x1, x2) = [2 0 0] x1 + [0 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]
g^#(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]
Complexity induced by the adequate RMI: YES(?,O(n^2))
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^#(x, x) -> c_0(f^#(g(x), x))}
Weak Rules: {g(x) -> s(x)}
Proof Output:
The input cannot be shown compatible
* Path {2}: 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(g) = {}, Uargs(s) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(g^#) = {}, Uargs(c_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]
g(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
s(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_0(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_1(x1) = [1 1 1] x1 + [0]
[0 0 0] [0]
[0 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_1(x)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(g^#) = {}, Uargs(c_1) = {}
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_1(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^#(g(x), x))
, 2: g^#(x) -> c_1(x)}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{2} [ YES(?,O(1)) ]
->{1} [ MAYBE ]
Sub-problems:
-------------
* Path {1}: MAYBE
---------------
The usable rules for this path are:
{g(x) -> s(x)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(g) = {}, Uargs(s) = {}, Uargs(f^#) = {1},
Uargs(c_0) = {1}, Uargs(g^#) = {}, Uargs(c_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]
g(x1) = [0 0] x1 + [3]
[3 3] [3]
s(x1) = [0 0] x1 + [2]
[0 1] [1]
f^#(x1, x2) = [2 0] x1 + [0 0] x2 + [0]
[3 3] [3 3] [0]
c_0(x1) = [1 0] x1 + [0]
[0 1] [0]
g^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(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: MAYBE
Input Problem: DP runtime-complexity with respect to
Strict Rules: {f^#(x, x) -> c_0(f^#(g(x), x))}
Weak Rules: {g(x) -> s(x)}
Proof Output:
The input cannot be shown compatible
* Path {2}: 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(g) = {}, Uargs(s) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(g^#) = {}, Uargs(c_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]
g(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 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]
g^#(x1) = [3 3] x1 + [0]
[0 0] [0]
c_1(x1) = [1 1] x1 + [0]
[0 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_1(x)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(g^#) = {}, Uargs(c_1) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
g^#(x1) = [7 7] x1 + [7]
[7 7] [7]
c_1(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^#(g(x), x))
, 2: g^#(x) -> c_1(x)}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{2} [ YES(?,O(1)) ]
->{1} [ MAYBE ]
Sub-problems:
-------------
* Path {1}: MAYBE
---------------
The usable rules for this path are:
{g(x) -> s(x)}
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(f) = {}, Uargs(g) = {}, Uargs(s) = {}, Uargs(f^#) = {1},
Uargs(c_0) = {1}, Uargs(g^#) = {}, Uargs(c_1) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0] x1 + [0] x2 + [0]
g(x1) = [1] x1 + [3]
s(x1) = [1] x1 + [1]
f^#(x1, x2) = [3] x1 + [3] x2 + [0]
c_0(x1) = [1] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [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^#(g(x), x))}
Weak Rules: {g(x) -> s(x)}
Proof Output:
The input cannot be shown compatible
* Path {2}: 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(g) = {}, Uargs(s) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(g^#) = {}, Uargs(c_1) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0] x1 + [0] x2 + [0]
g(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
f^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
g^#(x1) = [3] x1 + [0]
c_1(x1) = [1] x1 + [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_1(x)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(g^#) = {}, Uargs(c_1) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
g^#(x1) = [7] x1 + [7]
c_1(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.