Tool CaT
stdout:
MAYBE
Problem:
f(s(s(s(s(s(s(s(s(x)))))))),y,y) -> f(id(s(s(s(s(s(s(s(s(x))))))))),y,y)
id(s(x)) -> s(id(x))
id(0()) -> 0()
Proof:
Complexity Transformation Processor:
strict:
f(s(s(s(s(s(s(s(s(x)))))))),y,y) -> f(id(s(s(s(s(s(s(s(s(x))))))))),y,y)
id(s(x)) -> s(id(x))
id(0()) -> 0()
weak:
Matrix Interpretation Processor:
dimension: 1
max_matrix:
1
interpretation:
[0] = 64,
[id](x0) = x0 + 1,
[f](x0, x1, x2) = x0 + x1 + x2 + 11,
[s](x0) = x0 + 11
orientation:
f(s(s(s(s(s(s(s(s(x)))))))),y,y) = x + 2y + 99 >= x + 2y + 100 = f(id(s(s(s(s(s(s(s(s(x))))))))),y,y)
id(s(x)) = x + 12 >= x + 12 = s(id(x))
id(0()) = 65 >= 64 = 0()
problem:
strict:
f(s(s(s(s(s(s(s(s(x)))))))),y,y) -> f(id(s(s(s(s(s(s(s(s(x))))))))),y,y)
id(s(x)) -> s(id(x))
weak:
id(0()) -> 0()
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(s(s(s(s(s(s(s(s(x)))))))), y, y) ->
f(id(s(s(s(s(s(s(s(s(x))))))))), y, y)
, id(s(x)) -> s(id(x))
, id(0()) -> 0()}
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^#(s(s(s(s(s(s(s(s(x)))))))), y, y) ->
c_0(f^#(id(s(s(s(s(s(s(s(s(x))))))))), y, y))
, 2: id^#(s(x)) -> c_1(id^#(x))
, 3: id^#(0()) -> c_2()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{2} [ YES(?,O(n^1)) ]
|
`->{3} [ YES(?,O(n^1)) ]
->{1} [ NA ]
Sub-problems:
-------------
* Path {1}: NA
------------
The usable rules for this path are:
{ id(s(x)) -> s(id(x))
, id(0()) -> 0()}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {2}: 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(f) = {}, Uargs(s) = {}, Uargs(id) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(id^#) = {}, Uargs(c_1) = {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]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
id(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
f^#(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]
id^#(x1) = [3 3] x1 + [0]
[3 3] [0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
c_2() = [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: {id^#(s(x)) -> c_1(id^#(x))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(id^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 0] x1 + [0]
[0 1] [1]
id^#(x1) = [0 1] x1 + [1]
[0 0] [0]
c_1(x1) = [1 0] x1 + [0]
[0 0] [0]
* 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(f) = {}, Uargs(s) = {}, Uargs(id) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(id^#) = {}, Uargs(c_1) = {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]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
id(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
f^#(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]
id^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
c_2() = [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: {id^#(0()) -> c_2()}
Weak Rules: {id^#(s(x)) -> c_1(id^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(id^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [1]
[0 0] [3]
0() = [2]
[2]
id^#(x1) = [1 2] x1 + [2]
[6 1] [0]
c_1(x1) = [1 0] x1 + [5]
[2 0] [3]
c_2() = [1]
[0]
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: f^#(s(s(s(s(s(s(s(s(x)))))))), y, y) ->
c_0(f^#(id(s(s(s(s(s(s(s(s(x))))))))), y, y))
, 2: id^#(s(x)) -> c_1(id^#(x))
, 3: id^#(0()) -> c_2()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{2} [ YES(?,O(n^1)) ]
|
`->{3} [ YES(?,O(n^1)) ]
->{1} [ MAYBE ]
Sub-problems:
-------------
* Path {1}: MAYBE
---------------
The usable rules for this path are:
{ id(s(x)) -> s(id(x))
, id(0()) -> 0()}
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:
{ f^#(s(s(s(s(s(s(s(s(x)))))))), y, y) ->
c_0(f^#(id(s(s(s(s(s(s(s(s(x))))))))), y, y))
, id(s(x)) -> s(id(x))
, id(0()) -> 0()}
Proof Output:
The input cannot be shown compatible
* Path {2}: 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(f) = {}, Uargs(s) = {}, Uargs(id) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(id^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
s(x1) = [1] x1 + [0]
id(x1) = [0] x1 + [0]
0() = [0]
f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [0] x1 + [0]
id^#(x1) = [3] x1 + [0]
c_1(x1) = [1] x1 + [0]
c_2() = [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: {id^#(s(x)) -> c_1(id^#(x))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(id^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [4]
id^#(x1) = [2] x1 + [0]
c_1(x1) = [1] x1 + [7]
* 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(f) = {}, Uargs(s) = {}, Uargs(id) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(id^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
s(x1) = [0] x1 + [0]
id(x1) = [0] x1 + [0]
0() = [0]
f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [0] x1 + [0]
id^#(x1) = [0] x1 + [0]
c_1(x1) = [1] x1 + [0]
c_2() = [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: {id^#(0()) -> c_2()}
Weak Rules: {id^#(s(x)) -> c_1(id^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(id^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [0]
0() = [2]
id^#(x1) = [2] x1 + [0]
c_1(x1) = [1] x1 + [0]
c_2() = [1]
3) 'matrix-interpretation of dimension 1' failed due to the following reason:
The input cannot be shown compatible
4) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
5) '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(s(s(s(s(s(s(s(s(x)))))))), y, y) ->
f(id(s(s(s(s(s(s(s(s(x))))))))), y, y)
, id(s(x)) -> s(id(x))
, id(0()) -> 0()}
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^#(s(s(s(s(s(s(s(s(x)))))))), y, y) ->
c_0(f^#(id(s(s(s(s(s(s(s(s(x))))))))), y, y))
, 2: id^#(s(x)) -> c_1(id^#(x))
, 3: id^#(0()) -> c_2()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{2} [ YES(?,O(n^1)) ]
|
`->{3} [ YES(?,O(n^1)) ]
->{1} [ NA ]
Sub-problems:
-------------
* Path {1}: NA
------------
The usable rules for this path are:
{ id(s(x)) -> s(id(x))
, id(0()) -> 0()}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
* Path {2}: 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(f) = {}, Uargs(s) = {}, Uargs(id) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(id^#) = {}, Uargs(c_1) = {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]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
id(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
f^#(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]
id^#(x1) = [3 3] x1 + [0]
[3 3] [0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
c_2() = [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: {id^#(s(x)) -> c_1(id^#(x))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(id^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 0] x1 + [0]
[0 1] [1]
id^#(x1) = [0 1] x1 + [1]
[0 0] [0]
c_1(x1) = [1 0] x1 + [0]
[0 0] [0]
* 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(f) = {}, Uargs(s) = {}, Uargs(id) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(id^#) = {}, Uargs(c_1) = {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]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
id(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
f^#(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]
id^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1(x1) = [1 0] x1 + [0]
[0 1] [0]
c_2() = [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: {id^#(0()) -> c_2()}
Weak Rules: {id^#(s(x)) -> c_1(id^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(id^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [1]
[0 0] [3]
0() = [2]
[2]
id^#(x1) = [1 2] x1 + [2]
[6 1] [0]
c_1(x1) = [1 0] x1 + [5]
[2 0] [3]
c_2() = [1]
[0]
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: f^#(s(s(s(s(s(s(s(s(x)))))))), y, y) ->
c_0(f^#(id(s(s(s(s(s(s(s(s(x))))))))), y, y))
, 2: id^#(s(x)) -> c_1(id^#(x))
, 3: id^#(0()) -> c_2()}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{2} [ YES(?,O(n^1)) ]
|
`->{3} [ YES(?,O(n^1)) ]
->{1} [ MAYBE ]
Sub-problems:
-------------
* Path {1}: MAYBE
---------------
The usable rules for this path are:
{ id(s(x)) -> s(id(x))
, id(0()) -> 0()}
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:
{ f^#(s(s(s(s(s(s(s(s(x)))))))), y, y) ->
c_0(f^#(id(s(s(s(s(s(s(s(s(x))))))))), y, y))
, id(s(x)) -> s(id(x))
, id(0()) -> 0()}
Proof Output:
The input cannot be shown compatible
* Path {2}: 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(f) = {}, Uargs(s) = {}, Uargs(id) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(id^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
s(x1) = [1] x1 + [0]
id(x1) = [0] x1 + [0]
0() = [0]
f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [0] x1 + [0]
id^#(x1) = [3] x1 + [0]
c_1(x1) = [1] x1 + [0]
c_2() = [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: {id^#(s(x)) -> c_1(id^#(x))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(id^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [4]
id^#(x1) = [2] x1 + [0]
c_1(x1) = [1] x1 + [7]
* 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(f) = {}, Uargs(s) = {}, Uargs(id) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(id^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
s(x1) = [0] x1 + [0]
id(x1) = [0] x1 + [0]
0() = [0]
f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [0] x1 + [0]
id^#(x1) = [0] x1 + [0]
c_1(x1) = [1] x1 + [0]
c_2() = [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: {id^#(0()) -> c_2()}
Weak Rules: {id^#(s(x)) -> c_1(id^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(id^#) = {}, Uargs(c_1) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [0]
0() = [2]
id^#(x1) = [2] x1 + [0]
c_1(x1) = [1] x1 + [0]
c_2() = [1]
3) 'matrix-interpretation of dimension 1' failed due to the following reason:
The input cannot be shown compatible
4) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
5) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.