Tool CaT
stdout:
MAYBE
Problem:
f(s(x),x) -> f(s(x),round(s(x)))
round(0()) -> 0()
round(0()) -> s(0())
round(s(0())) -> s(0())
round(s(s(x))) -> s(s(round(x)))
Proof:
Complexity Transformation Processor:
strict:
f(s(x),x) -> f(s(x),round(s(x)))
round(0()) -> 0()
round(0()) -> s(0())
round(s(0())) -> s(0())
round(s(s(x))) -> s(s(round(x)))
weak:
Matrix Interpretation Processor:
dimension: 1
max_matrix:
1
interpretation:
[0] = 8,
[round](x0) = x0 + 102,
[f](x0, x1) = x0 + x1 + 127,
[s](x0) = x0 + 2
orientation:
f(s(x),x) = 2x + 129 >= 2x + 233 = f(s(x),round(s(x)))
round(0()) = 110 >= 8 = 0()
round(0()) = 110 >= 10 = s(0())
round(s(0())) = 112 >= 10 = s(0())
round(s(s(x))) = x + 106 >= x + 106 = s(s(round(x)))
problem:
strict:
f(s(x),x) -> f(s(x),round(s(x)))
round(s(s(x))) -> s(s(round(x)))
weak:
round(0()) -> 0()
round(0()) -> s(0())
round(s(0())) -> s(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(x), x) -> f(s(x), round(s(x)))
, round(0()) -> 0()
, round(0()) -> s(0())
, round(s(0())) -> s(0())
, round(s(s(x))) -> s(s(round(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^#(s(x), x) -> c_0(f^#(s(x), round(s(x))))
, 2: round^#(0()) -> c_1()
, 3: round^#(0()) -> c_2()
, 4: round^#(s(0())) -> c_3()
, 5: round^#(s(s(x))) -> c_4(round^#(x))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{5} [ YES(?,O(n^2)) ]
|
|->{2} [ YES(?,O(n^1)) ]
|
|->{3} [ YES(?,O(n^1)) ]
|
`->{4} [ YES(?,O(n^2)) ]
->{1} [ NA ]
Sub-problems:
-------------
* Path {1}: NA
------------
The usable rules for this path are:
{ round(0()) -> 0()
, round(0()) -> s(0())
, round(s(0())) -> s(0())
, round(s(s(x))) -> s(s(round(x)))}
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 {5}: 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(f) = {}, Uargs(s) = {}, Uargs(round) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(round^#) = {}, Uargs(c_4) = {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]
s(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
round(x1) = [0 0 0] x1 + [0]
[0 0 0] [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]
round^#(x1) = [0 0 0] x1 + [0]
[3 3 3] [0]
[3 3 3] [0]
c_1() = [0]
[0]
[0]
c_2() = [0]
[0]
[0]
c_3() = [0]
[0]
[0]
c_4(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {round^#(s(s(x))) -> c_4(round^#(x))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(round^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2 0] x1 + [0]
[0 0 1] [0]
[0 0 1] [2]
round^#(x1) = [2 0 2] x1 + [0]
[4 0 2] [0]
[0 0 0] [2]
c_4(x1) = [1 0 2] x1 + [3]
[2 0 0] [0]
[0 0 0] [2]
* Path {5}->{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(round) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(round^#) = {}, Uargs(c_4) = {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]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
round(x1) = [0 0 0] x1 + [0]
[0 0 0] [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]
round^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1() = [0]
[0]
[0]
c_2() = [0]
[0]
[0]
c_3() = [0]
[0]
[0]
c_4(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {round^#(0()) -> c_1()}
Weak Rules: {round^#(s(s(x))) -> c_4(round^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(round^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2 0] x1 + [0]
[0 0 0] [2]
[0 0 0] [2]
0() = [2]
[2]
[2]
round^#(x1) = [2 0 0] x1 + [0]
[0 0 2] [4]
[1 2 2] [0]
c_1() = [1]
[0]
[0]
c_4(x1) = [1 0 0] x1 + [3]
[0 0 0] [7]
[0 0 0] [6]
* Path {5}->{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(round) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(round^#) = {}, Uargs(c_4) = {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]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
round(x1) = [0 0 0] x1 + [0]
[0 0 0] [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]
round^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1() = [0]
[0]
[0]
c_2() = [0]
[0]
[0]
c_3() = [0]
[0]
[0]
c_4(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {round^#(0()) -> c_2()}
Weak Rules: {round^#(s(s(x))) -> c_4(round^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(round^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2 0] x1 + [0]
[0 0 0] [2]
[0 0 0] [2]
0() = [2]
[2]
[2]
round^#(x1) = [2 0 0] x1 + [0]
[0 0 2] [4]
[1 2 2] [0]
c_2() = [1]
[0]
[0]
c_4(x1) = [1 0 0] x1 + [3]
[0 0 0] [7]
[0 0 0] [6]
* Path {5}->{4}: 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(f) = {}, Uargs(s) = {}, Uargs(round) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(round^#) = {}, Uargs(c_4) = {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]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
round(x1) = [0 0 0] x1 + [0]
[0 0 0] [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]
round^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1() = [0]
[0]
[0]
c_2() = [0]
[0]
[0]
c_3() = [0]
[0]
[0]
c_4(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {round^#(s(0())) -> c_3()}
Weak Rules: {round^#(s(s(x))) -> c_4(round^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(round^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2 0] x1 + [2]
[0 1 2] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
round^#(x1) = [2 3 0] x1 + [0]
[0 0 0] [0]
[0 5 0] [0]
c_3() = [1]
[0]
[0]
c_4(x1) = [1 0 1] x1 + [0]
[0 0 0] [0]
[0 0 0] [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(x), x) -> c_0(f^#(s(x), round(s(x))))
, 2: round^#(0()) -> c_1()
, 3: round^#(0()) -> c_2()
, 4: round^#(s(0())) -> c_3()
, 5: round^#(s(s(x))) -> c_4(round^#(x))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{5} [ YES(?,O(n^1)) ]
|
|->{2} [ YES(?,O(n^1)) ]
|
|->{3} [ YES(?,O(n^1)) ]
|
`->{4} [ YES(?,O(n^2)) ]
->{1} [ MAYBE ]
Sub-problems:
-------------
* Path {1}: MAYBE
---------------
The usable rules for this path are:
{ round(0()) -> 0()
, round(0()) -> s(0())
, round(s(0())) -> s(0())
, round(s(s(x))) -> s(s(round(x)))}
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:
{ f^#(s(x), x) -> c_0(f^#(s(x), round(s(x))))
, round(0()) -> 0()
, round(0()) -> s(0())
, round(s(0())) -> s(0())
, round(s(s(x))) -> s(s(round(x)))}
Proof Output:
The input cannot be shown compatible
* Path {5}: 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(round) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(round^#) = {}, Uargs(c_4) = {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]
s(x1) = [1 0] x1 + [0]
[0 0] [0]
round(x1) = [0 0] x1 + [0]
[0 0] [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]
round^#(x1) = [0 0] x1 + [0]
[3 3] [0]
c_1() = [0]
[0]
c_2() = [0]
[0]
c_3() = [0]
[0]
c_4(x1) = [1 0] x1 + [0]
[0 1] [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: {round^#(s(s(x))) -> c_4(round^#(x))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(round^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [0]
[0 0] [1]
round^#(x1) = [2 2] x1 + [2]
[6 0] [0]
c_4(x1) = [1 0] x1 + [5]
[2 0] [7]
* Path {5}->{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(round) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(round^#) = {}, Uargs(c_4) = {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]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
round(x1) = [0 0] x1 + [0]
[0 0] [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]
round^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1() = [0]
[0]
c_2() = [0]
[0]
c_3() = [0]
[0]
c_4(x1) = [1 0] x1 + [0]
[0 1] [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: {round^#(0()) -> c_1()}
Weak Rules: {round^#(s(s(x))) -> c_4(round^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(round^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [1]
[0 0] [0]
0() = [0]
[2]
round^#(x1) = [2 2] x1 + [4]
[6 2] [0]
c_1() = [1]
[0]
c_4(x1) = [1 0] x1 + [3]
[2 0] [3]
* Path {5}->{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(round) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(round^#) = {}, Uargs(c_4) = {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]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
round(x1) = [0 0] x1 + [0]
[0 0] [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]
round^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1() = [0]
[0]
c_2() = [0]
[0]
c_3() = [0]
[0]
c_4(x1) = [1 0] x1 + [0]
[0 1] [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: {round^#(0()) -> c_2()}
Weak Rules: {round^#(s(s(x))) -> c_4(round^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(round^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [1]
[0 0] [0]
0() = [0]
[2]
round^#(x1) = [2 2] x1 + [4]
[6 2] [0]
c_2() = [1]
[0]
c_4(x1) = [1 0] x1 + [3]
[2 0] [3]
* Path {5}->{4}: 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(f) = {}, Uargs(s) = {}, Uargs(round) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(round^#) = {}, Uargs(c_4) = {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]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
round(x1) = [0 0] x1 + [0]
[0 0] [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]
round^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1() = [0]
[0]
c_2() = [0]
[0]
c_3() = [0]
[0]
c_4(x1) = [1 0] x1 + [0]
[0 1] [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: {round^#(s(0())) -> c_3()}
Weak Rules: {round^#(s(s(x))) -> c_4(round^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(round^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 1] x1 + [2]
[0 1] [1]
0() = [1]
[0]
round^#(x1) = [2 1] x1 + [1]
[0 0] [7]
c_3() = [1]
[1]
c_4(x1) = [1 1] x1 + [3]
[0 0] [3]
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: f^#(s(x), x) -> c_0(f^#(s(x), round(s(x))))
, 2: round^#(0()) -> c_1()
, 3: round^#(0()) -> c_2()
, 4: round^#(s(0())) -> c_3()
, 5: round^#(s(s(x))) -> c_4(round^#(x))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{5} [ YES(?,O(n^1)) ]
|
|->{2} [ YES(?,O(n^1)) ]
|
|->{3} [ YES(?,O(n^1)) ]
|
`->{4} [ YES(?,O(1)) ]
->{1} [ MAYBE ]
Sub-problems:
-------------
* Path {1}: MAYBE
---------------
The usable rules for this path are:
{ round(0()) -> 0()
, round(0()) -> s(0())
, round(s(0())) -> s(0())
, round(s(s(x))) -> s(s(round(x)))}
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(x), x) -> c_0(f^#(s(x), round(s(x))))
, round(0()) -> 0()
, round(0()) -> s(0())
, round(s(0())) -> s(0())
, round(s(s(x))) -> s(s(round(x)))}
Proof Output:
The input cannot be shown compatible
* Path {5}: 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(round) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(round^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [1] x1 + [0]
round(x1) = [0] x1 + [0]
0() = [0]
f^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
round^#(x1) = [3] x1 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [1] x1 + [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: {round^#(s(s(x))) -> c_4(round^#(x))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(round^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
round^#(x1) = [2] x1 + [0]
c_4(x1) = [1] x1 + [7]
* Path {5}->{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(round) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(round^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
round(x1) = [0] x1 + [0]
0() = [0]
f^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
round^#(x1) = [0] x1 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [1] x1 + [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: {round^#(0()) -> c_1()}
Weak Rules: {round^#(s(s(x))) -> c_4(round^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(round^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [0]
0() = [2]
round^#(x1) = [2] x1 + [4]
c_1() = [1]
c_4(x1) = [1] x1 + [0]
* Path {5}->{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(round) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(round^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
round(x1) = [0] x1 + [0]
0() = [0]
f^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
round^#(x1) = [0] x1 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [1] x1 + [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: {round^#(0()) -> c_2()}
Weak Rules: {round^#(s(s(x))) -> c_4(round^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(round^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [0]
0() = [2]
round^#(x1) = [2] x1 + [4]
c_2() = [1]
c_4(x1) = [1] x1 + [0]
* Path {5}->{4}: 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(s) = {}, Uargs(round) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(round^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
round(x1) = [0] x1 + [0]
0() = [0]
f^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
round^#(x1) = [0] x1 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4(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: innermost DP runtime-complexity with respect to
Strict Rules: {round^#(s(0())) -> c_3()}
Weak Rules: {round^#(s(s(x))) -> c_4(round^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(round^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [0] x1 + [0]
0() = [0]
round^#(x1) = [0] x1 + [1]
c_3() = [0]
c_4(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.
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(x), x) -> f(s(x), round(s(x)))
, round(0()) -> 0()
, round(0()) -> s(0())
, round(s(0())) -> s(0())
, round(s(s(x))) -> s(s(round(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^#(s(x), x) -> c_0(f^#(s(x), round(s(x))))
, 2: round^#(0()) -> c_1()
, 3: round^#(0()) -> c_2()
, 4: round^#(s(0())) -> c_3()
, 5: round^#(s(s(x))) -> c_4(round^#(x))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{5} [ YES(?,O(n^2)) ]
|
|->{2} [ YES(?,O(n^1)) ]
|
|->{3} [ YES(?,O(n^1)) ]
|
`->{4} [ YES(?,O(n^2)) ]
->{1} [ NA ]
Sub-problems:
-------------
* Path {1}: NA
------------
The usable rules for this path are:
{ round(0()) -> 0()
, round(0()) -> s(0())
, round(s(0())) -> s(0())
, round(s(s(x))) -> s(s(round(x)))}
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 {5}: 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(f) = {}, Uargs(s) = {}, Uargs(round) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(round^#) = {}, Uargs(c_4) = {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]
s(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
round(x1) = [0 0 0] x1 + [0]
[0 0 0] [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]
round^#(x1) = [0 0 0] x1 + [0]
[3 3 3] [0]
[3 3 3] [0]
c_1() = [0]
[0]
[0]
c_2() = [0]
[0]
[0]
c_3() = [0]
[0]
[0]
c_4(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {round^#(s(s(x))) -> c_4(round^#(x))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(round^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2 0] x1 + [0]
[0 0 1] [0]
[0 0 1] [2]
round^#(x1) = [2 0 2] x1 + [0]
[4 0 2] [0]
[0 0 0] [2]
c_4(x1) = [1 0 2] x1 + [3]
[2 0 0] [0]
[0 0 0] [2]
* Path {5}->{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(round) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(round^#) = {}, Uargs(c_4) = {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]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
round(x1) = [0 0 0] x1 + [0]
[0 0 0] [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]
round^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1() = [0]
[0]
[0]
c_2() = [0]
[0]
[0]
c_3() = [0]
[0]
[0]
c_4(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {round^#(0()) -> c_1()}
Weak Rules: {round^#(s(s(x))) -> c_4(round^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(round^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2 0] x1 + [0]
[0 0 0] [2]
[0 0 0] [2]
0() = [2]
[2]
[2]
round^#(x1) = [2 0 0] x1 + [0]
[0 0 2] [4]
[1 2 2] [0]
c_1() = [1]
[0]
[0]
c_4(x1) = [1 0 0] x1 + [3]
[0 0 0] [7]
[0 0 0] [6]
* Path {5}->{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(round) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(round^#) = {}, Uargs(c_4) = {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]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
round(x1) = [0 0 0] x1 + [0]
[0 0 0] [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]
round^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1() = [0]
[0]
[0]
c_2() = [0]
[0]
[0]
c_3() = [0]
[0]
[0]
c_4(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {round^#(0()) -> c_2()}
Weak Rules: {round^#(s(s(x))) -> c_4(round^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(round^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2 0] x1 + [0]
[0 0 0] [2]
[0 0 0] [2]
0() = [2]
[2]
[2]
round^#(x1) = [2 0 0] x1 + [0]
[0 0 2] [4]
[1 2 2] [0]
c_2() = [1]
[0]
[0]
c_4(x1) = [1 0 0] x1 + [3]
[0 0 0] [7]
[0 0 0] [6]
* Path {5}->{4}: 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(f) = {}, Uargs(s) = {}, Uargs(round) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(round^#) = {}, Uargs(c_4) = {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]
s(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
round(x1) = [0 0 0] x1 + [0]
[0 0 0] [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]
round^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1() = [0]
[0]
[0]
c_2() = [0]
[0]
[0]
c_3() = [0]
[0]
[0]
c_4(x1) = [1 0 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 3'
--------------------------------------
Answer: YES(?,O(n^2))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {round^#(s(0())) -> c_3()}
Weak Rules: {round^#(s(s(x))) -> c_4(round^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(round^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2 0] x1 + [2]
[0 1 2] [0]
[0 0 0] [0]
0() = [0]
[0]
[0]
round^#(x1) = [2 3 0] x1 + [0]
[0 0 0] [0]
[0 5 0] [0]
c_3() = [1]
[0]
[0]
c_4(x1) = [1 0 1] x1 + [0]
[0 0 0] [0]
[0 0 0] [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(x), x) -> c_0(f^#(s(x), round(s(x))))
, 2: round^#(0()) -> c_1()
, 3: round^#(0()) -> c_2()
, 4: round^#(s(0())) -> c_3()
, 5: round^#(s(s(x))) -> c_4(round^#(x))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{5} [ YES(?,O(n^1)) ]
|
|->{2} [ YES(?,O(n^1)) ]
|
|->{3} [ YES(?,O(n^1)) ]
|
`->{4} [ YES(?,O(n^2)) ]
->{1} [ MAYBE ]
Sub-problems:
-------------
* Path {1}: MAYBE
---------------
The usable rules for this path are:
{ round(0()) -> 0()
, round(0()) -> s(0())
, round(s(0())) -> s(0())
, round(s(s(x))) -> s(s(round(x)))}
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:
{ f^#(s(x), x) -> c_0(f^#(s(x), round(s(x))))
, round(0()) -> 0()
, round(0()) -> s(0())
, round(s(0())) -> s(0())
, round(s(s(x))) -> s(s(round(x)))}
Proof Output:
The input cannot be shown compatible
* Path {5}: 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(round) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(round^#) = {}, Uargs(c_4) = {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]
s(x1) = [1 0] x1 + [0]
[0 0] [0]
round(x1) = [0 0] x1 + [0]
[0 0] [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]
round^#(x1) = [0 0] x1 + [0]
[3 3] [0]
c_1() = [0]
[0]
c_2() = [0]
[0]
c_3() = [0]
[0]
c_4(x1) = [1 0] x1 + [0]
[0 1] [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: {round^#(s(s(x))) -> c_4(round^#(x))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(round^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [0]
[0 0] [1]
round^#(x1) = [2 2] x1 + [2]
[6 0] [0]
c_4(x1) = [1 0] x1 + [5]
[2 0] [7]
* Path {5}->{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(round) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(round^#) = {}, Uargs(c_4) = {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]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
round(x1) = [0 0] x1 + [0]
[0 0] [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]
round^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1() = [0]
[0]
c_2() = [0]
[0]
c_3() = [0]
[0]
c_4(x1) = [1 0] x1 + [0]
[0 1] [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: {round^#(0()) -> c_1()}
Weak Rules: {round^#(s(s(x))) -> c_4(round^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(round^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [1]
[0 0] [0]
0() = [0]
[2]
round^#(x1) = [2 2] x1 + [4]
[6 2] [0]
c_1() = [1]
[0]
c_4(x1) = [1 0] x1 + [3]
[2 0] [3]
* Path {5}->{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(round) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(round^#) = {}, Uargs(c_4) = {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]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
round(x1) = [0 0] x1 + [0]
[0 0] [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]
round^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1() = [0]
[0]
c_2() = [0]
[0]
c_3() = [0]
[0]
c_4(x1) = [1 0] x1 + [0]
[0 1] [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: {round^#(0()) -> c_2()}
Weak Rules: {round^#(s(s(x))) -> c_4(round^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(round^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 2] x1 + [1]
[0 0] [0]
0() = [0]
[2]
round^#(x1) = [2 2] x1 + [4]
[6 2] [0]
c_2() = [1]
[0]
c_4(x1) = [1 0] x1 + [3]
[2 0] [3]
* Path {5}->{4}: 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(f) = {}, Uargs(s) = {}, Uargs(round) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(round^#) = {}, Uargs(c_4) = {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]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
round(x1) = [0 0] x1 + [0]
[0 0] [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]
round^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_1() = [0]
[0]
c_2() = [0]
[0]
c_3() = [0]
[0]
c_4(x1) = [1 0] x1 + [0]
[0 1] [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: {round^#(s(0())) -> c_3()}
Weak Rules: {round^#(s(s(x))) -> c_4(round^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(round^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1 1] x1 + [2]
[0 1] [1]
0() = [1]
[0]
round^#(x1) = [2 1] x1 + [1]
[0 0] [7]
c_3() = [1]
[1]
c_4(x1) = [1 1] x1 + [3]
[0 0] [3]
3) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: f^#(s(x), x) -> c_0(f^#(s(x), round(s(x))))
, 2: round^#(0()) -> c_1()
, 3: round^#(0()) -> c_2()
, 4: round^#(s(0())) -> c_3()
, 5: round^#(s(s(x))) -> c_4(round^#(x))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{5} [ YES(?,O(n^1)) ]
|
|->{2} [ YES(?,O(n^1)) ]
|
|->{3} [ YES(?,O(n^1)) ]
|
`->{4} [ YES(?,O(1)) ]
->{1} [ MAYBE ]
Sub-problems:
-------------
* Path {1}: MAYBE
---------------
The usable rules for this path are:
{ round(0()) -> 0()
, round(0()) -> s(0())
, round(s(0())) -> s(0())
, round(s(s(x))) -> s(s(round(x)))}
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(x), x) -> c_0(f^#(s(x), round(s(x))))
, round(0()) -> 0()
, round(0()) -> s(0())
, round(s(0())) -> s(0())
, round(s(s(x))) -> s(s(round(x)))}
Proof Output:
The input cannot be shown compatible
* Path {5}: 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(round) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(round^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [1] x1 + [0]
round(x1) = [0] x1 + [0]
0() = [0]
f^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
round^#(x1) = [3] x1 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [1] x1 + [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: {round^#(s(s(x))) -> c_4(round^#(x))}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(round^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [2]
round^#(x1) = [2] x1 + [0]
c_4(x1) = [1] x1 + [7]
* Path {5}->{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(round) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(round^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
round(x1) = [0] x1 + [0]
0() = [0]
f^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
round^#(x1) = [0] x1 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [1] x1 + [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: {round^#(0()) -> c_1()}
Weak Rules: {round^#(s(s(x))) -> c_4(round^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(round^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [0]
0() = [2]
round^#(x1) = [2] x1 + [4]
c_1() = [1]
c_4(x1) = [1] x1 + [0]
* Path {5}->{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(round) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(round^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
round(x1) = [0] x1 + [0]
0() = [0]
f^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
round^#(x1) = [0] x1 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [1] x1 + [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: {round^#(0()) -> c_2()}
Weak Rules: {round^#(s(s(x))) -> c_4(round^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(round^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [1] x1 + [0]
0() = [2]
round^#(x1) = [2] x1 + [4]
c_2() = [1]
c_4(x1) = [1] x1 + [0]
* Path {5}->{4}: 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(s) = {}, Uargs(round) = {}, Uargs(f^#) = {},
Uargs(c_0) = {}, Uargs(round^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
f(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
round(x1) = [0] x1 + [0]
0() = [0]
f^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
round^#(x1) = [0] x1 + [0]
c_1() = [0]
c_2() = [0]
c_3() = [0]
c_4(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: {round^#(s(0())) -> c_3()}
Weak Rules: {round^#(s(s(x))) -> c_4(round^#(x))}
Proof Output:
The following argument positions are usable:
Uargs(s) = {}, Uargs(round^#) = {}, Uargs(c_4) = {1}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
s(x1) = [0] x1 + [0]
0() = [0]
round^#(x1) = [0] x1 + [1]
c_3() = [0]
c_4(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.