'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ 0(x1) -> 1(x1)
, 4(5(4(5(x1)))) -> 4(4(5(5(x1))))
, 5(5(5(5(5(5(4(4(4(4(4(4(x1)))))))))))) -> 2(x1)}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ 0^#(x1) -> c_0()
, 4^#(5(4(5(x1)))) -> c_1(4^#(4(5(5(x1)))))
, 5^#(5(5(5(5(5(4(4(4(4(4(4(x1)))))))))))) -> c_2()}
The usable rules are:
{ 4(5(4(5(x1)))) -> 4(4(5(5(x1))))
, 5(5(5(5(5(5(4(4(4(4(4(4(x1)))))))))))) -> 2(x1)}
The dependency graph contains no edges.
We consider the following path(s):
1) {4^#(5(4(5(x1)))) -> c_1(4^#(4(5(5(x1)))))}
The usable rules for this path are the following:
{ 4(5(4(5(x1)))) -> 4(4(5(5(x1))))
, 5(5(5(5(5(5(4(4(4(4(4(4(x1)))))))))))) -> 2(x1)}
We have applied the subprocessor on the union of usable rules and weak (innermost) dependency pairs.
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ 4(5(4(5(x1)))) -> 4(4(5(5(x1))))
, 5(5(5(5(5(5(4(4(4(4(4(4(x1)))))))))))) -> 2(x1)
, 4^#(5(4(5(x1)))) -> c_1(4^#(4(5(5(x1)))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{5(5(5(5(5(5(4(4(4(4(4(4(x1)))))))))))) -> 2(x1)}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{5(5(5(5(5(5(4(4(4(4(4(4(x1)))))))))))) -> 2(x1)}
Details:
Interpretation Functions:
0(x1) = [0] x1 + [0]
1(x1) = [0] x1 + [0]
4(x1) = [1] x1 + [8]
5(x1) = [1] x1 + [0]
2(x1) = [1] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_0() = [0]
4^#(x1) = [1] x1 + [8]
c_1(x1) = [1] x1 + [1]
5^#(x1) = [0] x1 + [0]
c_2() = [0]
Finally we apply the subprocessor
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ 4(5(4(5(x1)))) -> 4(4(5(5(x1))))
, 4^#(5(4(5(x1)))) -> c_1(4^#(4(5(5(x1)))))}
Weak Rules: {5(5(5(5(5(5(4(4(4(4(4(4(x1)))))))))))) -> 2(x1)}
Details:
The problem was solved by processor 'Bounds with default enrichment':
'Bounds with default enrichment'
--------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ 4(5(4(5(x1)))) -> 4(4(5(5(x1))))
, 4^#(5(4(5(x1)))) -> c_1(4^#(4(5(5(x1)))))}
Weak Rules: {5(5(5(5(5(5(4(4(4(4(4(4(x1)))))))))))) -> 2(x1)}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ 2_0(5) -> 5
, 4^#_0(5) -> 8}
2) {5^#(5(5(5(5(5(4(4(4(4(4(4(x1)))))))))))) -> c_2()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
0(x1) = [0] x1 + [0]
1(x1) = [0] x1 + [0]
4(x1) = [0] x1 + [0]
5(x1) = [0] x1 + [0]
2(x1) = [0] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_0() = [0]
4^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
5^#(x1) = [0] x1 + [0]
c_2() = [0]
We have applied the subprocessor on the resulting DP-problem:
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {5^#(5(5(5(5(5(4(4(4(4(4(4(x1)))))))))))) -> c_2()}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{5^#(5(5(5(5(5(4(4(4(4(4(4(x1)))))))))))) -> c_2()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{5^#(5(5(5(5(5(4(4(4(4(4(4(x1)))))))))))) -> c_2()}
Details:
Interpretation Functions:
0(x1) = [0] x1 + [0]
1(x1) = [0] x1 + [0]
4(x1) = [1] x1 + [0]
5(x1) = [1] x1 + [0]
2(x1) = [0] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_0() = [0]
4^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
5^#(x1) = [1] x1 + [1]
c_2() = [0]
Finally we apply the subprocessor
'Empty TRS'
-----------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {}
Weak Rules: {5^#(5(5(5(5(5(4(4(4(4(4(4(x1)))))))))))) -> c_2()}
Details:
The given problem does not contain any strict rules
3) {0^#(x1) -> c_0()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
0(x1) = [0] x1 + [0]
1(x1) = [0] x1 + [0]
4(x1) = [0] x1 + [0]
5(x1) = [0] x1 + [0]
2(x1) = [0] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_0() = [0]
4^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
5^#(x1) = [0] x1 + [0]
c_2() = [0]
We have applied the subprocessor on the resulting DP-problem:
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {0^#(x1) -> c_0()}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{0^#(x1) -> c_0()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{0^#(x1) -> c_0()}
Details:
Interpretation Functions:
0(x1) = [0] x1 + [0]
1(x1) = [0] x1 + [0]
4(x1) = [0] x1 + [0]
5(x1) = [0] x1 + [0]
2(x1) = [0] x1 + [0]
0^#(x1) = [1] x1 + [4]
c_0() = [0]
4^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
5^#(x1) = [0] x1 + [0]
c_2() = [0]
Finally we apply the subprocessor
'Empty TRS'
-----------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {}
Weak Rules: {0^#(x1) -> c_0()}
Details:
The given problem does not contain any strict rules