'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ b(a(b(x1))) -> a(b(a(x1)))
, b(b(a(x1))) -> b(b(b(x1)))
, c(a(x1)) -> a(b(c(x1)))
, c(b(x1)) -> b(a(c(x1)))}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ b^#(a(b(x1))) -> c_0(b^#(a(x1)))
, b^#(b(a(x1))) -> c_1(b^#(b(b(x1))))
, c^#(a(x1)) -> c_2(b^#(c(x1)))
, c^#(b(x1)) -> c_3(b^#(a(c(x1))))}
The usable rules are:
{ b(a(b(x1))) -> a(b(a(x1)))
, b(b(a(x1))) -> b(b(b(x1)))
, c(a(x1)) -> a(b(c(x1)))
, c(b(x1)) -> b(a(c(x1)))}
The estimated dependency graph contains the following edges:
{b^#(a(b(x1))) -> c_0(b^#(a(x1)))}
==> {b^#(a(b(x1))) -> c_0(b^#(a(x1)))}
{b^#(b(a(x1))) -> c_1(b^#(b(b(x1))))}
==> {b^#(b(a(x1))) -> c_1(b^#(b(b(x1))))}
{b^#(b(a(x1))) -> c_1(b^#(b(b(x1))))}
==> {b^#(a(b(x1))) -> c_0(b^#(a(x1)))}
{c^#(a(x1)) -> c_2(b^#(c(x1)))}
==> {b^#(b(a(x1))) -> c_1(b^#(b(b(x1))))}
{c^#(a(x1)) -> c_2(b^#(c(x1)))}
==> {b^#(a(b(x1))) -> c_0(b^#(a(x1)))}
{c^#(b(x1)) -> c_3(b^#(a(c(x1))))}
==> {b^#(a(b(x1))) -> c_0(b^#(a(x1)))}
We consider the following path(s):
1) { c^#(a(x1)) -> c_2(b^#(c(x1)))
, b^#(b(a(x1))) -> c_1(b^#(b(b(x1))))
, b^#(a(b(x1))) -> c_0(b^#(a(x1)))}
The usable rules for this path are the following:
{ b(a(b(x1))) -> a(b(a(x1)))
, b(b(a(x1))) -> b(b(b(x1)))
, c(a(x1)) -> a(b(c(x1)))
, c(b(x1)) -> b(a(c(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:
{ b(a(b(x1))) -> a(b(a(x1)))
, b(b(a(x1))) -> b(b(b(x1)))
, c(a(x1)) -> a(b(c(x1)))
, c(b(x1)) -> b(a(c(x1)))
, b^#(b(a(x1))) -> c_1(b^#(b(b(x1))))
, c^#(a(x1)) -> c_2(b^#(c(x1)))
, b^#(a(b(x1))) -> c_0(b^#(a(x1)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{c^#(a(x1)) -> c_2(b^#(c(x1)))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c^#(a(x1)) -> c_2(b^#(c(x1)))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
b^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [9]
c_2(x1) = [1] x1 + [6]
c_3(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ b(b(a(x1))) -> b(b(b(x1)))
, b^#(b(a(x1))) -> c_1(b^#(b(b(x1))))}
and weakly orienting the rules
{c^#(a(x1)) -> c_2(b^#(c(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ b(b(a(x1))) -> b(b(b(x1)))
, b^#(b(a(x1))) -> c_1(b^#(b(b(x1))))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [8]
c(x1) = [1] x1 + [8]
b^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [1]
c_3(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b^#(a(b(x1))) -> c_0(b^#(a(x1)))}
and weakly orienting the rules
{ b(b(a(x1))) -> b(b(b(x1)))
, b^#(b(a(x1))) -> c_1(b^#(b(b(x1))))
, c^#(a(x1)) -> c_2(b^#(c(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b^#(a(b(x1))) -> c_0(b^#(a(x1)))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [2]
a(x1) = [1] x1 + [4]
c(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [13]
c_2(x1) = [1] x1 + [1]
c_3(x1) = [0] x1 + [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:
{ b(a(b(x1))) -> a(b(a(x1)))
, c(a(x1)) -> a(b(c(x1)))
, c(b(x1)) -> b(a(c(x1)))}
Weak Rules:
{ b^#(a(b(x1))) -> c_0(b^#(a(x1)))
, b(b(a(x1))) -> b(b(b(x1)))
, b^#(b(a(x1))) -> c_1(b^#(b(b(x1))))
, c^#(a(x1)) -> c_2(b^#(c(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:
{ b(a(b(x1))) -> a(b(a(x1)))
, c(a(x1)) -> a(b(c(x1)))
, c(b(x1)) -> b(a(c(x1)))}
Weak Rules:
{ b^#(a(b(x1))) -> c_0(b^#(a(x1)))
, b(b(a(x1))) -> b(b(b(x1)))
, b^#(b(a(x1))) -> c_1(b^#(b(b(x1))))
, c^#(a(x1)) -> c_2(b^#(c(x1)))}
Details:
The problem is Match-bounded by 2.
The enriched problem is compatible with the following automaton:
{ b_1(11) -> 10
, b_2(16) -> 15
, a_0(2) -> 2
, a_1(10) -> 9
, a_1(10) -> 11
, a_1(11) -> 13
, a_2(11) -> 16
, a_2(15) -> 10
, c_0(2) -> 9
, c_1(2) -> 11
, b^#_0(2) -> 4
, b^#_0(9) -> 8
, b^#_1(11) -> 14
, b^#_1(13) -> 12
, c_0_1(12) -> 8
, c_0_1(12) -> 14
, c^#_0(2) -> 7
, c_2_0(8) -> 7
, c_2_1(14) -> 7}
2) { c^#(a(x1)) -> c_2(b^#(c(x1)))
, b^#(b(a(x1))) -> c_1(b^#(b(b(x1))))}
The usable rules for this path are the following:
{ b(a(b(x1))) -> a(b(a(x1)))
, b(b(a(x1))) -> b(b(b(x1)))
, c(a(x1)) -> a(b(c(x1)))
, c(b(x1)) -> b(a(c(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:
{ b(a(b(x1))) -> a(b(a(x1)))
, b(b(a(x1))) -> b(b(b(x1)))
, c(a(x1)) -> a(b(c(x1)))
, c(b(x1)) -> b(a(c(x1)))
, c^#(a(x1)) -> c_2(b^#(c(x1)))
, b^#(b(a(x1))) -> c_1(b^#(b(b(x1))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{c^#(a(x1)) -> c_2(b^#(c(x1)))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c^#(a(x1)) -> c_2(b^#(c(x1)))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
b^#(x1) = [1] x1 + [4]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [1] x1 + [6]
c^#(x1) = [1] x1 + [9]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ b(b(a(x1))) -> b(b(b(x1)))
, b^#(b(a(x1))) -> c_1(b^#(b(b(x1))))}
and weakly orienting the rules
{c^#(a(x1)) -> c_2(b^#(c(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ b(b(a(x1))) -> b(b(b(x1)))
, b^#(b(a(x1))) -> c_1(b^#(b(b(x1))))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [4]
c(x1) = [1] x1 + [2]
b^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [14]
c_2(x1) = [1] x1 + [1]
c_3(x1) = [0] x1 + [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:
{ b(a(b(x1))) -> a(b(a(x1)))
, c(a(x1)) -> a(b(c(x1)))
, c(b(x1)) -> b(a(c(x1)))}
Weak Rules:
{ b(b(a(x1))) -> b(b(b(x1)))
, b^#(b(a(x1))) -> c_1(b^#(b(b(x1))))
, c^#(a(x1)) -> c_2(b^#(c(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:
{ b(a(b(x1))) -> a(b(a(x1)))
, c(a(x1)) -> a(b(c(x1)))
, c(b(x1)) -> b(a(c(x1)))}
Weak Rules:
{ b(b(a(x1))) -> b(b(b(x1)))
, b^#(b(a(x1))) -> c_1(b^#(b(b(x1))))
, c^#(a(x1)) -> c_2(b^#(c(x1)))}
Details:
The problem is Match-bounded by 2.
The enriched problem is compatible with the following automaton:
{ b_1(11) -> 10
, b_2(14) -> 13
, a_0(2) -> 2
, a_1(10) -> 9
, a_1(10) -> 11
, a_2(11) -> 14
, a_2(13) -> 10
, c_0(2) -> 9
, c_1(2) -> 11
, b^#_0(2) -> 4
, b^#_0(9) -> 8
, b^#_1(11) -> 12
, c^#_0(2) -> 7
, c_2_0(8) -> 7
, c_2_1(12) -> 7}
3) { c^#(b(x1)) -> c_3(b^#(a(c(x1))))
, b^#(a(b(x1))) -> c_0(b^#(a(x1)))}
The usable rules for this path are the following:
{ c(a(x1)) -> a(b(c(x1)))
, c(b(x1)) -> b(a(c(x1)))
, b(a(b(x1))) -> a(b(a(x1)))
, b(b(a(x1))) -> b(b(b(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:
{ c(a(x1)) -> a(b(c(x1)))
, c(b(x1)) -> b(a(c(x1)))
, b(a(b(x1))) -> a(b(a(x1)))
, b(b(a(x1))) -> b(b(b(x1)))
, c^#(b(x1)) -> c_3(b^#(a(c(x1))))
, b^#(a(b(x1))) -> c_0(b^#(a(x1)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{c^#(b(x1)) -> c_3(b^#(a(c(x1))))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c^#(b(x1)) -> c_3(b^#(a(c(x1))))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
b^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [9]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [4]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ b(a(b(x1))) -> a(b(a(x1)))
, b^#(a(b(x1))) -> c_0(b^#(a(x1)))}
and weakly orienting the rules
{c^#(b(x1)) -> c_3(b^#(a(c(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ b(a(b(x1))) -> a(b(a(x1)))
, b^#(a(b(x1))) -> c_0(b^#(a(x1)))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [8]
a(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
b^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [1]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [1]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [1]
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:
{ c(a(x1)) -> a(b(c(x1)))
, c(b(x1)) -> b(a(c(x1)))
, b(b(a(x1))) -> b(b(b(x1)))}
Weak Rules:
{ b(a(b(x1))) -> a(b(a(x1)))
, b^#(a(b(x1))) -> c_0(b^#(a(x1)))
, c^#(b(x1)) -> c_3(b^#(a(c(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:
{ c(a(x1)) -> a(b(c(x1)))
, c(b(x1)) -> b(a(c(x1)))
, b(b(a(x1))) -> b(b(b(x1)))}
Weak Rules:
{ b(a(b(x1))) -> a(b(a(x1)))
, b^#(a(b(x1))) -> c_0(b^#(a(x1)))
, c^#(b(x1)) -> c_3(b^#(a(c(x1))))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ a_0(2) -> 2
, b^#_0(2) -> 4
, c^#_0(2) -> 7}
4) { c^#(a(x1)) -> c_2(b^#(c(x1)))
, b^#(a(b(x1))) -> c_0(b^#(a(x1)))}
The usable rules for this path are the following:
{ c(a(x1)) -> a(b(c(x1)))
, c(b(x1)) -> b(a(c(x1)))
, b(a(b(x1))) -> a(b(a(x1)))
, b(b(a(x1))) -> b(b(b(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:
{ c(a(x1)) -> a(b(c(x1)))
, c(b(x1)) -> b(a(c(x1)))
, b(a(b(x1))) -> a(b(a(x1)))
, b(b(a(x1))) -> b(b(b(x1)))
, c^#(a(x1)) -> c_2(b^#(c(x1)))
, b^#(a(b(x1))) -> c_0(b^#(a(x1)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{c^#(a(x1)) -> c_2(b^#(c(x1)))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c^#(a(x1)) -> c_2(b^#(c(x1)))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
b^#(x1) = [1] x1 + [4]
c_0(x1) = [1] x1 + [6]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [9]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ b(a(b(x1))) -> a(b(a(x1)))
, b^#(a(b(x1))) -> c_0(b^#(a(x1)))}
and weakly orienting the rules
{c^#(a(x1)) -> c_2(b^#(c(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ b(a(b(x1))) -> a(b(a(x1)))
, b^#(a(b(x1))) -> c_0(b^#(a(x1)))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [8]
a(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
b^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [4]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [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:
{ c(a(x1)) -> a(b(c(x1)))
, c(b(x1)) -> b(a(c(x1)))
, b(b(a(x1))) -> b(b(b(x1)))}
Weak Rules:
{ b(a(b(x1))) -> a(b(a(x1)))
, b^#(a(b(x1))) -> c_0(b^#(a(x1)))
, c^#(a(x1)) -> c_2(b^#(c(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:
{ c(a(x1)) -> a(b(c(x1)))
, c(b(x1)) -> b(a(c(x1)))
, b(b(a(x1))) -> b(b(b(x1)))}
Weak Rules:
{ b(a(b(x1))) -> a(b(a(x1)))
, b^#(a(b(x1))) -> c_0(b^#(a(x1)))
, c^#(a(x1)) -> c_2(b^#(c(x1)))}
Details:
The problem is Match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ b_1(11) -> 10
, b_1(13) -> 15
, a_0(2) -> 2
, a_1(10) -> 9
, a_1(10) -> 11
, a_1(11) -> 13
, a_1(15) -> 10
, c_0(2) -> 9
, c_1(2) -> 11
, b^#_0(2) -> 4
, b^#_0(9) -> 8
, b^#_1(11) -> 14
, b^#_1(13) -> 12
, c_0_1(12) -> 8
, c_0_1(12) -> 14
, c^#_0(2) -> 7
, c_2_0(8) -> 7
, c_2_1(14) -> 7}
5) {c^#(b(x1)) -> c_3(b^#(a(c(x1))))}
The usable rules for this path are the following:
{ c(a(x1)) -> a(b(c(x1)))
, c(b(x1)) -> b(a(c(x1)))
, b(a(b(x1))) -> a(b(a(x1)))
, b(b(a(x1))) -> b(b(b(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:
{ c(a(x1)) -> a(b(c(x1)))
, c(b(x1)) -> b(a(c(x1)))
, b(a(b(x1))) -> a(b(a(x1)))
, b(b(a(x1))) -> b(b(b(x1)))
, c^#(b(x1)) -> c_3(b^#(a(c(x1))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{c^#(b(x1)) -> c_3(b^#(a(c(x1))))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c^#(b(x1)) -> c_3(b^#(a(c(x1))))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
b^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [9]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b(a(b(x1))) -> a(b(a(x1)))}
and weakly orienting the rules
{c^#(b(x1)) -> c_3(b^#(a(c(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b(a(b(x1))) -> a(b(a(x1)))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [8]
a(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
b^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [9]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [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:
{ c(a(x1)) -> a(b(c(x1)))
, c(b(x1)) -> b(a(c(x1)))
, b(b(a(x1))) -> b(b(b(x1)))}
Weak Rules:
{ b(a(b(x1))) -> a(b(a(x1)))
, c^#(b(x1)) -> c_3(b^#(a(c(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:
{ c(a(x1)) -> a(b(c(x1)))
, c(b(x1)) -> b(a(c(x1)))
, b(b(a(x1))) -> b(b(b(x1)))}
Weak Rules:
{ b(a(b(x1))) -> a(b(a(x1)))
, c^#(b(x1)) -> c_3(b^#(a(c(x1))))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ a_0(2) -> 2
, b^#_0(2) -> 4
, c^#_0(2) -> 7}
6) {c^#(a(x1)) -> c_2(b^#(c(x1)))}
The usable rules for this path are the following:
{ c(a(x1)) -> a(b(c(x1)))
, c(b(x1)) -> b(a(c(x1)))
, b(a(b(x1))) -> a(b(a(x1)))
, b(b(a(x1))) -> b(b(b(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:
{ c(a(x1)) -> a(b(c(x1)))
, c(b(x1)) -> b(a(c(x1)))
, b(a(b(x1))) -> a(b(a(x1)))
, b(b(a(x1))) -> b(b(b(x1)))
, c^#(a(x1)) -> c_2(b^#(c(x1)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{c^#(a(x1)) -> c_2(b^#(c(x1)))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c^#(a(x1)) -> c_2(b^#(c(x1)))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
b^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [9]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b(a(b(x1))) -> a(b(a(x1)))}
and weakly orienting the rules
{c^#(a(x1)) -> c_2(b^#(c(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b(a(b(x1))) -> a(b(a(x1)))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [8]
a(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
b^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [9]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [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:
{ c(a(x1)) -> a(b(c(x1)))
, c(b(x1)) -> b(a(c(x1)))
, b(b(a(x1))) -> b(b(b(x1)))}
Weak Rules:
{ b(a(b(x1))) -> a(b(a(x1)))
, c^#(a(x1)) -> c_2(b^#(c(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:
{ c(a(x1)) -> a(b(c(x1)))
, c(b(x1)) -> b(a(c(x1)))
, b(b(a(x1))) -> b(b(b(x1)))}
Weak Rules:
{ b(a(b(x1))) -> a(b(a(x1)))
, c^#(a(x1)) -> c_2(b^#(c(x1)))}
Details:
The problem is Match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ b_1(11) -> 10
, b_1(14) -> 13
, a_0(2) -> 2
, a_1(10) -> 9
, a_1(10) -> 11
, a_1(11) -> 14
, a_1(13) -> 10
, c_0(2) -> 9
, c_1(2) -> 11
, b^#_0(2) -> 4
, b^#_0(9) -> 8
, b^#_1(11) -> 12
, c^#_0(2) -> 7
, c_2_0(8) -> 7
, c_2_1(12) -> 7}