'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ a(l(x1)) -> l(a(x1))
, r(a(a(x1))) -> a(a(r(x1)))
, b(l(x1)) -> b(a(r(x1)))
, r(b(x1)) -> l(b(x1))}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ a^#(l(x1)) -> c_0(a^#(x1))
, r^#(a(a(x1))) -> c_1(a^#(a(r(x1))))
, b^#(l(x1)) -> c_2(b^#(a(r(x1))))
, r^#(b(x1)) -> c_3(b^#(x1))}
The usable rules are:
{ a(l(x1)) -> l(a(x1))
, r(a(a(x1))) -> a(a(r(x1)))
, r(b(x1)) -> l(b(x1))
, b(l(x1)) -> b(a(r(x1)))}
The estimated dependency graph contains the following edges:
{a^#(l(x1)) -> c_0(a^#(x1))}
==> {a^#(l(x1)) -> c_0(a^#(x1))}
{r^#(a(a(x1))) -> c_1(a^#(a(r(x1))))}
==> {a^#(l(x1)) -> c_0(a^#(x1))}
{b^#(l(x1)) -> c_2(b^#(a(r(x1))))}
==> {b^#(l(x1)) -> c_2(b^#(a(r(x1))))}
{r^#(b(x1)) -> c_3(b^#(x1))}
==> {b^#(l(x1)) -> c_2(b^#(a(r(x1))))}
We consider the following path(s):
1) { r^#(a(a(x1))) -> c_1(a^#(a(r(x1))))
, a^#(l(x1)) -> c_0(a^#(x1))}
The usable rules for this path are the following:
{ a(l(x1)) -> l(a(x1))
, r(a(a(x1))) -> a(a(r(x1)))
, r(b(x1)) -> l(b(x1))
, b(l(x1)) -> b(a(r(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:
{ a(l(x1)) -> l(a(x1))
, r(a(a(x1))) -> a(a(r(x1)))
, r(b(x1)) -> l(b(x1))
, b(l(x1)) -> b(a(r(x1)))
, r^#(a(a(x1))) -> c_1(a^#(a(r(x1))))
, a^#(l(x1)) -> c_0(a^#(x1))}
Details:
We apply the weight gap principle, strictly orienting the rules
{r(b(x1)) -> l(b(x1))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{r(b(x1)) -> l(b(x1))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
l(x1) = [1] x1 + [0]
r(x1) = [1] x1 + [1]
b(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [8]
c_0(x1) = [1] x1 + [1]
r^#(x1) = [1] x1 + [7]
c_1(x1) = [1] x1 + [1]
b^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{r^#(a(a(x1))) -> c_1(a^#(a(r(x1))))}
and weakly orienting the rules
{r(b(x1)) -> l(b(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{r^#(a(a(x1))) -> c_1(a^#(a(r(x1))))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
l(x1) = [1] x1 + [0]
r(x1) = [1] x1 + [1]
b(x1) = [1] x1 + [1]
a^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [1]
r^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [0]
b^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a^#(l(x1)) -> c_0(a^#(x1))}
and weakly orienting the rules
{ r^#(a(a(x1))) -> c_1(a^#(a(r(x1))))
, r(b(x1)) -> l(b(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a^#(l(x1)) -> c_0(a^#(x1))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
l(x1) = [1] x1 + [8]
r(x1) = [1] x1 + [8]
b(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [8]
c_0(x1) = [1] x1 + [0]
r^#(x1) = [1] x1 + [15]
c_1(x1) = [1] x1 + [0]
b^#(x1) = [0] x1 + [0]
c_2(x1) = [0] 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:
{ a(l(x1)) -> l(a(x1))
, r(a(a(x1))) -> a(a(r(x1)))
, b(l(x1)) -> b(a(r(x1)))}
Weak Rules:
{ a^#(l(x1)) -> c_0(a^#(x1))
, r^#(a(a(x1))) -> c_1(a^#(a(r(x1))))
, r(b(x1)) -> l(b(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:
{ a(l(x1)) -> l(a(x1))
, r(a(a(x1))) -> a(a(r(x1)))
, b(l(x1)) -> b(a(r(x1)))}
Weak Rules:
{ a^#(l(x1)) -> c_0(a^#(x1))
, r^#(a(a(x1))) -> c_1(a^#(a(r(x1))))
, r(b(x1)) -> l(b(x1))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ l_0(2) -> 2
, a^#_0(2) -> 5
, c_0_0(5) -> 5
, r^#_0(2) -> 7}
2) { r^#(b(x1)) -> c_3(b^#(x1))
, b^#(l(x1)) -> c_2(b^#(a(r(x1))))}
The usable rules for this path are the following:
{ a(l(x1)) -> l(a(x1))
, r(a(a(x1))) -> a(a(r(x1)))
, r(b(x1)) -> l(b(x1))
, b(l(x1)) -> b(a(r(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:
{ a(l(x1)) -> l(a(x1))
, r(a(a(x1))) -> a(a(r(x1)))
, r(b(x1)) -> l(b(x1))
, b(l(x1)) -> b(a(r(x1)))
, r^#(b(x1)) -> c_3(b^#(x1))
, b^#(l(x1)) -> c_2(b^#(a(r(x1))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{r(b(x1)) -> l(b(x1))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{r(b(x1)) -> l(b(x1))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
l(x1) = [1] x1 + [0]
r(x1) = [1] x1 + [1]
b(x1) = [1] x1 + [0]
a^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
r^#(x1) = [1] x1 + [1]
c_1(x1) = [0] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{r^#(b(x1)) -> c_3(b^#(x1))}
and weakly orienting the rules
{r(b(x1)) -> l(b(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{r^#(b(x1)) -> c_3(b^#(x1))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
l(x1) = [1] x1 + [0]
r(x1) = [1] x1 + [1]
b(x1) = [1] x1 + [1]
a^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
r^#(x1) = [1] x1 + [15]
c_1(x1) = [0] x1 + [0]
b^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [4]
c_3(x1) = [1] x1 + [7]
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:
{ a(l(x1)) -> l(a(x1))
, r(a(a(x1))) -> a(a(r(x1)))
, b(l(x1)) -> b(a(r(x1)))
, b^#(l(x1)) -> c_2(b^#(a(r(x1))))}
Weak Rules:
{ r^#(b(x1)) -> c_3(b^#(x1))
, r(b(x1)) -> l(b(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:
{ a(l(x1)) -> l(a(x1))
, r(a(a(x1))) -> a(a(r(x1)))
, b(l(x1)) -> b(a(r(x1)))
, b^#(l(x1)) -> c_2(b^#(a(r(x1))))}
Weak Rules:
{ r^#(b(x1)) -> c_3(b^#(x1))
, r(b(x1)) -> l(b(x1))}
Details:
The problem is Match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ a_1(12) -> 11
, l_0(2) -> 2
, r_1(2) -> 12
, r^#_0(2) -> 7
, b^#_0(2) -> 9
, b^#_1(11) -> 10
, c_2_1(10) -> 9}
3) {r^#(a(a(x1))) -> c_1(a^#(a(r(x1))))}
The usable rules for this path are the following:
{ a(l(x1)) -> l(a(x1))
, r(a(a(x1))) -> a(a(r(x1)))
, r(b(x1)) -> l(b(x1))
, b(l(x1)) -> b(a(r(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:
{ a(l(x1)) -> l(a(x1))
, r(a(a(x1))) -> a(a(r(x1)))
, r(b(x1)) -> l(b(x1))
, b(l(x1)) -> b(a(r(x1)))
, r^#(a(a(x1))) -> c_1(a^#(a(r(x1))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ r(b(x1)) -> l(b(x1))
, r^#(a(a(x1))) -> c_1(a^#(a(r(x1))))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ r(b(x1)) -> l(b(x1))
, r^#(a(a(x1))) -> c_1(a^#(a(r(x1))))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
l(x1) = [1] x1 + [0]
r(x1) = [1] x1 + [1]
b(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
r^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [0]
b^#(x1) = [0] x1 + [0]
c_2(x1) = [0] 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:
{ a(l(x1)) -> l(a(x1))
, r(a(a(x1))) -> a(a(r(x1)))
, b(l(x1)) -> b(a(r(x1)))}
Weak Rules:
{ r(b(x1)) -> l(b(x1))
, r^#(a(a(x1))) -> c_1(a^#(a(r(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:
{ a(l(x1)) -> l(a(x1))
, r(a(a(x1))) -> a(a(r(x1)))
, b(l(x1)) -> b(a(r(x1)))}
Weak Rules:
{ r(b(x1)) -> l(b(x1))
, r^#(a(a(x1))) -> c_1(a^#(a(r(x1))))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ l_0(2) -> 2
, a^#_0(2) -> 5
, r^#_0(2) -> 7}
4) {r^#(b(x1)) -> c_3(b^#(x1))}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
a(x1) = [0] x1 + [0]
l(x1) = [0] x1 + [0]
r(x1) = [0] x1 + [0]
b(x1) = [0] x1 + [0]
a^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
r^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
b^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [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: {r^#(b(x1)) -> c_3(b^#(x1))}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{r^#(b(x1)) -> c_3(b^#(x1))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{r^#(b(x1)) -> c_3(b^#(x1))}
Details:
Interpretation Functions:
a(x1) = [0] x1 + [0]
l(x1) = [0] x1 + [0]
r(x1) = [0] x1 + [0]
b(x1) = [1] x1 + [0]
a^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
r^#(x1) = [1] x1 + [1]
c_1(x1) = [0] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [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: {r^#(b(x1)) -> c_3(b^#(x1))}
Details:
The given problem does not contain any strict rules