'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ 1(2(1(x1))) -> 2(0(2(x1)))
, 0(2(1(x1))) -> 1(0(2(x1)))
, L(2(1(x1))) -> L(1(0(2(x1))))
, 1(2(0(x1))) -> 2(0(1(x1)))
, 1(2(R(x1))) -> 2(0(1(R(x1))))
, 0(2(0(x1))) -> 1(0(1(x1)))
, L(2(0(x1))) -> L(1(0(1(x1))))
, 0(2(R(x1))) -> 1(0(1(R(x1))))}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ 1^#(2(1(x1))) -> c_0(0^#(2(x1)))
, 0^#(2(1(x1))) -> c_1(1^#(0(2(x1))))
, L^#(2(1(x1))) -> c_2(L^#(1(0(2(x1)))))
, 1^#(2(0(x1))) -> c_3(0^#(1(x1)))
, 1^#(2(R(x1))) -> c_4(0^#(1(R(x1))))
, 0^#(2(0(x1))) -> c_5(1^#(0(1(x1))))
, L^#(2(0(x1))) -> c_6(L^#(1(0(1(x1)))))
, 0^#(2(R(x1))) -> c_7(1^#(0(1(R(x1)))))}
The usable rules are:
{ 1(2(1(x1))) -> 2(0(2(x1)))
, 0(2(1(x1))) -> 1(0(2(x1)))
, 1(2(0(x1))) -> 2(0(1(x1)))
, 1(2(R(x1))) -> 2(0(1(R(x1))))
, 0(2(0(x1))) -> 1(0(1(x1)))
, 0(2(R(x1))) -> 1(0(1(R(x1))))}
The estimated dependency graph contains the following edges:
{1^#(2(1(x1))) -> c_0(0^#(2(x1)))}
==> {0^#(2(0(x1))) -> c_5(1^#(0(1(x1))))}
{1^#(2(1(x1))) -> c_0(0^#(2(x1)))}
==> {0^#(2(1(x1))) -> c_1(1^#(0(2(x1))))}
{1^#(2(1(x1))) -> c_0(0^#(2(x1)))}
==> {0^#(2(R(x1))) -> c_7(1^#(0(1(R(x1)))))}
{0^#(2(1(x1))) -> c_1(1^#(0(2(x1))))}
==> {1^#(2(0(x1))) -> c_3(0^#(1(x1)))}
{0^#(2(1(x1))) -> c_1(1^#(0(2(x1))))}
==> {1^#(2(1(x1))) -> c_0(0^#(2(x1)))}
{L^#(2(1(x1))) -> c_2(L^#(1(0(2(x1)))))}
==> {L^#(2(0(x1))) -> c_6(L^#(1(0(1(x1)))))}
{L^#(2(1(x1))) -> c_2(L^#(1(0(2(x1)))))}
==> {L^#(2(1(x1))) -> c_2(L^#(1(0(2(x1)))))}
{1^#(2(0(x1))) -> c_3(0^#(1(x1)))}
==> {0^#(2(0(x1))) -> c_5(1^#(0(1(x1))))}
{1^#(2(0(x1))) -> c_3(0^#(1(x1)))}
==> {0^#(2(1(x1))) -> c_1(1^#(0(2(x1))))}
{0^#(2(0(x1))) -> c_5(1^#(0(1(x1))))}
==> {1^#(2(0(x1))) -> c_3(0^#(1(x1)))}
{0^#(2(0(x1))) -> c_5(1^#(0(1(x1))))}
==> {1^#(2(1(x1))) -> c_0(0^#(2(x1)))}
{L^#(2(0(x1))) -> c_6(L^#(1(0(1(x1)))))}
==> {L^#(2(0(x1))) -> c_6(L^#(1(0(1(x1)))))}
{L^#(2(0(x1))) -> c_6(L^#(1(0(1(x1)))))}
==> {L^#(2(1(x1))) -> c_2(L^#(1(0(2(x1)))))}
We consider the following path(s):
1) { 1^#(2(1(x1))) -> c_0(0^#(2(x1)))
, 0^#(2(0(x1))) -> c_5(1^#(0(1(x1))))
, 1^#(2(0(x1))) -> c_3(0^#(1(x1)))
, 0^#(2(1(x1))) -> c_1(1^#(0(2(x1))))}
The usable rules for this path are the following:
{ 1(2(1(x1))) -> 2(0(2(x1)))
, 0(2(1(x1))) -> 1(0(2(x1)))
, 1(2(0(x1))) -> 2(0(1(x1)))
, 1(2(R(x1))) -> 2(0(1(R(x1))))
, 0(2(0(x1))) -> 1(0(1(x1)))
, 0(2(R(x1))) -> 1(0(1(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:
{ 1(2(1(x1))) -> 2(0(2(x1)))
, 0(2(1(x1))) -> 1(0(2(x1)))
, 1(2(0(x1))) -> 2(0(1(x1)))
, 1(2(R(x1))) -> 2(0(1(R(x1))))
, 0(2(0(x1))) -> 1(0(1(x1)))
, 0(2(R(x1))) -> 1(0(1(R(x1))))
, 1^#(2(1(x1))) -> c_0(0^#(2(x1)))
, 0^#(2(0(x1))) -> c_5(1^#(0(1(x1))))
, 1^#(2(0(x1))) -> c_3(0^#(1(x1)))
, 0^#(2(1(x1))) -> c_1(1^#(0(2(x1))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{1^#(2(1(x1))) -> c_0(0^#(2(x1)))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{1^#(2(1(x1))) -> c_0(0^#(2(x1)))}
Details:
Interpretation Functions:
1(x1) = [1] x1 + [0]
2(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [0]
L(x1) = [0] x1 + [0]
R(x1) = [1] x1 + [4]
1^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
L^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{1^#(2(0(x1))) -> c_3(0^#(1(x1)))}
and weakly orienting the rules
{1^#(2(1(x1))) -> c_0(0^#(2(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{1^#(2(0(x1))) -> c_3(0^#(1(x1)))}
Details:
Interpretation Functions:
1(x1) = [1] x1 + [0]
2(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [0]
L(x1) = [0] x1 + [0]
R(x1) = [1] x1 + [0]
1^#(x1) = [1] x1 + [8]
c_0(x1) = [1] x1 + [3]
0^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [1]
L^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [1]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{0(2(0(x1))) -> 1(0(1(x1)))}
and weakly orienting the rules
{ 1^#(2(0(x1))) -> c_3(0^#(1(x1)))
, 1^#(2(1(x1))) -> c_0(0^#(2(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{0(2(0(x1))) -> 1(0(1(x1)))}
Details:
Interpretation Functions:
1(x1) = [1] x1 + [0]
2(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [8]
L(x1) = [0] x1 + [0]
R(x1) = [1] x1 + [0]
1^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [1]
L^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [1]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ 0(2(R(x1))) -> 1(0(1(R(x1))))
, 0^#(2(0(x1))) -> c_5(1^#(0(1(x1))))}
and weakly orienting the rules
{ 0(2(0(x1))) -> 1(0(1(x1)))
, 1^#(2(0(x1))) -> c_3(0^#(1(x1)))
, 1^#(2(1(x1))) -> c_0(0^#(2(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 0(2(R(x1))) -> 1(0(1(R(x1))))
, 0^#(2(0(x1))) -> c_5(1^#(0(1(x1))))}
Details:
Interpretation Functions:
1(x1) = [1] x1 + [0]
2(x1) = [1] x1 + [8]
0(x1) = [1] x1 + [0]
L(x1) = [0] x1 + [0]
R(x1) = [1] x1 + [11]
1^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
L^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [2]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [1]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{0^#(2(1(x1))) -> c_1(1^#(0(2(x1))))}
and weakly orienting the rules
{ 0(2(R(x1))) -> 1(0(1(R(x1))))
, 0^#(2(0(x1))) -> c_5(1^#(0(1(x1))))
, 0(2(0(x1))) -> 1(0(1(x1)))
, 1^#(2(0(x1))) -> c_3(0^#(1(x1)))
, 1^#(2(1(x1))) -> c_0(0^#(2(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{0^#(2(1(x1))) -> c_1(1^#(0(2(x1))))}
Details:
Interpretation Functions:
1(x1) = [1] x1 + [2]
2(x1) = [1] x1 + [8]
0(x1) = [1] x1 + [0]
L(x1) = [0] x1 + [0]
R(x1) = [1] x1 + [0]
1^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [2]
c_1(x1) = [1] x1 + [1]
L^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(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:
{ 1(2(1(x1))) -> 2(0(2(x1)))
, 0(2(1(x1))) -> 1(0(2(x1)))
, 1(2(0(x1))) -> 2(0(1(x1)))
, 1(2(R(x1))) -> 2(0(1(R(x1))))}
Weak Rules:
{ 0^#(2(1(x1))) -> c_1(1^#(0(2(x1))))
, 0(2(R(x1))) -> 1(0(1(R(x1))))
, 0^#(2(0(x1))) -> c_5(1^#(0(1(x1))))
, 0(2(0(x1))) -> 1(0(1(x1)))
, 1^#(2(0(x1))) -> c_3(0^#(1(x1)))
, 1^#(2(1(x1))) -> c_0(0^#(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:
{ 1(2(1(x1))) -> 2(0(2(x1)))
, 0(2(1(x1))) -> 1(0(2(x1)))
, 1(2(0(x1))) -> 2(0(1(x1)))
, 1(2(R(x1))) -> 2(0(1(R(x1))))}
Weak Rules:
{ 0^#(2(1(x1))) -> c_1(1^#(0(2(x1))))
, 0(2(R(x1))) -> 1(0(1(R(x1))))
, 0^#(2(0(x1))) -> c_5(1^#(0(1(x1))))
, 0(2(0(x1))) -> 1(0(1(x1)))
, 1^#(2(0(x1))) -> c_3(0^#(1(x1)))
, 1^#(2(1(x1))) -> c_0(0^#(2(x1)))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ 2_0(2) -> 2
, R_0(2) -> 2
, 1^#_0(2) -> 1
, 0^#_0(2) -> 1}
2) { 1^#(2(1(x1))) -> c_0(0^#(2(x1)))
, 0^#(2(0(x1))) -> c_5(1^#(0(1(x1))))
, 1^#(2(0(x1))) -> c_3(0^#(1(x1)))
, 0^#(2(1(x1))) -> c_1(1^#(0(2(x1))))
, 0^#(2(R(x1))) -> c_7(1^#(0(1(R(x1)))))}
The usable rules for this path are the following:
{ 1(2(1(x1))) -> 2(0(2(x1)))
, 0(2(1(x1))) -> 1(0(2(x1)))
, 1(2(0(x1))) -> 2(0(1(x1)))
, 1(2(R(x1))) -> 2(0(1(R(x1))))
, 0(2(0(x1))) -> 1(0(1(x1)))
, 0(2(R(x1))) -> 1(0(1(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:
{ 1(2(1(x1))) -> 2(0(2(x1)))
, 0(2(1(x1))) -> 1(0(2(x1)))
, 1(2(0(x1))) -> 2(0(1(x1)))
, 1(2(R(x1))) -> 2(0(1(R(x1))))
, 0(2(0(x1))) -> 1(0(1(x1)))
, 0(2(R(x1))) -> 1(0(1(R(x1))))
, 1^#(2(1(x1))) -> c_0(0^#(2(x1)))
, 0^#(2(0(x1))) -> c_5(1^#(0(1(x1))))
, 1^#(2(0(x1))) -> c_3(0^#(1(x1)))
, 0^#(2(1(x1))) -> c_1(1^#(0(2(x1))))
, 0^#(2(R(x1))) -> c_7(1^#(0(1(R(x1)))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{1^#(2(0(x1))) -> c_3(0^#(1(x1)))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{1^#(2(0(x1))) -> c_3(0^#(1(x1)))}
Details:
Interpretation Functions:
1(x1) = [1] x1 + [0]
2(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [0]
L(x1) = [0] x1 + [0]
R(x1) = [1] x1 + [0]
1^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [1]
0^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
L^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{1^#(2(1(x1))) -> c_0(0^#(2(x1)))}
and weakly orienting the rules
{1^#(2(0(x1))) -> c_3(0^#(1(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{1^#(2(1(x1))) -> c_0(0^#(2(x1)))}
Details:
Interpretation Functions:
1(x1) = [1] x1 + [0]
2(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [0]
L(x1) = [0] x1 + [0]
R(x1) = [1] x1 + [0]
1^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
L^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ 0(2(0(x1))) -> 1(0(1(x1)))
, 0(2(R(x1))) -> 1(0(1(R(x1))))}
and weakly orienting the rules
{ 1^#(2(1(x1))) -> c_0(0^#(2(x1)))
, 1^#(2(0(x1))) -> c_3(0^#(1(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 0(2(0(x1))) -> 1(0(1(x1)))
, 0(2(R(x1))) -> 1(0(1(R(x1))))}
Details:
Interpretation Functions:
1(x1) = [1] x1 + [2]
2(x1) = [1] x1 + [7]
0(x1) = [1] x1 + [15]
L(x1) = [0] x1 + [0]
R(x1) = [1] x1 + [11]
1^#(x1) = [1] x1 + [15]
c_0(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [14]
c_1(x1) = [1] x1 + [8]
L^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [4]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [1] x1 + [8]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{0^#(2(0(x1))) -> c_5(1^#(0(1(x1))))}
and weakly orienting the rules
{ 0(2(0(x1))) -> 1(0(1(x1)))
, 0(2(R(x1))) -> 1(0(1(R(x1))))
, 1^#(2(1(x1))) -> c_0(0^#(2(x1)))
, 1^#(2(0(x1))) -> c_3(0^#(1(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{0^#(2(0(x1))) -> c_5(1^#(0(1(x1))))}
Details:
Interpretation Functions:
1(x1) = [1] x1 + [4]
2(x1) = [1] x1 + [15]
0(x1) = [1] x1 + [14]
L(x1) = [0] x1 + [0]
R(x1) = [1] x1 + [9]
1^#(x1) = [1] x1 + [9]
c_0(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [13]
c_1(x1) = [1] x1 + [9]
L^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [1] x1 + [3]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ 0^#(2(1(x1))) -> c_1(1^#(0(2(x1))))
, 0^#(2(R(x1))) -> c_7(1^#(0(1(R(x1)))))}
and weakly orienting the rules
{ 0^#(2(0(x1))) -> c_5(1^#(0(1(x1))))
, 0(2(0(x1))) -> 1(0(1(x1)))
, 0(2(R(x1))) -> 1(0(1(R(x1))))
, 1^#(2(1(x1))) -> c_0(0^#(2(x1)))
, 1^#(2(0(x1))) -> c_3(0^#(1(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 0^#(2(1(x1))) -> c_1(1^#(0(2(x1))))
, 0^#(2(R(x1))) -> c_7(1^#(0(1(R(x1)))))}
Details:
Interpretation Functions:
1(x1) = [1] x1 + [1]
2(x1) = [1] x1 + [9]
0(x1) = [1] x1 + [0]
L(x1) = [0] x1 + [0]
R(x1) = [1] x1 + [8]
1^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [0]
L^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(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:
{ 1(2(1(x1))) -> 2(0(2(x1)))
, 0(2(1(x1))) -> 1(0(2(x1)))
, 1(2(0(x1))) -> 2(0(1(x1)))
, 1(2(R(x1))) -> 2(0(1(R(x1))))}
Weak Rules:
{ 0^#(2(1(x1))) -> c_1(1^#(0(2(x1))))
, 0^#(2(R(x1))) -> c_7(1^#(0(1(R(x1)))))
, 0^#(2(0(x1))) -> c_5(1^#(0(1(x1))))
, 0(2(0(x1))) -> 1(0(1(x1)))
, 0(2(R(x1))) -> 1(0(1(R(x1))))
, 1^#(2(1(x1))) -> c_0(0^#(2(x1)))
, 1^#(2(0(x1))) -> c_3(0^#(1(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:
{ 1(2(1(x1))) -> 2(0(2(x1)))
, 0(2(1(x1))) -> 1(0(2(x1)))
, 1(2(0(x1))) -> 2(0(1(x1)))
, 1(2(R(x1))) -> 2(0(1(R(x1))))}
Weak Rules:
{ 0^#(2(1(x1))) -> c_1(1^#(0(2(x1))))
, 0^#(2(R(x1))) -> c_7(1^#(0(1(R(x1)))))
, 0^#(2(0(x1))) -> c_5(1^#(0(1(x1))))
, 0(2(0(x1))) -> 1(0(1(x1)))
, 0(2(R(x1))) -> 1(0(1(R(x1))))
, 1^#(2(1(x1))) -> c_0(0^#(2(x1)))
, 1^#(2(0(x1))) -> c_3(0^#(1(x1)))}
Details:
The problem is Match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ 1_0(2) -> 5
, 1_1(7) -> 11
, 1_1(8) -> 7
, 1_1(10) -> 4
, 2_0(2) -> 2
, 2_1(6) -> 5
, 0_0(5) -> 4
, 0_1(7) -> 6
, 0_1(11) -> 10
, R_0(2) -> 2
, R_1(2) -> 8
, 1^#_0(2) -> 1
, 1^#_0(4) -> 3
, 1^#_1(6) -> 9
, 0^#_0(2) -> 1
, c_7_0(3) -> 1
, c_7_1(9) -> 1}
3) { L^#(2(1(x1))) -> c_2(L^#(1(0(2(x1)))))
, L^#(2(0(x1))) -> c_6(L^#(1(0(1(x1)))))}
The usable rules for this path are the following:
{ 1(2(1(x1))) -> 2(0(2(x1)))
, 0(2(1(x1))) -> 1(0(2(x1)))
, 1(2(0(x1))) -> 2(0(1(x1)))
, 1(2(R(x1))) -> 2(0(1(R(x1))))
, 0(2(0(x1))) -> 1(0(1(x1)))
, 0(2(R(x1))) -> 1(0(1(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:
{ 1(2(1(x1))) -> 2(0(2(x1)))
, 0(2(1(x1))) -> 1(0(2(x1)))
, 1(2(0(x1))) -> 2(0(1(x1)))
, 1(2(R(x1))) -> 2(0(1(R(x1))))
, 0(2(0(x1))) -> 1(0(1(x1)))
, 0(2(R(x1))) -> 1(0(1(R(x1))))
, L^#(2(1(x1))) -> c_2(L^#(1(0(2(x1)))))
, L^#(2(0(x1))) -> c_6(L^#(1(0(1(x1)))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{0(2(0(x1))) -> 1(0(1(x1)))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{0(2(0(x1))) -> 1(0(1(x1)))}
Details:
Interpretation Functions:
1(x1) = [1] x1 + [0]
2(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [8]
L(x1) = [0] x1 + [0]
R(x1) = [1] x1 + [0]
1^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
L^#(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [12]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [8]
c_7(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ 0(2(R(x1))) -> 1(0(1(R(x1))))
, L^#(2(0(x1))) -> c_6(L^#(1(0(1(x1)))))}
and weakly orienting the rules
{0(2(0(x1))) -> 1(0(1(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 0(2(R(x1))) -> 1(0(1(R(x1))))
, L^#(2(0(x1))) -> c_6(L^#(1(0(1(x1)))))}
Details:
Interpretation Functions:
1(x1) = [1] x1 + [0]
2(x1) = [1] x1 + [8]
0(x1) = [1] x1 + [0]
L(x1) = [0] x1 + [0]
R(x1) = [1] x1 + [2]
1^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
L^#(x1) = [1] x1 + [4]
c_2(x1) = [1] x1 + [1]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [0]
c_7(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:
{ 1(2(1(x1))) -> 2(0(2(x1)))
, 0(2(1(x1))) -> 1(0(2(x1)))
, 1(2(0(x1))) -> 2(0(1(x1)))
, 1(2(R(x1))) -> 2(0(1(R(x1))))
, L^#(2(1(x1))) -> c_2(L^#(1(0(2(x1)))))}
Weak Rules:
{ 0(2(R(x1))) -> 1(0(1(R(x1))))
, L^#(2(0(x1))) -> c_6(L^#(1(0(1(x1)))))
, 0(2(0(x1))) -> 1(0(1(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:
{ 1(2(1(x1))) -> 2(0(2(x1)))
, 0(2(1(x1))) -> 1(0(2(x1)))
, 1(2(0(x1))) -> 2(0(1(x1)))
, 1(2(R(x1))) -> 2(0(1(R(x1))))
, L^#(2(1(x1))) -> c_2(L^#(1(0(2(x1)))))}
Weak Rules:
{ 0(2(R(x1))) -> 1(0(1(R(x1))))
, L^#(2(0(x1))) -> c_6(L^#(1(0(1(x1)))))
, 0(2(0(x1))) -> 1(0(1(x1)))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ 2_0(2) -> 2
, 2_0(5) -> 2
, R_0(2) -> 5
, R_0(5) -> 5
, L^#_0(2) -> 10
, L^#_0(5) -> 10}
4) {1^#(2(R(x1))) -> c_4(0^#(1(R(x1))))}
The usable rules for this path are the following:
{ 1(2(1(x1))) -> 2(0(2(x1)))
, 1(2(0(x1))) -> 2(0(1(x1)))
, 1(2(R(x1))) -> 2(0(1(R(x1))))
, 0(2(1(x1))) -> 1(0(2(x1)))
, 0(2(0(x1))) -> 1(0(1(x1)))
, 0(2(R(x1))) -> 1(0(1(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:
{ 1(2(1(x1))) -> 2(0(2(x1)))
, 1(2(0(x1))) -> 2(0(1(x1)))
, 1(2(R(x1))) -> 2(0(1(R(x1))))
, 0(2(1(x1))) -> 1(0(2(x1)))
, 0(2(0(x1))) -> 1(0(1(x1)))
, 0(2(R(x1))) -> 1(0(1(R(x1))))
, 1^#(2(R(x1))) -> c_4(0^#(1(R(x1))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{1^#(2(R(x1))) -> c_4(0^#(1(R(x1))))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{1^#(2(R(x1))) -> c_4(0^#(1(R(x1))))}
Details:
Interpretation Functions:
1(x1) = [1] x1 + [0]
2(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [0]
L(x1) = [0] x1 + [0]
R(x1) = [1] x1 + [0]
1^#(x1) = [1] x1 + [8]
c_0(x1) = [0] x1 + [0]
0^#(x1) = [1] x1 + [0]
c_1(x1) = [0] x1 + [0]
L^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [1]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{0(2(0(x1))) -> 1(0(1(x1)))}
and weakly orienting the rules
{1^#(2(R(x1))) -> c_4(0^#(1(R(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{0(2(0(x1))) -> 1(0(1(x1)))}
Details:
Interpretation Functions:
1(x1) = [1] x1 + [0]
2(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [4]
L(x1) = [0] x1 + [0]
R(x1) = [1] x1 + [0]
1^#(x1) = [1] x1 + [12]
c_0(x1) = [0] x1 + [0]
0^#(x1) = [1] x1 + [6]
c_1(x1) = [0] x1 + [0]
L^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [5]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{0(2(R(x1))) -> 1(0(1(R(x1))))}
and weakly orienting the rules
{ 0(2(0(x1))) -> 1(0(1(x1)))
, 1^#(2(R(x1))) -> c_4(0^#(1(R(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{0(2(R(x1))) -> 1(0(1(R(x1))))}
Details:
Interpretation Functions:
1(x1) = [1] x1 + [0]
2(x1) = [1] x1 + [8]
0(x1) = [1] x1 + [0]
L(x1) = [0] x1 + [0]
R(x1) = [1] x1 + [0]
1^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
0^#(x1) = [1] x1 + [1]
c_1(x1) = [0] x1 + [0]
L^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [8]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(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:
{ 1(2(1(x1))) -> 2(0(2(x1)))
, 1(2(0(x1))) -> 2(0(1(x1)))
, 1(2(R(x1))) -> 2(0(1(R(x1))))
, 0(2(1(x1))) -> 1(0(2(x1)))}
Weak Rules:
{ 0(2(R(x1))) -> 1(0(1(R(x1))))
, 0(2(0(x1))) -> 1(0(1(x1)))
, 1^#(2(R(x1))) -> c_4(0^#(1(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:
{ 1(2(1(x1))) -> 2(0(2(x1)))
, 1(2(0(x1))) -> 2(0(1(x1)))
, 1(2(R(x1))) -> 2(0(1(R(x1))))
, 0(2(1(x1))) -> 1(0(2(x1)))}
Weak Rules:
{ 0(2(R(x1))) -> 1(0(1(R(x1))))
, 0(2(0(x1))) -> 1(0(1(x1)))
, 1^#(2(R(x1))) -> c_4(0^#(1(R(x1))))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ 1_0(5) -> 10
, 2_0(2) -> 2
, 2_0(5) -> 2
, R_0(2) -> 5
, R_0(5) -> 5
, 1^#_0(2) -> 6
, 1^#_0(5) -> 6
, 0^#_0(2) -> 8
, 0^#_0(5) -> 8
, 0^#_0(10) -> 9
, c_4_0(9) -> 6}