'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ 1(q0(1(x1))) -> 0(1(q1(x1)))
, 1(q0(0(x1))) -> 0(0(q1(x1)))
, 1(q1(1(x1))) -> 1(1(q1(x1)))
, 1(q1(0(x1))) -> 1(0(q1(x1)))
, 0(q1(x1)) -> q2(1(x1))
, 1(q2(x1)) -> q2(1(x1))
, 0(q2(x1)) -> 0(q0(x1))}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ 1^#(q0(1(x1))) -> c_0(0^#(1(q1(x1))))
, 1^#(q0(0(x1))) -> c_1(0^#(0(q1(x1))))
, 1^#(q1(1(x1))) -> c_2(1^#(1(q1(x1))))
, 1^#(q1(0(x1))) -> c_3(1^#(0(q1(x1))))
, 0^#(q1(x1)) -> c_4(1^#(x1))
, 1^#(q2(x1)) -> c_5(1^#(x1))
, 0^#(q2(x1)) -> c_6(0^#(q0(x1)))}
The usable rules are:
{ 1(q0(1(x1))) -> 0(1(q1(x1)))
, 1(q0(0(x1))) -> 0(0(q1(x1)))
, 1(q1(1(x1))) -> 1(1(q1(x1)))
, 1(q1(0(x1))) -> 1(0(q1(x1)))
, 0(q1(x1)) -> q2(1(x1))
, 1(q2(x1)) -> q2(1(x1))
, 0(q2(x1)) -> 0(q0(x1))}
The estimated dependency graph contains the following edges:
{1^#(q0(1(x1))) -> c_0(0^#(1(q1(x1))))}
==> {0^#(q2(x1)) -> c_6(0^#(q0(x1)))}
{1^#(q0(0(x1))) -> c_1(0^#(0(q1(x1))))}
==> {0^#(q2(x1)) -> c_6(0^#(q0(x1)))}
{1^#(q1(1(x1))) -> c_2(1^#(1(q1(x1))))}
==> {1^#(q2(x1)) -> c_5(1^#(x1))}
{1^#(q1(0(x1))) -> c_3(1^#(0(q1(x1))))}
==> {1^#(q2(x1)) -> c_5(1^#(x1))}
{0^#(q1(x1)) -> c_4(1^#(x1))}
==> {1^#(q2(x1)) -> c_5(1^#(x1))}
{0^#(q1(x1)) -> c_4(1^#(x1))}
==> {1^#(q1(0(x1))) -> c_3(1^#(0(q1(x1))))}
{0^#(q1(x1)) -> c_4(1^#(x1))}
==> {1^#(q1(1(x1))) -> c_2(1^#(1(q1(x1))))}
{0^#(q1(x1)) -> c_4(1^#(x1))}
==> {1^#(q0(0(x1))) -> c_1(0^#(0(q1(x1))))}
{0^#(q1(x1)) -> c_4(1^#(x1))}
==> {1^#(q0(1(x1))) -> c_0(0^#(1(q1(x1))))}
{1^#(q2(x1)) -> c_5(1^#(x1))}
==> {1^#(q2(x1)) -> c_5(1^#(x1))}
{1^#(q2(x1)) -> c_5(1^#(x1))}
==> {1^#(q1(0(x1))) -> c_3(1^#(0(q1(x1))))}
{1^#(q2(x1)) -> c_5(1^#(x1))}
==> {1^#(q1(1(x1))) -> c_2(1^#(1(q1(x1))))}
{1^#(q2(x1)) -> c_5(1^#(x1))}
==> {1^#(q0(0(x1))) -> c_1(0^#(0(q1(x1))))}
{1^#(q2(x1)) -> c_5(1^#(x1))}
==> {1^#(q0(1(x1))) -> c_0(0^#(1(q1(x1))))}
We consider the following path(s):
1) { 0^#(q1(x1)) -> c_4(1^#(x1))
, 1^#(q1(1(x1))) -> c_2(1^#(1(q1(x1))))
, 1^#(q2(x1)) -> c_5(1^#(x1))
, 1^#(q1(0(x1))) -> c_3(1^#(0(q1(x1))))
, 1^#(q0(1(x1))) -> c_0(0^#(1(q1(x1))))}
The usable rules for this path are the following:
{ 1(q0(1(x1))) -> 0(1(q1(x1)))
, 1(q0(0(x1))) -> 0(0(q1(x1)))
, 1(q1(1(x1))) -> 1(1(q1(x1)))
, 1(q1(0(x1))) -> 1(0(q1(x1)))
, 0(q1(x1)) -> q2(1(x1))
, 1(q2(x1)) -> q2(1(x1))
, 0(q2(x1)) -> 0(q0(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(q0(1(x1))) -> 0(1(q1(x1)))
, 1(q0(0(x1))) -> 0(0(q1(x1)))
, 1(q1(1(x1))) -> 1(1(q1(x1)))
, 1(q1(0(x1))) -> 1(0(q1(x1)))
, 0(q1(x1)) -> q2(1(x1))
, 1(q2(x1)) -> q2(1(x1))
, 0(q2(x1)) -> 0(q0(x1))
, 1^#(q1(1(x1))) -> c_2(1^#(1(q1(x1))))
, 1^#(q2(x1)) -> c_5(1^#(x1))
, 1^#(q1(0(x1))) -> c_3(1^#(0(q1(x1))))
, 0^#(q1(x1)) -> c_4(1^#(x1))
, 1^#(q0(1(x1))) -> c_0(0^#(1(q1(x1))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{0(q2(x1)) -> 0(q0(x1))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{0(q2(x1)) -> 0(q0(x1))}
Details:
Interpretation Functions:
1(x1) = [1] x1 + [0]
q0(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [1]
q1(x1) = [1] x1 + [0]
q2(x1) = [1] x1 + [1]
1^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [1]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [1]
c_5(x1) = [1] x1 + [1]
c_6(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ 1^#(q2(x1)) -> c_5(1^#(x1))
, 0^#(q1(x1)) -> c_4(1^#(x1))}
and weakly orienting the rules
{0(q2(x1)) -> 0(q0(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 1^#(q2(x1)) -> c_5(1^#(x1))
, 0^#(q1(x1)) -> c_4(1^#(x1))}
Details:
Interpretation Functions:
1(x1) = [1] x1 + [0]
q0(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [1]
q1(x1) = [1] x1 + [0]
q2(x1) = [1] x1 + [1]
1^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [3]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [1]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{0(q1(x1)) -> q2(1(x1))}
and weakly orienting the rules
{ 1^#(q2(x1)) -> c_5(1^#(x1))
, 0^#(q1(x1)) -> c_4(1^#(x1))
, 0(q2(x1)) -> 0(q0(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{0(q1(x1)) -> q2(1(x1))}
Details:
Interpretation Functions:
1(x1) = [1] x1 + [0]
q0(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [1]
q1(x1) = [1] x1 + [0]
q2(x1) = [1] x1 + [0]
1^#(x1) = [1] x1 + [15]
c_0(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [15]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [1] x1 + [2]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{1^#(q0(1(x1))) -> c_0(0^#(1(q1(x1))))}
and weakly orienting the rules
{ 0(q1(x1)) -> q2(1(x1))
, 1^#(q2(x1)) -> c_5(1^#(x1))
, 0^#(q1(x1)) -> c_4(1^#(x1))
, 0(q2(x1)) -> 0(q0(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{1^#(q0(1(x1))) -> c_0(0^#(1(q1(x1))))}
Details:
Interpretation Functions:
1(x1) = [1] x1 + [0]
q0(x1) = [1] x1 + [13]
0(x1) = [1] x1 + [10]
q1(x1) = [1] x1 + [4]
q2(x1) = [1] x1 + [13]
1^#(x1) = [1] x1 + [3]
c_0(x1) = [1] x1 + [1]
0^#(x1) = [1] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [1] x1 + [12]
c_3(x1) = [1] x1 + [6]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [1] x1 + [5]
c_6(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(q0(1(x1))) -> 0(1(q1(x1)))
, 1(q0(0(x1))) -> 0(0(q1(x1)))
, 1(q1(1(x1))) -> 1(1(q1(x1)))
, 1(q1(0(x1))) -> 1(0(q1(x1)))
, 1(q2(x1)) -> q2(1(x1))
, 1^#(q1(1(x1))) -> c_2(1^#(1(q1(x1))))
, 1^#(q1(0(x1))) -> c_3(1^#(0(q1(x1))))}
Weak Rules:
{ 1^#(q0(1(x1))) -> c_0(0^#(1(q1(x1))))
, 0(q1(x1)) -> q2(1(x1))
, 1^#(q2(x1)) -> c_5(1^#(x1))
, 0^#(q1(x1)) -> c_4(1^#(x1))
, 0(q2(x1)) -> 0(q0(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(q0(1(x1))) -> 0(1(q1(x1)))
, 1(q0(0(x1))) -> 0(0(q1(x1)))
, 1(q1(1(x1))) -> 1(1(q1(x1)))
, 1(q1(0(x1))) -> 1(0(q1(x1)))
, 1(q2(x1)) -> q2(1(x1))
, 1^#(q1(1(x1))) -> c_2(1^#(1(q1(x1))))
, 1^#(q1(0(x1))) -> c_3(1^#(0(q1(x1))))}
Weak Rules:
{ 1^#(q0(1(x1))) -> c_0(0^#(1(q1(x1))))
, 0(q1(x1)) -> q2(1(x1))
, 1^#(q2(x1)) -> c_5(1^#(x1))
, 0^#(q1(x1)) -> c_4(1^#(x1))
, 0(q2(x1)) -> 0(q0(x1))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ q0_0(2) -> 2
, q0_0(4) -> 2
, q0_0(5) -> 2
, q1_0(2) -> 4
, q1_0(4) -> 4
, q1_0(5) -> 4
, q2_0(2) -> 5
, q2_0(4) -> 5
, q2_0(5) -> 5
, 1^#_0(2) -> 6
, 1^#_0(4) -> 6
, 1^#_0(5) -> 6
, 0^#_0(2) -> 8
, 0^#_0(4) -> 8
, 0^#_0(5) -> 8
, c_4_0(6) -> 8
, c_5_0(6) -> 6}
2) { 0^#(q1(x1)) -> c_4(1^#(x1))
, 1^#(q1(1(x1))) -> c_2(1^#(1(q1(x1))))
, 1^#(q2(x1)) -> c_5(1^#(x1))
, 1^#(q1(0(x1))) -> c_3(1^#(0(q1(x1))))
, 1^#(q0(0(x1))) -> c_1(0^#(0(q1(x1))))
, 0^#(q2(x1)) -> c_6(0^#(q0(x1)))}
The usable rules for this path are the following:
{ 1(q0(1(x1))) -> 0(1(q1(x1)))
, 1(q0(0(x1))) -> 0(0(q1(x1)))
, 1(q1(1(x1))) -> 1(1(q1(x1)))
, 1(q1(0(x1))) -> 1(0(q1(x1)))
, 0(q1(x1)) -> q2(1(x1))
, 1(q2(x1)) -> q2(1(x1))
, 0(q2(x1)) -> 0(q0(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(q0(1(x1))) -> 0(1(q1(x1)))
, 1(q0(0(x1))) -> 0(0(q1(x1)))
, 1(q1(1(x1))) -> 1(1(q1(x1)))
, 1(q1(0(x1))) -> 1(0(q1(x1)))
, 0(q1(x1)) -> q2(1(x1))
, 1(q2(x1)) -> q2(1(x1))
, 0(q2(x1)) -> 0(q0(x1))
, 1^#(q0(0(x1))) -> c_1(0^#(0(q1(x1))))
, 1^#(q1(1(x1))) -> c_2(1^#(1(q1(x1))))
, 1^#(q2(x1)) -> c_5(1^#(x1))
, 1^#(q1(0(x1))) -> c_3(1^#(0(q1(x1))))
, 0^#(q1(x1)) -> c_4(1^#(x1))
, 0^#(q2(x1)) -> c_6(0^#(q0(x1)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{0(q2(x1)) -> 0(q0(x1))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{0(q2(x1)) -> 0(q0(x1))}
Details:
Interpretation Functions:
1(x1) = [1] x1 + [0]
q0(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [1]
q1(x1) = [1] x1 + [0]
q2(x1) = [1] x1 + [1]
1^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
0^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [4]
c_4(x1) = [1] x1 + [1]
c_5(x1) = [1] x1 + [1]
c_6(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{0^#(q2(x1)) -> c_6(0^#(q0(x1)))}
and weakly orienting the rules
{0(q2(x1)) -> 0(q0(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{0^#(q2(x1)) -> c_6(0^#(q0(x1)))}
Details:
Interpretation Functions:
1(x1) = [1] x1 + [0]
q0(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [1]
q1(x1) = [1] x1 + [0]
q2(x1) = [1] x1 + [1]
1^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
0^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [1]
c_5(x1) = [1] x1 + [1]
c_6(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{1^#(q0(0(x1))) -> c_1(0^#(0(q1(x1))))}
and weakly orienting the rules
{ 0^#(q2(x1)) -> c_6(0^#(q0(x1)))
, 0(q2(x1)) -> 0(q0(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{1^#(q0(0(x1))) -> c_1(0^#(0(q1(x1))))}
Details:
Interpretation Functions:
1(x1) = [1] x1 + [0]
q0(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [1]
q1(x1) = [1] x1 + [0]
q2(x1) = [1] x1 + [1]
1^#(x1) = [1] x1 + [4]
c_0(x1) = [0] x1 + [0]
0^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [1]
c_5(x1) = [1] x1 + [1]
c_6(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{1^#(q2(x1)) -> c_5(1^#(x1))}
and weakly orienting the rules
{ 1^#(q0(0(x1))) -> c_1(0^#(0(q1(x1))))
, 0^#(q2(x1)) -> c_6(0^#(q0(x1)))
, 0(q2(x1)) -> 0(q0(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{1^#(q2(x1)) -> c_5(1^#(x1))}
Details:
Interpretation Functions:
1(x1) = [1] x1 + [0]
q0(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [1]
q1(x1) = [1] x1 + [0]
q2(x1) = [1] x1 + [1]
1^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
0^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [1]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [1] x1 + [0]
c_6(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{0(q1(x1)) -> q2(1(x1))}
and weakly orienting the rules
{ 1^#(q2(x1)) -> c_5(1^#(x1))
, 1^#(q0(0(x1))) -> c_1(0^#(0(q1(x1))))
, 0^#(q2(x1)) -> c_6(0^#(q0(x1)))
, 0(q2(x1)) -> 0(q0(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{0(q1(x1)) -> q2(1(x1))}
Details:
Interpretation Functions:
1(x1) = [1] x1 + [8]
q0(x1) = [1] x1 + [4]
0(x1) = [1] x1 + [9]
q1(x1) = [1] x1 + [12]
q2(x1) = [1] x1 + [8]
1^#(x1) = [1] x1 + [12]
c_0(x1) = [0] x1 + [0]
0^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [8]
c_3(x1) = [1] x1 + [1]
c_4(x1) = [1] x1 + [1]
c_5(x1) = [1] x1 + [4]
c_6(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{0^#(q1(x1)) -> c_4(1^#(x1))}
and weakly orienting the rules
{ 0(q1(x1)) -> q2(1(x1))
, 1^#(q2(x1)) -> c_5(1^#(x1))
, 1^#(q0(0(x1))) -> c_1(0^#(0(q1(x1))))
, 0^#(q2(x1)) -> c_6(0^#(q0(x1)))
, 0(q2(x1)) -> 0(q0(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{0^#(q1(x1)) -> c_4(1^#(x1))}
Details:
Interpretation Functions:
1(x1) = [1] x1 + [0]
q0(x1) = [1] x1 + [2]
0(x1) = [1] x1 + [0]
q1(x1) = [1] x1 + [8]
q2(x1) = [1] x1 + [4]
1^#(x1) = [1] x1 + [6]
c_0(x1) = [0] x1 + [0]
0^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [8]
c_4(x1) = [1] x1 + [1]
c_5(x1) = [1] x1 + [1]
c_6(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:
{ 1(q0(1(x1))) -> 0(1(q1(x1)))
, 1(q0(0(x1))) -> 0(0(q1(x1)))
, 1(q1(1(x1))) -> 1(1(q1(x1)))
, 1(q1(0(x1))) -> 1(0(q1(x1)))
, 1(q2(x1)) -> q2(1(x1))
, 1^#(q1(1(x1))) -> c_2(1^#(1(q1(x1))))
, 1^#(q1(0(x1))) -> c_3(1^#(0(q1(x1))))}
Weak Rules:
{ 0^#(q1(x1)) -> c_4(1^#(x1))
, 0(q1(x1)) -> q2(1(x1))
, 1^#(q2(x1)) -> c_5(1^#(x1))
, 1^#(q0(0(x1))) -> c_1(0^#(0(q1(x1))))
, 0^#(q2(x1)) -> c_6(0^#(q0(x1)))
, 0(q2(x1)) -> 0(q0(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(q0(1(x1))) -> 0(1(q1(x1)))
, 1(q0(0(x1))) -> 0(0(q1(x1)))
, 1(q1(1(x1))) -> 1(1(q1(x1)))
, 1(q1(0(x1))) -> 1(0(q1(x1)))
, 1(q2(x1)) -> q2(1(x1))
, 1^#(q1(1(x1))) -> c_2(1^#(1(q1(x1))))
, 1^#(q1(0(x1))) -> c_3(1^#(0(q1(x1))))}
Weak Rules:
{ 0^#(q1(x1)) -> c_4(1^#(x1))
, 0(q1(x1)) -> q2(1(x1))
, 1^#(q2(x1)) -> c_5(1^#(x1))
, 1^#(q0(0(x1))) -> c_1(0^#(0(q1(x1))))
, 0^#(q2(x1)) -> c_6(0^#(q0(x1)))
, 0(q2(x1)) -> 0(q0(x1))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ q0_0(2) -> 2
, q0_0(4) -> 2
, q0_0(5) -> 2
, q1_0(2) -> 4
, q1_0(4) -> 4
, q1_0(5) -> 4
, q2_0(2) -> 5
, q2_0(4) -> 5
, q2_0(5) -> 5
, 1^#_0(2) -> 6
, 1^#_0(4) -> 6
, 1^#_0(5) -> 6
, 0^#_0(2) -> 8
, 0^#_0(4) -> 8
, 0^#_0(5) -> 8
, c_4_0(6) -> 8
, c_5_0(6) -> 6
, c_6_0(8) -> 8}
3) { 0^#(q1(x1)) -> c_4(1^#(x1))
, 1^#(q1(1(x1))) -> c_2(1^#(1(q1(x1))))
, 1^#(q2(x1)) -> c_5(1^#(x1))
, 1^#(q1(0(x1))) -> c_3(1^#(0(q1(x1))))
, 1^#(q0(0(x1))) -> c_1(0^#(0(q1(x1))))}
The usable rules for this path are the following:
{ 1(q0(1(x1))) -> 0(1(q1(x1)))
, 1(q0(0(x1))) -> 0(0(q1(x1)))
, 1(q1(1(x1))) -> 1(1(q1(x1)))
, 1(q1(0(x1))) -> 1(0(q1(x1)))
, 0(q1(x1)) -> q2(1(x1))
, 1(q2(x1)) -> q2(1(x1))
, 0(q2(x1)) -> 0(q0(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(q0(1(x1))) -> 0(1(q1(x1)))
, 1(q0(0(x1))) -> 0(0(q1(x1)))
, 1(q1(1(x1))) -> 1(1(q1(x1)))
, 1(q1(0(x1))) -> 1(0(q1(x1)))
, 0(q1(x1)) -> q2(1(x1))
, 1(q2(x1)) -> q2(1(x1))
, 0(q2(x1)) -> 0(q0(x1))
, 1^#(q1(1(x1))) -> c_2(1^#(1(q1(x1))))
, 1^#(q2(x1)) -> c_5(1^#(x1))
, 1^#(q1(0(x1))) -> c_3(1^#(0(q1(x1))))
, 0^#(q1(x1)) -> c_4(1^#(x1))
, 1^#(q0(0(x1))) -> c_1(0^#(0(q1(x1))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{0(q2(x1)) -> 0(q0(x1))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{0(q2(x1)) -> 0(q0(x1))}
Details:
Interpretation Functions:
1(x1) = [1] x1 + [0]
q0(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [1]
q1(x1) = [1] x1 + [0]
q2(x1) = [1] x1 + [1]
1^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
0^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [1]
c_5(x1) = [1] x1 + [1]
c_6(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ 0(q1(x1)) -> q2(1(x1))
, 1^#(q0(0(x1))) -> c_1(0^#(0(q1(x1))))}
and weakly orienting the rules
{0(q2(x1)) -> 0(q0(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 0(q1(x1)) -> q2(1(x1))
, 1^#(q0(0(x1))) -> c_1(0^#(0(q1(x1))))}
Details:
Interpretation Functions:
1(x1) = [1] x1 + [0]
q0(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [1]
q1(x1) = [1] x1 + [0]
q2(x1) = [1] x1 + [0]
1^#(x1) = [1] x1 + [9]
c_0(x1) = [0] x1 + [0]
0^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [1]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{1^#(q2(x1)) -> c_5(1^#(x1))}
and weakly orienting the rules
{ 0(q1(x1)) -> q2(1(x1))
, 1^#(q0(0(x1))) -> c_1(0^#(0(q1(x1))))
, 0(q2(x1)) -> 0(q0(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{1^#(q2(x1)) -> c_5(1^#(x1))}
Details:
Interpretation Functions:
1(x1) = [1] x1 + [0]
q0(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [1]
q1(x1) = [1] x1 + [0]
q2(x1) = [1] x1 + [1]
1^#(x1) = [1] x1 + [15]
c_0(x1) = [0] x1 + [0]
0^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [2]
c_3(x1) = [1] x1 + [1]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{0^#(q1(x1)) -> c_4(1^#(x1))}
and weakly orienting the rules
{ 1^#(q2(x1)) -> c_5(1^#(x1))
, 0(q1(x1)) -> q2(1(x1))
, 1^#(q0(0(x1))) -> c_1(0^#(0(q1(x1))))
, 0(q2(x1)) -> 0(q0(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{0^#(q1(x1)) -> c_4(1^#(x1))}
Details:
Interpretation Functions:
1(x1) = [1] x1 + [0]
q0(x1) = [1] x1 + [1]
0(x1) = [1] x1 + [1]
q1(x1) = [1] x1 + [0]
q2(x1) = [1] x1 + [1]
1^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
0^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [2]
c_3(x1) = [1] x1 + [10]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [1] x1 + [1]
c_6(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(q0(1(x1))) -> 0(1(q1(x1)))
, 1(q0(0(x1))) -> 0(0(q1(x1)))
, 1(q1(1(x1))) -> 1(1(q1(x1)))
, 1(q1(0(x1))) -> 1(0(q1(x1)))
, 1(q2(x1)) -> q2(1(x1))
, 1^#(q1(1(x1))) -> c_2(1^#(1(q1(x1))))
, 1^#(q1(0(x1))) -> c_3(1^#(0(q1(x1))))}
Weak Rules:
{ 0^#(q1(x1)) -> c_4(1^#(x1))
, 1^#(q2(x1)) -> c_5(1^#(x1))
, 0(q1(x1)) -> q2(1(x1))
, 1^#(q0(0(x1))) -> c_1(0^#(0(q1(x1))))
, 0(q2(x1)) -> 0(q0(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(q0(1(x1))) -> 0(1(q1(x1)))
, 1(q0(0(x1))) -> 0(0(q1(x1)))
, 1(q1(1(x1))) -> 1(1(q1(x1)))
, 1(q1(0(x1))) -> 1(0(q1(x1)))
, 1(q2(x1)) -> q2(1(x1))
, 1^#(q1(1(x1))) -> c_2(1^#(1(q1(x1))))
, 1^#(q1(0(x1))) -> c_3(1^#(0(q1(x1))))}
Weak Rules:
{ 0^#(q1(x1)) -> c_4(1^#(x1))
, 1^#(q2(x1)) -> c_5(1^#(x1))
, 0(q1(x1)) -> q2(1(x1))
, 1^#(q0(0(x1))) -> c_1(0^#(0(q1(x1))))
, 0(q2(x1)) -> 0(q0(x1))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ q0_0(2) -> 2
, q1_0(2) -> 2
, q2_0(2) -> 2
, 1^#_0(2) -> 1
, 0^#_0(2) -> 1
, c_4_0(1) -> 1
, c_5_0(1) -> 1}
4) { 0^#(q1(x1)) -> c_4(1^#(x1))
, 1^#(q1(1(x1))) -> c_2(1^#(1(q1(x1))))
, 1^#(q2(x1)) -> c_5(1^#(x1))
, 1^#(q1(0(x1))) -> c_3(1^#(0(q1(x1))))}
The usable rules for this path are the following:
{ 1(q0(1(x1))) -> 0(1(q1(x1)))
, 1(q0(0(x1))) -> 0(0(q1(x1)))
, 1(q1(1(x1))) -> 1(1(q1(x1)))
, 1(q1(0(x1))) -> 1(0(q1(x1)))
, 0(q1(x1)) -> q2(1(x1))
, 1(q2(x1)) -> q2(1(x1))
, 0(q2(x1)) -> 0(q0(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(q0(1(x1))) -> 0(1(q1(x1)))
, 1(q0(0(x1))) -> 0(0(q1(x1)))
, 1(q1(1(x1))) -> 1(1(q1(x1)))
, 1(q1(0(x1))) -> 1(0(q1(x1)))
, 0(q1(x1)) -> q2(1(x1))
, 1(q2(x1)) -> q2(1(x1))
, 0(q2(x1)) -> 0(q0(x1))
, 0^#(q1(x1)) -> c_4(1^#(x1))
, 1^#(q1(1(x1))) -> c_2(1^#(1(q1(x1))))
, 1^#(q2(x1)) -> c_5(1^#(x1))
, 1^#(q1(0(x1))) -> c_3(1^#(0(q1(x1))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{0(q2(x1)) -> 0(q0(x1))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{0(q2(x1)) -> 0(q0(x1))}
Details:
Interpretation Functions:
1(x1) = [1] x1 + [0]
q0(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [1]
q1(x1) = [1] x1 + [0]
q2(x1) = [1] x1 + [1]
1^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
0^#(x1) = [1] x1 + [1]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [1]
c_5(x1) = [1] x1 + [1]
c_6(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{1^#(q2(x1)) -> c_5(1^#(x1))}
and weakly orienting the rules
{0(q2(x1)) -> 0(q0(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{1^#(q2(x1)) -> c_5(1^#(x1))}
Details:
Interpretation Functions:
1(x1) = [1] x1 + [0]
q0(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [1]
q1(x1) = [1] x1 + [0]
q2(x1) = [1] x1 + [1]
1^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
0^#(x1) = [1] x1 + [1]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [9]
c_5(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{0^#(q1(x1)) -> c_4(1^#(x1))}
and weakly orienting the rules
{ 1^#(q2(x1)) -> c_5(1^#(x1))
, 0(q2(x1)) -> 0(q0(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{0^#(q1(x1)) -> c_4(1^#(x1))}
Details:
Interpretation Functions:
1(x1) = [1] x1 + [0]
q0(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [1]
q1(x1) = [1] x1 + [0]
q2(x1) = [1] x1 + [1]
1^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
0^#(x1) = [1] x1 + [1]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [1] x1 + [9]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{0(q1(x1)) -> q2(1(x1))}
and weakly orienting the rules
{ 0^#(q1(x1)) -> c_4(1^#(x1))
, 1^#(q2(x1)) -> c_5(1^#(x1))
, 0(q2(x1)) -> 0(q0(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{0(q1(x1)) -> q2(1(x1))}
Details:
Interpretation Functions:
1(x1) = [1] x1 + [0]
q0(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [1]
q1(x1) = [1] x1 + [0]
q2(x1) = [1] x1 + [0]
1^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
0^#(x1) = [1] x1 + [9]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [1] x1 + [3]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [8]
c_5(x1) = [1] x1 + [0]
c_6(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(q0(1(x1))) -> 0(1(q1(x1)))
, 1(q0(0(x1))) -> 0(0(q1(x1)))
, 1(q1(1(x1))) -> 1(1(q1(x1)))
, 1(q1(0(x1))) -> 1(0(q1(x1)))
, 1(q2(x1)) -> q2(1(x1))
, 1^#(q1(1(x1))) -> c_2(1^#(1(q1(x1))))
, 1^#(q1(0(x1))) -> c_3(1^#(0(q1(x1))))}
Weak Rules:
{ 0(q1(x1)) -> q2(1(x1))
, 0^#(q1(x1)) -> c_4(1^#(x1))
, 1^#(q2(x1)) -> c_5(1^#(x1))
, 0(q2(x1)) -> 0(q0(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(q0(1(x1))) -> 0(1(q1(x1)))
, 1(q0(0(x1))) -> 0(0(q1(x1)))
, 1(q1(1(x1))) -> 1(1(q1(x1)))
, 1(q1(0(x1))) -> 1(0(q1(x1)))
, 1(q2(x1)) -> q2(1(x1))
, 1^#(q1(1(x1))) -> c_2(1^#(1(q1(x1))))
, 1^#(q1(0(x1))) -> c_3(1^#(0(q1(x1))))}
Weak Rules:
{ 0(q1(x1)) -> q2(1(x1))
, 0^#(q1(x1)) -> c_4(1^#(x1))
, 1^#(q2(x1)) -> c_5(1^#(x1))
, 0(q2(x1)) -> 0(q0(x1))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ q0_0(2) -> 2
, q0_0(4) -> 2
, q0_0(5) -> 2
, q1_0(2) -> 4
, q1_0(4) -> 4
, q1_0(5) -> 4
, q2_0(2) -> 5
, q2_0(4) -> 5
, q2_0(5) -> 5
, 1^#_0(2) -> 6
, 1^#_0(4) -> 6
, 1^#_0(5) -> 6
, 0^#_0(2) -> 8
, 0^#_0(4) -> 8
, 0^#_0(5) -> 8
, c_4_0(6) -> 8
, c_5_0(6) -> 6}
5) { 0^#(q1(x1)) -> c_4(1^#(x1))
, 1^#(q1(1(x1))) -> c_2(1^#(1(q1(x1))))
, 1^#(q2(x1)) -> c_5(1^#(x1))
, 1^#(q1(0(x1))) -> c_3(1^#(0(q1(x1))))
, 1^#(q0(1(x1))) -> c_0(0^#(1(q1(x1))))
, 0^#(q2(x1)) -> c_6(0^#(q0(x1)))}
The usable rules for this path are the following:
{ 1(q0(1(x1))) -> 0(1(q1(x1)))
, 1(q0(0(x1))) -> 0(0(q1(x1)))
, 1(q1(1(x1))) -> 1(1(q1(x1)))
, 1(q1(0(x1))) -> 1(0(q1(x1)))
, 0(q1(x1)) -> q2(1(x1))
, 1(q2(x1)) -> q2(1(x1))
, 0(q2(x1)) -> 0(q0(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(q0(1(x1))) -> 0(1(q1(x1)))
, 1(q0(0(x1))) -> 0(0(q1(x1)))
, 1(q1(1(x1))) -> 1(1(q1(x1)))
, 1(q1(0(x1))) -> 1(0(q1(x1)))
, 0(q1(x1)) -> q2(1(x1))
, 1(q2(x1)) -> q2(1(x1))
, 0(q2(x1)) -> 0(q0(x1))
, 1^#(q0(1(x1))) -> c_0(0^#(1(q1(x1))))
, 1^#(q1(1(x1))) -> c_2(1^#(1(q1(x1))))
, 1^#(q2(x1)) -> c_5(1^#(x1))
, 1^#(q1(0(x1))) -> c_3(1^#(0(q1(x1))))
, 0^#(q1(x1)) -> c_4(1^#(x1))
, 0^#(q2(x1)) -> c_6(0^#(q0(x1)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{0(q2(x1)) -> 0(q0(x1))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{0(q2(x1)) -> 0(q0(x1))}
Details:
Interpretation Functions:
1(x1) = [1] x1 + [0]
q0(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [1]
q1(x1) = [1] x1 + [0]
q2(x1) = [1] x1 + [1]
1^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [1]
0^#(x1) = [1] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [4]
c_4(x1) = [1] x1 + [1]
c_5(x1) = [1] x1 + [1]
c_6(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{1^#(q2(x1)) -> c_5(1^#(x1))}
and weakly orienting the rules
{0(q2(x1)) -> 0(q0(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{1^#(q2(x1)) -> c_5(1^#(x1))}
Details:
Interpretation Functions:
1(x1) = [1] x1 + [0]
q0(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [1]
q1(x1) = [1] x1 + [0]
q2(x1) = [1] x1 + [1]
1^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [1]
0^#(x1) = [1] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [1]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [1] x1 + [0]
c_6(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ 0^#(q1(x1)) -> c_4(1^#(x1))
, 0^#(q2(x1)) -> c_6(0^#(q0(x1)))}
and weakly orienting the rules
{ 1^#(q2(x1)) -> c_5(1^#(x1))
, 0(q2(x1)) -> 0(q0(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 0^#(q1(x1)) -> c_4(1^#(x1))
, 0^#(q2(x1)) -> c_6(0^#(q0(x1)))}
Details:
Interpretation Functions:
1(x1) = [1] x1 + [0]
q0(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [1]
q1(x1) = [1] x1 + [0]
q2(x1) = [1] x1 + [1]
1^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [4]
0^#(x1) = [1] x1 + [7]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [1]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [1] x1 + [0]
c_6(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ 0(q1(x1)) -> q2(1(x1))
, 1^#(q0(1(x1))) -> c_0(0^#(1(q1(x1))))}
and weakly orienting the rules
{ 0^#(q1(x1)) -> c_4(1^#(x1))
, 0^#(q2(x1)) -> c_6(0^#(q0(x1)))
, 1^#(q2(x1)) -> c_5(1^#(x1))
, 0(q2(x1)) -> 0(q0(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 0(q1(x1)) -> q2(1(x1))
, 1^#(q0(1(x1))) -> c_0(0^#(1(q1(x1))))}
Details:
Interpretation Functions:
1(x1) = [1] x1 + [0]
q0(x1) = [1] x1 + [6]
0(x1) = [1] x1 + [6]
q1(x1) = [1] x1 + [2]
q2(x1) = [1] x1 + [7]
1^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [1]
0^#(x1) = [1] x1 + [2]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [3]
c_4(x1) = [1] x1 + [1]
c_5(x1) = [1] x1 + [1]
c_6(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(q0(1(x1))) -> 0(1(q1(x1)))
, 1(q0(0(x1))) -> 0(0(q1(x1)))
, 1(q1(1(x1))) -> 1(1(q1(x1)))
, 1(q1(0(x1))) -> 1(0(q1(x1)))
, 1(q2(x1)) -> q2(1(x1))
, 1^#(q1(1(x1))) -> c_2(1^#(1(q1(x1))))
, 1^#(q1(0(x1))) -> c_3(1^#(0(q1(x1))))}
Weak Rules:
{ 0(q1(x1)) -> q2(1(x1))
, 1^#(q0(1(x1))) -> c_0(0^#(1(q1(x1))))
, 0^#(q1(x1)) -> c_4(1^#(x1))
, 0^#(q2(x1)) -> c_6(0^#(q0(x1)))
, 1^#(q2(x1)) -> c_5(1^#(x1))
, 0(q2(x1)) -> 0(q0(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(q0(1(x1))) -> 0(1(q1(x1)))
, 1(q0(0(x1))) -> 0(0(q1(x1)))
, 1(q1(1(x1))) -> 1(1(q1(x1)))
, 1(q1(0(x1))) -> 1(0(q1(x1)))
, 1(q2(x1)) -> q2(1(x1))
, 1^#(q1(1(x1))) -> c_2(1^#(1(q1(x1))))
, 1^#(q1(0(x1))) -> c_3(1^#(0(q1(x1))))}
Weak Rules:
{ 0(q1(x1)) -> q2(1(x1))
, 1^#(q0(1(x1))) -> c_0(0^#(1(q1(x1))))
, 0^#(q1(x1)) -> c_4(1^#(x1))
, 0^#(q2(x1)) -> c_6(0^#(q0(x1)))
, 1^#(q2(x1)) -> c_5(1^#(x1))
, 0(q2(x1)) -> 0(q0(x1))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ q0_0(2) -> 2
, q0_0(4) -> 2
, q0_0(5) -> 2
, q1_0(2) -> 4
, q1_0(4) -> 4
, q1_0(5) -> 4
, q2_0(2) -> 5
, q2_0(4) -> 5
, q2_0(5) -> 5
, 1^#_0(2) -> 6
, 1^#_0(4) -> 6
, 1^#_0(5) -> 6
, 0^#_0(2) -> 8
, 0^#_0(4) -> 8
, 0^#_0(5) -> 8
, c_4_0(6) -> 8
, c_5_0(6) -> 6
, c_6_0(8) -> 8}
6) { 0^#(q1(x1)) -> c_4(1^#(x1))
, 1^#(q0(1(x1))) -> c_0(0^#(1(q1(x1))))
, 0^#(q2(x1)) -> c_6(0^#(q0(x1)))}
The usable rules for this path are the following:
{ 1(q0(1(x1))) -> 0(1(q1(x1)))
, 1(q0(0(x1))) -> 0(0(q1(x1)))
, 1(q1(1(x1))) -> 1(1(q1(x1)))
, 1(q1(0(x1))) -> 1(0(q1(x1)))
, 1(q2(x1)) -> q2(1(x1))
, 0(q1(x1)) -> q2(1(x1))
, 0(q2(x1)) -> 0(q0(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(q0(1(x1))) -> 0(1(q1(x1)))
, 1(q0(0(x1))) -> 0(0(q1(x1)))
, 1(q1(1(x1))) -> 1(1(q1(x1)))
, 1(q1(0(x1))) -> 1(0(q1(x1)))
, 1(q2(x1)) -> q2(1(x1))
, 0(q1(x1)) -> q2(1(x1))
, 0(q2(x1)) -> 0(q0(x1))
, 1^#(q0(1(x1))) -> c_0(0^#(1(q1(x1))))
, 0^#(q1(x1)) -> c_4(1^#(x1))
, 0^#(q2(x1)) -> c_6(0^#(q0(x1)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{0(q1(x1)) -> q2(1(x1))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{0(q1(x1)) -> q2(1(x1))}
Details:
Interpretation Functions:
1(x1) = [1] x1 + [0]
q0(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [1]
q1(x1) = [1] x1 + [0]
q2(x1) = [1] x1 + [0]
1^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [1]
0^#(x1) = [1] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [7]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{0^#(q1(x1)) -> c_4(1^#(x1))}
and weakly orienting the rules
{0(q1(x1)) -> q2(1(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{0^#(q1(x1)) -> c_4(1^#(x1))}
Details:
Interpretation Functions:
1(x1) = [1] x1 + [0]
q0(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [1]
q1(x1) = [1] x1 + [0]
q2(x1) = [1] x1 + [0]
1^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [8]
0^#(x1) = [1] x1 + [9]
c_1(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) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{0(q2(x1)) -> 0(q0(x1))}
and weakly orienting the rules
{ 0^#(q1(x1)) -> c_4(1^#(x1))
, 0(q1(x1)) -> q2(1(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{0(q2(x1)) -> 0(q0(x1))}
Details:
Interpretation Functions:
1(x1) = [1] x1 + [0]
q0(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [1]
q1(x1) = [1] x1 + [0]
q2(x1) = [1] x1 + [1]
1^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [8]
0^#(x1) = [1] x1 + [9]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [8]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{0^#(q2(x1)) -> c_6(0^#(q0(x1)))}
and weakly orienting the rules
{ 0(q2(x1)) -> 0(q0(x1))
, 0^#(q1(x1)) -> c_4(1^#(x1))
, 0(q1(x1)) -> q2(1(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{0^#(q2(x1)) -> c_6(0^#(q0(x1)))}
Details:
Interpretation Functions:
1(x1) = [1] x1 + [0]
q0(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [1]
q1(x1) = [1] x1 + [0]
q2(x1) = [1] x1 + [1]
1^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [7]
c_1(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) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{1^#(q0(1(x1))) -> c_0(0^#(1(q1(x1))))}
and weakly orienting the rules
{ 0^#(q2(x1)) -> c_6(0^#(q0(x1)))
, 0(q2(x1)) -> 0(q0(x1))
, 0^#(q1(x1)) -> c_4(1^#(x1))
, 0(q1(x1)) -> q2(1(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{1^#(q0(1(x1))) -> c_0(0^#(1(q1(x1))))}
Details:
Interpretation Functions:
1(x1) = [1] x1 + [0]
q0(x1) = [1] x1 + [1]
0(x1) = [1] x1 + [8]
q1(x1) = [1] x1 + [8]
q2(x1) = [1] x1 + [12]
1^#(x1) = [1] x1 + [8]
c_0(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(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:
{ 1(q0(1(x1))) -> 0(1(q1(x1)))
, 1(q0(0(x1))) -> 0(0(q1(x1)))
, 1(q1(1(x1))) -> 1(1(q1(x1)))
, 1(q1(0(x1))) -> 1(0(q1(x1)))
, 1(q2(x1)) -> q2(1(x1))}
Weak Rules:
{ 1^#(q0(1(x1))) -> c_0(0^#(1(q1(x1))))
, 0^#(q2(x1)) -> c_6(0^#(q0(x1)))
, 0(q2(x1)) -> 0(q0(x1))
, 0^#(q1(x1)) -> c_4(1^#(x1))
, 0(q1(x1)) -> q2(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(q0(1(x1))) -> 0(1(q1(x1)))
, 1(q0(0(x1))) -> 0(0(q1(x1)))
, 1(q1(1(x1))) -> 1(1(q1(x1)))
, 1(q1(0(x1))) -> 1(0(q1(x1)))
, 1(q2(x1)) -> q2(1(x1))}
Weak Rules:
{ 1^#(q0(1(x1))) -> c_0(0^#(1(q1(x1))))
, 0^#(q2(x1)) -> c_6(0^#(q0(x1)))
, 0(q2(x1)) -> 0(q0(x1))
, 0^#(q1(x1)) -> c_4(1^#(x1))
, 0(q1(x1)) -> q2(1(x1))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ q0_0(2) -> 2
, q0_0(4) -> 2
, q0_0(5) -> 2
, q1_0(2) -> 4
, q1_0(4) -> 4
, q1_0(5) -> 4
, q2_0(2) -> 5
, q2_0(4) -> 5
, q2_0(5) -> 5
, 1^#_0(2) -> 6
, 1^#_0(4) -> 6
, 1^#_0(5) -> 6
, 0^#_0(2) -> 8
, 0^#_0(4) -> 8
, 0^#_0(5) -> 8
, c_4_0(6) -> 8
, c_6_0(8) -> 8}
7) { 0^#(q1(x1)) -> c_4(1^#(x1))
, 1^#(q0(0(x1))) -> c_1(0^#(0(q1(x1))))
, 0^#(q2(x1)) -> c_6(0^#(q0(x1)))}
The usable rules for this path are the following:
{ 0(q1(x1)) -> q2(1(x1))
, 0(q2(x1)) -> 0(q0(x1))
, 1(q0(1(x1))) -> 0(1(q1(x1)))
, 1(q0(0(x1))) -> 0(0(q1(x1)))
, 1(q1(1(x1))) -> 1(1(q1(x1)))
, 1(q1(0(x1))) -> 1(0(q1(x1)))
, 1(q2(x1)) -> q2(1(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:
{ 0(q1(x1)) -> q2(1(x1))
, 0(q2(x1)) -> 0(q0(x1))
, 1(q0(1(x1))) -> 0(1(q1(x1)))
, 1(q0(0(x1))) -> 0(0(q1(x1)))
, 1(q1(1(x1))) -> 1(1(q1(x1)))
, 1(q1(0(x1))) -> 1(0(q1(x1)))
, 1(q2(x1)) -> q2(1(x1))
, 1^#(q0(0(x1))) -> c_1(0^#(0(q1(x1))))
, 0^#(q1(x1)) -> c_4(1^#(x1))
, 0^#(q2(x1)) -> c_6(0^#(q0(x1)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{0(q2(x1)) -> 0(q0(x1))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{0(q2(x1)) -> 0(q0(x1))}
Details:
Interpretation Functions:
1(x1) = [1] x1 + [0]
q0(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [1]
q1(x1) = [1] x1 + [0]
q2(x1) = [1] x1 + [1]
1^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
0^#(x1) = [1] x1 + [0]
c_1(x1) = [1] 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) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{0^#(q1(x1)) -> c_4(1^#(x1))}
and weakly orienting the rules
{0(q2(x1)) -> 0(q0(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{0^#(q1(x1)) -> c_4(1^#(x1))}
Details:
Interpretation Functions:
1(x1) = [1] x1 + [0]
q0(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [1]
q1(x1) = [1] x1 + [0]
q2(x1) = [1] x1 + [1]
1^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
0^#(x1) = [1] x1 + [2]
c_1(x1) = [1] 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) = [1] x1 + [3]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{0^#(q2(x1)) -> c_6(0^#(q0(x1)))}
and weakly orienting the rules
{ 0^#(q1(x1)) -> c_4(1^#(x1))
, 0(q2(x1)) -> 0(q0(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{0^#(q2(x1)) -> c_6(0^#(q0(x1)))}
Details:
Interpretation Functions:
1(x1) = [1] x1 + [0]
q0(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [1]
q1(x1) = [1] x1 + [0]
q2(x1) = [1] x1 + [1]
1^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
0^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{1^#(q0(0(x1))) -> c_1(0^#(0(q1(x1))))}
and weakly orienting the rules
{ 0^#(q2(x1)) -> c_6(0^#(q0(x1)))
, 0^#(q1(x1)) -> c_4(1^#(x1))
, 0(q2(x1)) -> 0(q0(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{1^#(q0(0(x1))) -> c_1(0^#(0(q1(x1))))}
Details:
Interpretation Functions:
1(x1) = [1] x1 + [0]
q0(x1) = [1] x1 + [1]
0(x1) = [1] x1 + [1]
q1(x1) = [1] x1 + [0]
q2(x1) = [1] x1 + [1]
1^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
0^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ 1(q0(1(x1))) -> 0(1(q1(x1)))
, 1(q0(0(x1))) -> 0(0(q1(x1)))}
and weakly orienting the rules
{ 1^#(q0(0(x1))) -> c_1(0^#(0(q1(x1))))
, 0^#(q2(x1)) -> c_6(0^#(q0(x1)))
, 0^#(q1(x1)) -> c_4(1^#(x1))
, 0(q2(x1)) -> 0(q0(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 1(q0(1(x1))) -> 0(1(q1(x1)))
, 1(q0(0(x1))) -> 0(0(q1(x1)))}
Details:
Interpretation Functions:
1(x1) = [1] x1 + [0]
q0(x1) = [1] x1 + [4]
0(x1) = [1] x1 + [1]
q1(x1) = [1] x1 + [0]
q2(x1) = [1] x1 + [9]
1^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
0^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(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:
{ 0(q1(x1)) -> q2(1(x1))
, 1(q1(1(x1))) -> 1(1(q1(x1)))
, 1(q1(0(x1))) -> 1(0(q1(x1)))
, 1(q2(x1)) -> q2(1(x1))}
Weak Rules:
{ 1(q0(1(x1))) -> 0(1(q1(x1)))
, 1(q0(0(x1))) -> 0(0(q1(x1)))
, 1^#(q0(0(x1))) -> c_1(0^#(0(q1(x1))))
, 0^#(q2(x1)) -> c_6(0^#(q0(x1)))
, 0^#(q1(x1)) -> c_4(1^#(x1))
, 0(q2(x1)) -> 0(q0(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:
{ 0(q1(x1)) -> q2(1(x1))
, 1(q1(1(x1))) -> 1(1(q1(x1)))
, 1(q1(0(x1))) -> 1(0(q1(x1)))
, 1(q2(x1)) -> q2(1(x1))}
Weak Rules:
{ 1(q0(1(x1))) -> 0(1(q1(x1)))
, 1(q0(0(x1))) -> 0(0(q1(x1)))
, 1^#(q0(0(x1))) -> c_1(0^#(0(q1(x1))))
, 0^#(q2(x1)) -> c_6(0^#(q0(x1)))
, 0^#(q1(x1)) -> c_4(1^#(x1))
, 0(q2(x1)) -> 0(q0(x1))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ q0_0(2) -> 2
, q0_0(4) -> 2
, q0_0(5) -> 2
, q1_0(2) -> 4
, q1_0(4) -> 4
, q1_0(5) -> 4
, q2_0(2) -> 5
, q2_0(4) -> 5
, q2_0(5) -> 5
, 1^#_0(2) -> 6
, 1^#_0(4) -> 6
, 1^#_0(5) -> 6
, 0^#_0(2) -> 8
, 0^#_0(4) -> 8
, 0^#_0(5) -> 8
, c_4_0(6) -> 8
, c_6_0(8) -> 8}
8) { 0^#(q1(x1)) -> c_4(1^#(x1))
, 1^#(q0(1(x1))) -> c_0(0^#(1(q1(x1))))}
The usable rules for this path are the following:
{ 1(q0(1(x1))) -> 0(1(q1(x1)))
, 1(q0(0(x1))) -> 0(0(q1(x1)))
, 1(q1(1(x1))) -> 1(1(q1(x1)))
, 1(q1(0(x1))) -> 1(0(q1(x1)))
, 1(q2(x1)) -> q2(1(x1))
, 0(q1(x1)) -> q2(1(x1))
, 0(q2(x1)) -> 0(q0(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(q0(1(x1))) -> 0(1(q1(x1)))
, 1(q0(0(x1))) -> 0(0(q1(x1)))
, 1(q1(1(x1))) -> 1(1(q1(x1)))
, 1(q1(0(x1))) -> 1(0(q1(x1)))
, 1(q2(x1)) -> q2(1(x1))
, 0(q1(x1)) -> q2(1(x1))
, 0(q2(x1)) -> 0(q0(x1))
, 0^#(q1(x1)) -> c_4(1^#(x1))
, 1^#(q0(1(x1))) -> c_0(0^#(1(q1(x1))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{0(q1(x1)) -> q2(1(x1))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{0(q1(x1)) -> q2(1(x1))}
Details:
Interpretation Functions:
1(x1) = [1] x1 + [0]
q0(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [1]
q1(x1) = [1] x1 + [0]
q2(x1) = [1] x1 + [0]
1^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [1]
c_1(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]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{1^#(q0(1(x1))) -> c_0(0^#(1(q1(x1))))}
and weakly orienting the rules
{0(q1(x1)) -> q2(1(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{1^#(q0(1(x1))) -> c_0(0^#(1(q1(x1))))}
Details:
Interpretation Functions:
1(x1) = [1] x1 + [0]
q0(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [1]
q1(x1) = [1] x1 + [0]
q2(x1) = [1] x1 + [0]
1^#(x1) = [1] x1 + [8]
c_0(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [1]
c_1(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]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{0(q2(x1)) -> 0(q0(x1))}
and weakly orienting the rules
{ 1^#(q0(1(x1))) -> c_0(0^#(1(q1(x1))))
, 0(q1(x1)) -> q2(1(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{0(q2(x1)) -> 0(q0(x1))}
Details:
Interpretation Functions:
1(x1) = [1] x1 + [0]
q0(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [9]
q1(x1) = [1] x1 + [0]
q2(x1) = [1] x1 + [8]
1^#(x1) = [1] x1 + [4]
c_0(x1) = [1] x1 + [3]
0^#(x1) = [1] x1 + [1]
c_1(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]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{0^#(q1(x1)) -> c_4(1^#(x1))}
and weakly orienting the rules
{ 0(q2(x1)) -> 0(q0(x1))
, 1^#(q0(1(x1))) -> c_0(0^#(1(q1(x1))))
, 0(q1(x1)) -> q2(1(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{0^#(q1(x1)) -> c_4(1^#(x1))}
Details:
Interpretation Functions:
1(x1) = [1] x1 + [0]
q0(x1) = [1] x1 + [9]
0(x1) = [1] x1 + [0]
q1(x1) = [1] x1 + [9]
q2(x1) = [1] x1 + [9]
1^#(x1) = [1] x1 + [7]
c_0(x1) = [1] x1 + [0]
0^#(x1) = [1] x1 + [1]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [2]
c_5(x1) = [0] x1 + [0]
c_6(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(q0(1(x1))) -> 0(1(q1(x1)))
, 1(q0(0(x1))) -> 0(0(q1(x1)))
, 1(q1(1(x1))) -> 1(1(q1(x1)))
, 1(q1(0(x1))) -> 1(0(q1(x1)))
, 1(q2(x1)) -> q2(1(x1))}
Weak Rules:
{ 0^#(q1(x1)) -> c_4(1^#(x1))
, 0(q2(x1)) -> 0(q0(x1))
, 1^#(q0(1(x1))) -> c_0(0^#(1(q1(x1))))
, 0(q1(x1)) -> q2(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(q0(1(x1))) -> 0(1(q1(x1)))
, 1(q0(0(x1))) -> 0(0(q1(x1)))
, 1(q1(1(x1))) -> 1(1(q1(x1)))
, 1(q1(0(x1))) -> 1(0(q1(x1)))
, 1(q2(x1)) -> q2(1(x1))}
Weak Rules:
{ 0^#(q1(x1)) -> c_4(1^#(x1))
, 0(q2(x1)) -> 0(q0(x1))
, 1^#(q0(1(x1))) -> c_0(0^#(1(q1(x1))))
, 0(q1(x1)) -> q2(1(x1))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ q0_0(2) -> 2
, q0_0(4) -> 2
, q0_0(5) -> 2
, q1_0(2) -> 4
, q1_0(4) -> 4
, q1_0(5) -> 4
, q2_0(2) -> 5
, q2_0(4) -> 5
, q2_0(5) -> 5
, 1^#_0(2) -> 6
, 1^#_0(4) -> 6
, 1^#_0(5) -> 6
, 0^#_0(2) -> 8
, 0^#_0(4) -> 8
, 0^#_0(5) -> 8
, c_4_0(6) -> 8}
9) { 0^#(q1(x1)) -> c_4(1^#(x1))
, 1^#(q0(0(x1))) -> c_1(0^#(0(q1(x1))))}
The usable rules for this path are the following:
{ 0(q1(x1)) -> q2(1(x1))
, 0(q2(x1)) -> 0(q0(x1))
, 1(q0(1(x1))) -> 0(1(q1(x1)))
, 1(q0(0(x1))) -> 0(0(q1(x1)))
, 1(q1(1(x1))) -> 1(1(q1(x1)))
, 1(q1(0(x1))) -> 1(0(q1(x1)))
, 1(q2(x1)) -> q2(1(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:
{ 0(q1(x1)) -> q2(1(x1))
, 0(q2(x1)) -> 0(q0(x1))
, 1(q0(1(x1))) -> 0(1(q1(x1)))
, 1(q0(0(x1))) -> 0(0(q1(x1)))
, 1(q1(1(x1))) -> 1(1(q1(x1)))
, 1(q1(0(x1))) -> 1(0(q1(x1)))
, 1(q2(x1)) -> q2(1(x1))
, 0^#(q1(x1)) -> c_4(1^#(x1))
, 1^#(q0(0(x1))) -> c_1(0^#(0(q1(x1))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{0(q2(x1)) -> 0(q0(x1))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{0(q2(x1)) -> 0(q0(x1))}
Details:
Interpretation Functions:
1(x1) = [1] x1 + [0]
q0(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [1]
q1(x1) = [1] x1 + [0]
q2(x1) = [1] x1 + [1]
1^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
0^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [7]
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]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{1^#(q0(0(x1))) -> c_1(0^#(0(q1(x1))))}
and weakly orienting the rules
{0(q2(x1)) -> 0(q0(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{1^#(q0(0(x1))) -> c_1(0^#(0(q1(x1))))}
Details:
Interpretation Functions:
1(x1) = [1] x1 + [0]
q0(x1) = [1] x1 + [0]
0(x1) = [1] x1 + [1]
q1(x1) = [1] x1 + [0]
q2(x1) = [1] x1 + [1]
1^#(x1) = [1] x1 + [8]
c_0(x1) = [0] x1 + [0]
0^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [6]
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]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{0^#(q1(x1)) -> c_4(1^#(x1))}
and weakly orienting the rules
{ 1^#(q0(0(x1))) -> c_1(0^#(0(q1(x1))))
, 0(q2(x1)) -> 0(q0(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{0^#(q1(x1)) -> c_4(1^#(x1))}
Details:
Interpretation Functions:
1(x1) = [1] x1 + [0]
q0(x1) = [1] x1 + [1]
0(x1) = [1] x1 + [1]
q1(x1) = [1] x1 + [0]
q2(x1) = [1] x1 + [1]
1^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
0^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ 1(q0(1(x1))) -> 0(1(q1(x1)))
, 1(q0(0(x1))) -> 0(0(q1(x1)))}
and weakly orienting the rules
{ 0^#(q1(x1)) -> c_4(1^#(x1))
, 1^#(q0(0(x1))) -> c_1(0^#(0(q1(x1))))
, 0(q2(x1)) -> 0(q0(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ 1(q0(1(x1))) -> 0(1(q1(x1)))
, 1(q0(0(x1))) -> 0(0(q1(x1)))}
Details:
Interpretation Functions:
1(x1) = [1] x1 + [0]
q0(x1) = [1] x1 + [8]
0(x1) = [1] x1 + [1]
q1(x1) = [1] x1 + [0]
q2(x1) = [1] x1 + [9]
1^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
0^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [3]
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]
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:
{ 0(q1(x1)) -> q2(1(x1))
, 1(q1(1(x1))) -> 1(1(q1(x1)))
, 1(q1(0(x1))) -> 1(0(q1(x1)))
, 1(q2(x1)) -> q2(1(x1))}
Weak Rules:
{ 1(q0(1(x1))) -> 0(1(q1(x1)))
, 1(q0(0(x1))) -> 0(0(q1(x1)))
, 0^#(q1(x1)) -> c_4(1^#(x1))
, 1^#(q0(0(x1))) -> c_1(0^#(0(q1(x1))))
, 0(q2(x1)) -> 0(q0(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:
{ 0(q1(x1)) -> q2(1(x1))
, 1(q1(1(x1))) -> 1(1(q1(x1)))
, 1(q1(0(x1))) -> 1(0(q1(x1)))
, 1(q2(x1)) -> q2(1(x1))}
Weak Rules:
{ 1(q0(1(x1))) -> 0(1(q1(x1)))
, 1(q0(0(x1))) -> 0(0(q1(x1)))
, 0^#(q1(x1)) -> c_4(1^#(x1))
, 1^#(q0(0(x1))) -> c_1(0^#(0(q1(x1))))
, 0(q2(x1)) -> 0(q0(x1))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ q0_0(2) -> 2
, q0_0(4) -> 2
, q0_0(5) -> 2
, q1_0(2) -> 4
, q1_0(4) -> 4
, q1_0(5) -> 4
, q2_0(2) -> 5
, q2_0(4) -> 5
, q2_0(5) -> 5
, 1^#_0(2) -> 6
, 1^#_0(4) -> 6
, 1^#_0(5) -> 6
, 0^#_0(2) -> 8
, 0^#_0(4) -> 8
, 0^#_0(5) -> 8
, c_4_0(6) -> 8}
10)
{0^#(q1(x1)) -> c_4(1^#(x1))}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
1(x1) = [0] x1 + [0]
q0(x1) = [0] x1 + [0]
0(x1) = [0] x1 + [0]
q1(x1) = [0] x1 + [0]
q2(x1) = [0] x1 + [0]
1^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
0^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(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: {0^#(q1(x1)) -> c_4(1^#(x1))}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{0^#(q1(x1)) -> c_4(1^#(x1))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{0^#(q1(x1)) -> c_4(1^#(x1))}
Details:
Interpretation Functions:
1(x1) = [0] x1 + [0]
q0(x1) = [0] x1 + [0]
0(x1) = [0] x1 + [0]
q1(x1) = [1] x1 + [0]
q2(x1) = [0] x1 + [0]
1^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
0^#(x1) = [1] x1 + [1]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] 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: {0^#(q1(x1)) -> c_4(1^#(x1))}
Details:
The given problem does not contain any strict rules