'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ a(b(b(x1))) -> P(a(b(x1)))
, a(P(x1)) -> P(a(x(x1)))
, a(x(x1)) -> x(a(x1))
, b(P(x1)) -> b(Q(x1))
, Q(x(x1)) -> a(Q(x1))
, Q(a(x1)) -> b(b(a(x1)))}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ a^#(b(b(x1))) -> c_0(a^#(b(x1)))
, a^#(P(x1)) -> c_1(a^#(x(x1)))
, a^#(x(x1)) -> c_2(a^#(x1))
, b^#(P(x1)) -> c_3(b^#(Q(x1)))
, Q^#(x(x1)) -> c_4(a^#(Q(x1)))
, Q^#(a(x1)) -> c_5(b^#(b(a(x1))))}
The usable rules are:
{ a(b(b(x1))) -> P(a(b(x1)))
, a(P(x1)) -> P(a(x(x1)))
, a(x(x1)) -> x(a(x1))
, b(P(x1)) -> b(Q(x1))
, Q(x(x1)) -> a(Q(x1))
, Q(a(x1)) -> b(b(a(x1)))}
The estimated dependency graph contains the following edges:
{a^#(b(b(x1))) -> c_0(a^#(b(x1)))}
==> {a^#(x(x1)) -> c_2(a^#(x1))}
{a^#(b(b(x1))) -> c_0(a^#(b(x1)))}
==> {a^#(P(x1)) -> c_1(a^#(x(x1)))}
{a^#(b(b(x1))) -> c_0(a^#(b(x1)))}
==> {a^#(b(b(x1))) -> c_0(a^#(b(x1)))}
{a^#(P(x1)) -> c_1(a^#(x(x1)))}
==> {a^#(x(x1)) -> c_2(a^#(x1))}
{a^#(x(x1)) -> c_2(a^#(x1))}
==> {a^#(x(x1)) -> c_2(a^#(x1))}
{a^#(x(x1)) -> c_2(a^#(x1))}
==> {a^#(P(x1)) -> c_1(a^#(x(x1)))}
{a^#(x(x1)) -> c_2(a^#(x1))}
==> {a^#(b(b(x1))) -> c_0(a^#(b(x1)))}
{b^#(P(x1)) -> c_3(b^#(Q(x1)))}
==> {b^#(P(x1)) -> c_3(b^#(Q(x1)))}
{Q^#(x(x1)) -> c_4(a^#(Q(x1)))}
==> {a^#(x(x1)) -> c_2(a^#(x1))}
{Q^#(x(x1)) -> c_4(a^#(Q(x1)))}
==> {a^#(P(x1)) -> c_1(a^#(x(x1)))}
{Q^#(x(x1)) -> c_4(a^#(Q(x1)))}
==> {a^#(b(b(x1))) -> c_0(a^#(b(x1)))}
{Q^#(a(x1)) -> c_5(b^#(b(a(x1))))}
==> {b^#(P(x1)) -> c_3(b^#(Q(x1)))}
We consider the following path(s):
1) { Q^#(x(x1)) -> c_4(a^#(Q(x1)))
, a^#(b(b(x1))) -> c_0(a^#(b(x1)))
, a^#(x(x1)) -> c_2(a^#(x1))
, a^#(P(x1)) -> c_1(a^#(x(x1)))}
The usable rules for this path are the following:
{ b(P(x1)) -> b(Q(x1))
, Q(x(x1)) -> a(Q(x1))
, Q(a(x1)) -> b(b(a(x1)))
, a(b(b(x1))) -> P(a(b(x1)))
, a(P(x1)) -> P(a(x(x1)))
, a(x(x1)) -> x(a(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(P(x1)) -> b(Q(x1))
, Q(x(x1)) -> a(Q(x1))
, Q(a(x1)) -> b(b(a(x1)))
, a(b(b(x1))) -> P(a(b(x1)))
, a(P(x1)) -> P(a(x(x1)))
, a(x(x1)) -> x(a(x1))
, Q^#(x(x1)) -> c_4(a^#(Q(x1)))
, a^#(b(b(x1))) -> c_0(a^#(b(x1)))
, a^#(x(x1)) -> c_2(a^#(x1))
, a^#(P(x1)) -> c_1(a^#(x(x1)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ a(b(b(x1))) -> P(a(b(x1)))
, a^#(b(b(x1))) -> c_0(a^#(b(x1)))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ a(b(b(x1))) -> P(a(b(x1)))
, a^#(b(b(x1))) -> c_0(a^#(b(x1)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
b(x1) = [1] x1 + [1]
P(x1) = [1] x1 + [0]
x(x1) = [1] x1 + [0]
Q(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [1]
b^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
Q^#(x1) = [1] x1 + [1]
c_4(x1) = [1] x1 + [1]
c_5(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{Q^#(x(x1)) -> c_4(a^#(Q(x1)))}
and weakly orienting the rules
{ a(b(b(x1))) -> P(a(b(x1)))
, a^#(b(b(x1))) -> c_0(a^#(b(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{Q^#(x(x1)) -> c_4(a^#(Q(x1)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
b(x1) = [1] x1 + [1]
P(x1) = [1] x1 + [0]
x(x1) = [1] x1 + [0]
Q(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [1]
b^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
Q^#(x1) = [1] x1 + [1]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ Q(x(x1)) -> a(Q(x1))
, a^#(x(x1)) -> c_2(a^#(x1))}
and weakly orienting the rules
{ Q^#(x(x1)) -> c_4(a^#(Q(x1)))
, a(b(b(x1))) -> P(a(b(x1)))
, a^#(b(b(x1))) -> c_0(a^#(b(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ Q(x(x1)) -> a(Q(x1))
, a^#(x(x1)) -> c_2(a^#(x1))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [1]
P(x1) = [1] x1 + [0]
x(x1) = [1] x1 + [13]
Q(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [0]
b^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
Q^#(x1) = [1] x1 + [8]
c_4(x1) = [1] x1 + [1]
c_5(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ b(P(x1)) -> b(Q(x1))
, a^#(P(x1)) -> c_1(a^#(x(x1)))}
and weakly orienting the rules
{ Q(x(x1)) -> a(Q(x1))
, a^#(x(x1)) -> c_2(a^#(x1))
, Q^#(x(x1)) -> c_4(a^#(Q(x1)))
, a(b(b(x1))) -> P(a(b(x1)))
, a^#(b(b(x1))) -> c_0(a^#(b(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ b(P(x1)) -> b(Q(x1))
, a^#(P(x1)) -> c_1(a^#(x(x1)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [13]
P(x1) = [1] x1 + [13]
x(x1) = [1] x1 + [0]
Q(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [0]
b^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
Q^#(x1) = [1] x1 + [9]
c_4(x1) = [1] x1 + [0]
c_5(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:
{ Q(a(x1)) -> b(b(a(x1)))
, a(P(x1)) -> P(a(x(x1)))
, a(x(x1)) -> x(a(x1))}
Weak Rules:
{ b(P(x1)) -> b(Q(x1))
, a^#(P(x1)) -> c_1(a^#(x(x1)))
, Q(x(x1)) -> a(Q(x1))
, a^#(x(x1)) -> c_2(a^#(x1))
, Q^#(x(x1)) -> c_4(a^#(Q(x1)))
, a(b(b(x1))) -> P(a(b(x1)))
, a^#(b(b(x1))) -> c_0(a^#(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:
{ Q(a(x1)) -> b(b(a(x1)))
, a(P(x1)) -> P(a(x(x1)))
, a(x(x1)) -> x(a(x1))}
Weak Rules:
{ b(P(x1)) -> b(Q(x1))
, a^#(P(x1)) -> c_1(a^#(x(x1)))
, Q(x(x1)) -> a(Q(x1))
, a^#(x(x1)) -> c_2(a^#(x1))
, Q^#(x(x1)) -> c_4(a^#(Q(x1)))
, a(b(b(x1))) -> P(a(b(x1)))
, a^#(b(b(x1))) -> c_0(a^#(b(x1)))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ a_0(14) -> 14
, P_0(3) -> 3
, P_0(4) -> 3
, x_0(3) -> 4
, x_0(4) -> 4
, Q_0(3) -> 14
, Q_0(4) -> 14
, a^#_0(3) -> 6
, a^#_0(4) -> 6
, a^#_0(14) -> 13
, c_1_0(6) -> 6
, c_2_0(6) -> 6
, Q^#_0(3) -> 12
, Q^#_0(4) -> 12
, c_4_0(13) -> 12}
2) {Q^#(a(x1)) -> c_5(b^#(b(a(x1))))}
The usable rules for this path are the following:
{ a(b(b(x1))) -> P(a(b(x1)))
, a(P(x1)) -> P(a(x(x1)))
, a(x(x1)) -> x(a(x1))
, b(P(x1)) -> b(Q(x1))
, Q(x(x1)) -> a(Q(x1))
, Q(a(x1)) -> b(b(a(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(b(b(x1))) -> P(a(b(x1)))
, a(P(x1)) -> P(a(x(x1)))
, a(x(x1)) -> x(a(x1))
, b(P(x1)) -> b(Q(x1))
, Q(x(x1)) -> a(Q(x1))
, Q(a(x1)) -> b(b(a(x1)))
, Q^#(a(x1)) -> c_5(b^#(b(a(x1))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{Q^#(a(x1)) -> c_5(b^#(b(a(x1))))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{Q^#(a(x1)) -> c_5(b^#(b(a(x1))))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
b(x1) = [1] x1 + [0]
P(x1) = [1] x1 + [0]
x(x1) = [1] x1 + [0]
Q(x1) = [1] x1 + [0]
a^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
Q^#(x1) = [1] x1 + [8]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{Q(a(x1)) -> b(b(a(x1)))}
and weakly orienting the rules
{Q^#(a(x1)) -> c_5(b^#(b(a(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{Q(a(x1)) -> b(b(a(x1)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
b(x1) = [1] x1 + [0]
P(x1) = [1] x1 + [0]
x(x1) = [1] x1 + [0]
Q(x1) = [1] x1 + [8]
a^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
Q^#(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{Q(x(x1)) -> a(Q(x1))}
and weakly orienting the rules
{ Q(a(x1)) -> b(b(a(x1)))
, Q^#(a(x1)) -> c_5(b^#(b(a(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{Q(x(x1)) -> a(Q(x1))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
b(x1) = [1] x1 + [0]
P(x1) = [1] x1 + [0]
x(x1) = [1] x1 + [8]
Q(x1) = [1] x1 + [8]
a^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
b^#(x1) = [1] x1 + [2]
c_3(x1) = [0] x1 + [0]
Q^#(x1) = [1] x1 + [15]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [13]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b(P(x1)) -> b(Q(x1))}
and weakly orienting the rules
{ Q(x(x1)) -> a(Q(x1))
, Q(a(x1)) -> b(b(a(x1)))
, Q^#(a(x1)) -> c_5(b^#(b(a(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b(P(x1)) -> b(Q(x1))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
b(x1) = [1] x1 + [0]
P(x1) = [1] x1 + [8]
x(x1) = [1] x1 + [8]
Q(x1) = [1] x1 + [0]
a^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
Q^#(x1) = [1] x1 + [8]
c_4(x1) = [0] x1 + [0]
c_5(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:
{ a(b(b(x1))) -> P(a(b(x1)))
, a(P(x1)) -> P(a(x(x1)))
, a(x(x1)) -> x(a(x1))}
Weak Rules:
{ b(P(x1)) -> b(Q(x1))
, Q(x(x1)) -> a(Q(x1))
, Q(a(x1)) -> b(b(a(x1)))
, Q^#(a(x1)) -> c_5(b^#(b(a(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(b(b(x1))) -> P(a(b(x1)))
, a(P(x1)) -> P(a(x(x1)))
, a(x(x1)) -> x(a(x1))}
Weak Rules:
{ b(P(x1)) -> b(Q(x1))
, Q(x(x1)) -> a(Q(x1))
, Q(a(x1)) -> b(b(a(x1)))
, Q^#(a(x1)) -> c_5(b^#(b(a(x1))))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ P_0(3) -> 3
, P_0(4) -> 3
, x_0(3) -> 4
, x_0(4) -> 4
, b^#_0(3) -> 10
, b^#_0(4) -> 10
, Q^#_0(3) -> 12
, Q^#_0(4) -> 12}
3) {Q^#(x(x1)) -> c_4(a^#(Q(x1)))}
The usable rules for this path are the following:
{ Q(x(x1)) -> a(Q(x1))
, Q(a(x1)) -> b(b(a(x1)))
, a(b(b(x1))) -> P(a(b(x1)))
, a(P(x1)) -> P(a(x(x1)))
, a(x(x1)) -> x(a(x1))
, b(P(x1)) -> b(Q(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:
{ Q(x(x1)) -> a(Q(x1))
, Q(a(x1)) -> b(b(a(x1)))
, a(b(b(x1))) -> P(a(b(x1)))
, a(P(x1)) -> P(a(x(x1)))
, a(x(x1)) -> x(a(x1))
, b(P(x1)) -> b(Q(x1))
, Q^#(x(x1)) -> c_4(a^#(Q(x1)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{Q(a(x1)) -> b(b(a(x1)))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{Q(a(x1)) -> b(b(a(x1)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
P(x1) = [1] x1 + [1]
x(x1) = [1] x1 + [0]
Q(x1) = [1] x1 + [1]
a^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
b^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
Q^#(x1) = [1] x1 + [1]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{Q^#(x(x1)) -> c_4(a^#(Q(x1)))}
and weakly orienting the rules
{Q(a(x1)) -> b(b(a(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{Q^#(x(x1)) -> c_4(a^#(Q(x1)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
P(x1) = [1] x1 + [1]
x(x1) = [1] x1 + [0]
Q(x1) = [1] x1 + [1]
a^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
b^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
Q^#(x1) = [1] x1 + [9]
c_4(x1) = [1] x1 + [2]
c_5(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b(P(x1)) -> b(Q(x1))}
and weakly orienting the rules
{ Q^#(x(x1)) -> c_4(a^#(Q(x1)))
, Q(a(x1)) -> b(b(a(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b(P(x1)) -> b(Q(x1))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
P(x1) = [1] x1 + [9]
x(x1) = [1] x1 + [0]
Q(x1) = [1] x1 + [1]
a^#(x1) = [1] x1 + [2]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
b^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
Q^#(x1) = [1] x1 + [9]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{Q(x(x1)) -> a(Q(x1))}
and weakly orienting the rules
{ b(P(x1)) -> b(Q(x1))
, Q^#(x(x1)) -> c_4(a^#(Q(x1)))
, Q(a(x1)) -> b(b(a(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{Q(x(x1)) -> a(Q(x1))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
P(x1) = [1] x1 + [8]
x(x1) = [1] x1 + [4]
Q(x1) = [1] x1 + [8]
a^#(x1) = [1] x1 + [9]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
b^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
Q^#(x1) = [1] x1 + [13]
c_4(x1) = [1] x1 + [0]
c_5(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(b(b(x1))) -> P(a(b(x1)))
, a(P(x1)) -> P(a(x(x1)))
, a(x(x1)) -> x(a(x1))}
Weak Rules:
{ Q(x(x1)) -> a(Q(x1))
, b(P(x1)) -> b(Q(x1))
, Q^#(x(x1)) -> c_4(a^#(Q(x1)))
, Q(a(x1)) -> b(b(a(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(b(b(x1))) -> P(a(b(x1)))
, a(P(x1)) -> P(a(x(x1)))
, a(x(x1)) -> x(a(x1))}
Weak Rules:
{ Q(x(x1)) -> a(Q(x1))
, b(P(x1)) -> b(Q(x1))
, Q^#(x(x1)) -> c_4(a^#(Q(x1)))
, Q(a(x1)) -> b(b(a(x1)))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ a_0(14) -> 14
, P_0(3) -> 3
, P_0(4) -> 3
, x_0(3) -> 4
, x_0(4) -> 4
, Q_0(3) -> 14
, Q_0(4) -> 14
, a^#_0(3) -> 6
, a^#_0(4) -> 6
, a^#_0(14) -> 13
, Q^#_0(3) -> 12
, Q^#_0(4) -> 12
, c_4_0(13) -> 12}
4) { Q^#(a(x1)) -> c_5(b^#(b(a(x1))))
, b^#(P(x1)) -> c_3(b^#(Q(x1)))}
The usable rules for this path are the following:
{ a(b(b(x1))) -> P(a(b(x1)))
, a(P(x1)) -> P(a(x(x1)))
, a(x(x1)) -> x(a(x1))
, b(P(x1)) -> b(Q(x1))
, Q(x(x1)) -> a(Q(x1))
, Q(a(x1)) -> b(b(a(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(b(b(x1))) -> P(a(b(x1)))
, a(P(x1)) -> P(a(x(x1)))
, a(x(x1)) -> x(a(x1))
, b(P(x1)) -> b(Q(x1))
, Q(x(x1)) -> a(Q(x1))
, Q(a(x1)) -> b(b(a(x1)))
, Q^#(a(x1)) -> c_5(b^#(b(a(x1))))
, b^#(P(x1)) -> c_3(b^#(Q(x1)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{Q^#(a(x1)) -> c_5(b^#(b(a(x1))))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{Q^#(a(x1)) -> c_5(b^#(b(a(x1))))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
b(x1) = [1] x1 + [0]
P(x1) = [1] x1 + [0]
x(x1) = [1] x1 + [0]
Q(x1) = [1] x1 + [0]
a^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [8]
Q^#(x1) = [1] x1 + [15]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{Q(a(x1)) -> b(b(a(x1)))}
and weakly orienting the rules
{Q^#(a(x1)) -> c_5(b^#(b(a(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{Q(a(x1)) -> b(b(a(x1)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
b(x1) = [1] x1 + [0]
P(x1) = [1] x1 + [0]
x(x1) = [1] x1 + [0]
Q(x1) = [1] x1 + [4]
a^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
b^#(x1) = [1] x1 + [12]
c_3(x1) = [1] x1 + [2]
Q^#(x1) = [1] x1 + [12]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ b(P(x1)) -> b(Q(x1))
, Q(x(x1)) -> a(Q(x1))
, b^#(P(x1)) -> c_3(b^#(Q(x1)))}
and weakly orienting the rules
{ Q(a(x1)) -> b(b(a(x1)))
, Q^#(a(x1)) -> c_5(b^#(b(a(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ b(P(x1)) -> b(Q(x1))
, Q(x(x1)) -> a(Q(x1))
, b^#(P(x1)) -> c_3(b^#(Q(x1)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
b(x1) = [1] x1 + [0]
P(x1) = [1] x1 + [4]
x(x1) = [1] x1 + [10]
Q(x1) = [1] x1 + [2]
a^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
b^#(x1) = [1] x1 + [14]
c_3(x1) = [1] x1 + [0]
Q^#(x1) = [1] x1 + [15]
c_4(x1) = [0] x1 + [0]
c_5(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:
{ a(b(b(x1))) -> P(a(b(x1)))
, a(P(x1)) -> P(a(x(x1)))
, a(x(x1)) -> x(a(x1))}
Weak Rules:
{ b(P(x1)) -> b(Q(x1))
, Q(x(x1)) -> a(Q(x1))
, b^#(P(x1)) -> c_3(b^#(Q(x1)))
, Q(a(x1)) -> b(b(a(x1)))
, Q^#(a(x1)) -> c_5(b^#(b(a(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(b(b(x1))) -> P(a(b(x1)))
, a(P(x1)) -> P(a(x(x1)))
, a(x(x1)) -> x(a(x1))}
Weak Rules:
{ b(P(x1)) -> b(Q(x1))
, Q(x(x1)) -> a(Q(x1))
, b^#(P(x1)) -> c_3(b^#(Q(x1)))
, Q(a(x1)) -> b(b(a(x1)))
, Q^#(a(x1)) -> c_5(b^#(b(a(x1))))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ a_0(14) -> 14
, P_0(3) -> 3
, P_0(4) -> 3
, x_0(3) -> 4
, x_0(4) -> 4
, Q_0(3) -> 14
, Q_0(4) -> 14
, b^#_0(3) -> 10
, b^#_0(4) -> 10
, b^#_0(14) -> 13
, c_3_0(13) -> 10
, Q^#_0(3) -> 12
, Q^#_0(4) -> 12}