'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ b(a(a(x1))) -> a(b(c(x1)))
, c(a(x1)) -> a(c(x1))
, b(c(a(x1))) -> a(b(c(x1)))
, c(b(x1)) -> d(x1)
, a(d(x1)) -> d(a(x1))
, d(x1) -> b(a(x1))
, L(a(a(x1))) -> L(a(b(c(x1))))
, c(R(x1)) -> c(b(R(x1)))}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ b^#(a(a(x1))) -> c_0(a^#(b(c(x1))))
, c^#(a(x1)) -> c_1(a^#(c(x1)))
, b^#(c(a(x1))) -> c_2(a^#(b(c(x1))))
, c^#(b(x1)) -> c_3(d^#(x1))
, a^#(d(x1)) -> c_4(d^#(a(x1)))
, d^#(x1) -> c_5(b^#(a(x1)))
, L^#(a(a(x1))) -> c_6(L^#(a(b(c(x1)))))
, c^#(R(x1)) -> c_7(c^#(b(R(x1))))}
The usable rules are:
{ b(a(a(x1))) -> a(b(c(x1)))
, c(a(x1)) -> a(c(x1))
, b(c(a(x1))) -> a(b(c(x1)))
, c(b(x1)) -> d(x1)
, a(d(x1)) -> d(a(x1))
, c(R(x1)) -> c(b(R(x1)))
, d(x1) -> b(a(x1))}
The estimated dependency graph contains the following edges:
{b^#(a(a(x1))) -> c_0(a^#(b(c(x1))))}
==> {a^#(d(x1)) -> c_4(d^#(a(x1)))}
{c^#(a(x1)) -> c_1(a^#(c(x1)))}
==> {a^#(d(x1)) -> c_4(d^#(a(x1)))}
{b^#(c(a(x1))) -> c_2(a^#(b(c(x1))))}
==> {a^#(d(x1)) -> c_4(d^#(a(x1)))}
{c^#(b(x1)) -> c_3(d^#(x1))}
==> {d^#(x1) -> c_5(b^#(a(x1)))}
{a^#(d(x1)) -> c_4(d^#(a(x1)))}
==> {d^#(x1) -> c_5(b^#(a(x1)))}
{d^#(x1) -> c_5(b^#(a(x1)))}
==> {b^#(c(a(x1))) -> c_2(a^#(b(c(x1))))}
{d^#(x1) -> c_5(b^#(a(x1)))}
==> {b^#(a(a(x1))) -> c_0(a^#(b(c(x1))))}
{L^#(a(a(x1))) -> c_6(L^#(a(b(c(x1)))))}
==> {L^#(a(a(x1))) -> c_6(L^#(a(b(c(x1)))))}
{c^#(R(x1)) -> c_7(c^#(b(R(x1))))}
==> {c^#(b(x1)) -> c_3(d^#(x1))}
We consider the following path(s):
1) { c^#(a(x1)) -> c_1(a^#(c(x1)))
, b^#(a(a(x1))) -> c_0(a^#(b(c(x1))))
, d^#(x1) -> c_5(b^#(a(x1)))
, a^#(d(x1)) -> c_4(d^#(a(x1)))
, b^#(c(a(x1))) -> c_2(a^#(b(c(x1))))}
The usable rules for this path are the following:
{ b(a(a(x1))) -> a(b(c(x1)))
, c(a(x1)) -> a(c(x1))
, b(c(a(x1))) -> a(b(c(x1)))
, c(b(x1)) -> d(x1)
, a(d(x1)) -> d(a(x1))
, c(R(x1)) -> c(b(R(x1)))
, d(x1) -> 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:
{ b(a(a(x1))) -> a(b(c(x1)))
, c(a(x1)) -> a(c(x1))
, b(c(a(x1))) -> a(b(c(x1)))
, c(b(x1)) -> d(x1)
, a(d(x1)) -> d(a(x1))
, c(R(x1)) -> c(b(R(x1)))
, d(x1) -> b(a(x1))
, c^#(a(x1)) -> c_1(a^#(c(x1)))
, b^#(a(a(x1))) -> c_0(a^#(b(c(x1))))
, d^#(x1) -> c_5(b^#(a(x1)))
, a^#(d(x1)) -> c_4(d^#(a(x1)))
, b^#(c(a(x1))) -> c_2(a^#(b(c(x1))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ c(b(x1)) -> d(x1)
, d^#(x1) -> c_5(b^#(a(x1)))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ c(b(x1)) -> d(x1)
, d^#(x1) -> c_5(b^#(a(x1)))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
L(x1) = [0] x1 + [0]
R(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [4]
c_4(x1) = [1] x1 + [1]
c_5(x1) = [1] x1 + [0]
L^#(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
{ b^#(a(a(x1))) -> c_0(a^#(b(c(x1))))
, b^#(c(a(x1))) -> c_2(a^#(b(c(x1))))}
and weakly orienting the rules
{ c(b(x1)) -> d(x1)
, d^#(x1) -> c_5(b^#(a(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ b^#(a(a(x1))) -> c_0(a^#(b(c(x1))))
, b^#(c(a(x1))) -> c_2(a^#(b(c(x1))))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
L(x1) = [0] x1 + [0]
R(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [9]
c_0(x1) = [1] x1 + [7]
a^#(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [12]
c_4(x1) = [1] x1 + [13]
c_5(x1) = [1] x1 + [3]
L^#(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
{c^#(a(x1)) -> c_1(a^#(c(x1)))}
and weakly orienting the rules
{ b^#(a(a(x1))) -> c_0(a^#(b(c(x1))))
, b^#(c(a(x1))) -> c_2(a^#(b(c(x1))))
, c(b(x1)) -> d(x1)
, d^#(x1) -> c_5(b^#(a(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c^#(a(x1)) -> c_1(a^#(c(x1)))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
L(x1) = [0] x1 + [0]
R(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [2]
c_0(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [1]
c_3(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [5]
c_4(x1) = [1] x1 + [1]
c_5(x1) = [1] x1 + [1]
L^#(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
{b(a(a(x1))) -> a(b(c(x1)))}
and weakly orienting the rules
{ c^#(a(x1)) -> c_1(a^#(c(x1)))
, b^#(a(a(x1))) -> c_0(a^#(b(c(x1))))
, b^#(c(a(x1))) -> c_2(a^#(b(c(x1))))
, c(b(x1)) -> d(x1)
, d^#(x1) -> c_5(b^#(a(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b(a(a(x1))) -> a(b(c(x1)))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [9]
c(x1) = [1] x1 + [4]
d(x1) = [1] x1 + [5]
L(x1) = [0] x1 + [0]
R(x1) = [1] x1 + [6]
b^#(x1) = [1] x1 + [3]
c_0(x1) = [1] x1 + [2]
a^#(x1) = [1] x1 + [11]
c^#(x1) = [1] x1 + [8]
c_1(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [13]
c_4(x1) = [1] x1 + [1]
c_5(x1) = [1] x1 + [1]
L^#(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:
{ c(a(x1)) -> a(c(x1))
, b(c(a(x1))) -> a(b(c(x1)))
, a(d(x1)) -> d(a(x1))
, c(R(x1)) -> c(b(R(x1)))
, d(x1) -> b(a(x1))
, a^#(d(x1)) -> c_4(d^#(a(x1)))}
Weak Rules:
{ b(a(a(x1))) -> a(b(c(x1)))
, c^#(a(x1)) -> c_1(a^#(c(x1)))
, b^#(a(a(x1))) -> c_0(a^#(b(c(x1))))
, b^#(c(a(x1))) -> c_2(a^#(b(c(x1))))
, c(b(x1)) -> d(x1)
, d^#(x1) -> c_5(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:
{ c(a(x1)) -> a(c(x1))
, b(c(a(x1))) -> a(b(c(x1)))
, a(d(x1)) -> d(a(x1))
, c(R(x1)) -> c(b(R(x1)))
, d(x1) -> b(a(x1))
, a^#(d(x1)) -> c_4(d^#(a(x1)))}
Weak Rules:
{ b(a(a(x1))) -> a(b(c(x1)))
, c^#(a(x1)) -> c_1(a^#(c(x1)))
, b^#(a(a(x1))) -> c_0(a^#(b(c(x1))))
, b^#(c(a(x1))) -> c_2(a^#(b(c(x1))))
, c(b(x1)) -> d(x1)
, d^#(x1) -> c_5(b^#(a(x1)))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ a_0(6) -> 16
, R_0(6) -> 6
, b^#_0(6) -> 7
, b^#_0(16) -> 15
, a^#_0(6) -> 9
, c^#_0(6) -> 10
, d^#_0(6) -> 14
, c_5_0(15) -> 14}
2) { c^#(R(x1)) -> c_7(c^#(b(R(x1))))
, c^#(b(x1)) -> c_3(d^#(x1))
, b^#(a(a(x1))) -> c_0(a^#(b(c(x1))))
, d^#(x1) -> c_5(b^#(a(x1)))
, a^#(d(x1)) -> c_4(d^#(a(x1)))
, b^#(c(a(x1))) -> c_2(a^#(b(c(x1))))}
The usable rules for this path are the following:
{ b(a(a(x1))) -> a(b(c(x1)))
, c(a(x1)) -> a(c(x1))
, b(c(a(x1))) -> a(b(c(x1)))
, c(b(x1)) -> d(x1)
, a(d(x1)) -> d(a(x1))
, c(R(x1)) -> c(b(R(x1)))
, d(x1) -> 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:
{ b(a(a(x1))) -> a(b(c(x1)))
, c(a(x1)) -> a(c(x1))
, b(c(a(x1))) -> a(b(c(x1)))
, c(b(x1)) -> d(x1)
, a(d(x1)) -> d(a(x1))
, c(R(x1)) -> c(b(R(x1)))
, d(x1) -> b(a(x1))
, c^#(b(x1)) -> c_3(d^#(x1))
, c^#(R(x1)) -> c_7(c^#(b(R(x1))))
, b^#(a(a(x1))) -> c_0(a^#(b(c(x1))))
, d^#(x1) -> c_5(b^#(a(x1)))
, a^#(d(x1)) -> c_4(d^#(a(x1)))
, b^#(c(a(x1))) -> c_2(a^#(b(c(x1))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ c(b(x1)) -> d(x1)
, a^#(d(x1)) -> c_4(d^#(a(x1)))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ c(b(x1)) -> d(x1)
, a^#(d(x1)) -> c_4(d^#(a(x1)))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
L(x1) = [0] x1 + [0]
R(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [4]
c^#(x1) = [1] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [1] x1 + [3]
c_3(x1) = [1] x1 + [1]
d^#(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [1]
c_5(x1) = [1] x1 + [0]
L^#(x1) = [0] 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
{ b^#(a(a(x1))) -> c_0(a^#(b(c(x1))))
, b^#(c(a(x1))) -> c_2(a^#(b(c(x1))))}
and weakly orienting the rules
{ c(b(x1)) -> d(x1)
, a^#(d(x1)) -> c_4(d^#(a(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ b^#(a(a(x1))) -> c_0(a^#(b(c(x1))))
, b^#(c(a(x1))) -> c_2(a^#(b(c(x1))))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
L(x1) = [0] x1 + [0]
R(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [9]
c_0(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [1] x1 + [3]
c_3(x1) = [1] x1 + [1]
d^#(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [1] x1 + [3]
L^#(x1) = [0] 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
{c^#(b(x1)) -> c_3(d^#(x1))}
and weakly orienting the rules
{ b^#(a(a(x1))) -> c_0(a^#(b(c(x1))))
, b^#(c(a(x1))) -> c_2(a^#(b(c(x1))))
, c(b(x1)) -> d(x1)
, a^#(d(x1)) -> c_4(d^#(a(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c^#(b(x1)) -> c_3(d^#(x1))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
L(x1) = [0] x1 + [0]
R(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [4]
c_0(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [1] x1 + [5]
L^#(x1) = [0] 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
{d(x1) -> b(a(x1))}
and weakly orienting the rules
{ c^#(b(x1)) -> c_3(d^#(x1))
, b^#(a(a(x1))) -> c_0(a^#(b(c(x1))))
, b^#(c(a(x1))) -> c_2(a^#(b(c(x1))))
, c(b(x1)) -> d(x1)
, a^#(d(x1)) -> c_4(d^#(a(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d(x1) -> b(a(x1))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [2]
a(x1) = [1] x1 + [10]
c(x1) = [1] x1 + [14]
d(x1) = [1] x1 + [14]
L(x1) = [0] x1 + [0]
R(x1) = [1] x1 + [4]
b^#(x1) = [1] x1 + [8]
c_0(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [2]
c^#(x1) = [1] x1 + [4]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [1] x1 + [2]
c_3(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [6]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [1] x1 + [0]
L^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [1] x1 + [4]
Finally we apply the subprocessor
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ b(a(a(x1))) -> a(b(c(x1)))
, c(a(x1)) -> a(c(x1))
, b(c(a(x1))) -> a(b(c(x1)))
, a(d(x1)) -> d(a(x1))
, c(R(x1)) -> c(b(R(x1)))
, c^#(R(x1)) -> c_7(c^#(b(R(x1))))
, d^#(x1) -> c_5(b^#(a(x1)))}
Weak Rules:
{ d(x1) -> b(a(x1))
, c^#(b(x1)) -> c_3(d^#(x1))
, b^#(a(a(x1))) -> c_0(a^#(b(c(x1))))
, b^#(c(a(x1))) -> c_2(a^#(b(c(x1))))
, c(b(x1)) -> d(x1)
, a^#(d(x1)) -> c_4(d^#(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:
{ b(a(a(x1))) -> a(b(c(x1)))
, c(a(x1)) -> a(c(x1))
, b(c(a(x1))) -> a(b(c(x1)))
, a(d(x1)) -> d(a(x1))
, c(R(x1)) -> c(b(R(x1)))
, c^#(R(x1)) -> c_7(c^#(b(R(x1))))
, d^#(x1) -> c_5(b^#(a(x1)))}
Weak Rules:
{ d(x1) -> b(a(x1))
, c^#(b(x1)) -> c_3(d^#(x1))
, b^#(a(a(x1))) -> c_0(a^#(b(c(x1))))
, b^#(c(a(x1))) -> c_2(a^#(b(c(x1))))
, c(b(x1)) -> d(x1)
, a^#(d(x1)) -> c_4(d^#(a(x1)))}
Details:
The problem is Match-bounded by 2.
The enriched problem is compatible with the following automaton:
{ b_1(5) -> 4
, a_1(2) -> 7
, a_2(5) -> 10
, R_0(2) -> 2
, R_1(2) -> 5
, b^#_0(2) -> 1
, b^#_1(7) -> 6
, b^#_2(10) -> 9
, a^#_0(2) -> 1
, c^#_0(2) -> 1
, c^#_1(4) -> 3
, c_3_1(8) -> 3
, d^#_0(2) -> 1
, d^#_1(5) -> 8
, c_5_1(6) -> 1
, c_5_2(9) -> 8
, c_7_1(3) -> 1}
3) { c^#(R(x1)) -> c_7(c^#(b(R(x1))))
, c^#(b(x1)) -> c_3(d^#(x1))}
The usable rules for this path are the following:
{ b(a(a(x1))) -> a(b(c(x1)))
, b(c(a(x1))) -> a(b(c(x1)))
, c(a(x1)) -> a(c(x1))
, c(b(x1)) -> d(x1)
, a(d(x1)) -> d(a(x1))
, c(R(x1)) -> c(b(R(x1)))
, d(x1) -> 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:
{ b(a(a(x1))) -> a(b(c(x1)))
, b(c(a(x1))) -> a(b(c(x1)))
, c(a(x1)) -> a(c(x1))
, c(b(x1)) -> d(x1)
, a(d(x1)) -> d(a(x1))
, c(R(x1)) -> c(b(R(x1)))
, d(x1) -> b(a(x1))
, c^#(R(x1)) -> c_7(c^#(b(R(x1))))
, c^#(b(x1)) -> c_3(d^#(x1))}
Details:
We apply the weight gap principle, strictly orienting the rules
{d(x1) -> b(a(x1))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d(x1) -> b(a(x1))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [4]
L(x1) = [0] x1 + [0]
R(x1) = [1] x1 + [0]
b^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a^#(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [1]
d^#(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
L^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{c^#(b(x1)) -> c_3(d^#(x1))}
and weakly orienting the rules
{d(x1) -> b(a(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c^#(b(x1)) -> c_3(d^#(x1))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [8]
L(x1) = [0] x1 + [0]
R(x1) = [1] x1 + [0]
b^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a^#(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [1]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [1]
d^#(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
L^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b(a(a(x1))) -> a(b(c(x1)))}
and weakly orienting the rules
{ c^#(b(x1)) -> c_3(d^#(x1))
, d(x1) -> b(a(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b(a(a(x1))) -> a(b(c(x1)))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [10]
c(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [10]
L(x1) = [0] x1 + [0]
R(x1) = [1] x1 + [0]
b^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a^#(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
L^#(x1) = [0] 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:
{ b(c(a(x1))) -> a(b(c(x1)))
, c(a(x1)) -> a(c(x1))
, c(b(x1)) -> d(x1)
, a(d(x1)) -> d(a(x1))
, c(R(x1)) -> c(b(R(x1)))
, c^#(R(x1)) -> c_7(c^#(b(R(x1))))}
Weak Rules:
{ b(a(a(x1))) -> a(b(c(x1)))
, c^#(b(x1)) -> c_3(d^#(x1))
, d(x1) -> 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:
{ b(c(a(x1))) -> a(b(c(x1)))
, c(a(x1)) -> a(c(x1))
, c(b(x1)) -> d(x1)
, a(d(x1)) -> d(a(x1))
, c(R(x1)) -> c(b(R(x1)))
, c^#(R(x1)) -> c_7(c^#(b(R(x1))))}
Weak Rules:
{ b(a(a(x1))) -> a(b(c(x1)))
, c^#(b(x1)) -> c_3(d^#(x1))
, d(x1) -> b(a(x1))}
Details:
The problem is Match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ b_1(17) -> 16
, R_0(6) -> 6
, R_1(6) -> 17
, c^#_0(6) -> 10
, c^#_1(16) -> 15
, c_3_1(18) -> 15
, d^#_0(6) -> 14
, d^#_1(17) -> 18
, c_7_1(15) -> 10}
4) {c^#(a(x1)) -> c_1(a^#(c(x1)))}
The usable rules for this path are the following:
{ c(a(x1)) -> a(c(x1))
, c(b(x1)) -> d(x1)
, c(R(x1)) -> c(b(R(x1)))
, b(a(a(x1))) -> a(b(c(x1)))
, b(c(a(x1))) -> a(b(c(x1)))
, a(d(x1)) -> d(a(x1))
, d(x1) -> 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:
{ c(a(x1)) -> a(c(x1))
, c(b(x1)) -> d(x1)
, c(R(x1)) -> c(b(R(x1)))
, b(a(a(x1))) -> a(b(c(x1)))
, b(c(a(x1))) -> a(b(c(x1)))
, a(d(x1)) -> d(a(x1))
, d(x1) -> b(a(x1))
, c^#(a(x1)) -> c_1(a^#(c(x1)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{c(b(x1)) -> d(x1)}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c(b(x1)) -> d(x1)}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [0]
L(x1) = [0] x1 + [0]
R(x1) = [1] x1 + [0]
b^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a^#(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [3]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
d^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
L^#(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
{c^#(a(x1)) -> c_1(a^#(c(x1)))}
and weakly orienting the rules
{c(b(x1)) -> d(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c^#(a(x1)) -> c_1(a^#(c(x1)))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [0]
L(x1) = [0] x1 + [0]
R(x1) = [1] x1 + [0]
b^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a^#(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [9]
c_1(x1) = [1] x1 + [2]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
d^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
L^#(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
{d(x1) -> b(a(x1))}
and weakly orienting the rules
{ c^#(a(x1)) -> c_1(a^#(c(x1)))
, c(b(x1)) -> d(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d(x1) -> b(a(x1))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [1]
L(x1) = [0] x1 + [0]
R(x1) = [1] x1 + [0]
b^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a^#(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
d^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
L^#(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:
{ c(a(x1)) -> a(c(x1))
, c(R(x1)) -> c(b(R(x1)))
, b(a(a(x1))) -> a(b(c(x1)))
, b(c(a(x1))) -> a(b(c(x1)))
, a(d(x1)) -> d(a(x1))}
Weak Rules:
{ d(x1) -> b(a(x1))
, c^#(a(x1)) -> c_1(a^#(c(x1)))
, c(b(x1)) -> d(x1)}
Details:
The problem was solved by processor 'Bounds with default enrichment':
'Bounds with default enrichment'
--------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ c(a(x1)) -> a(c(x1))
, c(R(x1)) -> c(b(R(x1)))
, b(a(a(x1))) -> a(b(c(x1)))
, b(c(a(x1))) -> a(b(c(x1)))
, a(d(x1)) -> d(a(x1))}
Weak Rules:
{ d(x1) -> b(a(x1))
, c^#(a(x1)) -> c_1(a^#(c(x1)))
, c(b(x1)) -> d(x1)}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ R_0(6) -> 6
, a^#_0(6) -> 9
, c^#_0(6) -> 10}
5) {c^#(R(x1)) -> c_7(c^#(b(R(x1))))}
The usable rules for this path are the following:
{ b(a(a(x1))) -> a(b(c(x1)))
, b(c(a(x1))) -> a(b(c(x1)))
, c(a(x1)) -> a(c(x1))
, c(b(x1)) -> d(x1)
, a(d(x1)) -> d(a(x1))
, c(R(x1)) -> c(b(R(x1)))
, d(x1) -> 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:
{ b(a(a(x1))) -> a(b(c(x1)))
, b(c(a(x1))) -> a(b(c(x1)))
, c(a(x1)) -> a(c(x1))
, c(b(x1)) -> d(x1)
, a(d(x1)) -> d(a(x1))
, c(R(x1)) -> c(b(R(x1)))
, d(x1) -> b(a(x1))
, c^#(R(x1)) -> c_7(c^#(b(R(x1))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{c(b(x1)) -> d(x1)}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c(b(x1)) -> d(x1)}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
L(x1) = [0] x1 + [0]
R(x1) = [1] x1 + [0]
b^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a^#(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
d^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
L^#(x1) = [0] 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
{b(a(a(x1))) -> a(b(c(x1)))}
and weakly orienting the rules
{c(b(x1)) -> d(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b(a(a(x1))) -> a(b(c(x1)))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [8]
c(x1) = [1] x1 + [4]
d(x1) = [1] x1 + [0]
L(x1) = [0] x1 + [0]
R(x1) = [1] x1 + [0]
b^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a^#(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [1]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
d^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
L^#(x1) = [0] 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:
{ b(c(a(x1))) -> a(b(c(x1)))
, c(a(x1)) -> a(c(x1))
, a(d(x1)) -> d(a(x1))
, c(R(x1)) -> c(b(R(x1)))
, d(x1) -> b(a(x1))
, c^#(R(x1)) -> c_7(c^#(b(R(x1))))}
Weak Rules:
{ b(a(a(x1))) -> a(b(c(x1)))
, c(b(x1)) -> d(x1)}
Details:
The problem was solved by processor 'Bounds with default enrichment':
'Bounds with default enrichment'
--------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ b(c(a(x1))) -> a(b(c(x1)))
, c(a(x1)) -> a(c(x1))
, a(d(x1)) -> d(a(x1))
, c(R(x1)) -> c(b(R(x1)))
, d(x1) -> b(a(x1))
, c^#(R(x1)) -> c_7(c^#(b(R(x1))))}
Weak Rules:
{ b(a(a(x1))) -> a(b(c(x1)))
, c(b(x1)) -> d(x1)}
Details:
The problem is Match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ b_1(13) -> 12
, R_0(6) -> 6
, R_1(6) -> 13
, c^#_0(6) -> 10
, c^#_1(12) -> 11
, c_7_1(11) -> 10}
6) {L^#(a(a(x1))) -> c_6(L^#(a(b(c(x1)))))}
The usable rules for this path are the following:
{ b(a(a(x1))) -> a(b(c(x1)))
, c(a(x1)) -> a(c(x1))
, b(c(a(x1))) -> a(b(c(x1)))
, c(b(x1)) -> d(x1)
, a(d(x1)) -> d(a(x1))
, c(R(x1)) -> c(b(R(x1)))
, d(x1) -> 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:
{ b(a(a(x1))) -> a(b(c(x1)))
, c(a(x1)) -> a(c(x1))
, b(c(a(x1))) -> a(b(c(x1)))
, c(b(x1)) -> d(x1)
, a(d(x1)) -> d(a(x1))
, c(R(x1)) -> c(b(R(x1)))
, d(x1) -> b(a(x1))
, L^#(a(a(x1))) -> c_6(L^#(a(b(c(x1)))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{c(b(x1)) -> d(x1)}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c(b(x1)) -> d(x1)}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
L(x1) = [0] x1 + [0]
R(x1) = [1] x1 + [8]
b^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a^#(x1) = [0] x1 + [0]
c^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
d^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
L^#(x1) = [1] x1 + [1]
c_6(x1) = [1] x1 + [1]
c_7(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ b(a(a(x1))) -> a(b(c(x1)))
, L^#(a(a(x1))) -> c_6(L^#(a(b(c(x1)))))}
and weakly orienting the rules
{c(b(x1)) -> d(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ b(a(a(x1))) -> a(b(c(x1)))
, L^#(a(a(x1))) -> c_6(L^#(a(b(c(x1)))))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [2]
a(x1) = [1] x1 + [14]
c(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [1]
L(x1) = [0] x1 + [0]
R(x1) = [1] x1 + [8]
b^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a^#(x1) = [0] x1 + [0]
c^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
d^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
L^#(x1) = [1] x1 + [0]
c_6(x1) = [1] x1 + [5]
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:
{ c(a(x1)) -> a(c(x1))
, b(c(a(x1))) -> a(b(c(x1)))
, a(d(x1)) -> d(a(x1))
, c(R(x1)) -> c(b(R(x1)))
, d(x1) -> b(a(x1))}
Weak Rules:
{ b(a(a(x1))) -> a(b(c(x1)))
, L^#(a(a(x1))) -> c_6(L^#(a(b(c(x1)))))
, c(b(x1)) -> d(x1)}
Details:
The problem was solved by processor 'Bounds with default enrichment':
'Bounds with default enrichment'
--------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ c(a(x1)) -> a(c(x1))
, b(c(a(x1))) -> a(b(c(x1)))
, a(d(x1)) -> d(a(x1))
, c(R(x1)) -> c(b(R(x1)))
, d(x1) -> b(a(x1))}
Weak Rules:
{ b(a(a(x1))) -> a(b(c(x1)))
, L^#(a(a(x1))) -> c_6(L^#(a(b(c(x1)))))
, c(b(x1)) -> d(x1)}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ R_0(6) -> 6
, L^#_0(6) -> 17}