'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ a(c(a(x1))) -> c(a(c(x1)))
, a(a(b(x1))) -> a(d(b(x1)))
, a(b(x1)) -> b(a(a(x1)))
, d(d(x1)) -> a(d(b(x1)))
, b(b(x1)) -> b(c(x1))
, a(d(c(x1))) -> c(a(x1))
, b(c(x1)) -> a(a(a(x1)))}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ a^#(c(a(x1))) -> c_0(a^#(c(x1)))
, a^#(a(b(x1))) -> c_1(a^#(d(b(x1))))
, a^#(b(x1)) -> c_2(b^#(a(a(x1))))
, d^#(d(x1)) -> c_3(a^#(d(b(x1))))
, b^#(b(x1)) -> c_4(b^#(c(x1)))
, a^#(d(c(x1))) -> c_5(a^#(x1))
, b^#(c(x1)) -> c_6(a^#(a(a(x1))))}
The usable rules are:
{ a(c(a(x1))) -> c(a(c(x1)))
, a(a(b(x1))) -> a(d(b(x1)))
, a(b(x1)) -> b(a(a(x1)))
, d(d(x1)) -> a(d(b(x1)))
, b(b(x1)) -> b(c(x1))
, a(d(c(x1))) -> c(a(x1))
, b(c(x1)) -> a(a(a(x1)))}
The estimated dependency graph contains the following edges:
{a^#(c(a(x1))) -> c_0(a^#(c(x1)))}
==> {a^#(c(a(x1))) -> c_0(a^#(c(x1)))}
{a^#(a(b(x1))) -> c_1(a^#(d(b(x1))))}
==> {a^#(d(c(x1))) -> c_5(a^#(x1))}
{a^#(a(b(x1))) -> c_1(a^#(d(b(x1))))}
==> {a^#(b(x1)) -> c_2(b^#(a(a(x1))))}
{a^#(a(b(x1))) -> c_1(a^#(d(b(x1))))}
==> {a^#(a(b(x1))) -> c_1(a^#(d(b(x1))))}
{a^#(a(b(x1))) -> c_1(a^#(d(b(x1))))}
==> {a^#(c(a(x1))) -> c_0(a^#(c(x1)))}
{a^#(b(x1)) -> c_2(b^#(a(a(x1))))}
==> {b^#(c(x1)) -> c_6(a^#(a(a(x1))))}
{a^#(b(x1)) -> c_2(b^#(a(a(x1))))}
==> {b^#(b(x1)) -> c_4(b^#(c(x1)))}
{d^#(d(x1)) -> c_3(a^#(d(b(x1))))}
==> {a^#(d(c(x1))) -> c_5(a^#(x1))}
{d^#(d(x1)) -> c_3(a^#(d(b(x1))))}
==> {a^#(b(x1)) -> c_2(b^#(a(a(x1))))}
{d^#(d(x1)) -> c_3(a^#(d(b(x1))))}
==> {a^#(a(b(x1))) -> c_1(a^#(d(b(x1))))}
{d^#(d(x1)) -> c_3(a^#(d(b(x1))))}
==> {a^#(c(a(x1))) -> c_0(a^#(c(x1)))}
{b^#(b(x1)) -> c_4(b^#(c(x1)))}
==> {b^#(c(x1)) -> c_6(a^#(a(a(x1))))}
{a^#(d(c(x1))) -> c_5(a^#(x1))}
==> {a^#(d(c(x1))) -> c_5(a^#(x1))}
{a^#(d(c(x1))) -> c_5(a^#(x1))}
==> {a^#(b(x1)) -> c_2(b^#(a(a(x1))))}
{a^#(d(c(x1))) -> c_5(a^#(x1))}
==> {a^#(a(b(x1))) -> c_1(a^#(d(b(x1))))}
{a^#(d(c(x1))) -> c_5(a^#(x1))}
==> {a^#(c(a(x1))) -> c_0(a^#(c(x1)))}
{b^#(c(x1)) -> c_6(a^#(a(a(x1))))}
==> {a^#(b(x1)) -> c_2(b^#(a(a(x1))))}
{b^#(c(x1)) -> c_6(a^#(a(a(x1))))}
==> {a^#(a(b(x1))) -> c_1(a^#(d(b(x1))))}
{b^#(c(x1)) -> c_6(a^#(a(a(x1))))}
==> {a^#(c(a(x1))) -> c_0(a^#(c(x1)))}
We consider the following path(s):
1) { d^#(d(x1)) -> c_3(a^#(d(b(x1))))
, a^#(a(b(x1))) -> c_1(a^#(d(b(x1))))
, b^#(c(x1)) -> c_6(a^#(a(a(x1))))
, b^#(b(x1)) -> c_4(b^#(c(x1)))
, a^#(b(x1)) -> c_2(b^#(a(a(x1))))
, a^#(d(c(x1))) -> c_5(a^#(x1))
, a^#(c(a(x1))) -> c_0(a^#(c(x1)))}
The usable rules for this path are the following:
{ a(c(a(x1))) -> c(a(c(x1)))
, a(a(b(x1))) -> a(d(b(x1)))
, a(b(x1)) -> b(a(a(x1)))
, d(d(x1)) -> a(d(b(x1)))
, b(b(x1)) -> b(c(x1))
, a(d(c(x1))) -> c(a(x1))
, b(c(x1)) -> a(a(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(c(a(x1))) -> c(a(c(x1)))
, a(a(b(x1))) -> a(d(b(x1)))
, a(b(x1)) -> b(a(a(x1)))
, d(d(x1)) -> a(d(b(x1)))
, b(b(x1)) -> b(c(x1))
, a(d(c(x1))) -> c(a(x1))
, b(c(x1)) -> a(a(a(x1)))
, a^#(a(b(x1))) -> c_1(a^#(d(b(x1))))
, b^#(c(x1)) -> c_6(a^#(a(a(x1))))
, b^#(b(x1)) -> c_4(b^#(c(x1)))
, a^#(b(x1)) -> c_2(b^#(a(a(x1))))
, a^#(d(c(x1))) -> c_5(a^#(x1))
, d^#(d(x1)) -> c_3(a^#(d(b(x1))))
, a^#(c(a(x1))) -> c_0(a^#(c(x1)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ d(d(x1)) -> a(d(b(x1)))
, a(d(c(x1))) -> c(a(x1))
, a^#(d(c(x1))) -> c_5(a^#(x1))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ d(d(x1)) -> a(d(b(x1)))
, a(d(c(x1))) -> c(a(x1))
, a^#(d(c(x1))) -> c_5(a^#(x1))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [8]
a^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [1]
b^#(x1) = [1] x1 + [1]
d^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
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
{d^#(d(x1)) -> c_3(a^#(d(b(x1))))}
and weakly orienting the rules
{ d(d(x1)) -> a(d(b(x1)))
, a(d(c(x1))) -> c(a(x1))
, a^#(d(c(x1))) -> c_5(a^#(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d^#(d(x1)) -> c_3(a^#(d(b(x1))))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [7]
c_1(x1) = [1] x1 + [3]
c_2(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [1]
d^#(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) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a^#(b(x1)) -> c_2(b^#(a(a(x1))))}
and weakly orienting the rules
{ d^#(d(x1)) -> c_3(a^#(d(b(x1))))
, d(d(x1)) -> a(d(b(x1)))
, a(d(c(x1))) -> c(a(x1))
, a^#(d(c(x1))) -> c_5(a^#(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a^#(b(x1)) -> c_2(b^#(a(a(x1))))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
d(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) = [1] x1 + [0]
d^#(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) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ b(b(x1)) -> b(c(x1))
, b(c(x1)) -> a(a(a(x1)))
, b^#(b(x1)) -> c_4(b^#(c(x1)))}
and weakly orienting the rules
{ a^#(b(x1)) -> c_2(b^#(a(a(x1))))
, d^#(d(x1)) -> c_3(a^#(d(b(x1))))
, d(d(x1)) -> a(d(b(x1)))
, a(d(c(x1))) -> c(a(x1))
, a^#(d(c(x1))) -> c_5(a^#(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ b(b(x1)) -> b(c(x1))
, b(c(x1)) -> a(a(a(x1)))
, b^#(b(x1)) -> c_4(b^#(c(x1)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [1]
b(x1) = [1] x1 + [8]
d(x1) = [1] x1 + [15]
a^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [4]
c_1(x1) = [1] x1 + [5]
c_2(x1) = [1] x1 + [1]
b^#(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [12]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [1] x1 + [9]
c_6(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b^#(c(x1)) -> c_6(a^#(a(a(x1))))}
and weakly orienting the rules
{ b(b(x1)) -> b(c(x1))
, b(c(x1)) -> a(a(a(x1)))
, b^#(b(x1)) -> c_4(b^#(c(x1)))
, a^#(b(x1)) -> c_2(b^#(a(a(x1))))
, d^#(d(x1)) -> c_3(a^#(d(b(x1))))
, d(d(x1)) -> a(d(b(x1)))
, a(d(c(x1))) -> c(a(x1))
, a^#(d(c(x1))) -> c_5(a^#(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b^#(c(x1)) -> c_6(a^#(a(a(x1))))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [4]
b(x1) = [1] x1 + [8]
d(x1) = [1] x1 + [8]
a^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [2]
c_1(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [5]
d^#(x1) = [1] x1 + [9]
c_3(x1) = [1] x1 + [1]
c_4(x1) = [1] x1 + [1]
c_5(x1) = [1] x1 + [4]
c_6(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a^#(c(a(x1))) -> c_0(a^#(c(x1)))}
and weakly orienting the rules
{ b^#(c(x1)) -> c_6(a^#(a(a(x1))))
, b(b(x1)) -> b(c(x1))
, b(c(x1)) -> a(a(a(x1)))
, b^#(b(x1)) -> c_4(b^#(c(x1)))
, a^#(b(x1)) -> c_2(b^#(a(a(x1))))
, d^#(d(x1)) -> c_3(a^#(d(b(x1))))
, d(d(x1)) -> a(d(b(x1)))
, a(d(c(x1))) -> c(a(x1))
, a^#(d(c(x1))) -> c_5(a^#(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a^#(c(a(x1))) -> c_0(a^#(c(x1)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [2]
c(x1) = [1] x1 + [4]
b(x1) = [1] x1 + [4]
d(x1) = [1] x1 + [9]
a^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [7]
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
{a(c(a(x1))) -> c(a(c(x1)))}
and weakly orienting the rules
{ a^#(c(a(x1))) -> c_0(a^#(c(x1)))
, b^#(c(x1)) -> c_6(a^#(a(a(x1))))
, b(b(x1)) -> b(c(x1))
, b(c(x1)) -> a(a(a(x1)))
, b^#(b(x1)) -> c_4(b^#(c(x1)))
, a^#(b(x1)) -> c_2(b^#(a(a(x1))))
, d^#(d(x1)) -> c_3(a^#(d(b(x1))))
, d(d(x1)) -> a(d(b(x1)))
, a(d(c(x1))) -> c(a(x1))
, a^#(d(c(x1))) -> c_5(a^#(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a(c(a(x1))) -> c(a(c(x1)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [2]
c(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [8]
d(x1) = [1] x1 + [10]
a^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [4]
d^#(x1) = [1] x1 + [8]
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
'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(a(b(x1))) -> a(d(b(x1)))
, a(b(x1)) -> b(a(a(x1)))
, a^#(a(b(x1))) -> c_1(a^#(d(b(x1))))}
Weak Rules:
{ a(c(a(x1))) -> c(a(c(x1)))
, a^#(c(a(x1))) -> c_0(a^#(c(x1)))
, b^#(c(x1)) -> c_6(a^#(a(a(x1))))
, b(b(x1)) -> b(c(x1))
, b(c(x1)) -> a(a(a(x1)))
, b^#(b(x1)) -> c_4(b^#(c(x1)))
, a^#(b(x1)) -> c_2(b^#(a(a(x1))))
, d^#(d(x1)) -> c_3(a^#(d(b(x1))))
, d(d(x1)) -> a(d(b(x1)))
, a(d(c(x1))) -> c(a(x1))
, a^#(d(c(x1))) -> c_5(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(a(b(x1))) -> a(d(b(x1)))
, a(b(x1)) -> b(a(a(x1)))
, a^#(a(b(x1))) -> c_1(a^#(d(b(x1))))}
Weak Rules:
{ a(c(a(x1))) -> c(a(c(x1)))
, a^#(c(a(x1))) -> c_0(a^#(c(x1)))
, b^#(c(x1)) -> c_6(a^#(a(a(x1))))
, b(b(x1)) -> b(c(x1))
, b(c(x1)) -> a(a(a(x1)))
, b^#(b(x1)) -> c_4(b^#(c(x1)))
, a^#(b(x1)) -> c_2(b^#(a(a(x1))))
, d^#(d(x1)) -> c_3(a^#(d(b(x1))))
, d(d(x1)) -> a(d(b(x1)))
, a(d(c(x1))) -> c(a(x1))
, a^#(d(c(x1))) -> c_5(a^#(x1))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ a_0(2) -> 13
, a_0(13) -> 12
, c_0(2) -> 2
, a^#_0(2) -> 5
, a^#_0(12) -> 11
, b^#_0(2) -> 9
, d^#_0(2) -> 10
, c_6_0(11) -> 9}
2) { d^#(d(x1)) -> c_3(a^#(d(b(x1))))
, a^#(a(b(x1))) -> c_1(a^#(d(b(x1))))
, b^#(c(x1)) -> c_6(a^#(a(a(x1))))
, b^#(b(x1)) -> c_4(b^#(c(x1)))
, a^#(b(x1)) -> c_2(b^#(a(a(x1))))
, a^#(d(c(x1))) -> c_5(a^#(x1))}
The usable rules for this path are the following:
{ a(c(a(x1))) -> c(a(c(x1)))
, a(a(b(x1))) -> a(d(b(x1)))
, a(b(x1)) -> b(a(a(x1)))
, d(d(x1)) -> a(d(b(x1)))
, b(b(x1)) -> b(c(x1))
, a(d(c(x1))) -> c(a(x1))
, b(c(x1)) -> a(a(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(c(a(x1))) -> c(a(c(x1)))
, a(a(b(x1))) -> a(d(b(x1)))
, a(b(x1)) -> b(a(a(x1)))
, d(d(x1)) -> a(d(b(x1)))
, b(b(x1)) -> b(c(x1))
, a(d(c(x1))) -> c(a(x1))
, b(c(x1)) -> a(a(a(x1)))
, d^#(d(x1)) -> c_3(a^#(d(b(x1))))
, a^#(a(b(x1))) -> c_1(a^#(d(b(x1))))
, b^#(c(x1)) -> c_6(a^#(a(a(x1))))
, b^#(b(x1)) -> c_4(b^#(c(x1)))
, a^#(b(x1)) -> c_2(b^#(a(a(x1))))
, a^#(d(c(x1))) -> c_5(a^#(x1))}
Details:
We apply the weight gap principle, strictly orienting the rules
{b^#(c(x1)) -> c_6(a^#(a(a(x1))))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b^#(c(x1)) -> c_6(a^#(a(a(x1))))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [1]
d^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [1]
c_4(x1) = [1] x1 + [7]
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
{d^#(d(x1)) -> c_3(a^#(d(b(x1))))}
and weakly orienting the rules
{b^#(c(x1)) -> c_6(a^#(a(a(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d^#(d(x1)) -> c_3(a^#(d(b(x1))))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [1]
b^#(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [8]
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
{ a(a(b(x1))) -> a(d(b(x1)))
, a(d(c(x1))) -> c(a(x1))
, b(c(x1)) -> a(a(a(x1)))
, a^#(a(b(x1))) -> c_1(a^#(d(b(x1))))
, a^#(b(x1)) -> c_2(b^#(a(a(x1))))
, a^#(d(c(x1))) -> c_5(a^#(x1))}
and weakly orienting the rules
{ d^#(d(x1)) -> c_3(a^#(d(b(x1))))
, b^#(c(x1)) -> c_6(a^#(a(a(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ a(a(b(x1))) -> a(d(b(x1)))
, a(d(c(x1))) -> c(a(x1))
, b(c(x1)) -> a(a(a(x1)))
, a^#(a(b(x1))) -> c_1(a^#(d(b(x1))))
, a^#(b(x1)) -> c_2(b^#(a(a(x1))))
, a^#(d(c(x1))) -> c_5(a^#(x1))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [3]
c(x1) = [1] x1 + [13]
b(x1) = [1] x1 + [11]
d(x1) = [1] x1 + [1]
a^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [1]
d^#(x1) = [1] x1 + [15]
c_3(x1) = [1] x1 + [4]
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
{ b(b(x1)) -> b(c(x1))
, b^#(b(x1)) -> c_4(b^#(c(x1)))}
and weakly orienting the rules
{ a(a(b(x1))) -> a(d(b(x1)))
, a(d(c(x1))) -> c(a(x1))
, b(c(x1)) -> a(a(a(x1)))
, a^#(a(b(x1))) -> c_1(a^#(d(b(x1))))
, a^#(b(x1)) -> c_2(b^#(a(a(x1))))
, a^#(d(c(x1))) -> c_5(a^#(x1))
, d^#(d(x1)) -> c_3(a^#(d(b(x1))))
, b^#(c(x1)) -> c_6(a^#(a(a(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ b(b(x1)) -> b(c(x1))
, b^#(b(x1)) -> c_4(b^#(c(x1)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [8]
d(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [1]
b^#(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [9]
c_3(x1) = [1] x1 + [1]
c_4(x1) = [1] x1 + [1]
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
{a(c(a(x1))) -> c(a(c(x1)))}
and weakly orienting the rules
{ b(b(x1)) -> b(c(x1))
, b^#(b(x1)) -> c_4(b^#(c(x1)))
, a(a(b(x1))) -> a(d(b(x1)))
, a(d(c(x1))) -> c(a(x1))
, b(c(x1)) -> a(a(a(x1)))
, a^#(a(b(x1))) -> c_1(a^#(d(b(x1))))
, a^#(b(x1)) -> c_2(b^#(a(a(x1))))
, a^#(d(c(x1))) -> c_5(a^#(x1))
, d^#(d(x1)) -> c_3(a^#(d(b(x1))))
, b^#(c(x1)) -> c_6(a^#(a(a(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a(c(a(x1))) -> c(a(c(x1)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [4]
c(x1) = [1] x1 + [2]
b(x1) = [1] x1 + [14]
d(x1) = [1] x1 + [2]
a^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [6]
d^#(x1) = [1] x1 + [15]
c_3(x1) = [1] x1 + [1]
c_4(x1) = [1] x1 + [0]
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:
{ a(b(x1)) -> b(a(a(x1)))
, d(d(x1)) -> a(d(b(x1)))}
Weak Rules:
{ a(c(a(x1))) -> c(a(c(x1)))
, b(b(x1)) -> b(c(x1))
, b^#(b(x1)) -> c_4(b^#(c(x1)))
, a(a(b(x1))) -> a(d(b(x1)))
, a(d(c(x1))) -> c(a(x1))
, b(c(x1)) -> a(a(a(x1)))
, a^#(a(b(x1))) -> c_1(a^#(d(b(x1))))
, a^#(b(x1)) -> c_2(b^#(a(a(x1))))
, a^#(d(c(x1))) -> c_5(a^#(x1))
, d^#(d(x1)) -> c_3(a^#(d(b(x1))))
, b^#(c(x1)) -> c_6(a^#(a(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(x1)) -> b(a(a(x1)))
, d(d(x1)) -> a(d(b(x1)))}
Weak Rules:
{ a(c(a(x1))) -> c(a(c(x1)))
, b(b(x1)) -> b(c(x1))
, b^#(b(x1)) -> c_4(b^#(c(x1)))
, a(a(b(x1))) -> a(d(b(x1)))
, a(d(c(x1))) -> c(a(x1))
, b(c(x1)) -> a(a(a(x1)))
, a^#(a(b(x1))) -> c_1(a^#(d(b(x1))))
, a^#(b(x1)) -> c_2(b^#(a(a(x1))))
, a^#(d(c(x1))) -> c_5(a^#(x1))
, d^#(d(x1)) -> c_3(a^#(d(b(x1))))
, b^#(c(x1)) -> c_6(a^#(a(a(x1))))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ a_0(2) -> 13
, a_0(13) -> 12
, c_0(2) -> 2
, a^#_0(2) -> 5
, a^#_0(12) -> 11
, b^#_0(2) -> 9
, d^#_0(2) -> 10
, c_6_0(11) -> 9}
3) { d^#(d(x1)) -> c_3(a^#(d(b(x1))))
, a^#(c(a(x1))) -> c_0(a^#(c(x1)))}
The usable rules for this path are the following:
{ d(d(x1)) -> a(d(b(x1)))
, b(b(x1)) -> b(c(x1))
, b(c(x1)) -> a(a(a(x1)))
, a(c(a(x1))) -> c(a(c(x1)))
, a(a(b(x1))) -> a(d(b(x1)))
, a(b(x1)) -> b(a(a(x1)))
, a(d(c(x1))) -> c(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:
{ d(d(x1)) -> a(d(b(x1)))
, b(b(x1)) -> b(c(x1))
, b(c(x1)) -> a(a(a(x1)))
, a(c(a(x1))) -> c(a(c(x1)))
, a(a(b(x1))) -> a(d(b(x1)))
, a(b(x1)) -> b(a(a(x1)))
, a(d(c(x1))) -> c(a(x1))
, d^#(d(x1)) -> c_3(a^#(d(b(x1))))
, a^#(c(a(x1))) -> c_0(a^#(c(x1)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ a(c(a(x1))) -> c(a(c(x1)))
, a(a(b(x1))) -> a(d(b(x1)))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ a(c(a(x1))) -> c(a(c(x1)))
, a(a(b(x1))) -> a(d(b(x1)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [1]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
b^#(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [1]
c_4(x1) = [0] 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
{a^#(c(a(x1))) -> c_0(a^#(c(x1)))}
and weakly orienting the rules
{ a(c(a(x1))) -> c(a(c(x1)))
, a(a(b(x1))) -> a(d(b(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a^#(c(a(x1))) -> c_0(a^#(c(x1)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
b^#(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [1]
c_4(x1) = [0] 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
{d^#(d(x1)) -> c_3(a^#(d(b(x1))))}
and weakly orienting the rules
{ a^#(c(a(x1))) -> c_0(a^#(c(x1)))
, a(c(a(x1))) -> c(a(c(x1)))
, a(a(b(x1))) -> a(d(b(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d^#(d(x1)) -> c_3(a^#(d(b(x1))))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
b^#(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [9]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [0] 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
{ b(b(x1)) -> b(c(x1))
, b(c(x1)) -> a(a(a(x1)))}
and weakly orienting the rules
{ d^#(d(x1)) -> c_3(a^#(d(b(x1))))
, a^#(c(a(x1))) -> c_0(a^#(c(x1)))
, a(c(a(x1))) -> c(a(c(x1)))
, a(a(b(x1))) -> a(d(b(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ b(b(x1)) -> b(c(x1))
, b(c(x1)) -> a(a(a(x1)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [7]
d(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
b^#(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [9]
c_3(x1) = [1] x1 + [1]
c_4(x1) = [0] 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
{a(d(c(x1))) -> c(a(x1))}
and weakly orienting the rules
{ b(b(x1)) -> b(c(x1))
, b(c(x1)) -> a(a(a(x1)))
, d^#(d(x1)) -> c_3(a^#(d(b(x1))))
, a^#(c(a(x1))) -> c_0(a^#(c(x1)))
, a(c(a(x1))) -> c(a(c(x1)))
, a(a(b(x1))) -> a(d(b(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a(d(c(x1))) -> c(a(x1))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [1]
b(x1) = [1] x1 + [8]
d(x1) = [1] x1 + [1]
a^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
b^#(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [15]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
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:
{ d(d(x1)) -> a(d(b(x1)))
, a(b(x1)) -> b(a(a(x1)))}
Weak Rules:
{ a(d(c(x1))) -> c(a(x1))
, b(b(x1)) -> b(c(x1))
, b(c(x1)) -> a(a(a(x1)))
, d^#(d(x1)) -> c_3(a^#(d(b(x1))))
, a^#(c(a(x1))) -> c_0(a^#(c(x1)))
, a(c(a(x1))) -> c(a(c(x1)))
, a(a(b(x1))) -> a(d(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:
{ d(d(x1)) -> a(d(b(x1)))
, a(b(x1)) -> b(a(a(x1)))}
Weak Rules:
{ a(d(c(x1))) -> c(a(x1))
, b(b(x1)) -> b(c(x1))
, b(c(x1)) -> a(a(a(x1)))
, d^#(d(x1)) -> c_3(a^#(d(b(x1))))
, a^#(c(a(x1))) -> c_0(a^#(c(x1)))
, a(c(a(x1))) -> c(a(c(x1)))
, a(a(b(x1))) -> a(d(b(x1)))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ c_0(2) -> 2
, a^#_0(2) -> 5
, d^#_0(2) -> 10}
4) {d^#(d(x1)) -> c_3(a^#(d(b(x1))))}
The usable rules for this path are the following:
{ d(d(x1)) -> a(d(b(x1)))
, b(b(x1)) -> b(c(x1))
, b(c(x1)) -> a(a(a(x1)))
, a(c(a(x1))) -> c(a(c(x1)))
, a(a(b(x1))) -> a(d(b(x1)))
, a(b(x1)) -> b(a(a(x1)))
, a(d(c(x1))) -> c(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:
{ d(d(x1)) -> a(d(b(x1)))
, b(b(x1)) -> b(c(x1))
, b(c(x1)) -> a(a(a(x1)))
, a(c(a(x1))) -> c(a(c(x1)))
, a(a(b(x1))) -> a(d(b(x1)))
, a(b(x1)) -> b(a(a(x1)))
, a(d(c(x1))) -> c(a(x1))
, d^#(d(x1)) -> c_3(a^#(d(b(x1))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ a(c(a(x1))) -> c(a(c(x1)))
, a(a(b(x1))) -> a(d(b(x1)))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ a(c(a(x1))) -> c(a(c(x1)))
, a(a(b(x1))) -> a(d(b(x1)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
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]
d^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [1]
c_4(x1) = [0] 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
{d^#(d(x1)) -> c_3(a^#(d(b(x1))))}
and weakly orienting the rules
{ a(c(a(x1))) -> c(a(c(x1)))
, a(a(b(x1))) -> a(d(b(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d^#(d(x1)) -> c_3(a^#(d(b(x1))))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
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]
d^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [0] 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
{ b(b(x1)) -> b(c(x1))
, b(c(x1)) -> a(a(a(x1)))}
and weakly orienting the rules
{ d^#(d(x1)) -> c_3(a^#(d(b(x1))))
, a(c(a(x1))) -> c(a(c(x1)))
, a(a(b(x1))) -> a(d(b(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ b(b(x1)) -> b(c(x1))
, b(c(x1)) -> a(a(a(x1)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [3]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
b^#(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [7]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [0] 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
{a(d(c(x1))) -> c(a(x1))}
and weakly orienting the rules
{ b(b(x1)) -> b(c(x1))
, b(c(x1)) -> a(a(a(x1)))
, d^#(d(x1)) -> c_3(a^#(d(b(x1))))
, a(c(a(x1))) -> c(a(c(x1)))
, a(a(b(x1))) -> a(d(b(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a(d(c(x1))) -> c(a(x1))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [8]
d(x1) = [1] x1 + [1]
a^#(x1) = [1] x1 + [3]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
b^#(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [15]
c_3(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
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:
{ d(d(x1)) -> a(d(b(x1)))
, a(b(x1)) -> b(a(a(x1)))}
Weak Rules:
{ a(d(c(x1))) -> c(a(x1))
, b(b(x1)) -> b(c(x1))
, b(c(x1)) -> a(a(a(x1)))
, d^#(d(x1)) -> c_3(a^#(d(b(x1))))
, a(c(a(x1))) -> c(a(c(x1)))
, a(a(b(x1))) -> a(d(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:
{ d(d(x1)) -> a(d(b(x1)))
, a(b(x1)) -> b(a(a(x1)))}
Weak Rules:
{ a(d(c(x1))) -> c(a(x1))
, b(b(x1)) -> b(c(x1))
, b(c(x1)) -> a(a(a(x1)))
, d^#(d(x1)) -> c_3(a^#(d(b(x1))))
, a(c(a(x1))) -> c(a(c(x1)))
, a(a(b(x1))) -> a(d(b(x1)))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ c_0(2) -> 2
, a^#_0(2) -> 5
, d^#_0(2) -> 10}