'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ t(o(x1)) -> m(a(x1))
, t(e(x1)) -> n(s(x1))
, a(l(x1)) -> a(t(x1))
, o(m(a(x1))) -> t(e(n(x1)))
, s(a(x1)) -> l(a(t(o(m(a(t(e(x1))))))))
, n(s(x1)) -> a(l(a(t(x1))))}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ t^#(o(x1)) -> c_0(a^#(x1))
, t^#(e(x1)) -> c_1(n^#(s(x1)))
, a^#(l(x1)) -> c_2(a^#(t(x1)))
, o^#(m(a(x1))) -> c_3(t^#(e(n(x1))))
, s^#(a(x1)) -> c_4(a^#(t(o(m(a(t(e(x1))))))))
, n^#(s(x1)) -> c_5(a^#(l(a(t(x1)))))}
The usable rules are:
{ t(o(x1)) -> m(a(x1))
, t(e(x1)) -> n(s(x1))
, a(l(x1)) -> a(t(x1))
, o(m(a(x1))) -> t(e(n(x1)))
, s(a(x1)) -> l(a(t(o(m(a(t(e(x1))))))))
, n(s(x1)) -> a(l(a(t(x1))))}
The estimated dependency graph contains the following edges:
{t^#(o(x1)) -> c_0(a^#(x1))}
==> {a^#(l(x1)) -> c_2(a^#(t(x1)))}
{t^#(e(x1)) -> c_1(n^#(s(x1)))}
==> {n^#(s(x1)) -> c_5(a^#(l(a(t(x1)))))}
{a^#(l(x1)) -> c_2(a^#(t(x1)))}
==> {a^#(l(x1)) -> c_2(a^#(t(x1)))}
{o^#(m(a(x1))) -> c_3(t^#(e(n(x1))))}
==> {t^#(e(x1)) -> c_1(n^#(s(x1)))}
{s^#(a(x1)) -> c_4(a^#(t(o(m(a(t(e(x1))))))))}
==> {a^#(l(x1)) -> c_2(a^#(t(x1)))}
{n^#(s(x1)) -> c_5(a^#(l(a(t(x1)))))}
==> {a^#(l(x1)) -> c_2(a^#(t(x1)))}
We consider the following path(s):
1) { o^#(m(a(x1))) -> c_3(t^#(e(n(x1))))
, t^#(e(x1)) -> c_1(n^#(s(x1)))
, n^#(s(x1)) -> c_5(a^#(l(a(t(x1)))))
, a^#(l(x1)) -> c_2(a^#(t(x1)))}
The usable rules for this path are the following:
{ t(o(x1)) -> m(a(x1))
, t(e(x1)) -> n(s(x1))
, a(l(x1)) -> a(t(x1))
, s(a(x1)) -> l(a(t(o(m(a(t(e(x1))))))))
, n(s(x1)) -> a(l(a(t(x1))))
, o(m(a(x1))) -> t(e(n(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:
{ t(o(x1)) -> m(a(x1))
, t(e(x1)) -> n(s(x1))
, a(l(x1)) -> a(t(x1))
, s(a(x1)) -> l(a(t(o(m(a(t(e(x1))))))))
, n(s(x1)) -> a(l(a(t(x1))))
, o(m(a(x1))) -> t(e(n(x1)))
, n^#(s(x1)) -> c_5(a^#(l(a(t(x1)))))
, t^#(e(x1)) -> c_1(n^#(s(x1)))
, o^#(m(a(x1))) -> c_3(t^#(e(n(x1))))
, a^#(l(x1)) -> c_2(a^#(t(x1)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{o^#(m(a(x1))) -> c_3(t^#(e(n(x1))))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{o^#(m(a(x1))) -> c_3(t^#(e(n(x1))))}
Details:
Interpretation Functions:
t(x1) = [1] x1 + [1]
o(x1) = [1] x1 + [0]
m(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [0]
e(x1) = [1] x1 + [0]
n(x1) = [1] x1 + [1]
s(x1) = [1] x1 + [0]
l(x1) = [1] x1 + [0]
t^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
n^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [0]
o^#(x1) = [1] x1 + [15]
c_3(x1) = [1] x1 + [1]
s^#(x1) = [0] 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
{t^#(e(x1)) -> c_1(n^#(s(x1)))}
and weakly orienting the rules
{o^#(m(a(x1))) -> c_3(t^#(e(n(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{t^#(e(x1)) -> c_1(n^#(s(x1)))}
Details:
Interpretation Functions:
t(x1) = [1] x1 + [1]
o(x1) = [1] x1 + [0]
m(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [0]
e(x1) = [1] x1 + [0]
n(x1) = [1] x1 + [1]
s(x1) = [1] x1 + [0]
l(x1) = [1] x1 + [0]
t^#(x1) = [1] x1 + [5]
c_0(x1) = [0] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
n^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [0]
o^#(x1) = [1] x1 + [8]
c_3(x1) = [1] x1 + [3]
s^#(x1) = [0] 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
{n^#(s(x1)) -> c_5(a^#(l(a(t(x1)))))}
and weakly orienting the rules
{ t^#(e(x1)) -> c_1(n^#(s(x1)))
, o^#(m(a(x1))) -> c_3(t^#(e(n(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{n^#(s(x1)) -> c_5(a^#(l(a(t(x1)))))}
Details:
Interpretation Functions:
t(x1) = [1] x1 + [1]
o(x1) = [1] x1 + [0]
m(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [0]
e(x1) = [1] x1 + [0]
n(x1) = [1] x1 + [1]
s(x1) = [1] x1 + [0]
l(x1) = [1] x1 + [0]
t^#(x1) = [1] x1 + [12]
c_0(x1) = [0] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
n^#(x1) = [1] x1 + [9]
c_2(x1) = [1] x1 + [0]
o^#(x1) = [1] x1 + [12]
c_3(x1) = [1] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [7]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ a(l(x1)) -> a(t(x1))
, a^#(l(x1)) -> c_2(a^#(t(x1)))}
and weakly orienting the rules
{ n^#(s(x1)) -> c_5(a^#(l(a(t(x1)))))
, t^#(e(x1)) -> c_1(n^#(s(x1)))
, o^#(m(a(x1))) -> c_3(t^#(e(n(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ a(l(x1)) -> a(t(x1))
, a^#(l(x1)) -> c_2(a^#(t(x1)))}
Details:
Interpretation Functions:
t(x1) = [1] x1 + [1]
o(x1) = [1] x1 + [0]
m(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [0]
e(x1) = [1] x1 + [0]
n(x1) = [1] x1 + [1]
s(x1) = [1] x1 + [0]
l(x1) = [1] x1 + [8]
t^#(x1) = [1] x1 + [12]
c_0(x1) = [0] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
n^#(x1) = [1] x1 + [12]
c_2(x1) = [1] x1 + [0]
o^#(x1) = [1] x1 + [15]
c_3(x1) = [1] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [3]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{n(s(x1)) -> a(l(a(t(x1))))}
and weakly orienting the rules
{ a(l(x1)) -> a(t(x1))
, a^#(l(x1)) -> c_2(a^#(t(x1)))
, n^#(s(x1)) -> c_5(a^#(l(a(t(x1)))))
, t^#(e(x1)) -> c_1(n^#(s(x1)))
, o^#(m(a(x1))) -> c_3(t^#(e(n(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{n(s(x1)) -> a(l(a(t(x1))))}
Details:
Interpretation Functions:
t(x1) = [1] x1 + [1]
o(x1) = [1] x1 + [0]
m(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [0]
e(x1) = [1] x1 + [0]
n(x1) = [1] x1 + [9]
s(x1) = [1] x1 + [0]
l(x1) = [1] x1 + [1]
t^#(x1) = [1] x1 + [6]
c_0(x1) = [0] x1 + [0]
a^#(x1) = [1] x1 + [2]
c_1(x1) = [1] x1 + [0]
n^#(x1) = [1] x1 + [5]
c_2(x1) = [1] x1 + [0]
o^#(x1) = [1] x1 + [15]
c_3(x1) = [1] x1 + [0]
s^#(x1) = [0] 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
{s(a(x1)) -> l(a(t(o(m(a(t(e(x1))))))))}
and weakly orienting the rules
{ n(s(x1)) -> a(l(a(t(x1))))
, a(l(x1)) -> a(t(x1))
, a^#(l(x1)) -> c_2(a^#(t(x1)))
, n^#(s(x1)) -> c_5(a^#(l(a(t(x1)))))
, t^#(e(x1)) -> c_1(n^#(s(x1)))
, o^#(m(a(x1))) -> c_3(t^#(e(n(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{s(a(x1)) -> l(a(t(o(m(a(t(e(x1))))))))}
Details:
Interpretation Functions:
t(x1) = [1] x1 + [1]
o(x1) = [1] x1 + [0]
m(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [0]
e(x1) = [1] x1 + [0]
n(x1) = [1] x1 + [1]
s(x1) = [1] x1 + [8]
l(x1) = [1] x1 + [4]
t^#(x1) = [1] x1 + [9]
c_0(x1) = [0] x1 + [0]
a^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [0]
n^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [1]
o^#(x1) = [1] x1 + [12]
c_3(x1) = [1] x1 + [3]
s^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{o(m(a(x1))) -> t(e(n(x1)))}
and weakly orienting the rules
{ s(a(x1)) -> l(a(t(o(m(a(t(e(x1))))))))
, n(s(x1)) -> a(l(a(t(x1))))
, a(l(x1)) -> a(t(x1))
, a^#(l(x1)) -> c_2(a^#(t(x1)))
, n^#(s(x1)) -> c_5(a^#(l(a(t(x1)))))
, t^#(e(x1)) -> c_1(n^#(s(x1)))
, o^#(m(a(x1))) -> c_3(t^#(e(n(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{o(m(a(x1))) -> t(e(n(x1)))}
Details:
Interpretation Functions:
t(x1) = [1] x1 + [1]
o(x1) = [1] x1 + [0]
m(x1) = [1] x1 + [9]
a(x1) = [1] x1 + [0]
e(x1) = [1] x1 + [0]
n(x1) = [1] x1 + [0]
s(x1) = [1] x1 + [12]
l(x1) = [1] x1 + [1]
t^#(x1) = [1] x1 + [13]
c_0(x1) = [0] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
n^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [0]
o^#(x1) = [1] x1 + [4]
c_3(x1) = [1] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [3]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{t(o(x1)) -> m(a(x1))}
and weakly orienting the rules
{ o(m(a(x1))) -> t(e(n(x1)))
, s(a(x1)) -> l(a(t(o(m(a(t(e(x1))))))))
, n(s(x1)) -> a(l(a(t(x1))))
, a(l(x1)) -> a(t(x1))
, a^#(l(x1)) -> c_2(a^#(t(x1)))
, n^#(s(x1)) -> c_5(a^#(l(a(t(x1)))))
, t^#(e(x1)) -> c_1(n^#(s(x1)))
, o^#(m(a(x1))) -> c_3(t^#(e(n(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{t(o(x1)) -> m(a(x1))}
Details:
Interpretation Functions:
t(x1) = [1] x1 + [1]
o(x1) = [1] x1 + [4]
m(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [2]
e(x1) = [1] x1 + [0]
n(x1) = [1] x1 + [0]
s(x1) = [1] x1 + [14]
l(x1) = [1] x1 + [6]
t^#(x1) = [1] x1 + [14]
c_0(x1) = [0] x1 + [0]
a^#(x1) = [1] x1 + [3]
c_1(x1) = [1] x1 + [0]
n^#(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [0]
o^#(x1) = [1] x1 + [14]
c_3(x1) = [1] x1 + [2]
s^#(x1) = [0] x1 + [0]
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: {t(e(x1)) -> n(s(x1))}
Weak Rules:
{ t(o(x1)) -> m(a(x1))
, o(m(a(x1))) -> t(e(n(x1)))
, s(a(x1)) -> l(a(t(o(m(a(t(e(x1))))))))
, n(s(x1)) -> a(l(a(t(x1))))
, a(l(x1)) -> a(t(x1))
, a^#(l(x1)) -> c_2(a^#(t(x1)))
, n^#(s(x1)) -> c_5(a^#(l(a(t(x1)))))
, t^#(e(x1)) -> c_1(n^#(s(x1)))
, o^#(m(a(x1))) -> c_3(t^#(e(n(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: {t(e(x1)) -> n(s(x1))}
Weak Rules:
{ t(o(x1)) -> m(a(x1))
, o(m(a(x1))) -> t(e(n(x1)))
, s(a(x1)) -> l(a(t(o(m(a(t(e(x1))))))))
, n(s(x1)) -> a(l(a(t(x1))))
, a(l(x1)) -> a(t(x1))
, a^#(l(x1)) -> c_2(a^#(t(x1)))
, n^#(s(x1)) -> c_5(a^#(l(a(t(x1)))))
, t^#(e(x1)) -> c_1(n^#(s(x1)))
, o^#(m(a(x1))) -> c_3(t^#(e(n(x1))))}
Details:
The problem is Match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ t_0(2) -> 4
, t_1(2) -> 10
, t_1(9) -> 8
, m_0(2) -> 2
, a_1(8) -> 4
, a_1(8) -> 10
, a_1(10) -> 9
, e_0(2) -> 2
, n_1(7) -> 4
, n_1(7) -> 10
, s_0(2) -> 6
, s_1(2) -> 7
, l_0(2) -> 2
, l_1(9) -> 8
, t^#_0(2) -> 1
, a^#_0(2) -> 1
, a^#_0(4) -> 3
, a^#_1(8) -> 12
, a^#_1(10) -> 11
, c_1_0(5) -> 1
, c_1_1(13) -> 1
, n^#_0(2) -> 1
, n^#_0(6) -> 5
, n^#_1(7) -> 13
, c_2_0(3) -> 1
, c_2_1(11) -> 1
, c_2_1(12) -> 12
, o^#_0(2) -> 1
, c_5_1(12) -> 5
, c_5_1(12) -> 13}
2) { o^#(m(a(x1))) -> c_3(t^#(e(n(x1))))
, t^#(e(x1)) -> c_1(n^#(s(x1)))
, n^#(s(x1)) -> c_5(a^#(l(a(t(x1)))))}
The usable rules for this path are the following:
{ t(o(x1)) -> m(a(x1))
, t(e(x1)) -> n(s(x1))
, a(l(x1)) -> a(t(x1))
, s(a(x1)) -> l(a(t(o(m(a(t(e(x1))))))))
, n(s(x1)) -> a(l(a(t(x1))))
, o(m(a(x1))) -> t(e(n(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:
{ t(o(x1)) -> m(a(x1))
, t(e(x1)) -> n(s(x1))
, a(l(x1)) -> a(t(x1))
, s(a(x1)) -> l(a(t(o(m(a(t(e(x1))))))))
, n(s(x1)) -> a(l(a(t(x1))))
, o(m(a(x1))) -> t(e(n(x1)))
, t^#(e(x1)) -> c_1(n^#(s(x1)))
, o^#(m(a(x1))) -> c_3(t^#(e(n(x1))))
, n^#(s(x1)) -> c_5(a^#(l(a(t(x1)))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{o^#(m(a(x1))) -> c_3(t^#(e(n(x1))))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{o^#(m(a(x1))) -> c_3(t^#(e(n(x1))))}
Details:
Interpretation Functions:
t(x1) = [1] x1 + [1]
o(x1) = [1] x1 + [0]
m(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [0]
e(x1) = [1] x1 + [0]
n(x1) = [1] x1 + [1]
s(x1) = [1] x1 + [0]
l(x1) = [1] x1 + [0]
t^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [1]
n^#(x1) = [1] x1 + [0]
c_2(x1) = [0] x1 + [0]
o^#(x1) = [1] x1 + [8]
c_3(x1) = [1] x1 + [0]
s^#(x1) = [0] 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
{t^#(e(x1)) -> c_1(n^#(s(x1)))}
and weakly orienting the rules
{o^#(m(a(x1))) -> c_3(t^#(e(n(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{t^#(e(x1)) -> c_1(n^#(s(x1)))}
Details:
Interpretation Functions:
t(x1) = [1] x1 + [1]
o(x1) = [1] x1 + [0]
m(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [0]
e(x1) = [1] x1 + [0]
n(x1) = [1] x1 + [1]
s(x1) = [1] x1 + [0]
l(x1) = [1] x1 + [0]
t^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
n^#(x1) = [1] x1 + [0]
c_2(x1) = [0] x1 + [0]
o^#(x1) = [1] x1 + [8]
c_3(x1) = [1] x1 + [7]
s^#(x1) = [0] 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
{n^#(s(x1)) -> c_5(a^#(l(a(t(x1)))))}
and weakly orienting the rules
{ t^#(e(x1)) -> c_1(n^#(s(x1)))
, o^#(m(a(x1))) -> c_3(t^#(e(n(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{n^#(s(x1)) -> c_5(a^#(l(a(t(x1)))))}
Details:
Interpretation Functions:
t(x1) = [1] x1 + [1]
o(x1) = [1] x1 + [0]
m(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [0]
e(x1) = [1] x1 + [0]
n(x1) = [1] x1 + [1]
s(x1) = [1] x1 + [0]
l(x1) = [1] x1 + [0]
t^#(x1) = [1] x1 + [12]
c_0(x1) = [0] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
n^#(x1) = [1] x1 + [9]
c_2(x1) = [0] x1 + [0]
o^#(x1) = [1] x1 + [15]
c_3(x1) = [1] x1 + [3]
s^#(x1) = [0] 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
{a(l(x1)) -> a(t(x1))}
and weakly orienting the rules
{ n^#(s(x1)) -> c_5(a^#(l(a(t(x1)))))
, t^#(e(x1)) -> c_1(n^#(s(x1)))
, o^#(m(a(x1))) -> c_3(t^#(e(n(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a(l(x1)) -> a(t(x1))}
Details:
Interpretation Functions:
t(x1) = [1] x1 + [1]
o(x1) = [1] x1 + [0]
m(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [0]
e(x1) = [1] x1 + [0]
n(x1) = [1] x1 + [1]
s(x1) = [1] x1 + [0]
l(x1) = [1] x1 + [4]
t^#(x1) = [1] x1 + [12]
c_0(x1) = [0] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [1]
n^#(x1) = [1] x1 + [9]
c_2(x1) = [0] x1 + [0]
o^#(x1) = [1] x1 + [15]
c_3(x1) = [1] x1 + [0]
s^#(x1) = [0] 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
{n(s(x1)) -> a(l(a(t(x1))))}
and weakly orienting the rules
{ a(l(x1)) -> a(t(x1))
, n^#(s(x1)) -> c_5(a^#(l(a(t(x1)))))
, t^#(e(x1)) -> c_1(n^#(s(x1)))
, o^#(m(a(x1))) -> c_3(t^#(e(n(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{n(s(x1)) -> a(l(a(t(x1))))}
Details:
Interpretation Functions:
t(x1) = [1] x1 + [1]
o(x1) = [1] x1 + [0]
m(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [0]
e(x1) = [1] x1 + [0]
n(x1) = [1] x1 + [9]
s(x1) = [1] x1 + [0]
l(x1) = [1] x1 + [4]
t^#(x1) = [1] x1 + [7]
c_0(x1) = [0] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
n^#(x1) = [1] x1 + [5]
c_2(x1) = [0] x1 + [0]
o^#(x1) = [1] x1 + [15]
c_3(x1) = [1] x1 + [0]
s^#(x1) = [0] 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
{s(a(x1)) -> l(a(t(o(m(a(t(e(x1))))))))}
and weakly orienting the rules
{ n(s(x1)) -> a(l(a(t(x1))))
, a(l(x1)) -> a(t(x1))
, n^#(s(x1)) -> c_5(a^#(l(a(t(x1)))))
, t^#(e(x1)) -> c_1(n^#(s(x1)))
, o^#(m(a(x1))) -> c_3(t^#(e(n(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{s(a(x1)) -> l(a(t(o(m(a(t(e(x1))))))))}
Details:
Interpretation Functions:
t(x1) = [1] x1 + [1]
o(x1) = [1] x1 + [0]
m(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [0]
e(x1) = [1] x1 + [0]
n(x1) = [1] x1 + [1]
s(x1) = [1] x1 + [8]
l(x1) = [1] x1 + [2]
t^#(x1) = [1] x1 + [13]
c_0(x1) = [0] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
n^#(x1) = [1] x1 + [3]
c_2(x1) = [0] x1 + [0]
o^#(x1) = [1] x1 + [15]
c_3(x1) = [1] x1 + [1]
s^#(x1) = [0] 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
{o(m(a(x1))) -> t(e(n(x1)))}
and weakly orienting the rules
{ s(a(x1)) -> l(a(t(o(m(a(t(e(x1))))))))
, n(s(x1)) -> a(l(a(t(x1))))
, a(l(x1)) -> a(t(x1))
, n^#(s(x1)) -> c_5(a^#(l(a(t(x1)))))
, t^#(e(x1)) -> c_1(n^#(s(x1)))
, o^#(m(a(x1))) -> c_3(t^#(e(n(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{o(m(a(x1))) -> t(e(n(x1)))}
Details:
Interpretation Functions:
t(x1) = [1] x1 + [1]
o(x1) = [1] x1 + [0]
m(x1) = [1] x1 + [9]
a(x1) = [1] x1 + [0]
e(x1) = [1] x1 + [0]
n(x1) = [1] x1 + [0]
s(x1) = [1] x1 + [13]
l(x1) = [1] x1 + [2]
t^#(x1) = [1] x1 + [14]
c_0(x1) = [0] x1 + [0]
a^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [0]
n^#(x1) = [1] x1 + [0]
c_2(x1) = [0] x1 + [0]
o^#(x1) = [1] x1 + [10]
c_3(x1) = [1] x1 + [1]
s^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [4]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{t(o(x1)) -> m(a(x1))}
and weakly orienting the rules
{ o(m(a(x1))) -> t(e(n(x1)))
, s(a(x1)) -> l(a(t(o(m(a(t(e(x1))))))))
, n(s(x1)) -> a(l(a(t(x1))))
, a(l(x1)) -> a(t(x1))
, n^#(s(x1)) -> c_5(a^#(l(a(t(x1)))))
, t^#(e(x1)) -> c_1(n^#(s(x1)))
, o^#(m(a(x1))) -> c_3(t^#(e(n(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{t(o(x1)) -> m(a(x1))}
Details:
Interpretation Functions:
t(x1) = [1] x1 + [0]
o(x1) = [1] x1 + [3]
m(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [2]
e(x1) = [1] x1 + [0]
n(x1) = [1] x1 + [0]
s(x1) = [1] x1 + [15]
l(x1) = [1] x1 + [10]
t^#(x1) = [1] x1 + [15]
c_0(x1) = [0] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
n^#(x1) = [1] x1 + [0]
c_2(x1) = [0] x1 + [0]
o^#(x1) = [1] x1 + [13]
c_3(x1) = [1] x1 + [0]
s^#(x1) = [0] x1 + [0]
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: {t(e(x1)) -> n(s(x1))}
Weak Rules:
{ t(o(x1)) -> m(a(x1))
, o(m(a(x1))) -> t(e(n(x1)))
, s(a(x1)) -> l(a(t(o(m(a(t(e(x1))))))))
, n(s(x1)) -> a(l(a(t(x1))))
, a(l(x1)) -> a(t(x1))
, n^#(s(x1)) -> c_5(a^#(l(a(t(x1)))))
, t^#(e(x1)) -> c_1(n^#(s(x1)))
, o^#(m(a(x1))) -> c_3(t^#(e(n(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: {t(e(x1)) -> n(s(x1))}
Weak Rules:
{ t(o(x1)) -> m(a(x1))
, o(m(a(x1))) -> t(e(n(x1)))
, s(a(x1)) -> l(a(t(o(m(a(t(e(x1))))))))
, n(s(x1)) -> a(l(a(t(x1))))
, a(l(x1)) -> a(t(x1))
, n^#(s(x1)) -> c_5(a^#(l(a(t(x1)))))
, t^#(e(x1)) -> c_1(n^#(s(x1)))
, o^#(m(a(x1))) -> c_3(t^#(e(n(x1))))}
Details:
The problem is Match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ t_0(2) -> 8
, t_1(2) -> 12
, t_1(11) -> 10
, m_0(2) -> 2
, a_0(8) -> 7
, a_1(10) -> 8
, a_1(10) -> 12
, a_1(12) -> 11
, e_0(2) -> 2
, n_1(9) -> 8
, n_1(9) -> 12
, s_0(2) -> 4
, s_1(2) -> 9
, l_0(2) -> 2
, l_0(7) -> 6
, l_1(11) -> 10
, t^#_0(2) -> 1
, a^#_0(2) -> 1
, a^#_0(6) -> 5
, a^#_1(10) -> 13
, c_1_0(3) -> 1
, c_1_1(14) -> 1
, n^#_0(2) -> 1
, n^#_0(4) -> 3
, n^#_1(9) -> 14
, o^#_0(2) -> 1
, c_5_0(5) -> 3
, c_5_1(13) -> 3
, c_5_1(13) -> 14}
3) { s^#(a(x1)) -> c_4(a^#(t(o(m(a(t(e(x1))))))))
, a^#(l(x1)) -> c_2(a^#(t(x1)))}
The usable rules for this path are the following:
{ t(o(x1)) -> m(a(x1))
, t(e(x1)) -> n(s(x1))
, a(l(x1)) -> a(t(x1))
, o(m(a(x1))) -> t(e(n(x1)))
, s(a(x1)) -> l(a(t(o(m(a(t(e(x1))))))))
, n(s(x1)) -> a(l(a(t(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:
{ t(o(x1)) -> m(a(x1))
, t(e(x1)) -> n(s(x1))
, a(l(x1)) -> a(t(x1))
, o(m(a(x1))) -> t(e(n(x1)))
, s(a(x1)) -> l(a(t(o(m(a(t(e(x1))))))))
, n(s(x1)) -> a(l(a(t(x1))))
, s^#(a(x1)) -> c_4(a^#(t(o(m(a(t(e(x1))))))))
, a^#(l(x1)) -> c_2(a^#(t(x1)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{s^#(a(x1)) -> c_4(a^#(t(o(m(a(t(e(x1))))))))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{s^#(a(x1)) -> c_4(a^#(t(o(m(a(t(e(x1))))))))}
Details:
Interpretation Functions:
t(x1) = [1] x1 + [1]
o(x1) = [1] x1 + [0]
m(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [0]
e(x1) = [1] x1 + [0]
n(x1) = [1] x1 + [1]
s(x1) = [1] x1 + [0]
l(x1) = [1] x1 + [0]
t^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_1(x1) = [0] x1 + [0]
n^#(x1) = [0] x1 + [0]
c_2(x1) = [1] x1 + [1]
o^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
s^#(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
{ t(e(x1)) -> n(s(x1))
, a(l(x1)) -> a(t(x1))}
and weakly orienting the rules
{s^#(a(x1)) -> c_4(a^#(t(o(m(a(t(e(x1))))))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ t(e(x1)) -> n(s(x1))
, a(l(x1)) -> a(t(x1))}
Details:
Interpretation Functions:
t(x1) = [1] x1 + [1]
o(x1) = [1] x1 + [0]
m(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [0]
e(x1) = [1] x1 + [4]
n(x1) = [1] x1 + [1]
s(x1) = [1] x1 + [1]
l(x1) = [1] x1 + [2]
t^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a^#(x1) = [1] x1 + [7]
c_1(x1) = [0] x1 + [0]
n^#(x1) = [0] x1 + [0]
c_2(x1) = [1] x1 + [7]
o^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
s^#(x1) = [1] x1 + [14]
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
{a^#(l(x1)) -> c_2(a^#(t(x1)))}
and weakly orienting the rules
{ t(e(x1)) -> n(s(x1))
, a(l(x1)) -> a(t(x1))
, s^#(a(x1)) -> c_4(a^#(t(o(m(a(t(e(x1))))))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a^#(l(x1)) -> c_2(a^#(t(x1)))}
Details:
Interpretation Functions:
t(x1) = [1] x1 + [1]
o(x1) = [1] x1 + [0]
m(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [0]
e(x1) = [1] x1 + [0]
n(x1) = [1] x1 + [0]
s(x1) = [1] x1 + [1]
l(x1) = [1] x1 + [8]
t^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a^#(x1) = [1] x1 + [3]
c_1(x1) = [0] x1 + [0]
n^#(x1) = [0] x1 + [0]
c_2(x1) = [1] x1 + [1]
o^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
s^#(x1) = [1] x1 + [9]
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
{n(s(x1)) -> a(l(a(t(x1))))}
and weakly orienting the rules
{ a^#(l(x1)) -> c_2(a^#(t(x1)))
, t(e(x1)) -> n(s(x1))
, a(l(x1)) -> a(t(x1))
, s^#(a(x1)) -> c_4(a^#(t(o(m(a(t(e(x1))))))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{n(s(x1)) -> a(l(a(t(x1))))}
Details:
Interpretation Functions:
t(x1) = [1] x1 + [1]
o(x1) = [1] x1 + [0]
m(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [0]
e(x1) = [1] x1 + [8]
n(x1) = [1] x1 + [4]
s(x1) = [1] x1 + [0]
l(x1) = [1] x1 + [2]
t^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a^#(x1) = [1] x1 + [2]
c_1(x1) = [0] x1 + [0]
n^#(x1) = [0] x1 + [0]
c_2(x1) = [1] x1 + [0]
o^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
s^#(x1) = [1] x1 + [14]
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
{o(m(a(x1))) -> t(e(n(x1)))}
and weakly orienting the rules
{ n(s(x1)) -> a(l(a(t(x1))))
, a^#(l(x1)) -> c_2(a^#(t(x1)))
, t(e(x1)) -> n(s(x1))
, a(l(x1)) -> a(t(x1))
, s^#(a(x1)) -> c_4(a^#(t(o(m(a(t(e(x1))))))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{o(m(a(x1))) -> t(e(n(x1)))}
Details:
Interpretation Functions:
t(x1) = [1] x1 + [1]
o(x1) = [1] x1 + [0]
m(x1) = [1] x1 + [9]
a(x1) = [1] x1 + [0]
e(x1) = [1] x1 + [1]
n(x1) = [1] x1 + [2]
s(x1) = [1] x1 + [0]
l(x1) = [1] x1 + [1]
t^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a^#(x1) = [1] x1 + [1]
c_1(x1) = [0] x1 + [0]
n^#(x1) = [0] x1 + [0]
c_2(x1) = [1] x1 + [0]
o^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
s^#(x1) = [1] x1 + [15]
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
{t(o(x1)) -> m(a(x1))}
and weakly orienting the rules
{ o(m(a(x1))) -> t(e(n(x1)))
, n(s(x1)) -> a(l(a(t(x1))))
, a^#(l(x1)) -> c_2(a^#(t(x1)))
, t(e(x1)) -> n(s(x1))
, a(l(x1)) -> a(t(x1))
, s^#(a(x1)) -> c_4(a^#(t(o(m(a(t(e(x1))))))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{t(o(x1)) -> m(a(x1))}
Details:
Interpretation Functions:
t(x1) = [1] x1 + [0]
o(x1) = [1] x1 + [2]
m(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
e(x1) = [1] x1 + [0]
n(x1) = [1] x1 + [0]
s(x1) = [1] x1 + [0]
l(x1) = [1] x1 + [0]
t^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a^#(x1) = [1] x1 + [1]
c_1(x1) = [0] x1 + [0]
n^#(x1) = [0] x1 + [0]
c_2(x1) = [1] x1 + [0]
o^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
s^#(x1) = [1] x1 + [4]
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: {s(a(x1)) -> l(a(t(o(m(a(t(e(x1))))))))}
Weak Rules:
{ t(o(x1)) -> m(a(x1))
, o(m(a(x1))) -> t(e(n(x1)))
, n(s(x1)) -> a(l(a(t(x1))))
, a^#(l(x1)) -> c_2(a^#(t(x1)))
, t(e(x1)) -> n(s(x1))
, a(l(x1)) -> a(t(x1))
, s^#(a(x1)) -> c_4(a^#(t(o(m(a(t(e(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: {s(a(x1)) -> l(a(t(o(m(a(t(e(x1))))))))}
Weak Rules:
{ t(o(x1)) -> m(a(x1))
, o(m(a(x1))) -> t(e(n(x1)))
, n(s(x1)) -> a(l(a(t(x1))))
, a^#(l(x1)) -> c_2(a^#(t(x1)))
, t(e(x1)) -> n(s(x1))
, a(l(x1)) -> a(t(x1))
, s^#(a(x1)) -> c_4(a^#(t(o(m(a(t(e(x1))))))))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ t_0(3) -> 19
, t_0(5) -> 19
, t_0(8) -> 19
, t_0(22) -> 21
, m_0(3) -> 3
, m_0(5) -> 3
, m_0(8) -> 3
, a_0(19) -> 22
, a_0(21) -> 19
, e_0(3) -> 5
, e_0(5) -> 5
, e_0(8) -> 5
, n_0(20) -> 19
, s_0(3) -> 20
, s_0(5) -> 20
, s_0(8) -> 20
, l_0(3) -> 8
, l_0(5) -> 8
, l_0(8) -> 8
, l_0(22) -> 21
, a^#_0(3) -> 11
, a^#_0(5) -> 11
, a^#_0(8) -> 11
, a^#_0(19) -> 18
, c_2_0(18) -> 11
, s^#_0(3) -> 17
, s^#_0(5) -> 17
, s^#_0(8) -> 17}
4) {s^#(a(x1)) -> c_4(a^#(t(o(m(a(t(e(x1))))))))}
The usable rules for this path are the following:
{ t(o(x1)) -> m(a(x1))
, t(e(x1)) -> n(s(x1))
, a(l(x1)) -> a(t(x1))
, o(m(a(x1))) -> t(e(n(x1)))
, s(a(x1)) -> l(a(t(o(m(a(t(e(x1))))))))
, n(s(x1)) -> a(l(a(t(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:
{ t(o(x1)) -> m(a(x1))
, t(e(x1)) -> n(s(x1))
, a(l(x1)) -> a(t(x1))
, o(m(a(x1))) -> t(e(n(x1)))
, s(a(x1)) -> l(a(t(o(m(a(t(e(x1))))))))
, n(s(x1)) -> a(l(a(t(x1))))
, s^#(a(x1)) -> c_4(a^#(t(o(m(a(t(e(x1))))))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{s^#(a(x1)) -> c_4(a^#(t(o(m(a(t(e(x1))))))))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{s^#(a(x1)) -> c_4(a^#(t(o(m(a(t(e(x1))))))))}
Details:
Interpretation Functions:
t(x1) = [1] x1 + [1]
o(x1) = [1] x1 + [0]
m(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [0]
e(x1) = [1] x1 + [0]
n(x1) = [1] x1 + [1]
s(x1) = [1] x1 + [0]
l(x1) = [1] x1 + [0]
t^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_1(x1) = [0] x1 + [0]
n^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
o^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
s^#(x1) = [1] x1 + [9]
c_4(x1) = [1] x1 + [4]
c_5(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a(l(x1)) -> a(t(x1))}
and weakly orienting the rules
{s^#(a(x1)) -> c_4(a^#(t(o(m(a(t(e(x1))))))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a(l(x1)) -> a(t(x1))}
Details:
Interpretation Functions:
t(x1) = [1] x1 + [1]
o(x1) = [1] x1 + [0]
m(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [0]
e(x1) = [1] x1 + [0]
n(x1) = [1] x1 + [1]
s(x1) = [1] x1 + [0]
l(x1) = [1] x1 + [8]
t^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_1(x1) = [0] x1 + [0]
n^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
o^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
s^#(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
{n(s(x1)) -> a(l(a(t(x1))))}
and weakly orienting the rules
{ a(l(x1)) -> a(t(x1))
, s^#(a(x1)) -> c_4(a^#(t(o(m(a(t(e(x1))))))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{n(s(x1)) -> a(l(a(t(x1))))}
Details:
Interpretation Functions:
t(x1) = [1] x1 + [1]
o(x1) = [1] x1 + [0]
m(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [0]
e(x1) = [1] x1 + [0]
n(x1) = [1] x1 + [9]
s(x1) = [1] x1 + [0]
l(x1) = [1] x1 + [1]
t^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_1(x1) = [0] x1 + [0]
n^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
o^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
s^#(x1) = [1] x1 + [5]
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
{s(a(x1)) -> l(a(t(o(m(a(t(e(x1))))))))}
and weakly orienting the rules
{ n(s(x1)) -> a(l(a(t(x1))))
, a(l(x1)) -> a(t(x1))
, s^#(a(x1)) -> c_4(a^#(t(o(m(a(t(e(x1))))))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{s(a(x1)) -> l(a(t(o(m(a(t(e(x1))))))))}
Details:
Interpretation Functions:
t(x1) = [1] x1 + [1]
o(x1) = [1] x1 + [0]
m(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [0]
e(x1) = [1] x1 + [0]
n(x1) = [1] x1 + [1]
s(x1) = [1] x1 + [8]
l(x1) = [1] x1 + [1]
t^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_1(x1) = [0] x1 + [0]
n^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
o^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
s^#(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
{o(m(a(x1))) -> t(e(n(x1)))}
and weakly orienting the rules
{ s(a(x1)) -> l(a(t(o(m(a(t(e(x1))))))))
, n(s(x1)) -> a(l(a(t(x1))))
, a(l(x1)) -> a(t(x1))
, s^#(a(x1)) -> c_4(a^#(t(o(m(a(t(e(x1))))))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{o(m(a(x1))) -> t(e(n(x1)))}
Details:
Interpretation Functions:
t(x1) = [1] x1 + [1]
o(x1) = [1] x1 + [0]
m(x1) = [1] x1 + [9]
a(x1) = [1] x1 + [0]
e(x1) = [1] x1 + [3]
n(x1) = [1] x1 + [0]
s(x1) = [1] x1 + [15]
l(x1) = [1] x1 + [1]
t^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a^#(x1) = [1] x1 + [1]
c_1(x1) = [0] x1 + [0]
n^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
o^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
s^#(x1) = [1] x1 + [15]
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
{t(o(x1)) -> m(a(x1))}
and weakly orienting the rules
{ o(m(a(x1))) -> t(e(n(x1)))
, s(a(x1)) -> l(a(t(o(m(a(t(e(x1))))))))
, n(s(x1)) -> a(l(a(t(x1))))
, a(l(x1)) -> a(t(x1))
, s^#(a(x1)) -> c_4(a^#(t(o(m(a(t(e(x1))))))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{t(o(x1)) -> m(a(x1))}
Details:
Interpretation Functions:
t(x1) = [1] x1 + [0]
o(x1) = [1] x1 + [2]
m(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
e(x1) = [1] x1 + [0]
n(x1) = [1] x1 + [0]
s(x1) = [1] x1 + [2]
l(x1) = [1] x1 + [0]
t^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_1(x1) = [0] x1 + [0]
n^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
o^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
s^#(x1) = [1] x1 + [8]
c_4(x1) = [1] x1 + [1]
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: {t(e(x1)) -> n(s(x1))}
Weak Rules:
{ t(o(x1)) -> m(a(x1))
, o(m(a(x1))) -> t(e(n(x1)))
, s(a(x1)) -> l(a(t(o(m(a(t(e(x1))))))))
, n(s(x1)) -> a(l(a(t(x1))))
, a(l(x1)) -> a(t(x1))
, s^#(a(x1)) -> c_4(a^#(t(o(m(a(t(e(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: {t(e(x1)) -> n(s(x1))}
Weak Rules:
{ t(o(x1)) -> m(a(x1))
, o(m(a(x1))) -> t(e(n(x1)))
, s(a(x1)) -> l(a(t(o(m(a(t(e(x1))))))))
, n(s(x1)) -> a(l(a(t(x1))))
, a(l(x1)) -> a(t(x1))
, s^#(a(x1)) -> c_4(a^#(t(o(m(a(t(e(x1))))))))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ m_0(3) -> 3
, m_0(5) -> 3
, m_0(8) -> 3
, e_0(3) -> 5
, e_0(5) -> 5
, e_0(8) -> 5
, l_0(3) -> 8
, l_0(5) -> 8
, l_0(8) -> 8
, a^#_0(3) -> 11
, a^#_0(5) -> 11
, a^#_0(8) -> 11
, s^#_0(3) -> 17
, s^#_0(5) -> 17
, s^#_0(8) -> 17}
5) {o^#(m(a(x1))) -> c_3(t^#(e(n(x1))))}
The usable rules for this path are the following:
{ n(s(x1)) -> a(l(a(t(x1))))
, t(o(x1)) -> m(a(x1))
, t(e(x1)) -> n(s(x1))
, a(l(x1)) -> a(t(x1))
, s(a(x1)) -> l(a(t(o(m(a(t(e(x1))))))))
, o(m(a(x1))) -> t(e(n(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:
{ n(s(x1)) -> a(l(a(t(x1))))
, t(o(x1)) -> m(a(x1))
, t(e(x1)) -> n(s(x1))
, a(l(x1)) -> a(t(x1))
, s(a(x1)) -> l(a(t(o(m(a(t(e(x1))))))))
, o(m(a(x1))) -> t(e(n(x1)))
, o^#(m(a(x1))) -> c_3(t^#(e(n(x1))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{n(s(x1)) -> a(l(a(t(x1))))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{n(s(x1)) -> a(l(a(t(x1))))}
Details:
Interpretation Functions:
t(x1) = [1] x1 + [0]
o(x1) = [1] x1 + [0]
m(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [0]
e(x1) = [1] x1 + [1]
n(x1) = [1] x1 + [1]
s(x1) = [1] x1 + [0]
l(x1) = [1] x1 + [0]
t^#(x1) = [1] x1 + [8]
c_0(x1) = [0] x1 + [0]
a^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
n^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
o^#(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [1]
s^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{o^#(m(a(x1))) -> c_3(t^#(e(n(x1))))}
and weakly orienting the rules
{n(s(x1)) -> a(l(a(t(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{o^#(m(a(x1))) -> c_3(t^#(e(n(x1))))}
Details:
Interpretation Functions:
t(x1) = [1] x1 + [1]
o(x1) = [1] x1 + [0]
m(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [0]
e(x1) = [1] x1 + [0]
n(x1) = [1] x1 + [1]
s(x1) = [1] x1 + [0]
l(x1) = [1] x1 + [0]
t^#(x1) = [1] x1 + [3]
c_0(x1) = [0] x1 + [0]
a^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
n^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
o^#(x1) = [1] x1 + [8]
c_3(x1) = [1] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a(l(x1)) -> a(t(x1))}
and weakly orienting the rules
{ o^#(m(a(x1))) -> c_3(t^#(e(n(x1))))
, n(s(x1)) -> a(l(a(t(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a(l(x1)) -> a(t(x1))}
Details:
Interpretation Functions:
t(x1) = [1] x1 + [1]
o(x1) = [1] x1 + [0]
m(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [0]
e(x1) = [1] x1 + [0]
n(x1) = [1] x1 + [9]
s(x1) = [1] x1 + [0]
l(x1) = [1] x1 + [2]
t^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
a^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
n^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
o^#(x1) = [1] x1 + [8]
c_3(x1) = [1] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{s(a(x1)) -> l(a(t(o(m(a(t(e(x1))))))))}
and weakly orienting the rules
{ a(l(x1)) -> a(t(x1))
, o^#(m(a(x1))) -> c_3(t^#(e(n(x1))))
, n(s(x1)) -> a(l(a(t(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{s(a(x1)) -> l(a(t(o(m(a(t(e(x1))))))))}
Details:
Interpretation Functions:
t(x1) = [1] x1 + [1]
o(x1) = [1] x1 + [0]
m(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [0]
e(x1) = [1] x1 + [0]
n(x1) = [1] x1 + [1]
s(x1) = [1] x1 + [8]
l(x1) = [1] x1 + [4]
t^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
a^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
n^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
o^#(x1) = [1] x1 + [8]
c_3(x1) = [1] x1 + [1]
s^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{o(m(a(x1))) -> t(e(n(x1)))}
and weakly orienting the rules
{ s(a(x1)) -> l(a(t(o(m(a(t(e(x1))))))))
, a(l(x1)) -> a(t(x1))
, o^#(m(a(x1))) -> c_3(t^#(e(n(x1))))
, n(s(x1)) -> a(l(a(t(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{o(m(a(x1))) -> t(e(n(x1)))}
Details:
Interpretation Functions:
t(x1) = [1] x1 + [0]
o(x1) = [1] x1 + [0]
m(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [1]
e(x1) = [1] x1 + [0]
n(x1) = [1] x1 + [0]
s(x1) = [1] x1 + [15]
l(x1) = [1] x1 + [0]
t^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
a^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
n^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
o^#(x1) = [1] x1 + [3]
c_3(x1) = [1] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{t(o(x1)) -> m(a(x1))}
and weakly orienting the rules
{ o(m(a(x1))) -> t(e(n(x1)))
, s(a(x1)) -> l(a(t(o(m(a(t(e(x1))))))))
, a(l(x1)) -> a(t(x1))
, o^#(m(a(x1))) -> c_3(t^#(e(n(x1))))
, n(s(x1)) -> a(l(a(t(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{t(o(x1)) -> m(a(x1))}
Details:
Interpretation Functions:
t(x1) = [1] x1 + [0]
o(x1) = [1] x1 + [2]
m(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [1]
e(x1) = [1] x1 + [0]
n(x1) = [1] x1 + [0]
s(x1) = [1] x1 + [7]
l(x1) = [1] x1 + [0]
t^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
a^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
n^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
o^#(x1) = [1] x1 + [5]
c_3(x1) = [1] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_4(x1) = [0] 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: {t(e(x1)) -> n(s(x1))}
Weak Rules:
{ t(o(x1)) -> m(a(x1))
, o(m(a(x1))) -> t(e(n(x1)))
, s(a(x1)) -> l(a(t(o(m(a(t(e(x1))))))))
, a(l(x1)) -> a(t(x1))
, o^#(m(a(x1))) -> c_3(t^#(e(n(x1))))
, n(s(x1)) -> a(l(a(t(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: {t(e(x1)) -> n(s(x1))}
Weak Rules:
{ t(o(x1)) -> m(a(x1))
, o(m(a(x1))) -> t(e(n(x1)))
, s(a(x1)) -> l(a(t(o(m(a(t(e(x1))))))))
, a(l(x1)) -> a(t(x1))
, o^#(m(a(x1))) -> c_3(t^#(e(n(x1))))
, n(s(x1)) -> a(l(a(t(x1))))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ m_0(3) -> 3
, m_0(5) -> 3
, m_0(8) -> 3
, e_0(3) -> 5
, e_0(5) -> 5
, e_0(8) -> 5
, l_0(3) -> 8
, l_0(5) -> 8
, l_0(8) -> 8
, t^#_0(3) -> 9
, t^#_0(5) -> 9
, t^#_0(8) -> 9
, o^#_0(3) -> 15
, o^#_0(5) -> 15
, o^#_0(8) -> 15}
6) { t^#(o(x1)) -> c_0(a^#(x1))
, a^#(l(x1)) -> c_2(a^#(t(x1)))}
The usable rules for this path are the following:
{ t(o(x1)) -> m(a(x1))
, t(e(x1)) -> n(s(x1))
, a(l(x1)) -> a(t(x1))
, s(a(x1)) -> l(a(t(o(m(a(t(e(x1))))))))
, n(s(x1)) -> a(l(a(t(x1))))
, o(m(a(x1))) -> t(e(n(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:
{ t(o(x1)) -> m(a(x1))
, t(e(x1)) -> n(s(x1))
, a(l(x1)) -> a(t(x1))
, s(a(x1)) -> l(a(t(o(m(a(t(e(x1))))))))
, n(s(x1)) -> a(l(a(t(x1))))
, o(m(a(x1))) -> t(e(n(x1)))
, t^#(o(x1)) -> c_0(a^#(x1))
, a^#(l(x1)) -> c_2(a^#(t(x1)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{t^#(o(x1)) -> c_0(a^#(x1))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{t^#(o(x1)) -> c_0(a^#(x1))}
Details:
Interpretation Functions:
t(x1) = [1] x1 + [1]
o(x1) = [1] x1 + [0]
m(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [0]
e(x1) = [1] x1 + [0]
n(x1) = [1] x1 + [1]
s(x1) = [1] x1 + [0]
l(x1) = [1] x1 + [0]
t^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_1(x1) = [0] x1 + [0]
n^#(x1) = [0] x1 + [0]
c_2(x1) = [1] x1 + [0]
o^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ a(l(x1)) -> a(t(x1))
, a^#(l(x1)) -> c_2(a^#(t(x1)))}
and weakly orienting the rules
{t^#(o(x1)) -> c_0(a^#(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ a(l(x1)) -> a(t(x1))
, a^#(l(x1)) -> c_2(a^#(t(x1)))}
Details:
Interpretation Functions:
t(x1) = [1] x1 + [1]
o(x1) = [1] x1 + [0]
m(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [0]
e(x1) = [1] x1 + [0]
n(x1) = [1] x1 + [1]
s(x1) = [1] x1 + [0]
l(x1) = [1] x1 + [8]
t^#(x1) = [1] x1 + [7]
c_0(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [7]
c_1(x1) = [0] x1 + [0]
n^#(x1) = [0] x1 + [0]
c_2(x1) = [1] x1 + [1]
o^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{n(s(x1)) -> a(l(a(t(x1))))}
and weakly orienting the rules
{ a(l(x1)) -> a(t(x1))
, a^#(l(x1)) -> c_2(a^#(t(x1)))
, t^#(o(x1)) -> c_0(a^#(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{n(s(x1)) -> a(l(a(t(x1))))}
Details:
Interpretation Functions:
t(x1) = [1] x1 + [1]
o(x1) = [1] x1 + [0]
m(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [0]
e(x1) = [1] x1 + [0]
n(x1) = [1] x1 + [9]
s(x1) = [1] x1 + [0]
l(x1) = [1] x1 + [4]
t^#(x1) = [1] x1 + [15]
c_0(x1) = [1] x1 + [1]
a^#(x1) = [1] x1 + [12]
c_1(x1) = [0] x1 + [0]
n^#(x1) = [0] x1 + [0]
c_2(x1) = [1] x1 + [0]
o^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{s(a(x1)) -> l(a(t(o(m(a(t(e(x1))))))))}
and weakly orienting the rules
{ n(s(x1)) -> a(l(a(t(x1))))
, a(l(x1)) -> a(t(x1))
, a^#(l(x1)) -> c_2(a^#(t(x1)))
, t^#(o(x1)) -> c_0(a^#(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{s(a(x1)) -> l(a(t(o(m(a(t(e(x1))))))))}
Details:
Interpretation Functions:
t(x1) = [1] x1 + [1]
o(x1) = [1] x1 + [0]
m(x1) = [1] x1 + [1]
a(x1) = [1] x1 + [0]
e(x1) = [1] x1 + [0]
n(x1) = [1] x1 + [9]
s(x1) = [1] x1 + [8]
l(x1) = [1] x1 + [4]
t^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [1]
a^#(x1) = [1] x1 + [0]
c_1(x1) = [0] x1 + [0]
n^#(x1) = [0] x1 + [0]
c_2(x1) = [1] x1 + [0]
o^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{o(m(a(x1))) -> t(e(n(x1)))}
and weakly orienting the rules
{ s(a(x1)) -> l(a(t(o(m(a(t(e(x1))))))))
, n(s(x1)) -> a(l(a(t(x1))))
, a(l(x1)) -> a(t(x1))
, a^#(l(x1)) -> c_2(a^#(t(x1)))
, t^#(o(x1)) -> c_0(a^#(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{o(m(a(x1))) -> t(e(n(x1)))}
Details:
Interpretation Functions:
t(x1) = [1] x1 + [1]
o(x1) = [1] x1 + [0]
m(x1) = [1] x1 + [9]
a(x1) = [1] x1 + [0]
e(x1) = [1] x1 + [0]
n(x1) = [1] x1 + [0]
s(x1) = [1] x1 + [15]
l(x1) = [1] x1 + [4]
t^#(x1) = [1] x1 + [13]
c_0(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [1]
c_1(x1) = [0] x1 + [0]
n^#(x1) = [0] x1 + [0]
c_2(x1) = [1] x1 + [0]
o^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{t(o(x1)) -> m(a(x1))}
and weakly orienting the rules
{ o(m(a(x1))) -> t(e(n(x1)))
, s(a(x1)) -> l(a(t(o(m(a(t(e(x1))))))))
, n(s(x1)) -> a(l(a(t(x1))))
, a(l(x1)) -> a(t(x1))
, a^#(l(x1)) -> c_2(a^#(t(x1)))
, t^#(o(x1)) -> c_0(a^#(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{t(o(x1)) -> m(a(x1))}
Details:
Interpretation Functions:
t(x1) = [1] x1 + [0]
o(x1) = [1] x1 + [4]
m(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
e(x1) = [1] x1 + [0]
n(x1) = [1] x1 + [0]
s(x1) = [1] x1 + [4]
l(x1) = [1] x1 + [0]
t^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [1]
a^#(x1) = [1] x1 + [0]
c_1(x1) = [0] x1 + [0]
n^#(x1) = [0] x1 + [0]
c_2(x1) = [1] x1 + [0]
o^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_4(x1) = [0] 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: {t(e(x1)) -> n(s(x1))}
Weak Rules:
{ t(o(x1)) -> m(a(x1))
, o(m(a(x1))) -> t(e(n(x1)))
, s(a(x1)) -> l(a(t(o(m(a(t(e(x1))))))))
, n(s(x1)) -> a(l(a(t(x1))))
, a(l(x1)) -> a(t(x1))
, a^#(l(x1)) -> c_2(a^#(t(x1)))
, t^#(o(x1)) -> c_0(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: {t(e(x1)) -> n(s(x1))}
Weak Rules:
{ t(o(x1)) -> m(a(x1))
, o(m(a(x1))) -> t(e(n(x1)))
, s(a(x1)) -> l(a(t(o(m(a(t(e(x1))))))))
, n(s(x1)) -> a(l(a(t(x1))))
, a(l(x1)) -> a(t(x1))
, a^#(l(x1)) -> c_2(a^#(t(x1)))
, t^#(o(x1)) -> c_0(a^#(x1))}
Details:
The problem is Match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ t_0(3) -> 13
, t_0(5) -> 13
, t_0(8) -> 13
, t_1(3) -> 17
, t_1(5) -> 17
, t_1(8) -> 17
, t_1(16) -> 15
, m_0(3) -> 3
, m_0(5) -> 3
, m_0(8) -> 3
, a_1(15) -> 13
, a_1(15) -> 17
, a_1(17) -> 16
, e_0(3) -> 5
, e_0(5) -> 5
, e_0(8) -> 5
, n_1(14) -> 13
, n_1(14) -> 17
, s_1(3) -> 14
, s_1(5) -> 14
, s_1(8) -> 14
, l_0(3) -> 8
, l_0(5) -> 8
, l_0(8) -> 8
, l_1(16) -> 15
, t^#_0(3) -> 9
, t^#_0(5) -> 9
, t^#_0(8) -> 9
, a^#_0(3) -> 11
, a^#_0(5) -> 11
, a^#_0(8) -> 11
, a^#_0(13) -> 12
, a^#_1(17) -> 18
, c_2_0(12) -> 11
, c_2_1(18) -> 11}
7) { o^#(m(a(x1))) -> c_3(t^#(e(n(x1))))
, t^#(e(x1)) -> c_1(n^#(s(x1)))}
The usable rules for this path are the following:
{ s(a(x1)) -> l(a(t(o(m(a(t(e(x1))))))))
, n(s(x1)) -> a(l(a(t(x1))))
, t(o(x1)) -> m(a(x1))
, t(e(x1)) -> n(s(x1))
, a(l(x1)) -> a(t(x1))
, o(m(a(x1))) -> t(e(n(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:
{ s(a(x1)) -> l(a(t(o(m(a(t(e(x1))))))))
, n(s(x1)) -> a(l(a(t(x1))))
, t(o(x1)) -> m(a(x1))
, t(e(x1)) -> n(s(x1))
, a(l(x1)) -> a(t(x1))
, o(m(a(x1))) -> t(e(n(x1)))
, o^#(m(a(x1))) -> c_3(t^#(e(n(x1))))
, t^#(e(x1)) -> c_1(n^#(s(x1)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ a(l(x1)) -> a(t(x1))
, t^#(e(x1)) -> c_1(n^#(s(x1)))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ a(l(x1)) -> a(t(x1))
, t^#(e(x1)) -> c_1(n^#(s(x1)))}
Details:
Interpretation Functions:
t(x1) = [1] x1 + [0]
o(x1) = [1] x1 + [0]
m(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
e(x1) = [1] x1 + [0]
n(x1) = [1] x1 + [0]
s(x1) = [1] x1 + [1]
l(x1) = [1] x1 + [1]
t^#(x1) = [1] x1 + [2]
c_0(x1) = [0] x1 + [0]
a^#(x1) = [0] x1 + [0]
c_1(x1) = [1] x1 + [0]
n^#(x1) = [1] x1 + [0]
c_2(x1) = [0] x1 + [0]
o^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [1]
s^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ o(m(a(x1))) -> t(e(n(x1)))
, o^#(m(a(x1))) -> c_3(t^#(e(n(x1))))}
and weakly orienting the rules
{ a(l(x1)) -> a(t(x1))
, t^#(e(x1)) -> c_1(n^#(s(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ o(m(a(x1))) -> t(e(n(x1)))
, o^#(m(a(x1))) -> c_3(t^#(e(n(x1))))}
Details:
Interpretation Functions:
t(x1) = [1] x1 + [0]
o(x1) = [1] x1 + [8]
m(x1) = [1] x1 + [8]
a(x1) = [1] x1 + [0]
e(x1) = [1] x1 + [0]
n(x1) = [1] x1 + [0]
s(x1) = [1] x1 + [1]
l(x1) = [1] x1 + [1]
t^#(x1) = [1] x1 + [8]
c_0(x1) = [0] x1 + [0]
a^#(x1) = [0] x1 + [0]
c_1(x1) = [1] x1 + [0]
n^#(x1) = [1] x1 + [0]
c_2(x1) = [0] x1 + [0]
o^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ s(a(x1)) -> l(a(t(o(m(a(t(e(x1))))))))
, n(s(x1)) -> a(l(a(t(x1))))}
and weakly orienting the rules
{ o(m(a(x1))) -> t(e(n(x1)))
, o^#(m(a(x1))) -> c_3(t^#(e(n(x1))))
, a(l(x1)) -> a(t(x1))
, t^#(e(x1)) -> c_1(n^#(s(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ s(a(x1)) -> l(a(t(o(m(a(t(e(x1))))))))
, n(s(x1)) -> a(l(a(t(x1))))}
Details:
Interpretation Functions:
t(x1) = [1] x1 + [0]
o(x1) = [1] x1 + [0]
m(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
e(x1) = [1] x1 + [0]
n(x1) = [1] x1 + [0]
s(x1) = [1] x1 + [1]
l(x1) = [1] x1 + [0]
t^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
a^#(x1) = [0] x1 + [0]
c_1(x1) = [1] x1 + [0]
n^#(x1) = [1] x1 + [0]
c_2(x1) = [0] x1 + [0]
o^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{t(o(x1)) -> m(a(x1))}
and weakly orienting the rules
{ s(a(x1)) -> l(a(t(o(m(a(t(e(x1))))))))
, n(s(x1)) -> a(l(a(t(x1))))
, o(m(a(x1))) -> t(e(n(x1)))
, o^#(m(a(x1))) -> c_3(t^#(e(n(x1))))
, a(l(x1)) -> a(t(x1))
, t^#(e(x1)) -> c_1(n^#(s(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{t(o(x1)) -> m(a(x1))}
Details:
Interpretation Functions:
t(x1) = [1] x1 + [0]
o(x1) = [1] x1 + [4]
m(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
e(x1) = [1] x1 + [0]
n(x1) = [1] x1 + [0]
s(x1) = [1] x1 + [4]
l(x1) = [1] x1 + [0]
t^#(x1) = [1] x1 + [8]
c_0(x1) = [0] x1 + [0]
a^#(x1) = [0] x1 + [0]
c_1(x1) = [1] x1 + [0]
n^#(x1) = [1] x1 + [4]
c_2(x1) = [0] x1 + [0]
o^#(x1) = [1] x1 + [13]
c_3(x1) = [1] x1 + [1]
s^#(x1) = [0] x1 + [0]
c_4(x1) = [0] 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: {t(e(x1)) -> n(s(x1))}
Weak Rules:
{ t(o(x1)) -> m(a(x1))
, s(a(x1)) -> l(a(t(o(m(a(t(e(x1))))))))
, n(s(x1)) -> a(l(a(t(x1))))
, o(m(a(x1))) -> t(e(n(x1)))
, o^#(m(a(x1))) -> c_3(t^#(e(n(x1))))
, a(l(x1)) -> a(t(x1))
, t^#(e(x1)) -> c_1(n^#(s(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: {t(e(x1)) -> n(s(x1))}
Weak Rules:
{ t(o(x1)) -> m(a(x1))
, s(a(x1)) -> l(a(t(o(m(a(t(e(x1))))))))
, n(s(x1)) -> a(l(a(t(x1))))
, o(m(a(x1))) -> t(e(n(x1)))
, o^#(m(a(x1))) -> c_3(t^#(e(n(x1))))
, a(l(x1)) -> a(t(x1))
, t^#(e(x1)) -> c_1(n^#(s(x1)))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ m_0(3) -> 3
, m_0(5) -> 3
, m_0(8) -> 3
, e_0(3) -> 5
, e_0(5) -> 5
, e_0(8) -> 5
, s_0(3) -> 17
, s_0(5) -> 17
, s_0(8) -> 17
, l_0(3) -> 8
, l_0(5) -> 8
, l_0(8) -> 8
, t^#_0(3) -> 9
, t^#_0(5) -> 9
, t^#_0(8) -> 9
, c_1_0(16) -> 9
, n^#_0(3) -> 13
, n^#_0(5) -> 13
, n^#_0(8) -> 13
, n^#_0(17) -> 16
, o^#_0(3) -> 15
, o^#_0(5) -> 15
, o^#_0(8) -> 15}
8) {t^#(o(x1)) -> c_0(a^#(x1))}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
t(x1) = [0] x1 + [0]
o(x1) = [0] x1 + [0]
m(x1) = [0] x1 + [0]
a(x1) = [0] x1 + [0]
e(x1) = [0] x1 + [0]
n(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
l(x1) = [0] x1 + [0]
t^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
n^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
o^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(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: {t^#(o(x1)) -> c_0(a^#(x1))}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{t^#(o(x1)) -> c_0(a^#(x1))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{t^#(o(x1)) -> c_0(a^#(x1))}
Details:
Interpretation Functions:
t(x1) = [0] x1 + [0]
o(x1) = [1] x1 + [0]
m(x1) = [0] x1 + [0]
a(x1) = [0] x1 + [0]
e(x1) = [0] x1 + [0]
n(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
l(x1) = [0] x1 + [0]
t^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_1(x1) = [0] x1 + [0]
n^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
o^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
s^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(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: {t^#(o(x1)) -> c_0(a^#(x1))}
Details:
The given problem does not contain any strict rules