'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ a(x1) -> g(d(x1))
, b(b(b(x1))) -> c(d(c(x1)))
, b(b(x1)) -> a(g(g(x1)))
, c(d(x1)) -> g(g(x1))
, g(g(g(x1))) -> b(b(x1))}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ a^#(x1) -> c_0(g^#(d(x1)))
, b^#(b(b(x1))) -> c_1(c^#(d(c(x1))))
, b^#(b(x1)) -> c_2(a^#(g(g(x1))))
, c^#(d(x1)) -> c_3(g^#(g(x1)))
, g^#(g(g(x1))) -> c_4(b^#(b(x1)))}
The usable rules are:
{ b(b(b(x1))) -> c(d(c(x1)))
, b(b(x1)) -> a(g(g(x1)))
, c(d(x1)) -> g(g(x1))
, g(g(g(x1))) -> b(b(x1))
, a(x1) -> g(d(x1))}
The estimated dependency graph contains the following edges:
{b^#(b(b(x1))) -> c_1(c^#(d(c(x1))))}
==> {c^#(d(x1)) -> c_3(g^#(g(x1)))}
{b^#(b(x1)) -> c_2(a^#(g(g(x1))))}
==> {a^#(x1) -> c_0(g^#(d(x1)))}
{c^#(d(x1)) -> c_3(g^#(g(x1)))}
==> {g^#(g(g(x1))) -> c_4(b^#(b(x1)))}
{g^#(g(g(x1))) -> c_4(b^#(b(x1)))}
==> {b^#(b(x1)) -> c_2(a^#(g(g(x1))))}
{g^#(g(g(x1))) -> c_4(b^#(b(x1)))}
==> {b^#(b(b(x1))) -> c_1(c^#(d(c(x1))))}
We consider the following path(s):
1) { b^#(b(b(x1))) -> c_1(c^#(d(c(x1))))
, g^#(g(g(x1))) -> c_4(b^#(b(x1)))
, c^#(d(x1)) -> c_3(g^#(g(x1)))
, b^#(b(x1)) -> c_2(a^#(g(g(x1))))
, a^#(x1) -> c_0(g^#(d(x1)))}
The usable rules for this path are the following:
{ b(b(b(x1))) -> c(d(c(x1)))
, b(b(x1)) -> a(g(g(x1)))
, c(d(x1)) -> g(g(x1))
, g(g(g(x1))) -> b(b(x1))
, a(x1) -> g(d(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(b(b(x1))) -> c(d(c(x1)))
, b(b(x1)) -> a(g(g(x1)))
, c(d(x1)) -> g(g(x1))
, g(g(g(x1))) -> b(b(x1))
, a(x1) -> g(d(x1))
, b^#(b(x1)) -> c_2(a^#(g(g(x1))))
, b^#(b(b(x1))) -> c_1(c^#(d(c(x1))))
, g^#(g(g(x1))) -> c_4(b^#(b(x1)))
, c^#(d(x1)) -> c_3(g^#(g(x1)))
, a^#(x1) -> c_0(g^#(d(x1)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{a(x1) -> g(d(x1))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a(x1) -> g(d(x1))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
g^#(x1) = [1] x1 + [1]
b^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [1]
c^#(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [15]
c_4(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{c^#(d(x1)) -> c_3(g^#(g(x1)))}
and weakly orienting the rules
{a(x1) -> g(d(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c^#(d(x1)) -> c_3(g^#(g(x1)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
g^#(x1) = [1] x1 + [1]
b^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [1]
c^#(x1) = [1] x1 + [2]
c_2(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b^#(b(b(x1))) -> c_1(c^#(d(c(x1))))}
and weakly orienting the rules
{ c^#(d(x1)) -> c_3(g^#(g(x1)))
, a(x1) -> g(d(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b^#(b(b(x1))) -> c_1(c^#(d(c(x1))))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [1]
g^#(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a^#(x1) -> c_0(g^#(d(x1)))}
and weakly orienting the rules
{ b^#(b(b(x1))) -> c_1(c^#(d(c(x1))))
, c^#(d(x1)) -> c_3(g^#(g(x1)))
, a(x1) -> g(d(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a^#(x1) -> c_0(g^#(d(x1)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [0]
g^#(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [1]
c^#(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b^#(b(x1)) -> c_2(a^#(g(g(x1))))}
and weakly orienting the rules
{ a^#(x1) -> c_0(g^#(d(x1)))
, b^#(b(b(x1))) -> c_1(c^#(d(c(x1))))
, c^#(d(x1)) -> c_3(g^#(g(x1)))
, a(x1) -> g(d(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b^#(b(x1)) -> c_2(a^#(g(g(x1))))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
g^#(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [1]
c^#(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ b(b(b(x1))) -> c(d(c(x1)))
, b(b(x1)) -> a(g(g(x1)))}
and weakly orienting the rules
{ b^#(b(x1)) -> c_2(a^#(g(g(x1))))
, a^#(x1) -> c_0(g^#(d(x1)))
, b^#(b(b(x1))) -> c_1(c^#(d(c(x1))))
, c^#(d(x1)) -> c_3(g^#(g(x1)))
, a(x1) -> g(d(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ b(b(b(x1))) -> c(d(c(x1)))
, b(b(x1)) -> a(g(g(x1)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [8]
g(x1) = [1] x1 + [2]
d(x1) = [1] x1 + [2]
b(x1) = [1] x1 + [10]
c(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [12]
c_0(x1) = [1] x1 + [0]
g^#(x1) = [1] x1 + [6]
b^#(x1) = [1] x1 + [10]
c_1(x1) = [1] x1 + [9]
c^#(x1) = [1] x1 + [14]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [8]
c_4(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{c(d(x1)) -> g(g(x1))}
and weakly orienting the rules
{ b(b(b(x1))) -> c(d(c(x1)))
, b(b(x1)) -> a(g(g(x1)))
, b^#(b(x1)) -> c_2(a^#(g(g(x1))))
, a^#(x1) -> c_0(g^#(d(x1)))
, b^#(b(b(x1))) -> c_1(c^#(d(c(x1))))
, c^#(d(x1)) -> c_3(g^#(g(x1)))
, a(x1) -> g(d(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c(d(x1)) -> g(g(x1))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [8]
c(x1) = [1] x1 + [1]
a^#(x1) = [1] x1 + [8]
c_0(x1) = [1] x1 + [7]
g^#(x1) = [1] x1 + [1]
b^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [8]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [3]
c_4(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:
{ g(g(g(x1))) -> b(b(x1))
, g^#(g(g(x1))) -> c_4(b^#(b(x1)))}
Weak Rules:
{ c(d(x1)) -> g(g(x1))
, b(b(b(x1))) -> c(d(c(x1)))
, b(b(x1)) -> a(g(g(x1)))
, b^#(b(x1)) -> c_2(a^#(g(g(x1))))
, a^#(x1) -> c_0(g^#(d(x1)))
, b^#(b(b(x1))) -> c_1(c^#(d(c(x1))))
, c^#(d(x1)) -> c_3(g^#(g(x1)))
, a(x1) -> g(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:
{ g(g(g(x1))) -> b(b(x1))
, g^#(g(g(x1))) -> c_4(b^#(b(x1)))}
Weak Rules:
{ c(d(x1)) -> g(g(x1))
, b(b(b(x1))) -> c(d(c(x1)))
, b(b(x1)) -> a(g(g(x1)))
, b^#(b(x1)) -> c_2(a^#(g(g(x1))))
, a^#(x1) -> c_0(g^#(d(x1)))
, b^#(b(b(x1))) -> c_1(c^#(d(c(x1))))
, c^#(d(x1)) -> c_3(g^#(g(x1)))
, a(x1) -> g(d(x1))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ g_0(3) -> 13
, d_0(3) -> 3
, a^#_0(3) -> 6
, c_0_0(8) -> 6
, g^#_0(3) -> 8
, g^#_0(13) -> 12
, b^#_0(3) -> 9
, c^#_0(3) -> 11
, c_3_0(12) -> 11}
2) { b^#(b(b(x1))) -> c_1(c^#(d(c(x1))))
, g^#(g(g(x1))) -> c_4(b^#(b(x1)))
, c^#(d(x1)) -> c_3(g^#(g(x1)))}
The usable rules for this path are the following:
{ b(b(b(x1))) -> c(d(c(x1)))
, b(b(x1)) -> a(g(g(x1)))
, c(d(x1)) -> g(g(x1))
, g(g(g(x1))) -> b(b(x1))
, a(x1) -> g(d(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(b(b(x1))) -> c(d(c(x1)))
, b(b(x1)) -> a(g(g(x1)))
, c(d(x1)) -> g(g(x1))
, g(g(g(x1))) -> b(b(x1))
, a(x1) -> g(d(x1))
, b^#(b(b(x1))) -> c_1(c^#(d(c(x1))))
, g^#(g(g(x1))) -> c_4(b^#(b(x1)))
, c^#(d(x1)) -> c_3(g^#(g(x1)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{a(x1) -> g(d(x1))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a(x1) -> g(d(x1))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
a^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
g^#(x1) = [1] x1 + [1]
b^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [1]
c^#(x1) = [1] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [7]
c_4(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{g^#(g(g(x1))) -> c_4(b^#(b(x1)))}
and weakly orienting the rules
{a(x1) -> g(d(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{g^#(g(g(x1))) -> c_4(b^#(b(x1)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
a^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
g^#(x1) = [1] x1 + [5]
b^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [1]
c^#(x1) = [1] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [3]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{c^#(d(x1)) -> c_3(g^#(g(x1)))}
and weakly orienting the rules
{ g^#(g(g(x1))) -> c_4(b^#(b(x1)))
, a(x1) -> g(d(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c^#(d(x1)) -> c_3(g^#(g(x1)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
a^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
g^#(x1) = [1] x1 + [1]
b^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [1]
c^#(x1) = [1] x1 + [2]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{g(g(g(x1))) -> b(b(x1))}
and weakly orienting the rules
{ c^#(d(x1)) -> c_3(g^#(g(x1)))
, g^#(g(g(x1))) -> c_4(b^#(b(x1)))
, a(x1) -> g(d(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{g(g(g(x1))) -> b(b(x1))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [12]
g(x1) = [1] x1 + [8]
d(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
a^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
g^#(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [1]
c^#(x1) = [1] x1 + [8]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b^#(b(b(x1))) -> c_1(c^#(d(c(x1))))}
and weakly orienting the rules
{ g(g(g(x1))) -> b(b(x1))
, c^#(d(x1)) -> c_3(g^#(g(x1)))
, g^#(g(g(x1))) -> c_4(b^#(b(x1)))
, a(x1) -> g(d(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b^#(b(b(x1))) -> c_1(c^#(d(c(x1))))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [8]
g(x1) = [1] x1 + [5]
d(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
a^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
g^#(x1) = [1] x1 + [3]
b^#(x1) = [1] x1 + [12]
c_1(x1) = [1] x1 + [1]
c^#(x1) = [1] x1 + [8]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{c(d(x1)) -> g(g(x1))}
and weakly orienting the rules
{ b^#(b(b(x1))) -> c_1(c^#(d(c(x1))))
, g(g(g(x1))) -> b(b(x1))
, c^#(d(x1)) -> c_3(g^#(g(x1)))
, g^#(g(g(x1))) -> c_4(b^#(b(x1)))
, a(x1) -> g(d(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c(d(x1)) -> g(g(x1))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [9]
g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [8]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
a^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
g^#(x1) = [1] x1 + [8]
b^#(x1) = [1] x1 + [8]
c_1(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b(b(b(x1))) -> c(d(c(x1)))}
and weakly orienting the rules
{ c(d(x1)) -> g(g(x1))
, b^#(b(b(x1))) -> c_1(c^#(d(c(x1))))
, g(g(g(x1))) -> b(b(x1))
, c^#(d(x1)) -> c_3(g^#(g(x1)))
, g^#(g(g(x1))) -> c_4(b^#(b(x1)))
, a(x1) -> g(d(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b(b(b(x1))) -> c(d(c(x1)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [15]
g(x1) = [1] x1 + [6]
d(x1) = [1] x1 + [9]
b(x1) = [1] x1 + [8]
c(x1) = [1] x1 + [3]
a^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
g^#(x1) = [1] x1 + [8]
b^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [5]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [0]
c_4(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: {b(b(x1)) -> a(g(g(x1)))}
Weak Rules:
{ b(b(b(x1))) -> c(d(c(x1)))
, c(d(x1)) -> g(g(x1))
, b^#(b(b(x1))) -> c_1(c^#(d(c(x1))))
, g(g(g(x1))) -> b(b(x1))
, c^#(d(x1)) -> c_3(g^#(g(x1)))
, g^#(g(g(x1))) -> c_4(b^#(b(x1)))
, a(x1) -> g(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(b(x1)) -> a(g(g(x1)))}
Weak Rules:
{ b(b(b(x1))) -> c(d(c(x1)))
, c(d(x1)) -> g(g(x1))
, b^#(b(b(x1))) -> c_1(c^#(d(c(x1))))
, g(g(g(x1))) -> b(b(x1))
, c^#(d(x1)) -> c_3(g^#(g(x1)))
, g^#(g(g(x1))) -> c_4(b^#(b(x1)))
, a(x1) -> g(d(x1))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ g_0(3) -> 13
, d_0(3) -> 3
, g^#_0(3) -> 8
, g^#_0(13) -> 12
, b^#_0(3) -> 9
, c^#_0(3) -> 11
, c_3_0(12) -> 11}
3) { b^#(b(b(x1))) -> c_1(c^#(d(c(x1))))
, g^#(g(g(x1))) -> c_4(b^#(b(x1)))
, c^#(d(x1)) -> c_3(g^#(g(x1)))
, b^#(b(x1)) -> c_2(a^#(g(g(x1))))}
The usable rules for this path are the following:
{ b(b(b(x1))) -> c(d(c(x1)))
, b(b(x1)) -> a(g(g(x1)))
, c(d(x1)) -> g(g(x1))
, g(g(g(x1))) -> b(b(x1))
, a(x1) -> g(d(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(b(b(x1))) -> c(d(c(x1)))
, b(b(x1)) -> a(g(g(x1)))
, c(d(x1)) -> g(g(x1))
, g(g(g(x1))) -> b(b(x1))
, a(x1) -> g(d(x1))
, b^#(b(b(x1))) -> c_1(c^#(d(c(x1))))
, g^#(g(g(x1))) -> c_4(b^#(b(x1)))
, c^#(d(x1)) -> c_3(g^#(g(x1)))
, b^#(b(x1)) -> c_2(a^#(g(g(x1))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{a(x1) -> g(d(x1))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a(x1) -> g(d(x1))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
g^#(x1) = [1] x1 + [1]
b^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [1]
c^#(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b^#(b(x1)) -> c_2(a^#(g(g(x1))))}
and weakly orienting the rules
{a(x1) -> g(d(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b^#(b(x1)) -> c_2(a^#(g(g(x1))))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
g^#(x1) = [1] x1 + [1]
b^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [1]
c^#(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{c^#(d(x1)) -> c_3(g^#(g(x1)))}
and weakly orienting the rules
{ b^#(b(x1)) -> c_2(a^#(g(g(x1))))
, a(x1) -> g(d(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c^#(d(x1)) -> c_3(g^#(g(x1)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
g^#(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [1]
c^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ g(g(g(x1))) -> b(b(x1))
, b^#(b(b(x1))) -> c_1(c^#(d(c(x1))))}
and weakly orienting the rules
{ c^#(d(x1)) -> c_3(g^#(g(x1)))
, b^#(b(x1)) -> c_2(a^#(g(g(x1))))
, a(x1) -> g(d(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ g(g(g(x1))) -> b(b(x1))
, b^#(b(b(x1))) -> c_1(c^#(d(c(x1))))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [1]
g(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
g^#(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [13]
c_1(x1) = [1] x1 + [4]
c^#(x1) = [1] x1 + [8]
c_2(x1) = [1] x1 + [8]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{g^#(g(g(x1))) -> c_4(b^#(b(x1)))}
and weakly orienting the rules
{ g(g(g(x1))) -> b(b(x1))
, b^#(b(b(x1))) -> c_1(c^#(d(c(x1))))
, c^#(d(x1)) -> c_3(g^#(g(x1)))
, b^#(b(x1)) -> c_2(a^#(g(g(x1))))
, a(x1) -> g(d(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{g^#(g(g(x1))) -> c_4(b^#(b(x1)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [9]
g(x1) = [1] x1 + [3]
d(x1) = [1] x1 + [0]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
g^#(x1) = [1] x1 + [2]
b^#(x1) = [1] x1 + [6]
c_1(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [5]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{c(d(x1)) -> g(g(x1))}
and weakly orienting the rules
{ g^#(g(g(x1))) -> c_4(b^#(b(x1)))
, g(g(g(x1))) -> b(b(x1))
, b^#(b(b(x1))) -> c_1(c^#(d(c(x1))))
, c^#(d(x1)) -> c_3(g^#(g(x1)))
, b^#(b(x1)) -> c_2(a^#(g(g(x1))))
, a(x1) -> g(d(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c(d(x1)) -> g(g(x1))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [9]
g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [8]
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
a^#(x1) = [1] x1 + [4]
c_0(x1) = [0] x1 + [0]
g^#(x1) = [1] x1 + [9]
b^#(x1) = [1] x1 + [9]
c_1(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b(b(b(x1))) -> c(d(c(x1)))}
and weakly orienting the rules
{ c(d(x1)) -> g(g(x1))
, g^#(g(g(x1))) -> c_4(b^#(b(x1)))
, g(g(g(x1))) -> b(b(x1))
, b^#(b(b(x1))) -> c_1(c^#(d(c(x1))))
, c^#(d(x1)) -> c_3(g^#(g(x1)))
, b^#(b(x1)) -> c_2(a^#(g(g(x1))))
, a(x1) -> g(d(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b(b(b(x1))) -> c(d(c(x1)))}
Details:
Interpretation Functions:
a(x1) = [1] x1 + [7]
g(x1) = [1] x1 + [6]
d(x1) = [1] x1 + [1]
b(x1) = [1] x1 + [8]
c(x1) = [1] x1 + [11]
a^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
g^#(x1) = [1] x1 + [5]
b^#(x1) = [1] x1 + [9]
c_1(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [13]
c_2(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
c_4(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(b(x1)) -> a(g(g(x1)))}
Weak Rules:
{ b(b(b(x1))) -> c(d(c(x1)))
, c(d(x1)) -> g(g(x1))
, g^#(g(g(x1))) -> c_4(b^#(b(x1)))
, g(g(g(x1))) -> b(b(x1))
, b^#(b(b(x1))) -> c_1(c^#(d(c(x1))))
, c^#(d(x1)) -> c_3(g^#(g(x1)))
, b^#(b(x1)) -> c_2(a^#(g(g(x1))))
, a(x1) -> g(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(b(x1)) -> a(g(g(x1)))}
Weak Rules:
{ b(b(b(x1))) -> c(d(c(x1)))
, c(d(x1)) -> g(g(x1))
, g^#(g(g(x1))) -> c_4(b^#(b(x1)))
, g(g(g(x1))) -> b(b(x1))
, b^#(b(b(x1))) -> c_1(c^#(d(c(x1))))
, c^#(d(x1)) -> c_3(g^#(g(x1)))
, b^#(b(x1)) -> c_2(a^#(g(g(x1))))
, a(x1) -> g(d(x1))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ g_0(3) -> 13
, d_0(3) -> 3
, a^#_0(3) -> 6
, g^#_0(3) -> 8
, g^#_0(13) -> 12
, b^#_0(3) -> 9
, c^#_0(3) -> 11
, c_3_0(12) -> 11}