'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ b(b(x1)) -> c(d(x1))
, c(c(x1)) -> d(d(d(x1)))
, c(x1) -> g(x1)
, d(d(x1)) -> c(f(x1))
, d(d(d(x1))) -> g(c(x1))
, f(x1) -> a(g(x1))
, g(x1) -> d(a(b(x1)))
, g(g(x1)) -> b(c(x1))}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ b^#(b(x1)) -> c_0(c^#(d(x1)))
, c^#(c(x1)) -> c_1(d^#(d(d(x1))))
, c^#(x1) -> c_2(g^#(x1))
, d^#(d(x1)) -> c_3(c^#(f(x1)))
, d^#(d(d(x1))) -> c_4(g^#(c(x1)))
, f^#(x1) -> c_5(g^#(x1))
, g^#(x1) -> c_6(d^#(a(b(x1))))
, g^#(g(x1)) -> c_7(b^#(c(x1)))}
The usable rules are:
{ b(b(x1)) -> c(d(x1))
, c(c(x1)) -> d(d(d(x1)))
, c(x1) -> g(x1)
, d(d(x1)) -> c(f(x1))
, d(d(d(x1))) -> g(c(x1))
, f(x1) -> a(g(x1))
, g(x1) -> d(a(b(x1)))
, g(g(x1)) -> b(c(x1))}
The estimated dependency graph contains the following edges:
{b^#(b(x1)) -> c_0(c^#(d(x1)))}
==> {c^#(x1) -> c_2(g^#(x1))}
{b^#(b(x1)) -> c_0(c^#(d(x1)))}
==> {c^#(c(x1)) -> c_1(d^#(d(d(x1))))}
{c^#(c(x1)) -> c_1(d^#(d(d(x1))))}
==> {d^#(d(d(x1))) -> c_4(g^#(c(x1)))}
{c^#(c(x1)) -> c_1(d^#(d(d(x1))))}
==> {d^#(d(x1)) -> c_3(c^#(f(x1)))}
{c^#(x1) -> c_2(g^#(x1))}
==> {g^#(g(x1)) -> c_7(b^#(c(x1)))}
{c^#(x1) -> c_2(g^#(x1))}
==> {g^#(x1) -> c_6(d^#(a(b(x1))))}
{d^#(d(x1)) -> c_3(c^#(f(x1)))}
==> {c^#(x1) -> c_2(g^#(x1))}
{d^#(d(x1)) -> c_3(c^#(f(x1)))}
==> {c^#(c(x1)) -> c_1(d^#(d(d(x1))))}
{d^#(d(d(x1))) -> c_4(g^#(c(x1)))}
==> {g^#(g(x1)) -> c_7(b^#(c(x1)))}
{d^#(d(d(x1))) -> c_4(g^#(c(x1)))}
==> {g^#(x1) -> c_6(d^#(a(b(x1))))}
{f^#(x1) -> c_5(g^#(x1))}
==> {g^#(g(x1)) -> c_7(b^#(c(x1)))}
{f^#(x1) -> c_5(g^#(x1))}
==> {g^#(x1) -> c_6(d^#(a(b(x1))))}
{g^#(g(x1)) -> c_7(b^#(c(x1)))}
==> {b^#(b(x1)) -> c_0(c^#(d(x1)))}
We consider the following path(s):
1) { f^#(x1) -> c_5(g^#(x1))
, b^#(b(x1)) -> c_0(c^#(d(x1)))
, g^#(g(x1)) -> c_7(b^#(c(x1)))
, d^#(d(d(x1))) -> c_4(g^#(c(x1)))
, c^#(c(x1)) -> c_1(d^#(d(d(x1))))
, d^#(d(x1)) -> c_3(c^#(f(x1)))
, c^#(x1) -> c_2(g^#(x1))}
The usable rules for this path are the following:
{ c(c(x1)) -> d(d(d(x1)))
, c(x1) -> g(x1)
, d(d(x1)) -> c(f(x1))
, d(d(d(x1))) -> g(c(x1))
, f(x1) -> a(g(x1))
, g(x1) -> d(a(b(x1)))
, g(g(x1)) -> b(c(x1))
, b(b(x1)) -> c(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:
{ c(c(x1)) -> d(d(d(x1)))
, c(x1) -> g(x1)
, d(d(x1)) -> c(f(x1))
, d(d(d(x1))) -> g(c(x1))
, f(x1) -> a(g(x1))
, g(x1) -> d(a(b(x1)))
, g(g(x1)) -> b(c(x1))
, b(b(x1)) -> c(d(x1))
, f^#(x1) -> c_5(g^#(x1))
, b^#(b(x1)) -> c_0(c^#(d(x1)))
, g^#(g(x1)) -> c_7(b^#(c(x1)))
, d^#(d(d(x1))) -> c_4(g^#(c(x1)))
, c^#(c(x1)) -> c_1(d^#(d(d(x1))))
, d^#(d(x1)) -> c_3(c^#(f(x1)))
, c^#(x1) -> c_2(g^#(x1))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ b(b(x1)) -> c(d(x1))
, b^#(b(x1)) -> c_0(c^#(d(x1)))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ b(b(x1)) -> c(d(x1))
, b^#(b(x1)) -> c_0(c^#(d(x1)))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [8]
c(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [1]
f(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [1]
c^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [1]
g^#(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [1]
c_4(x1) = [1] x1 + [1]
f^#(x1) = [1] x1 + [0]
c_5(x1) = [1] x1 + [1]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ g(x1) -> d(a(b(x1)))
, g(g(x1)) -> b(c(x1))
, d^#(d(d(x1))) -> c_4(g^#(c(x1)))}
and weakly orienting the rules
{ b(b(x1)) -> c(d(x1))
, b^#(b(x1)) -> c_0(c^#(d(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ g(x1) -> d(a(b(x1)))
, g(g(x1)) -> b(c(x1))
, d^#(d(d(x1))) -> c_4(g^#(c(x1)))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [1]
f(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [1]
g^#(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [1]
c_4(x1) = [1] x1 + [0]
f^#(x1) = [1] x1 + [0]
c_5(x1) = [1] x1 + [1]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{f^#(x1) -> c_5(g^#(x1))}
and weakly orienting the rules
{ g(x1) -> d(a(b(x1)))
, g(g(x1)) -> b(c(x1))
, d^#(d(d(x1))) -> c_4(g^#(c(x1)))
, b(b(x1)) -> c(d(x1))
, b^#(b(x1)) -> c_0(c^#(d(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{f^#(x1) -> c_5(g^#(x1))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [1]
f(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [1]
d^#(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [1]
g^#(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [0]
f^#(x1) = [1] x1 + [8]
c_5(x1) = [1] x1 + [1]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{c^#(x1) -> c_2(g^#(x1))}
and weakly orienting the rules
{ f^#(x1) -> c_5(g^#(x1))
, g(x1) -> d(a(b(x1)))
, g(g(x1)) -> b(c(x1))
, d^#(d(d(x1))) -> c_4(g^#(c(x1)))
, b(b(x1)) -> c(d(x1))
, b^#(b(x1)) -> c_0(c^#(d(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c^#(x1) -> c_2(g^#(x1))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [1]
f(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [1]
d^#(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [0]
g^#(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [0]
f^#(x1) = [1] x1 + [8]
c_5(x1) = [1] x1 + [1]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{c^#(c(x1)) -> c_1(d^#(d(d(x1))))}
and weakly orienting the rules
{ c^#(x1) -> c_2(g^#(x1))
, f^#(x1) -> c_5(g^#(x1))
, g(x1) -> d(a(b(x1)))
, g(g(x1)) -> b(c(x1))
, d^#(d(d(x1))) -> c_4(g^#(c(x1)))
, b(b(x1)) -> c(d(x1))
, b^#(b(x1)) -> c_0(c^#(d(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c^#(c(x1)) -> c_1(d^#(d(d(x1))))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [1]
f(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [1]
g^#(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [0]
f^#(x1) = [1] x1 + [8]
c_5(x1) = [1] x1 + [1]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{g^#(g(x1)) -> c_7(b^#(c(x1)))}
and weakly orienting the rules
{ c^#(c(x1)) -> c_1(d^#(d(d(x1))))
, c^#(x1) -> c_2(g^#(x1))
, f^#(x1) -> c_5(g^#(x1))
, g(x1) -> d(a(b(x1)))
, g(g(x1)) -> b(c(x1))
, d^#(d(d(x1))) -> c_4(g^#(c(x1)))
, b(b(x1)) -> c(d(x1))
, b^#(b(x1)) -> c_0(c^#(d(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{g^#(g(x1)) -> c_7(b^#(c(x1)))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [1]
f(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [1]
d^#(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [1]
g^#(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [3]
c_4(x1) = [1] x1 + [0]
f^#(x1) = [1] x1 + [8]
c_5(x1) = [1] x1 + [1]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{f(x1) -> a(g(x1))}
and weakly orienting the rules
{ g^#(g(x1)) -> c_7(b^#(c(x1)))
, c^#(c(x1)) -> c_1(d^#(d(d(x1))))
, c^#(x1) -> c_2(g^#(x1))
, f^#(x1) -> c_5(g^#(x1))
, g(x1) -> d(a(b(x1)))
, g(g(x1)) -> b(c(x1))
, d^#(d(d(x1))) -> c_4(g^#(c(x1)))
, b(b(x1)) -> c(d(x1))
, b^#(b(x1)) -> c_0(c^#(d(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{f(x1) -> a(g(x1))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [3]
c(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [3]
f(x1) = [1] x1 + [8]
a(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [4]
c_1(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [2]
g^#(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [5]
c_4(x1) = [1] x1 + [0]
f^#(x1) = [1] x1 + [8]
c_5(x1) = [1] x1 + [1]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{c(c(x1)) -> d(d(d(x1)))}
and weakly orienting the rules
{ f(x1) -> a(g(x1))
, g^#(g(x1)) -> c_7(b^#(c(x1)))
, c^#(c(x1)) -> c_1(d^#(d(d(x1))))
, c^#(x1) -> c_2(g^#(x1))
, f^#(x1) -> c_5(g^#(x1))
, g(x1) -> d(a(b(x1)))
, g(g(x1)) -> b(c(x1))
, d^#(d(d(x1))) -> c_4(g^#(c(x1)))
, b(b(x1)) -> c(d(x1))
, b^#(b(x1)) -> c_0(c^#(d(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c(c(x1)) -> d(d(d(x1)))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [8]
c(x1) = [1] x1 + [13]
d(x1) = [1] x1 + [1]
g(x1) = [1] x1 + [13]
f(x1) = [1] x1 + [14]
a(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [2]
c_0(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [9]
c_1(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [15]
c_2(x1) = [1] x1 + [1]
g^#(x1) = [1] x1 + [2]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [0]
f^#(x1) = [1] x1 + [8]
c_5(x1) = [1] x1 + [1]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{c(x1) -> g(x1)}
and weakly orienting the rules
{ c(c(x1)) -> d(d(d(x1)))
, f(x1) -> a(g(x1))
, g^#(g(x1)) -> c_7(b^#(c(x1)))
, c^#(c(x1)) -> c_1(d^#(d(d(x1))))
, c^#(x1) -> c_2(g^#(x1))
, f^#(x1) -> c_5(g^#(x1))
, g(x1) -> d(a(b(x1)))
, g(g(x1)) -> b(c(x1))
, d^#(d(d(x1))) -> c_4(g^#(c(x1)))
, b(b(x1)) -> c(d(x1))
, b^#(b(x1)) -> c_0(c^#(d(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c(x1) -> g(x1)}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [4]
c(x1) = [1] x1 + [8]
d(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [6]
f(x1) = [1] x1 + [6]
a(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [2]
c_1(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [10]
c_2(x1) = [1] x1 + [0]
g^#(x1) = [1] x1 + [2]
c_3(x1) = [1] x1 + [8]
c_4(x1) = [1] x1 + [0]
f^#(x1) = [1] x1 + [4]
c_5(x1) = [1] x1 + [1]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{d^#(d(x1)) -> c_3(c^#(f(x1)))}
and weakly orienting the rules
{ c(x1) -> g(x1)
, c(c(x1)) -> d(d(d(x1)))
, f(x1) -> a(g(x1))
, g^#(g(x1)) -> c_7(b^#(c(x1)))
, c^#(c(x1)) -> c_1(d^#(d(d(x1))))
, c^#(x1) -> c_2(g^#(x1))
, f^#(x1) -> c_5(g^#(x1))
, g(x1) -> d(a(b(x1)))
, g(g(x1)) -> b(c(x1))
, d^#(d(d(x1))) -> c_4(g^#(c(x1)))
, b(b(x1)) -> c(d(x1))
, b^#(b(x1)) -> c_0(c^#(d(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d^#(d(x1)) -> c_3(c^#(f(x1)))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [3]
c(x1) = [1] x1 + [6]
d(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [5]
f(x1) = [1] x1 + [5]
a(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [7]
c_2(x1) = [1] x1 + [0]
g^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [0]
f^#(x1) = [1] x1 + [8]
c_5(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [1] x1 + [0]
Finally we apply the subprocessor
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ d(d(x1)) -> c(f(x1))
, d(d(d(x1))) -> g(c(x1))}
Weak Rules:
{ d^#(d(x1)) -> c_3(c^#(f(x1)))
, c(x1) -> g(x1)
, c(c(x1)) -> d(d(d(x1)))
, f(x1) -> a(g(x1))
, g^#(g(x1)) -> c_7(b^#(c(x1)))
, c^#(c(x1)) -> c_1(d^#(d(d(x1))))
, c^#(x1) -> c_2(g^#(x1))
, f^#(x1) -> c_5(g^#(x1))
, g(x1) -> d(a(b(x1)))
, g(g(x1)) -> b(c(x1))
, d^#(d(d(x1))) -> c_4(g^#(c(x1)))
, b(b(x1)) -> c(d(x1))
, b^#(b(x1)) -> c_0(c^#(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:
{ d(d(x1)) -> c(f(x1))
, d(d(d(x1))) -> g(c(x1))}
Weak Rules:
{ d^#(d(x1)) -> c_3(c^#(f(x1)))
, c(x1) -> g(x1)
, c(c(x1)) -> d(d(d(x1)))
, f(x1) -> a(g(x1))
, g^#(g(x1)) -> c_7(b^#(c(x1)))
, c^#(c(x1)) -> c_1(d^#(d(d(x1))))
, c^#(x1) -> c_2(g^#(x1))
, f^#(x1) -> c_5(g^#(x1))
, g(x1) -> d(a(b(x1)))
, g(g(x1)) -> b(c(x1))
, d^#(d(d(x1))) -> c_4(g^#(c(x1)))
, b(b(x1)) -> c(d(x1))
, b^#(b(x1)) -> c_0(c^#(d(x1)))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ a_0(2) -> 2
, b^#_0(2) -> 1
, c^#_0(2) -> 1
, d^#_0(2) -> 1
, c_2_0(1) -> 1
, g^#_0(2) -> 1
, f^#_0(2) -> 1
, c_5_0(1) -> 1}
2) { f^#(x1) -> c_5(g^#(x1))
, b^#(b(x1)) -> c_0(c^#(d(x1)))
, g^#(g(x1)) -> c_7(b^#(c(x1)))
, d^#(d(d(x1))) -> c_4(g^#(c(x1)))
, c^#(c(x1)) -> c_1(d^#(d(d(x1))))
, d^#(d(x1)) -> c_3(c^#(f(x1)))
, c^#(x1) -> c_2(g^#(x1))
, g^#(x1) -> c_6(d^#(a(b(x1))))}
The usable rules for this path are the following:
{ b(b(x1)) -> c(d(x1))
, c(c(x1)) -> d(d(d(x1)))
, c(x1) -> g(x1)
, d(d(x1)) -> c(f(x1))
, d(d(d(x1))) -> g(c(x1))
, f(x1) -> a(g(x1))
, g(x1) -> d(a(b(x1)))
, g(g(x1)) -> b(c(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(x1)) -> c(d(x1))
, c(c(x1)) -> d(d(d(x1)))
, c(x1) -> g(x1)
, d(d(x1)) -> c(f(x1))
, d(d(d(x1))) -> g(c(x1))
, f(x1) -> a(g(x1))
, g(x1) -> d(a(b(x1)))
, g(g(x1)) -> b(c(x1))
, b^#(b(x1)) -> c_0(c^#(d(x1)))
, g^#(g(x1)) -> c_7(b^#(c(x1)))
, d^#(d(d(x1))) -> c_4(g^#(c(x1)))
, c^#(c(x1)) -> c_1(d^#(d(d(x1))))
, d^#(d(x1)) -> c_3(c^#(f(x1)))
, c^#(x1) -> c_2(g^#(x1))
, f^#(x1) -> c_5(g^#(x1))
, g^#(x1) -> c_6(d^#(a(b(x1))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ c(c(x1)) -> d(d(d(x1)))
, c(x1) -> g(x1)
, c^#(c(x1)) -> c_1(d^#(d(d(x1))))
, c^#(x1) -> c_2(g^#(x1))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ c(c(x1)) -> d(d(d(x1)))
, c(x1) -> g(x1)
, c^#(c(x1)) -> c_1(d^#(d(d(x1))))
, c^#(x1) -> c_2(g^#(x1))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [1]
b^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [1]
c^#(x1) = [1] x1 + [8]
c_1(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [0]
g^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [1]
c_4(x1) = [1] x1 + [1]
f^#(x1) = [1] x1 + [0]
c_5(x1) = [1] x1 + [0]
c_6(x1) = [1] x1 + [1]
c_7(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b^#(b(x1)) -> c_0(c^#(d(x1)))}
and weakly orienting the rules
{ c(c(x1)) -> d(d(d(x1)))
, c(x1) -> g(x1)
, c^#(c(x1)) -> c_1(d^#(d(d(x1))))
, c^#(x1) -> c_2(g^#(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b^#(b(x1)) -> c_0(c^#(d(x1)))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [1]
b^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [0]
g^#(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [1]
c_4(x1) = [1] x1 + [7]
f^#(x1) = [1] x1 + [0]
c_5(x1) = [1] x1 + [1]
c_6(x1) = [1] x1 + [1]
c_7(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{f^#(x1) -> c_5(g^#(x1))}
and weakly orienting the rules
{ b^#(b(x1)) -> c_0(c^#(d(x1)))
, c(c(x1)) -> d(d(d(x1)))
, c(x1) -> g(x1)
, c^#(c(x1)) -> c_1(d^#(d(d(x1))))
, c^#(x1) -> c_2(g^#(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{f^#(x1) -> c_5(g^#(x1))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [1]
b^#(x1) = [1] x1 + [8]
c_0(x1) = [1] x1 + [1]
c^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [0]
g^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [1]
f^#(x1) = [1] x1 + [8]
c_5(x1) = [1] x1 + [0]
c_6(x1) = [1] x1 + [1]
c_7(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{f(x1) -> a(g(x1))}
and weakly orienting the rules
{ f^#(x1) -> c_5(g^#(x1))
, b^#(b(x1)) -> c_0(c^#(d(x1)))
, c(c(x1)) -> d(d(d(x1)))
, c(x1) -> g(x1)
, c^#(c(x1)) -> c_1(d^#(d(d(x1))))
, c^#(x1) -> c_2(g^#(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{f(x1) -> a(g(x1))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [8]
a(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [0]
g^#(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [1]
c_4(x1) = [1] x1 + [1]
f^#(x1) = [1] x1 + [8]
c_5(x1) = [1] x1 + [1]
c_6(x1) = [1] x1 + [1]
c_7(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ g(x1) -> d(a(b(x1)))
, g(g(x1)) -> b(c(x1))}
and weakly orienting the rules
{ f(x1) -> a(g(x1))
, f^#(x1) -> c_5(g^#(x1))
, b^#(b(x1)) -> c_0(c^#(d(x1)))
, c(c(x1)) -> d(d(d(x1)))
, c(x1) -> g(x1)
, c^#(c(x1)) -> c_1(d^#(d(d(x1))))
, c^#(x1) -> c_2(g^#(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ g(x1) -> d(a(b(x1)))
, g(g(x1)) -> b(c(x1))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [1]
f(x1) = [1] x1 + [4]
a(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [8]
c_0(x1) = [1] x1 + [8]
c^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [0]
g^#(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [13]
c_4(x1) = [1] x1 + [1]
f^#(x1) = [1] x1 + [8]
c_5(x1) = [1] x1 + [1]
c_6(x1) = [1] x1 + [8]
c_7(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{d^#(d(d(x1))) -> c_4(g^#(c(x1)))}
and weakly orienting the rules
{ g(x1) -> d(a(b(x1)))
, g(g(x1)) -> b(c(x1))
, f(x1) -> a(g(x1))
, f^#(x1) -> c_5(g^#(x1))
, b^#(b(x1)) -> c_0(c^#(d(x1)))
, c(c(x1)) -> d(d(d(x1)))
, c(x1) -> g(x1)
, c^#(c(x1)) -> c_1(d^#(d(d(x1))))
, c^#(x1) -> c_2(g^#(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d^#(d(d(x1))) -> c_4(g^#(c(x1)))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [1]
f(x1) = [1] x1 + [4]
a(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [8]
c_0(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [8]
c_1(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [2]
c_2(x1) = [1] x1 + [0]
g^#(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [5]
c_4(x1) = [1] x1 + [0]
f^#(x1) = [1] x1 + [8]
c_5(x1) = [1] x1 + [1]
c_6(x1) = [1] x1 + [1]
c_7(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{d^#(d(x1)) -> c_3(c^#(f(x1)))}
and weakly orienting the rules
{ d^#(d(d(x1))) -> c_4(g^#(c(x1)))
, g(x1) -> d(a(b(x1)))
, g(g(x1)) -> b(c(x1))
, f(x1) -> a(g(x1))
, f^#(x1) -> c_5(g^#(x1))
, b^#(b(x1)) -> c_0(c^#(d(x1)))
, c(c(x1)) -> d(d(d(x1)))
, c(x1) -> g(x1)
, c^#(c(x1)) -> c_1(d^#(d(d(x1))))
, c^#(x1) -> c_2(g^#(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d^#(d(x1)) -> c_3(c^#(f(x1)))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [12]
d(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [8]
f(x1) = [1] x1 + [8]
a(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [12]
c_2(x1) = [1] x1 + [0]
g^#(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [0]
f^#(x1) = [1] x1 + [8]
c_5(x1) = [1] x1 + [2]
c_6(x1) = [1] x1 + [4]
c_7(x1) = [1] x1 + [9]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{b(b(x1)) -> c(d(x1))}
and weakly orienting the rules
{ d^#(d(x1)) -> c_3(c^#(f(x1)))
, d^#(d(d(x1))) -> c_4(g^#(c(x1)))
, g(x1) -> d(a(b(x1)))
, g(g(x1)) -> b(c(x1))
, f(x1) -> a(g(x1))
, f^#(x1) -> c_5(g^#(x1))
, b^#(b(x1)) -> c_0(c^#(d(x1)))
, c(c(x1)) -> d(d(d(x1)))
, c(x1) -> g(x1)
, c^#(c(x1)) -> c_1(d^#(d(d(x1))))
, c^#(x1) -> c_2(g^#(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{b(b(x1)) -> c(d(x1))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [7]
c(x1) = [1] x1 + [12]
d(x1) = [1] x1 + [1]
g(x1) = [1] x1 + [11]
f(x1) = [1] x1 + [11]
a(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [1]
c^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [11]
c_2(x1) = [1] x1 + [0]
g^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [0]
f^#(x1) = [1] x1 + [8]
c_5(x1) = [1] x1 + [1]
c_6(x1) = [1] x1 + [1]
c_7(x1) = [1] x1 + [9]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{g^#(g(x1)) -> c_7(b^#(c(x1)))}
and weakly orienting the rules
{ b(b(x1)) -> c(d(x1))
, d^#(d(x1)) -> c_3(c^#(f(x1)))
, d^#(d(d(x1))) -> c_4(g^#(c(x1)))
, g(x1) -> d(a(b(x1)))
, g(g(x1)) -> b(c(x1))
, f(x1) -> a(g(x1))
, f^#(x1) -> c_5(g^#(x1))
, b^#(b(x1)) -> c_0(c^#(d(x1)))
, c(c(x1)) -> d(d(d(x1)))
, c(x1) -> g(x1)
, c^#(c(x1)) -> c_1(d^#(d(d(x1))))
, c^#(x1) -> c_2(g^#(x1))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{g^#(g(x1)) -> c_7(b^#(c(x1)))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [2]
d(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [2]
f(x1) = [1] x1 + [2]
a(x1) = [1] x1 + [0]
b^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [3]
c_2(x1) = [1] x1 + [0]
g^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [0]
f^#(x1) = [1] x1 + [8]
c_5(x1) = [1] x1 + [2]
c_6(x1) = [1] x1 + [8]
c_7(x1) = [1] x1 + [0]
Finally we apply the subprocessor
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ d(d(x1)) -> c(f(x1))
, d(d(d(x1))) -> g(c(x1))
, g^#(x1) -> c_6(d^#(a(b(x1))))}
Weak Rules:
{ g^#(g(x1)) -> c_7(b^#(c(x1)))
, b(b(x1)) -> c(d(x1))
, d^#(d(x1)) -> c_3(c^#(f(x1)))
, d^#(d(d(x1))) -> c_4(g^#(c(x1)))
, g(x1) -> d(a(b(x1)))
, g(g(x1)) -> b(c(x1))
, f(x1) -> a(g(x1))
, f^#(x1) -> c_5(g^#(x1))
, b^#(b(x1)) -> c_0(c^#(d(x1)))
, c(c(x1)) -> d(d(d(x1)))
, c(x1) -> g(x1)
, c^#(c(x1)) -> c_1(d^#(d(d(x1))))
, c^#(x1) -> c_2(g^#(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)) -> c(f(x1))
, d(d(d(x1))) -> g(c(x1))
, g^#(x1) -> c_6(d^#(a(b(x1))))}
Weak Rules:
{ g^#(g(x1)) -> c_7(b^#(c(x1)))
, b(b(x1)) -> c(d(x1))
, d^#(d(x1)) -> c_3(c^#(f(x1)))
, d^#(d(d(x1))) -> c_4(g^#(c(x1)))
, g(x1) -> d(a(b(x1)))
, g(g(x1)) -> b(c(x1))
, f(x1) -> a(g(x1))
, f^#(x1) -> c_5(g^#(x1))
, b^#(b(x1)) -> c_0(c^#(d(x1)))
, c(c(x1)) -> d(d(d(x1)))
, c(x1) -> g(x1)
, c^#(c(x1)) -> c_1(d^#(d(d(x1))))
, c^#(x1) -> c_2(g^#(x1))}
Details:
The problem is Match-bounded by 2.
The enriched problem is compatible with the following automaton:
{ b_1(6) -> 19
, b_2(6) -> 23
, a_0(6) -> 6
, a_1(19) -> 18
, a_2(23) -> 22
, b^#_0(6) -> 7
, c^#_0(6) -> 9
, d^#_0(6) -> 11
, d^#_1(18) -> 17
, d^#_2(22) -> 21
, c_2_0(13) -> 9
, c_2_1(20) -> 9
, g^#_0(6) -> 13
, g^#_1(6) -> 20
, f^#_0(6) -> 16
, c_5_0(13) -> 16
, c_5_1(20) -> 16
, c_6_1(17) -> 13
, c_6_2(21) -> 20}
3) { f^#(x1) -> c_5(g^#(x1))
, g^#(x1) -> c_6(d^#(a(b(x1))))}
The usable rules for this path are the following:
{ b(b(x1)) -> c(d(x1))
, c(c(x1)) -> d(d(d(x1)))
, c(x1) -> g(x1)
, d(d(x1)) -> c(f(x1))
, d(d(d(x1))) -> g(c(x1))
, f(x1) -> a(g(x1))
, g(x1) -> d(a(b(x1)))
, g(g(x1)) -> b(c(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(x1)) -> c(d(x1))
, c(c(x1)) -> d(d(d(x1)))
, c(x1) -> g(x1)
, d(d(x1)) -> c(f(x1))
, d(d(d(x1))) -> g(c(x1))
, f(x1) -> a(g(x1))
, g(x1) -> d(a(b(x1)))
, g(g(x1)) -> b(c(x1))
, f^#(x1) -> c_5(g^#(x1))
, g^#(x1) -> c_6(d^#(a(b(x1))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ c(c(x1)) -> d(d(d(x1)))
, c(x1) -> g(x1)}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ c(c(x1)) -> d(d(d(x1)))
, c(x1) -> g(x1)}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [1]
b^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [0]
c_2(x1) = [0] x1 + [0]
g^#(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
f^#(x1) = [1] x1 + [0]
c_5(x1) = [1] x1 + [1]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{f^#(x1) -> c_5(g^#(x1))}
and weakly orienting the rules
{ c(c(x1)) -> d(d(d(x1)))
, c(x1) -> g(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{f^#(x1) -> c_5(g^#(x1))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [1]
b^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [0]
c_2(x1) = [0] x1 + [0]
g^#(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
f^#(x1) = [1] x1 + [8]
c_5(x1) = [1] x1 + [1]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{g^#(x1) -> c_6(d^#(a(b(x1))))}
and weakly orienting the rules
{ f^#(x1) -> c_5(g^#(x1))
, c(c(x1)) -> d(d(d(x1)))
, c(x1) -> g(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{g^#(x1) -> c_6(d^#(a(b(x1))))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [9]
b^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [0]
c_2(x1) = [0] x1 + [0]
g^#(x1) = [1] x1 + [12]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
f^#(x1) = [1] x1 + [14]
c_5(x1) = [1] x1 + [1]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ g(x1) -> d(a(b(x1)))
, g(g(x1)) -> b(c(x1))}
and weakly orienting the rules
{ g^#(x1) -> c_6(d^#(a(b(x1))))
, f^#(x1) -> c_5(g^#(x1))
, c(c(x1)) -> d(d(d(x1)))
, c(x1) -> g(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ g(x1) -> d(a(b(x1)))
, g(g(x1)) -> b(c(x1))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [1]
f(x1) = [1] x1 + [0]
a(x1) = [1] x1 + [0]
b^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [0]
c_2(x1) = [0] x1 + [0]
g^#(x1) = [1] x1 + [8]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
f^#(x1) = [1] x1 + [12]
c_5(x1) = [1] x1 + [1]
c_6(x1) = [1] x1 + [1]
c_7(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{f(x1) -> a(g(x1))}
and weakly orienting the rules
{ g(x1) -> d(a(b(x1)))
, g(g(x1)) -> b(c(x1))
, g^#(x1) -> c_6(d^#(a(b(x1))))
, f^#(x1) -> c_5(g^#(x1))
, c(c(x1)) -> d(d(d(x1)))
, c(x1) -> g(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{f(x1) -> a(g(x1))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [1]
f(x1) = [1] x1 + [8]
a(x1) = [1] x1 + [1]
b^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [3]
c_2(x1) = [0] x1 + [0]
g^#(x1) = [1] x1 + [9]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
f^#(x1) = [1] x1 + [14]
c_5(x1) = [1] x1 + [3]
c_6(x1) = [1] x1 + [1]
c_7(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{d(d(d(x1))) -> g(c(x1))}
and weakly orienting the rules
{ f(x1) -> a(g(x1))
, g(x1) -> d(a(b(x1)))
, g(g(x1)) -> b(c(x1))
, g^#(x1) -> c_6(d^#(a(b(x1))))
, f^#(x1) -> c_5(g^#(x1))
, c(c(x1)) -> d(d(d(x1)))
, c(x1) -> g(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d(d(d(x1))) -> g(c(x1))}
Details:
Interpretation Functions:
b(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [13]
d(x1) = [1] x1 + [8]
g(x1) = [1] x1 + [8]
f(x1) = [1] x1 + [8]
a(x1) = [1] x1 + [0]
b^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [0]
c_2(x1) = [0] x1 + [0]
g^#(x1) = [1] x1 + [8]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
f^#(x1) = [1] x1 + [12]
c_5(x1) = [1] x1 + [1]
c_6(x1) = [1] x1 + [1]
c_7(x1) = [0] x1 + [0]
Finally we apply the subprocessor
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ b(b(x1)) -> c(d(x1))
, d(d(x1)) -> c(f(x1))}
Weak Rules:
{ d(d(d(x1))) -> g(c(x1))
, f(x1) -> a(g(x1))
, g(x1) -> d(a(b(x1)))
, g(g(x1)) -> b(c(x1))
, g^#(x1) -> c_6(d^#(a(b(x1))))
, f^#(x1) -> c_5(g^#(x1))
, c(c(x1)) -> d(d(d(x1)))
, c(x1) -> g(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)) -> c(d(x1))
, d(d(x1)) -> c(f(x1))}
Weak Rules:
{ d(d(d(x1))) -> g(c(x1))
, f(x1) -> a(g(x1))
, g(x1) -> d(a(b(x1)))
, g(g(x1)) -> b(c(x1))
, g^#(x1) -> c_6(d^#(a(b(x1))))
, f^#(x1) -> c_5(g^#(x1))
, c(c(x1)) -> d(d(d(x1)))
, c(x1) -> g(x1)}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ b_0(6) -> 19
, a_0(6) -> 6
, a_0(19) -> 18
, d^#_0(6) -> 11
, d^#_0(18) -> 17
, g^#_0(6) -> 13
, f^#_0(6) -> 16
, c_5_0(13) -> 16
, c_6_0(17) -> 13}
4) {f^#(x1) -> c_5(g^#(x1))}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
b(x1) = [0] x1 + [0]
c(x1) = [0] x1 + [0]
d(x1) = [0] x1 + [0]
g(x1) = [0] x1 + [0]
f(x1) = [0] x1 + [0]
a(x1) = [0] x1 + [0]
b^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
d^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(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: {f^#(x1) -> c_5(g^#(x1))}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{f^#(x1) -> c_5(g^#(x1))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{f^#(x1) -> c_5(g^#(x1))}
Details:
Interpretation Functions:
b(x1) = [0] x1 + [0]
c(x1) = [0] x1 + [0]
d(x1) = [0] x1 + [0]
g(x1) = [0] x1 + [0]
f(x1) = [0] x1 + [0]
a(x1) = [0] x1 + [0]
b^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
d^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
g^#(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
f^#(x1) = [1] x1 + [8]
c_5(x1) = [1] x1 + [1]
c_6(x1) = [0] x1 + [0]
c_7(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: {f^#(x1) -> c_5(g^#(x1))}
Details:
The given problem does not contain any strict rules