'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ g(s(x)) -> f(x)
, f(0()) -> s(0())
, f(s(x)) -> s(s(g(x)))
, g(0()) -> 0()}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ g^#(s(x)) -> c_0(f^#(x))
, f^#(0()) -> c_1()
, f^#(s(x)) -> c_2(g^#(x))
, g^#(0()) -> c_3()}
The usable rules are:
{}
The estimated dependency graph contains the following edges:
{g^#(s(x)) -> c_0(f^#(x))}
==> {f^#(s(x)) -> c_2(g^#(x))}
{g^#(s(x)) -> c_0(f^#(x))}
==> {f^#(0()) -> c_1()}
{f^#(s(x)) -> c_2(g^#(x))}
==> {g^#(0()) -> c_3()}
{f^#(s(x)) -> c_2(g^#(x))}
==> {g^#(s(x)) -> c_0(f^#(x))}
We consider the following path(s):
1) { g^#(s(x)) -> c_0(f^#(x))
, f^#(s(x)) -> c_2(g^#(x))}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
g(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
f(x1) = [0] x1 + [0]
0() = [0]
g^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
c_3() = [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:
{ g^#(s(x)) -> c_0(f^#(x))
, f^#(s(x)) -> c_2(g^#(x))}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{f^#(s(x)) -> c_2(g^#(x))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{f^#(s(x)) -> c_2(g^#(x))}
Details:
Interpretation Functions:
g(x1) = [0] x1 + [0]
s(x1) = [1] x1 + [0]
f(x1) = [0] x1 + [0]
0() = [0]
g^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [1]
f^#(x1) = [1] x1 + [8]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
c_3() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{g^#(s(x)) -> c_0(f^#(x))}
and weakly orienting the rules
{f^#(s(x)) -> c_2(g^#(x))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{g^#(s(x)) -> c_0(f^#(x))}
Details:
Interpretation Functions:
g(x1) = [0] x1 + [0]
s(x1) = [1] x1 + [8]
f(x1) = [0] x1 + [0]
0() = [0]
g^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [1]
f^#(x1) = [1] x1 + [0]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
c_3() = [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:
{ g^#(s(x)) -> c_0(f^#(x))
, f^#(s(x)) -> c_2(g^#(x))}
Details:
The given problem does not contain any strict rules
2) { g^#(s(x)) -> c_0(f^#(x))
, f^#(s(x)) -> c_2(g^#(x))
, f^#(0()) -> c_1()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
g(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
f(x1) = [0] x1 + [0]
0() = [0]
g^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
c_3() = [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^#(0()) -> c_1()}
Weak Rules:
{ g^#(s(x)) -> c_0(f^#(x))
, f^#(s(x)) -> c_2(g^#(x))}
Details:
We apply the weight gap principle, strictly orienting the rules
{f^#(0()) -> c_1()}
and weakly orienting the rules
{ g^#(s(x)) -> c_0(f^#(x))
, f^#(s(x)) -> c_2(g^#(x))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{f^#(0()) -> c_1()}
Details:
Interpretation Functions:
g(x1) = [0] x1 + [0]
s(x1) = [1] x1 + [0]
f(x1) = [0] x1 + [0]
0() = [0]
g^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [0]
f^#(x1) = [1] x1 + [1]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
c_3() = [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^#(0()) -> c_1()
, g^#(s(x)) -> c_0(f^#(x))
, f^#(s(x)) -> c_2(g^#(x))}
Details:
The given problem does not contain any strict rules
3) { g^#(s(x)) -> c_0(f^#(x))
, f^#(s(x)) -> c_2(g^#(x))
, g^#(0()) -> c_3()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
g(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
f(x1) = [0] x1 + [0]
0() = [0]
g^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
c_3() = [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: {g^#(0()) -> c_3()}
Weak Rules:
{ g^#(s(x)) -> c_0(f^#(x))
, f^#(s(x)) -> c_2(g^#(x))}
Details:
We apply the weight gap principle, strictly orienting the rules
{g^#(0()) -> c_3()}
and weakly orienting the rules
{ g^#(s(x)) -> c_0(f^#(x))
, f^#(s(x)) -> c_2(g^#(x))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{g^#(0()) -> c_3()}
Details:
Interpretation Functions:
g(x1) = [0] x1 + [0]
s(x1) = [1] x1 + [0]
f(x1) = [0] x1 + [0]
0() = [0]
g^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [0]
f^#(x1) = [1] x1 + [1]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
c_3() = [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:
{ g^#(0()) -> c_3()
, g^#(s(x)) -> c_0(f^#(x))
, f^#(s(x)) -> c_2(g^#(x))}
Details:
The given problem does not contain any strict rules