'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ g(0(), f(x, x)) -> x
, g(x, s(y)) -> g(f(x, y), 0())
, g(s(x), y) -> g(f(x, y), 0())
, g(f(x, y), 0()) -> f(g(x, 0()), g(y, 0()))}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ g^#(0(), f(x, x)) -> c_0()
, g^#(x, s(y)) -> c_1(g^#(f(x, y), 0()))
, g^#(s(x), y) -> c_2(g^#(f(x, y), 0()))
, g^#(f(x, y), 0()) -> c_3(g^#(x, 0()), g^#(y, 0()))}
The usable rules are:
{}
The estimated dependency graph contains the following edges:
{g^#(x, s(y)) -> c_1(g^#(f(x, y), 0()))}
==> {g^#(f(x, y), 0()) -> c_3(g^#(x, 0()), g^#(y, 0()))}
{g^#(s(x), y) -> c_2(g^#(f(x, y), 0()))}
==> {g^#(f(x, y), 0()) -> c_3(g^#(x, 0()), g^#(y, 0()))}
{g^#(f(x, y), 0()) -> c_3(g^#(x, 0()), g^#(y, 0()))}
==> {g^#(f(x, y), 0()) -> c_3(g^#(x, 0()), g^#(y, 0()))}
{g^#(f(x, y), 0()) -> c_3(g^#(x, 0()), g^#(y, 0()))}
==> {g^#(s(x), y) -> c_2(g^#(f(x, y), 0()))}
We consider the following path(s):
1) { g^#(x, s(y)) -> c_1(g^#(f(x, y), 0()))
, g^#(f(x, y), 0()) -> c_3(g^#(x, 0()), g^#(y, 0()))
, g^#(s(x), y) -> c_2(g^#(f(x, y), 0()))}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
g(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
f(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
g^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1, x2) = [0] x1 + [0] x2 + [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^#(f(x, y), 0()) -> c_3(g^#(x, 0()), g^#(y, 0()))
, g^#(s(x), y) -> c_2(g^#(f(x, y), 0()))}
Weak Rules: {g^#(x, s(y)) -> c_1(g^#(f(x, y), 0()))}
Details:
We apply the weight gap principle, strictly orienting the rules
{g^#(s(x), y) -> c_2(g^#(f(x, y), 0()))}
and weakly orienting the rules
{g^#(x, s(y)) -> c_1(g^#(f(x, y), 0()))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{g^#(s(x), y) -> c_2(g^#(f(x, y), 0()))}
Details:
Interpretation Functions:
g(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
f(x1, x2) = [1] x1 + [1] x2 + [0]
s(x1) = [1] x1 + [8]
g^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_0() = [0]
c_1(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [7]
c_3(x1, x2) = [1] x1 + [1] x2 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{g^#(f(x, y), 0()) -> c_3(g^#(x, 0()), g^#(y, 0()))}
and weakly orienting the rules
{ g^#(s(x), y) -> c_2(g^#(f(x, y), 0()))
, g^#(x, s(y)) -> c_1(g^#(f(x, y), 0()))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{g^#(f(x, y), 0()) -> c_3(g^#(x, 0()), g^#(y, 0()))}
Details:
Interpretation Functions:
g(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
f(x1, x2) = [1] x1 + [1] x2 + [8]
s(x1) = [1] x1 + [8]
g^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_0() = [0]
c_1(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [0]
c_3(x1, x2) = [1] x1 + [1] x2 + [7]
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^#(f(x, y), 0()) -> c_3(g^#(x, 0()), g^#(y, 0()))
, g^#(s(x), y) -> c_2(g^#(f(x, y), 0()))
, g^#(x, s(y)) -> c_1(g^#(f(x, y), 0()))}
Details:
The given problem does not contain any strict rules
2) {g^#(x, s(y)) -> c_1(g^#(f(x, y), 0()))}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
g(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
f(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
g^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1, x2) = [0] x1 + [0] x2 + [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^#(x, s(y)) -> c_1(g^#(f(x, y), 0()))}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{g^#(x, s(y)) -> c_1(g^#(f(x, y), 0()))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{g^#(x, s(y)) -> c_1(g^#(f(x, y), 0()))}
Details:
Interpretation Functions:
g(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
f(x1, x2) = [1] x1 + [1] x2 + [0]
s(x1) = [1] x1 + [1]
g^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_0() = [0]
c_1(x1) = [1] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1, x2) = [0] x1 + [0] x2 + [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^#(x, s(y)) -> c_1(g^#(f(x, y), 0()))}
Details:
The given problem does not contain any strict rules
3) {g^#(0(), f(x, x)) -> c_0()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
g(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
f(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
g^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1, x2) = [0] x1 + [0] x2 + [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(), f(x, x)) -> c_0()}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{g^#(0(), f(x, x)) -> c_0()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{g^#(0(), f(x, x)) -> c_0()}
Details:
Interpretation Functions:
g(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
f(x1, x2) = [1] x1 + [1] x2 + [0]
s(x1) = [0] x1 + [0]
g^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1, x2) = [0] x1 + [0] x2 + [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(), f(x, x)) -> c_0()}
Details:
The given problem does not contain any strict rules