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