'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ f(X) -> if(X, c(), n__f(true()))
, if(true(), X, Y) -> X
, if(false(), X, Y) -> activate(Y)
, f(X) -> n__f(X)
, activate(n__f(X)) -> f(X)
, activate(X) -> X}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ f^#(X) -> c_0(if^#(X, c(), n__f(true())))
, if^#(true(), X, Y) -> c_1()
, if^#(false(), X, Y) -> c_2(activate^#(Y))
, f^#(X) -> c_3()
, activate^#(n__f(X)) -> c_4(f^#(X))
, activate^#(X) -> c_5()}
The usable rules are:
{}
The estimated dependency graph contains the following edges:
{f^#(X) -> c_0(if^#(X, c(), n__f(true())))}
==> {if^#(false(), X, Y) -> c_2(activate^#(Y))}
{f^#(X) -> c_0(if^#(X, c(), n__f(true())))}
==> {if^#(true(), X, Y) -> c_1()}
{if^#(false(), X, Y) -> c_2(activate^#(Y))}
==> {activate^#(n__f(X)) -> c_4(f^#(X))}
{if^#(false(), X, Y) -> c_2(activate^#(Y))}
==> {activate^#(X) -> c_5()}
{activate^#(n__f(X)) -> c_4(f^#(X))}
==> {f^#(X) -> c_3()}
{activate^#(n__f(X)) -> c_4(f^#(X))}
==> {f^#(X) -> c_0(if^#(X, c(), n__f(true())))}
We consider the following path(s):
1) { f^#(X) -> c_0(if^#(X, c(), n__f(true())))
, activate^#(n__f(X)) -> c_4(f^#(X))
, if^#(false(), X, Y) -> c_2(activate^#(Y))}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
f(x1) = [0] x1 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c() = [0]
n__f(x1) = [0] x1 + [0]
true() = [0]
false() = [0]
activate(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
activate^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
c_5() = [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^#(X) -> c_0(if^#(X, c(), n__f(true())))
, activate^#(n__f(X)) -> c_4(f^#(X))
, if^#(false(), X, Y) -> c_2(activate^#(Y))}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{if^#(false(), X, Y) -> c_2(activate^#(Y))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{if^#(false(), X, Y) -> c_2(activate^#(Y))}
Details:
Interpretation Functions:
f(x1) = [0] x1 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c() = [0]
n__f(x1) = [1] x1 + [0]
true() = [0]
false() = [8]
activate(x1) = [0] x1 + [0]
f^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [1]
if^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
activate^#(x1) = [1] x1 + [1]
c_3() = [0]
c_4(x1) = [1] x1 + [1]
c_5() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{f^#(X) -> c_0(if^#(X, c(), n__f(true())))}
and weakly orienting the rules
{if^#(false(), X, Y) -> c_2(activate^#(Y))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{f^#(X) -> c_0(if^#(X, c(), n__f(true())))}
Details:
Interpretation Functions:
f(x1) = [0] x1 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c() = [0]
n__f(x1) = [1] x1 + [0]
true() = [0]
false() = [8]
activate(x1) = [0] x1 + [0]
f^#(x1) = [1] x1 + [8]
c_0(x1) = [1] x1 + [1]
if^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
activate^#(x1) = [1] x1 + [1]
c_3() = [0]
c_4(x1) = [1] x1 + [1]
c_5() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{activate^#(n__f(X)) -> c_4(f^#(X))}
and weakly orienting the rules
{ f^#(X) -> c_0(if^#(X, c(), n__f(true())))
, if^#(false(), X, Y) -> c_2(activate^#(Y))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{activate^#(n__f(X)) -> c_4(f^#(X))}
Details:
Interpretation Functions:
f(x1) = [0] x1 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c() = [0]
n__f(x1) = [1] x1 + [0]
true() = [0]
false() = [8]
activate(x1) = [0] x1 + [0]
f^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
if^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
c_1() = [0]
c_2(x1) = [1] x1 + [7]
activate^#(x1) = [1] x1 + [1]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
c_5() = [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^#(n__f(X)) -> c_4(f^#(X))
, f^#(X) -> c_0(if^#(X, c(), n__f(true())))
, if^#(false(), X, Y) -> c_2(activate^#(Y))}
Details:
The given problem does not contain any strict rules
2) { f^#(X) -> c_0(if^#(X, c(), n__f(true())))
, activate^#(n__f(X)) -> c_4(f^#(X))
, if^#(false(), X, Y) -> c_2(activate^#(Y))
, if^#(true(), X, Y) -> c_1()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
f(x1) = [0] x1 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c() = [0]
n__f(x1) = [0] x1 + [0]
true() = [0]
false() = [0]
activate(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
activate^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
c_5() = [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: {if^#(true(), X, Y) -> c_1()}
Weak Rules:
{ f^#(X) -> c_0(if^#(X, c(), n__f(true())))
, activate^#(n__f(X)) -> c_4(f^#(X))
, if^#(false(), X, Y) -> c_2(activate^#(Y))}
Details:
We apply the weight gap principle, strictly orienting the rules
{if^#(true(), X, Y) -> c_1()}
and weakly orienting the rules
{ f^#(X) -> c_0(if^#(X, c(), n__f(true())))
, activate^#(n__f(X)) -> c_4(f^#(X))
, if^#(false(), X, Y) -> c_2(activate^#(Y))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{if^#(true(), X, Y) -> c_1()}
Details:
Interpretation Functions:
f(x1) = [0] x1 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c() = [0]
n__f(x1) = [1] x1 + [0]
true() = [0]
false() = [8]
activate(x1) = [0] x1 + [0]
f^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [0]
if^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [1]
c_1() = [0]
c_2(x1) = [1] x1 + [7]
activate^#(x1) = [1] x1 + [1]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
c_5() = [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:
{ if^#(true(), X, Y) -> c_1()
, f^#(X) -> c_0(if^#(X, c(), n__f(true())))
, activate^#(n__f(X)) -> c_4(f^#(X))
, if^#(false(), X, Y) -> c_2(activate^#(Y))}
Details:
The given problem does not contain any strict rules
3) { f^#(X) -> c_0(if^#(X, c(), n__f(true())))
, activate^#(n__f(X)) -> c_4(f^#(X))
, if^#(false(), X, Y) -> c_2(activate^#(Y))
, f^#(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:
f(x1) = [0] x1 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c() = [0]
n__f(x1) = [0] x1 + [0]
true() = [0]
false() = [0]
activate(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
activate^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
c_5() = [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^#(X) -> c_3()}
Weak Rules:
{ f^#(X) -> c_0(if^#(X, c(), n__f(true())))
, activate^#(n__f(X)) -> c_4(f^#(X))
, if^#(false(), X, Y) -> c_2(activate^#(Y))}
Details:
We apply the weight gap principle, strictly orienting the rules
{f^#(X) -> c_3()}
and weakly orienting the rules
{ f^#(X) -> c_0(if^#(X, c(), n__f(true())))
, activate^#(n__f(X)) -> c_4(f^#(X))
, if^#(false(), X, Y) -> c_2(activate^#(Y))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{f^#(X) -> c_3()}
Details:
Interpretation Functions:
f(x1) = [0] x1 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c() = [0]
n__f(x1) = [1] x1 + [0]
true() = [0]
false() = [8]
activate(x1) = [0] x1 + [0]
f^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [1]
if^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
c_1() = [0]
c_2(x1) = [1] x1 + [4]
activate^#(x1) = [1] x1 + [1]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
c_5() = [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^#(X) -> c_3()
, f^#(X) -> c_0(if^#(X, c(), n__f(true())))
, activate^#(n__f(X)) -> c_4(f^#(X))
, if^#(false(), X, Y) -> c_2(activate^#(Y))}
Details:
The given problem does not contain any strict rules
4) { f^#(X) -> c_0(if^#(X, c(), n__f(true())))
, activate^#(n__f(X)) -> c_4(f^#(X))
, if^#(false(), X, Y) -> c_2(activate^#(Y))
, activate^#(X) -> c_5()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
f(x1) = [0] x1 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c() = [0]
n__f(x1) = [0] x1 + [0]
true() = [0]
false() = [0]
activate(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
activate^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
c_5() = [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_5()}
Weak Rules:
{ f^#(X) -> c_0(if^#(X, c(), n__f(true())))
, activate^#(n__f(X)) -> c_4(f^#(X))
, if^#(false(), X, Y) -> c_2(activate^#(Y))}
Details:
We apply the weight gap principle, strictly orienting the rules
{activate^#(X) -> c_5()}
and weakly orienting the rules
{ f^#(X) -> c_0(if^#(X, c(), n__f(true())))
, activate^#(n__f(X)) -> c_4(f^#(X))
, if^#(false(), X, Y) -> c_2(activate^#(Y))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{activate^#(X) -> c_5()}
Details:
Interpretation Functions:
f(x1) = [0] x1 + [0]
if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c() = [0]
n__f(x1) = [1] x1 + [0]
true() = [0]
false() = [8]
activate(x1) = [0] x1 + [0]
f^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [1]
if^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
activate^#(x1) = [1] x1 + [1]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
c_5() = [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_5()
, f^#(X) -> c_0(if^#(X, c(), n__f(true())))
, activate^#(n__f(X)) -> c_4(f^#(X))
, if^#(false(), X, Y) -> c_2(activate^#(Y))}
Details:
The given problem does not contain any strict rules