'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ f(f(a())) -> c(n__f(n__g(n__f(n__a()))))
, f(X) -> n__f(X)
, g(X) -> n__g(X)
, a() -> n__a()
, activate(n__f(X)) -> f(activate(X))
, activate(n__g(X)) -> g(activate(X))
, activate(n__a()) -> a()
, activate(X) -> X}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ f^#(f(a())) -> c_0()
, f^#(X) -> c_1()
, g^#(X) -> c_2()
, a^#() -> c_3()
, activate^#(n__f(X)) -> c_4(f^#(activate(X)))
, activate^#(n__g(X)) -> c_5(g^#(activate(X)))
, activate^#(n__a()) -> c_6(a^#())
, activate^#(X) -> c_7()}
The usable rules are:
{ activate(n__f(X)) -> f(activate(X))
, activate(n__g(X)) -> g(activate(X))
, activate(n__a()) -> a()
, activate(X) -> X
, f(f(a())) -> c(n__f(n__g(n__f(n__a()))))
, f(X) -> n__f(X)
, g(X) -> n__g(X)
, a() -> n__a()}
The estimated dependency graph contains the following edges:
{activate^#(n__f(X)) -> c_4(f^#(activate(X)))}
==> {f^#(X) -> c_1()}
{activate^#(n__f(X)) -> c_4(f^#(activate(X)))}
==> {f^#(f(a())) -> c_0()}
{activate^#(n__g(X)) -> c_5(g^#(activate(X)))}
==> {g^#(X) -> c_2()}
{activate^#(n__a()) -> c_6(a^#())}
==> {a^#() -> c_3()}
We consider the following path(s):
1) { activate^#(n__f(X)) -> c_4(f^#(activate(X)))
, f^#(f(a())) -> c_0()}
The usable rules for this path are the following:
{ activate(n__f(X)) -> f(activate(X))
, activate(n__g(X)) -> g(activate(X))
, activate(n__a()) -> a()
, activate(X) -> X
, f(f(a())) -> c(n__f(n__g(n__f(n__a()))))
, f(X) -> n__f(X)
, g(X) -> n__g(X)
, a() -> n__a()}
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:
{ activate(n__f(X)) -> f(activate(X))
, activate(n__g(X)) -> g(activate(X))
, activate(n__a()) -> a()
, activate(X) -> X
, f(f(a())) -> c(n__f(n__g(n__f(n__a()))))
, f(X) -> n__f(X)
, g(X) -> n__g(X)
, a() -> n__a()
, activate^#(n__f(X)) -> c_4(f^#(activate(X)))
, f^#(f(a())) -> c_0()}
Details:
We apply the weight gap principle, strictly orienting the rules
{ activate(n__a()) -> a()
, activate(X) -> X}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ activate(n__a()) -> a()
, activate(X) -> X}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
a() = [0]
c(x1) = [1] x1 + [5]
n__f(x1) = [1] x1 + [0]
n__g(x1) = [1] x1 + [0]
n__a() = [0]
g(x1) = [1] x1 + [0]
activate(x1) = [1] x1 + [1]
f^#(x1) = [1] x1 + [0]
c_0() = [0]
c_1() = [0]
g^#(x1) = [0] x1 + [0]
c_2() = [0]
a^#() = [0]
c_3() = [0]
activate^#(x1) = [1] x1 + [1]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{f^#(f(a())) -> c_0()}
and weakly orienting the rules
{ activate(n__a()) -> a()
, activate(X) -> X}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{f^#(f(a())) -> c_0()}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
a() = [0]
c(x1) = [1] x1 + [1]
n__f(x1) = [1] x1 + [0]
n__g(x1) = [1] x1 + [0]
n__a() = [0]
g(x1) = [1] x1 + [0]
activate(x1) = [1] x1 + [1]
f^#(x1) = [1] x1 + [8]
c_0() = [0]
c_1() = [0]
g^#(x1) = [0] x1 + [0]
c_2() = [0]
a^#() = [0]
c_3() = [0]
activate^#(x1) = [1] x1 + [1]
c_4(x1) = [1] x1 + [3]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{activate^#(n__f(X)) -> c_4(f^#(activate(X)))}
and weakly orienting the rules
{ f^#(f(a())) -> c_0()
, activate(n__a()) -> a()
, activate(X) -> X}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{activate^#(n__f(X)) -> c_4(f^#(activate(X)))}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
a() = [0]
c(x1) = [1] x1 + [1]
n__f(x1) = [1] x1 + [0]
n__g(x1) = [1] x1 + [0]
n__a() = [0]
g(x1) = [1] x1 + [0]
activate(x1) = [1] x1 + [1]
f^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1() = [0]
g^#(x1) = [0] x1 + [0]
c_2() = [0]
a^#() = [0]
c_3() = [0]
activate^#(x1) = [1] x1 + [9]
c_4(x1) = [1] x1 + [1]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ f(f(a())) -> c(n__f(n__g(n__f(n__a()))))
, a() -> n__a()}
and weakly orienting the rules
{ activate^#(n__f(X)) -> c_4(f^#(activate(X)))
, f^#(f(a())) -> c_0()
, activate(n__a()) -> a()
, activate(X) -> X}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ f(f(a())) -> c(n__f(n__g(n__f(n__a()))))
, a() -> n__a()}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
a() = [8]
c(x1) = [1] x1 + [0]
n__f(x1) = [1] x1 + [0]
n__g(x1) = [1] x1 + [0]
n__a() = [7]
g(x1) = [1] x1 + [0]
activate(x1) = [1] x1 + [1]
f^#(x1) = [1] x1 + [2]
c_0() = [0]
c_1() = [0]
g^#(x1) = [0] x1 + [0]
c_2() = [0]
a^#() = [0]
c_3() = [0]
activate^#(x1) = [1] x1 + [9]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{g(X) -> n__g(X)}
and weakly orienting the rules
{ f(f(a())) -> c(n__f(n__g(n__f(n__a()))))
, a() -> n__a()
, activate^#(n__f(X)) -> c_4(f^#(activate(X)))
, f^#(f(a())) -> c_0()
, activate(n__a()) -> a()
, activate(X) -> X}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{g(X) -> n__g(X)}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
a() = [0]
c(x1) = [1] x1 + [0]
n__f(x1) = [1] x1 + [0]
n__g(x1) = [1] x1 + [0]
n__a() = [0]
g(x1) = [1] x1 + [8]
activate(x1) = [1] x1 + [1]
f^#(x1) = [1] x1 + [0]
c_0() = [0]
c_1() = [0]
g^#(x1) = [0] x1 + [0]
c_2() = [0]
a^#() = [0]
c_3() = [0]
activate^#(x1) = [1] x1 + [1]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{f(X) -> n__f(X)}
and weakly orienting the rules
{ g(X) -> n__g(X)
, f(f(a())) -> c(n__f(n__g(n__f(n__a()))))
, a() -> n__a()
, activate^#(n__f(X)) -> c_4(f^#(activate(X)))
, f^#(f(a())) -> c_0()
, activate(n__a()) -> a()
, activate(X) -> X}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{f(X) -> n__f(X)}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [8]
a() = [4]
c(x1) = [1] x1 + [0]
n__f(x1) = [1] x1 + [0]
n__g(x1) = [1] x1 + [0]
n__a() = [3]
g(x1) = [1] x1 + [0]
activate(x1) = [1] x1 + [1]
f^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1() = [0]
g^#(x1) = [0] x1 + [0]
c_2() = [0]
a^#() = [0]
c_3() = [0]
activate^#(x1) = [1] x1 + [9]
c_4(x1) = [1] x1 + [7]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7() = [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:
{ activate(n__f(X)) -> f(activate(X))
, activate(n__g(X)) -> g(activate(X))}
Weak Rules:
{ f(X) -> n__f(X)
, g(X) -> n__g(X)
, f(f(a())) -> c(n__f(n__g(n__f(n__a()))))
, a() -> n__a()
, activate^#(n__f(X)) -> c_4(f^#(activate(X)))
, f^#(f(a())) -> c_0()
, activate(n__a()) -> a()
, activate(X) -> X}
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:
{ activate(n__f(X)) -> f(activate(X))
, activate(n__g(X)) -> g(activate(X))}
Weak Rules:
{ f(X) -> n__f(X)
, g(X) -> n__g(X)
, f(f(a())) -> c(n__f(n__g(n__f(n__a()))))
, a() -> n__a()
, activate^#(n__f(X)) -> c_4(f^#(activate(X)))
, f^#(f(a())) -> c_0()
, activate(n__a()) -> a()
, activate(X) -> X}
Details:
The problem is Match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ f_1(5) -> 4
, f_1(5) -> 5
, a_0() -> 4
, a_1() -> 5
, c_0(2) -> 2
, c_0(2) -> 4
, c_0(2) -> 5
, c_1(7) -> 4
, c_1(7) -> 5
, n__f_0(2) -> 2
, n__f_0(2) -> 4
, n__f_0(2) -> 5
, n__f_1(5) -> 4
, n__f_1(5) -> 5
, n__f_1(8) -> 7
, n__f_1(10) -> 9
, n__g_0(2) -> 2
, n__g_0(2) -> 4
, n__g_0(2) -> 5
, n__g_1(5) -> 4
, n__g_1(5) -> 5
, n__g_1(9) -> 8
, n__a_0() -> 2
, n__a_0() -> 4
, n__a_0() -> 5
, n__a_1() -> 5
, n__a_1() -> 10
, g_1(5) -> 4
, g_1(5) -> 5
, activate_0(2) -> 4
, activate_1(2) -> 5
, f^#_0(2) -> 1
, f^#_0(4) -> 3
, f^#_1(5) -> 6
, c_0_1() -> 3
, c_0_1() -> 6
, activate^#_0(2) -> 1
, c_4_0(3) -> 1
, c_4_1(6) -> 1}
2) { activate^#(n__f(X)) -> c_4(f^#(activate(X)))
, f^#(X) -> c_1()}
The usable rules for this path are the following:
{ activate(n__f(X)) -> f(activate(X))
, activate(n__g(X)) -> g(activate(X))
, activate(n__a()) -> a()
, activate(X) -> X
, f(f(a())) -> c(n__f(n__g(n__f(n__a()))))
, f(X) -> n__f(X)
, g(X) -> n__g(X)
, a() -> n__a()}
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:
{ activate(n__f(X)) -> f(activate(X))
, activate(n__g(X)) -> g(activate(X))
, activate(n__a()) -> a()
, activate(X) -> X
, f(f(a())) -> c(n__f(n__g(n__f(n__a()))))
, f(X) -> n__f(X)
, g(X) -> n__g(X)
, a() -> n__a()
, activate^#(n__f(X)) -> c_4(f^#(activate(X)))
, f^#(X) -> c_1()}
Details:
We apply the weight gap principle, strictly orienting the rules
{ activate(n__a()) -> a()
, activate(X) -> X}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ activate(n__a()) -> a()
, activate(X) -> X}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
a() = [0]
c(x1) = [1] x1 + [1]
n__f(x1) = [1] x1 + [0]
n__g(x1) = [1] x1 + [0]
n__a() = [0]
g(x1) = [1] x1 + [0]
activate(x1) = [1] x1 + [1]
f^#(x1) = [1] x1 + [0]
c_0() = [0]
c_1() = [0]
g^#(x1) = [0] x1 + [0]
c_2() = [0]
a^#() = [0]
c_3() = [0]
activate^#(x1) = [1] x1 + [1]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{f^#(X) -> c_1()}
and weakly orienting the rules
{ activate(n__a()) -> a()
, activate(X) -> X}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{f^#(X) -> c_1()}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
a() = [0]
c(x1) = [1] x1 + [1]
n__f(x1) = [1] x1 + [0]
n__g(x1) = [1] x1 + [0]
n__a() = [4]
g(x1) = [1] x1 + [0]
activate(x1) = [1] x1 + [1]
f^#(x1) = [1] x1 + [8]
c_0() = [0]
c_1() = [0]
g^#(x1) = [0] x1 + [0]
c_2() = [0]
a^#() = [0]
c_3() = [0]
activate^#(x1) = [1] x1 + [1]
c_4(x1) = [1] x1 + [3]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{activate^#(n__f(X)) -> c_4(f^#(activate(X)))}
and weakly orienting the rules
{ f^#(X) -> c_1()
, activate(n__a()) -> a()
, activate(X) -> X}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{activate^#(n__f(X)) -> c_4(f^#(activate(X)))}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
a() = [0]
c(x1) = [1] x1 + [1]
n__f(x1) = [1] x1 + [0]
n__g(x1) = [1] x1 + [0]
n__a() = [4]
g(x1) = [1] x1 + [0]
activate(x1) = [1] x1 + [1]
f^#(x1) = [1] x1 + [7]
c_0() = [0]
c_1() = [0]
g^#(x1) = [0] x1 + [0]
c_2() = [0]
a^#() = [0]
c_3() = [0]
activate^#(x1) = [1] x1 + [9]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ f(f(a())) -> c(n__f(n__g(n__f(n__a()))))
, a() -> n__a()}
and weakly orienting the rules
{ activate^#(n__f(X)) -> c_4(f^#(activate(X)))
, f^#(X) -> c_1()
, activate(n__a()) -> a()
, activate(X) -> X}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ f(f(a())) -> c(n__f(n__g(n__f(n__a()))))
, a() -> n__a()}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
a() = [8]
c(x1) = [1] x1 + [0]
n__f(x1) = [1] x1 + [0]
n__g(x1) = [1] x1 + [0]
n__a() = [7]
g(x1) = [1] x1 + [0]
activate(x1) = [1] x1 + [1]
f^#(x1) = [1] x1 + [0]
c_0() = [0]
c_1() = [0]
g^#(x1) = [0] x1 + [0]
c_2() = [0]
a^#() = [0]
c_3() = [0]
activate^#(x1) = [1] x1 + [1]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{g(X) -> n__g(X)}
and weakly orienting the rules
{ f(f(a())) -> c(n__f(n__g(n__f(n__a()))))
, a() -> n__a()
, activate^#(n__f(X)) -> c_4(f^#(activate(X)))
, f^#(X) -> c_1()
, activate(n__a()) -> a()
, activate(X) -> X}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{g(X) -> n__g(X)}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
a() = [0]
c(x1) = [1] x1 + [0]
n__f(x1) = [1] x1 + [0]
n__g(x1) = [1] x1 + [0]
n__a() = [0]
g(x1) = [1] x1 + [8]
activate(x1) = [1] x1 + [1]
f^#(x1) = [1] x1 + [0]
c_0() = [0]
c_1() = [0]
g^#(x1) = [0] x1 + [0]
c_2() = [0]
a^#() = [0]
c_3() = [0]
activate^#(x1) = [1] x1 + [1]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{f(X) -> n__f(X)}
and weakly orienting the rules
{ g(X) -> n__g(X)
, f(f(a())) -> c(n__f(n__g(n__f(n__a()))))
, a() -> n__a()
, activate^#(n__f(X)) -> c_4(f^#(activate(X)))
, f^#(X) -> c_1()
, activate(n__a()) -> a()
, activate(X) -> X}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{f(X) -> n__f(X)}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [2]
a() = [7]
c(x1) = [1] x1 + [1]
n__f(x1) = [1] x1 + [0]
n__g(x1) = [1] x1 + [0]
n__a() = [7]
g(x1) = [1] x1 + [0]
activate(x1) = [1] x1 + [1]
f^#(x1) = [1] x1 + [3]
c_0() = [0]
c_1() = [0]
g^#(x1) = [0] x1 + [0]
c_2() = [0]
a^#() = [0]
c_3() = [0]
activate^#(x1) = [1] x1 + [9]
c_4(x1) = [1] x1 + [1]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7() = [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:
{ activate(n__f(X)) -> f(activate(X))
, activate(n__g(X)) -> g(activate(X))}
Weak Rules:
{ f(X) -> n__f(X)
, g(X) -> n__g(X)
, f(f(a())) -> c(n__f(n__g(n__f(n__a()))))
, a() -> n__a()
, activate^#(n__f(X)) -> c_4(f^#(activate(X)))
, f^#(X) -> c_1()
, activate(n__a()) -> a()
, activate(X) -> X}
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:
{ activate(n__f(X)) -> f(activate(X))
, activate(n__g(X)) -> g(activate(X))}
Weak Rules:
{ f(X) -> n__f(X)
, g(X) -> n__g(X)
, f(f(a())) -> c(n__f(n__g(n__f(n__a()))))
, a() -> n__a()
, activate^#(n__f(X)) -> c_4(f^#(activate(X)))
, f^#(X) -> c_1()
, activate(n__a()) -> a()
, activate(X) -> X}
Details:
The problem is Match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ f_1(5) -> 4
, f_1(5) -> 5
, a_0() -> 4
, a_1() -> 5
, c_0(2) -> 2
, c_0(2) -> 4
, c_0(2) -> 5
, c_1(7) -> 4
, c_1(7) -> 5
, n__f_0(2) -> 2
, n__f_0(2) -> 4
, n__f_0(2) -> 5
, n__f_1(5) -> 4
, n__f_1(5) -> 5
, n__f_1(8) -> 7
, n__f_1(10) -> 9
, n__g_0(2) -> 2
, n__g_0(2) -> 4
, n__g_0(2) -> 5
, n__g_1(5) -> 4
, n__g_1(5) -> 5
, n__g_1(9) -> 8
, n__a_0() -> 2
, n__a_0() -> 4
, n__a_0() -> 5
, n__a_1() -> 5
, n__a_1() -> 10
, g_1(5) -> 4
, g_1(5) -> 5
, activate_0(2) -> 4
, activate_1(2) -> 5
, f^#_0(2) -> 1
, f^#_0(4) -> 3
, f^#_1(5) -> 6
, c_1_0() -> 1
, c_1_0() -> 3
, c_1_1() -> 6
, activate^#_0(2) -> 1
, c_4_0(3) -> 1
, c_4_1(6) -> 1}
3) { activate^#(n__g(X)) -> c_5(g^#(activate(X)))
, g^#(X) -> c_2()}
The usable rules for this path are the following:
{ activate(n__f(X)) -> f(activate(X))
, activate(n__g(X)) -> g(activate(X))
, activate(n__a()) -> a()
, activate(X) -> X
, f(f(a())) -> c(n__f(n__g(n__f(n__a()))))
, f(X) -> n__f(X)
, g(X) -> n__g(X)
, a() -> n__a()}
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:
{ activate(n__f(X)) -> f(activate(X))
, activate(n__g(X)) -> g(activate(X))
, activate(n__a()) -> a()
, activate(X) -> X
, f(f(a())) -> c(n__f(n__g(n__f(n__a()))))
, f(X) -> n__f(X)
, g(X) -> n__g(X)
, a() -> n__a()
, activate^#(n__g(X)) -> c_5(g^#(activate(X)))
, g^#(X) -> c_2()}
Details:
We apply the weight gap principle, strictly orienting the rules
{ activate(n__a()) -> a()
, activate(X) -> X}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ activate(n__a()) -> a()
, activate(X) -> X}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
a() = [0]
c(x1) = [1] x1 + [1]
n__f(x1) = [1] x1 + [0]
n__g(x1) = [1] x1 + [0]
n__a() = [0]
g(x1) = [1] x1 + [0]
activate(x1) = [1] x1 + [1]
f^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
g^#(x1) = [1] x1 + [0]
c_2() = [0]
a^#() = [0]
c_3() = [0]
activate^#(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{g^#(X) -> c_2()}
and weakly orienting the rules
{ activate(n__a()) -> a()
, activate(X) -> X}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{g^#(X) -> c_2()}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
a() = [0]
c(x1) = [1] x1 + [1]
n__f(x1) = [1] x1 + [0]
n__g(x1) = [1] x1 + [0]
n__a() = [4]
g(x1) = [1] x1 + [0]
activate(x1) = [1] x1 + [1]
f^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
g^#(x1) = [1] x1 + [8]
c_2() = [0]
a^#() = [0]
c_3() = [0]
activate^#(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [3]
c_6(x1) = [0] x1 + [0]
c_7() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{activate^#(n__g(X)) -> c_5(g^#(activate(X)))}
and weakly orienting the rules
{ g^#(X) -> c_2()
, activate(n__a()) -> a()
, activate(X) -> X}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{activate^#(n__g(X)) -> c_5(g^#(activate(X)))}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
a() = [0]
c(x1) = [1] x1 + [1]
n__f(x1) = [1] x1 + [0]
n__g(x1) = [1] x1 + [0]
n__a() = [4]
g(x1) = [1] x1 + [0]
activate(x1) = [1] x1 + [1]
f^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
g^#(x1) = [1] x1 + [7]
c_2() = [0]
a^#() = [0]
c_3() = [0]
activate^#(x1) = [1] x1 + [9]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ f(f(a())) -> c(n__f(n__g(n__f(n__a()))))
, a() -> n__a()}
and weakly orienting the rules
{ activate^#(n__g(X)) -> c_5(g^#(activate(X)))
, g^#(X) -> c_2()
, activate(n__a()) -> a()
, activate(X) -> X}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ f(f(a())) -> c(n__f(n__g(n__f(n__a()))))
, a() -> n__a()}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
a() = [8]
c(x1) = [1] x1 + [0]
n__f(x1) = [1] x1 + [0]
n__g(x1) = [1] x1 + [0]
n__a() = [7]
g(x1) = [1] x1 + [0]
activate(x1) = [1] x1 + [1]
f^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
g^#(x1) = [1] x1 + [0]
c_2() = [0]
a^#() = [0]
c_3() = [0]
activate^#(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{g(X) -> n__g(X)}
and weakly orienting the rules
{ f(f(a())) -> c(n__f(n__g(n__f(n__a()))))
, a() -> n__a()
, activate^#(n__g(X)) -> c_5(g^#(activate(X)))
, g^#(X) -> c_2()
, activate(n__a()) -> a()
, activate(X) -> X}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{g(X) -> n__g(X)}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
a() = [0]
c(x1) = [1] x1 + [0]
n__f(x1) = [1] x1 + [0]
n__g(x1) = [1] x1 + [0]
n__a() = [0]
g(x1) = [1] x1 + [8]
activate(x1) = [1] x1 + [1]
f^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
g^#(x1) = [1] x1 + [0]
c_2() = [0]
a^#() = [0]
c_3() = [0]
activate^#(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{f(X) -> n__f(X)}
and weakly orienting the rules
{ g(X) -> n__g(X)
, f(f(a())) -> c(n__f(n__g(n__f(n__a()))))
, a() -> n__a()
, activate^#(n__g(X)) -> c_5(g^#(activate(X)))
, g^#(X) -> c_2()
, activate(n__a()) -> a()
, activate(X) -> X}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{f(X) -> n__f(X)}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [2]
a() = [7]
c(x1) = [1] x1 + [1]
n__f(x1) = [1] x1 + [0]
n__g(x1) = [1] x1 + [0]
n__a() = [7]
g(x1) = [1] x1 + [0]
activate(x1) = [1] x1 + [1]
f^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
g^#(x1) = [1] x1 + [3]
c_2() = [0]
a^#() = [0]
c_3() = [0]
activate^#(x1) = [1] x1 + [9]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [1]
c_6(x1) = [0] x1 + [0]
c_7() = [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:
{ activate(n__f(X)) -> f(activate(X))
, activate(n__g(X)) -> g(activate(X))}
Weak Rules:
{ f(X) -> n__f(X)
, g(X) -> n__g(X)
, f(f(a())) -> c(n__f(n__g(n__f(n__a()))))
, a() -> n__a()
, activate^#(n__g(X)) -> c_5(g^#(activate(X)))
, g^#(X) -> c_2()
, activate(n__a()) -> a()
, activate(X) -> X}
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:
{ activate(n__f(X)) -> f(activate(X))
, activate(n__g(X)) -> g(activate(X))}
Weak Rules:
{ f(X) -> n__f(X)
, g(X) -> n__g(X)
, f(f(a())) -> c(n__f(n__g(n__f(n__a()))))
, a() -> n__a()
, activate^#(n__g(X)) -> c_5(g^#(activate(X)))
, g^#(X) -> c_2()
, activate(n__a()) -> a()
, activate(X) -> X}
Details:
The problem is Match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ f_1(5) -> 4
, f_1(5) -> 5
, a_0() -> 4
, a_1() -> 5
, c_0(2) -> 2
, c_0(2) -> 4
, c_0(2) -> 5
, c_1(7) -> 4
, c_1(7) -> 5
, n__f_0(2) -> 2
, n__f_0(2) -> 4
, n__f_0(2) -> 5
, n__f_1(5) -> 4
, n__f_1(5) -> 5
, n__f_1(8) -> 7
, n__f_1(10) -> 9
, n__g_0(2) -> 2
, n__g_0(2) -> 4
, n__g_0(2) -> 5
, n__g_1(5) -> 4
, n__g_1(5) -> 5
, n__g_1(9) -> 8
, n__a_0() -> 2
, n__a_0() -> 4
, n__a_0() -> 5
, n__a_1() -> 5
, n__a_1() -> 10
, g_1(5) -> 4
, g_1(5) -> 5
, activate_0(2) -> 4
, activate_1(2) -> 5
, g^#_0(2) -> 1
, g^#_0(4) -> 3
, g^#_1(5) -> 6
, c_2_0() -> 1
, c_2_0() -> 3
, c_2_1() -> 6
, activate^#_0(2) -> 1
, c_5_0(3) -> 1
, c_5_1(6) -> 1}
4) {activate^#(n__g(X)) -> c_5(g^#(activate(X)))}
The usable rules for this path are the following:
{ activate(n__f(X)) -> f(activate(X))
, activate(n__g(X)) -> g(activate(X))
, activate(n__a()) -> a()
, activate(X) -> X
, f(f(a())) -> c(n__f(n__g(n__f(n__a()))))
, f(X) -> n__f(X)
, g(X) -> n__g(X)
, a() -> n__a()}
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:
{ activate(n__f(X)) -> f(activate(X))
, activate(n__g(X)) -> g(activate(X))
, activate(n__a()) -> a()
, activate(X) -> X
, f(f(a())) -> c(n__f(n__g(n__f(n__a()))))
, f(X) -> n__f(X)
, g(X) -> n__g(X)
, a() -> n__a()
, activate^#(n__g(X)) -> c_5(g^#(activate(X)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ activate(n__a()) -> a()
, activate(X) -> X}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ activate(n__a()) -> a()
, activate(X) -> X}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
a() = [0]
c(x1) = [1] x1 + [1]
n__f(x1) = [1] x1 + [0]
n__g(x1) = [1] x1 + [0]
n__a() = [2]
g(x1) = [1] x1 + [0]
activate(x1) = [1] x1 + [1]
f^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
g^#(x1) = [1] x1 + [0]
c_2() = [0]
a^#() = [0]
c_3() = [0]
activate^#(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{activate^#(n__g(X)) -> c_5(g^#(activate(X)))}
and weakly orienting the rules
{ activate(n__a()) -> a()
, activate(X) -> X}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{activate^#(n__g(X)) -> c_5(g^#(activate(X)))}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
a() = [0]
c(x1) = [1] x1 + [1]
n__f(x1) = [1] x1 + [0]
n__g(x1) = [1] x1 + [0]
n__a() = [1]
g(x1) = [1] x1 + [0]
activate(x1) = [1] x1 + [1]
f^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
g^#(x1) = [1] x1 + [0]
c_2() = [0]
a^#() = [0]
c_3() = [0]
activate^#(x1) = [1] x1 + [9]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ f(f(a())) -> c(n__f(n__g(n__f(n__a()))))
, a() -> n__a()}
and weakly orienting the rules
{ activate^#(n__g(X)) -> c_5(g^#(activate(X)))
, activate(n__a()) -> a()
, activate(X) -> X}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ f(f(a())) -> c(n__f(n__g(n__f(n__a()))))
, a() -> n__a()}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
a() = [4]
c(x1) = [1] x1 + [0]
n__f(x1) = [1] x1 + [0]
n__g(x1) = [1] x1 + [0]
n__a() = [3]
g(x1) = [1] x1 + [0]
activate(x1) = [1] x1 + [1]
f^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
g^#(x1) = [1] x1 + [0]
c_2() = [0]
a^#() = [0]
c_3() = [0]
activate^#(x1) = [1] x1 + [9]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [3]
c_6(x1) = [0] x1 + [0]
c_7() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{g(X) -> n__g(X)}
and weakly orienting the rules
{ f(f(a())) -> c(n__f(n__g(n__f(n__a()))))
, a() -> n__a()
, activate^#(n__g(X)) -> c_5(g^#(activate(X)))
, activate(n__a()) -> a()
, activate(X) -> X}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{g(X) -> n__g(X)}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
a() = [0]
c(x1) = [1] x1 + [0]
n__f(x1) = [1] x1 + [0]
n__g(x1) = [1] x1 + [0]
n__a() = [0]
g(x1) = [1] x1 + [8]
activate(x1) = [1] x1 + [1]
f^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
g^#(x1) = [1] x1 + [0]
c_2() = [0]
a^#() = [0]
c_3() = [0]
activate^#(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{f(X) -> n__f(X)}
and weakly orienting the rules
{ g(X) -> n__g(X)
, f(f(a())) -> c(n__f(n__g(n__f(n__a()))))
, a() -> n__a()
, activate^#(n__g(X)) -> c_5(g^#(activate(X)))
, activate(n__a()) -> a()
, activate(X) -> X}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{f(X) -> n__f(X)}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [1]
a() = [6]
c(x1) = [1] x1 + [1]
n__f(x1) = [1] x1 + [0]
n__g(x1) = [1] x1 + [0]
n__a() = [5]
g(x1) = [1] x1 + [0]
activate(x1) = [1] x1 + [1]
f^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
g^#(x1) = [1] x1 + [0]
c_2() = [0]
a^#() = [0]
c_3() = [0]
activate^#(x1) = [1] x1 + [11]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [7]
c_6(x1) = [0] x1 + [0]
c_7() = [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:
{ activate(n__f(X)) -> f(activate(X))
, activate(n__g(X)) -> g(activate(X))}
Weak Rules:
{ f(X) -> n__f(X)
, g(X) -> n__g(X)
, f(f(a())) -> c(n__f(n__g(n__f(n__a()))))
, a() -> n__a()
, activate^#(n__g(X)) -> c_5(g^#(activate(X)))
, activate(n__a()) -> a()
, activate(X) -> X}
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:
{ activate(n__f(X)) -> f(activate(X))
, activate(n__g(X)) -> g(activate(X))}
Weak Rules:
{ f(X) -> n__f(X)
, g(X) -> n__g(X)
, f(f(a())) -> c(n__f(n__g(n__f(n__a()))))
, a() -> n__a()
, activate^#(n__g(X)) -> c_5(g^#(activate(X)))
, activate(n__a()) -> a()
, activate(X) -> X}
Details:
The problem is Match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ f_1(5) -> 4
, f_1(5) -> 5
, a_0() -> 4
, a_1() -> 5
, c_0(2) -> 2
, c_0(2) -> 4
, c_0(2) -> 5
, c_1(7) -> 4
, c_1(7) -> 5
, n__f_0(2) -> 2
, n__f_0(2) -> 4
, n__f_0(2) -> 5
, n__f_1(5) -> 4
, n__f_1(5) -> 5
, n__f_1(8) -> 7
, n__f_1(10) -> 9
, n__g_0(2) -> 2
, n__g_0(2) -> 4
, n__g_0(2) -> 5
, n__g_1(5) -> 4
, n__g_1(5) -> 5
, n__g_1(9) -> 8
, n__a_0() -> 2
, n__a_0() -> 4
, n__a_0() -> 5
, n__a_1() -> 5
, n__a_1() -> 10
, g_1(5) -> 4
, g_1(5) -> 5
, activate_0(2) -> 4
, activate_1(2) -> 5
, g^#_0(2) -> 1
, g^#_0(4) -> 3
, g^#_1(5) -> 6
, activate^#_0(2) -> 1
, c_5_0(3) -> 1
, c_5_1(6) -> 1}
5) {activate^#(n__f(X)) -> c_4(f^#(activate(X)))}
The usable rules for this path are the following:
{ activate(n__f(X)) -> f(activate(X))
, activate(n__g(X)) -> g(activate(X))
, activate(n__a()) -> a()
, activate(X) -> X
, f(f(a())) -> c(n__f(n__g(n__f(n__a()))))
, f(X) -> n__f(X)
, g(X) -> n__g(X)
, a() -> n__a()}
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:
{ activate(n__f(X)) -> f(activate(X))
, activate(n__g(X)) -> g(activate(X))
, activate(n__a()) -> a()
, activate(X) -> X
, f(f(a())) -> c(n__f(n__g(n__f(n__a()))))
, f(X) -> n__f(X)
, g(X) -> n__g(X)
, a() -> n__a()
, activate^#(n__f(X)) -> c_4(f^#(activate(X)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ activate(n__a()) -> a()
, activate(X) -> X}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ activate(n__a()) -> a()
, activate(X) -> X}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
a() = [0]
c(x1) = [1] x1 + [1]
n__f(x1) = [1] x1 + [0]
n__g(x1) = [1] x1 + [0]
n__a() = [2]
g(x1) = [1] x1 + [0]
activate(x1) = [1] x1 + [1]
f^#(x1) = [1] x1 + [0]
c_0() = [0]
c_1() = [0]
g^#(x1) = [0] x1 + [0]
c_2() = [0]
a^#() = [0]
c_3() = [0]
activate^#(x1) = [1] x1 + [1]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{activate^#(n__f(X)) -> c_4(f^#(activate(X)))}
and weakly orienting the rules
{ activate(n__a()) -> a()
, activate(X) -> X}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{activate^#(n__f(X)) -> c_4(f^#(activate(X)))}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
a() = [0]
c(x1) = [1] x1 + [1]
n__f(x1) = [1] x1 + [0]
n__g(x1) = [1] x1 + [0]
n__a() = [1]
g(x1) = [1] x1 + [0]
activate(x1) = [1] x1 + [1]
f^#(x1) = [1] x1 + [0]
c_0() = [0]
c_1() = [0]
g^#(x1) = [0] x1 + [0]
c_2() = [0]
a^#() = [0]
c_3() = [0]
activate^#(x1) = [1] x1 + [9]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ f(f(a())) -> c(n__f(n__g(n__f(n__a()))))
, a() -> n__a()}
and weakly orienting the rules
{ activate^#(n__f(X)) -> c_4(f^#(activate(X)))
, activate(n__a()) -> a()
, activate(X) -> X}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ f(f(a())) -> c(n__f(n__g(n__f(n__a()))))
, a() -> n__a()}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
a() = [4]
c(x1) = [1] x1 + [0]
n__f(x1) = [1] x1 + [0]
n__g(x1) = [1] x1 + [0]
n__a() = [3]
g(x1) = [1] x1 + [0]
activate(x1) = [1] x1 + [1]
f^#(x1) = [1] x1 + [0]
c_0() = [0]
c_1() = [0]
g^#(x1) = [0] x1 + [0]
c_2() = [0]
a^#() = [0]
c_3() = [0]
activate^#(x1) = [1] x1 + [9]
c_4(x1) = [1] x1 + [3]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{g(X) -> n__g(X)}
and weakly orienting the rules
{ f(f(a())) -> c(n__f(n__g(n__f(n__a()))))
, a() -> n__a()
, activate^#(n__f(X)) -> c_4(f^#(activate(X)))
, activate(n__a()) -> a()
, activate(X) -> X}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{g(X) -> n__g(X)}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
a() = [0]
c(x1) = [1] x1 + [0]
n__f(x1) = [1] x1 + [0]
n__g(x1) = [1] x1 + [0]
n__a() = [0]
g(x1) = [1] x1 + [8]
activate(x1) = [1] x1 + [1]
f^#(x1) = [1] x1 + [0]
c_0() = [0]
c_1() = [0]
g^#(x1) = [0] x1 + [0]
c_2() = [0]
a^#() = [0]
c_3() = [0]
activate^#(x1) = [1] x1 + [1]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{f(X) -> n__f(X)}
and weakly orienting the rules
{ g(X) -> n__g(X)
, f(f(a())) -> c(n__f(n__g(n__f(n__a()))))
, a() -> n__a()
, activate^#(n__f(X)) -> c_4(f^#(activate(X)))
, activate(n__a()) -> a()
, activate(X) -> X}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{f(X) -> n__f(X)}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [1]
a() = [6]
c(x1) = [1] x1 + [1]
n__f(x1) = [1] x1 + [0]
n__g(x1) = [1] x1 + [0]
n__a() = [5]
g(x1) = [1] x1 + [0]
activate(x1) = [1] x1 + [1]
f^#(x1) = [1] x1 + [0]
c_0() = [0]
c_1() = [0]
g^#(x1) = [0] x1 + [0]
c_2() = [0]
a^#() = [0]
c_3() = [0]
activate^#(x1) = [1] x1 + [11]
c_4(x1) = [1] x1 + [7]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7() = [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:
{ activate(n__f(X)) -> f(activate(X))
, activate(n__g(X)) -> g(activate(X))}
Weak Rules:
{ f(X) -> n__f(X)
, g(X) -> n__g(X)
, f(f(a())) -> c(n__f(n__g(n__f(n__a()))))
, a() -> n__a()
, activate^#(n__f(X)) -> c_4(f^#(activate(X)))
, activate(n__a()) -> a()
, activate(X) -> X}
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:
{ activate(n__f(X)) -> f(activate(X))
, activate(n__g(X)) -> g(activate(X))}
Weak Rules:
{ f(X) -> n__f(X)
, g(X) -> n__g(X)
, f(f(a())) -> c(n__f(n__g(n__f(n__a()))))
, a() -> n__a()
, activate^#(n__f(X)) -> c_4(f^#(activate(X)))
, activate(n__a()) -> a()
, activate(X) -> X}
Details:
The problem is Match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ f_1(5) -> 4
, f_1(5) -> 5
, a_0() -> 4
, a_1() -> 5
, c_0(2) -> 2
, c_0(2) -> 4
, c_0(2) -> 5
, c_1(7) -> 4
, c_1(7) -> 5
, n__f_0(2) -> 2
, n__f_0(2) -> 4
, n__f_0(2) -> 5
, n__f_1(5) -> 4
, n__f_1(5) -> 5
, n__f_1(8) -> 7
, n__f_1(10) -> 9
, n__g_0(2) -> 2
, n__g_0(2) -> 4
, n__g_0(2) -> 5
, n__g_1(5) -> 4
, n__g_1(5) -> 5
, n__g_1(9) -> 8
, n__a_0() -> 2
, n__a_0() -> 4
, n__a_0() -> 5
, n__a_1() -> 5
, n__a_1() -> 10
, g_1(5) -> 4
, g_1(5) -> 5
, activate_0(2) -> 4
, activate_1(2) -> 5
, f^#_0(2) -> 1
, f^#_0(4) -> 3
, f^#_1(5) -> 6
, activate^#_0(2) -> 1
, c_4_0(3) -> 1
, c_4_1(6) -> 1}
6) {activate^#(X) -> c_7()}
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]
a() = [0]
c(x1) = [0] x1 + [0]
n__f(x1) = [0] x1 + [0]
n__g(x1) = [0] x1 + [0]
n__a() = [0]
g(x1) = [0] x1 + [0]
activate(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
g^#(x1) = [0] x1 + [0]
c_2() = [0]
a^#() = [0]
c_3() = [0]
activate^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7() = [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_7()}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{activate^#(X) -> c_7()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{activate^#(X) -> c_7()}
Details:
Interpretation Functions:
f(x1) = [0] x1 + [0]
a() = [0]
c(x1) = [0] x1 + [0]
n__f(x1) = [0] x1 + [0]
n__g(x1) = [0] x1 + [0]
n__a() = [0]
g(x1) = [0] x1 + [0]
activate(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
g^#(x1) = [0] x1 + [0]
c_2() = [0]
a^#() = [0]
c_3() = [0]
activate^#(x1) = [1] x1 + [4]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7() = [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_7()}
Details:
The given problem does not contain any strict rules
7) {activate^#(n__a()) -> c_6(a^#())}
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]
a() = [0]
c(x1) = [0] x1 + [0]
n__f(x1) = [0] x1 + [0]
n__g(x1) = [0] x1 + [0]
n__a() = [0]
g(x1) = [0] x1 + [0]
activate(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
g^#(x1) = [0] x1 + [0]
c_2() = [0]
a^#() = [0]
c_3() = [0]
activate^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7() = [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^#(n__a()) -> c_6(a^#())}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{activate^#(n__a()) -> c_6(a^#())}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{activate^#(n__a()) -> c_6(a^#())}
Details:
Interpretation Functions:
f(x1) = [0] x1 + [0]
a() = [0]
c(x1) = [0] x1 + [0]
n__f(x1) = [0] x1 + [0]
n__g(x1) = [0] x1 + [0]
n__a() = [0]
g(x1) = [0] x1 + [0]
activate(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
g^#(x1) = [0] x1 + [0]
c_2() = [0]
a^#() = [0]
c_3() = [0]
activate^#(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [0]
c_7() = [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__a()) -> c_6(a^#())}
Details:
The given problem does not contain any strict rules
8) { activate^#(n__a()) -> c_6(a^#())
, a^#() -> 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]
a() = [0]
c(x1) = [0] x1 + [0]
n__f(x1) = [0] x1 + [0]
n__g(x1) = [0] x1 + [0]
n__a() = [0]
g(x1) = [0] x1 + [0]
activate(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
g^#(x1) = [0] x1 + [0]
c_2() = [0]
a^#() = [0]
c_3() = [0]
activate^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7() = [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: {a^#() -> c_3()}
Weak Rules: {activate^#(n__a()) -> c_6(a^#())}
Details:
We apply the weight gap principle, strictly orienting the rules
{a^#() -> c_3()}
and weakly orienting the rules
{activate^#(n__a()) -> c_6(a^#())}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a^#() -> c_3()}
Details:
Interpretation Functions:
f(x1) = [0] x1 + [0]
a() = [0]
c(x1) = [0] x1 + [0]
n__f(x1) = [0] x1 + [0]
n__g(x1) = [0] x1 + [0]
n__a() = [0]
g(x1) = [0] x1 + [0]
activate(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
g^#(x1) = [0] x1 + [0]
c_2() = [0]
a^#() = [1]
c_3() = [0]
activate^#(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [0]
c_7() = [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:
{ a^#() -> c_3()
, activate^#(n__a()) -> c_6(a^#())}
Details:
The given problem does not contain any strict rules