* Step 1: Sum WORST_CASE(Omega(n^1),O(n^1))
+ Considered Problem:
- Strict TRS:
activate(X) -> X
activate(n__d(X)) -> d(X)
activate(n__f(X)) -> f(activate(X))
activate(n__g(X)) -> g(X)
c(X) -> d(activate(X))
d(X) -> n__d(X)
f(X) -> n__f(X)
f(f(X)) -> c(n__f(n__g(n__f(X))))
g(X) -> n__g(X)
h(X) -> c(n__d(X))
- Signature:
{activate/1,c/1,d/1,f/1,g/1,h/1} / {n__d/1,n__f/1,n__g/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate,c,d,f,g,h} and constructors {n__d,n__f,n__g}
+ Applied Processor:
Sum {left = someStrategy, right = someStrategy}
+ Details:
()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
activate(X) -> X
activate(n__d(X)) -> d(X)
activate(n__f(X)) -> f(activate(X))
activate(n__g(X)) -> g(X)
c(X) -> d(activate(X))
d(X) -> n__d(X)
f(X) -> n__f(X)
f(f(X)) -> c(n__f(n__g(n__f(X))))
g(X) -> n__g(X)
h(X) -> c(n__d(X))
- Signature:
{activate/1,c/1,d/1,f/1,g/1,h/1} / {n__d/1,n__f/1,n__g/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate,c,d,f,g,h} and constructors {n__d,n__f,n__g}
+ Applied Processor:
DecreasingLoops {bound = AnyLoop, narrow = 10}
+ Details:
The system has following decreasing Loops:
activate(x){x -> n__f(x)} =
activate(n__f(x)) ->^+ f(activate(x))
= C[activate(x) = activate(x){}]
** Step 1.b:1: InnermostRuleRemoval WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
activate(X) -> X
activate(n__d(X)) -> d(X)
activate(n__f(X)) -> f(activate(X))
activate(n__g(X)) -> g(X)
c(X) -> d(activate(X))
d(X) -> n__d(X)
f(X) -> n__f(X)
f(f(X)) -> c(n__f(n__g(n__f(X))))
g(X) -> n__g(X)
h(X) -> c(n__d(X))
- Signature:
{activate/1,c/1,d/1,f/1,g/1,h/1} / {n__d/1,n__f/1,n__g/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate,c,d,f,g,h} and constructors {n__d,n__f,n__g}
+ Applied Processor:
InnermostRuleRemoval
+ Details:
Arguments of following rules are not normal-forms.
f(f(X)) -> c(n__f(n__g(n__f(X))))
All above mentioned rules can be savely removed.
** Step 1.b:2: Bounds WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
activate(X) -> X
activate(n__d(X)) -> d(X)
activate(n__f(X)) -> f(activate(X))
activate(n__g(X)) -> g(X)
c(X) -> d(activate(X))
d(X) -> n__d(X)
f(X) -> n__f(X)
g(X) -> n__g(X)
h(X) -> c(n__d(X))
- Signature:
{activate/1,c/1,d/1,f/1,g/1,h/1} / {n__d/1,n__f/1,n__g/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate,c,d,f,g,h} and constructors {n__d,n__f,n__g}
+ Applied Processor:
Bounds {initialAutomaton = minimal, enrichment = match}
+ Details:
The problem is match-bounded by 4.
The enriched problem is compatible with follwoing automaton.
activate_0(2) -> 1
activate_0(2) -> 4
activate_1(2) -> 3
activate_1(2) -> 6
activate_2(1) -> 4
activate_2(2) -> 5
activate_3(3) -> 6
c_0(2) -> 1
c_0(2) -> 4
c_1(1) -> 1
c_1(1) -> 4
d_0(2) -> 1
d_0(2) -> 4
d_1(2) -> 1
d_1(2) -> 3
d_1(2) -> 4
d_1(2) -> 5
d_1(2) -> 6
d_1(3) -> 1
d_1(3) -> 4
d_2(2) -> 4
d_2(4) -> 1
d_2(4) -> 4
d_3(2) -> 4
d_3(2) -> 6
d_3(3) -> 4
d_3(4) -> 4
f_0(2) -> 1
f_0(2) -> 4
f_1(3) -> 1
f_1(3) -> 3
f_1(3) -> 4
f_1(3) -> 6
f_1(6) -> 5
f_2(5) -> 4
f_3(6) -> 4
f_3(6) -> 6
g_0(2) -> 1
g_0(2) -> 4
g_1(2) -> 1
g_1(2) -> 3
g_1(2) -> 4
g_1(2) -> 5
g_1(2) -> 6
g_2(2) -> 4
g_3(2) -> 4
g_3(2) -> 6
h_0(2) -> 1
h_0(2) -> 4
n__d_0(2) -> 1
n__d_0(2) -> 2
n__d_0(2) -> 3
n__d_0(2) -> 4
n__d_0(2) -> 5
n__d_0(2) -> 6
n__d_1(2) -> 1
n__d_1(2) -> 4
n__d_2(2) -> 1
n__d_2(2) -> 3
n__d_2(2) -> 4
n__d_2(2) -> 5
n__d_2(2) -> 6
n__d_2(3) -> 1
n__d_2(3) -> 4
n__d_3(2) -> 4
n__d_3(4) -> 1
n__d_3(4) -> 4
n__d_4(2) -> 4
n__d_4(2) -> 6
n__d_4(3) -> 4
n__d_4(4) -> 4
n__f_0(2) -> 1
n__f_0(2) -> 2
n__f_0(2) -> 3
n__f_0(2) -> 4
n__f_0(2) -> 5
n__f_0(2) -> 6
n__f_1(2) -> 1
n__f_1(2) -> 4
n__f_2(3) -> 1
n__f_2(3) -> 3
n__f_2(3) -> 4
n__f_2(3) -> 6
n__f_2(6) -> 5
n__f_3(5) -> 4
n__f_4(6) -> 4
n__f_4(6) -> 6
n__g_0(2) -> 1
n__g_0(2) -> 2
n__g_0(2) -> 3
n__g_0(2) -> 4
n__g_0(2) -> 5
n__g_0(2) -> 6
n__g_1(2) -> 1
n__g_1(2) -> 4
n__g_2(2) -> 1
n__g_2(2) -> 3
n__g_2(2) -> 4
n__g_2(2) -> 5
n__g_2(2) -> 6
n__g_3(2) -> 4
n__g_4(2) -> 4
n__g_4(2) -> 6
1 -> 4
2 -> 1
2 -> 3
2 -> 4
2 -> 5
2 -> 6
3 -> 6
** Step 1.b:3: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
activate(X) -> X
activate(n__d(X)) -> d(X)
activate(n__f(X)) -> f(activate(X))
activate(n__g(X)) -> g(X)
c(X) -> d(activate(X))
d(X) -> n__d(X)
f(X) -> n__f(X)
g(X) -> n__g(X)
h(X) -> c(n__d(X))
- Signature:
{activate/1,c/1,d/1,f/1,g/1,h/1} / {n__d/1,n__f/1,n__g/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate,c,d,f,g,h} and constructors {n__d,n__f,n__g}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(Omega(n^1),O(n^1))