* Step 1: Sum WORST_CASE(Omega(n^1),O(n^1))
+ Considered Problem:
- Strict TRS:
active(cons(X1,X2)) -> cons(active(X1),X2)
active(f(X)) -> f(active(X))
active(f(0())) -> mark(cons(0(),f(s(0()))))
active(f(s(0()))) -> mark(f(p(s(0()))))
active(p(X)) -> p(active(X))
active(p(s(X))) -> mark(X)
active(s(X)) -> s(active(X))
cons(mark(X1),X2) -> mark(cons(X1,X2))
cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
p(mark(X)) -> mark(p(X))
p(ok(X)) -> ok(p(X))
proper(0()) -> ok(0())
proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
proper(f(X)) -> f(proper(X))
proper(p(X)) -> p(proper(X))
proper(s(X)) -> s(proper(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
- Signature:
{active/1,cons/2,f/1,p/1,proper/1,s/1,top/1} / {0/0,mark/1,ok/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {active,cons,f,p,proper,s,top} and constructors {0,mark
,ok}
+ Applied Processor:
Sum {left = someStrategy, right = someStrategy}
+ Details:
()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
active(cons(X1,X2)) -> cons(active(X1),X2)
active(f(X)) -> f(active(X))
active(f(0())) -> mark(cons(0(),f(s(0()))))
active(f(s(0()))) -> mark(f(p(s(0()))))
active(p(X)) -> p(active(X))
active(p(s(X))) -> mark(X)
active(s(X)) -> s(active(X))
cons(mark(X1),X2) -> mark(cons(X1,X2))
cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
p(mark(X)) -> mark(p(X))
p(ok(X)) -> ok(p(X))
proper(0()) -> ok(0())
proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
proper(f(X)) -> f(proper(X))
proper(p(X)) -> p(proper(X))
proper(s(X)) -> s(proper(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
- Signature:
{active/1,cons/2,f/1,p/1,proper/1,s/1,top/1} / {0/0,mark/1,ok/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {active,cons,f,p,proper,s,top} and constructors {0,mark
,ok}
+ Applied Processor:
DecreasingLoops {bound = AnyLoop, narrow = 10}
+ Details:
The system has following decreasing Loops:
cons(x,y){x -> mark(x)} =
cons(mark(x),y) ->^+ mark(cons(x,y))
= C[cons(x,y) = cons(x,y){}]
** Step 1.b:1: Bounds WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
active(cons(X1,X2)) -> cons(active(X1),X2)
active(f(X)) -> f(active(X))
active(f(0())) -> mark(cons(0(),f(s(0()))))
active(f(s(0()))) -> mark(f(p(s(0()))))
active(p(X)) -> p(active(X))
active(p(s(X))) -> mark(X)
active(s(X)) -> s(active(X))
cons(mark(X1),X2) -> mark(cons(X1,X2))
cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
p(mark(X)) -> mark(p(X))
p(ok(X)) -> ok(p(X))
proper(0()) -> ok(0())
proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
proper(f(X)) -> f(proper(X))
proper(p(X)) -> p(proper(X))
proper(s(X)) -> s(proper(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
- Signature:
{active/1,cons/2,f/1,p/1,proper/1,s/1,top/1} / {0/0,mark/1,ok/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {active,cons,f,p,proper,s,top} and constructors {0,mark
,ok}
+ Applied Processor:
Bounds {initialAutomaton = perSymbol, enrichment = match}
+ Details:
The problem is match-bounded by 2.
The enriched problem is compatible with follwoing automaton.
0_0() -> 1
0_1() -> 14
active_0(1) -> 2
active_0(5) -> 2
active_0(6) -> 2
active_1(1) -> 16
active_1(5) -> 16
active_1(6) -> 16
active_2(14) -> 17
cons_0(1,1) -> 3
cons_0(1,5) -> 3
cons_0(1,6) -> 3
cons_0(5,1) -> 3
cons_0(5,5) -> 3
cons_0(5,6) -> 3
cons_0(6,1) -> 3
cons_0(6,5) -> 3
cons_0(6,6) -> 3
cons_1(1,1) -> 11
cons_1(1,5) -> 11
cons_1(1,6) -> 11
cons_1(5,1) -> 11
cons_1(5,5) -> 11
cons_1(5,6) -> 11
cons_1(6,1) -> 11
cons_1(6,5) -> 11
cons_1(6,6) -> 11
f_0(1) -> 4
f_0(5) -> 4
f_0(6) -> 4
f_1(1) -> 12
f_1(5) -> 12
f_1(6) -> 12
mark_0(1) -> 5
mark_0(5) -> 5
mark_0(6) -> 5
mark_1(11) -> 3
mark_1(11) -> 11
mark_1(12) -> 4
mark_1(12) -> 12
mark_1(13) -> 7
mark_1(13) -> 13
mark_1(15) -> 9
mark_1(15) -> 15
ok_0(1) -> 6
ok_0(5) -> 6
ok_0(6) -> 6
ok_1(11) -> 3
ok_1(11) -> 11
ok_1(12) -> 4
ok_1(12) -> 12
ok_1(13) -> 7
ok_1(13) -> 13
ok_1(14) -> 8
ok_1(14) -> 16
ok_1(15) -> 9
ok_1(15) -> 15
p_0(1) -> 7
p_0(5) -> 7
p_0(6) -> 7
p_1(1) -> 13
p_1(5) -> 13
p_1(6) -> 13
proper_0(1) -> 8
proper_0(5) -> 8
proper_0(6) -> 8
proper_1(1) -> 16
proper_1(5) -> 16
proper_1(6) -> 16
s_0(1) -> 9
s_0(5) -> 9
s_0(6) -> 9
s_1(1) -> 15
s_1(5) -> 15
s_1(6) -> 15
top_0(1) -> 10
top_0(5) -> 10
top_0(6) -> 10
top_1(16) -> 10
top_2(17) -> 10
** Step 1.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
active(cons(X1,X2)) -> cons(active(X1),X2)
active(f(X)) -> f(active(X))
active(f(0())) -> mark(cons(0(),f(s(0()))))
active(f(s(0()))) -> mark(f(p(s(0()))))
active(p(X)) -> p(active(X))
active(p(s(X))) -> mark(X)
active(s(X)) -> s(active(X))
cons(mark(X1),X2) -> mark(cons(X1,X2))
cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
p(mark(X)) -> mark(p(X))
p(ok(X)) -> ok(p(X))
proper(0()) -> ok(0())
proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
proper(f(X)) -> f(proper(X))
proper(p(X)) -> p(proper(X))
proper(s(X)) -> s(proper(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
- Signature:
{active/1,cons/2,f/1,p/1,proper/1,s/1,top/1} / {0/0,mark/1,ok/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {active,cons,f,p,proper,s,top} and constructors {0,mark
,ok}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(Omega(n^1),O(n^1))