* Step 1: Sum WORST_CASE(Omega(n^1),O(n^1))
+ Considered Problem:
- Strict TRS:
@(Cons(x,xs),ys) -> Cons(x,@(xs,ys))
@(Nil(),ys) -> ys
equal(Capture(),Capture()) -> True()
equal(Capture(),Swap()) -> False()
equal(Swap(),Capture()) -> False()
equal(Swap(),Swap()) -> True()
game(p1,p2,Cons(Swap(),xs)) -> game(p2,p1,xs)
game(p1,p2,Nil()) -> @(p1,p2)
game(p1,Cons(x',xs'),Cons(Capture(),xs)) -> game(Cons(x',p1),xs',xs)
goal(p1,p2,moves) -> game(p1,p2,moves)
- Signature:
{@/2,equal/2,game/3,goal/3} / {Capture/0,Cons/2,False/0,Nil/0,Swap/0,True/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {@,equal,game,goal} and constructors {Capture,Cons,False
,Nil,Swap,True}
+ Applied Processor:
Sum {left = someStrategy, right = someStrategy}
+ Details:
()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
@(Cons(x,xs),ys) -> Cons(x,@(xs,ys))
@(Nil(),ys) -> ys
equal(Capture(),Capture()) -> True()
equal(Capture(),Swap()) -> False()
equal(Swap(),Capture()) -> False()
equal(Swap(),Swap()) -> True()
game(p1,p2,Cons(Swap(),xs)) -> game(p2,p1,xs)
game(p1,p2,Nil()) -> @(p1,p2)
game(p1,Cons(x',xs'),Cons(Capture(),xs)) -> game(Cons(x',p1),xs',xs)
goal(p1,p2,moves) -> game(p1,p2,moves)
- Signature:
{@/2,equal/2,game/3,goal/3} / {Capture/0,Cons/2,False/0,Nil/0,Swap/0,True/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {@,equal,game,goal} and constructors {Capture,Cons,False
,Nil,Swap,True}
+ Applied Processor:
DecreasingLoops {bound = AnyLoop, narrow = 10}
+ Details:
The system has following decreasing Loops:
@(y,z){y -> Cons(x,y)} =
@(Cons(x,y),z) ->^+ Cons(x,@(y,z))
= C[@(y,z) = @(y,z){}]
** Step 1.b:1: Bounds WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
@(Cons(x,xs),ys) -> Cons(x,@(xs,ys))
@(Nil(),ys) -> ys
equal(Capture(),Capture()) -> True()
equal(Capture(),Swap()) -> False()
equal(Swap(),Capture()) -> False()
equal(Swap(),Swap()) -> True()
game(p1,p2,Cons(Swap(),xs)) -> game(p2,p1,xs)
game(p1,p2,Nil()) -> @(p1,p2)
game(p1,Cons(x',xs'),Cons(Capture(),xs)) -> game(Cons(x',p1),xs',xs)
goal(p1,p2,moves) -> game(p1,p2,moves)
- Signature:
{@/2,equal/2,game/3,goal/3} / {Capture/0,Cons/2,False/0,Nil/0,Swap/0,True/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {@,equal,game,goal} and constructors {Capture,Cons,False
,Nil,Swap,True}
+ Applied Processor:
Bounds {initialAutomaton = minimal, enrichment = match}
+ Details:
The problem is match-bounded by 2.
The enriched problem is compatible with follwoing automaton.
@_0(2,2) -> 1
@_1(2,2) -> 1
@_1(2,2) -> 3
@_2(2,2) -> 4
Capture_0() -> 1
Capture_0() -> 2
Capture_0() -> 3
Capture_0() -> 4
Cons_0(2,2) -> 1
Cons_0(2,2) -> 2
Cons_0(2,2) -> 3
Cons_0(2,2) -> 4
Cons_1(2,2) -> 1
Cons_1(2,2) -> 2
Cons_1(2,2) -> 3
Cons_1(2,2) -> 4
Cons_1(2,3) -> 1
Cons_1(2,3) -> 3
Cons_1(2,3) -> 4
Cons_2(2,4) -> 1
Cons_2(2,4) -> 3
Cons_2(2,4) -> 4
False_0() -> 1
False_0() -> 2
False_0() -> 3
False_0() -> 4
False_1() -> 1
Nil_0() -> 1
Nil_0() -> 2
Nil_0() -> 3
Nil_0() -> 4
Swap_0() -> 1
Swap_0() -> 2
Swap_0() -> 3
Swap_0() -> 4
True_0() -> 1
True_0() -> 2
True_0() -> 3
True_0() -> 4
True_1() -> 1
equal_0(2,2) -> 1
game_0(2,2,2) -> 1
game_1(2,2,2) -> 1
goal_0(2,2,2) -> 1
2 -> 1
2 -> 3
2 -> 4
** Step 1.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
@(Cons(x,xs),ys) -> Cons(x,@(xs,ys))
@(Nil(),ys) -> ys
equal(Capture(),Capture()) -> True()
equal(Capture(),Swap()) -> False()
equal(Swap(),Capture()) -> False()
equal(Swap(),Swap()) -> True()
game(p1,p2,Cons(Swap(),xs)) -> game(p2,p1,xs)
game(p1,p2,Nil()) -> @(p1,p2)
game(p1,Cons(x',xs'),Cons(Capture(),xs)) -> game(Cons(x',p1),xs',xs)
goal(p1,p2,moves) -> game(p1,p2,moves)
- Signature:
{@/2,equal/2,game/3,goal/3} / {Capture/0,Cons/2,False/0,Nil/0,Swap/0,True/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {@,equal,game,goal} and constructors {Capture,Cons,False
,Nil,Swap,True}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(Omega(n^1),O(n^1))