WORST_CASE(Omega(n^1),O(n^2))
* Step 1: Sum WORST_CASE(Omega(n^1),O(n^2))
+ Considered Problem:
- Strict TRS:
*(0(),y) -> 0()
*(s(x),y) -> +(y,*(x,y))
-(x,0()) -> x
-(0(),y) -> 0()
-(s(x),s(y)) -> -(x,y)
exp(x,0()) -> s(0())
exp(x,s(y)) -> *(x,exp(x,y))
- Signature:
{*/2,-/2,exp/2} / {+/2,0/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {*,-,exp} and constructors {+,0,s}
+ Applied Processor:
Sum {left = someStrategy, right = someStrategy}
+ Details:
()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
*(0(),y) -> 0()
*(s(x),y) -> +(y,*(x,y))
-(x,0()) -> x
-(0(),y) -> 0()
-(s(x),s(y)) -> -(x,y)
exp(x,0()) -> s(0())
exp(x,s(y)) -> *(x,exp(x,y))
- Signature:
{*/2,-/2,exp/2} / {+/2,0/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {*,-,exp} and constructors {+,0,s}
+ Applied Processor:
DecreasingLoops {bound = AnyLoop, narrow = 10}
+ Details:
The system has following decreasing Loops:
*(x,y){x -> s(x)} =
*(s(x),y) ->^+ +(y,*(x,y))
= C[*(x,y) = *(x,y){}]
** Step 1.b:1: NaturalPI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
*(0(),y) -> 0()
*(s(x),y) -> +(y,*(x,y))
-(x,0()) -> x
-(0(),y) -> 0()
-(s(x),s(y)) -> -(x,y)
exp(x,0()) -> s(0())
exp(x,s(y)) -> *(x,exp(x,y))
- Signature:
{*/2,-/2,exp/2} / {+/2,0/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {*,-,exp} and constructors {+,0,s}
+ Applied Processor:
NaturalPI {shape = Linear, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules}
+ Details:
We apply a polynomial interpretation of kind constructor-based(linear):
The following argument positions are considered usable:
uargs(*) = {2},
uargs(+) = {2}
Following symbols are considered usable:
{*,-,exp}
TcT has computed the following interpretation:
p(*) = 1 + x2
p(+) = x2
p(-) = x1 + 9*x2
p(0) = 1
p(exp) = 8 + x1 + 2*x2
p(s) = 2 + x1
Following rules are strictly oriented:
-(x,0()) = 9 + x
> x
= x
-(s(x),s(y)) = 20 + x + 9*y
> x + 9*y
= -(x,y)
exp(x,0()) = 10 + x
> 3
= s(0())
exp(x,s(y)) = 12 + x + 2*y
> 9 + x + 2*y
= *(x,exp(x,y))
Following rules are (at-least) weakly oriented:
*(0(),y) = 1 + y
>= 1
= 0()
*(s(x),y) = 1 + y
>= 1 + y
= +(y,*(x,y))
-(0(),y) = 1 + 9*y
>= 1
= 0()
** Step 1.b:2: NaturalPI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
*(0(),y) -> 0()
*(s(x),y) -> +(y,*(x,y))
-(0(),y) -> 0()
- Weak TRS:
-(x,0()) -> x
-(s(x),s(y)) -> -(x,y)
exp(x,0()) -> s(0())
exp(x,s(y)) -> *(x,exp(x,y))
- Signature:
{*/2,-/2,exp/2} / {+/2,0/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {*,-,exp} and constructors {+,0,s}
+ Applied Processor:
NaturalPI {shape = Linear, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules}
+ Details:
We apply a polynomial interpretation of kind constructor-based(linear):
The following argument positions are considered usable:
uargs(*) = {2},
uargs(+) = {2}
Following symbols are considered usable:
{*,-,exp}
TcT has computed the following interpretation:
p(*) = x2
p(+) = x2
p(-) = 7 + 2*x1
p(0) = 0
p(exp) = 12
p(s) = 10 + x1
Following rules are strictly oriented:
-(0(),y) = 7
> 0
= 0()
Following rules are (at-least) weakly oriented:
*(0(),y) = y
>= 0
= 0()
*(s(x),y) = y
>= y
= +(y,*(x,y))
-(x,0()) = 7 + 2*x
>= x
= x
-(s(x),s(y)) = 27 + 2*x
>= 7 + 2*x
= -(x,y)
exp(x,0()) = 12
>= 10
= s(0())
exp(x,s(y)) = 12
>= 12
= *(x,exp(x,y))
** Step 1.b:3: NaturalPI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
*(0(),y) -> 0()
*(s(x),y) -> +(y,*(x,y))
- Weak TRS:
-(x,0()) -> x
-(0(),y) -> 0()
-(s(x),s(y)) -> -(x,y)
exp(x,0()) -> s(0())
exp(x,s(y)) -> *(x,exp(x,y))
- Signature:
{*/2,-/2,exp/2} / {+/2,0/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {*,-,exp} and constructors {+,0,s}
+ Applied Processor:
NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules}
+ Details:
We apply a polynomial interpretation of kind constructor-based(mixed(2)):
The following argument positions are considered usable:
uargs(*) = {2},
uargs(+) = {2}
Following symbols are considered usable:
{*,-,exp}
TcT has computed the following interpretation:
p(*) = 3 + x2
p(+) = x2
p(-) = 2 + 4*x1
p(0) = 2
p(exp) = x1^2 + 2*x2 + x2^2
p(s) = 2 + x1
Following rules are strictly oriented:
*(0(),y) = 3 + y
> 2
= 0()
Following rules are (at-least) weakly oriented:
*(s(x),y) = 3 + y
>= 3 + y
= +(y,*(x,y))
-(x,0()) = 2 + 4*x
>= x
= x
-(0(),y) = 10
>= 2
= 0()
-(s(x),s(y)) = 10 + 4*x
>= 2 + 4*x
= -(x,y)
exp(x,0()) = 8 + x^2
>= 4
= s(0())
exp(x,s(y)) = 8 + x^2 + 6*y + y^2
>= 3 + x^2 + 2*y + y^2
= *(x,exp(x,y))
** Step 1.b:4: NaturalPI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
*(s(x),y) -> +(y,*(x,y))
- Weak TRS:
*(0(),y) -> 0()
-(x,0()) -> x
-(0(),y) -> 0()
-(s(x),s(y)) -> -(x,y)
exp(x,0()) -> s(0())
exp(x,s(y)) -> *(x,exp(x,y))
- Signature:
{*/2,-/2,exp/2} / {+/2,0/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {*,-,exp} and constructors {+,0,s}
+ Applied Processor:
NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules}
+ Details:
We apply a polynomial interpretation of kind constructor-based(mixed(2)):
The following argument positions are considered usable:
uargs(*) = {2},
uargs(+) = {2}
Following symbols are considered usable:
{*,-,exp}
TcT has computed the following interpretation:
p(*) = 2 + x1 + x2
p(+) = x2
p(-) = x1*x2 + 2*x1^2 + x2
p(0) = 2
p(exp) = x1*x2 + 2*x2
p(s) = 1 + x1
Following rules are strictly oriented:
*(s(x),y) = 3 + x + y
> 2 + x + y
= +(y,*(x,y))
Following rules are (at-least) weakly oriented:
*(0(),y) = 4 + y
>= 2
= 0()
-(x,0()) = 2 + 2*x + 2*x^2
>= x
= x
-(0(),y) = 8 + 3*y
>= 2
= 0()
-(s(x),s(y)) = 4 + 5*x + x*y + 2*x^2 + 2*y
>= x*y + 2*x^2 + y
= -(x,y)
exp(x,0()) = 4 + 2*x
>= 3
= s(0())
exp(x,s(y)) = 2 + x + x*y + 2*y
>= 2 + x + x*y + 2*y
= *(x,exp(x,y))
** Step 1.b:5: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
*(0(),y) -> 0()
*(s(x),y) -> +(y,*(x,y))
-(x,0()) -> x
-(0(),y) -> 0()
-(s(x),s(y)) -> -(x,y)
exp(x,0()) -> s(0())
exp(x,s(y)) -> *(x,exp(x,y))
- Signature:
{*/2,-/2,exp/2} / {+/2,0/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {*,-,exp} and constructors {+,0,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(Omega(n^1),O(n^2))