* Step 1: Sum WORST_CASE(Omega(n^1),O(n^1))
+ Considered Problem:
- Strict TRS:
half(0()) -> 0()
half(s(s(x))) -> s(half(x))
log(s(0())) -> 0()
log(s(s(x))) -> s(log(s(half(x))))
- Signature:
{half/1,log/1} / {0/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {half,log} 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:
half(0()) -> 0()
half(s(s(x))) -> s(half(x))
log(s(0())) -> 0()
log(s(s(x))) -> s(log(s(half(x))))
- Signature:
{half/1,log/1} / {0/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {half,log} and constructors {0,s}
+ Applied Processor:
DecreasingLoops {bound = AnyLoop, narrow = 10}
+ Details:
The system has following decreasing Loops:
half(x){x -> s(s(x))} =
half(s(s(x))) ->^+ s(half(x))
= C[half(x) = half(x){}]
** Step 1.b:1: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
half(0()) -> 0()
half(s(s(x))) -> s(half(x))
log(s(0())) -> 0()
log(s(s(x))) -> s(log(s(half(x))))
- Signature:
{half/1,log/1} / {0/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {half,log} and constructors {0,s}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
+ Details:
The weightgap principle applies using the following nonconstant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(log) = {1},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [1]
p(half) = [8]
p(log) = [1] x1 + [3]
p(s) = [1] x1 + [0]
Following rules are strictly oriented:
half(0()) = [8]
> [1]
= 0()
log(s(0())) = [4]
> [1]
= 0()
Following rules are (at-least) weakly oriented:
half(s(s(x))) = [8]
>= [8]
= s(half(x))
log(s(s(x))) = [1] x + [3]
>= [11]
= s(log(s(half(x))))
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:2: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
half(s(s(x))) -> s(half(x))
log(s(s(x))) -> s(log(s(half(x))))
- Weak TRS:
half(0()) -> 0()
log(s(0())) -> 0()
- Signature:
{half/1,log/1} / {0/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {half,log} and constructors {0,s}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
+ Details:
The weightgap principle applies using the following nonconstant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(log) = {1},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [10]
p(half) = [1] x1 + [4]
p(log) = [1] x1 + [0]
p(s) = [1] x1 + [9]
Following rules are strictly oriented:
half(s(s(x))) = [1] x + [22]
> [1] x + [13]
= s(half(x))
Following rules are (at-least) weakly oriented:
half(0()) = [14]
>= [10]
= 0()
log(s(0())) = [19]
>= [10]
= 0()
log(s(s(x))) = [1] x + [18]
>= [1] x + [22]
= s(log(s(half(x))))
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:3: MI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
log(s(s(x))) -> s(log(s(half(x))))
- Weak TRS:
half(0()) -> 0()
half(s(s(x))) -> s(half(x))
log(s(0())) -> 0()
- Signature:
{half/1,log/1} / {0/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {half,log} and constructors {0,s}
+ Applied Processor:
MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity Nothing)), miDimension = 1, miUArgs = UArgs, miURules = URules, miSelector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind MaximalMatrix (UpperTriangular (Multiplicity Nothing)):
The following argument positions are considered usable:
uargs(log) = {1},
uargs(s) = {1}
Following symbols are considered usable:
{half,log}
TcT has computed the following interpretation:
p(0) = [2]
p(half) = [1] x_1 + [0]
p(log) = [4] x_1 + [13]
p(s) = [1] x_1 + [2]
Following rules are strictly oriented:
log(s(s(x))) = [4] x + [29]
> [4] x + [23]
= s(log(s(half(x))))
Following rules are (at-least) weakly oriented:
half(0()) = [2]
>= [2]
= 0()
half(s(s(x))) = [1] x + [4]
>= [1] x + [2]
= s(half(x))
log(s(0())) = [29]
>= [2]
= 0()
** Step 1.b:4: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
half(0()) -> 0()
half(s(s(x))) -> s(half(x))
log(s(0())) -> 0()
log(s(s(x))) -> s(log(s(half(x))))
- Signature:
{half/1,log/1} / {0/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {half,log} 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^1))