* Step 1: WeightGap WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
log(s(0())) -> 0()
log(s(s(X))) -> s(log(s(quot(X,s(s(0()))))))
min(X,0()) -> X
min(s(X),s(Y)) -> min(X,Y)
quot(0(),s(Y)) -> 0()
quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y)))
- Signature:
{log/1,min/2,quot/2} / {0/0,s/1}
- Obligation:
runtime complexity wrt. defined symbols {log,min,quot} 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(min) = {1},
uargs(quot) = {1},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [7]
p(log) = [1] x1 + [0]
p(min) = [1] x1 + [2]
p(quot) = [1] x1 + [9]
p(s) = [1] x1 + [0]
Following rules are strictly oriented:
min(X,0()) = [1] X + [2]
> [1] X + [0]
= X
quot(0(),s(Y)) = [16]
> [7]
= 0()
Following rules are (at-least) weakly oriented:
log(s(0())) = [7]
>= [7]
= 0()
log(s(s(X))) = [1] X + [0]
>= [1] X + [9]
= s(log(s(quot(X,s(s(0()))))))
min(s(X),s(Y)) = [1] X + [2]
>= [1] X + [2]
= min(X,Y)
quot(s(X),s(Y)) = [1] X + [9]
>= [1] X + [11]
= s(quot(min(X,Y),s(Y)))
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 2: WeightGap WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
log(s(0())) -> 0()
log(s(s(X))) -> s(log(s(quot(X,s(s(0()))))))
min(s(X),s(Y)) -> min(X,Y)
quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y)))
- Weak TRS:
min(X,0()) -> X
quot(0(),s(Y)) -> 0()
- Signature:
{log/1,min/2,quot/2} / {0/0,s/1}
- Obligation:
runtime complexity wrt. defined symbols {log,min,quot} 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(min) = {1},
uargs(quot) = {1},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(log) = [1] x1 + [0]
p(min) = [1] x1 + [7]
p(quot) = [1] x1 + [8]
p(s) = [1] x1 + [1]
Following rules are strictly oriented:
log(s(0())) = [1]
> [0]
= 0()
min(s(X),s(Y)) = [1] X + [8]
> [1] X + [7]
= min(X,Y)
Following rules are (at-least) weakly oriented:
log(s(s(X))) = [1] X + [2]
>= [1] X + [10]
= s(log(s(quot(X,s(s(0()))))))
min(X,0()) = [1] X + [7]
>= [1] X + [0]
= X
quot(0(),s(Y)) = [8]
>= [0]
= 0()
quot(s(X),s(Y)) = [1] X + [9]
>= [1] X + [16]
= s(quot(min(X,Y),s(Y)))
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 3: MI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
log(s(s(X))) -> s(log(s(quot(X,s(s(0()))))))
quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y)))
- Weak TRS:
log(s(0())) -> 0()
min(X,0()) -> X
min(s(X),s(Y)) -> min(X,Y)
quot(0(),s(Y)) -> 0()
- Signature:
{log/1,min/2,quot/2} / {0/0,s/1}
- Obligation:
runtime complexity wrt. defined symbols {log,min,quot} 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(min) = {1},
uargs(quot) = {1},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [2]
p(log) = [8] x_1 + [6]
p(min) = [1] x_1 + [0]
p(quot) = [1] x_1 + [0]
p(s) = [1] x_1 + [1]
Following rules are strictly oriented:
log(s(s(X))) = [8] X + [22]
> [8] X + [15]
= s(log(s(quot(X,s(s(0()))))))
Following rules are (at-least) weakly oriented:
log(s(0())) = [30]
>= [2]
= 0()
min(X,0()) = [1] X + [0]
>= [1] X + [0]
= X
min(s(X),s(Y)) = [1] X + [1]
>= [1] X + [0]
= min(X,Y)
quot(0(),s(Y)) = [2]
>= [2]
= 0()
quot(s(X),s(Y)) = [1] X + [1]
>= [1] X + [1]
= s(quot(min(X,Y),s(Y)))
* Step 4: MI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y)))
- Weak TRS:
log(s(0())) -> 0()
log(s(s(X))) -> s(log(s(quot(X,s(s(0()))))))
min(X,0()) -> X
min(s(X),s(Y)) -> min(X,Y)
quot(0(),s(Y)) -> 0()
- Signature:
{log/1,min/2,quot/2} / {0/0,s/1}
- Obligation:
runtime complexity wrt. defined symbols {log,min,quot} and constructors {0,s}
+ Applied Processor:
MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity Nothing)), miDimension = 2, 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(min) = {1},
uargs(quot) = {1},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [2]
[1]
p(log) = [2 0] x_1 + [4]
[0 1] [1]
p(min) = [1 0] x_1 + [0]
[0 1] [0]
p(quot) = [1 1] x_1 + [0]
[0 1] [0]
p(s) = [1 2] x_1 + [0]
[0 1] [1]
Following rules are strictly oriented:
quot(s(X),s(Y)) = [1 3] X + [1]
[0 1] [1]
> [1 3] X + [0]
[0 1] [1]
= s(quot(min(X,Y),s(Y)))
Following rules are (at-least) weakly oriented:
log(s(0())) = [12]
[3]
>= [2]
[1]
= 0()
log(s(s(X))) = [2 8] X + [8]
[0 1] [3]
>= [2 8] X + [8]
[0 1] [3]
= s(log(s(quot(X,s(s(0()))))))
min(X,0()) = [1 0] X + [0]
[0 1] [0]
>= [1 0] X + [0]
[0 1] [0]
= X
min(s(X),s(Y)) = [1 2] X + [0]
[0 1] [1]
>= [1 0] X + [0]
[0 1] [0]
= min(X,Y)
quot(0(),s(Y)) = [3]
[1]
>= [2]
[1]
= 0()
* Step 5: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
log(s(0())) -> 0()
log(s(s(X))) -> s(log(s(quot(X,s(s(0()))))))
min(X,0()) -> X
min(s(X),s(Y)) -> min(X,Y)
quot(0(),s(Y)) -> 0()
quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y)))
- Signature:
{log/1,min/2,quot/2} / {0/0,s/1}
- Obligation:
runtime complexity wrt. defined symbols {log,min,quot} and constructors {0,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(?,O(n^2))