* Step 1: WeightGap WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
f(0()) -> true()
f(1()) -> false()
f(s(x)) -> f(x)
g(x,c(y)) -> g(x,g(s(c(y)),y))
g(s(x),s(y)) -> if(f(x),s(x),s(y))
if(false(),x,y) -> y
if(true(),x,y) -> x
- Signature:
{f/1,g/2,if/3} / {0/0,1/0,c/1,false/0,s/1,true/0}
- Obligation:
runtime complexity wrt. defined symbols {f,g,if} and constructors {0,1,c,false,s,true}
+ 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(g) = {2},
uargs(if) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [2]
p(1) = [2]
p(c) = [1] x1 + [0]
p(f) = [0]
p(false) = [8]
p(g) = [1] x2 + [15]
p(if) = [1] x1 + [7] x2 + [4] x3 + [1]
p(s) = [1]
p(true) = [3]
Following rules are strictly oriented:
g(s(x),s(y)) = [16]
> [12]
= if(f(x),s(x),s(y))
if(false(),x,y) = [7] x + [4] y + [9]
> [1] y + [0]
= y
if(true(),x,y) = [7] x + [4] y + [4]
> [1] x + [0]
= x
Following rules are (at-least) weakly oriented:
f(0()) = [0]
>= [3]
= true()
f(1()) = [0]
>= [8]
= false()
f(s(x)) = [0]
>= [0]
= f(x)
g(x,c(y)) = [1] y + [15]
>= [1] y + [30]
= g(x,g(s(c(y)),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:
f(0()) -> true()
f(1()) -> false()
f(s(x)) -> f(x)
g(x,c(y)) -> g(x,g(s(c(y)),y))
- Weak TRS:
g(s(x),s(y)) -> if(f(x),s(x),s(y))
if(false(),x,y) -> y
if(true(),x,y) -> x
- Signature:
{f/1,g/2,if/3} / {0/0,1/0,c/1,false/0,s/1,true/0}
- Obligation:
runtime complexity wrt. defined symbols {f,g,if} and constructors {0,1,c,false,s,true}
+ 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(g) = {2},
uargs(if) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(1) = [1]
p(c) = [1] x1 + [10]
p(f) = [4]
p(false) = [4]
p(g) = [1] x2 + [9]
p(if) = [1] x1 + [1] x2 + [2] x3 + [5]
p(s) = [0]
p(true) = [0]
Following rules are strictly oriented:
f(0()) = [4]
> [0]
= true()
g(x,c(y)) = [1] y + [19]
> [1] y + [18]
= g(x,g(s(c(y)),y))
Following rules are (at-least) weakly oriented:
f(1()) = [4]
>= [4]
= false()
f(s(x)) = [4]
>= [4]
= f(x)
g(s(x),s(y)) = [9]
>= [9]
= if(f(x),s(x),s(y))
if(false(),x,y) = [1] x + [2] y + [9]
>= [1] y + [0]
= y
if(true(),x,y) = [1] x + [2] y + [5]
>= [1] x + [0]
= x
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 3: WeightGap WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
f(1()) -> false()
f(s(x)) -> f(x)
- Weak TRS:
f(0()) -> true()
g(x,c(y)) -> g(x,g(s(c(y)),y))
g(s(x),s(y)) -> if(f(x),s(x),s(y))
if(false(),x,y) -> y
if(true(),x,y) -> x
- Signature:
{f/1,g/2,if/3} / {0/0,1/0,c/1,false/0,s/1,true/0}
- Obligation:
runtime complexity wrt. defined symbols {f,g,if} and constructors {0,1,c,false,s,true}
+ 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(g) = {2},
uargs(if) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(1) = [0]
p(c) = [1] x1 + [8]
p(f) = [4]
p(false) = [0]
p(g) = [1] x2 + [8]
p(if) = [1] x1 + [2] x2 + [2] x3 + [0]
p(s) = [0]
p(true) = [4]
Following rules are strictly oriented:
f(1()) = [4]
> [0]
= false()
Following rules are (at-least) weakly oriented:
f(0()) = [4]
>= [4]
= true()
f(s(x)) = [4]
>= [4]
= f(x)
g(x,c(y)) = [1] y + [16]
>= [1] y + [16]
= g(x,g(s(c(y)),y))
g(s(x),s(y)) = [8]
>= [4]
= if(f(x),s(x),s(y))
if(false(),x,y) = [2] x + [2] y + [0]
>= [1] y + [0]
= y
if(true(),x,y) = [2] x + [2] y + [4]
>= [1] x + [0]
= x
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 4: MI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
f(s(x)) -> f(x)
- Weak TRS:
f(0()) -> true()
f(1()) -> false()
g(x,c(y)) -> g(x,g(s(c(y)),y))
g(s(x),s(y)) -> if(f(x),s(x),s(y))
if(false(),x,y) -> y
if(true(),x,y) -> x
- Signature:
{f/1,g/2,if/3} / {0/0,1/0,c/1,false/0,s/1,true/0}
- Obligation:
runtime complexity wrt. defined symbols {f,g,if} and constructors {0,1,c,false,s,true}
+ 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(g) = {2},
uargs(if) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [2]
[2]
p(1) = [0]
[1]
p(c) = [1 0] x_1 + [2]
[0 0] [0]
p(f) = [0 2] x_1 + [0]
[0 1] [2]
p(false) = [2]
[3]
p(g) = [3 0] x_1 + [1 0] x_2 + [0]
[4 3] [4 0] [2]
p(if) = [2 0] x_1 + [1 0] x_2 + [1 0] x_3 + [0]
[4 1] [0 1] [0 2] [0]
p(s) = [0 2] x_1 + [0]
[0 1] [4]
p(true) = [2]
[0]
Following rules are strictly oriented:
f(s(x)) = [0 2] x + [8]
[0 1] [6]
> [0 2] x + [0]
[0 1] [2]
= f(x)
Following rules are (at-least) weakly oriented:
f(0()) = [4]
[4]
>= [2]
[0]
= true()
f(1()) = [2]
[3]
>= [2]
[3]
= false()
g(x,c(y)) = [3 0] x + [1 0] y + [2]
[4 3] [4 0] [10]
>= [3 0] x + [1 0] y + [0]
[4 3] [4 0] [2]
= g(x,g(s(c(y)),y))
g(s(x),s(y)) = [0 6] x + [0 2] y + [0]
[0 11] [0 8] [14]
>= [0 6] x + [0 2] y + [0]
[0 10] [0 2] [14]
= if(f(x),s(x),s(y))
if(false(),x,y) = [1 0] x + [1 0] y + [4]
[0 1] [0 2] [11]
>= [1 0] y + [0]
[0 1] [0]
= y
if(true(),x,y) = [1 0] x + [1 0] y + [4]
[0 1] [0 2] [8]
>= [1 0] x + [0]
[0 1] [0]
= x
* Step 5: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
f(0()) -> true()
f(1()) -> false()
f(s(x)) -> f(x)
g(x,c(y)) -> g(x,g(s(c(y)),y))
g(s(x),s(y)) -> if(f(x),s(x),s(y))
if(false(),x,y) -> y
if(true(),x,y) -> x
- Signature:
{f/1,g/2,if/3} / {0/0,1/0,c/1,false/0,s/1,true/0}
- Obligation:
runtime complexity wrt. defined symbols {f,g,if} and constructors {0,1,c,false,s,true}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(?,O(n^2))