WORST_CASE(?,O(n^2))
* Step 1: NaturalMI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
a__f(X1,X2) -> f(X1,X2)
a__f(g(X),Y) -> a__f(mark(X),f(g(X),Y))
mark(f(X1,X2)) -> a__f(mark(X1),X2)
mark(g(X)) -> g(mark(X))
- Signature:
{a__f/2,mark/1} / {f/2,g/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__f,mark} and constructors {f,g}
+ Applied Processor:
NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(a__f) = {1},
uargs(g) = {1}
Following symbols are considered usable:
{a__f,mark}
TcT has computed the following interpretation:
p(a__f) = [1] x1 + [8]
p(f) = [1] x1 + [8]
p(g) = [1] x1 + [4]
p(mark) = [1] x1 + [0]
Following rules are strictly oriented:
a__f(g(X),Y) = [1] X + [12]
> [1] X + [8]
= a__f(mark(X),f(g(X),Y))
Following rules are (at-least) weakly oriented:
a__f(X1,X2) = [1] X1 + [8]
>= [1] X1 + [8]
= f(X1,X2)
mark(f(X1,X2)) = [1] X1 + [8]
>= [1] X1 + [8]
= a__f(mark(X1),X2)
mark(g(X)) = [1] X + [4]
>= [1] X + [4]
= g(mark(X))
* Step 2: WeightGap WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
a__f(X1,X2) -> f(X1,X2)
mark(f(X1,X2)) -> a__f(mark(X1),X2)
mark(g(X)) -> g(mark(X))
- Weak TRS:
a__f(g(X),Y) -> a__f(mark(X),f(g(X),Y))
- Signature:
{a__f/2,mark/1} / {f/2,g/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__f,mark} and constructors {f,g}
+ 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(a__f) = {1},
uargs(g) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(a__f) = [1] x1 + [1]
p(f) = [1] x1 + [2]
p(g) = [1] x1 + [2]
p(mark) = [1] x1 + [1]
Following rules are strictly oriented:
mark(f(X1,X2)) = [1] X1 + [3]
> [1] X1 + [2]
= a__f(mark(X1),X2)
Following rules are (at-least) weakly oriented:
a__f(X1,X2) = [1] X1 + [1]
>= [1] X1 + [2]
= f(X1,X2)
a__f(g(X),Y) = [1] X + [3]
>= [1] X + [2]
= a__f(mark(X),f(g(X),Y))
mark(g(X)) = [1] X + [3]
>= [1] X + [3]
= g(mark(X))
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 3: Ara WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
a__f(X1,X2) -> f(X1,X2)
mark(g(X)) -> g(mark(X))
- Weak TRS:
a__f(g(X),Y) -> a__f(mark(X),f(g(X),Y))
mark(f(X1,X2)) -> a__f(mark(X1),X2)
- Signature:
{a__f/2,mark/1} / {f/2,g/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__f,mark} and constructors {f,g}
+ Applied Processor:
Ara {heuristics_ = NoHeuristics, minDegree = 2, maxDegree = 2, araTimeout = 15, araFindStrictRules = Just 1}
+ Details:
Signatures used:
----------------
a__f :: [A(8, 7) x A(0, 0)] -(2)-> A(1, 7)
f :: [A(15, 7) x A(0, 0)] -(8)-> A(8, 7)
f :: [A(8, 7) x A(0, 0)] -(1)-> A(1, 7)
f :: [A(0, 0) x A(0, 0)] -(0)-> A(0, 0)
g :: [A(15, 7)] -(8)-> A(8, 7)
g :: [A(11, 7)] -(4)-> A(4, 7)
g :: [A(0, 0)] -(0)-> A(0, 0)
mark :: [A(8, 7)] -(7)-> A(4, 7)
Cost-free Signatures used:
--------------------------
a__f :: [A_cf(7, 0) x A_cf(0, 0)] -(7)-> A_cf(7, 0)
a__f :: [A_cf(0, 0) x A_cf(0, 0)] -(0)-> A_cf(0, 0)
a__f :: [A_cf(4, 0) x A_cf(0, 0)] -(4)-> A_cf(4, 0)
a__f :: [A_cf(3, 0) x A_cf(0, 0)] -(3)-> A_cf(3, 0)
f :: [A_cf(7, 0) x A_cf(0, 0)] -(7)-> A_cf(7, 0)
f :: [A_cf(0, 0) x A_cf(0, 0)] -(0)-> A_cf(0, 0)
f :: [A_cf(4, 0) x A_cf(0, 0)] -(4)-> A_cf(4, 0)
f :: [A_cf(3, 0) x A_cf(0, 0)] -(3)-> A_cf(3, 0)
g :: [A_cf(7, 0)] -(7)-> A_cf(7, 0)
g :: [A_cf(0, 0)] -(0)-> A_cf(0, 0)
g :: [A_cf(4, 0)] -(4)-> A_cf(4, 0)
g :: [A_cf(3, 0)] -(3)-> A_cf(3, 0)
mark :: [A_cf(7, 0)] -(0)-> A_cf(7, 0)
mark :: [A_cf(0, 0)] -(0)-> A_cf(0, 0)
mark :: [A_cf(4, 0)] -(0)-> A_cf(4, 0)
mark :: [A_cf(3, 0)] -(0)-> A_cf(3, 0)
mark :: [A_cf(7, 0)] -(1)-> A_cf(7, 0)
Base Constructor Signatures used:
---------------------------------
f_A :: [A(1, 0) x A(0, 0)] -(1)-> A(1, 0)
f_A :: [A(1, 1) x A(0, 0)] -(0)-> A(0, 1)
g_A :: [A(1, 0)] -(1)-> A(1, 0)
g_A :: [A(1, 1)] -(0)-> A(0, 1)
Following Still Strict Rules were Typed as:
-------------------------------------------
1. Strict:
a__f(X1,X2) -> f(X1,X2)
2. Weak:
mark(g(X)) -> g(mark(X))
* Step 4: Ara WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
mark(g(X)) -> g(mark(X))
- Weak TRS:
a__f(X1,X2) -> f(X1,X2)
a__f(g(X),Y) -> a__f(mark(X),f(g(X),Y))
mark(f(X1,X2)) -> a__f(mark(X1),X2)
- Signature:
{a__f/2,mark/1} / {f/2,g/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__f,mark} and constructors {f,g}
+ Applied Processor:
Ara {heuristics_ = NoHeuristics, minDegree = 2, maxDegree = 2, araTimeout = 15, araFindStrictRules = Just 1}
+ Details:
Signatures used:
----------------
a__f :: [A(12, 2) x A(0, 0)] -(2)-> A(12, 2)
f :: [A(13, 2) x A(0, 0)] -(2)-> A(13, 2)
f :: [A(12, 2) x A(0, 0)] -(2)-> A(12, 2)
f :: [A(0, 0) x A(0, 0)] -(0)-> A(0, 0)
g :: [A(15, 2)] -(13)-> A(13, 2)
g :: [A(14, 2)] -(12)-> A(12, 2)
g :: [A(0, 0)] -(0)-> A(0, 0)
mark :: [A(13, 2)] -(7)-> A(12, 2)
Cost-free Signatures used:
--------------------------
a__f :: [A_cf(2, 0) x A_cf(0, 0)] -(0)-> A_cf(2, 0)
a__f :: [A_cf(0, 0) x A_cf(0, 0)] -(0)-> A_cf(0, 0)
f :: [A_cf(2, 0) x A_cf(0, 0)] -(0)-> A_cf(2, 0)
f :: [A_cf(0, 0) x A_cf(0, 0)] -(0)-> A_cf(0, 0)
g :: [A_cf(2, 0)] -(2)-> A_cf(2, 0)
g :: [A_cf(0, 0)] -(0)-> A_cf(0, 0)
mark :: [A_cf(2, 0)] -(0)-> A_cf(2, 0)
mark :: [A_cf(0, 0)] -(0)-> A_cf(0, 0)
Base Constructor Signatures used:
---------------------------------
f_A :: [A(1, 0) x A(0, 0)] -(0)-> A(1, 0)
f_A :: [A(0, 1) x A(0, 0)] -(1)-> A(0, 1)
g_A :: [A(1, 0)] -(1)-> A(1, 0)
g_A :: [A(1, 1)] -(0)-> A(0, 1)
Following Still Strict Rules were Typed as:
-------------------------------------------
1. Strict:
mark(g(X)) -> g(mark(X))
2. Weak:
* Step 5: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
a__f(X1,X2) -> f(X1,X2)
a__f(g(X),Y) -> a__f(mark(X),f(g(X),Y))
mark(f(X1,X2)) -> a__f(mark(X1),X2)
mark(g(X)) -> g(mark(X))
- Signature:
{a__f/2,mark/1} / {f/2,g/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__f,mark} and constructors {f,g}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(?,O(n^2))