WORST_CASE(?,O(n^1))
* Step 1: DependencyPairs WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
:(z,+(x,f(y))) -> :(g(z,y),+(x,a()))
:(+(x,y),z) -> +(:(x,z),:(y,z))
:(:(x,y),z) -> :(x,:(y,z))
- Signature:
{:/2} / {+/2,a/0,f/1,g/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {:} and constructors {+,a,f,g}
+ Applied Processor:
DependencyPairs {dpKind_ = DT}
+ Details:
We add the following dependency tuples:
Strict DPs
:#(z,+(x,f(y))) -> c_1(:#(g(z,y),+(x,a())))
:#(+(x,y),z) -> c_2(:#(x,z),:#(y,z))
:#(:(x,y),z) -> c_3(:#(x,:(y,z)),:#(y,z))
Weak DPs
and mark the set of starting terms.
* Step 2: PredecessorEstimation WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
:#(z,+(x,f(y))) -> c_1(:#(g(z,y),+(x,a())))
:#(+(x,y),z) -> c_2(:#(x,z),:#(y,z))
:#(:(x,y),z) -> c_3(:#(x,:(y,z)),:#(y,z))
- Weak TRS:
:(z,+(x,f(y))) -> :(g(z,y),+(x,a()))
:(+(x,y),z) -> +(:(x,z),:(y,z))
:(:(x,y),z) -> :(x,:(y,z))
- Signature:
{:/2,:#/2} / {+/2,a/0,f/1,g/2,c_1/1,c_2/2,c_3/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {:#} and constructors {+,a,f,g}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{1}
by application of
Pre({1}) = {2,3}.
Here rules are labelled as follows:
1: :#(z,+(x,f(y))) -> c_1(:#(g(z,y),+(x,a())))
2: :#(+(x,y),z) -> c_2(:#(x,z),:#(y,z))
3: :#(:(x,y),z) -> c_3(:#(x,:(y,z)),:#(y,z))
* Step 3: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
:#(+(x,y),z) -> c_2(:#(x,z),:#(y,z))
:#(:(x,y),z) -> c_3(:#(x,:(y,z)),:#(y,z))
- Weak DPs:
:#(z,+(x,f(y))) -> c_1(:#(g(z,y),+(x,a())))
- Weak TRS:
:(z,+(x,f(y))) -> :(g(z,y),+(x,a()))
:(+(x,y),z) -> +(:(x,z),:(y,z))
:(:(x,y),z) -> :(x,:(y,z))
- Signature:
{:/2,:#/2} / {+/2,a/0,f/1,g/2,c_1/1,c_2/2,c_3/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {:#} and constructors {+,a,f,g}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S::#(+(x,y),z) -> c_2(:#(x,z),:#(y,z))
-->_2 :#(:(x,y),z) -> c_3(:#(x,:(y,z)),:#(y,z)):2
-->_1 :#(:(x,y),z) -> c_3(:#(x,:(y,z)),:#(y,z)):2
-->_2 :#(z,+(x,f(y))) -> c_1(:#(g(z,y),+(x,a()))):3
-->_1 :#(z,+(x,f(y))) -> c_1(:#(g(z,y),+(x,a()))):3
-->_2 :#(+(x,y),z) -> c_2(:#(x,z),:#(y,z)):1
-->_1 :#(+(x,y),z) -> c_2(:#(x,z),:#(y,z)):1
2:S::#(:(x,y),z) -> c_3(:#(x,:(y,z)),:#(y,z))
-->_2 :#(z,+(x,f(y))) -> c_1(:#(g(z,y),+(x,a()))):3
-->_1 :#(z,+(x,f(y))) -> c_1(:#(g(z,y),+(x,a()))):3
-->_2 :#(:(x,y),z) -> c_3(:#(x,:(y,z)),:#(y,z)):2
-->_1 :#(:(x,y),z) -> c_3(:#(x,:(y,z)),:#(y,z)):2
-->_2 :#(+(x,y),z) -> c_2(:#(x,z),:#(y,z)):1
-->_1 :#(+(x,y),z) -> c_2(:#(x,z),:#(y,z)):1
3:W::#(z,+(x,f(y))) -> c_1(:#(g(z,y),+(x,a())))
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
3: :#(z,+(x,f(y))) -> c_1(:#(g(z,y),+(x,a())))
* Step 4: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
:#(+(x,y),z) -> c_2(:#(x,z),:#(y,z))
:#(:(x,y),z) -> c_3(:#(x,:(y,z)),:#(y,z))
- Weak TRS:
:(z,+(x,f(y))) -> :(g(z,y),+(x,a()))
:(+(x,y),z) -> +(:(x,z),:(y,z))
:(:(x,y),z) -> :(x,:(y,z))
- Signature:
{:/2,:#/2} / {+/2,a/0,f/1,g/2,c_1/1,c_2/2,c_3/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {:#} and constructors {+,a,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(c_2) = {1,2},
uargs(c_3) = {1,2}
Following symbols are considered usable:
{:#}
TcT has computed the following interpretation:
p(+) = [1] x1 + [1] x2 + [8]
p(:) = [1] x1 + [2] x2 + [0]
p(a) = [1]
p(f) = [1] x1 + [0]
p(g) = [1] x1 + [1] x2 + [0]
p(:#) = [1] x1 + [0]
p(c_1) = [1]
p(c_2) = [1] x1 + [1] x2 + [4]
p(c_3) = [1] x1 + [2] x2 + [0]
Following rules are strictly oriented:
:#(+(x,y),z) = [1] x + [1] y + [8]
> [1] x + [1] y + [4]
= c_2(:#(x,z),:#(y,z))
Following rules are (at-least) weakly oriented:
:#(:(x,y),z) = [1] x + [2] y + [0]
>= [1] x + [2] y + [0]
= c_3(:#(x,:(y,z)),:#(y,z))
* Step 5: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
:#(:(x,y),z) -> c_3(:#(x,:(y,z)),:#(y,z))
- Weak DPs:
:#(+(x,y),z) -> c_2(:#(x,z),:#(y,z))
- Weak TRS:
:(z,+(x,f(y))) -> :(g(z,y),+(x,a()))
:(+(x,y),z) -> +(:(x,z),:(y,z))
:(:(x,y),z) -> :(x,:(y,z))
- Signature:
{:/2,:#/2} / {+/2,a/0,f/1,g/2,c_1/1,c_2/2,c_3/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {:#} and constructors {+,a,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(c_2) = {1,2},
uargs(c_3) = {1,2}
Following symbols are considered usable:
{:#}
TcT has computed the following interpretation:
p(+) = [1] x1 + [1] x2 + [1]
p(:) = [2] x1 + [2] x2 + [8]
p(a) = [0]
p(f) = [0]
p(g) = [4]
p(:#) = [2] x1 + [0]
p(c_1) = [2] x1 + [0]
p(c_2) = [1] x1 + [1] x2 + [2]
p(c_3) = [2] x1 + [1] x2 + [1]
Following rules are strictly oriented:
:#(:(x,y),z) = [4] x + [4] y + [16]
> [4] x + [2] y + [1]
= c_3(:#(x,:(y,z)),:#(y,z))
Following rules are (at-least) weakly oriented:
:#(+(x,y),z) = [2] x + [2] y + [2]
>= [2] x + [2] y + [2]
= c_2(:#(x,z),:#(y,z))
* Step 6: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
:#(+(x,y),z) -> c_2(:#(x,z),:#(y,z))
:#(:(x,y),z) -> c_3(:#(x,:(y,z)),:#(y,z))
- Weak TRS:
:(z,+(x,f(y))) -> :(g(z,y),+(x,a()))
:(+(x,y),z) -> +(:(x,z),:(y,z))
:(:(x,y),z) -> :(x,:(y,z))
- Signature:
{:/2,:#/2} / {+/2,a/0,f/1,g/2,c_1/1,c_2/2,c_3/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {:#} and constructors {+,a,f,g}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(?,O(n^1))