Certification Problem

Input (TPDB Runtime_Complexity_Innermost_Rewriting/Frederiksen_Glenstrup/naiverev)

The rewrite relation of the following TRS is considered.

naiverev(Cons(x,xs))	→	app(naiverev(xs),Cons(x,Nil))	(1)
app(Cons(x,xs),ys)	→	Cons(x,app(xs,ys))	(2)
notEmpty(Cons(x,xs))	→	True	(3)
notEmpty(Nil)	→	False	(4)
naiverev(Nil)	→	Nil	(5)
app(Nil,ys)	→	ys	(6)
goal(xs)	→	naiverev(xs)	(7)

The evaluation strategy is innermost.

Property / Task

Determine bounds on the runtime complexity.

Answer / Result

An upperbound for the complexity is O(n²).

Proof (by AProVE @ termCOMP 2023)

1 Dependency Tuples

We get the following set of dependency tuples:

naiverev^#(Cons(z0,z1))

→

c(app^#(naiverev(z1),Cons(z0,Nil)),naiverev^#(z1))

(9)

originates from

naiverev(Cons(z0,z1))

→

app(naiverev(z1),Cons(z0,Nil))

(8)

naiverev^#(Nil)

→

(10)

originates from

naiverev(Nil)

→

Nil

(5)

app^#(Cons(z0,z1),z2)

→

c2(app^#(z1,z2))

(12)

originates from

app(Cons(z0,z1),z2)

→

Cons(z0,app(z1,z2))

(11)

app^#(Nil,z0)

→

(14)

originates from

app(Nil,z0)

→

(13)

notEmpty^#(Cons(z0,z1))

→

(16)

originates from

notEmpty(Cons(z0,z1))

→

True

(15)

notEmpty^#(Nil)

→

(17)

originates from

notEmpty(Nil)

→

False

(4)

goal^#(z0)

→

c6(naiverev^#(z0))

(19)

originates from

goal(z0)

→

naiverev(z0)

(18)

Moreover, we add the following terms to the innermost strategy.

naiverev^#(Cons(z0,z1))

naiverev^#(Nil)

app^#(Cons(z0,z1),z2)

app^#(Nil,z0)

notEmpty^#(Cons(z0,z1))

notEmpty^#(Nil)

goal^#(z0)

1.1 Usable Rules

We remove the following rules since they are not usable.

notEmpty(Cons(z0,z1))	→	True	(15)
notEmpty(Nil)	→	False	(4)
goal(z0)	→	naiverev(z0)	(18)

1.1.1 Rule Shifting

The rules

goal^#(z0)

→

c6(naiverev^#(z0))

(19)

are strictly oriented by the following non-linear polynomial interpretation over the naturals

[c(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c1]	=	0
[c2(x₁)]	=	1 · x₁ + 0
[c3]	=	0
[c4]	=	0
[c5]	=	0
[c6(x₁)]	=	1 · x₁ + 0
[naiverev(x₁)]	=	0
[app(x₁, x₂)]	=	1
[naiverev^#(x₁)]	=	0
[app^#(x₁, x₂)]	=	0
[notEmpty^#(x₁)]	=	0
[goal^#(x₁)]	=	2 + 1 · x₁ + 2 · x₁ · x₁
[Cons(x₁, x₂)]	=	0
[Nil]	=	0

which has the intended complexity. Here, only the following usable rules have been considered:

naiverev^#(Cons(z0,z1))	→	c(app^#(naiverev(z1),Cons(z0,Nil)),naiverev^#(z1))	(9)
naiverev^#(Nil)	→	c1	(10)
app^#(Cons(z0,z1),z2)	→	c2(app^#(z1,z2))	(12)
app^#(Nil,z0)	→	c3	(14)
notEmpty^#(Cons(z0,z1))	→	c4	(16)
notEmpty^#(Nil)	→	c5	(17)
goal^#(z0)	→	c6(naiverev^#(z0))	(19)

1.1.1.1 Rule Shifting

The rules

naiverev^#(Nil)	→	c1	(10)
app^#(Nil,z0)	→	c3	(14)
notEmpty^#(Cons(z0,z1))	→	c4	(16)
notEmpty^#(Nil)	→	c5	(17)

are strictly oriented by the following linear polynomial interpretation over the naturals

[c(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c1]	=	0
[c2(x₁)]	=	1 · x₁ + 0
[c3]	=	0
[c4]	=	0
[c5]	=	0
[c6(x₁)]	=	1 · x₁ + 0
[naiverev(x₁)]	=	1 + 1 · x₁
[app(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[naiverev^#(x₁)]	=	1 + 1 · x₁
[app^#(x₁, x₂)]	=	1
[notEmpty^#(x₁)]	=	1 + 1 · x₁
[goal^#(x₁)]	=	1 + 1 · x₁
[Cons(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[Nil]	=	0

which has the intended complexity. Here, only the following usable rules have been considered:

naiverev^#(Cons(z0,z1))	→	c(app^#(naiverev(z1),Cons(z0,Nil)),naiverev^#(z1))	(9)
naiverev^#(Nil)	→	c1	(10)
app^#(Cons(z0,z1),z2)	→	c2(app^#(z1,z2))	(12)
app^#(Nil,z0)	→	c3	(14)
notEmpty^#(Cons(z0,z1))	→	c4	(16)
notEmpty^#(Nil)	→	c5	(17)
goal^#(z0)	→	c6(naiverev^#(z0))	(19)

1.1.1.1.1 Rule Shifting

The rules

naiverev^#(Cons(z0,z1))

→

c(app^#(naiverev(z1),Cons(z0,Nil)),naiverev^#(z1))

(9)

are strictly oriented by the following linear polynomial interpretation over the naturals

[c(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c1]	=	0
[c2(x₁)]	=	1 · x₁ + 0
[c3]	=	0
[c4]	=	0
[c5]	=	0
[c6(x₁)]	=	1 · x₁ + 0
[naiverev(x₁)]	=	3
[app(x₁, x₂)]	=	3
[naiverev^#(x₁)]	=	3 + 1 · x₁
[app^#(x₁, x₂)]	=	0
[notEmpty^#(x₁)]	=	0
[goal^#(x₁)]	=	3 + 1 · x₁
[Cons(x₁, x₂)]	=	2 + 1 · x₁ + 1 · x₂
[Nil]	=	0

which has the intended complexity. Here, only the following usable rules have been considered:

naiverev^#(Cons(z0,z1))	→	c(app^#(naiverev(z1),Cons(z0,Nil)),naiverev^#(z1))	(9)
naiverev^#(Nil)	→	c1	(10)
app^#(Cons(z0,z1),z2)	→	c2(app^#(z1,z2))	(12)
app^#(Nil,z0)	→	c3	(14)
notEmpty^#(Cons(z0,z1))	→	c4	(16)
notEmpty^#(Nil)	→	c5	(17)
goal^#(z0)	→	c6(naiverev^#(z0))	(19)

1.1.1.1.1.1 Rule Shifting

The rules

app^#(Cons(z0,z1),z2)

→

c2(app^#(z1,z2))

(12)

are strictly oriented by the following non-linear polynomial interpretation over the naturals

[c(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c1]	=	0
[c2(x₁)]	=	1 · x₁ + 0
[c3]	=	0
[c4]	=	0
[c5]	=	0
[c6(x₁)]	=	1 · x₁ + 0
[naiverev(x₁)]	=	1 · x₁ + 0
[app(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[naiverev^#(x₁)]	=	1 + 1 · x₁ + 1 · x₁ · x₁
[app^#(x₁, x₂)]	=	1 · x₁ + 0
[notEmpty^#(x₁)]	=	0
[goal^#(x₁)]	=	1 + 1 · x₁ + 1 · x₁ · x₁
[Cons(x₁, x₂)]	=	1 + 1 · x₂
[Nil]	=	0

which has the intended complexity. Here, only the following usable rules have been considered:

naiverev^#(Cons(z0,z1))	→	c(app^#(naiverev(z1),Cons(z0,Nil)),naiverev^#(z1))	(9)
naiverev^#(Nil)	→	c1	(10)
app^#(Cons(z0,z1),z2)	→	c2(app^#(z1,z2))	(12)
app^#(Nil,z0)	→	c3	(14)
notEmpty^#(Cons(z0,z1))	→	c4	(16)
notEmpty^#(Nil)	→	c5	(17)
goal^#(z0)	→	c6(naiverev^#(z0))	(19)
naiverev(Nil)	→	Nil	(5)
naiverev(Cons(z0,z1))	→	app(naiverev(z1),Cons(z0,Nil))	(8)
app(Nil,z0)	→	z0	(13)
app(Cons(z0,z1),z2)	→	Cons(z0,app(z1,z2))	(11)

1.1.1.1.1.1.1 R is empty

There are no rules in the TRS R. Hence, R/S has complexity O(1).