Certification Problem

Input (TPDB Runtime_Complexity_Innermost_Rewriting/AG01/#3.12)

The rewrite relation of the following TRS is considered.

app(nil,y)	→	y	(1)
app(add(n,x),y)	→	add(n,app(x,y))	(2)
reverse(nil)	→	nil	(3)
reverse(add(n,x))	→	app(reverse(x),add(n,nil))	(4)
shuffle(nil)	→	nil	(5)
shuffle(add(n,x))	→	add(n,shuffle(reverse(x)))	(6)

The evaluation strategy is innermost.

Property / Task

Determine bounds on the runtime complexity.

Answer / Result

An upperbound for the complexity is O(n³).

Proof (by AProVE @ termCOMP 2023)

1 Dependency Tuples

We get the following set of dependency tuples:

app^#(nil,z0)

→

(8)

originates from

app(nil,z0)

→

(7)

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

→

c1(app^#(z1,z2))

(10)

originates from

app(add(z0,z1),z2)

→

add(z0,app(z1,z2))

(9)

reverse^#(nil)

→

(11)

originates from

reverse(nil)

→

nil

(3)

reverse^#(add(z0,z1))

→

c3(app^#(reverse(z1),add(z0,nil)),reverse^#(z1))

(13)

originates from

reverse(add(z0,z1))

→

app(reverse(z1),add(z0,nil))

(12)

shuffle^#(nil)

→

(14)

originates from

shuffle(nil)

→

nil

(5)

shuffle^#(add(z0,z1))

→

c5(shuffle^#(reverse(z1)),reverse^#(z1))

(16)

originates from

shuffle(add(z0,z1))

→

add(z0,shuffle(reverse(z1)))

(15)

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

app^#(nil,z0)

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

reverse^#(nil)

reverse^#(add(z0,z1))

shuffle^#(nil)

shuffle^#(add(z0,z1))

1.1 Usable Rules

We remove the following rules since they are not usable.

shuffle(nil)	→	nil	(5)
shuffle(add(z0,z1))	→	add(z0,shuffle(reverse(z1)))	(15)

1.1.1 Rule Shifting

The rules

shuffle^#(nil)

→

(14)

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

[c]	=	0
[c1(x₁)]	=	1 · x₁ + 0
[c2]	=	0
[c3(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c4]	=	0
[c5(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[reverse(x₁)]	=	1
[app(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[app^#(x₁, x₂)]	=	0
[reverse^#(x₁)]	=	0
[shuffle^#(x₁)]	=	1
[nil]	=	1
[add(x₁, x₂)]	=	1 + 1 · x₁

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

app^#(nil,z0)	→	c	(8)
app^#(add(z0,z1),z2)	→	c1(app^#(z1,z2))	(10)
reverse^#(nil)	→	c2	(11)
reverse^#(add(z0,z1))	→	c3(app^#(reverse(z1),add(z0,nil)),reverse^#(z1))	(13)
shuffle^#(nil)	→	c4	(14)
shuffle^#(add(z0,z1))	→	c5(shuffle^#(reverse(z1)),reverse^#(z1))	(16)

1.1.1.1 Rule Shifting

The rules

reverse^#(nil)

→

(11)

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

[c]	=	0
[c1(x₁)]	=	1 · x₁ + 0
[c2]	=	0
[c3(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c4]	=	0
[c5(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[reverse(x₁)]	=	1 · x₁ + 0
[app(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[app^#(x₁, x₂)]	=	0
[reverse^#(x₁)]	=	1
[shuffle^#(x₁)]	=	1 · x₁ + 0
[nil]	=	0
[add(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂

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

app^#(nil,z0)	→	c	(8)
app^#(add(z0,z1),z2)	→	c1(app^#(z1,z2))	(10)
reverse^#(nil)	→	c2	(11)
reverse^#(add(z0,z1))	→	c3(app^#(reverse(z1),add(z0,nil)),reverse^#(z1))	(13)
shuffle^#(nil)	→	c4	(14)
shuffle^#(add(z0,z1))	→	c5(shuffle^#(reverse(z1)),reverse^#(z1))	(16)
reverse(nil)	→	nil	(3)
app(add(z0,z1),z2)	→	add(z0,app(z1,z2))	(9)
app(nil,z0)	→	z0	(7)
reverse(add(z0,z1))	→	app(reverse(z1),add(z0,nil))	(12)

1.1.1.1.1 Rule Shifting

The rules

shuffle^#(add(z0,z1))

→

c5(shuffle^#(reverse(z1)),reverse^#(z1))

(16)

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

[c]	=	0
[c1(x₁)]	=	1 · x₁ + 0
[c2]	=	0
[c3(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c4]	=	0
[c5(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[reverse(x₁)]	=	1 · x₁ + 0
[app(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[app^#(x₁, x₂)]	=	0
[reverse^#(x₁)]	=	0
[shuffle^#(x₁)]	=	1 · x₁ + 0
[nil]	=	0
[add(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂

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

app^#(nil,z0)	→	c	(8)
app^#(add(z0,z1),z2)	→	c1(app^#(z1,z2))	(10)
reverse^#(nil)	→	c2	(11)
reverse^#(add(z0,z1))	→	c3(app^#(reverse(z1),add(z0,nil)),reverse^#(z1))	(13)
shuffle^#(nil)	→	c4	(14)
shuffle^#(add(z0,z1))	→	c5(shuffle^#(reverse(z1)),reverse^#(z1))	(16)
reverse(nil)	→	nil	(3)
app(add(z0,z1),z2)	→	add(z0,app(z1,z2))	(9)
app(nil,z0)	→	z0	(7)
reverse(add(z0,z1))	→	app(reverse(z1),add(z0,nil))	(12)

1.1.1.1.1.1 Rule Shifting

The rules

reverse^#(add(z0,z1))

→

c3(app^#(reverse(z1),add(z0,nil)),reverse^#(z1))

(13)

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

[c]	=	0
[c1(x₁)]	=	1 · x₁ + 0
[c2]	=	0
[c3(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c4]	=	0
[c5(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[reverse(x₁)]	=	1 · x₁ + 0
[app(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[app^#(x₁, x₂)]	=	0
[reverse^#(x₁)]	=	1 · x₁ + 0
[shuffle^#(x₁)]	=	2 · x₁ · x₁ + 0
[nil]	=	0
[add(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂

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

app^#(nil,z0)	→	c	(8)
app^#(add(z0,z1),z2)	→	c1(app^#(z1,z2))	(10)
reverse^#(nil)	→	c2	(11)
reverse^#(add(z0,z1))	→	c3(app^#(reverse(z1),add(z0,nil)),reverse^#(z1))	(13)
shuffle^#(nil)	→	c4	(14)
shuffle^#(add(z0,z1))	→	c5(shuffle^#(reverse(z1)),reverse^#(z1))	(16)
reverse(nil)	→	nil	(3)
app(add(z0,z1),z2)	→	add(z0,app(z1,z2))	(9)
app(nil,z0)	→	z0	(7)
reverse(add(z0,z1))	→	app(reverse(z1),add(z0,nil))	(12)

1.1.1.1.1.1.1 Rule Shifting

The rules

app^#(nil,z0)

→

(8)

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

[c]	=	0
[c1(x₁)]	=	1 · x₁ + 0
[c2]	=	0
[c3(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c4]	=	0
[c5(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[reverse(x₁)]	=	1 · x₁ + 0
[app(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[app^#(x₁, x₂)]	=	2
[reverse^#(x₁)]	=	2 + 1 · x₁
[shuffle^#(x₁)]	=	2 · x₁ · x₁ + 0
[nil]	=	0
[add(x₁, x₂)]	=	2 + 1 · x₂

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

app^#(nil,z0)	→	c	(8)
app^#(add(z0,z1),z2)	→	c1(app^#(z1,z2))	(10)
reverse^#(nil)	→	c2	(11)
reverse^#(add(z0,z1))	→	c3(app^#(reverse(z1),add(z0,nil)),reverse^#(z1))	(13)
shuffle^#(nil)	→	c4	(14)
shuffle^#(add(z0,z1))	→	c5(shuffle^#(reverse(z1)),reverse^#(z1))	(16)
reverse(nil)	→	nil	(3)
app(add(z0,z1),z2)	→	add(z0,app(z1,z2))	(9)
app(nil,z0)	→	z0	(7)
reverse(add(z0,z1))	→	app(reverse(z1),add(z0,nil))	(12)

1.1.1.1.1.1.1.1 Rule Shifting

The rules

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

→

c1(app^#(z1,z2))

(10)

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

[c]	=	0
[c1(x₁)]	=	1 · x₁ + 0
[c2]	=	0
[c3(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c4]	=	0
[c5(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[reverse(x₁)]	=	1 · x₁ + 0
[app(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[app^#(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂ · x₂ · x₁
[reverse^#(x₁)]	=	1 · x₁ · x₁ + 0
[shuffle^#(x₁)]	=	1 · x₁ · x₁ · x₁ + 0
[nil]	=	0
[add(x₁, x₂)]	=	1 + 1 · x₂

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

app^#(nil,z0)	→	c	(8)
app^#(add(z0,z1),z2)	→	c1(app^#(z1,z2))	(10)
reverse^#(nil)	→	c2	(11)
reverse^#(add(z0,z1))	→	c3(app^#(reverse(z1),add(z0,nil)),reverse^#(z1))	(13)
shuffle^#(nil)	→	c4	(14)
shuffle^#(add(z0,z1))	→	c5(shuffle^#(reverse(z1)),reverse^#(z1))	(16)
reverse(nil)	→	nil	(3)
app(add(z0,z1),z2)	→	add(z0,app(z1,z2))	(9)
app(nil,z0)	→	z0	(7)
reverse(add(z0,z1))	→	app(reverse(z1),add(z0,nil))	(12)

1.1.1.1.1.1.1.1.1 R is empty

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