Certification Problem

Input (TPDB Runtime_Complexity_Innermost_Rewriting/Transformed_CSR_04/Ex1_Luc02b_Z)

The rewrite relation of the following TRS is considered.

from(X)	→	cons(X,n__from(s(X)))	(1)
first(0,Z)	→	nil	(2)
first(s(X),cons(Y,Z))	→	cons(Y,n__first(X,activate(Z)))	(3)
sel(0,cons(X,Z))	→	X	(4)
sel(s(X),cons(Y,Z))	→	sel(X,activate(Z))	(5)
from(X)	→	n__from(X)	(6)
first(X1,X2)	→	n__first(X1,X2)	(7)
activate(n__from(X))	→	from(X)	(8)
activate(n__first(X1,X2))	→	first(X1,X2)	(9)
activate(X)	→	X	(10)

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:

from^#(z0)

→

(12)

originates from

from(z0)

→

cons(z0,n__from(s(z0)))

(11)

from^#(z0)

→

(14)

originates from

from(z0)

→

n__from(z0)

(13)

first^#(0,z0)

→

(16)

originates from

first(0,z0)

→

nil

(15)

first^#(s(z0),cons(z1,z2))

→

c3(activate^#(z2))

(18)

originates from

first(s(z0),cons(z1,z2))

→

cons(z1,n__first(z0,activate(z2)))

(17)

first^#(z0,z1)

→

(20)

originates from

first(z0,z1)

→

n__first(z0,z1)

(19)

sel^#(0,cons(z0,z1))

→

(22)

originates from

sel(0,cons(z0,z1))

→

(21)

sel^#(s(z0),cons(z1,z2))

→

c6(sel^#(z0,activate(z2)),activate^#(z2))

(24)

originates from

sel(s(z0),cons(z1,z2))

→

sel(z0,activate(z2))

(23)

activate^#(n__from(z0))

→

c7(from^#(z0))

(26)

originates from

activate(n__from(z0))

→

from(z0)

(25)

activate^#(n__first(z0,z1))

→

c8(first^#(z0,z1))

(28)

originates from

activate(n__first(z0,z1))

→

first(z0,z1)

(27)

activate^#(z0)

→

(30)

originates from

activate(z0)

→

(29)

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

from^#(z0)

first^#(0,z0)

first^#(s(z0),cons(z1,z2))

first^#(z0,z1)

sel^#(0,cons(z0,z1))

sel^#(s(z0),cons(z1,z2))

activate^#(n__from(z0))

activate^#(n__first(z0,z1))

activate^#(z0)

1.1 Usable Rules

We remove the following rules since they are not usable.

sel(0,cons(z0,z1))	→	z0	(21)
sel(s(z0),cons(z1,z2))	→	sel(z0,activate(z2))	(23)

1.1.1 Rule Shifting

The rules

sel^#(0,cons(z0,z1))

→

(22)

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

[c]	=	0
[c1]	=	0
[c2]	=	0
[c3(x₁)]	=	1 · x₁ + 0
[c4]	=	0
[c5]	=	0
[c6(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c7(x₁)]	=	1 · x₁ + 0
[c8(x₁)]	=	1 · x₁ + 0
[c9]	=	0
[activate(x₁)]	=	3 + 3 · x₁
[from(x₁)]	=	0
[first(x₁, x₂)]	=	1 + 3 · x₁ + 3 · x₂
[from^#(x₁)]	=	0
[first^#(x₁, x₂)]	=	0
[sel^#(x₁, x₂)]	=	1
[activate^#(x₁)]	=	0
[n__from(x₁)]	=	0
[n__first(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[0]	=	3
[nil]	=	0
[s(x₁)]	=	3 + 1 · x₁
[cons(x₁, x₂)]	=	1 · x₂ + 0

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

from^#(z0)	→	c	(12)
from^#(z0)	→	c1	(14)
first^#(0,z0)	→	c2	(16)
first^#(s(z0),cons(z1,z2))	→	c3(activate^#(z2))	(18)
first^#(z0,z1)	→	c4	(20)
sel^#(0,cons(z0,z1))	→	c5	(22)
sel^#(s(z0),cons(z1,z2))	→	c6(sel^#(z0,activate(z2)),activate^#(z2))	(24)
activate^#(n__from(z0))	→	c7(from^#(z0))	(26)
activate^#(n__first(z0,z1))	→	c8(first^#(z0,z1))	(28)
activate^#(z0)	→	c9	(30)

1.1.1.1 Rule Shifting

The rules

from^#(z0)	→	c	(12)
from^#(z0)	→	c1	(14)
first^#(0,z0)	→	c2	(16)
first^#(z0,z1)	→	c4	(20)
sel^#(s(z0),cons(z1,z2))	→	c6(sel^#(z0,activate(z2)),activate^#(z2))	(24)
activate^#(z0)	→	c9	(30)

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

[c]	=	0
[c1]	=	0
[c2]	=	0
[c3(x₁)]	=	1 · x₁ + 0
[c4]	=	0
[c5]	=	0
[c6(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c7(x₁)]	=	1 · x₁ + 0
[c8(x₁)]	=	1 · x₁ + 0
[c9]	=	0
[activate(x₁)]	=	3 + 3 · x₁
[from(x₁)]	=	3
[first(x₁, x₂)]	=	2 + 3 · x₁ + 3 · x₂
[from^#(x₁)]	=	1
[first^#(x₁, x₂)]	=	1
[sel^#(x₁, x₂)]	=	1 · x₁ + 0
[activate^#(x₁)]	=	1
[n__from(x₁)]	=	0
[n__first(x₁, x₂)]	=	2 + 1 · x₁ + 1 · x₂
[0]	=	3
[nil]	=	3
[s(x₁)]	=	3 + 1 · x₁
[cons(x₁, x₂)]	=	3 + 1 · x₂

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

from^#(z0)	→	c	(12)
from^#(z0)	→	c1	(14)
first^#(0,z0)	→	c2	(16)
first^#(s(z0),cons(z1,z2))	→	c3(activate^#(z2))	(18)
first^#(z0,z1)	→	c4	(20)
sel^#(0,cons(z0,z1))	→	c5	(22)
sel^#(s(z0),cons(z1,z2))	→	c6(sel^#(z0,activate(z2)),activate^#(z2))	(24)
activate^#(n__from(z0))	→	c7(from^#(z0))	(26)
activate^#(n__first(z0,z1))	→	c8(first^#(z0,z1))	(28)
activate^#(z0)	→	c9	(30)

1.1.1.1.1 Rule Shifting

The rules

activate^#(n__from(z0))

→

c7(from^#(z0))

(26)

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

[c]	=	0
[c1]	=	0
[c2]	=	0
[c3(x₁)]	=	1 · x₁ + 0
[c4]	=	0
[c5]	=	0
[c6(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c7(x₁)]	=	1 · x₁ + 0
[c8(x₁)]	=	1 · x₁ + 0
[c9]	=	0
[activate(x₁)]	=	3
[from(x₁)]	=	3 + 3 · x₁
[first(x₁, x₂)]	=	3
[from^#(x₁)]	=	0
[first^#(x₁, x₂)]	=	1
[sel^#(x₁, x₂)]	=	2 · x₁ + 0
[activate^#(x₁)]	=	1
[n__from(x₁)]	=	3 + 1 · x₁
[n__first(x₁, x₂)]	=	1 · x₁ + 0
[0]	=	3
[nil]	=	3
[s(x₁)]	=	2 + 1 · x₁
[cons(x₁, x₂)]	=	1 · x₁ + 0

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

from^#(z0)	→	c	(12)
from^#(z0)	→	c1	(14)
first^#(0,z0)	→	c2	(16)
first^#(s(z0),cons(z1,z2))	→	c3(activate^#(z2))	(18)
first^#(z0,z1)	→	c4	(20)
sel^#(0,cons(z0,z1))	→	c5	(22)
sel^#(s(z0),cons(z1,z2))	→	c6(sel^#(z0,activate(z2)),activate^#(z2))	(24)
activate^#(n__from(z0))	→	c7(from^#(z0))	(26)
activate^#(n__first(z0,z1))	→	c8(first^#(z0,z1))	(28)
activate^#(z0)	→	c9	(30)

1.1.1.1.1.1 Rule Shifting

The rules

activate^#(n__first(z0,z1))

→

c8(first^#(z0,z1))

(28)

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

[c]	=	0
[c1]	=	0
[c2]	=	0
[c3(x₁)]	=	1 · x₁ + 0
[c4]	=	0
[c5]	=	0
[c6(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c7(x₁)]	=	1 · x₁ + 0
[c8(x₁)]	=	1 · x₁ + 0
[c9]	=	0
[activate(x₁)]	=	1 · x₁ + 0
[from(x₁)]	=	1
[first(x₁, x₂)]	=	1 + 1 · x₂
[from^#(x₁)]	=	1
[first^#(x₁, x₂)]	=	1 · x₂ + 0
[sel^#(x₁, x₂)]	=	1 · x₂ · x₁ · x₁ + 0
[activate^#(x₁)]	=	1 · x₁ + 0
[n__from(x₁)]	=	1
[n__first(x₁, x₂)]	=	1 + 1 · x₂
[0]	=	1
[nil]	=	1
[s(x₁)]	=	1 + 1 · x₁
[cons(x₁, x₂)]	=	1 · x₂ + 0

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

from^#(z0)	→	c	(12)
from^#(z0)	→	c1	(14)
first^#(0,z0)	→	c2	(16)
first^#(s(z0),cons(z1,z2))	→	c3(activate^#(z2))	(18)
first^#(z0,z1)	→	c4	(20)
sel^#(0,cons(z0,z1))	→	c5	(22)
sel^#(s(z0),cons(z1,z2))	→	c6(sel^#(z0,activate(z2)),activate^#(z2))	(24)
activate^#(n__from(z0))	→	c7(from^#(z0))	(26)
activate^#(n__first(z0,z1))	→	c8(first^#(z0,z1))	(28)
activate^#(z0)	→	c9	(30)
first(z0,z1)	→	n__first(z0,z1)	(19)
from(z0)	→	cons(z0,n__from(s(z0)))	(11)
first(s(z0),cons(z1,z2))	→	cons(z1,n__first(z0,activate(z2)))	(17)
first(0,z0)	→	nil	(15)
from(z0)	→	n__from(z0)	(13)
activate(n__from(z0))	→	from(z0)	(25)
activate(z0)	→	z0	(29)
activate(n__first(z0,z1))	→	first(z0,z1)	(27)

1.1.1.1.1.1.1 Rule Shifting

The rules

first^#(s(z0),cons(z1,z2))

→

c3(activate^#(z2))

(18)

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

[c]	=	0
[c1]	=	0
[c2]	=	0
[c3(x₁)]	=	1 · x₁ + 0
[c4]	=	0
[c5]	=	0
[c6(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c7(x₁)]	=	1 · x₁ + 0
[c8(x₁)]	=	1 · x₁ + 0
[c9]	=	0
[activate(x₁)]	=	2 + 1 · x₁
[from(x₁)]	=	0
[first(x₁, x₂)]	=	2 + 1 · x₁ + 1 · x₂
[from^#(x₁)]	=	0
[first^#(x₁, x₂)]	=	1 + 2 · x₂
[sel^#(x₁, x₂)]	=	1 · x₁ · x₂ + 0 + 2 · x₁ · x₁
[activate^#(x₁)]	=	2 · x₁ + 0
[n__from(x₁)]	=	0
[n__first(x₁, x₂)]	=	2 + 1 · x₁ + 1 · x₂
[0]	=	2
[nil]	=	1
[s(x₁)]	=	2 + 1 · x₁
[cons(x₁, x₂)]	=	1 · x₂ + 0

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

from^#(z0)	→	c	(12)
from^#(z0)	→	c1	(14)
first^#(0,z0)	→	c2	(16)
first^#(s(z0),cons(z1,z2))	→	c3(activate^#(z2))	(18)
first^#(z0,z1)	→	c4	(20)
sel^#(0,cons(z0,z1))	→	c5	(22)
sel^#(s(z0),cons(z1,z2))	→	c6(sel^#(z0,activate(z2)),activate^#(z2))	(24)
activate^#(n__from(z0))	→	c7(from^#(z0))	(26)
activate^#(n__first(z0,z1))	→	c8(first^#(z0,z1))	(28)
activate^#(z0)	→	c9	(30)
first(z0,z1)	→	n__first(z0,z1)	(19)
from(z0)	→	cons(z0,n__from(s(z0)))	(11)
first(s(z0),cons(z1,z2))	→	cons(z1,n__first(z0,activate(z2)))	(17)
first(0,z0)	→	nil	(15)
from(z0)	→	n__from(z0)	(13)
activate(n__from(z0))	→	from(z0)	(25)
activate(z0)	→	z0	(29)
activate(n__first(z0,z1))	→	first(z0,z1)	(27)

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).