Certification Problem

Input (TPDB Runtime_Complexity_Innermost_Rewriting/SK90/2.39)

The rewrite relation of the following TRS is considered.

rev(nil)	→	nil	(1)
rev(.(x,y))	→	++(rev(y),.(x,nil))	(2)
car(.(x,y))	→	x	(3)
cdr(.(x,y))	→	y	(4)
null(nil)	→	true	(5)
null(.(x,y))	→	false	(6)
++(nil,y)	→	y	(7)
++(.(x,y),z)	→	.(x,++(y,z))	(8)

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:

rev^#(nil)

→

(9)

originates from

rev(nil)

→

nil

(1)

rev^#(.(z0,z1))

→

c1(++^#(rev(z1),.(z0,nil)),rev^#(z1))

(11)

originates from

rev(.(z0,z1))

→

++(rev(z1),.(z0,nil))

(10)

car^#(.(z0,z1))

→

(13)

originates from

car(.(z0,z1))

→

(12)

cdr^#(.(z0,z1))

→

(15)

originates from

cdr(.(z0,z1))

→

(14)

null^#(nil)

→

(16)

originates from

null(nil)

→

true

(5)

null^#(.(z0,z1))

→

(18)

originates from

null(.(z0,z1))

→

false

(17)

++^#(nil,z0)

→

(20)

originates from

++(nil,z0)

→

(19)

++^#(.(z0,z1),z2)

→

c7(++^#(z1,z2))

(22)

originates from

++(.(z0,z1),z2)

→

.(z0,++(z1,z2))

(21)

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

rev^#(nil)

rev^#(.(z0,z1))

car^#(.(z0,z1))

cdr^#(.(z0,z1))

null^#(nil)

null^#(.(z0,z1))

++^#(nil,z0)

++^#(.(z0,z1),z2)

1.1 Usable Rules

We remove the following rules since they are not usable.

car(.(z0,z1))	→	z0	(12)
cdr(.(z0,z1))	→	z1	(14)
null(nil)	→	true	(5)
null(.(z0,z1))	→	false	(17)

1.1.1 Rule Shifting

The rules

rev^#(nil)	→	c	(9)
car^#(.(z0,z1))	→	c2	(13)
cdr^#(.(z0,z1))	→	c3	(15)
null^#(nil)	→	c4	(16)
null^#(.(z0,z1))	→	c5	(18)

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

[c]	=	0
[c1(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c2]	=	0
[c3]	=	0
[c4]	=	0
[c5]	=	0
[c6]	=	0
[c7(x₁)]	=	1 · x₁ + 0
[rev(x₁)]	=	1 + 1 · x₁
[++(x₁, x₂)]	=	1 + 1 · x₂
[rev^#(x₁)]	=	1
[car^#(x₁)]	=	1 · x₁ + 0
[cdr^#(x₁)]	=	1 · x₁ + 0
[null^#(x₁)]	=	1 + 1 · x₁
[++^#(x₁, x₂)]	=	0
[nil]	=	1
[.(x₁, x₂)]	=	1

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

rev^#(nil)	→	c	(9)
rev^#(.(z0,z1))	→	c1(++^#(rev(z1),.(z0,nil)),rev^#(z1))	(11)
car^#(.(z0,z1))	→	c2	(13)
cdr^#(.(z0,z1))	→	c3	(15)
null^#(nil)	→	c4	(16)
null^#(.(z0,z1))	→	c5	(18)
++^#(nil,z0)	→	c6	(20)
++^#(.(z0,z1),z2)	→	c7(++^#(z1,z2))	(22)

1.1.1.1 Rule Shifting

The rules

rev^#(.(z0,z1))

→

c1(++^#(rev(z1),.(z0,nil)),rev^#(z1))

(11)

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

[c]	=	0
[c1(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c2]	=	0
[c3]	=	0
[c4]	=	0
[c5]	=	0
[c6]	=	0
[c7(x₁)]	=	1 · x₁ + 0
[rev(x₁)]	=	3
[++(x₁, x₂)]	=	3
[rev^#(x₁)]	=	1 · x₁ + 0
[car^#(x₁)]	=	0
[cdr^#(x₁)]	=	0
[null^#(x₁)]	=	0
[++^#(x₁, x₂)]	=	0
[nil]	=	0
[.(x₁, x₂)]	=	2 + 1 · x₁ + 1 · x₂

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

rev^#(nil)	→	c	(9)
rev^#(.(z0,z1))	→	c1(++^#(rev(z1),.(z0,nil)),rev^#(z1))	(11)
car^#(.(z0,z1))	→	c2	(13)
cdr^#(.(z0,z1))	→	c3	(15)
null^#(nil)	→	c4	(16)
null^#(.(z0,z1))	→	c5	(18)
++^#(nil,z0)	→	c6	(20)
++^#(.(z0,z1),z2)	→	c7(++^#(z1,z2))	(22)

1.1.1.1.1 Rule Shifting

The rules

++^#(nil,z0)

→

(20)

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

[c]	=	0
[c1(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c2]	=	0
[c3]	=	0
[c4]	=	0
[c5]	=	0
[c6]	=	0
[c7(x₁)]	=	1 · x₁ + 0
[rev(x₁)]	=	1 + 1 · x₁
[++(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[rev^#(x₁)]	=	1 · x₁ + 0
[car^#(x₁)]	=	0
[cdr^#(x₁)]	=	0
[null^#(x₁)]	=	0
[++^#(x₁, x₂)]	=	1
[nil]	=	0
[.(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂

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

rev^#(nil)	→	c	(9)
rev^#(.(z0,z1))	→	c1(++^#(rev(z1),.(z0,nil)),rev^#(z1))	(11)
car^#(.(z0,z1))	→	c2	(13)
cdr^#(.(z0,z1))	→	c3	(15)
null^#(nil)	→	c4	(16)
null^#(.(z0,z1))	→	c5	(18)
++^#(nil,z0)	→	c6	(20)
++^#(.(z0,z1),z2)	→	c7(++^#(z1,z2))	(22)

1.1.1.1.1.1 Rule Shifting

The rules

++^#(.(z0,z1),z2)

→

c7(++^#(z1,z2))

(22)

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

[c]	=	0
[c1(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c2]	=	0
[c3]	=	0
[c4]	=	0
[c5]	=	0
[c6]	=	0
[c7(x₁)]	=	1 · x₁ + 0
[rev(x₁)]	=	1 · x₁ + 0
[++(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[rev^#(x₁)]	=	1 · x₁ · x₁ + 0
[car^#(x₁)]	=	0
[cdr^#(x₁)]	=	0
[null^#(x₁)]	=	0
[++^#(x₁, x₂)]	=	1 · x₁ + 0
[nil]	=	0
[.(x₁, x₂)]	=	2 + 1 · x₁ + 1 · x₂

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

rev^#(nil)	→	c	(9)
rev^#(.(z0,z1))	→	c1(++^#(rev(z1),.(z0,nil)),rev^#(z1))	(11)
car^#(.(z0,z1))	→	c2	(13)
cdr^#(.(z0,z1))	→	c3	(15)
null^#(nil)	→	c4	(16)
null^#(.(z0,z1))	→	c5	(18)
++^#(nil,z0)	→	c6	(20)
++^#(.(z0,z1),z2)	→	c7(++^#(z1,z2))	(22)
rev(.(z0,z1))	→	++(rev(z1),.(z0,nil))	(10)
++(.(z0,z1),z2)	→	.(z0,++(z1,z2))	(21)
rev(nil)	→	nil	(1)
++(nil,z0)	→	z0	(19)

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).