Certification Problem

Input (TPDB Runtime_Complexity_Innermost_Rewriting/Frederiksen_Glenstrup/map0)

The relative rewrite relation R/S is considered where R is the following TRS

map(Cons(x,xs))	→	Cons(f(x),map(xs))	(1)
map(Nil)	→	Nil	(2)
goal(xs)	→	map(xs)	(3)
f(x)	→	*(x,x)	(4)
+Full(S(x),y)	→	+Full(x,S(y))	(5)
+Full(0,y)	→	y	(6)

and S is the following TRS.

*(x,S(S(y)))	→	+(x,*(x,S(y)))	(7)
*(x,S(0))	→	x	(8)
*(x,0)	→	0	(9)
*(0,y)	→	0	(10)

The evaluation strategy is innermost.

Property / Task

Determine bounds on the runtime complexity.

Answer / Result

An upperbound for the complexity is O(n).

Proof (by AProVE @ termCOMP 2023)

1 Dependency Tuples

We get the following set of dependency tuples:

map^#(Cons(z0,z1))

→

c4(f^#(z0),map^#(z1))

(12)

originates from

map(Cons(z0,z1))

→

Cons(f(z0),map(z1))

(11)

map^#(Nil)

→

(13)

originates from

map(Nil)

→

Nil

(2)

goal^#(z0)

→

c6(map^#(z0))

(15)

originates from

goal(z0)

→

map(z0)

(14)

f^#(z0)

→

c7(*^#(z0,z0))

(17)

originates from

f(z0)

→

*(z0,z0)

(16)

+Full^#(S(z0),z1)

→

c8(+Full^#(z0,S(z1)))

(19)

originates from

+Full(S(z0),z1)

→

+Full(z0,S(z1))

(18)

+Full^#(0,z0)

→

(21)

originates from

+Full(0,z0)

→

(20)

*^#(z0,S(S(z1)))

→

c(*^#(z0,S(z1)))

(23)

originates from

*(z0,S(S(z1)))

→

+(z0,*(z0,S(z1)))

(22)

*^#(z0,S(0))

→

(25)

originates from

*(z0,S(0))

→

(24)

*^#(z0,0)

→

(27)

originates from

*(z0,0)

→

(26)

*^#(0,z0)

→

(29)

originates from

*(0,z0)

→

(28)

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

*^#(z0,S(S(z1)))

*^#(z0,S(0))

*^#(z0,0)

*^#(0,z0)

map^#(Cons(z0,z1))

map^#(Nil)

goal^#(z0)

f^#(z0)

+Full^#(S(z0),z1)

+Full^#(0,z0)

1.1 Usable Rules

We remove the following rules since they are not usable.

*(z0,S(S(z1)))	→	+(z0,*(z0,S(z1)))	(22)
*(z0,S(0))	→	z0	(24)
*(z0,0)	→	0	(26)
*(0,z0)	→	0	(28)
map(Cons(z0,z1))	→	Cons(f(z0),map(z1))	(11)
map(Nil)	→	Nil	(2)
goal(z0)	→	map(z0)	(14)
f(z0)	→	*(z0,z0)	(16)
+Full(S(z0),z1)	→	+Full(z0,S(z1))	(18)
+Full(0,z0)	→	z0	(20)

1.1.1 Rule Shifting

The rules

map^#(Cons(z0,z1))	→	c4(f^#(z0),map^#(z1))	(12)
goal^#(z0)	→	c6(map^#(z0))	(15)
+Full^#(S(z0),z1)	→	c8(+Full^#(z0,S(z1)))	(19)
+Full^#(0,z0)	→	c9	(21)

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

[c(x₁)]	=	1 · x₁ + 0
[c1]	=	0
[c2]	=	0
[c3]	=	0
[c4(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c5]	=	0
[c6(x₁)]	=	1 · x₁ + 0
[c7(x₁)]	=	1 · x₁ + 0
[c8(x₁)]	=	1 · x₁ + 0
[c9]	=	0
[*^#(x₁, x₂)]	=	1 · x₂ + 0
[map^#(x₁)]	=	1 · x₁ + 0
[goal^#(x₁)]	=	1 + 1 · x₁
[f^#(x₁)]	=	1 · x₁ + 0
[+Full^#(x₁, x₂)]	=	1 + 1 · x₁
[S(x₁)]	=	1 + 1 · x₁
[0]	=	1
[Cons(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[Nil]	=	0

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

*^#(z0,S(S(z1)))	→	c(*^#(z0,S(z1)))	(23)
*^#(z0,S(0))	→	c1	(25)
*^#(z0,0)	→	c2	(27)
*^#(0,z0)	→	c3	(29)
map^#(Cons(z0,z1))	→	c4(f^#(z0),map^#(z1))	(12)
map^#(Nil)	→	c5	(13)
goal^#(z0)	→	c6(map^#(z0))	(15)
f^#(z0)	→	c7(*^#(z0,z0))	(17)
+Full^#(S(z0),z1)	→	c8(+Full^#(z0,S(z1)))	(19)
+Full^#(0,z0)	→	c9	(21)

1.1.1.1 Rule Shifting

The rules

map^#(Nil)

→

(13)

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

[c(x₁)]	=	1 · x₁ + 0
[c1]	=	0
[c2]	=	0
[c3]	=	0
[c4(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c5]	=	0
[c6(x₁)]	=	1 · x₁ + 0
[c7(x₁)]	=	1 · x₁ + 0
[c8(x₁)]	=	1 · x₁ + 0
[c9]	=	0
[*^#(x₁, x₂)]	=	1 + 1 · x₂
[map^#(x₁)]	=	1 · x₁ + 0
[goal^#(x₁)]	=	1 · x₁ + 0
[f^#(x₁)]	=	1 + 1 · x₁
[+Full^#(x₁, x₂)]	=	1 · x₂ + 0
[S(x₁)]	=	0
[0]	=	0
[Cons(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[Nil]	=	1

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

*^#(z0,S(S(z1)))	→	c(*^#(z0,S(z1)))	(23)
*^#(z0,S(0))	→	c1	(25)
*^#(z0,0)	→	c2	(27)
*^#(0,z0)	→	c3	(29)
map^#(Cons(z0,z1))	→	c4(f^#(z0),map^#(z1))	(12)
map^#(Nil)	→	c5	(13)
goal^#(z0)	→	c6(map^#(z0))	(15)
f^#(z0)	→	c7(*^#(z0,z0))	(17)
+Full^#(S(z0),z1)	→	c8(+Full^#(z0,S(z1)))	(19)
+Full^#(0,z0)	→	c9	(21)

1.1.1.1.1 Rule Shifting

The rules

f^#(z0)

→

c7(*^#(z0,z0))

(17)

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

[c(x₁)]	=	1 · x₁ + 0
[c1]	=	0
[c2]	=	0
[c3]	=	0
[c4(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c5]	=	0
[c6(x₁)]	=	1 · x₁ + 0
[c7(x₁)]	=	1 · x₁ + 0
[c8(x₁)]	=	1 · x₁ + 0
[c9]	=	0
[*^#(x₁, x₂)]	=	2 + 1 · x₁ + 1 · x₂
[map^#(x₁)]	=	3 + 3 · x₁
[goal^#(x₁)]	=	3 + 3 · x₁
[f^#(x₁)]	=	3 + 2 · x₁
[+Full^#(x₁, x₂)]	=	2 · x₁ + 0
[S(x₁)]	=	1 + 1 · x₁
[0]	=	0
[Cons(x₁, x₂)]	=	2 + 1 · x₁ + 1 · x₂
[Nil]	=	0

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

*^#(z0,S(S(z1)))	→	c(*^#(z0,S(z1)))	(23)
*^#(z0,S(0))	→	c1	(25)
*^#(z0,0)	→	c2	(27)
*^#(0,z0)	→	c3	(29)
map^#(Cons(z0,z1))	→	c4(f^#(z0),map^#(z1))	(12)
map^#(Nil)	→	c5	(13)
goal^#(z0)	→	c6(map^#(z0))	(15)
f^#(z0)	→	c7(*^#(z0,z0))	(17)
+Full^#(S(z0),z1)	→	c8(+Full^#(z0,S(z1)))	(19)
+Full^#(0,z0)	→	c9	(21)

1.1.1.1.1.1 R is empty

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