Certification Problem

Input (TPDB Runtime_Complexity_Innermost_Rewriting/AG01/#3.15)

The rewrite relation of the following TRS is considered.

average(s(x),y)	→	average(x,s(y))	(1)
average(x,s(s(s(y))))	→	s(average(s(x),y))	(2)
average(0,0)	→	0	(3)
average(0,s(0))	→	0	(4)
average(0,s(s(0)))	→	s(0)	(5)

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:

average^#(s(z0),z1)

→

c(average^#(z0,s(z1)))

(7)

originates from

average(s(z0),z1)

→

average(z0,s(z1))

(6)

average^#(z0,s(s(s(z1))))

→

c1(average^#(s(z0),z1))

(9)

originates from

average(z0,s(s(s(z1))))

→

s(average(s(z0),z1))

(8)

average^#(0,0)

→

(10)

originates from

average(0,0)

→

(3)

average^#(0,s(0))

→

(11)

originates from

average(0,s(0))

→

(4)

average^#(0,s(s(0)))

→

(12)

originates from

average(0,s(s(0)))

→

s(0)

(5)

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

average^#(s(z0),z1)

average^#(z0,s(s(s(z1))))

average^#(0,0)

average^#(0,s(0))

average^#(0,s(s(0)))

1.1 Usable Rules

We remove the following rules since they are not usable.

average(s(z0),z1)	→	average(z0,s(z1))	(6)
average(z0,s(s(s(z1))))	→	s(average(s(z0),z1))	(8)
average(0,0)	→	0	(3)
average(0,s(0))	→	0	(4)
average(0,s(s(0)))	→	s(0)	(5)

1.1.1 Rule Shifting

The rules

average^#(z0,s(s(s(z1))))	→	c1(average^#(s(z0),z1))	(9)
average^#(0,0)	→	c2	(10)
average^#(0,s(0))	→	c3	(11)
average^#(0,s(s(0)))	→	c4	(12)

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

[c(x₁)]	=	1 · x₁ + 0
[c1(x₁)]	=	1 · x₁ + 0
[c2]	=	0
[c3]	=	0
[c4]	=	0
[average^#(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[s(x₁)]	=	1 + 1 · x₁
[0]	=	0

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

average^#(s(z0),z1)	→	c(average^#(z0,s(z1)))	(7)
average^#(z0,s(s(s(z1))))	→	c1(average^#(s(z0),z1))	(9)
average^#(0,0)	→	c2	(10)
average^#(0,s(0))	→	c3	(11)
average^#(0,s(s(0)))	→	c4	(12)

1.1.1.1 Rule Shifting

The rules

average^#(s(z0),z1)

→

c(average^#(z0,s(z1)))

(7)

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

[c(x₁)]	=	1 · x₁ + 0
[c1(x₁)]	=	1 · x₁ + 0
[c2]	=	0
[c3]	=	0
[c4]	=	0
[average^#(x₁, x₂)]	=	2 · x₁ + 0 + 1 · x₂
[s(x₁)]	=	1 + 1 · x₁
[0]	=	0

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

average^#(s(z0),z1)	→	c(average^#(z0,s(z1)))	(7)
average^#(z0,s(s(s(z1))))	→	c1(average^#(s(z0),z1))	(9)
average^#(0,0)	→	c2	(10)
average^#(0,s(0))	→	c3	(11)
average^#(0,s(s(0)))	→	c4	(12)

1.1.1.1.1 R is empty

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