Certification Problem

Input (TPDB Runtime_Complexity_Innermost_Rewriting/SK90/2.29)

The rewrite relation of the following TRS is considered.

prime(0)	→	false	(1)
prime(s(0))	→	false	(2)
prime(s(s(x)))	→	prime1(s(s(x)),s(x))	(3)
prime1(x,0)	→	false	(4)
prime1(x,s(0))	→	true	(5)
prime1(x,s(s(y)))	→	and(not(divp(s(s(y)),x)),prime1(x,s(y)))	(6)
divp(x,y)	→	=(rem(x,y),0)	(7)

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:

prime^#(0)

→

(8)

originates from

prime(0)

→

false

(1)

prime^#(s(0))

→

(9)

originates from

prime(s(0))

→

false

(2)

prime^#(s(s(z0)))

→

c2(prime1^#(s(s(z0)),s(z0)))

(11)

originates from

prime(s(s(z0)))

→

prime1(s(s(z0)),s(z0))

(10)

prime1^#(z0,0)

→

(13)

originates from

prime1(z0,0)

→

false

(12)

prime1^#(z0,s(0))

→

(15)

originates from

prime1(z0,s(0))

→

true

(14)

prime1^#(z0,s(s(z1)))

→

c5(divp^#(s(s(z1)),z0),prime1^#(z0,s(z1)))

(17)

originates from

prime1(z0,s(s(z1)))

→

and(not(divp(s(s(z1)),z0)),prime1(z0,s(z1)))

(16)

divp^#(z0,z1)

→

(19)

originates from

divp(z0,z1)

→

=(rem(z0,z1),0)

(18)

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

prime^#(0)

prime^#(s(0))

prime^#(s(s(z0)))

prime1^#(z0,0)

prime1^#(z0,s(0))

prime1^#(z0,s(s(z1)))

divp^#(z0,z1)

1.1 Usable Rules

We remove the following rules since they are not usable.

prime(0)	→	false	(1)
prime(s(0))	→	false	(2)
prime(s(s(z0)))	→	prime1(s(s(z0)),s(z0))	(10)
prime1(z0,0)	→	false	(12)
prime1(z0,s(0))	→	true	(14)
prime1(z0,s(s(z1)))	→	and(not(divp(s(s(z1)),z0)),prime1(z0,s(z1)))	(16)
divp(z0,z1)	→	=(rem(z0,z1),0)	(18)

1.1.1 Rule Shifting

The rules

prime^#(0)	→	c	(8)
prime^#(s(0))	→	c1	(9)
prime^#(s(s(z0)))	→	c2(prime1^#(s(s(z0)),s(z0)))	(11)

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

[c]	=	0
[c1]	=	0
[c2(x₁)]	=	1 · x₁ + 0
[c3]	=	0
[c4]	=	0
[c5(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c6]	=	0
[prime^#(x₁)]	=	1
[prime1^#(x₁, x₂)]	=	0
[divp^#(x₁, x₂)]	=	0
[0]	=	0
[s(x₁)]	=	0

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

prime^#(0)	→	c	(8)
prime^#(s(0))	→	c1	(9)
prime^#(s(s(z0)))	→	c2(prime1^#(s(s(z0)),s(z0)))	(11)
prime1^#(z0,0)	→	c3	(13)
prime1^#(z0,s(0))	→	c4	(15)
prime1^#(z0,s(s(z1)))	→	c5(divp^#(s(s(z1)),z0),prime1^#(z0,s(z1)))	(17)
divp^#(z0,z1)	→	c6	(19)

1.1.1.1 Rule Shifting

The rules

prime1^#(z0,s(0))	→	c4	(15)
prime1^#(z0,s(s(z1)))	→	c5(divp^#(s(s(z1)),z0),prime1^#(z0,s(z1)))	(17)

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

[c]	=	0
[c1]	=	0
[c2(x₁)]	=	1 · x₁ + 0
[c3]	=	0
[c4]	=	0
[c5(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c6]	=	0
[prime^#(x₁)]	=	2 · x₁ · x₁ + 0
[prime1^#(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂ + 2 · x₂ · x₂
[divp^#(x₁, x₂)]	=	1 · x₁ + 0
[0]	=	0
[s(x₁)]	=	2 + 1 · x₁

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

prime^#(0)	→	c	(8)
prime^#(s(0))	→	c1	(9)
prime^#(s(s(z0)))	→	c2(prime1^#(s(s(z0)),s(z0)))	(11)
prime1^#(z0,0)	→	c3	(13)
prime1^#(z0,s(0))	→	c4	(15)
prime1^#(z0,s(s(z1)))	→	c5(divp^#(s(s(z1)),z0),prime1^#(z0,s(z1)))	(17)
divp^#(z0,z1)	→	c6	(19)

1.1.1.1.1 Rule Shifting

The rules

prime1^#(z0,0)

→

(13)

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

[c]	=	0
[c1]	=	0
[c2(x₁)]	=	1 · x₁ + 0
[c3]	=	0
[c4]	=	0
[c5(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c6]	=	0
[prime^#(x₁)]	=	1 + 1 · x₁
[prime1^#(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[divp^#(x₁, x₂)]	=	0
[0]	=	1
[s(x₁)]	=	1

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

prime^#(0)	→	c	(8)
prime^#(s(0))	→	c1	(9)
prime^#(s(s(z0)))	→	c2(prime1^#(s(s(z0)),s(z0)))	(11)
prime1^#(z0,0)	→	c3	(13)
prime1^#(z0,s(0))	→	c4	(15)
prime1^#(z0,s(s(z1)))	→	c5(divp^#(s(s(z1)),z0),prime1^#(z0,s(z1)))	(17)
divp^#(z0,z1)	→	c6	(19)

1.1.1.1.1.1 Rule Shifting

The rules

divp^#(z0,z1)

→

(19)

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

[c]	=	0
[c1]	=	0
[c2(x₁)]	=	1 · x₁ + 0
[c3]	=	0
[c4]	=	0
[c5(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c6]	=	0
[prime^#(x₁)]	=	2 + 3 · x₁
[prime1^#(x₁, x₂)]	=	1 · x₂ + 0
[divp^#(x₁, x₂)]	=	1
[0]	=	0
[s(x₁)]	=	2 + 1 · x₁

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

prime^#(0)	→	c	(8)
prime^#(s(0))	→	c1	(9)
prime^#(s(s(z0)))	→	c2(prime1^#(s(s(z0)),s(z0)))	(11)
prime1^#(z0,0)	→	c3	(13)
prime1^#(z0,s(0))	→	c4	(15)
prime1^#(z0,s(s(z1)))	→	c5(divp^#(s(s(z1)),z0),prime1^#(z0,s(z1)))	(17)
divp^#(z0,z1)	→	c6	(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).