Certification Problem

Input (TPDB Runtime_Complexity_Innermost_Rewriting/Rubio_04/logarquot)

The rewrite relation of the following TRS is considered.

min(X,0)	→	X	(1)
min(s(X),s(Y))	→	min(X,Y)	(2)
quot(0,s(Y))	→	0	(3)
quot(s(X),s(Y))	→	s(quot(min(X,Y),s(Y)))	(4)
log(s(0))	→	0	(5)
log(s(s(X)))	→	s(log(s(quot(X,s(s(0))))))	(6)

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:

min^#(z0,0)

→

(8)

originates from

min(z0,0)

→

(7)

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

→

c1(min^#(z0,z1))

(10)

originates from

min(s(z0),s(z1))

→

min(z0,z1)

(9)

quot^#(0,s(z0))

→

(12)

originates from

quot(0,s(z0))

→

(11)

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

→

c3(quot^#(min(z0,z1),s(z1)),min^#(z0,z1))

(14)

originates from

quot(s(z0),s(z1))

→

s(quot(min(z0,z1),s(z1)))

(13)

log^#(s(0))

→

(15)

originates from

log(s(0))

→

(5)

log^#(s(s(z0)))

→

c5(log^#(s(quot(z0,s(s(0))))),quot^#(z0,s(s(0))))

(17)

originates from

log(s(s(z0)))

→

s(log(s(quot(z0,s(s(0))))))

(16)

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

min^#(z0,0)

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

quot^#(0,s(z0))

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

log^#(s(0))

log^#(s(s(z0)))

1.1 Usable Rules

We remove the following rules since they are not usable.

log(s(0))	→	0	(5)
log(s(s(z0)))	→	s(log(s(quot(z0,s(s(0))))))	(16)

1.1.1 Rule Shifting

The rules

log^#(s(0))

→

(15)

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

[c]	=	0
[c1(x₁)]	=	1 · x₁ + 0
[c2]	=	0
[c3(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c4]	=	0
[c5(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[min(x₁, x₂)]	=	1 · x₁ + 0
[quot(x₁, x₂)]	=	1 · x₂ + 0
[min^#(x₁, x₂)]	=	0
[quot^#(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[log^#(x₁)]	=	1 + 1 · x₁
[0]	=	0
[s(x₁)]	=	1 · x₁ + 0

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

min^#(z0,0)	→	c	(8)
min^#(s(z0),s(z1))	→	c1(min^#(z0,z1))	(10)
quot^#(0,s(z0))	→	c2	(12)
quot^#(s(z0),s(z1))	→	c3(quot^#(min(z0,z1),s(z1)),min^#(z0,z1))	(14)
log^#(s(0))	→	c4	(15)
log^#(s(s(z0)))	→	c5(log^#(s(quot(z0,s(s(0))))),quot^#(z0,s(s(0))))	(17)
min(z0,0)	→	z0	(7)
min(s(z0),s(z1))	→	min(z0,z1)	(9)
quot(s(z0),s(z1))	→	s(quot(min(z0,z1),s(z1)))	(13)
quot(0,s(z0))	→	0	(11)

1.1.1.1 Rule Shifting

The rules

quot^#(0,s(z0))

→

(12)

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

[c]	=	0
[c1(x₁)]	=	1 · x₁ + 0
[c2]	=	0
[c3(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c4]	=	0
[c5(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[min(x₁, x₂)]	=	1 · x₁ + 0
[quot(x₁, x₂)]	=	1 · x₁ + 0
[min^#(x₁, x₂)]	=	0
[quot^#(x₁, x₂)]	=	1
[log^#(x₁)]	=	1 · x₁ + 0
[0]	=	1
[s(x₁)]	=	1 + 1 · x₁

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

min^#(z0,0)	→	c	(8)
min^#(s(z0),s(z1))	→	c1(min^#(z0,z1))	(10)
quot^#(0,s(z0))	→	c2	(12)
quot^#(s(z0),s(z1))	→	c3(quot^#(min(z0,z1),s(z1)),min^#(z0,z1))	(14)
log^#(s(0))	→	c4	(15)
log^#(s(s(z0)))	→	c5(log^#(s(quot(z0,s(s(0))))),quot^#(z0,s(s(0))))	(17)
min(z0,0)	→	z0	(7)
min(s(z0),s(z1))	→	min(z0,z1)	(9)
quot(s(z0),s(z1))	→	s(quot(min(z0,z1),s(z1)))	(13)
quot(0,s(z0))	→	0	(11)

1.1.1.1.1 Rule Shifting

The rules

log^#(s(s(z0)))

→

c5(log^#(s(quot(z0,s(s(0))))),quot^#(z0,s(s(0))))

(17)

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

[c]	=	0
[c1(x₁)]	=	1 · x₁ + 0
[c2]	=	0
[c3(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c4]	=	0
[c5(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[min(x₁, x₂)]	=	1 · x₁ + 0
[quot(x₁, x₂)]	=	1 · x₁ + 0
[min^#(x₁, x₂)]	=	0
[quot^#(x₁, x₂)]	=	0
[log^#(x₁)]	=	1 · x₁ + 0
[0]	=	1
[s(x₁)]	=	1 + 1 · x₁

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

min^#(z0,0)	→	c	(8)
min^#(s(z0),s(z1))	→	c1(min^#(z0,z1))	(10)
quot^#(0,s(z0))	→	c2	(12)
quot^#(s(z0),s(z1))	→	c3(quot^#(min(z0,z1),s(z1)),min^#(z0,z1))	(14)
log^#(s(0))	→	c4	(15)
log^#(s(s(z0)))	→	c5(log^#(s(quot(z0,s(s(0))))),quot^#(z0,s(s(0))))	(17)
min(z0,0)	→	z0	(7)
min(s(z0),s(z1))	→	min(z0,z1)	(9)
quot(s(z0),s(z1))	→	s(quot(min(z0,z1),s(z1)))	(13)
quot(0,s(z0))	→	0	(11)

1.1.1.1.1.1 Rule Shifting

The rules

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

→

c3(quot^#(min(z0,z1),s(z1)),min^#(z0,z1))

(14)

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

[c]	=	0
[c1(x₁)]	=	1 · x₁ + 0
[c2]	=	0
[c3(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c4]	=	0
[c5(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[min(x₁, x₂)]	=	1 · x₁ + 0
[quot(x₁, x₂)]	=	1 · x₁ + 0
[min^#(x₁, x₂)]	=	0
[quot^#(x₁, x₂)]	=	2 · x₂ + 0 + 2 · x₁ · x₂
[log^#(x₁)]	=	2 · x₁ · x₁ + 0
[0]	=	0
[s(x₁)]	=	2 + 1 · x₁

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

min^#(z0,0)	→	c	(8)
min^#(s(z0),s(z1))	→	c1(min^#(z0,z1))	(10)
quot^#(0,s(z0))	→	c2	(12)
quot^#(s(z0),s(z1))	→	c3(quot^#(min(z0,z1),s(z1)),min^#(z0,z1))	(14)
log^#(s(0))	→	c4	(15)
log^#(s(s(z0)))	→	c5(log^#(s(quot(z0,s(s(0))))),quot^#(z0,s(s(0))))	(17)
min(z0,0)	→	z0	(7)
min(s(z0),s(z1))	→	min(z0,z1)	(9)
quot(s(z0),s(z1))	→	s(quot(min(z0,z1),s(z1)))	(13)
quot(0,s(z0))	→	0	(11)

1.1.1.1.1.1.1 Rule Shifting

The rules

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

→

c1(min^#(z0,z1))

(10)

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

[c]	=	0
[c1(x₁)]	=	1 · x₁ + 0
[c2]	=	0
[c3(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c4]	=	0
[c5(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[min(x₁, x₂)]	=	1 · x₁ + 0
[quot(x₁, x₂)]	=	1 + 1 · x₁
[min^#(x₁, x₂)]	=	2 · x₂ + 0
[quot^#(x₁, x₂)]	=	1 · x₁ · x₂ + 0
[log^#(x₁)]	=	2 · x₁ · x₁ + 0
[0]	=	0
[s(x₁)]	=	2 + 1 · x₁

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

min^#(z0,0)	→	c	(8)
min^#(s(z0),s(z1))	→	c1(min^#(z0,z1))	(10)
quot^#(0,s(z0))	→	c2	(12)
quot^#(s(z0),s(z1))	→	c3(quot^#(min(z0,z1),s(z1)),min^#(z0,z1))	(14)
log^#(s(0))	→	c4	(15)
log^#(s(s(z0)))	→	c5(log^#(s(quot(z0,s(s(0))))),quot^#(z0,s(s(0))))	(17)
min(z0,0)	→	z0	(7)
min(s(z0),s(z1))	→	min(z0,z1)	(9)
quot(s(z0),s(z1))	→	s(quot(min(z0,z1),s(z1)))	(13)
quot(0,s(z0))	→	0	(11)

1.1.1.1.1.1.1.1 Rule Shifting

The rules

min^#(z0,0)

→

(8)

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

[c]	=	0
[c1(x₁)]	=	1 · x₁ + 0
[c2]	=	0
[c3(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c4]	=	0
[c5(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[min(x₁, x₂)]	=	1 · x₁ + 0
[quot(x₁, x₂)]	=	1 · x₁ + 0
[min^#(x₁, x₂)]	=	2
[quot^#(x₁, x₂)]	=	2 · x₁ · x₂ + 0
[log^#(x₁)]	=	2 · x₁ + 0 + 2 · x₁ · x₁
[0]	=	0
[s(x₁)]	=	2 + 1 · x₁

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

min^#(z0,0)	→	c	(8)
min^#(s(z0),s(z1))	→	c1(min^#(z0,z1))	(10)
quot^#(0,s(z0))	→	c2	(12)
quot^#(s(z0),s(z1))	→	c3(quot^#(min(z0,z1),s(z1)),min^#(z0,z1))	(14)
log^#(s(0))	→	c4	(15)
log^#(s(s(z0)))	→	c5(log^#(s(quot(z0,s(s(0))))),quot^#(z0,s(s(0))))	(17)
min(z0,0)	→	z0	(7)
min(s(z0),s(z1))	→	min(z0,z1)	(9)
quot(s(z0),s(z1))	→	s(quot(min(z0,z1),s(z1)))	(13)
quot(0,s(z0))	→	0	(11)

1.1.1.1.1.1.1.1.1 R is empty

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