Certification Problem

Input (TPDB Runtime_Complexity_Innermost_Rewriting/AG01/#3.16)

The rewrite relation of the following TRS is considered.

times(x,0)	→	0	(1)
times(x,s(y))	→	plus(times(x,y),x)	(2)
plus(x,0)	→	x	(3)
plus(0,x)	→	x	(4)
plus(x,s(y))	→	s(plus(x,y))	(5)
plus(s(x),y)	→	s(plus(x,y))	(6)

The evaluation strategy is innermost.

Property / Task

Determine bounds on the runtime complexity.

Answer / Result

An upperbound for the complexity is O(n³).

Proof (by AProVE @ termCOMP 2023)

1 Dependency Tuples

We get the following set of dependency tuples:

times^#(z0,0)

→

(8)

originates from

times(z0,0)

→

(7)

times^#(z0,s(z1))

→

c1(plus^#(times(z0,z1),z0),times^#(z0,z1))

(10)

originates from

times(z0,s(z1))

→

plus(times(z0,z1),z0)

(9)

plus^#(z0,0)

→

(12)

originates from

plus(z0,0)

→

(11)

plus^#(0,z0)

→

(14)

originates from

plus(0,z0)

→

(13)

plus^#(z0,s(z1))

→

c4(plus^#(z0,z1))

(16)

originates from

plus(z0,s(z1))

→

s(plus(z0,z1))

(15)

plus^#(s(z0),z1)

→

c5(plus^#(z0,z1))

(18)

originates from

plus(s(z0),z1)

→

s(plus(z0,z1))

(17)

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

times^#(z0,0)

times^#(z0,s(z1))

plus^#(z0,0)

plus^#(0,z0)

plus^#(z0,s(z1))

plus^#(s(z0),z1)

1.1 Rule Shifting

The rules

times^#(z0,0)	→	c	(8)
times^#(z0,s(z1))	→	c1(plus^#(times(z0,z1),z0),times^#(z0,z1))	(10)

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(x₁)]	=	1 · x₁ + 0
[c5(x₁)]	=	1 · x₁ + 0
[times(x₁, x₂)]	=	3 + 3 · x₁
[plus(x₁, x₂)]	=	3
[times^#(x₁, x₂)]	=	3 + 2 · x₂
[plus^#(x₁, x₂)]	=	0
[0]	=	3
[s(x₁)]	=	2 + 1 · x₁

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

times^#(z0,0)	→	c	(8)
times^#(z0,s(z1))	→	c1(plus^#(times(z0,z1),z0),times^#(z0,z1))	(10)
plus^#(z0,0)	→	c2	(12)
plus^#(0,z0)	→	c3	(14)
plus^#(z0,s(z1))	→	c4(plus^#(z0,z1))	(16)
plus^#(s(z0),z1)	→	c5(plus^#(z0,z1))	(18)

1.1.1 Rule Shifting

The rules

plus^#(z0,0)	→	c2	(12)
plus^#(0,z0)	→	c3	(14)

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(x₁)]	=	1 · x₁ + 0
[c5(x₁)]	=	1 · x₁ + 0
[times(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[plus(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[times^#(x₁, x₂)]	=	1 · x₂ + 0
[plus^#(x₁, x₂)]	=	1
[0]	=	1
[s(x₁)]	=	1 + 1 · x₁

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

times^#(z0,0)	→	c	(8)
times^#(z0,s(z1))	→	c1(plus^#(times(z0,z1),z0),times^#(z0,z1))	(10)
plus^#(z0,0)	→	c2	(12)
plus^#(0,z0)	→	c3	(14)
plus^#(z0,s(z1))	→	c4(plus^#(z0,z1))	(16)
plus^#(s(z0),z1)	→	c5(plus^#(z0,z1))	(18)

1.1.1.1 Rule Shifting

The rules

plus^#(z0,s(z1))

→

c4(plus^#(z0,z1))

(16)

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(x₁)]	=	1 · x₁ + 0
[c5(x₁)]	=	1 · x₁ + 0
[times(x₁, x₂)]	=	2 · x₁ · x₁ + 0
[plus(x₁, x₂)]	=	2 + 1 · x₂ · x₂
[times^#(x₁, x₂)]	=	1 · x₂ · x₂ + 0 + 2 · x₁ · x₂
[plus^#(x₁, x₂)]	=	1 + 2 · x₂
[0]	=	0
[s(x₁)]	=	1 + 1 · x₁

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

times^#(z0,0)	→	c	(8)
times^#(z0,s(z1))	→	c1(plus^#(times(z0,z1),z0),times^#(z0,z1))	(10)
plus^#(z0,0)	→	c2	(12)
plus^#(0,z0)	→	c3	(14)
plus^#(z0,s(z1))	→	c4(plus^#(z0,z1))	(16)
plus^#(s(z0),z1)	→	c5(plus^#(z0,z1))	(18)

1.1.1.1.1 Rule Shifting

The rules

plus^#(s(z0),z1)

→

c5(plus^#(z0,z1))

(18)

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(x₁)]	=	1 · x₁ + 0
[c5(x₁)]	=	1 · x₁ + 0
[times(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₁ · x₂
[plus(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[times^#(x₁, x₂)]	=	1 · x₂ · x₁ · x₁ + 0 + 1 · x₂ · x₂ · x₁
[plus^#(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂ · x₂
[0]	=	0
[s(x₁)]	=	1 + 1 · x₁

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

times^#(z0,0)	→	c	(8)
times^#(z0,s(z1))	→	c1(plus^#(times(z0,z1),z0),times^#(z0,z1))	(10)
plus^#(z0,0)	→	c2	(12)
plus^#(0,z0)	→	c3	(14)
plus^#(z0,s(z1))	→	c4(plus^#(z0,z1))	(16)
plus^#(s(z0),z1)	→	c5(plus^#(z0,z1))	(18)
times(z0,s(z1))	→	plus(times(z0,z1),z0)	(9)
plus(z0,s(z1))	→	s(plus(z0,z1))	(15)
plus(s(z0),z1)	→	s(plus(z0,z1))	(17)
plus(z0,0)	→	z0	(11)
plus(0,z0)	→	z0	(13)
times(z0,0)	→	0	(7)

1.1.1.1.1.1 R is empty

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