Certification Problem

Input (TPDB Runtime_Complexity_Innermost_Rewriting/SK90/2.22)

The rewrite relation of the following TRS is considered.

exp(x,0)	→	s(0)	(1)
exp(x,s(y))	→	*(x,exp(x,y))	(2)
*(0,y)	→	0	(3)
*(s(x),y)	→	+(y,*(x,y))	(4)
-(0,y)	→	0	(5)
-(x,0)	→	x	(6)
-(s(x),s(y))	→	-(x,y)	(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:

exp^#(z0,0)

→

(9)

originates from

exp(z0,0)

→

s(0)

(8)

exp^#(z0,s(z1))

→

c1(*^#(z0,exp(z0,z1)),exp^#(z0,z1))

(11)

originates from

exp(z0,s(z1))

→

*(z0,exp(z0,z1))

(10)

*^#(0,z0)

→

(13)

originates from

*(0,z0)

→

(12)

*^#(s(z0),z1)

→

c3(*^#(z0,z1))

(15)

originates from

*(s(z0),z1)

→

+(z1,*(z0,z1))

(14)

-^#(0,z0)

→

(17)

originates from

-(0,z0)

→

(16)

-^#(z0,0)

→

(19)

originates from

-(z0,0)

→

(18)

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

→

c6(-^#(z0,z1))

(21)

originates from

-(s(z0),s(z1))

→

-(z0,z1)

(20)

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

exp^#(z0,0)

exp^#(z0,s(z1))

*^#(0,z0)

*^#(s(z0),z1)

-^#(0,z0)

-^#(z0,0)

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

1.1 Usable Rules

We remove the following rules since they are not usable.

-(0,z0)	→	0	(16)
-(z0,0)	→	z0	(18)
-(s(z0),s(z1))	→	-(z0,z1)	(20)

1.1.1 Rule Shifting

The rules

exp^#(z0,0)	→	c	(9)
exp^#(z0,s(z1))	→	c1(*^#(z0,exp(z0,z1)),exp^#(z0,z1))	(11)
-^#(0,z0)	→	c4	(17)
-^#(z0,0)	→	c5	(19)
-^#(s(z0),s(z1))	→	c6(-^#(z0,z1))	(21)

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(x₁)]	=	1 · x₁ + 0
[c4]	=	0
[c5]	=	0
[c6(x₁)]	=	1 · x₁ + 0
[exp(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[*(x₁, x₂)]	=	1 + 1 · x₁
[exp^#(x₁, x₂)]	=	1 + 1 · x₂
[*^#(x₁, x₂)]	=	0
[-^#(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[0]	=	0
[s(x₁)]	=	1 + 1 · x₁
[+(x₁, x₂)]	=	1 + 1 · x₂

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

exp^#(z0,0)	→	c	(9)
exp^#(z0,s(z1))	→	c1(*^#(z0,exp(z0,z1)),exp^#(z0,z1))	(11)
*^#(0,z0)	→	c2	(13)
*^#(s(z0),z1)	→	c3(*^#(z0,z1))	(15)
-^#(0,z0)	→	c4	(17)
-^#(z0,0)	→	c5	(19)
-^#(s(z0),s(z1))	→	c6(-^#(z0,z1))	(21)

1.1.1.1 Rule Shifting

The rules

*^#(0,z0)

→

(13)

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(x₁)]	=	1 · x₁ + 0
[c4]	=	0
[c5]	=	0
[c6(x₁)]	=	1 · x₁ + 0
[exp(x₁, x₂)]	=	3 · x₁ + 0 + 2 · x₂
[*(x₁, x₂)]	=	2 · x₁ + 0
[exp^#(x₁, x₂)]	=	1 · x₂ + 0
[*^#(x₁, x₂)]	=	1
[-^#(x₁, x₂)]	=	3 · x₁ + 0 + 3 · x₂
[0]	=	2
[s(x₁)]	=	2 + 1 · x₁
[+(x₁, x₂)]	=	2

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

exp^#(z0,0)	→	c	(9)
exp^#(z0,s(z1))	→	c1(*^#(z0,exp(z0,z1)),exp^#(z0,z1))	(11)
*^#(0,z0)	→	c2	(13)
*^#(s(z0),z1)	→	c3(*^#(z0,z1))	(15)
-^#(0,z0)	→	c4	(17)
-^#(z0,0)	→	c5	(19)
-^#(s(z0),s(z1))	→	c6(-^#(z0,z1))	(21)

1.1.1.1.1 Rule Shifting

The rules

*^#(s(z0),z1)

→

c3(*^#(z0,z1))

(15)

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(x₁)]	=	1 · x₁ + 0
[c4]	=	0
[c5]	=	0
[c6(x₁)]	=	1 · x₁ + 0
[exp(x₁, x₂)]	=	2 · x₁ · x₁ + 0
[*(x₁, x₂)]	=	1 + 1 · x₁ · x₁
[exp^#(x₁, x₂)]	=	1 · x₂ + 0 + 2 · x₁ · x₂
[*^#(x₁, x₂)]	=	2 · x₁ + 0
[-^#(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[0]	=	0
[s(x₁)]	=	2 + 1 · x₁
[+(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂

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

exp^#(z0,0)	→	c	(9)
exp^#(z0,s(z1))	→	c1(*^#(z0,exp(z0,z1)),exp^#(z0,z1))	(11)
*^#(0,z0)	→	c2	(13)
*^#(s(z0),z1)	→	c3(*^#(z0,z1))	(15)
-^#(0,z0)	→	c4	(17)
-^#(z0,0)	→	c5	(19)
-^#(s(z0),s(z1))	→	c6(-^#(z0,z1))	(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).