Certification Problem

Input (TPDB Runtime_Complexity_Innermost_Rewriting/Der95/11)

The rewrite relation of the following TRS is considered.

D(t)	→	1	(1)
D(constant)	→	0	(2)
D(+(x,y))	→	+(D(x),D(y))	(3)
D(*(x,y))	→	+((y,D(x)),(x,D(y)))	(4)
D(-(x,y))	→	-(D(x),D(y))	(5)
D(minus(x))	→	minus(D(x))	(6)
D(div(x,y))	→	-(div(D(x),y),div(*(x,D(y)),pow(y,2)))	(7)
D(ln(x))	→	div(D(x),x)	(8)
D(pow(x,y))	→	+(((y,pow(x,-(y,1))),D(x)),((pow(x,y),ln(x)),D(y)))	(9)

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:

D^#(t)

→

(10)

originates from

D(t)

→

(1)

D^#(constant)

→

(11)

originates from

D(constant)

→

(2)

D^#(+(z0,z1))

→

c2(D^#(z0),D^#(z1))

(13)

originates from

D(+(z0,z1))

→

+(D(z0),D(z1))

(12)

D^#(*(z0,z1))

→

c3(D^#(z0),D^#(z1))

(15)

originates from

D(*(z0,z1))

→

+(*(z1,D(z0)),*(z0,D(z1)))

(14)

D^#(-(z0,z1))

→

c4(D^#(z0),D^#(z1))

(17)

originates from

D(-(z0,z1))

→

-(D(z0),D(z1))

(16)

D^#(minus(z0))

→

c5(D^#(z0))

(19)

originates from

D(minus(z0))

→

minus(D(z0))

(18)

D^#(div(z0,z1))

→

c6(D^#(z0),D^#(z1))

(21)

originates from

D(div(z0,z1))

→

-(div(D(z0),z1),div(*(z0,D(z1)),pow(z1,2)))

(20)

D^#(ln(z0))

→

c7(D^#(z0))

(23)

originates from

D(ln(z0))

→

div(D(z0),z0)

(22)

D^#(pow(z0,z1))

→

c8(D^#(z0),D^#(z1))

(25)

originates from

D(pow(z0,z1))

→

+(*(*(z1,pow(z0,-(z1,1))),D(z0)),*(*(pow(z0,z1),ln(z0)),D(z1)))

(24)

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

D^#(t)

D^#(constant)

D^#(+(z0,z1))

D^#(*(z0,z1))

D^#(-(z0,z1))

D^#(minus(z0))

D^#(div(z0,z1))

D^#(ln(z0))

D^#(pow(z0,z1))

1.1 Usable Rules

We remove the following rules since they are not usable.

D(t)	→	1	(1)
D(constant)	→	0	(2)
D(+(z0,z1))	→	+(D(z0),D(z1))	(12)
D(*(z0,z1))	→	+((z1,D(z0)),(z0,D(z1)))	(14)
D(-(z0,z1))	→	-(D(z0),D(z1))	(16)
D(minus(z0))	→	minus(D(z0))	(18)
D(div(z0,z1))	→	-(div(D(z0),z1),div(*(z0,D(z1)),pow(z1,2)))	(20)
D(ln(z0))	→	div(D(z0),z0)	(22)
D(pow(z0,z1))	→	+(((z1,pow(z0,-(z1,1))),D(z0)),((pow(z0,z1),ln(z0)),D(z1)))	(24)

1.1.1 Rule Shifting

The rules

D^#(t)	→	c	(10)
D^#(constant)	→	c1	(11)
D^#(+(z0,z1))	→	c2(D^#(z0),D^#(z1))	(13)
D^#(*(z0,z1))	→	c3(D^#(z0),D^#(z1))	(15)
D^#(-(z0,z1))	→	c4(D^#(z0),D^#(z1))	(17)
D^#(minus(z0))	→	c5(D^#(z0))	(19)
D^#(div(z0,z1))	→	c6(D^#(z0),D^#(z1))	(21)
D^#(ln(z0))	→	c7(D^#(z0))	(23)
D^#(pow(z0,z1))	→	c8(D^#(z0),D^#(z1))	(25)

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

[c]	=	0
[c1]	=	0
[c2(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c3(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c4(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c5(x₁)]	=	1 · x₁ + 0
[c6(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c7(x₁)]	=	1 · x₁ + 0
[c8(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[D^#(x₁)]	=	2 · x₁ + 0
[t]	=	3
[constant]	=	3
[+(x₁, x₂)]	=	2 + 1 · x₁ + 1 · x₂
[*(x₁, x₂)]	=	3 + 1 · x₁ + 1 · x₂
[-(x₁, x₂)]	=	2 + 1 · x₁ + 1 · x₂
[minus(x₁)]	=	3 + 1 · x₁
[div(x₁, x₂)]	=	3 + 1 · x₁ + 1 · x₂
[ln(x₁)]	=	3 + 1 · x₁
[pow(x₁, x₂)]	=	2 + 1 · x₁ + 1 · x₂

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

D^#(t)	→	c	(10)
D^#(constant)	→	c1	(11)
D^#(+(z0,z1))	→	c2(D^#(z0),D^#(z1))	(13)
D^#(*(z0,z1))	→	c3(D^#(z0),D^#(z1))	(15)
D^#(-(z0,z1))	→	c4(D^#(z0),D^#(z1))	(17)
D^#(minus(z0))	→	c5(D^#(z0))	(19)
D^#(div(z0,z1))	→	c6(D^#(z0),D^#(z1))	(21)
D^#(ln(z0))	→	c7(D^#(z0))	(23)
D^#(pow(z0,z1))	→	c8(D^#(z0),D^#(z1))	(25)

1.1.1.1 R is empty

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