Certification Problem

Input (TPDB Runtime_Complexity_Innermost_Rewriting/Frederiksen_Glenstrup/mul_better)

The rewrite relation of the following TRS is considered.

mul0(C(x,y),y')	→	add0(mul0(y,y'),y')	(1)
mul0(Z,y)	→	Z	(2)
add0(C(x,y),y')	→	add0(y,C(S,y'))	(3)
add0(Z,y)	→	y	(4)
second(C(x,y))	→	y	(5)
isZero(C(x,y))	→	False	(6)
isZero(Z)	→	True	(7)
goal(xs,ys)	→	mul0(xs,ys)	(8)

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:

mul0^#(C(z0,z1),z2)

→

c(add0^#(mul0(z1,z2),z2),mul0^#(z1,z2))

(10)

originates from

mul0(C(z0,z1),z2)

→

add0(mul0(z1,z2),z2)

(9)

mul0^#(Z,z0)

→

(12)

originates from

mul0(Z,z0)

→

(11)

add0^#(C(z0,z1),z2)

→

c2(add0^#(z1,C(S,z2)))

(14)

originates from

add0(C(z0,z1),z2)

→

add0(z1,C(S,z2))

(13)

add0^#(Z,z0)

→

(16)

originates from

add0(Z,z0)

→

(15)

second^#(C(z0,z1))

→

(18)

originates from

second(C(z0,z1))

→

(17)

isZero^#(C(z0,z1))

→

(20)

originates from

isZero(C(z0,z1))

→

False

(19)

isZero^#(Z)

→

(21)

originates from

isZero(Z)

→

True

(7)

goal^#(z0,z1)

→

c7(mul0^#(z0,z1))

(23)

originates from

goal(z0,z1)

→

mul0(z0,z1)

(22)

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

mul0^#(C(z0,z1),z2)

mul0^#(Z,z0)

add0^#(C(z0,z1),z2)

add0^#(Z,z0)

second^#(C(z0,z1))

isZero^#(C(z0,z1))

isZero^#(Z)

goal^#(z0,z1)

1.1 Usable Rules

We remove the following rules since they are not usable.

second(C(z0,z1))	→	z1	(17)
isZero(C(z0,z1))	→	False	(19)
isZero(Z)	→	True	(7)
goal(z0,z1)	→	mul0(z0,z1)	(22)

1.1.1 Rule Shifting

The rules

mul0^#(Z,z0)	→	c1	(12)
add0^#(Z,z0)	→	c3	(16)
second^#(C(z0,z1))	→	c4	(18)
isZero^#(C(z0,z1))	→	c5	(20)
isZero^#(Z)	→	c6	(21)

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

[c(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c1]	=	0
[c2(x₁)]	=	1 · x₁ + 0
[c3]	=	0
[c4]	=	0
[c5]	=	0
[c6]	=	0
[c7(x₁)]	=	1 · x₁ + 0
[mul0(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[add0(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[mul0^#(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[add0^#(x₁, x₂)]	=	1
[second^#(x₁)]	=	1 · x₁ + 0
[isZero^#(x₁)]	=	1 · x₁ + 0
[goal^#(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[C(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[Z]	=	1
[S]	=	1

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

mul0^#(C(z0,z1),z2)	→	c(add0^#(mul0(z1,z2),z2),mul0^#(z1,z2))	(10)
mul0^#(Z,z0)	→	c1	(12)
add0^#(C(z0,z1),z2)	→	c2(add0^#(z1,C(S,z2)))	(14)
add0^#(Z,z0)	→	c3	(16)
second^#(C(z0,z1))	→	c4	(18)
isZero^#(C(z0,z1))	→	c5	(20)
isZero^#(Z)	→	c6	(21)
goal^#(z0,z1)	→	c7(mul0^#(z0,z1))	(23)

1.1.1.1 Rule Shifting

The rules

goal^#(z0,z1)

→

c7(mul0^#(z0,z1))

(23)

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

[c(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c1]	=	0
[c2(x₁)]	=	1 · x₁ + 0
[c3]	=	0
[c4]	=	0
[c5]	=	0
[c6]	=	0
[c7(x₁)]	=	1 · x₁ + 0
[mul0(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[add0(x₁, x₂)]	=	1 · x₂ + 0
[mul0^#(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[add0^#(x₁, x₂)]	=	0
[second^#(x₁)]	=	0
[isZero^#(x₁)]	=	0
[goal^#(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[C(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[Z]	=	1
[S]	=	0

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

mul0^#(C(z0,z1),z2)	→	c(add0^#(mul0(z1,z2),z2),mul0^#(z1,z2))	(10)
mul0^#(Z,z0)	→	c1	(12)
add0^#(C(z0,z1),z2)	→	c2(add0^#(z1,C(S,z2)))	(14)
add0^#(Z,z0)	→	c3	(16)
second^#(C(z0,z1))	→	c4	(18)
isZero^#(C(z0,z1))	→	c5	(20)
isZero^#(Z)	→	c6	(21)
goal^#(z0,z1)	→	c7(mul0^#(z0,z1))	(23)

1.1.1.1.1 Rule Shifting

The rules

mul0^#(C(z0,z1),z2)

→

c(add0^#(mul0(z1,z2),z2),mul0^#(z1,z2))

(10)

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

[c(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c1]	=	0
[c2(x₁)]	=	1 · x₁ + 0
[c3]	=	0
[c4]	=	0
[c5]	=	0
[c6]	=	0
[c7(x₁)]	=	1 · x₁ + 0
[mul0(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[add0(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[mul0^#(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[add0^#(x₁, x₂)]	=	0
[second^#(x₁)]	=	0
[isZero^#(x₁)]	=	0
[goal^#(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[C(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[Z]	=	1
[S]	=	1

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

mul0^#(C(z0,z1),z2)	→	c(add0^#(mul0(z1,z2),z2),mul0^#(z1,z2))	(10)
mul0^#(Z,z0)	→	c1	(12)
add0^#(C(z0,z1),z2)	→	c2(add0^#(z1,C(S,z2)))	(14)
add0^#(Z,z0)	→	c3	(16)
second^#(C(z0,z1))	→	c4	(18)
isZero^#(C(z0,z1))	→	c5	(20)
isZero^#(Z)	→	c6	(21)
goal^#(z0,z1)	→	c7(mul0^#(z0,z1))	(23)

1.1.1.1.1.1 Rule Shifting

The rules

add0^#(C(z0,z1),z2)

→

c2(add0^#(z1,C(S,z2)))

(14)

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

[c(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c1]	=	0
[c2(x₁)]	=	1 · x₁ + 0
[c3]	=	0
[c4]	=	0
[c5]	=	0
[c6]	=	0
[c7(x₁)]	=	1 · x₁ + 0
[mul0(x₁, x₂)]	=	1 · x₁ · x₂ + 0
[add0(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[mul0^#(x₁, x₂)]	=	1 + 1 · x₂ + 1 · x₂ · x₂ + 1 · x₁ · x₂ + 1 · x₂ · x₁ · x₁ + 1 · x₂ · x₂ · x₁ + 1 · x₂ · x₂ · x₂
[add0^#(x₁, x₂)]	=	1 · x₁ + 0
[second^#(x₁)]	=	0
[isZero^#(x₁)]	=	0
[goal^#(x₁, x₂)]	=	1 + 1 · x₂ + 1 · x₂ · x₂ + 1 · x₁ · x₂ + 1 · x₂ · x₁ · x₁ + 1 · x₂ · x₂ · x₁ + 1 · x₂ · x₂ · x₂
[C(x₁, x₂)]	=	1 + 1 · x₂
[Z]	=	0
[S]	=	0

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

mul0^#(C(z0,z1),z2)	→	c(add0^#(mul0(z1,z2),z2),mul0^#(z1,z2))	(10)
mul0^#(Z,z0)	→	c1	(12)
add0^#(C(z0,z1),z2)	→	c2(add0^#(z1,C(S,z2)))	(14)
add0^#(Z,z0)	→	c3	(16)
second^#(C(z0,z1))	→	c4	(18)
isZero^#(C(z0,z1))	→	c5	(20)
isZero^#(Z)	→	c6	(21)
goal^#(z0,z1)	→	c7(mul0^#(z0,z1))	(23)
mul0(Z,z0)	→	Z	(11)
add0(Z,z0)	→	z0	(15)
add0(C(z0,z1),z2)	→	add0(z1,C(S,z2))	(13)
mul0(C(z0,z1),z2)	→	add0(mul0(z1,z2),z2)	(9)

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).