Certification Problem

Input (TPDB Runtime_Complexity_Innermost_Rewriting/SK90/2.12)

The rewrite relation of the following TRS is considered.

+(0,y)	→	y	(1)
+(s(x),y)	→	s(+(x,y))	(2)
+(p(x),y)	→	p(+(x,y))	(3)
minus(0)	→	0	(4)
minus(s(x))	→	p(minus(x))	(5)
minus(p(x))	→	s(minus(x))	(6)
*(0,y)	→	0	(7)
*(s(x),y)	→	+(*(x,y),y)	(8)
*(p(x),y)	→	+(*(x,y),minus(y))	(9)

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:

+^#(0,z0)

→

(11)

originates from

+(0,z0)

→

(10)

+^#(s(z0),z1)

→

c1(+^#(z0,z1))

(13)

originates from

+(s(z0),z1)

→

s(+(z0,z1))

(12)

+^#(p(z0),z1)

→

c2(+^#(z0,z1))

(15)

originates from

+(p(z0),z1)

→

p(+(z0,z1))

(14)

minus^#(0)

→

(16)

originates from

minus(0)

→

(4)

minus^#(s(z0))

→

c4(minus^#(z0))

(18)

originates from

minus(s(z0))

→

p(minus(z0))

(17)

minus^#(p(z0))

→

c5(minus^#(z0))

(20)

originates from

minus(p(z0))

→

s(minus(z0))

(19)

*^#(0,z0)

→

(22)

originates from

*(0,z0)

→

(21)

*^#(s(z0),z1)

→

c7(+^#(*(z0,z1),z1),*^#(z0,z1))

(24)

originates from

*(s(z0),z1)

→

+(*(z0,z1),z1)

(23)

*^#(p(z0),z1)

→

c8(+^#(*(z0,z1),minus(z1)),*^#(z0,z1),minus^#(z1))

(26)

originates from

*(p(z0),z1)

→

+(*(z0,z1),minus(z1))

(25)

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

+^#(0,z0)

+^#(s(z0),z1)

+^#(p(z0),z1)

minus^#(0)

minus^#(s(z0))

minus^#(p(z0))

*^#(0,z0)

*^#(s(z0),z1)

*^#(p(z0),z1)

1.1 Rule Shifting

The rules

minus^#(0)	→	c3	(16)
*^#(0,z0)	→	c6	(22)
*^#(s(z0),z1)	→	c7(+^#((z0,z1),z1),^#(z0,z1))	(24)

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

[c]	=	0
[c1(x₁)]	=	1 · x₁ + 0
[c2(x₁)]	=	1 · x₁ + 0
[c3]	=	0
[c4(x₁)]	=	1 · x₁ + 0
[c5(x₁)]	=	1 · x₁ + 0
[c6]	=	0
[c7(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c8(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[+(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[minus(x₁)]	=	1 + 1 · x₁
[*(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[+^#(x₁, x₂)]	=	0
[minus^#(x₁)]	=	1
[*^#(x₁, x₂)]	=	1 + 1 · x₁
[0]	=	1
[s(x₁)]	=	1 + 1 · x₁
[p(x₁)]	=	1 + 1 · x₁

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

+^#(0,z0)	→	c	(11)
+^#(s(z0),z1)	→	c1(+^#(z0,z1))	(13)
+^#(p(z0),z1)	→	c2(+^#(z0,z1))	(15)
minus^#(0)	→	c3	(16)
minus^#(s(z0))	→	c4(minus^#(z0))	(18)
minus^#(p(z0))	→	c5(minus^#(z0))	(20)
*^#(0,z0)	→	c6	(22)
*^#(s(z0),z1)	→	c7(+^#((z0,z1),z1),^#(z0,z1))	(24)
*^#(p(z0),z1)	→	c8(+^#((z0,z1),minus(z1)),^#(z0,z1),minus^#(z1))	(26)

1.1.1 Rule Shifting

The rules

+^#(0,z0)

→

(11)

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

[c]	=	0
[c1(x₁)]	=	1 · x₁ + 0
[c2(x₁)]	=	1 · x₁ + 0
[c3]	=	0
[c4(x₁)]	=	1 · x₁ + 0
[c5(x₁)]	=	1 · x₁ + 0
[c6]	=	0
[c7(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c8(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[+(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[minus(x₁)]	=	1 + 1 · x₁
[*(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[+^#(x₁, x₂)]	=	1
[minus^#(x₁)]	=	0
[*^#(x₁, x₂)]	=	1 · x₁ + 0
[0]	=	1
[s(x₁)]	=	1 + 1 · x₁
[p(x₁)]	=	1 + 1 · x₁

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

+^#(0,z0)	→	c	(11)
+^#(s(z0),z1)	→	c1(+^#(z0,z1))	(13)
+^#(p(z0),z1)	→	c2(+^#(z0,z1))	(15)
minus^#(0)	→	c3	(16)
minus^#(s(z0))	→	c4(minus^#(z0))	(18)
minus^#(p(z0))	→	c5(minus^#(z0))	(20)
*^#(0,z0)	→	c6	(22)
*^#(s(z0),z1)	→	c7(+^#((z0,z1),z1),^#(z0,z1))	(24)
*^#(p(z0),z1)	→	c8(+^#((z0,z1),minus(z1)),^#(z0,z1),minus^#(z1))	(26)

1.1.1.1 Rule Shifting

The rules

*^#(p(z0),z1)

→

c8(+^#(*(z0,z1),minus(z1)),*^#(z0,z1),minus^#(z1))

(26)

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

[c]	=	0
[c1(x₁)]	=	1 · x₁ + 0
[c2(x₁)]	=	1 · x₁ + 0
[c3]	=	0
[c4(x₁)]	=	1 · x₁ + 0
[c5(x₁)]	=	1 · x₁ + 0
[c6]	=	0
[c7(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c8(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[+(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[minus(x₁)]	=	1 + 1 · x₁
[*(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[+^#(x₁, x₂)]	=	0
[minus^#(x₁)]	=	0
[*^#(x₁, x₂)]	=	1 · x₁ + 0
[0]	=	1
[s(x₁)]	=	1 + 1 · x₁
[p(x₁)]	=	1 + 1 · x₁

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

+^#(0,z0)	→	c	(11)
+^#(s(z0),z1)	→	c1(+^#(z0,z1))	(13)
+^#(p(z0),z1)	→	c2(+^#(z0,z1))	(15)
minus^#(0)	→	c3	(16)
minus^#(s(z0))	→	c4(minus^#(z0))	(18)
minus^#(p(z0))	→	c5(minus^#(z0))	(20)
*^#(0,z0)	→	c6	(22)
*^#(s(z0),z1)	→	c7(+^#((z0,z1),z1),^#(z0,z1))	(24)
*^#(p(z0),z1)	→	c8(+^#((z0,z1),minus(z1)),^#(z0,z1),minus^#(z1))	(26)

1.1.1.1.1 Rule Shifting

The rules

minus^#(s(z0))	→	c4(minus^#(z0))	(18)
minus^#(p(z0))	→	c5(minus^#(z0))	(20)

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

[c]	=	0
[c1(x₁)]	=	1 · x₁ + 0
[c2(x₁)]	=	1 · x₁ + 0
[c3]	=	0
[c4(x₁)]	=	1 · x₁ + 0
[c5(x₁)]	=	1 · x₁ + 0
[c6]	=	0
[c7(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c8(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[+(x₁, x₂)]	=	1
[minus(x₁)]	=	0
[*(x₁, x₂)]	=	1 · x₂ · x₂ + 0
[+^#(x₁, x₂)]	=	0
[minus^#(x₁)]	=	1 · x₁ + 0
[*^#(x₁, x₂)]	=	1 · x₁ · x₂ + 0
[0]	=	0
[s(x₁)]	=	1 + 1 · x₁
[p(x₁)]	=	2 + 1 · x₁

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

+^#(0,z0)	→	c	(11)
+^#(s(z0),z1)	→	c1(+^#(z0,z1))	(13)
+^#(p(z0),z1)	→	c2(+^#(z0,z1))	(15)
minus^#(0)	→	c3	(16)
minus^#(s(z0))	→	c4(minus^#(z0))	(18)
minus^#(p(z0))	→	c5(minus^#(z0))	(20)
*^#(0,z0)	→	c6	(22)
*^#(s(z0),z1)	→	c7(+^#((z0,z1),z1),^#(z0,z1))	(24)
*^#(p(z0),z1)	→	c8(+^#((z0,z1),minus(z1)),^#(z0,z1),minus^#(z1))	(26)

1.1.1.1.1.1 Rule Shifting

The rules

+^#(s(z0),z1)	→	c1(+^#(z0,z1))	(13)
+^#(p(z0),z1)	→	c2(+^#(z0,z1))	(15)

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

[c]	=	0
[c1(x₁)]	=	1 · x₁ + 0
[c2(x₁)]	=	1 · x₁ + 0
[c3]	=	0
[c4(x₁)]	=	1 · x₁ + 0
[c5(x₁)]	=	1 · x₁ + 0
[c6]	=	0
[c7(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c8(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[+(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[minus(x₁)]	=	1 · x₁ + 0
[*(x₁, x₂)]	=	1 · x₁ · x₂ + 0
[+^#(x₁, x₂)]	=	1 · x₁ + 0
[minus^#(x₁)]	=	0
[*^#(x₁, x₂)]	=	1 · x₂ · x₁ · x₁ + 0
[0]	=	0
[s(x₁)]	=	1 + 1 · x₁
[p(x₁)]	=	1 + 1 · x₁

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

+^#(0,z0)	→	c	(11)
+^#(s(z0),z1)	→	c1(+^#(z0,z1))	(13)
+^#(p(z0),z1)	→	c2(+^#(z0,z1))	(15)
minus^#(0)	→	c3	(16)
minus^#(s(z0))	→	c4(minus^#(z0))	(18)
minus^#(p(z0))	→	c5(minus^#(z0))	(20)
*^#(0,z0)	→	c6	(22)
*^#(s(z0),z1)	→	c7(+^#((z0,z1),z1),^#(z0,z1))	(24)
*^#(p(z0),z1)	→	c8(+^#((z0,z1),minus(z1)),^#(z0,z1),minus^#(z1))	(26)
+(s(z0),z1)	→	s(+(z0,z1))	(12)
*(p(z0),z1)	→	+(*(z0,z1),minus(z1))	(25)
minus(0)	→	0	(4)
*(s(z0),z1)	→	+(*(z0,z1),z1)	(23)
minus(p(z0))	→	s(minus(z0))	(19)
*(0,z0)	→	0	(21)
minus(s(z0))	→	p(minus(z0))	(17)
+(p(z0),z1)	→	p(+(z0,z1))	(14)
+(0,z0)	→	z0	(10)

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).