Certification Problem

Input (TPDB Runtime_Complexity_Innermost_Rewriting/AG01/#3.6a)

The rewrite relation of the following TRS is considered.

le(0,y)	→	true	(1)
le(s(x),0)	→	false	(2)
le(s(x),s(y))	→	le(x,y)	(3)
minus(x,0)	→	x	(4)
minus(s(x),s(y))	→	minus(x,y)	(5)
gcd(0,y)	→	y	(6)
gcd(s(x),0)	→	s(x)	(7)
gcd(s(x),s(y))	→	if_gcd(le(y,x),s(x),s(y))	(8)
if_gcd(true,s(x),s(y))	→	gcd(minus(x,y),s(y))	(9)
if_gcd(false,s(x),s(y))	→	gcd(minus(y,x),s(x))	(10)

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:

le^#(0,z0)

→

(12)

originates from

le(0,z0)

→

true

(11)

le^#(s(z0),0)

→

(14)

originates from

le(s(z0),0)

→

false

(13)

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

→

c2(le^#(z0,z1))

(16)

originates from

le(s(z0),s(z1))

→

le(z0,z1)

(15)

minus^#(z0,0)

→

(18)

originates from

minus(z0,0)

→

(17)

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

→

c4(minus^#(z0,z1))

(20)

originates from

minus(s(z0),s(z1))

→

minus(z0,z1)

(19)

gcd^#(0,z0)

→

(22)

originates from

gcd(0,z0)

→

(21)

gcd^#(s(z0),0)

→

(24)

originates from

gcd(s(z0),0)

→

s(z0)

(23)

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

→

c7(if_gcd^#(le(z1,z0),s(z0),s(z1)),le^#(z1,z0))

(26)

originates from

gcd(s(z0),s(z1))

→

if_gcd(le(z1,z0),s(z0),s(z1))

(25)

if_gcd^#(true,s(z0),s(z1))

→

c8(gcd^#(minus(z0,z1),s(z1)),minus^#(z0,z1))

(28)

originates from

if_gcd(true,s(z0),s(z1))

→

gcd(minus(z0,z1),s(z1))

(27)

if_gcd^#(false,s(z0),s(z1))

→

c9(gcd^#(minus(z1,z0),s(z0)),minus^#(z1,z0))

(30)

originates from

if_gcd(false,s(z0),s(z1))

→

gcd(minus(z1,z0),s(z0))

(29)

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

le^#(0,z0)

le^#(s(z0),0)

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

minus^#(z0,0)

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

gcd^#(0,z0)

gcd^#(s(z0),0)

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

if_gcd^#(true,s(z0),s(z1))

if_gcd^#(false,s(z0),s(z1))

1.1 Usable Rules

We remove the following rules since they are not usable.

gcd(0,z0)	→	z0	(21)
gcd(s(z0),0)	→	s(z0)	(23)
gcd(s(z0),s(z1))	→	if_gcd(le(z1,z0),s(z0),s(z1))	(25)
if_gcd(true,s(z0),s(z1))	→	gcd(minus(z0,z1),s(z1))	(27)
if_gcd(false,s(z0),s(z1))	→	gcd(minus(z1,z0),s(z0))	(29)

1.1.1 Rule Shifting

The rules

gcd^#(0,z0)	→	c5	(22)
gcd^#(s(z0),0)	→	c6	(24)

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

[c]	=	0
[c1]	=	0
[c2(x₁)]	=	1 · x₁ + 0
[c3]	=	0
[c4(x₁)]	=	1 · x₁ + 0
[c5]	=	0
[c6]	=	0
[c7(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c8(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c9(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[le(x₁, x₂)]	=	0
[minus(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[le^#(x₁, x₂)]	=	0
[minus^#(x₁, x₂)]	=	0
[gcd^#(x₁, x₂)]	=	1
[if_gcd^#(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₃
[0]	=	0
[s(x₁)]	=	1
[true]	=	0
[false]	=	0

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

le^#(0,z0)	→	c	(12)
le^#(s(z0),0)	→	c1	(14)
le^#(s(z0),s(z1))	→	c2(le^#(z0,z1))	(16)
minus^#(z0,0)	→	c3	(18)
minus^#(s(z0),s(z1))	→	c4(minus^#(z0,z1))	(20)
gcd^#(0,z0)	→	c5	(22)
gcd^#(s(z0),0)	→	c6	(24)
gcd^#(s(z0),s(z1))	→	c7(if_gcd^#(le(z1,z0),s(z0),s(z1)),le^#(z1,z0))	(26)
if_gcd^#(true,s(z0),s(z1))	→	c8(gcd^#(minus(z0,z1),s(z1)),minus^#(z0,z1))	(28)
if_gcd^#(false,s(z0),s(z1))	→	c9(gcd^#(minus(z1,z0),s(z0)),minus^#(z1,z0))	(30)
le(s(z0),s(z1))	→	le(z0,z1)	(15)
le(s(z0),0)	→	false	(13)
le(0,z0)	→	true	(11)

1.1.1.1 Rule Shifting

The rules

le^#(0,z0)	→	c	(12)
le^#(s(z0),0)	→	c1	(14)
minus^#(z0,0)	→	c3	(18)
gcd^#(s(z0),s(z1))	→	c7(if_gcd^#(le(z1,z0),s(z0),s(z1)),le^#(z1,z0))	(26)
if_gcd^#(true,s(z0),s(z1))	→	c8(gcd^#(minus(z0,z1),s(z1)),minus^#(z0,z1))	(28)
if_gcd^#(false,s(z0),s(z1))	→	c9(gcd^#(minus(z1,z0),s(z0)),minus^#(z1,z0))	(30)

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

[c]	=	0
[c1]	=	0
[c2(x₁)]	=	1 · x₁ + 0
[c3]	=	0
[c4(x₁)]	=	1 · x₁ + 0
[c5]	=	0
[c6]	=	0
[c7(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c8(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c9(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[le(x₁, x₂)]	=	3 + 3 · x₁
[minus(x₁, x₂)]	=	1 · x₁ + 0
[le^#(x₁, x₂)]	=	1
[minus^#(x₁, x₂)]	=	1
[gcd^#(x₁, x₂)]	=	2 + 3 · x₁ + 3 · x₂
[if_gcd^#(x₁, x₂, x₃)]	=	3 · x₂ + 0 + 3 · x₃
[0]	=	0
[s(x₁)]	=	3 + 1 · x₁
[true]	=	3
[false]	=	3

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

le^#(0,z0)	→	c	(12)
le^#(s(z0),0)	→	c1	(14)
le^#(s(z0),s(z1))	→	c2(le^#(z0,z1))	(16)
minus^#(z0,0)	→	c3	(18)
minus^#(s(z0),s(z1))	→	c4(minus^#(z0,z1))	(20)
gcd^#(0,z0)	→	c5	(22)
gcd^#(s(z0),0)	→	c6	(24)
gcd^#(s(z0),s(z1))	→	c7(if_gcd^#(le(z1,z0),s(z0),s(z1)),le^#(z1,z0))	(26)
if_gcd^#(true,s(z0),s(z1))	→	c8(gcd^#(minus(z0,z1),s(z1)),minus^#(z0,z1))	(28)
if_gcd^#(false,s(z0),s(z1))	→	c9(gcd^#(minus(z1,z0),s(z0)),minus^#(z1,z0))	(30)
minus(s(z0),s(z1))	→	minus(z0,z1)	(19)
minus(z0,0)	→	z0	(17)

1.1.1.1.1 Rule Shifting

The rules

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

→

c4(minus^#(z0,z1))

(20)

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

[c]	=	0
[c1]	=	0
[c2(x₁)]	=	1 · x₁ + 0
[c3]	=	0
[c4(x₁)]	=	1 · x₁ + 0
[c5]	=	0
[c6]	=	0
[c7(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c8(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c9(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[le(x₁, x₂)]	=	0
[minus(x₁, x₂)]	=	1 · x₁ + 0
[le^#(x₁, x₂)]	=	0
[minus^#(x₁, x₂)]	=	2 + 1 · x₂
[gcd^#(x₁, x₂)]	=	2 · x₁ + 0 + 2 · x₂ + 1 · x₁ · x₂
[if_gcd^#(x₁, x₂, x₃)]	=	2 · x₂ + 0 + 2 · x₃ + 1 · x₂ · x₃
[0]	=	0
[s(x₁)]	=	2 + 1 · x₁
[true]	=	0
[false]	=	0

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

le^#(0,z0)	→	c	(12)
le^#(s(z0),0)	→	c1	(14)
le^#(s(z0),s(z1))	→	c2(le^#(z0,z1))	(16)
minus^#(z0,0)	→	c3	(18)
minus^#(s(z0),s(z1))	→	c4(minus^#(z0,z1))	(20)
gcd^#(0,z0)	→	c5	(22)
gcd^#(s(z0),0)	→	c6	(24)
gcd^#(s(z0),s(z1))	→	c7(if_gcd^#(le(z1,z0),s(z0),s(z1)),le^#(z1,z0))	(26)
if_gcd^#(true,s(z0),s(z1))	→	c8(gcd^#(minus(z0,z1),s(z1)),minus^#(z0,z1))	(28)
if_gcd^#(false,s(z0),s(z1))	→	c9(gcd^#(minus(z1,z0),s(z0)),minus^#(z1,z0))	(30)
minus(s(z0),s(z1))	→	minus(z0,z1)	(19)
minus(z0,0)	→	z0	(17)

1.1.1.1.1.1 Rule Shifting

The rules

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

→

c2(le^#(z0,z1))

(16)

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

[c]	=	0
[c1]	=	0
[c2(x₁)]	=	1 · x₁ + 0
[c3]	=	0
[c4(x₁)]	=	1 · x₁ + 0
[c5]	=	0
[c6]	=	0
[c7(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c8(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c9(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[le(x₁, x₂)]	=	0
[minus(x₁, x₂)]	=	1 · x₁ + 0
[le^#(x₁, x₂)]	=	1 + 1 · x₂
[minus^#(x₁, x₂)]	=	1 · x₁ + 0
[gcd^#(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂ · x₂ + 1 · x₁ · x₁
[if_gcd^#(x₁, x₂, x₃)]	=	1 · x₃ · x₃ + 0 + 1 · x₂ · x₂
[0]	=	0
[s(x₁)]	=	1 + 1 · x₁
[true]	=	0
[false]	=	0

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

le^#(0,z0)	→	c	(12)
le^#(s(z0),0)	→	c1	(14)
le^#(s(z0),s(z1))	→	c2(le^#(z0,z1))	(16)
minus^#(z0,0)	→	c3	(18)
minus^#(s(z0),s(z1))	→	c4(minus^#(z0,z1))	(20)
gcd^#(0,z0)	→	c5	(22)
gcd^#(s(z0),0)	→	c6	(24)
gcd^#(s(z0),s(z1))	→	c7(if_gcd^#(le(z1,z0),s(z0),s(z1)),le^#(z1,z0))	(26)
if_gcd^#(true,s(z0),s(z1))	→	c8(gcd^#(minus(z0,z1),s(z1)),minus^#(z0,z1))	(28)
if_gcd^#(false,s(z0),s(z1))	→	c9(gcd^#(minus(z1,z0),s(z0)),minus^#(z1,z0))	(30)
minus(s(z0),s(z1))	→	minus(z0,z1)	(19)
minus(z0,0)	→	z0	(17)

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).