Certification Problem

Input (TPDB Runtime_Complexity_Innermost_Rewriting/Rubio_04/gcd)

The rewrite relation of the following TRS is considered.

minus(X,s(Y))	→	pred(minus(X,Y))	(1)
minus(X,0)	→	X	(2)
pred(s(X))	→	X	(3)
le(s(X),s(Y))	→	le(X,Y)	(4)
le(s(X),0)	→	false	(5)
le(0,Y)	→	true	(6)
gcd(0,Y)	→	0	(7)
gcd(s(X),0)	→	s(X)	(8)
gcd(s(X),s(Y))	→	if(le(Y,X),s(X),s(Y))	(9)
if(true,s(X),s(Y))	→	gcd(minus(X,Y),s(Y))	(10)
if(false,s(X),s(Y))	→	gcd(minus(Y,X),s(X))	(11)

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:

minus^#(z0,s(z1))

→

c(pred^#(minus(z0,z1)),minus^#(z0,z1))

(13)

originates from

minus(z0,s(z1))

→

pred(minus(z0,z1))

(12)

minus^#(z0,0)

→

(15)

originates from

minus(z0,0)

→

(14)

pred^#(s(z0))

→

(17)

originates from

pred(s(z0))

→

(16)

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

→

c3(le^#(z0,z1))

(19)

originates from

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

→

le(z0,z1)

(18)

le^#(s(z0),0)

→

(21)

originates from

le(s(z0),0)

→

false

(20)

le^#(0,z0)

→

(23)

originates from

le(0,z0)

→

true

(22)

gcd^#(0,z0)

→

(25)

originates from

gcd(0,z0)

→

(24)

gcd^#(s(z0),0)

→

(27)

originates from

gcd(s(z0),0)

→

s(z0)

(26)

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

→

c8(if^#(le(z1,z0),s(z0),s(z1)),le^#(z1,z0))

(29)

originates from

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

→

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

(28)

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

→

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

(31)

originates from

if(true,s(z0),s(z1))

→

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

(30)

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

→

c10(gcd^#(minus(z1,z0),s(z0)),minus^#(z1,z0))

(33)

originates from

if(false,s(z0),s(z1))

→

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

(32)

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

minus^#(z0,s(z1))

minus^#(z0,0)

pred^#(s(z0))

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

le^#(s(z0),0)

le^#(0,z0)

gcd^#(0,z0)

gcd^#(s(z0),0)

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

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

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

1.1 Usable Rules

We remove the following rules since they are not usable.

gcd(0,z0)	→	0	(24)
gcd(s(z0),0)	→	s(z0)	(26)
gcd(s(z0),s(z1))	→	if(le(z1,z0),s(z0),s(z1))	(28)
if(true,s(z0),s(z1))	→	gcd(minus(z0,z1),s(z1))	(30)
if(false,s(z0),s(z1))	→	gcd(minus(z1,z0),s(z0))	(32)

1.1.1 Rule Shifting

The rules

le^#(s(z0),0)	→	c4	(21)
le^#(0,z0)	→	c5	(23)
gcd^#(0,z0)	→	c6	(25)
gcd^#(s(z0),0)	→	c7	(27)

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

[c(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c1]	=	0
[c2]	=	0
[c3(x₁)]	=	1 · x₁ + 0
[c4]	=	0
[c5]	=	0
[c6]	=	0
[c7]	=	0
[c8(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c9(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c10(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[minus(x₁, x₂)]	=	1 · x₁ + 0
[pred(x₁)]	=	1 · x₁ + 0
[le(x₁, x₂)]	=	0
[minus^#(x₁, x₂)]	=	0
[pred^#(x₁)]	=	0
[le^#(x₁, x₂)]	=	1
[gcd^#(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[if^#(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[s(x₁)]	=	1 + 1 · x₁
[0]	=	1
[false]	=	0
[true]	=	0

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

minus^#(z0,s(z1))	→	c(pred^#(minus(z0,z1)),minus^#(z0,z1))	(13)
minus^#(z0,0)	→	c1	(15)
pred^#(s(z0))	→	c2	(17)
le^#(s(z0),s(z1))	→	c3(le^#(z0,z1))	(19)
le^#(s(z0),0)	→	c4	(21)
le^#(0,z0)	→	c5	(23)
gcd^#(0,z0)	→	c6	(25)
gcd^#(s(z0),0)	→	c7	(27)
gcd^#(s(z0),s(z1))	→	c8(if^#(le(z1,z0),s(z0),s(z1)),le^#(z1,z0))	(29)
if^#(true,s(z0),s(z1))	→	c9(gcd^#(minus(z0,z1),s(z1)),minus^#(z0,z1))	(31)
if^#(false,s(z0),s(z1))	→	c10(gcd^#(minus(z1,z0),s(z0)),minus^#(z1,z0))	(33)
minus(z0,s(z1))	→	pred(minus(z0,z1))	(12)
le(s(z0),s(z1))	→	le(z0,z1)	(18)
le(s(z0),0)	→	false	(20)
le(0,z0)	→	true	(22)
minus(z0,0)	→	z0	(14)
pred(s(z0))	→	z0	(16)

1.1.1.1 Rule Shifting

The rules

if^#(true,s(z0),s(z1))	→	c9(gcd^#(minus(z0,z1),s(z1)),minus^#(z0,z1))	(31)
if^#(false,s(z0),s(z1))	→	c10(gcd^#(minus(z1,z0),s(z0)),minus^#(z1,z0))	(33)

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

[c(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c1]	=	0
[c2]	=	0
[c3(x₁)]	=	1 · x₁ + 0
[c4]	=	0
[c5]	=	0
[c6]	=	0
[c7]	=	0
[c8(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c9(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c10(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[minus(x₁, x₂)]	=	1 · x₁ + 0
[pred(x₁)]	=	1 · x₁ + 0
[le(x₁, x₂)]	=	1 + 1 · x₂
[minus^#(x₁, x₂)]	=	0
[pred^#(x₁)]	=	0
[le^#(x₁, x₂)]	=	0
[gcd^#(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[if^#(x₁, x₂, x₃)]	=	1 · x₂ + 0 + 1 · x₃
[s(x₁)]	=	1 + 1 · x₁
[0]	=	1
[false]	=	1
[true]	=	1

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

minus^#(z0,s(z1))	→	c(pred^#(minus(z0,z1)),minus^#(z0,z1))	(13)
minus^#(z0,0)	→	c1	(15)
pred^#(s(z0))	→	c2	(17)
le^#(s(z0),s(z1))	→	c3(le^#(z0,z1))	(19)
le^#(s(z0),0)	→	c4	(21)
le^#(0,z0)	→	c5	(23)
gcd^#(0,z0)	→	c6	(25)
gcd^#(s(z0),0)	→	c7	(27)
gcd^#(s(z0),s(z1))	→	c8(if^#(le(z1,z0),s(z0),s(z1)),le^#(z1,z0))	(29)
if^#(true,s(z0),s(z1))	→	c9(gcd^#(minus(z0,z1),s(z1)),minus^#(z0,z1))	(31)
if^#(false,s(z0),s(z1))	→	c10(gcd^#(minus(z1,z0),s(z0)),minus^#(z1,z0))	(33)
minus(z0,s(z1))	→	pred(minus(z0,z1))	(12)
minus(z0,0)	→	z0	(14)
pred(s(z0))	→	z0	(16)

1.1.1.1.1 Rule Shifting

The rules

minus^#(z0,0)

→

(15)

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

[c(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c1]	=	0
[c2]	=	0
[c3(x₁)]	=	1 · x₁ + 0
[c4]	=	0
[c5]	=	0
[c6]	=	0
[c7]	=	0
[c8(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c9(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c10(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[minus(x₁, x₂)]	=	1 · x₁ + 0
[pred(x₁)]	=	1 · x₁ + 0
[le(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[minus^#(x₁, x₂)]	=	1
[pred^#(x₁)]	=	0
[le^#(x₁, x₂)]	=	0
[gcd^#(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[if^#(x₁, x₂, x₃)]	=	1 · x₂ + 0 + 1 · x₃
[s(x₁)]	=	1 + 1 · x₁
[0]	=	1
[false]	=	1
[true]	=	1

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

minus^#(z0,s(z1))	→	c(pred^#(minus(z0,z1)),minus^#(z0,z1))	(13)
minus^#(z0,0)	→	c1	(15)
pred^#(s(z0))	→	c2	(17)
le^#(s(z0),s(z1))	→	c3(le^#(z0,z1))	(19)
le^#(s(z0),0)	→	c4	(21)
le^#(0,z0)	→	c5	(23)
gcd^#(0,z0)	→	c6	(25)
gcd^#(s(z0),0)	→	c7	(27)
gcd^#(s(z0),s(z1))	→	c8(if^#(le(z1,z0),s(z0),s(z1)),le^#(z1,z0))	(29)
if^#(true,s(z0),s(z1))	→	c9(gcd^#(minus(z0,z1),s(z1)),minus^#(z0,z1))	(31)
if^#(false,s(z0),s(z1))	→	c10(gcd^#(minus(z1,z0),s(z0)),minus^#(z1,z0))	(33)
minus(z0,s(z1))	→	pred(minus(z0,z1))	(12)
minus(z0,0)	→	z0	(14)
pred(s(z0))	→	z0	(16)

1.1.1.1.1.1 Rule Shifting

The rules

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

→

c8(if^#(le(z1,z0),s(z0),s(z1)),le^#(z1,z0))

(29)

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

[c(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c1]	=	0
[c2]	=	0
[c3(x₁)]	=	1 · x₁ + 0
[c4]	=	0
[c5]	=	0
[c6]	=	0
[c7]	=	0
[c8(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c9(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c10(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[minus(x₁, x₂)]	=	1 · x₁ + 0
[pred(x₁)]	=	1 · x₁ + 0
[le(x₁, x₂)]	=	0
[minus^#(x₁, x₂)]	=	0
[pred^#(x₁)]	=	0
[le^#(x₁, x₂)]	=	0
[gcd^#(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[if^#(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[s(x₁)]	=	1 + 1 · x₁
[0]	=	1
[false]	=	0
[true]	=	0

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

minus^#(z0,s(z1))	→	c(pred^#(minus(z0,z1)),minus^#(z0,z1))	(13)
minus^#(z0,0)	→	c1	(15)
pred^#(s(z0))	→	c2	(17)
le^#(s(z0),s(z1))	→	c3(le^#(z0,z1))	(19)
le^#(s(z0),0)	→	c4	(21)
le^#(0,z0)	→	c5	(23)
gcd^#(0,z0)	→	c6	(25)
gcd^#(s(z0),0)	→	c7	(27)
gcd^#(s(z0),s(z1))	→	c8(if^#(le(z1,z0),s(z0),s(z1)),le^#(z1,z0))	(29)
if^#(true,s(z0),s(z1))	→	c9(gcd^#(minus(z0,z1),s(z1)),minus^#(z0,z1))	(31)
if^#(false,s(z0),s(z1))	→	c10(gcd^#(minus(z1,z0),s(z0)),minus^#(z1,z0))	(33)
minus(z0,s(z1))	→	pred(minus(z0,z1))	(12)
le(s(z0),s(z1))	→	le(z0,z1)	(18)
le(s(z0),0)	→	false	(20)
le(0,z0)	→	true	(22)
minus(z0,0)	→	z0	(14)
pred(s(z0))	→	z0	(16)

1.1.1.1.1.1.1 Rule Shifting

The rules

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

→

c3(le^#(z0,z1))

(19)

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

[c(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c1]	=	0
[c2]	=	0
[c3(x₁)]	=	1 · x₁ + 0
[c4]	=	0
[c5]	=	0
[c6]	=	0
[c7]	=	0
[c8(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c9(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c10(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[minus(x₁, x₂)]	=	1 · x₁ + 0
[pred(x₁)]	=	1 · x₁ + 0
[le(x₁, x₂)]	=	0
[minus^#(x₁, x₂)]	=	0
[pred^#(x₁)]	=	0
[le^#(x₁, x₂)]	=	1 · x₂ + 0
[gcd^#(x₁, x₂)]	=	1 · x₁ + 0 + 2 · x₂ · x₂ + 2 · x₁ · x₁
[if^#(x₁, x₂, x₃)]	=	2 · x₃ · x₃ + 0 + 2 · x₂ · x₂
[s(x₁)]	=	2 + 1 · x₁
[0]	=	0
[false]	=	0
[true]	=	0

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

minus^#(z0,s(z1))	→	c(pred^#(minus(z0,z1)),minus^#(z0,z1))	(13)
minus^#(z0,0)	→	c1	(15)
pred^#(s(z0))	→	c2	(17)
le^#(s(z0),s(z1))	→	c3(le^#(z0,z1))	(19)
le^#(s(z0),0)	→	c4	(21)
le^#(0,z0)	→	c5	(23)
gcd^#(0,z0)	→	c6	(25)
gcd^#(s(z0),0)	→	c7	(27)
gcd^#(s(z0),s(z1))	→	c8(if^#(le(z1,z0),s(z0),s(z1)),le^#(z1,z0))	(29)
if^#(true,s(z0),s(z1))	→	c9(gcd^#(minus(z0,z1),s(z1)),minus^#(z0,z1))	(31)
if^#(false,s(z0),s(z1))	→	c10(gcd^#(minus(z1,z0),s(z0)),minus^#(z1,z0))	(33)
minus(z0,s(z1))	→	pred(minus(z0,z1))	(12)
minus(z0,0)	→	z0	(14)
pred(s(z0))	→	z0	(16)

1.1.1.1.1.1.1.1 Rule Shifting

The rules

minus^#(z0,s(z1))

→

c(pred^#(minus(z0,z1)),minus^#(z0,z1))

(13)

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

[c(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c1]	=	0
[c2]	=	0
[c3(x₁)]	=	1 · x₁ + 0
[c4]	=	0
[c5]	=	0
[c6]	=	0
[c7]	=	0
[c8(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c9(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c10(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[minus(x₁, x₂)]	=	1 · x₁ + 0
[pred(x₁)]	=	1 · x₁ + 0
[le(x₁, x₂)]	=	1
[minus^#(x₁, x₂)]	=	1 · x₂ + 0
[pred^#(x₁)]	=	0
[le^#(x₁, x₂)]	=	0
[gcd^#(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₁ · x₂
[if^#(x₁, x₂, x₃)]	=	1 · x₂ + 0 + 1 · x₂ · x₃ + 1 · x₁ · x₃
[s(x₁)]	=	1 + 1 · x₁
[0]	=	0
[false]	=	1
[true]	=	1

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

minus^#(z0,s(z1))	→	c(pred^#(minus(z0,z1)),minus^#(z0,z1))	(13)
minus^#(z0,0)	→	c1	(15)
pred^#(s(z0))	→	c2	(17)
le^#(s(z0),s(z1))	→	c3(le^#(z0,z1))	(19)
le^#(s(z0),0)	→	c4	(21)
le^#(0,z0)	→	c5	(23)
gcd^#(0,z0)	→	c6	(25)
gcd^#(s(z0),0)	→	c7	(27)
gcd^#(s(z0),s(z1))	→	c8(if^#(le(z1,z0),s(z0),s(z1)),le^#(z1,z0))	(29)
if^#(true,s(z0),s(z1))	→	c9(gcd^#(minus(z0,z1),s(z1)),minus^#(z0,z1))	(31)
if^#(false,s(z0),s(z1))	→	c10(gcd^#(minus(z1,z0),s(z0)),minus^#(z1,z0))	(33)
minus(z0,s(z1))	→	pred(minus(z0,z1))	(12)
le(s(z0),s(z1))	→	le(z0,z1)	(18)
le(s(z0),0)	→	false	(20)
le(0,z0)	→	true	(22)
minus(z0,0)	→	z0	(14)
pred(s(z0))	→	z0	(16)

1.1.1.1.1.1.1.1.1 Rule Shifting

The rules

pred^#(s(z0))

→

(17)

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

[c(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c1]	=	0
[c2]	=	0
[c3(x₁)]	=	1 · x₁ + 0
[c4]	=	0
[c5]	=	0
[c6]	=	0
[c7]	=	0
[c8(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c9(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c10(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[minus(x₁, x₂)]	=	1 · x₁ + 0
[pred(x₁)]	=	1 · x₁ + 0
[le(x₁, x₂)]	=	0
[minus^#(x₁, x₂)]	=	2 + 1 · x₂
[pred^#(x₁)]	=	1
[le^#(x₁, x₂)]	=	0
[gcd^#(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₁ · x₂
[if^#(x₁, x₂, x₃)]	=	1 · x₂ + 0 + 1 · x₃ + 1 · x₂ · x₃
[s(x₁)]	=	1 + 1 · x₁
[0]	=	0
[false]	=	0
[true]	=	0

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

minus^#(z0,s(z1))	→	c(pred^#(minus(z0,z1)),minus^#(z0,z1))	(13)
minus^#(z0,0)	→	c1	(15)
pred^#(s(z0))	→	c2	(17)
le^#(s(z0),s(z1))	→	c3(le^#(z0,z1))	(19)
le^#(s(z0),0)	→	c4	(21)
le^#(0,z0)	→	c5	(23)
gcd^#(0,z0)	→	c6	(25)
gcd^#(s(z0),0)	→	c7	(27)
gcd^#(s(z0),s(z1))	→	c8(if^#(le(z1,z0),s(z0),s(z1)),le^#(z1,z0))	(29)
if^#(true,s(z0),s(z1))	→	c9(gcd^#(minus(z0,z1),s(z1)),minus^#(z0,z1))	(31)
if^#(false,s(z0),s(z1))	→	c10(gcd^#(minus(z1,z0),s(z0)),minus^#(z1,z0))	(33)
minus(z0,s(z1))	→	pred(minus(z0,z1))	(12)
minus(z0,0)	→	z0	(14)
pred(s(z0))	→	z0	(16)

1.1.1.1.1.1.1.1.1.1 R is empty

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