Certification Problem

Input (TPDB Runtime_Complexity_Innermost_Rewriting/Strategy_removed_AG01/#4.34)

The rewrite relation of the following TRS is considered.

f(0)	→	true	(1)
f(1)	→	false	(2)
f(s(x))	→	f(x)	(3)
if(true,x,y)	→	x	(4)
if(false,x,y)	→	y	(5)
g(s(x),s(y))	→	if(f(x),s(x),s(y))	(6)
g(x,c(y))	→	g(x,g(s(c(y)),y))	(7)

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:

f^#(0)

→

(8)

originates from

f(0)

→

true

(1)

f^#(1)

→

(9)

originates from

f(1)

→

false

(2)

f^#(s(z0))

→

c3(f^#(z0))

(11)

originates from

f(s(z0))

→

f(z0)

(10)

if^#(true,z0,z1)

→

(13)

originates from

if(true,z0,z1)

→

(12)

if^#(false,z0,z1)

→

(15)

originates from

if(false,z0,z1)

→

(14)

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

→

c6(if^#(f(z0),s(z0),s(z1)),f^#(z0))

(17)

originates from

g(s(z0),s(z1))

→

if(f(z0),s(z0),s(z1))

(16)

g^#(z0,c(z1))

→

c7(g^#(z0,g(s(c(z1)),z1)),g^#(s(c(z1)),z1))

(19)

originates from

g(z0,c(z1))

→

g(z0,g(s(c(z1)),z1))

(18)

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

f^#(0)

f^#(1)

f^#(s(z0))

if^#(true,z0,z1)

if^#(false,z0,z1)

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

g^#(z0,c(z1))

1.1 Rule Shifting

The rules

f^#(0)	→	c1	(8)
f^#(1)	→	c2	(9)

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

[c1]	=	0
[c2]	=	0
[c3(x₁)]	=	1 · x₁ + 0
[c4]	=	0
[c5]	=	0
[c6(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c7(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[f(x₁)]	=	0
[if(x₁, x₂, x₃)]	=	3
[g(x₁, x₂)]	=	3
[f^#(x₁)]	=	1 · x₁ + 0
[if^#(x₁, x₂, x₃)]	=	0
[g^#(x₁, x₂)]	=	2 · x₁ + 0
[s(x₁)]	=	1 · x₁ + 0
[c(x₁)]	=	0
[0]	=	3
[true]	=	3
[1]	=	1
[false]	=	3

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

f^#(0)	→	c1	(8)
f^#(1)	→	c2	(9)
f^#(s(z0))	→	c3(f^#(z0))	(11)
if^#(true,z0,z1)	→	c4	(13)
if^#(false,z0,z1)	→	c5	(15)
g^#(s(z0),s(z1))	→	c6(if^#(f(z0),s(z0),s(z1)),f^#(z0))	(17)
g^#(z0,c(z1))	→	c7(g^#(z0,g(s(c(z1)),z1)),g^#(s(c(z1)),z1))	(19)

1.1.1 Rule Shifting

The rules

if^#(true,z0,z1)	→	c4	(13)
if^#(false,z0,z1)	→	c5	(15)

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

[c1]	=	0
[c2]	=	0
[c3(x₁)]	=	1 · x₁ + 0
[c4]	=	0
[c5]	=	0
[c6(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c7(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[f(x₁)]	=	1 · x₁ + 0
[if(x₁, x₂, x₃)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[g(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[f^#(x₁)]	=	0
[if^#(x₁, x₂, x₃)]	=	1 · x₁ + 0
[g^#(x₁, x₂)]	=	1 · x₁ + 0
[s(x₁)]	=	1 · x₁ + 0
[c(x₁)]	=	0
[0]	=	1
[true]	=	1
[1]	=	1
[false]	=	1

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

f^#(0)	→	c1	(8)
f^#(1)	→	c2	(9)
f^#(s(z0))	→	c3(f^#(z0))	(11)
if^#(true,z0,z1)	→	c4	(13)
if^#(false,z0,z1)	→	c5	(15)
g^#(s(z0),s(z1))	→	c6(if^#(f(z0),s(z0),s(z1)),f^#(z0))	(17)
g^#(z0,c(z1))	→	c7(g^#(z0,g(s(c(z1)),z1)),g^#(s(c(z1)),z1))	(19)
f(1)	→	false	(2)
f(0)	→	true	(1)
f(s(z0))	→	f(z0)	(10)

1.1.1.1 Rule Shifting

The rules

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

→

c6(if^#(f(z0),s(z0),s(z1)),f^#(z0))

(17)

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

[c1]	=	0
[c2]	=	0
[c3(x₁)]	=	1 · x₁ + 0
[c4]	=	0
[c5]	=	0
[c6(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c7(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[f(x₁)]	=	0
[if(x₁, x₂, x₃)]	=	2 · x₂ + 0 + 2 · x₃
[g(x₁, x₂)]	=	0
[f^#(x₁)]	=	0
[if^#(x₁, x₂, x₃)]	=	3 · x₂ + 0 + 3 · x₃
[g^#(x₁, x₂)]	=	2 + 3 · x₁ + 1 · x₂
[s(x₁)]	=	0
[c(x₁)]	=	2 + 1 · x₁
[0]	=	3
[true]	=	3
[1]	=	3
[false]	=	3

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

f^#(0)	→	c1	(8)
f^#(1)	→	c2	(9)
f^#(s(z0))	→	c3(f^#(z0))	(11)
if^#(true,z0,z1)	→	c4	(13)
if^#(false,z0,z1)	→	c5	(15)
g^#(s(z0),s(z1))	→	c6(if^#(f(z0),s(z0),s(z1)),f^#(z0))	(17)
g^#(z0,c(z1))	→	c7(g^#(z0,g(s(c(z1)),z1)),g^#(s(c(z1)),z1))	(19)
g(s(z0),s(z1))	→	if(f(z0),s(z0),s(z1))	(16)
if(true,z0,z1)	→	z0	(12)
g(z0,c(z1))	→	g(z0,g(s(c(z1)),z1))	(18)
if(false,z0,z1)	→	z1	(14)

1.1.1.1.1 Rule Shifting

The rules

g^#(z0,c(z1))

→

c7(g^#(z0,g(s(c(z1)),z1)),g^#(s(c(z1)),z1))

(19)

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

[c1]	=	0
[c2]	=	0
[c3(x₁)]	=	1 · x₁ + 0
[c4]	=	0
[c5]	=	0
[c6(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c7(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[f(x₁)]	=	1
[if(x₁, x₂, x₃)]	=	1 · x₂ + 0 + 1 · x₃
[g(x₁, x₂)]	=	0
[f^#(x₁)]	=	0
[if^#(x₁, x₂, x₃)]	=	1 · x₂ + 0 + 1 · x₃
[g^#(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[s(x₁)]	=	0
[c(x₁)]	=	1 + 1 · x₁
[0]	=	1
[true]	=	1
[1]	=	1
[false]	=	1

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

f^#(0)	→	c1	(8)
f^#(1)	→	c2	(9)
f^#(s(z0))	→	c3(f^#(z0))	(11)
if^#(true,z0,z1)	→	c4	(13)
if^#(false,z0,z1)	→	c5	(15)
g^#(s(z0),s(z1))	→	c6(if^#(f(z0),s(z0),s(z1)),f^#(z0))	(17)
g^#(z0,c(z1))	→	c7(g^#(z0,g(s(c(z1)),z1)),g^#(s(c(z1)),z1))	(19)
g(s(z0),s(z1))	→	if(f(z0),s(z0),s(z1))	(16)
if(true,z0,z1)	→	z0	(12)
g(z0,c(z1))	→	g(z0,g(s(c(z1)),z1))	(18)
if(false,z0,z1)	→	z1	(14)

1.1.1.1.1.1 Rule Shifting

The rules

f^#(s(z0))

→

c3(f^#(z0))

(11)

are strictly oriented by the following linear polynomial interpretation over (2 x 2)-matrices with strict dimension 1 over the rationals with delta = 1

[g^#(x₁, x₂)]

0	0
0	0

1	0
0	0

· x₁ +

0	0
0	0

· x₂

[c1]

0	0
0	0

[c5]

0	0
0	0

[c6(x₁, x₂)]

0	0
0	0

1	0
0	0

· x₁ +

1	0
0	0

· x₂

[c3(x₁)]

0	0
0	0

1	0
0	0

· x₁

[if^#(x₁, x₂, x₃)]

0	0
0	0

0	0
0	0

· x₁ +

0	0
0	0

· x₂ +

0	0
0	0

· x₃

[c4]

0	0
0	0

[f^#(x₁)]

0	0
0	0

0	2
0	0

· x₁

[c2]

0	0
0	0

[c7(x₁, x₂)]

0	0
0	0

1	0
0	0

· x₁ +

1	0
0	0

· x₂

[s(x₁)]

0	0
4	0

0	2
0	1

· x₁

[true]

0	0
0	0

[c(x₁)]

0	0
0	0

0	0
0	0

· x₁

[f(x₁)]

0	0
0	0

0	0
0	0

· x₁

[if(x₁, x₂, x₃)]

0	0
0	0

0	0
0	0

· x₁ +

0	0
0	0

· x₂ +

0	0
0	0

· x₃

[0]

0	0
0	0

[1]

0	0
0	0

[g(x₁, x₂)]

0	0
0	0

0	0
0	0

· x₁ +

0	0
0	0

· x₂

[false]

0	0
0	0

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

f^#(0)	→	c1	(8)
f^#(1)	→	c2	(9)
f^#(s(z0))	→	c3(f^#(z0))	(11)
if^#(true,z0,z1)	→	c4	(13)
if^#(false,z0,z1)	→	c5	(15)
g^#(s(z0),s(z1))	→	c6(if^#(f(z0),s(z0),s(z1)),f^#(z0))	(17)
g^#(z0,c(z1))	→	c7(g^#(z0,g(s(c(z1)),z1)),g^#(s(c(z1)),z1))	(19)

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).