Certification Problem

Input (TPDB Runtime_Complexity_Innermost_Rewriting/AG01/#3.48)

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,s(x),s(y))	→	s(x)	(4)
if(false,s(x),s(y))	→	s(y)	(5)
g(x,c(y))	→	c(g(x,y))	(6)
g(x,c(y))	→	g(x,if(f(x),c(g(s(x),y)),c(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,s(z0),s(z1))

→

(13)

originates from

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

→

s(z0)

(12)

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

→

(15)

originates from

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

→

s(z1)

(14)

g^#(z0,c(z1))

→

c6(g^#(z0,z1))

(17)

originates from

g(z0,c(z1))

→

c(g(z0,z1))

(16)

g^#(z0,c(z1))

→

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

(19)

originates from

g(z0,c(z1))

→

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

(18)

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

f^#(0)

f^#(1)

f^#(s(z0))

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

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

g^#(z0,c(z1))

1.1 Usable Rules

We remove the following rules since they are not usable.

if(true,s(z0),s(z1))	→	s(z0)	(12)
if(false,s(z0),s(z1))	→	s(z1)	(14)

1.1.1 Rule Shifting

The rules

if^#(true,s(z0),s(z1))	→	c4	(13)
if^#(false,s(z0),s(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)]	=	1 · x₁ + 0
[c7(x₁,...,x₄)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃ + 1 · x₄
[f(x₁)]	=	0
[g(x₁, x₂)]	=	0
[f^#(x₁)]	=	0
[if^#(x₁, x₂, x₃)]	=	1 · x₃ + 0
[g^#(x₁, x₂)]	=	0
[0]	=	1
[true]	=	1
[1]	=	1
[false]	=	1
[s(x₁)]	=	1 + 1 · x₁
[c(x₁)]	=	0
[if(x₁, x₂, x₃)]	=	1 + 1 · x₁ + 1 · x₃

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,s(z0),s(z1))	→	c4	(13)
if^#(false,s(z0),s(z1))	→	c5	(15)
g^#(z0,c(z1))	→	c6(g^#(z0,z1))	(17)
g^#(z0,c(z1))	→	c7(g^#(z0,if(f(z0),c(g(s(z0),z1)),c(z1))),if^#(f(z0),c(g(s(z0),z1)),c(z1)),f^#(z0),g^#(s(z0),z1))	(19)

1.1.1.1 Rule Shifting

The rules

g^#(z0,c(z1))

→

c6(g^#(z0,z1))

(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)]	=	1 · x₁ + 0
[c7(x₁,...,x₄)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃ + 1 · x₄
[f(x₁)]	=	0
[g(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[f^#(x₁)]	=	0
[if^#(x₁, x₂, x₃)]	=	1 · x₁ + 0
[g^#(x₁, x₂)]	=	1 · x₂ + 0
[0]	=	1
[true]	=	0
[1]	=	1
[false]	=	0
[s(x₁)]	=	1 + 1 · x₁
[c(x₁)]	=	1 + 1 · x₁
[if(x₁, x₂, x₃)]	=	1 + 1 · x₁

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

1.1.1.1.1 Rule Shifting

The rules

g^#(z0,c(z1))

→

c7(g^#(z0,if(f(z0),c(g(s(z0),z1)),c(z1))),if^#(f(z0),c(g(s(z0),z1)),c(z1)),f^#(z0),g^#(s(z0),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)]	=	1 · x₁ + 0
[c7(x₁,...,x₄)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃ + 1 · x₄
[f(x₁)]	=	0
[g(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[f^#(x₁)]	=	0
[if^#(x₁, x₂, x₃)]	=	1 · x₁ + 0
[g^#(x₁, x₂)]	=	1 · x₂ + 0
[0]	=	1
[true]	=	0
[1]	=	1
[false]	=	0
[s(x₁)]	=	1 + 1 · x₁
[c(x₁)]	=	1 + 1 · x₁
[if(x₁, x₂, x₃)]	=	1 · x₁ + 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,s(z0),s(z1))	→	c4	(13)
if^#(false,s(z0),s(z1))	→	c5	(15)
g^#(z0,c(z1))	→	c6(g^#(z0,z1))	(17)
g^#(z0,c(z1))	→	c7(g^#(z0,if(f(z0),c(g(s(z0),z1)),c(z1))),if^#(f(z0),c(g(s(z0),z1)),c(z1)),f^#(z0),g^#(s(z0),z1))	(19)
f(1)	→	false	(2)
f(0)	→	true	(1)
f(s(z0))	→	f(z0)	(10)

1.1.1.1.1.1 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)]	=	1 · x₁ + 0
[c7(x₁,...,x₄)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃ + 1 · x₄
[f(x₁)]	=	0
[g(x₁, x₂)]	=	1 · x₁ + 0
[f^#(x₁)]	=	1
[if^#(x₁, x₂, x₃)]	=	1 · x₁ + 0
[g^#(x₁, x₂)]	=	1 · x₂ + 0
[0]	=	1
[true]	=	0
[1]	=	1
[false]	=	0
[s(x₁)]	=	1 + 1 · x₁
[c(x₁)]	=	1 + 1 · x₁
[if(x₁, x₂, x₃)]	=	1 · x₁ + 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,s(z0),s(z1))	→	c4	(13)
if^#(false,s(z0),s(z1))	→	c5	(15)
g^#(z0,c(z1))	→	c6(g^#(z0,z1))	(17)
g^#(z0,c(z1))	→	c7(g^#(z0,if(f(z0),c(g(s(z0),z1)),c(z1))),if^#(f(z0),c(g(s(z0),z1)),c(z1)),f^#(z0),g^#(s(z0),z1))	(19)
f(1)	→	false	(2)
f(0)	→	true	(1)
f(s(z0))	→	f(z0)	(10)

1.1.1.1.1.1.1 Rule Shifting

The rules

f^#(s(z0))

→

c3(f^#(z0))

(11)

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

[c1]	=	0
[c2]	=	0
[c3(x₁)]	=	1 · x₁ + 0
[c4]	=	0
[c5]	=	0
[c6(x₁)]	=	1 · x₁ + 0
[c7(x₁,...,x₄)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃ + 1 · x₄
[f(x₁)]	=	0
[g(x₁, x₂)]	=	2 + 1 · x₂ + 1 · x₂ · x₂ + 2 · x₁ · x₂
[f^#(x₁)]	=	2 · x₁ + 0
[if^#(x₁, x₂, x₃)]	=	1
[g^#(x₁, x₂)]	=	1 + 2 · x₁ + 1 · x₂ · x₂ + 2 · x₁ · x₂
[0]	=	2
[true]	=	1
[1]	=	1
[false]	=	1
[s(x₁)]	=	1 + 1 · x₁
[c(x₁)]	=	2 + 1 · x₁
[if(x₁, x₂, x₃)]	=	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,s(z0),s(z1))	→	c4	(13)
if^#(false,s(z0),s(z1))	→	c5	(15)
g^#(z0,c(z1))	→	c6(g^#(z0,z1))	(17)
g^#(z0,c(z1))	→	c7(g^#(z0,if(f(z0),c(g(s(z0),z1)),c(z1))),if^#(f(z0),c(g(s(z0),z1)),c(z1)),f^#(z0),g^#(s(z0),z1))	(19)

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).