Certification Problem

Input (TPDB Runtime_Complexity_Innermost_Rewriting/Der95/18)

The rewrite relation of the following TRS is considered.

*(x,+(y,z))

→

+(*(x,y),*(x,z))

(1)

The evaluation strategy is innermost.

Property / Task

Determine bounds on the runtime complexity.

Answer / Result

An upperbound for the complexity is O(n).

Proof (by AProVE @ termCOMP 2023)

1 Dependency Tuples

We get the following set of dependency tuples:

*^#(z0,+(z1,z2))

→

c(*^#(z0,z1),*^#(z0,z2))

(3)

originates from

*(z0,+(z1,z2))

→

+(*(z0,z1),*(z0,z2))

(2)

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

*^#(z0,+(z1,z2))

1.1 Usable Rules

We remove the following rules since they are not usable.

*(z0,+(z1,z2))

→

+(*(z0,z1),*(z0,z2))

(2)

1.1.1 Rule Shifting

The rules

*^#(z0,+(z1,z2))

→

c(*^#(z0,z1),*^#(z0,z2))

(3)

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

[c(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[*^#(x₁, x₂)]	=	2 · x₂ + 0
[+(x₁, x₂)]	=	3 + 1 · x₁ + 1 · x₂

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

*^#(z0,+(z1,z2))

→

c(*^#(z0,z1),*^#(z0,z2))

(3)

1.1.1.1 R is empty

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