Certification Problem
Input (TPDB Runtime_Complexity_Innermost_Rewriting/Transformed_CSR_04/Ex6_Luc98_L)
The rewrite relation of the following TRS is considered.
|
first(0,X) |
→ |
nil |
(1) |
|
first(s(X),cons(Y)) |
→ |
cons(Y) |
(2) |
|
from(X) |
→ |
cons(X) |
(3) |
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 Bounds
The given TRS is
match-(raise)-bounded by 1.
This is shown by the following automaton.
-
final states:
{0, 1, 2}
-
transitions:
|
00
|
→ |
0 |
|
nil0
|
→ |
0 |
|
s0(0) |
→ |
0 |
|
cons0(0) |
→ |
0 |
|
first0(0,0) |
→ |
1 |
|
from0(0) |
→ |
2 |
|
nil1
|
→ |
1 |
|
cons1(0) |
→ |
1 |
|
cons1(0) |
→ |
2 |