Certification Problem
Input (TPDB Runtime_Complexity_Innermost_Rewriting/Various_04/15)
The rewrite relation of the following TRS is considered.
|
f(s(x)) |
→ |
s(s(f(p(s(x))))) |
(1) |
|
f(0) |
→ |
0 |
(2) |
|
p(s(x)) |
→ |
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:
|
s0(0) |
→ |
0 |
|
00
|
→ |
0 |
|
f0(0) |
→ |
1 |
|
p0(0) |
→ |
2 |
|
s1(0) |
→ |
6 |
|
p1(6) |
→ |
5 |
|
f1(5) |
→ |
4 |
|
s1(4) |
→ |
3 |
|
s1(3) |
→ |
1 |
|
01
|
→ |
1 |
|
s1(3) |
→ |
4 |
|
01
|
→ |
4 |
| 0 |
→ |
2 |
| 0 |
→ |
5 |