Certification Problem
Input (TPDB Runtime_Complexity_Innermost_Rewriting/Frederiksen_Others/oddeven)
The rewrite relation of the following TRS is considered.
|
odd(S(x)) |
→ |
even(x) |
(1) |
|
even(S(x)) |
→ |
odd(x) |
(2) |
|
odd(0) |
→ |
0 |
(3) |
|
even(0) |
→ |
S(0) |
(4) |
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:
{1, 2}
-
transitions:
|
odd0(2) |
→ |
1 |
|
even0(2) |
→ |
1 |
|
even1(2) |
→ |
1 |
|
01
|
→ |
1 |
|
odd1(2) |
→ |
1 |
|
S1(7) |
→ |
1 |
|
S0(2) |
→ |
2 |
|
00
|
→ |
2 |
|
01
|
→ |
7 |