Certification Problem
Input (TPDB Runtime_Complexity_Innermost_Rewriting/Transformed_CSR_04/PEANO_nosorts_FR)
The rewrite relation of the following TRS is considered.
|
and(tt,X) |
→ |
activate(X) |
(1) |
|
plus(N,0) |
→ |
N |
(2) |
|
plus(N,s(M)) |
→ |
s(plus(N,M)) |
(3) |
|
activate(X) |
→ |
X |
(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:
{0, 1, 2, 3}
-
transitions:
|
tt0
|
→ |
0 |
|
00
|
→ |
0 |
|
s0(0) |
→ |
0 |
|
and0(0,0) |
→ |
1 |
|
plus0(0,0) |
→ |
2 |
|
activate0(0) |
→ |
3 |
|
activate1(0) |
→ |
1 |
|
plus1(0,0) |
→ |
4 |
|
s1(4) |
→ |
2 |
|
s1(4) |
→ |
4 |
| 0 |
→ |
2 |
| 0 |
→ |
3 |
| 0 |
→ |
4 |
| 0 |
→ |
1 |