Certification Problem
Input (TPDB Runtime_Complexity_Innermost_Rewriting/Frederiksen_Others/ocall-unsafe)
The rewrite relation of the following TRS is considered.
|
dbl(S(0),S(0)) |
→ |
S(S(S(S(0)))) |
(1) |
|
unsafe(S(x)) |
→ |
dbl(unsafe(x),0) |
(2) |
|
unsafe(0) |
→ |
0 |
(3) |
|
dbl(0,y) |
→ |
y |
(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}
-
transitions:
|
S0(0) |
→ |
0 |
|
00
|
→ |
0 |
|
dbl0(0,0) |
→ |
1 |
|
unsafe0(0) |
→ |
2 |
|
01
|
→ |
5 |
|
S1(5) |
→ |
4 |
|
S1(4) |
→ |
3 |
|
S1(3) |
→ |
3 |
|
S1(3) |
→ |
1 |
|
unsafe1(0) |
→ |
6 |
|
01
|
→ |
7 |
|
dbl1(6,7) |
→ |
2 |
|
01
|
→ |
2 |
|
dbl1(6,7) |
→ |
6 |
|
01
|
→ |
6 |
| 0 |
→ |
1 |
| 7 |
→ |
2 |
| 7 |
→ |
6 |