Certification Problem
Input (TPDB Runtime_Complexity_Innermost_Rewriting/SK90/2.48)
The rewrite relation of the following TRS is considered.
|
d(x) |
→ |
e(u(x)) |
(1) |
|
d(u(x)) |
→ |
c(x) |
(2) |
|
c(u(x)) |
→ |
b(x) |
(3) |
|
v(e(x)) |
→ |
x |
(4) |
|
b(u(x)) |
→ |
a(e(x)) |
(5) |
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, 4}
-
transitions:
|
e0(0) |
→ |
0 |
|
u0(0) |
→ |
0 |
|
a0(0) |
→ |
0 |
|
d0(0) |
→ |
1 |
|
c0(0) |
→ |
2 |
|
v0(0) |
→ |
3 |
|
b0(0) |
→ |
4 |
|
u1(0) |
→ |
5 |
|
e1(5) |
→ |
1 |
|
c1(0) |
→ |
1 |
|
b1(0) |
→ |
2 |
|
e1(0) |
→ |
6 |
|
a1(6) |
→ |
4 |
|
b1(0) |
→ |
1 |
|
a1(6) |
→ |
2 |
|
a1(6) |
→ |
1 |
| 0 |
→ |
3 |