Certification Problem
Input (TPDB Runtime_Complexity_Innermost_Rewriting/Frederiksen_Glenstrup/append)
The rewrite relation of the following TRS is considered.
|
append(Cons(x,xs),ys) |
→ |
Cons(x,append(xs,ys)) |
(1) |
|
append(Nil,ys) |
→ |
ys |
(2) |
|
goal(x,y) |
→ |
append(x,y) |
(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:
|
Cons0(0,0) |
→ |
0 |
|
Nil0
|
→ |
0 |
|
append0(0,0) |
→ |
1 |
|
goal0(0,0) |
→ |
2 |
|
append1(0,0) |
→ |
3 |
|
Cons1(0,3) |
→ |
1 |
|
append1(0,0) |
→ |
2 |
|
Cons1(0,3) |
→ |
2 |
|
Cons1(0,3) |
→ |
3 |
| 0 |
→ |
1 |
| 0 |
→ |
2 |
| 0 |
→ |
3 |