Certification Problem
Input (TPDB Runtime_Complexity_Innermost_Rewriting/SK90/2.11)
The rewrite relation of the following TRS is considered.
|
+(0,y) |
→ |
y |
(1) |
|
+(s(x),y) |
→ |
s(+(x,y)) |
(2) |
|
-(0,y) |
→ |
0 |
(3) |
|
-(x,0) |
→ |
x |
(4) |
|
-(s(x),s(y)) |
→ |
-(x,y) |
(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}
-
transitions:
|
00
|
→ |
0 |
|
s0(0) |
→ |
0 |
|
+0(0,0) |
→ |
1 |
|
-0(0,0) |
→ |
2 |
|
+1(0,0) |
→ |
3 |
|
s1(3) |
→ |
1 |
|
01
|
→ |
2 |
|
-1(0,0) |
→ |
2 |
|
s1(3) |
→ |
3 |
| 0 |
→ |
1 |
| 0 |
→ |
2 |
| 0 |
→ |
3 |