Certification Problem
Input (TPDB Runtime_Complexity_Innermost_Rewriting/SK90/2.53)
The rewrite relation of the following TRS is considered.
|
f(x,y) |
→ |
g(x,y) |
(1) |
|
g(h(x),y) |
→ |
h(f(x,y)) |
(2) |
|
g(h(x),y) |
→ |
h(g(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 2.
This is shown by the following automaton.
-
final states:
{0, 1, 2}
-
transitions:
|
h0(0) |
→ |
0 |
|
f0(0,0) |
→ |
1 |
|
g0(0,0) |
→ |
2 |
|
g1(0,0) |
→ |
1 |
|
f1(0,0) |
→ |
3 |
|
h1(3) |
→ |
2 |
|
g1(0,0) |
→ |
4 |
|
h1(4) |
→ |
2 |
|
g2(0,0) |
→ |
3 |
|
h1(3) |
→ |
1 |
|
h1(3) |
→ |
4 |
|
h1(4) |
→ |
1 |
|
h1(4) |
→ |
4 |
|
h1(3) |
→ |
3 |
|
h1(4) |
→ |
3 |