Certification Problem
Input (TPDB Runtime_Complexity_Innermost_Rewriting/Waldmann_06/jwmatchb2)
The rewrite relation of the following TRS is considered.
|
h(f(x,y)) |
→ |
f(f(a,h(h(y))),x) |
(1) |
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}
-
transitions:
|
f0(0,0) |
→ |
0 |
|
a0
|
→ |
0 |
|
h0(0) |
→ |
1 |
|
a1
|
→ |
3 |
|
h1(0) |
→ |
5 |
|
h1(5) |
→ |
4 |
|
f1(3,4) |
→ |
2 |
|
f1(2,0) |
→ |
1 |
|
f1(2,0) |
→ |
5 |
|
a2
|
→ |
7 |
|
h2(0) |
→ |
9 |
|
h2(9) |
→ |
8 |
|
f2(7,8) |
→ |
6 |
|
f2(6,2) |
→ |
4 |
|
f1(2,0) |
→ |
9 |
|
f2(6,2) |
→ |
8 |