Certification Problem
Input (TPDB Runtime_Complexity_Innermost_Rewriting/AG01/#3.33)
The rewrite relation of the following TRS is considered.
|
p(f(f(x))) |
→ |
q(f(g(x))) |
(1) |
|
p(g(g(x))) |
→ |
q(g(f(x))) |
(2) |
|
q(f(f(x))) |
→ |
p(f(g(x))) |
(3) |
|
q(g(g(x))) |
→ |
p(g(f(x))) |
(4) |
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:
{1, 2}
-
transitions:
|
p0(2) |
→ |
1 |
|
q0(2) |
→ |
1 |
|
q1(3) |
→ |
1 |
|
q1(5) |
→ |
1 |
|
p1(7) |
→ |
1 |
|
p1(9) |
→ |
1 |
|
f0(2) |
→ |
2 |
|
g0(2) |
→ |
2 |
|
f1(4) |
→ |
3 |
|
g1(2) |
→ |
4 |
|
g1(6) |
→ |
5 |
|
f1(2) |
→ |
6 |
|
f1(8) |
→ |
7 |
|
g1(2) |
→ |
8 |
|
g1(10) |
→ |
9 |
|
f1(2) |
→ |
10 |