Certification Problem
Input (TPDB Runtime_Complexity_Innermost_Rewriting/Strategy_removed_mixed_05/test830)
The rewrite relation of the following TRS is considered.
|
f(s(X)) |
→ |
f(X) |
(1) |
|
g(cons(0,Y)) |
→ |
g(Y) |
(2) |
|
g(cons(s(X),Y)) |
→ |
s(X) |
(3) |
|
h(cons(X,Y)) |
→ |
h(g(cons(X,Y))) |
(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:
{0, 1, 2, 3}
-
transitions:
|
s0(0) |
→ |
0 |
|
cons0(0,0) |
→ |
0 |
|
00
|
→ |
0 |
|
f0(0) |
→ |
1 |
|
g0(0) |
→ |
2 |
|
h0(0) |
→ |
3 |
|
f1(0) |
→ |
1 |
|
g1(0) |
→ |
2 |
|
s1(0) |
→ |
2 |
|
cons1(0,0) |
→ |
5 |
|
g1(5) |
→ |
4 |
|
h1(4) |
→ |
3 |
|
g1(0) |
→ |
4 |
|
s1(0) |
→ |
4 |