Certification Problem
Input (COPS 1279)
We consider the TRS containing the following rules:
|
f(g(x)) |
→ |
g(f(f(x))) |
(1) |
|
g(x) |
→ |
x |
(2) |
The underlying signature is as follows:
{f/1, g/1}Property / Task
Prove or disprove confluence.Answer / Result
No.Proof (by csi @ CoCo 2023)
1 Non-Joinable Fork
The system is not confluent due to the following forking derivations.
| t0
|
= |
f(g(f2)) |
|
→
|
f(f2) |
|
= |
t1
|
| t0
|
= |
f(g(f2)) |
|
→
|
g(f(f(f2))) |
|
= |
t1
|
The two resulting terms cannot be joined for the following reason:
-
The reachable terms of these two terms are approximated via the following two tree automata,
and the tree automata have an empty intersection.
-
Automaton 1
-
final states:
{1}
-
transitions:
The automaton is closed under rewriting as it is compatible.
-
Automaton 2
-
final states:
{3}
-
transitions:
| 6 |
→ |
3 |
| f2 |
→ |
4 |
|
f(5) |
→ |
6 |
|
f(4) |
→ |
5 |
|
g(6) |
→ |
3 |
The automaton is closed under rewriting as it is compatible.