Certification Problem
Input (COPS 9)
We consider the TRS containing the following rules:
The underlying signature is as follows:
{f/1, g/1}Property / Task
Prove or disprove confluence.Answer / Result
No.Proof (by csi @ CoCo 2020)
1 Non-Joinable Fork
The system is not confluent due to the following forking derivations.
| t0
|
= |
f(g(f(g(f(f2))))) |
|
→
|
f(g(g(f2))) |
|
= |
t1
|
| t0
|
= |
f(g(f(g(f(f2))))) |
|
→
|
g(g(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:
| f2 |
→ |
2 |
|
g(2) |
→ |
3 |
|
g(3) |
→ |
4 |
|
f(4) |
→ |
1 |
The automaton is closed under rewriting as it is compatible.
-
Automaton 2
-
final states:
{5}
-
transitions:
| f2 |
→ |
6 |
|
g(7) |
→ |
8 |
|
g(8) |
→ |
5 |
|
f(6) |
→ |
7 |
The automaton is closed under rewriting as it is compatible.