Certification Problem
Input (COPS 80)
We consider the TRS containing the following rules:
| a |
→ |
f(a,b) |
(1) |
|
f(a,b) |
→ |
f(b,a) |
(2) |
The underlying signature is as follows:
{a/0, f/2, b/0}Property / Task
Prove or disprove confluence.Answer / Result
No.Proof (by csi @ CoCo 2022)
1 Non-Joinable Fork
The system is not confluent due to the following forking derivations.
| t0
|
= |
f(a,b) |
|
→
|
f(f(a,b),b) |
|
→
|
f(f(b,a),b) |
|
→
|
f(f(b,f(a,b)),b) |
|
= |
t3
|
| t0
|
= |
f(a,b) |
|
→
|
f(b,a) |
|
→
|
f(b,f(a,b)) |
|
→
|
f(b,f(b,a)) |
|
= |
t3
|
The two resulting terms cannot be joined for the following reason:
- When applying the cap-function on both terms (where variables may be treated like constants)
then the resulting terms do not unify.