Certification Problem
Input (COPS 226)
We consider the TRS containing the following rules:
| c |
→ |
f(c,c) |
(1) |
| c |
→ |
g(c,c) |
(2) |
|
f(x,f(x,x)) |
→ |
c |
(3) |
The underlying signature is as follows:
{c/0, f/2, g/2}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
|
= |
c |
|
→
|
f(c,c) |
|
→
|
f(g(c,c),c) |
|
→
|
f(g(c,c),g(c,c)) |
|
= |
t3
|
| t0
|
= |
c |
|
→
|
g(c,c) |
|
→
|
g(g(c,c),c) |
|
→
|
g(g(g(c,c),c),c) |
|
= |
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.