Certification Problem
Input (COPS 682)
We consider the TRS containing the following rules:
|
h(a,a) |
→ |
b |
(1) |
| b |
→ |
c |
(2) |
|
f(f(h(b,a))) |
→ |
a |
(3) |
| a |
→ |
b |
(4) |
The underlying signature is as follows:
{h/2, a/0, b/0, c/0, f/1}Property / Task
Prove or disprove confluence.Answer / Result
No.Proof (by csi @ CoCo 2021)
1 Non-Joinable Fork
The system is not confluent due to the following forking derivations.
| t0
|
= |
f(f(h(b,a))) |
|
→
|
f(f(h(c,a))) |
|
→
|
f(f(h(c,b))) |
|
→
|
f(f(h(c,c))) |
|
= |
t3
|
| t0
|
= |
f(f(h(b,a))) |
|
→
|
a |
|
→
|
b |
|
→
|
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.