Certification Problem
Input (COPS 238)
We consider the TRS containing the following rules:
|
a(x) |
→ |
g(g(b(x))) |
(1) |
|
a(x) |
→ |
g(c(x)) |
(2) |
|
b(x) |
→ |
g(b(x)) |
(3) |
The underlying signature is as follows:
{a/1, g/1, b/1, c/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
|
= |
a(a(x10)) |
|
→
|
a(g(g(b(x10)))) |
|
→
|
a(g(g(g(b(x10))))) |
|
→
|
g(c(g(g(g(b(x10)))))) |
|
= |
t3
|
| t0
|
= |
a(a(x10)) |
|
→
|
g(g(b(a(x10)))) |
|
→
|
g(g(g(b(a(x10))))) |
|
→
|
g(g(g(g(b(a(x10)))))) |
|
= |
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.