Certification Problem
Input (COPS 606)
We consider the TRS containing the following rules:
|
not(true) |
→ |
false |
(1) |
|
not(false) |
→ |
true |
(2) |
|
or(true,y) |
→ |
true |
(3) |
|
or(x,true) |
→ |
true |
(4) |
|
or(false,false) |
→ |
false |
(5) |
|
implies(true,y) |
→ |
y |
(6) |
|
implies(false,y) |
→ |
true |
(7) |
|
implies(x,y) |
→ |
or(not(x),y) |
(8) |
The underlying signature is as follows:
{not/1, true/0, false/0, or/2, implies/2}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
|
= |
implies(not(false),y) |
|
→
|
implies(true,y) |
|
→
|
y |
|
= |
t2
|
| t0
|
= |
implies(not(false),y) |
|
→
|
or(not(not(false)),y) |
|
→
|
or(not(true),y) |
|
→
|
or(false,y) |
|
= |
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.