Certification Problem
Input (COPS 592)
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) |
|
and(true,true) |
→ |
true |
(6) |
|
and(x,true) |
→ |
x |
(7) |
|
and(true,y) |
→ |
y |
(8) |
|
and(false,false) |
→ |
false |
(9) |
|
not(and(x,y)) |
→ |
or(not(x),not(y)) |
(10) |
|
not(or(x,y)) |
→ |
and(not(x),not(y)) |
(11) |
The underlying signature is as follows:
{not/1, true/0, false/0, or/2, and/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
|
= |
not(and(not(false),y)) |
|
→
|
not(and(true,y)) |
|
→
|
not(y) |
|
= |
t2
|
| t0
|
= |
not(and(not(false),y)) |
|
→
|
or(not(not(false)),not(y)) |
|
→
|
or(not(true),not(y)) |
|
→
|
or(false,not(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.