Certification Problem
Input (COPS 511)
We consider the TRS containing the following rules:
| b |
→ |
a |
(1) |
| b |
→ |
c |
(2) |
| c |
→ |
c |
(3) |
| d |
→ |
c |
(4) |
| d |
→ |
e |
(5) |
|
f(x,a) |
→ |
A |
(6) |
|
f(x,e) |
→ |
A |
(7) |
|
f(x,A) |
→ |
A |
(8) |
|
f(c,x) |
→ |
A |
(9) |
The underlying signature is as follows:
{b/0, a/0, c/0, d/0, e/0, f/2, A/0}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.
The two resulting terms cannot be joined for the following reason:
-
The reachable terms of these two terms are approximated via the following two tree automata,
and the tree automata have an empty intersection.
-
Automaton 1
-
final states:
{2}
-
transitions:
The automaton is closed under rewriting as it is compatible.
-
Automaton 2
-
final states:
{3}
-
transitions:
The automaton is closed under rewriting as it is compatible.