Certification Problem
Input (COPS 657)
We consider the TRS containing the following rules:
|
h(f(b),a) |
→ |
h(f(h(c,a)),f(a)) |
(1) |
| c |
→ |
f(c) |
(2) |
| c |
→ |
h(f(b),c) |
(3) |
| a |
→ |
c |
(4) |
|
f(a) |
→ |
b |
(5) |
The underlying signature is as follows:
{h/2, f/1, b/0, a/0, c/0}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.
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:
{1}
-
transitions:
| c |
→ |
2 |
|
f(2) |
→ |
1 |
|
f(2) |
→ |
2 |
|
f(61) |
→ |
62 |
| b |
→ |
61 |
|
h(62,2) |
→ |
2 |
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.