Certification Problem
Input (COPS 719)
We consider the TRS containing the following rules:
| b |
→ |
f(b) |
(1) |
| c |
→ |
b |
(2) |
|
f(f(f(c))) |
→ |
a |
(3) |
|
f(h(b,b)) |
→ |
h(h(c,a),b) |
(4) |
The underlying signature is as follows:
{b/0, f/1, c/0, a/0, h/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
|
= |
f(f(f(c))) |
|
→
|
f(f(f(b))) |
|
= |
t1
|
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:
|
f(2) |
→ |
3 |
|
f(2) |
→ |
2 |
|
f(3) |
→ |
4 |
|
f(4) |
→ |
1 |
| b |
→ |
2 |
The automaton is closed under rewriting as it is compatible.
-
Automaton 2
-
final states:
{5}
-
transitions:
The automaton is closed under rewriting as it is compatible.