Certification Problem
Input (COPS 255)
We consider the TRS containing the following rules:
| c |
→ |
f(c) |
(1) |
| c |
→ |
h(c) |
(2) |
|
f(x) |
→ |
f(g(x)) |
(3) |
|
h(x) |
→ |
h(g(x)) |
(4) |
|
g(x) |
→ |
g(f(h(x))) |
(5) |
The underlying signature is as follows:
{c/0, f/1, h/1, g/1}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.
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:
{3}
-
transitions:
|
h(4) |
→ |
4 |
|
f(4) |
→ |
3 |
|
f(4) |
→ |
4 |
| c |
→ |
4 |
|
g(4) |
→ |
4 |
The automaton is closed under rewriting as it is compatible.
-
Automaton 2
-
final states:
{1}
-
transitions:
|
h(2) |
→ |
1 |
|
h(2) |
→ |
2 |
|
f(2) |
→ |
2 |
| c |
→ |
2 |
|
g(2) |
→ |
2 |
The automaton is closed under rewriting as it is compatible.