Certification Problem
Input (COPS 246)
We consider the TRS containing the following rules:
| a |
→ |
f(a) |
(1) |
| a |
→ |
g(h(a)) |
(2) |
|
f(x) |
→ |
h(g(x)) |
(3) |
|
h(x) |
→ |
f(g(x)) |
(4) |
The underlying signature is as follows:
{a/0, f/1, g/1, h/1}Property / Task
Prove or disprove confluence.Answer / Result
No.Proof (by csi @ CoCo 2020)
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:
|
g(3) |
→ |
1 |
|
g(3) |
→ |
2 |
|
g(28) |
→ |
2 |
|
g(2) |
→ |
28 |
|
h(28) |
→ |
2 |
|
h(2) |
→ |
3 |
|
f(28) |
→ |
3 |
|
f(2) |
→ |
2 |
| a |
→ |
2 |
The automaton is closed under rewriting as it is compatible.
-
Automaton 2
-
final states:
{4}
-
transitions:
|
g(16) |
→ |
5 |
|
g(27) |
→ |
5 |
|
g(5) |
→ |
27 |
|
h(27) |
→ |
4 |
|
h(27) |
→ |
5 |
|
h(5) |
→ |
16 |
|
f(27) |
→ |
16 |
|
f(5) |
→ |
4 |
|
f(5) |
→ |
5 |
| a |
→ |
5 |
The automaton is closed under rewriting as it is compatible.