Certification Problem
Input (COPS 682)
We consider the TRS containing the following rules:
|
h(a,a) |
→ |
b |
(1) |
| b |
→ |
c |
(2) |
|
f(f(h(b,a))) |
→ |
a |
(3) |
| a |
→ |
b |
(4) |
The underlying signature is as follows:
{h/2, a/0, b/0, c/0, f/1}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.
| t0
|
= |
f(f(h(b,a))) |
|
→
|
f(f(h(b,b))) |
|
= |
t1
|
| t0
|
= |
f(f(h(b,a))) |
|
→
|
a |
|
= |
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:
| b |
→ |
2 |
| b |
→ |
3 |
|
f(4) |
→ |
5 |
|
f(5) |
→ |
1 |
|
h(3,2) |
→ |
4 |
| c |
→ |
2 |
| c |
→ |
3 |
The automaton is closed under rewriting as it is compatible.
-
Automaton 2
-
final states:
{6}
-
transitions:
The automaton is closed under rewriting as it is compatible.