Certification Problem
Input (COPS 734)
We consider the TRS containing the following rules:
|
f(a,f(a,x)) |
→ |
f(f(a,a),a) |
(1) |
The underlying signature is as follows:
{f/2, a/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.
| t0
|
= |
f(a,f(a,f(a,f2))) |
|
→
|
f(a,f(f(a,a),a)) |
|
= |
t1
|
| t0
|
= |
f(a,f(a,f(a,f2))) |
|
→
|
f(f(a,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:
|
f(7,6) |
→ |
1 |
|
f(5,2) |
→ |
6 |
|
f(4,3) |
→ |
5 |
| a |
→ |
2 |
| a |
→ |
3 |
| a |
→ |
4 |
| a |
→ |
7 |
The automaton is closed under rewriting as it is compatible.
-
Automaton 2
-
final states:
{8}
-
transitions:
|
f(12,9) |
→ |
8 |
|
f(11,10) |
→ |
12 |
| a |
→ |
9 |
| a |
→ |
10 |
| a |
→ |
11 |
The automaton is closed under rewriting as it is compatible.