Certification Problem
Input (COPS 963)
We consider the TRS containing the following rules:
|
b(a(a(x))) |
→ |
a(x) |
(1) |
|
a(a(a(x))) |
→ |
b(b(b(x))) |
(2) |
|
b(b(x)) |
→ |
a(b(a(x))) |
(3) |
The underlying signature is as follows:
{b/1, a/1}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
|
= |
b(b(b(f7))) |
|
→
|
b(a(b(a(f7)))) |
|
= |
t1
|
| t0
|
= |
b(b(b(f7))) |
|
→
|
a(b(a(b(f7)))) |
|
= |
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:
| f7 |
→ |
2 |
|
b(3) |
→ |
4 |
|
b(5) |
→ |
1 |
|
a(2) |
→ |
3 |
|
a(4) |
→ |
5 |
The automaton is closed under rewriting as it is compatible.
-
Automaton 2
-
final states:
{6}
-
transitions:
| f7 |
→ |
7 |
|
b(7) |
→ |
8 |
|
b(9) |
→ |
10 |
|
a(8) |
→ |
9 |
|
a(10) |
→ |
6 |
The automaton is closed under rewriting as it is compatible.