Certification Problem
Input (COPS 508)
We consider the TRS containing the following rules:
| a |
→ |
b |
(1) |
| b |
→ |
f(a,a) |
(2) |
|
f(x,a) |
→ |
b |
(3) |
The underlying signature is as follows:
{a/0, b/0, f/2}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(f3,a) |
|
→
|
f(f3,b) |
|
= |
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(3,2) |
→ |
1 |
|
f(9,9) |
→ |
10 |
|
f(9,9) |
→ |
9 |
|
f(10,9) |
→ |
2 |
| b |
→ |
2 |
| b |
→ |
9 |
| b |
→ |
10 |
| a |
→ |
9 |
| a |
→ |
10 |
| f3 |
→ |
3 |
The automaton is closed under rewriting as it is compatible.
-
Automaton 2
-
final states:
{4}
-
transitions:
|
f(8,7) |
→ |
4 |
|
f(7,7) |
→ |
8 |
|
f(7,7) |
→ |
7 |
| b |
→ |
4 |
| b |
→ |
7 |
| b |
→ |
8 |
| a |
→ |
7 |
| a |
→ |
8 |
The automaton is closed under rewriting as it is compatible.