Certification Problem
Input (COPS 694)
We consider the TRS containing the following rules:
| c |
→ |
b |
(1) |
|
h(b,a) |
→ |
b |
(2) |
| a |
→ |
f(a) |
(3) |
|
h(a,h(c,c)) |
→ |
f(h(b,f(c))) |
(4) |
|
f(h(b,f(c))) |
→ |
f(a) |
(5) |
The underlying signature is as follows:
{c/0, b/0, h/2, a/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(h(b,f(c))) |
|
→
|
f(h(b,f(b))) |
|
= |
t1
|
| t0
|
= |
f(h(b,f(c))) |
|
→
|
f(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:
{17}
-
transitions:
|
f(21) |
→ |
17 |
|
f(18) |
→ |
19 |
| b |
→ |
18 |
| b |
→ |
20 |
|
h(20,19) |
→ |
21 |
The automaton is closed under rewriting as it is compatible.
-
Automaton 2
-
final states:
{22}
-
transitions:
| a |
→ |
23 |
|
f(23) |
→ |
22 |
|
f(23) |
→ |
23 |
The automaton is closed under rewriting as it is compatible.