Certification Problem
Input (COPS 568)
We consider the TRS containing the following rules:
|
f(x,x) |
→ |
x |
(1) |
|
f(x,e) |
→ |
x |
(2) |
|
f(e,x) |
→ |
x |
(3) |
|
f(x,i(x)) |
→ |
e |
(4) |
|
f(i(x),x) |
→ |
e |
(5) |
The underlying signature is as follows:
{f/2, e/0, i/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
|
= |
f(i(e),e) |
|
→
|
i(e) |
|
= |
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:
The automaton is closed under rewriting as it is state-compatible w.r.t. the following relation.
-
Automaton 2
-
final states:
{2}
-
transitions:
The automaton is closed under rewriting as it is state-compatible w.r.t. the following relation.