Certification Problem
Input (COPS 31)
We consider the TRS containing the following rules:
|
f(g(x),h(x,y)) |
→ |
a |
(1) |
|
g(b) |
→ |
c |
(2) |
|
h(x,d) |
→ |
e |
(3) |
The underlying signature is as follows:
{f/2, g/1, h/2, a/0, b/0, c/0, d/0, e/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(g(f9),h(f9,d)) |
|
→
|
f(g(f9),e) |
|
= |
t1
|
| t0
|
= |
f(g(f9),h(f9,d)) |
|
→
|
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:
| e |
→ |
2 |
| f9 |
→ |
3 |
|
g(3) |
→ |
4 |
|
f(4,2) |
→ |
1 |
The automaton is closed under rewriting as it is state-compatible w.r.t. the following relation.
-
Automaton 2
-
final states:
{5}
-
transitions:
The automaton is closed under rewriting as it is state-compatible w.r.t. the following relation.