Certification Problem
Input (COPS 251)
We consider the TRS containing the following rules:
| a |
→ |
h(a) |
(1) |
| a |
→ |
f(c) |
(2) |
|
f(x) |
→ |
h(g(x)) |
(3) |
|
h(x) |
→ |
f(g(x)) |
(4) |
The underlying signature is as follows:
{a/0, h/1, f/1, c/0, g/1}Property / Task
Prove or disprove confluence.Answer / Result
No.Proof (by csi @ CoCo 2020)
1 Non-Joinable Fork
The system is not confluent due to the following forking derivations.
| t0
|
= |
a |
|
→
|
h(a) |
|
→
|
h(f(c)) |
|
= |
t2
|
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:
|
g(2) |
→ |
57 |
|
g(57) |
→ |
2 |
|
f(2) |
→ |
1 |
| c |
→ |
2 |
|
h(57) |
→ |
1 |
The automaton is closed under rewriting as it is compatible.
-
Automaton 2
-
final states:
{8}
-
transitions:
|
g(9) |
→ |
55 |
|
g(142) |
→ |
10 |
|
g(10) |
→ |
142 |
|
g(55) |
→ |
9 |
|
f(9) |
→ |
10 |
|
f(142) |
→ |
8 |
| c |
→ |
9 |
|
h(10) |
→ |
8 |
|
h(55) |
→ |
10 |
The automaton is closed under rewriting as it is compatible.