Certification Problem
Input (COPS 717)
We consider the TRS containing the following rules:
| a |
→ |
c |
(1) |
|
f(f(c)) |
→ |
h(a,h(h(c,c),b)) |
(2) |
|
f(f(a)) |
→ |
b |
(3) |
| b |
→ |
f(f(b)) |
(4) |
The underlying signature is as follows:
{a/0, c/0, f/1, h/2, b/0}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
|
= |
f(f(a)) |
|
→
|
f(f(c)) |
|
= |
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:
| b |
→ |
6 |
|
h(7,6) |
→ |
8 |
|
h(2,2) |
→ |
7 |
|
h(9,8) |
→ |
1 |
|
f(3) |
→ |
1 |
|
f(6) |
→ |
25 |
|
f(2) |
→ |
3 |
|
f(25) |
→ |
6 |
| c |
→ |
2 |
| c |
→ |
9 |
| a |
→ |
9 |
The automaton is closed under rewriting as it is compatible.
-
Automaton 2
-
final states:
{4}
-
transitions:
| b |
→ |
4 |
|
f(4) |
→ |
10 |
|
f(10) |
→ |
4 |
The automaton is closed under rewriting as it is compatible.