Certification Problem
Input (COPS 655)
We consider the TRS containing the following rules:
| b |
→ |
a |
(1) |
|
f(h(b,f(a))) |
→ |
f(c) |
(2) |
| c |
→ |
h(c,c) |
(3) |
|
h(f(f(b)),a) |
→ |
c |
(4) |
|
h(a,a) |
→ |
a |
(5) |
The underlying signature is as follows:
{b/0, a/0, f/1, h/2, c/0}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(a))) |
|
→
|
f(h(a,f(a))) |
|
= |
t1
|
| t0
|
= |
f(h(b,f(a))) |
|
→
|
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:
{7}
-
transitions:
|
f(8) |
→ |
9 |
|
f(11) |
→ |
7 |
| a |
→ |
8 |
| a |
→ |
10 |
|
h(10,9) |
→ |
11 |
The automaton is closed under rewriting as it is compatible.
-
Automaton 2
-
final states:
{12}
-
transitions:
|
f(13) |
→ |
12 |
| c |
→ |
13 |
|
h(13,13) |
→ |
13 |
The automaton is closed under rewriting as it is compatible.