Certification Problem
Input (COPS 947)
We consider the TRS containing the following rules:
|
a(a(b(b(x)))) |
→ |
b(b(b(a(a(a(x)))))) |
(1) |
|
c(x) |
→ |
c(a(x)) |
(2) |
|
c(x) |
→ |
c(b(x)) |
(3) |
The underlying signature is as follows:
{a/1, b/1, c/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
|
= |
c(f15) |
|
→
|
c(b(f15)) |
|
→
|
c(a(b(f15))) |
|
= |
t2
|
| t0
|
= |
c(f15) |
|
→
|
c(a(f15)) |
|
= |
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:
{15}
-
transitions:
| f15 |
→ |
16 |
|
c(18) |
→ |
15 |
|
a(18) |
→ |
18 |
|
a(17) |
→ |
18 |
|
b(18) |
→ |
18 |
|
b(16) |
→ |
17 |
The automaton is closed under rewriting as it is compatible.
-
Automaton 2
-
final states:
{4}
-
transitions:
| f15 |
→ |
5 |
|
c(6) |
→ |
4 |
|
a(6) |
→ |
6 |
|
a(5) |
→ |
6 |
|
b(6) |
→ |
6 |
The automaton is closed under rewriting as it is compatible.