Certification Problem
Input (COPS 526)
We consider the TRS containing the following rules:
|
f(x,x) |
→ |
g |
(1) |
| a |
→ |
b |
(2) |
| a |
→ |
c |
(3) |
| d |
→ |
c |
(4) |
| d |
→ |
e |
(5) |
| c |
→ |
c |
(6) |
|
f(a,x) |
→ |
g |
(7) |
|
f(b,x) |
→ |
g |
(8) |
|
f(c,x) |
→ |
g |
(9) |
|
f(d,x) |
→ |
g |
(10) |
|
f(e,x) |
→ |
g |
(11) |
|
f(x,a) |
→ |
g |
(12) |
|
f(x,b) |
→ |
g |
(13) |
|
f(x,c) |
→ |
g |
(14) |
|
f(x,d) |
→ |
g |
(15) |
|
f(x,e) |
→ |
g |
(16) |
The underlying signature is as follows:
{f/2, g/0, a/0, b/0, c/0, d/0, e/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.
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:
{14}
-
transitions:
The automaton is closed under rewriting as it is state-compatible w.r.t. the following relation.
-
Automaton 2
-
final states:
{15}
-
transitions:
The automaton is closed under rewriting as it is state-compatible w.r.t. the following relation.