Certification Problem
Input (COPS 580)
We consider the TRS containing the following rules:
|
+(0,0) |
→ |
0 |
(1) |
|
+(s(0),y) |
→ |
s(+(0,y)) |
(2) |
|
+(x,s(y)) |
→ |
s(+(y,x)) |
(3) |
|
s(s(x)) |
→ |
x |
(4) |
The underlying signature is as follows:
{+/2, 0/0, s/1}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
|
= |
+(f5,s(s(f6))) |
|
→
|
+(f5,f6) |
|
= |
t1
|
| t0
|
= |
+(f5,s(s(f6))) |
|
→
|
s(+(s(f6),f5)) |
|
= |
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:
{3}
-
transitions:
The automaton is closed under rewriting as it is compatible.
-
Automaton 2
-
final states:
{6}
-
transitions:
| f5 |
→ |
7 |
|
s(10) |
→ |
6 |
|
s(8) |
→ |
9 |
|
+(9,7) |
→ |
10 |
| f6 |
→ |
8 |
The automaton is closed under rewriting as it is compatible.