Certification Problem
Input (COPS 613)
We consider the TRS containing the following rules:
|
+(0,y) |
→ |
y |
(1) |
|
+(s(x),y) |
→ |
s(+(x,y)) |
(2) |
|
+(p(x),y) |
→ |
p(+(y,x)) |
(3) |
|
p(s(x)) |
→ |
s(p(x)) |
(4) |
|
s(p(x)) |
→ |
x |
(5) |
The underlying signature is as follows:
{+/2, 0/0, s/1, p/1}Property / Task
Prove or disprove confluence.Answer / Result
No.Proof (by csi @ CoCo 2022)
1 Non-Joinable Fork
The system is not confluent due to the following forking derivations.
| t0
|
= |
+(s(p(f7)),f5) |
|
→
|
+(f7,f5) |
|
= |
t1
|
| t0
|
= |
+(s(p(f7)),f5) |
|
→
|
s(+(p(f7),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:
{7}
-
transitions:
The automaton is closed under rewriting as it is compatible.
-
Automaton 2
-
final states:
{10}
-
transitions:
| 57 |
→ |
10 |
| f7 |
→ |
12 |
| f5 |
→ |
11 |
|
s(14) |
→ |
10 |
|
+(13,11) |
→ |
14 |
|
+(11,12) |
→ |
57 |
|
p(57) |
→ |
14 |
|
p(12) |
→ |
13 |
The automaton is closed under rewriting as it is compatible.