Certification Problem

Input

We consider two TRSs R and S where R contains the rules

b(b(x)) a(a(x))
a(a(x)) b(c(x))
a(c(x)) b(a(x))
b(a(x)) a(c(x))
a(b(x)) a(b(x))
b(c(x)) a(a(x))
a(b(x)) b(c(x))

and S contains the following rules:

a(x) b(b(b(x)))
a(x) c(x)
b(x) b(x)

The underlying signature is as follows:

{a/1, b/1, c/1}

Property / Task

Prove or disprove commutation.

Answer / Result

No.

Proof (by ACP)

1 Non-Joinable Fork

The systems are not commuting due to the following forking derivations.
t0 = a(a(x))
S b(b(b(a(x))))
= s1
and
t0 = a(a(x))
R b(c(x))
= t1
There is no possibility to join s1R*·←S* t1 for the following reason: