Certification Problem

Input (COPS 43)

We consider the TRS containing the following rules:

F(x)	→	A	(1)
F(x)	→	G(F(x))	(2)
G(F(x))	→	F(H(x))	(3)
G(F(x))	→	B	(4)

The underlying signature is as follows:

{F/1, A/0, G/1, H/1, B/0}

Property / Task

Prove or disprove confluence.

Answer / Result

No.

Proof (by csi @ CoCo 2021)

1 Non-Joinable Fork

The system is not confluent due to the following forking derivations.

t₀	=	G(F(f5))
	→	G(A)
	=	t₁

t₀	=	G(F(f5))
	→	B
	=	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:
    
    {13}
  - transitions:
    
    G(14) → 13
    
    A → 14
  The automaton is closed under rewriting as it is compatible.
- Automaton 2
  - final states:
    
    {1}
  - transitions:
    
    B → 1
  The automaton is closed under rewriting as it is compatible.