Certification Problem

Input (COPS 237)

We consider the TRS containing the following rules:

a(x)	→	g(b(x))	(1)
a(x)	→	c(x)	(2)
b(x)	→	g(b(x))	(3)

The underlying signature is as follows:

{a/1, g/1, b/1, c/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	=	a(f4)
	→	g(b(f4))
	=	t₁

t₀	=	a(f4)
	→	c(f4)
	=	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:
    
    g(5) → 3
    
    g(5) → 5
    
    f4 → 4
    
    b(4) → 5
  The automaton is closed under rewriting as it is compatible.
- Automaton 2
  - final states:
    
    {1}
  - transitions:
    
    f4 → 2
    
    c(2) → 1
  The automaton is closed under rewriting as it is compatible.