Certification Problem

Input (COPS 739)

We consider the TRS containing the following rules:

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

The underlying signature is as follows:

{a/0, g/1, b/0}

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	=	g(g(a))
	→	b
	=	t₁

t₀	=	g(g(a))
	→	g(g(g(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:
    
    {10}
  - transitions:
    
    b → 10
  The automaton is closed under rewriting as it is compatible.
- Automaton 2
  - final states:
    
    {6}
  - transitions:
    
    b → 7
    
    g(7) → 8
    
    g(9) → 6
    
    g(9) → 9
    
    g(8) → 9
  The automaton is closed under rewriting as it is compatible.