Certification Problem

Input (COPS 800)

We consider the TRS containing the following rules:

a	→	b	(1)
f(g(a))	→	f(a)	(2)

The underlying signature is as follows:

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

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	=	f(g(a))
	→	f(g(b))
	=	t₁

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