Certification Problem

Input (COPS 212)

We consider the TRS containing the following rules:

a	→	b	(1)
a	→	f(a)	(2)
b	→	f(f(b))	(3)

The underlying signature is as follows:

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

Property / Task

Prove or disprove confluence.

Answer / Result

No.

Proof (by csi @ CoCo 2023)

1 Non-Joinable Fork

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

t₀	=	a
	→	b
	→	f(f(b))
	=	t₂

t₀	=	a
	→	f(a)
	→	f(b)
	=	t₂

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:
    
    {4}
  - transitions:
    
    f(5) → 6
    
    f(6) → 4
    
    f(6) → 5
    
    b → 5
  The automaton is closed under rewriting as it is compatible.
- Automaton 2
  - final states:
    
    {7}
  - transitions:
    
    f(8) → 7
    
    f(7) → 8
    
    b → 8
  The automaton is closed under rewriting as it is compatible.