Certification Problem

Input (COPS 697)

We consider the TRS containing the following rules:

h(h(c,a),b)	→	b	(1)
a	→	h(a,b)	(2)
a	→	a	(3)

The underlying signature is as follows:

{h/2, c/0, a/0, b/0}

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	=	h(h(c,a),b)
	→	h(h(c,h(a,b)),b)
	=	t₁

t₀	=	h(h(c,a),b)
	→	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:
    
    {11}
  - transitions:
    
    h(14,12) → 14
    
    h(17,12) → 11
    
    h(14,13) → 15
    
    h(16,15) → 17
    
    a → 14
    
    c → 16
    
    b → 12
    
    b → 13
  The automaton is closed under rewriting as it is compatible.
- Automaton 2
  - final states:
    
    {10}
  - transitions:
    
    b → 10
  The automaton is closed under rewriting as it is compatible.