Certification Problem

Input (COPS 661)

We consider the TRS containing the following rules:

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

The underlying signature is as follows:

{b/0, f/1, h/2, c/0, a/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	=	h(h(c,c),b)
	→	h(h(c,c),f(h(f(f(b)),c)))
	=	t₁

t₀	=	h(h(c,c),b)
	→	c
	=	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:
    
    c → 12
    
    c → 18
    
    c → 19
    
    f(14) → 15
    
    f(13) → 14
    
    f(16) → 17
    
    f(16) → 13
    
    b → 13
    
    h(20,17) → 11
    
    h(15,12) → 16
    
    h(19,18) → 20
  The automaton is closed under rewriting as it is compatible.
- Automaton 2
  - final states:
    
    {21}
  - transitions:
    
    c → 21
  The automaton is closed under rewriting as it is compatible.