Certification Problem

Input (COPS 660)

We consider the TRS containing the following rules:

b	→	f(b)	(1)
f(a)	→	f(h(h(h(h(c,b),a),f(c)),c))	(2)
c	→	h(f(c),f(h(c,a)))	(3)
c	→	f(b)	(4)

The underlying signature is as follows:

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

t₀	=	c
	→	f(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:
    
    {3}
  - transitions:
    
    a → 4
    
    h(9,7) → 3
    
    h(9,7) → 8
    
    h(9,7) → 5
    
    h(5,4) → 6
    
    f(48) → 8
    
    f(48) → 5
    
    f(48) → 48
    
    f(6) → 7
    
    f(8) → 9
    
    b → 48
    
    c → 5
    
    c → 8
  The automaton is closed under rewriting as it is compatible.
- Automaton 2
  - final states:
    
    {1}
  - transitions:
    
    f(2) → 1
    
    f(2) → 2
    
    b → 2
  The automaton is closed under rewriting as it is compatible.