Certification Problem

Input (COPS 664)

We consider the TRS containing the following rules:

b	→	f(h(c,b))	(1)
f(f(f(c)))	→	b	(2)
c	→	a	(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 2023)

1 Non-Joinable Fork

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

t₀	=	f(f(f(c)))
	→	f(f(f(a)))
	=	t₁

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