Certification Problem

Input (COPS 253)

We consider the TRS containing the following rules:

c	→	f(c)	(1)
c	→	f(h(a))	(2)
f(x)	→	h(f(x))	(3)

The underlying signature is as follows:

{c/0, f/1, h/1, 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	=	c
	→	f(h(a))
	=	t₁

t₀	=	c
	→	f(c)
	→	f(f(h(a)))
	=	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:
    
    {1}
  - transitions:
    
    a → 2
    
    f(3) → 1
    
    h(1) → 1
    
    h(2) → 3
  The automaton is closed under rewriting as it is compatible.
- Automaton 2
  - final states:
    
    {9}
  - transitions:
    
    a → 10
    
    f(11) → 12
    
    f(12) → 9
    
    h(10) → 11
    
    h(9) → 9
    
    h(12) → 12
  The automaton is closed under rewriting as it is compatible.