Certification Problem

Input (COPS 258)

We consider the TRS containing the following rules:

c	→	f(c)	(1)
c	→	h(c)	(2)
f(x)	→	f(f(x))	(3)
h(x)	→	h(h(x))	(4)

The underlying signature is as follows:

{c/0, f/1, h/1}

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(c)
	=	t₁

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