Certification Problem

Input (COPS 21)

We consider the TRS containing the following rules:

b	→	a	(1)
b	→	c	(2)
c	→	h(b)	(3)
c	→	d	(4)
a	→	h(a)	(5)
d	→	h(d)	(6)

The underlying signature is as follows:

{b/0, a/0, c/0, h/1, d/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	=	b
	→	a
	→	h(a)
	=	t₂

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