Certification Problem

Input (COPS 673)

We consider the TRS containing the following rules:

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

The underlying signature is as follows:

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

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