Certification Problem

Input (COPS 688)

We consider the TRS containing the following rules:

c	→	h(c,c)	(1)
f(c)	→	c	(2)
f(b)	→	b	(3)
a	→	b	(4)
a	→	h(h(a,a),h(c,b))	(5)

The underlying signature is as follows:

{c/0, h/2, f/1, b/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(c)
	→	f(h(c,c))
	=	t₁

t₀	=	f(c)
	→	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:
    
    {9}
  - transitions:
    
    f(12) → 9
    
    h(10,10) → 10
    
    h(10,10) → 11
    
    h(11,10) → 12
    
    c → 10
    
    c → 11
  The automaton is closed under rewriting as it is compatible.
- Automaton 2
  - final states:
    
    {13}
  - transitions:
    
    h(13,13) → 13
    
    c → 13
  The automaton is closed under rewriting as it is compatible.