Certification Problem

Input (COPS 653)

We consider the TRS containing the following rules:

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

The underlying signature is as follows:

{c/0, a/0, b/0, f/1, h/2}

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	=	h(a,b)
	→	h(a,f(h(a,a)))
	=	t₁

t₀	=	h(a,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:
    
    {1}
  - transitions:
    
    f(4) → 5
    
    a → 2
    
    a → 3
    
    a → 6
    
    h(6,5) → 1
    
    h(3,2) → 4
  The automaton is closed under rewriting as it is compatible.
- Automaton 2
  - final states:
    
    {7}
  - transitions:
    
    a → 7
  The automaton is closed under rewriting as it is compatible.