Certification Problem

Input (COPS 683)

We consider the TRS containing the following rules:

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

The underlying signature is as follows:

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

t₀	=	f(h(b,a))
	→	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:
    
    {11}
  - transitions:
    
    h(13,12) → 14
    
    c → 13
    
    a → 12
    
    f(14) → 11
  The automaton is closed under rewriting as it is compatible.
- Automaton 2
  - final states:
    
    {5}
  - transitions:
    
    a → 5
  The automaton is closed under rewriting as it is compatible.