Certification Problem

Input (COPS 706)

We consider the TRS containing the following rules:

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

The underlying signature is as follows:

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

Property / Task

Prove or disprove confluence.

Answer / Result

No.

Proof (by csi @ CoCo 2021)

1 Non-Joinable Fork

The system is not confluent due to the following forking derivations.

t₀	=	f(h(c,c))
	→	f(h(c,f(c)))
	=	t₁

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