Certification Problem

Input (COPS 225)

We consider the TRS containing the following rules:

f(a)	→	b	(1)
a	→	a'	(2)
f(b)	→	c	(3)

The underlying signature is as follows:

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

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