Certification Problem

Input (COPS 670)

We consider the TRS containing the following rules:

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

The underlying signature is as follows:

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