Certification Problem

Input (COPS 543)

We consider the TRS containing the following rules:

f(c)	→	g(c)	(1)
g(c)	→	f(c)	(2)
c	→	d	(3)

The underlying signature is as follows:

{f/1, c/0, g/1, d/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	=	g(c)
	→	g(d)
	=	t₁

t₀	=	g(c)
	→	f(c)
	→	f(d)
	=	t₂

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:
    
    g(2) → 1
    
    d → 2
  The automaton is closed under rewriting as it is compatible.
- Automaton 2
  - final states:
    
    {7}
  - transitions:
    
    f(8) → 7
    
    d → 8
  The automaton is closed under rewriting as it is compatible.