Certification Problem

Input (COPS 1650)

We consider the TRS containing the following rules:

f(f(x))	→	f(g(f(x)))	(1)
f(x)	→	x	(2)

The underlying signature is as follows:

{f/1, g/1}

Property / Task

Prove or disprove confluence.

Answer / Result

No.

Proof (by csi @ CoCo 2022)

1 Non-Joinable Fork

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

t₀	=	f(f(f4))
	→	f(f4)
	=	t₁

t₀	=	f(f(f4))
	→	f(g(f(f4)))
	=	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:
    
    2 → 1
    
    f4 → 2
    
    f(2) → 1
  The automaton is closed under rewriting as it is compatible.
- Automaton 2
  - final states:
    
    {3}
  - transitions:
    
    6 → 3
    
    4 → 5
    
    f4 → 4
    
    g(5) → 6
    
    f(6) → 3
    
    f(4) → 5
  The automaton is closed under rewriting as it is compatible.