Certification Problem

Input (COPS 509)

We consider the TRS containing the following rules:

a	→	b	(1)
f(b,b)	→	f(a,a)	(2)
f(x,a)	→	f(a,a)	(3)

The underlying signature is as follows:

{a/0, b/0, f/2}

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(f3,a)
	→	f(f3,b)
	=	t₁

t₀	=	f(f3,a)
	→	f(a,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(3,2) → 1
    
    b → 2
    
    f3 → 3
  The automaton is closed under rewriting as it is compatible.
- Automaton 2
  - final states:
    
    {4}
  - transitions:
    
    f(6,5) → 4
    
    b → 5
    
    b → 6
    
    a → 5
    
    a → 6
  The automaton is closed under rewriting as it is compatible.