Certification Problem

Input (COPS 44)

We consider the TRS containing the following rules:

H(H(x))

→

K(x)

(1)

The underlying signature is as follows:

{H/1, K/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	=	H(H(H(f2)))
	→	H(K(f2))
	=	t₁

t₀	=	H(H(H(f2)))
	→	K(H(f2))
	=	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:
    
    f2 → 2
    
    K(2) → 3
    
    H(3) → 1
  The automaton is closed under rewriting as it is compatible.
- Automaton 2
  - final states:
    
    {4}
  - transitions:
    
    f2 → 5
    
    K(6) → 4
    
    H(5) → 6
  The automaton is closed under rewriting as it is compatible.