Certification Problem

Input (COPS 647)

We consider the TRS containing the following rules:

f(b)	→	b	(1)
b	→	h(h(c,f(b)),a)	(2)

The underlying signature is as follows:

{f/1, b/0, h/2, c/0, a/0}

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(b)
	→	f(h(h(c,f(b)),a))
	=	t₁

t₀	=	f(b)
	→	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:
    
    c → 5
    
    a → 2
    
    f(3) → 4
    
    f(7) → 1
    
    b → 3
    
    b → 4
    
    h(5,4) → 6
    
    h(6,2) → 7
    
    h(6,2) → 3
    
    h(6,2) → 4
  The automaton is closed under rewriting as it is compatible.
- Automaton 2
  - final states:
    
    {8}
  - transitions:
    
    c → 13
    
    a → 11
    
    f(8) → 12
    
    b → 8
    
    b → 12
    
    h(13,12) → 14
    
    h(14,11) → 8
    
    h(14,11) → 12
  The automaton is closed under rewriting as it is compatible.