Certification Problem

Input (COPS 239)

We consider the TRS containing the following rules:

a(x)	→	g(h(b(x)))	(1)
a(x)	→	g(c(x))	(2)
b(x)	→	g(b(x))	(3)
g(a(x))	→	g(h(a(x)))	(4)

The underlying signature is as follows:

{a/1, g/1, h/1, b/1, c/1}

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	=	a(f5)
	→	g(h(b(f5)))
	=	t₁

t₀	=	a(f5)
	→	g(c(f5))
	=	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:
    
    {12}
  - transitions:
    
    f5 → 13
    
    h(14) → 15
    
    b(13) → 14
    
    g(14) → 14
    
    g(15) → 12
  The automaton is closed under rewriting as it is compatible.
- Automaton 2
  - final states:
    
    {9}
  - transitions:
    
    f5 → 10
    
    c(10) → 11
    
    g(11) → 9
  The automaton is closed under rewriting as it is compatible.