Certification Problem

Input (COPS 245)

We consider the TRS containing the following rules:

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

The underlying signature is as follows:

{a/0, h/1, c/0, f/1, g/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
	→	h(f(c))
	=	t₁

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