Certification Problem

Input (COPS 31)

We consider the TRS containing the following rules:

f(g(x),h(x,y))	→	a	(1)
g(b)	→	c	(2)
h(x,d)	→	e	(3)

The underlying signature is as follows:

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

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	=	f(g(f9),h(f9,d))
	→	f(g(f9),e)
	=	t₁

t₀	=	f(g(f9),h(f9,d))
	→	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:
    
    e → 2
    
    f9 → 3
    
    g(3) → 4
    
    f(4,2) → 1
  The automaton is closed under rewriting as it is state-compatible w.r.t. the following relation.
  
  2 » 2
  
  3 » 3
  
  4 » 4
  
  1 » 1
- Automaton 2
  - final states:
    
    {5}
  - transitions:
    
    a → 5
  The automaton is closed under rewriting as it is state-compatible w.r.t. the following relation.
  5 » 5