Certification Problem

Input (COPS 32)

We consider the TRS containing the following rules:

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

The underlying signature is as follows:

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

Property / Task

Prove or disprove confluence.

Answer / Result

No.

Proof (by csi @ CoCo 2021)

1 Non-Joinable Fork

The system is not confluent due to the following forking derivations.

t₀	=	f(g(c),h(c))
	→	f(g(c),d)
	=	t₁

t₀	=	f(g(c),h(c))
	→	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:
    
    d → 2
    
    c → 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
  
  4 » 4
  
  1 » 1
  
  3 » 3
- 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