Certification Problem

Input (COPS 708)

We consider the TRS containing the following rules:

h(f(f(c)),h(c,h(c,h(h(f(b),f(f(f(f(b))))),b))))	→	h(h(a,a),f(c))	(1)
h(b,h(c,h(b,h(a,b))))	→	a	(2)
f(c)	→	f(f(a))	(3)
a	→	f(f(b))	(4)

The underlying signature is as follows:

{h/2, f/1, c/0, b/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	=	h(b,h(c,h(b,h(a,b))))
	→	h(b,h(c,h(b,h(f(f(b)),b))))
	=	t₁

t₀	=	h(b,h(c,h(b,h(a,b))))
	→	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:
    
    b → 2
    
    b → 3
    
    b → 7
    
    b → 11
    
    f(3) → 4
    
    f(4) → 5
    
    c → 9
    
    h(11,10) → 1
    
    h(7,6) → 8
    
    h(5,2) → 6
    
    h(9,8) → 10
  The automaton is closed under rewriting as it is compatible.
- Automaton 2
  - final states:
    
    {12}
  - transitions:
    
    b → 87
    
    a → 12
    
    f(87) → 88
    
    f(88) → 12
  The automaton is closed under rewriting as it is compatible.