Certification Problem

Input (COPS 687)

We consider the TRS containing the following rules:

b	→	f(b)	(1)
h(a,c)	→	b	(2)
f(a)	→	c	(3)
c	→	a	(4)
c	→	b	(5)

The underlying signature is as follows:

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

t₀	=	h(a,c)
	→	b
	=	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:
    
    {6}
  - transitions:
    
    a → 7
    
    a → 8
    
    h(8,7) → 6
  The automaton is closed under rewriting as it is compatible.
- Automaton 2
  - final states:
    
    {1}
  - transitions:
    
    f(1) → 1
    
    b → 1
  The automaton is closed under rewriting as it is compatible.