Certification Problem

Input (COPS 525)

We consider the TRS containing the following rules:

f(x,x)	→	g	(1)
a	→	b	(2)
a	→	c	(3)
b	→	b	(4)
c	→	c	(5)
f(a,x)	→	g	(6)
f(b,x)	→	g	(7)
f(c,x)	→	g	(8)
f(x,a)	→	g	(9)
f(x,b)	→	g	(10)
f(x,c)	→	g	(11)

The underlying signature is as follows:

{f/2, g/0, a/0, b/0, c/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	=	a
	→	c
	=	t₁

t₀	=	a
	→	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:
    
    {14}
  - transitions:
    
    c → 14
  The automaton is closed under rewriting as it is state-compatible w.r.t. the following relation.
  14 » 14
- Automaton 2
  - final states:
    
    {15}
  - transitions:
    
    b → 15
  The automaton is closed under rewriting as it is state-compatible w.r.t. the following relation.
  15 » 15