Certification Problem

Input (COPS 675)

We consider the TRS containing the following rules:

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

The underlying signature is as follows:

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

t₀	=	h(h(b,b),f(f(h(h(c,c),c))))
	→	c
	=	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:
    
    {16}
  - transitions:
    
    h(26,23) → 16
    
    h(19,18) → 20
    
    h(25,24) → 26
    
    h(20,17) → 21
    
    c → 17
    
    c → 18
    
    c → 19
    
    c → 25
    
    c → 24
    
    b → 24
    
    f(21) → 22
    
    f(22) → 23
  The automaton is closed under rewriting as it is compatible.
- Automaton 2
  - final states:
    
    {15}
  - transitions:
    
    c → 15
  The automaton is closed under rewriting as it is compatible.