Certification Problem

Input (COPS 606)

We consider the TRS containing the following rules:

not(true)	→	false	(1)
not(false)	→	true	(2)
or(true,y)	→	true	(3)
or(x,true)	→	true	(4)
or(false,false)	→	false	(5)
implies(true,y)	→	y	(6)
implies(false,y)	→	true	(7)
implies(x,y)	→	or(not(x),y)	(8)

The underlying signature is as follows:

{not/1, true/0, false/0, or/2, implies/2}

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	=	implies(true,f5)
	→	f5
	=	t₁

t₀	=	implies(true,f5)
	→	or(not(true),f5)
	=	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:
    
    {10}
  - transitions:
    
    f5 → 10
  The automaton is closed under rewriting as it is compatible.
- Automaton 2
  - final states:
    
    {6}
  - transitions:
    
    f5 → 7
    
    false → 9
    
    not(8) → 9
    
    true → 8
    
    or(9,7) → 6
  The automaton is closed under rewriting as it is compatible.