## Boolean functions: time-reversal symmetry and the generalized technical condition of proper operation, Applied Sciences Volume 19, pp. 132-141, 2017

**Mathematical Subject Classification** (2010): 06E30, 94C10

**Keywords and phrases**: Boolean function; time-reversal symmetry; the generalized technical condition of proper operation; predecessor; successor

The asynchronous flows model the behavior of the digital devices from electronics.
They are generated by Boolean functions Φ:{0,1}^{n}→{0,1}^{n} that iterate their
coordinates Φ_{1},...,Φ_{n} independently on each other. The time instants and the order
in which Φ_{1},...,Φ_{n} are computed are not known and the generalized technical condition
of proper operation (the generalization of race-freedom) states particular
circumstances when the flow behaves ’almost’ deterministically.
Time-reversal symmetry is one of the fundamental symmetries discussed in
natural science. Consequently, it arises in many physically motivated dynamical systems,
in particular in classical and quantum mechanics. Our aim is to relate the
time-reversal symmetry, adapted to electronics and asynchronicity, with a strengthened form of generalized
technical condition of proper operation, restricting our attention to Boolean
functions - not to flows, for reasons of brevity. This is possible since, in discrete time,
reasoning goes on recurrently (and real time equivalent constructions exist also). Even
if time does not occur in the paper, we have kept the terminology of ’time-reversal
symmetry’, since other symmetries of the Boolean functions exist also.