Defining the Symmetry of the Universal Semi-Regular Autonomous Asynchronous Systems

The regular autonomous asynchronous systems are the non-deterministic Boolean dynamical systems and universality means the greatest in the sense of the inclusion. The paper gives four definitions of symmetry of these systems in a slightly more general framework, called semi-regularity and also many examples.


Introduction
Switching theory has developed in the 50's and the 60's as a common effort of the mathematicians and the engineers of studying the switching circuits (=asynchronous circuits) from digital electrical engineering. After 1970 we do not know to exist any mathematical published work in what we call switching theory. The published works are written by engineers and their approach is always descriptive and unacceptable for the mathematicians. The label of switching theory has changed to asynchronous systems (or circuits) theory. One of the possible motivations of the situation consists in the fact that the important producers of digital equipments have stopped the dissemination of such researches.
Our interest in asynchronous systems had bibliography coming from the 50's and the 60's, as well as engineering works giving intuition, as well as mathematical works giving analogies. An interesting rendezvous has happened when the asynchronous systems theory has met the dynamical systems theory, resulting the so called regular autonomous sys-tems=Boolean dynamical systems; the vector field is Φ : {0, 1} n → {0, 1} n , time is discrete or real and we obtain the unbounded delay model We use the same notations for the laws that are induced from B on other sets, for example ∀x ∈ B n , ∀y ∈ B n , x ∪ y = (x 1 ∪ y 1 , ..., x n ∪ y n ) etc. In Figure 1 we have drawn at a) the logical gate NOT, i.e. the circuit that computes the logical complement and at b) a circuit that makes use of logical gates NOT. The asynchronous system that models the circuit from b) has the state portrait drawn at c). In the state portraits, the arrows show the increase of (the discrete or continuous) time. The underlined coordinates µ i are these coordinates for which Φ i (µ i ) = µ i and they are called excited, or enabled, or unstable. The coordinates µ i that are not underlined fulfill by definition Φ i (µ i ) = µ i and they are called not excited, or not enabled, or stable. The existence of two underlined coordinates in (0, 0) shows that Φ 1 (0, 0) = 1 may be computed first, Φ 2 (0, 0) = 1 may be computed first, or Φ 1 (0, 0), Φ 2 (0, 0) may be computed simultaneously, thus when the system is in (0, 0), it may run in three different directions, non-determinism.
Our present purpose is to define the symmetry of these systems.
Notation 4 The set of the real sequences t 0 < t 1 < ... < t k < ... that are unbounded from above is denoted with Seq.
Definition 7 Let be α ∈ Π n . The function Φ α : B n × N → B n defined by ∀µ ∈ B n , ∀k ∈ N , is called discrete time α−semi-orbit of µ. We consider also the sequence (t k ) ∈ Seq and the function ρ ∈ P n from (1), for which the function Φ ρ : B n × R → B n is defined by: ∀µ ∈ B n , ∀t ∈ R, Remark 9 Ξ Φ , Ξ Φ and Φ are usually identified.
Example 10 In Figure 2 we have drawn at a) the AND gate that computes the logical intersection, at b) a circuit with two gates and at c) the state portrait of Φ : B 2 → B 2 , ∀(µ 1 , µ 2 ) ∈ B 2 , Φ(µ 1 , µ 2 ) = (0, 1). We conclude that since the first coordinate might finally decrease its value and the second coordinate might finally increase its value, but the order and the time instant when these things happen are arbitrary.

Remark 12
We compare the semi-orbits and the anti-semi-orbits now and see that they run both from the past to the future, but the relation cause-effect is different: in Φ α , Φ ρ the cause is in the past and the effect is in the future, while in * Φ α , * Φ ρ the cause is in the future and the effect is in the past.

Definition 13
The discrete time and the continuous time universal anti-semi-regular autonomous asynchronous systems are defined by Example 14 In Figure 3 we have drawn at a) the circuit and at b) the state portrait of Ψ : The arrows in Figures 2 c) and 3 b) are the same, but with a different sense and we note that Ξ Ψ = * Ξ Φ , where Φ is the one from Example 10.

Definition 17
We consider the functions Φ, Ψ : B n → B n . If g, g ′ : B n → B n bijective exist such that one of the equivalent properties a), b), c) from Theorem 16 is satisfied, then we say that the couple (g, g ′ ) defines an isomorphism from Ξ Φ to Ξ Ψ , or from Ξ Φ to Ξ Ψ , or from Φ to Ψ. We use the notation Iso(Φ, Ψ) for the set of these couples and we also denote with Aut(Φ) = Iso(Φ, Φ) the set of the automorphisms of Theorem 18 For Φ, Ψ, g, g ′ : B n → B n , the following statements are equivalent: a) ∀ν ∈ B n , the diagram c) ∀µ ∈ B n , g(µ) = Ψ g ′ (0,...,0) (g(µ)) and ∀µ ∈ B n , ∀ρ ∈ P n , ∀t ∈ R, Proof. a)=⇒b) We fix arbitrarily µ ∈ B n , α ∈ Π n and we use the induction on k ≥ −1. In the case k = −1 the equality to be proved is satisfied thus we presume that the statement is true for k and we prove it for k + 1. We have: µ, k)))), k) = * Ψ g ′ (α) (g( Φ α (µ, k + 1)), k + 1).
The proof is similar with the proof of Theorem 16.

Definition 19
Let be the functions Φ, Ψ : B n → B n . If g, g ′ : B n → B n bijective exist such that one of the equivalent properties a), b), c) from Theorem 18 is fulfilled, we say that the couple (g, g ′ ) defines an antiisomorphism from Ξ Φ to * Ξ Ψ , or from Ξ Φ to * Ξ Ψ , or from Φ to Ψ. We use the notation * Iso(Φ, Ψ) for these couples and we also denote with * Aut(Φ) = * Iso(Φ, Φ) the set of the anti-automorphisms of Ξ Φ , Ξ Φ or Φ.
Theorem 23 Let be Φ, Ψ : Proof. a) The hypothesis states that ∀ν ∈ B n , the diagram commutes, with g, g ′ bijective. We fix arbitrarily ν ∈ B n , µ ∈ B n . We denote µ ′ = g(µ), ν ′ = g ′ (ν) and we note that As ν, µ were chosen arbitrarily and on the other hand, when ν runs in B n , ν ′ runs in B n and when µ runs in B n , µ ′ runs in B n , we infer that (8) is equivalent with the commutativity of the diagram for any ν ′ ∈ B n . We have proved that (g −1 , g ′−1 ) ∈ Iso(Ψ, Φ). b) By hypothesis ∀ν ∈ B n , the diagram is commutative, g, g ′ bijective and we prove that ∀ν ′ ∈ B n , the diagram Theorem 24 Aut(Φ) is a group relative to the law: Proof. The fact that ∀(g, the fact that (1 B n , 1 B n ) ∈ Aut(Φ) was mentioned before; and the fact that ∀(g, g ′ ) ∈ Aut(Φ), (g −1 , g ′−1 ) ∈ Aut(Φ) was shown at Theorem 23 a).

Conclusions
The paper defines the universal semi-regular autonomous asynchronous systems and the universal anti-semi-regular autonomous asynchronous systems. It also defines and characterizes the isomorphisms (automorphisms) and the anti-isormorphisms (anti-automorphisms) of these systems. Symmetry is defined as the existence of such isomorphisms (automorphisms), while anti-symmetry is defined as the existence of such anti-isomorphisms (anti-automorphisms). Many examples are given. A by-pass product in this study is anti-symmetry, that is related with systems having the cause in the future and the effect in the present. Another by-pass product consists in semi-regularity, since important examples of isomorphisms (automorphisms) are of semi-regular systems only, they do not keep progressiveness and regularity [2], [3].