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Symmetry 2012, 4(1), 116-128; https://doi.org/10.3390/sym4010116
Defining the Symmetry of the Universal Semi-Regular Autonomous Asynchronous Systems
Street Zimbrului, Nr. 3, Bl. PB68, Ap. 11, 410430, Oradea, Romania
Received: 1 November 2011; in revised form: 8 February 2012 / Accepted: 9 February 2012 / Published: 15 February 2012
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.
Keywords:asynchronous system; symmetry; semi-regularity
Switching theory has developed in the 1950s and the 1960s as a common effort of the mathematicians and the engineers of studying the switching circuits (a.k.a. asynchronous circuits) from digital electrical engineering. We are unaware of any existent mathematical work published after 1970 on 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 1950s and the 1960s, as well as engineering works giving intuition, as well as mathematical works giving analogies. An interesting rendez-vous has happened when the asynchronous systems theory has met the dynamical systems theory, resulting in the so-called regular autonomous systems (a.k.a Boolean dynamical systems) where the vector field is and time is discrete or real, and we obtain the unbounded delay model of computation of suggested by the engineers. The synchronous iterations of of the dynamical systems are replaced by asynchronous iterations in which each coordinate is iterated independently on the others, in arbitrary finite time.
We denote with the binary Boolean algebra, together with the discrete topology and with the usual algebraic laws:
We use the same notations for the laws that are induced from on other sets, for example , ,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 are these coordinates for which and they are called excited, or enabled, or unstable. The coordinates that are not underlined fulfill by definition and they are called not excited, or not enabled, or stable. The existence of two underlined coordinates in shows that may be computed first, may be computed first, or , may be computed simultaneously, thus when the system is in , it may run in three different directions, which results in non-determinism.
Our present purpose is to define the symmetry of these systems.
2. Semi-Regular Systems
We denote .
is the notation of the characteristic function of the set :
We denote with the set of the sequences
The set of the real sequences that are unbounded from above is denoted with .
We use the notation for the set of the functions having the property that and exist with
Let be a function. For we define the function by
For any and if then i.e., is not computed and if then i.e., is computed. This is the meaning of asynchronicity.
Let The function defined byis called discrete time semi-orbit of We consider also the sequence and the function from Equation (2), for which the function is defined by:is called continuous time semi-orbit of
The discrete time and the continuous time universal semi-regular autonomous asynchronous systems associated to Φ are defined by
and Φ are usually identified.
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 We conclude thatsince 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.
3. Anti-Semi-Regular Systems
Let and from Equation (2). The function that satisfiesis called discrete time anti-semi-orbit of μ and the function that satisfiesis called continuous time anti-semi-orbit of
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 cause-effect relation is different: in the cause is in the past and the effect is in the future, while in the cause is in the future and the effect is in the past.
The discrete time and the continuous time universal anti-semi-regular autonomous asynchronous systems associated to Φ are defined by
4. Isomorphisms and Anti-Isomorphisms
Let It defines the functionsand
Let The following statements are equivalent:
- the diagram
(a)⟹(b): We fix arbitrarily and we use the induction on . For , (b) becomes , thus we suppose that it is true for k and we prove it for :
(b)⟹(c): The first statement results from (b) if we take and In order to prove the second statement, let and be arbitrary, thus Equation (2) holds with If the statement to prove takes the form so that we can suppose now that a finite or an infinite number of are In the case that does not restrict the generality of the proof, we have thatis an element of and
(c)⟹(a): Let be arbitrary and fixed and we consider given by Equation (2), with fixed, and . We have
Case (i) , the commutativity of the diagram is equivalent with the first statement of (c).
Case(ii) ,and from Equation (8), for we obtain
We consider the functions If bijective exist such that one of the equivalent properties (a), (b) or (c) from Theorem 17 is satisfied, then we say that the couple defines an isomorphism from to or from to or from Φ to We use the notation for the set of these couples and we also denote with the set of the automorphisms of or
For , the following statements are equivalent:
(a) the diagram is commutative;
(a)⟹(b): We fix arbitrarily and we use the induction on In the case the equality to be proved is satisfiedthus we presume that the statement is true for k and we prove it for We have:
The proof is similar with the proof of Theorem 17.
Let If bijective exist such that one of the equivalent properties (a), (b) or (c) from Theorem 19 is fulfilled, we say that the couple defines an anti-isomorphism from to or from to or from Φ to We use the notation for these couples and we also denote with the set of the anti-automorphisms of or Φ.
5. Symmetry and Anti-Symmetry
The fact that implies but all of and may be empty.
Let If , then are called symmetrical, or conjugated; if , then are called anti-symmetrical, or anti-conjugated.
If then and Φ are called symmetrical and if then and Φ are called anti-symmetrical.
The symmetry of means that maps the transfers in transfers the situation when Φ is symmetrical and is similar. Anti-symmetry may be understood as mirroring: maps the transfers (or arrows) in transfers and similarly for
(a) If , then
(b) If , then
(a): The hypothesis states that the diagramcommutes, with bijective. We fix arbitrarily We denote and we note thatAs were chosen arbitrarily and on the other hand, when runs in runs in and when runs in runs in , we infer that Equation (9) is equivalent with the commutativity of the diagramfor any We have proved that
(b): By hypothesis , the diagramis commutative, bijective and we prove that , the diagramis commutative.
is a group relative to the law:
The fact that is proved like this:the fact that was mentioned before; and the fact that was shown at Theorem 24(a).
Any subgroup with is called a group of symmetry of of or of
The function defined by fulfills forthus and Φ is anti-symmetrical. The state portrait of Φ was drawn in Figure 1(c).
Let be a bijection. We use the notation for the bijection given by
Any of and is called symmetrical relative to the coordinates if the bijection σ exists, such that
We consider the function defined by and the permutation A group of symmetry of is represented by We have given in Figure 6 the state portrait of
For we denote by the translation of vector
If holds for some , we say that any of and Φ is symmetrical relative to translations.
In Figure 7
we have the system with Φ given by Equation (11)and as resulting from the state portrait.
The system from Figure 9 is symmetrical relative to translations, since it has the group of symmetry Φ is self-dual where the dual of Φ is defined by
Functions exist, see Figure 10, that are symmetrical relative to the translations with any thus their group of symmetry is The fact that shows that all these functions: are self-dual,
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-isomorphisms (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, which is related with systems having the cause in the future and the effect in the past. Another by-pass product consists in semi-regularity, since important examples of isomorphisms (automorphisms) are of semi-regular systems only and do not keep progressiveness and regularity [2,3].
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Figure 1. (a) the logical gate NOT; (b) circuit with logical gates NOT; (c) state portrait.
Figure 2. The semi-regular system from Example 11.
Figure 3. The semi-regular system from Example 15.
Figure 4. Symmetrical systems, Example 27.
Figure 5. Symmetrical system, Example 28.
Figure 6. System that is symmetrical relative to the coordinates, Example 32.
Figure 7. Φ has the automorphism Example 35.
Figure 8. Φ is symmetrical relative to translations with , Example 36.
Figure 9. Function Φ that is self dual, , Example 37.
Figure 10. Functions Φ that are self dual, , Example 38.
Figure 11. Symmetry including symmetry relative to translations, Example 39.
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