aa r X i v : . [ c s . G L ] A p r The Non-Anticipation of the AsynchronousSystems
Serban E. VladOradea City Hall, Piata Unirii, Nr. 1, 410100, Oradea, Romaniaserban e [email protected]
Abstract
The asynchronous systems are the models of the asynchronous cir-cuits from the digital electrical engineering and non-anticipation is one ofthe most important properties in systems theory. Our present purposeis to introduce several concepts of non-anticipation of the asynchronoussystems.
The asynchronous systems are the mathematical models of the asynchronouscircuits from the digital electrical engineering and our challenge is the construc-tion of an asynchronous systems theory. The insufficient bibliography that wehave at disposal is probably influenced by the great importance of the topic(researchers do not publish) and it consists of:a) mathematical studies in switching theory from the 60’s and we mentionhere the name of Grigore Moisil that used the discrete time modeling of theasynchronous circuits;b) engineering studies, which are always non-formalized and dedicated toapplications. Such a literature creates intuition, but it does not give acceptablemodels or tools of investigation;c) mathematical literature that can produce analogies.The study of the asynchronous systems is closely connected to the notionof signal, meaning a ’nice’ R → { , } n function. In this context we mentionthe apparent total absence of the mathematicians’ interest in the study of the R → { , } functions, that should be an interesting direction of investigation intemporal logic too.The paper is dedicated to non-anticipation, one of the most important prop-erties of the systems. We exemplify its use by showing how 0 can be chosen asinitial time, how the transfers of the systems are composed and how the asyn-chronous real time non-deterministic systems behave in certain circumstancesin a synchronous discrete time deterministic way.1 Preliminaries
Definition 2.1 B = { , } endowed with the laws , ∪ , · , ⊕ is called the binaryBoole algebra . Notation 2.2
Seq = { ( t k ) | t k ∈ R , k ∈ N , t < t < t < ... unbounded fromabove } . Notation 2.3
The restriction of the function x : R → B n at the interval I ⊂ R is denoted by x | I . Definition 2.4
The initial value x ( −∞ +0) ∈ B n and the final value x ( ∞− ∈ B n of x : R → B n are defined by ∃ t ∈ R , ∀ t < t , x ( t ) = x ( −∞ + 0) , ∃ t f ∈ R , ∀ t > t f , x ( t ) = x ( ∞ − . Notation 2.5
Denote by χ A : R → B the characteristic function of the set A ⊂ R . Definition 2.6
The function x : R → B n is called signal if ( t k ) ∈ Seq existssuch that ∀ t ∈ R ,x ( t ) = x ( −∞ + 0) · χ ( −∞ ,t ) ( t ) ⊕ x ( t ) · χ [ t ,t ) ( t ) ⊕ ... ⊕ x ( t k ) · χ [ t k ,t k +1 ) ( t ) ⊕ ... We have used the same symbols · , ⊕ for the laws that are induced by those of B .The set of the signals is denoted by S ( n ) and instead of S (1) we usually write S . Notation 2.7 P ∗ ( S ( n ) ) = { X | X ⊂ S ( n ) , X = ∅} . Definition 2.8
The left limit x ( t − and the left derivative Dx ( t ) of x ∈ S ( n ) expressed like in Definition 2.6 are defined by x ( t −
0) = x ( −∞ + 0) · χ ( −∞ ,t ] ( t ) ⊕ x ( t ) · χ ( t ,t ] ( t ) ⊕ ... ⊕ x ( t k ) · χ ( t k ,t k +1 ] ( t ) ⊕ ...Dx ( t ) = x ( t − ⊕ x ( t ) . Definition 2.9
A multi-valued function f : U → P ∗ ( S ( n ) ) , U ∈ P ∗ ( S ( m ) ) iscalled ( asynchronous ) system . Any u ∈ U is called ( admissible ) input andthe functions x ∈ f ( u ) are called ( possible ) states . Definition 2.10 If ∀ u ∈ U, f ( u ) has exactly one element, then f is called de-terministic and we use the notation f : U → S ( n ) of the uni-valued functions. Definition 2.11 If g : V → P ∗ ( S ( n ) ) , V ∈ P ∗ ( S ( m ) ) is a system, then anysystem f : U → P ∗ ( S ( n ) ) , U ∈ P ∗ ( S ( m ) ) with U ⊂ V and ∀ u ∈ U, f ( u ) ⊂ g ( u ) is called a subsystem of g and the notation is f ⊂ g. emark 2.12 The concept of system originates in the modeling of the asyn-chronous circuits. The multi-valued character of the cause-effect association isdue to statistical fluctuations in the fabrication process, the variations in theambiental temperature, the power supply etc.Sometimes the systems are given by equations and/or inequalities. In thiscase, determinism means that their solution is unique.If the system g models a circuit, then the system f ⊂ g models the samecircuit more precisely, by restricting the set of the inputs perhaps. Definition 3.1
The system f : U → P ∗ ( S ( n ) ) , U ∈ P ∗ ( S ( m ) ) is non-anticipatory if for all u ∈ U and all x ∈ f ( u ) it satisfies one of the following statements:a) x is constant;b) u, x are both variable and we have min { t | u ( t − = u ( t ) } ≤ min { t | x ( t − = x ( t ) } , (1) i.e. the first input switch is prior to the first output switch. Remark 3.2
The non-anticipation means that the system f is in equilibrium,as represented by the existence of the time interval ( −∞ , t ) , where u and x are constant: u | ( −∞ ,t ) = u ( t − and x | ( −∞ ,t ) = x ( t − then the onlypossibility to get out of this situation is the switch of the input.Moisil presumes implicitly in his works [1], [2] that the models are non-anticipatory in the sense of Definition 3.1. However his ’equilibrium’, called’rest position’, is defined in the presence of a ’network function’ that does notexist here. Example 3.3
Any Boolean function F : B m → B n defines for d ≥ a system F d : S ( m ) → S ( n ) , called ’ideal combinational’: ∀ u ∈ S ( m ) , ∀ t ∈ R , F d ( u )( t ) = F ( u ( t − d )) which is non-anticipatory. First, ∀ u ∈ S ( m ) we have F d ( u ) ∈ S ( n ) indeed . Second, for any d, u, t , u | ( −∞ ,t ) = u ( t − implies F d ( u ) | ( −∞ ,t ) = F d ( u )( t − (we have in this situation u ( t −
0) = u ( −∞ + 0) and F d ( u )( t −
0) = F d ( u )( −∞ + 0) ) and if u ( t − = u ( t ) , then F ( u ( t − d − , F ( u ( t − d )) represent two values that may be equal or different. We infer that x = F d ( u ) fulfills one of a), b) from Definition 3.1. Example 3.4
The system f : S → S , ∀ u ∈ S, f ( u ) = χ [0 , ∞ ) is anticipatory. emma 3.5 If g : V → P ∗ ( S ( n ) ) , V ∈ P ∗ ( S ( m ) ) is a non-anticipatory sys-tem, then any system f : U → P ∗ ( S ( n ) ) , U ∈ P ∗ ( S ( m ) ) with f ⊂ g is non-anticipatory. Proof.
Let be u ∈ U. We have the following possibilities:i) u is constant. From Definition 3.1 we have that ∀ x ∈ g ( u ) , x is constant,in particular ∀ x ∈ f ( u ) , x is constant. Therefore, f is non-anticipatory;ii) u is variable. Let x ∈ f ( u ) be arbitrary. Thenii.1) x is constant implies that f is non-anticipatory, by Definition 3.1 a);ii.2) x is variable. As element of g ( u ) , x satisfies (1) and, by Definition 3.1b), f is non-anticipatory. Notation 4.1 ∀ d ∈ R , the function τ d : R → R is the translation with d , thusfor any x ∈ S ( n ) we denote by x ◦ τ d the function ∀ t ∈ R , ( x ◦ τ d )( t ) = x ( t − d ) . Definition 4.2
The system f is time invariant if ∀ d ∈ R , ∀ u ∈ U,u ◦ τ d ∈ U, ∀ x ∈ f ( u ) , x ◦ τ d ∈ f ( u ◦ τ d ) . Notation 4.3
We use the notation S ( m )0 = { u | u ∈ S ( m ) , ∀ t < , u ( t ) = u ( −∞ + 0) } . Theorem 4.4
We state the following properties relative to some system b f : b U → P ∗ ( S ( n ) ) , b U ∈ P ∗ ( S ( m ) ) : i) b U ⊂ S ( m )0 ; ii) ∀ u ∈ b U , b f ( u ) ⊂ S ( n )0 ; iii) ∀ d ∈ R , ∀ u ∈ b U , ∀ x, ( x ∈ b f ( u ) and u ◦ τ d ∈ b U ) = ⇒ x ◦ τ d ∈ b f ( u ◦ τ d ) . a) The time-invariant non-anticipatory system f : U → P ∗ ( S ( n ) ) , U ∈ P ∗ ( S ( m ) ) is given. We define the system b f : b U → P ∗ ( S ( n ) ) by b U = { u | u ∈ U ∩ S ( m )0 and f ( u ) ∩ S ( n )0 = ∅} , ∀ u ∈ b U , b f ( u ) = f ( u ) ∩ S ( n )0 . (2) The system b f fulfills i), ii), iii) and is also non-anticipatory. ) Let be the system b f : b U → P ∗ ( S ( n ) ) satisfying the properties i), ii), iii)and non-anticipation. The system f : U → P ∗ ( S ( n ) ) , U ∈ P ∗ ( S ( m ) ) defined by U = { u ◦ τ d | d ∈ R , u ∈ b U } , ∀ d ∈ R , ∀ u ∈ b U , f ( u ◦ τ d ) = { x ◦ τ d | x ∈ b f ( u ) } is time invariant and non-anticipatory. Proof. a) We show that U ∩ S ( m )0 = ∅ . Let be u ∈ U. We have the possibil-ities:1) u is constant. Then u ∈ S ( m )0 , thus u ∈ U ∩ S ( m )0 ;2) u is variable.We denote d = min { t | u ( t − = u ( t ) } . If d ≥
0, then u ∈ S ( m )0 and u ∈ U ∩ S ( m )0 are true. If d <
0, then for any d ′ ≥ − d , u ◦ τ d ′ ∈ U is true because U is invariant to translations and u ◦ τ d ′ ∈ S ( m )0 holds true also, making u ◦ τ d ′ ∈ U ∩ S ( m )0 true.We show that b U = ∅ . We take some arbitrary u ∈ U ∩ S ( m )0 . If f ( u ) ∩ S ( n )0 = ∅ , the property is true, otherwise let be some x ∈ f ( u ). The fact that x / ∈ S ( n )0 shows that it is variable and if we denote by d = min { t | x ( t − = x ( t ) } , we have d <
0. Remark that for all d ′ ≥ − d, u ◦ τ d ′ ∈ U, u ◦ τ d ′ ∈ S ( m )0 , x ◦ τ d ′ ∈ f ( u ◦ τ d ′ )and x ◦ τ d ′ ∈ S ( n )0 take place. In other words u ◦ τ d ′ ∈ b U .
This shows that b f is well defined, in the sense that b U = ∅ and ∀ u ∈ b U , b f ( u ) = ∅ . Moreover, i) and ii) are obviously satisfied.We show now the truth of iii). We take d ∈ R , u ∈ b U , x arbitrary with x ∈ b f ( u ) and u ◦ τ d ∈ b U true. We have the possibilities:j) x is constant. Then x ◦ τ d = x is constant and x ◦ τ d ∈ S ( n )0 ;jj) x is variable. Because x ∈ f ( u ) , from the non-anticipation of f we havethat u is variable and0 ≤ min { t | ( u ◦ τ d )( t − = ( u ◦ τ d )( t ) } ≤ min { t | ( x ◦ τ d )( t − = ( x ◦ τ d )( t ) } , showing that x ◦ τ d ∈ S ( n )0 .In both cases j), jj), x ∈ b f ( u ) has implied x ∈ f ( u ) and, furthermore, x ◦ τ d ∈ f ( u ◦ τ d ) from the time invariance of f and, eventually, x ◦ τ d ∈ b f ( u ◦ τ d )(= f ( u ◦ τ d ) ∩ S ( n )0 ).Because b f ⊂ f, the non-anticipation of b f is a consequence of Lemma 3.5.b) We show that f is well defined in the sense that if d, d ′ ∈ R and u, v ∈ b U satisfy u ◦ τ d = v ◦ τ d ′ , we get f ( u ◦ τ d ) = f ( v ◦ τ d ′ ) . Let be x ◦ τ d ∈ f ( u ◦ τ d ) . Weinfer that x ∈ b f ( u ) and v = u ◦ τ d − d ′ ∈ b U .
From iii) we have that x ◦ τ d − d ′ ∈ b f ( v ) , i.e. x ◦ τ d = x ◦ τ d − d ′ ◦ τ d ′ ∈ f ( v ◦ τ d ′ ) . We have obtained that f ( u ◦ τ d ) ⊂ f ( v ◦ τ d ′ )and the inverse inclusion is shown similarly.We show that U is invariant to translations. Let be v ∈ U . Then there aresome d ∈ R and u ∈ b U such that v = u ◦ τ d . For an arbitrary d ′ ∈ R , as v ◦ τ d ′ = u ◦ τ d + d ′ , we infer v ◦ τ d ′ ∈ U .5e show that f is time invariant. Let be v ∈ U and y ∈ f ( v ), meaning theexistence of u ∈ b U and d ∈ R with v = u ◦ τ d . We get y ∈ f ( u ◦ τ d ) = { x ◦ τ d | x ∈ b f ( u ) } . In other words ∃ x, y = x ◦ τ d and x ∈ b f ( u ). We take an arbitrary d ′ ∈ R for which y ◦ τ d ′ = x ◦ τ d + d ′ , y ◦ τ d ′ ∈ { x ◦ τ d + d ′ | x ∈ b f ( u ) } = f ( u ◦ τ d + d ′ ) = f ( v ◦ τ d ′ ).We show now that f is non-anticipatory. Let us take, like previously, v ∈ U and y ∈ f ( v ), for which there are u ∈ b U , x ∈ b f ( u ) and d ∈ R such that v = u ◦ τ d and y = x ◦ τ d . We have the possibilities:I) y is constant. Then f is non-anticipatory;II) y is variable. Then x ∈ b f ( u ) is variable and the hypothesis concerningthe non-anticipation of b f states that u is variable andmin { t | u ( t − = u ( t ) } ≤ min { t | x ( t − = x ( t ) } . We obtainmin { t | v ( t − = v ( t ) } = min { t | ( u ◦ τ d )( t − = ( u ◦ τ d )( t ) } == d + min { t | u ( t − = u ( t ) } ≤ d + min { t | x ( t − = x ( t ) } == min { t | ( x ◦ τ d )( t − = ( x ◦ τ d )( t ) } = min { t | y ( t − = y ( t ) } . Remark 4.5 S ( m )0 consists in these signals u ∈ S ( m ) that accept the ’initialtime instant’ t be 0. Items i), ii) of Theorem 4.4 mean that the inputs andthe states of b f accept the initial time instant be and item iii) of that theoremrepresents time invariance adapted to the situation when b U is not closed totranslations (any b U ⊂ S ( m )0 that contains non-constant signals is not invariantto translations).The possibility of choosing as initial time instant simplifies a little the studyof the asynchronous systems. Definition 5.1
Let the system f : U → P ∗ ( S ( n ) ) be given , U ∈ P ∗ ( S ( m ) ) . Itis called non-anticipatory if ∀ t ∈ R , ∀ u ∈ U, ∀ v ∈ U,u | ( −∞ ,t ) = v | ( −∞ ,t ) = ⇒ { x | ( −∞ ,t ] | x ∈ f ( u ) } = { y | ( −∞ ,t ] | y ∈ f ( v ) } . Remark 5.2
Definition 5.1 states that the history of all the possible states untilthe present moment, including the present depends only on the history of theinput and it does not depend on the present and the future values of the input.The definition means that ∀ t ∈ R a function f t exists that associates ∀ u ∈ U to u | ( −∞ ,t ) the set f t ( u | ( −∞ ,t ) ) = { x | ( −∞ ,t ] | x ∈ f ( u ) } . Definition 5.1 represents a perspective of non-anticipation, other than theprevious one and the two properties are logically independent. xample 5.3 The deterministic system f : S ( m ) → S, ∀ u ∈ S ( m ) , f ( u ) = χ [0 , ⊕ ( u ◦ τ ) · χ [1 , ∞ ) is non-anticipatory in the sense of Definition 5.1. The system f is anticipatoryin the sense of Definition 3.1 because for u = χ [2 , ∞ ) , u = ... = u m = 0 the contradiction min { t | u ( t − = u ( t ) } = 2 > { t | x ( t − = x ( t ) } isobtained. Example 5.4
The deterministic system f : S → S, ∀ u ∈ S, f ( u ) = (cid:26) , if u = χ [0 , ∞ ) u, otherwise , is anticipatory in the sense of Definition 5.1 because for t = 1 , u = χ [0 , ∞ ) ,v = χ [0 , we have u | ( −∞ , = v | ( −∞ , but | ( −∞ , = χ [0 , | ( −∞ , . However it isnon-anticipatory in the sense of Definition 3.1. Example 5.5
The deterministic system f : S → S ∀ u ∈ S, f ( u ) = (cid:26) , if u = χ [0 , ∞ ) u ◦ τ − , otherwise is anticipatory in the sense of both Definitions 3.1 and 5.1. Example 5.6
The deterministic system Dx ( t ) = ( x ( t − ⊕ u ( t − · [ ξ ∈ ( t − d,t ) Du ( ξ ) u, x ∈ S, d > is non-anticipatory in the sense of both Definitions 3.1, 5.1.The idea expressed by such an equation is: x switches ( Dx ( t ) = 1 ) at these timeinstants when u has indicated the necessity of such a switch ( x ( t − ⊕ u ( t −
0) =1) for d time units ( u | [ t − d,t ) is the constant function, with null derivative in theinterval ( t − d, t ) ). This equation models the delay circuit. ∗ Definition 6.1
Let be the system f : U → P ∗ ( S ( n ) ) , U ∈ P ∗ ( S ( m ) ) . It iscalled non-anticipatory if it satisfies one of the following conditions, calledconditions of non-anticipation:i) ∀ t ∈ R , ∀ u ∈ U, ∀ v ∈ U,u | ( −∞ ,t ) = v | ( −∞ ,t ) = ⇒ { x ( t ) | x ∈ f ( u ) } = { y ( t ) | y ∈ f ( v ) } ; ii) ∀ t ∈ R , ∀ u ∈ U, ∀ v ∈ U, ∃ d > ,u | [ t − d,t ) = v | [ t − d,t ) = ⇒ { x ( t ) | x ∈ f ( u ) } = { y ( t ) | y ∈ f ( v ) } ;7 ii) ∀ t ∈ R , ∃ d > , ∀ u ∈ U, ∀ v ∈ U,u | [ t − d,t ) = v | [ t − d,t ) = ⇒ { x ( t ) | x ∈ f ( u ) } = { y ( t ) | y ∈ f ( v ) } ; iv) ∃ d > , ∀ t ∈ R , ∀ u ∈ U, ∀ v ∈ U,u | [ t − d,t ) = v | [ t − d,t ) = ⇒ { x ( t ) | x ∈ f ( u ) } = { y ( t ) | y ∈ f ( v ) } ; v) ∀ t ∈ R , ∀ u ∈ U, ∀ v ∈ U,u | ( −∞ ,t ] = v | ( −∞ ,t ] = ⇒ { x | ( −∞ ,t ] | x ∈ f ( u ) } = { y | ( −∞ ,t ] | y ∈ f ( v ) } ; vi) ∀ t ∈ R , ∀ u ∈ U, ∀ v ∈ U,u | ( −∞ ,t ] = v | ( −∞ ,t ] = ⇒ { x ( t ) | x ∈ f ( u ) } = { y ( t ) | y ∈ f ( v ) } ; vii) ∀ t ∈ R , ∀ u ∈ U, ∀ v ∈ U , ∃ d, ∃ d ′ , ≤ d ≤ d ′ and u | [ t − d ′ ,t − d ] = v | [ t − d ′ ,t − d ] = ⇒ { x ( t ) | x ∈ f ( u ) } = { y ( t ) | y ∈ f ( v ) } ; viii) ∀ t ∈ R , ∃ d, ∃ d ′ , ≤ d ≤ d ′ and ∀ u ∈ U, ∀ v ∈ U , u | [ t − d ′ ,t − d ] = v | [ t − d ′ ,t − d ] = ⇒ { x ( t ) | x ∈ f ( u ) } = { y ( t ) | y ∈ f ( v ) } ; ix) ∃ d, ∃ d ′ , ≤ d ≤ d ′ and ∀ t ∈ R , ∀ u ∈ U, ∀ v ∈ U,u | [ t − d ′ ,t − d ] = v | [ t − d ′ ,t − d ] = ⇒ { x ( t ) | x ∈ f ( u ) } = { y ( t ) | y ∈ f ( v ) } . Theorem 6.2 If f : U → S ( n ) is a deterministic system, then Definition 6.1 v)and Definition 6.1 vi) are equivalent. We have that Definition 5.1 and Definition6.1 i) are equivalent in this case too. Proof.
We prove the first statement. Because v)= ⇒ vi) is obvious, weprove vi)= ⇒ v). Let us suppose against all reason that v) is not true, i.e. ∃ t ∈ R , ∃ u ∈ U, ∃ v ∈ U, u | ( −∞ ,t ] = v | ( −∞ ,t ] and f ( u ) | ( −∞ ,t ] = f ( v ) | ( −∞ ,t ] . Thismeans the existence of t ′ ≤ t such that u | ( −∞ ,t ′ ] = v | ( −∞ ,t ′ ] and f ( u )( t ′ ) = f ( v )( t ′ ) , contradiction with vi). Remark 6.3
In Definition 6.1, all of i),...,ix) express the same idea like Defini-tion 5.1, namely that the present depends on the past only and it is independenton the future. The implications are: iv ) = ⇒ iii ) = ⇒ ii ) = ⇒ i ) ⇐ = Def inition ⇓ ⇓ ix ) = ⇒ viii ) = ⇒ vii ) = ⇒ vi ) ⇐ = v ) In ii),...,iv), vii),...,ix) the boundness of the memory occurs: these are sys-tems whose states do not depend on all the input segment u | ( −∞ ,t ) , but on thelast d time units u | [ t − d,t ) only and similarly for u | ( −∞ ,t ] and u | [ t − d ′ ,t − d ] . ow have a look at the non-anticipation property iv). We note that if d > is a number for which it is fulfilled, then any number d ′ ≥ d fulfills it also: ∀ t ∈ R , ∀ u ∈ U, ∀ v ∈ U,u | [ t − d ′ ,t ) = v | [ t − d ′ ,t ) = ⇒ { x ( t ) | x ∈ f ( u ) } = { y ( t ) | y ∈ f ( v ) } . Our problem is whether, for a system f , the set of those d satisfying implicationiv) is bounded from below by some d ′′ > , because we have a non-anticipationproperty ∀ t ∈ R , ∀ u ∈ U, ∀ v ∈ U,u ( t −
0) = v ( t −
0) = ⇒ { x ( t ) | x ∈ f ( u ) } = { y ( t ) | y ∈ f ( v ) } also, like in the example u ( t − · x ( t ) = 0 where u, x ∈ S . If this lower bound exists, we obtain a new shading of thatconcept of non-anticipation. The problem of the existence of such bounds is, inprinciple, the same if d is variable like in ii), iii) or if instead of one parameter d we have two parameters d, d ′ and two bounds, like in vii), viii), ix).Remark that the reasoning of Theorem 6.2 is impossible to use if f is non-deterministic. We suppose, for this, that the system f : S → P ∗ ( S ) satisfies f (0) = { , } , f ( χ [2 , ∞ ) ) = { χ ( −∞ , , χ [0 , ∞ ) } , where , ∈ S are the constantfunctions. We have ∀ t ∈ [0 , , | ( −∞ ,t ] = χ [2 , ∞ ) | ( −∞ ,t ] andand { | ( −∞ ,t ] , | ( −∞ ,t ] } 6 = { χ ( −∞ , | ( −∞ ,t ] , χ [0 , ∞ ) | ( −∞ ,t ] } andand { x ( t ) | x ∈ f (0) } = { , } = { y ( t ) | y ∈ f ( χ [2 , ∞ ) ) } . Example 6.4
The system I d : S → S called the ’pure delay model’ of the delaycircuit, defined by ∀ u ∈ S, x ( t ) = I d ( u )( t ) = u ( t − d ) , satisfies for d > all thenon-anticipation properties i),...,ix) from Definition 6.1. Example 6.5
Let the system f : S → P ∗ ( S ) (version of the ’bounded delaymodel’ of the delay circuit) be defined by the inequalities \ ξ ∈ [ t − d r ,t ) u ( ξ ) ≤ x ( t ) ≤ [ ξ ∈ [ t − d f ,t ) u ( ξ ) , where d r > , d f > . It satisfies all the non-anticipation properties i),...,ix)from Definition 6.1.
Example 6.6
The system f : S → P ∗ ( S ) (version of the ’bounded delay model’of the delay circuit) described by the inequalities \ ξ ∈ [ t − d ′ ,t − d ] u ( ξ ) ≤ x ( t ) ≤ [ ξ ∈ [ t − d ′ ,t − d ] u ( ξ ) , where ≤ d ≤ d ′ satisfies the non-anticipation properties v),...,ix) from Defini-tion 6.1. xample 6.7 The system f : S → P ∗ ( S ) defined by t Z −∞ Du ≤ x ( t ) where t Z −∞ Du = (cid:26) , if | suppDu ∩ ( −∞ , t ] | is odd , if | suppDu ∩ ( −∞ , t ] | is even satisfies the non-anticipation properties v), vi) of Definition 6.1. We have de-noted by | suppDu ∩ ( −∞ , t ] | the number of elements of the finite set { ξ | ξ ∈ R , Du ( ξ ) = 1 } ∩ ( −∞ , t ] and we have supposed that is an even number. Example 6.8
Denote by ϕ : S ( m ) → [0 , ∞ ) the function ∀ u ∈ S ( m ) ,ϕ ( u ) = (cid:26) , if u is constant max {− min { t | u ( t − = u ( t ) } , min { t | u ( t − = u ( t ) }} , if u is variableThe deterministic system x ( t ) = \ ξ ∈ [ t − ϕ ( u ) ,t − ϕ ( u )] u ( ξ ) ,u, x ∈ S, satisfies the non-anticipation property vii) of Definition 6.1. Definition 6.9
The system f is called non-anticipatory ∗ if it satisfies oneof the following conditions, called conditions of non-anticipation ∗ : i) ∀ t ∈ R , ∀ u ∈ U, ∀ v ∈ U, ( u | [ t, ∞ ) = v | [ t, ∞ ) and { x ( t ) | x ∈ f ( u ) } = { y ( t ) | y ∈ f ( v ) } ) = ⇒ = ⇒ { x | [ t, ∞ ) | x ∈ f ( u ) } = { y | [ t, ∞ ) | y ∈ f ( v ) } ; ii) ∀ t ∈ R , ∀ u ∈ U, ∀ v ∈ U,u | [ t, ∞ ) = v | [ t, ∞ ) = ⇒ ∃ t ′ ∈ R , { x | [ t ′ , ∞ ) | x ∈ f ( u ) } = { y | [ t ′ , ∞ ) | y ∈ f ( v ) } ; iii) ∀ t ∈ R , ∀ u ∈ U, ∀ v ∈ U, ( u | [ t, ∞ ) = v | [ t, ∞ ) and { x | ( −∞ ,t ] | x ∈ f ( u ) } = { y | ( −∞ ,t ] | y ∈ f ( v ) } ) = ⇒ = ⇒ ∃ t ′ ∈ R , { x | [ t ′ , ∞ ) | x ∈ f ( u ) } = { y | [ t ′ , ∞ ) | y ∈ f ( v ) } . Remark 6.10
We remark that property i) resembles somehow with fixing theinitial conditions in a differential equation ( { x ( t ) | x ∈ f ( u ) } = { y ( t ) | y ∈ f ( v ) } ).The consequence is that the solution is unique ( { x | [ t, ∞ ) | x ∈ f ( u ) } = { y | [ t, ∞ ) | y ∈ f ( v ) } ) under an arbitrary given input ( u | [ t, ∞ ) = v | [ t, ∞ ) ).The reader is invited to write other similar properties of non-anticipationand non-anticipation ∗ . The Transfers of the Non-Anticipatory Sys-tems
Theorem 7.1
Let the system f satisfy the conditions:a) U is closed under ’concatenation’ ∀ t ∈ R , ∀ u ∈ U, ∀ v ∈ U,u · χ ( −∞ ,t ) ⊕ v · χ [ t, ∞ ) ∈ U ; b) non-anticipation ∀ t ∈ R , ∀ u ∈ U, ∀ v ∈ U,u | ( −∞ ,t ) = v | ( −∞ ,t ) = ⇒ { x | ( −∞ ,t ] | x ∈ f ( u ) } = { y | ( −∞ ,t ] | y ∈ f ( v ) } ; c) non-anticipation ∗ ∀ t ∈ R , ∀ u ∈ U, ∀ v ∈ U, ( u | [ t, ∞ ) = v | [ t, ∞ ) and { x ( t ) | x ∈ f ( u ) } = { y ( t ) | y ∈ f ( v ) } ) = ⇒ = ⇒ { x | [ t, ∞ ) | x ∈ f ( u ) } = { y | [ t, ∞ ) | y ∈ f ( v ) } ; d) time invariance ∀ d ∈ R , ∀ u ∈ U,u ◦ τ d ∈ U, ∀ x ∈ f ( u ) , x ◦ τ d ∈ f ( u ◦ τ d ); e) t , t ∈ R , u , u ∈ U and µ, µ ′ , µ ′′ ∈ B n are given such that ∀ x ∈ f ( u ) , ∃ t < t , x ( t ) = µ, (3) ∀ x ∈ f ( u ) , x ( t ) = µ ′ , (4) ∀ x ′ ∈ f ( u ) , x ′ ( t ) = µ ′ , (5) ∀ x ′ ∈ f ( u ) , ∃ t > t , x ′ ( t ) = µ ′′ . (6) Put d = t − t . Then e u ∈ U defined as e u = u · χ ( −∞ ,t ) ⊕ ( u ◦ τ d ) · χ [ t , ∞ ) , (7) satisfies ∀ e x ∈ f ( e u ) , ∃ t < t , e x ( t ) = µ, (8) ∀ e x ∈ f ( e u ) , ∃ t ′ > t , e x ( t ′ ) = µ ′′ . (9) Proof. e u belongs to U indeed, because of a) and d). We remark that wehave e u | ( −∞ ,t ) = u | ( −∞ ,t ) . (10)From (10) and b) we infer { e x | ( −∞ ,t ] | e x ∈ f ( e u ) } = { x | ( −∞ ,t ] | x ∈ f ( u ) } (11)11nd if, in addition, we take into account (3), (4), then we get the truth of (8)and of ∀ e x ∈ f ( e u ) , e x ( t ) = µ ′ . (12)Let be now some arbitrary x ′′ ∈ f ( u ◦ τ d ) . From d) we obtain the existenceof x ′ ∈ f ( u ) , such that x ′′ = x ′ ◦ τ d (namely x ′ = x ′′ ◦ τ − d ) and we have x ′′ ( t ) = ( x ′ ◦ τ d )( t ) = x ′ ( t ) = µ ′ (we have taken into account (5)) thus ∀ x ′′ ∈ f ( u ◦ τ d ) , x ′′ ( t ) = µ ′ (13)and, similarly, ∀ x ′′ ∈ f ( u ◦ τ d ) , ∃ t ′ > t , x ′′ ( t ′ ) = µ ′′ . (14)We see that e u | [ t , ∞ ) = ( u ◦ τ d ) | [ t , ∞ ) . (15)The hypothesis of c) is fulfilled by t , e u and u ◦ τ d , as follows from (12), (13)and (15). The conclusion of c) expresses the fact that { e x | [ t , ∞ ) | e x ∈ f ( e u ) } = { x ′′| [ t , ∞ ) | x ′′ ∈ f ( u ◦ τ d ) } (16)and, by (14), we get the truth of (9). Remark 7.2
The relations (3), (6), (8), (9) show the asynchronous access(weaker, the time instant when the access happens depends on x ) of the statesof f to the values µ, µ ′′ and the relations (4), (5) represent the synchronousaccess (stronger, the time instant when the access happens is the same for all x ;these two accesses must match) of the states of f to the value µ ′ . The theoremstates that if f ( u ) transfers µ in µ ′ and f ( u ) transfers µ ′ in µ ′′ , then f ( e u ) transfers µ in µ ′′ . Several versions of this theorem are obtained if we take t = t (then time in-variance disappears from the hypothesis), if we have in the hypothesis countablemany transfers (instead of two; these transfers must have synchronous accesses),if we state in the hypothesis a controllability/accessibility request etc. Definition 8.1
Consider the system f supposed to be non-anticipatory (Defi-nition 5.1) and let u ∈ U be a fixed input. If there are ( t k ) ∈ Seq, ( u k ) ∈ U and ( µ k ) ∈ B n such that ∀ x ∈ f ( u ) , x | ( −∞ ,t ) = µ and x | [ t , ∞ ) = µ ,u | ( −∞ ,t ) = u | ( −∞ ,t ) , u | ( −∞ ,t ) = u | ( −∞ ,t ) , u | ( −∞ ,t ) = u | ( −∞ ,t ) , ... ∀ x ∈ f ( u ) , x | [ t , ∞ ) = µ , ∀ x ∈ f ( u ) , x | [ t , ∞ ) = µ , ∀ x ∈ f ( u ) , x | [ t , ∞ ) = µ , ... then the input u is called a fundamental ( operating ) mode ( of f ). emark 8.2 The evolution of f under the fundamental mode u may be inter-preted as the discrete time evolution of a deterministic system of the form µ = x (0) u → µ = x (1) u → ... u k → µ k +1 = x ( k + 1) u k +1 → ... To be remarked the appearance of the next state partial function ∀ k ∈ N , B n × U ∋ ( µ k , u k ) → µ k +1 ∈ B n . If ∃ k ∈ N such that u k = u k +1 = ... and µ k +1 = µ k +2 = ..., then the evolution may be considered to be given by a finite sequence µ = x (0) u → µ = x (1) u → ... u k → µ k +1 = x ( k + 1) . Example 8.3
We get back to the system f from the Example 6.5, that fulfillsthe non-anticipation property from Definition 5.1. We suppose that the sequence ( t k ) ∈ Seq satisfies ∀ k ∈ N ,t k +1 ≥ t k + d r , t k +2 ≥ t k +1 + d f and let be the sequences ( u k ) ∈ S, ( µ k ) ∈ B ,u ( t ) = χ [ t , ∞ ) ( t ) , u ( t ) = χ [ t ,t ) ( t ) ⊕ χ [ t , ∞ ) ( t ) ,u ( t ) = χ [ t ,t ) ( t ) ⊕ χ [ t ,t ) ( t ) ⊕ χ [ t , ∞ ) ( t ) , ...u ( t ) = χ [ t ,t ) ( t ) , u ( t ) = χ [ t ,t ) ( t ) ⊕ χ [ t ,t ) ( t ) , ...µ = µ = µ = ... = 0 , µ = µ = µ = ... = 1 . For u ( t ) = χ [ t ,t ) ( t ) ⊕ χ [ t ,t ) ( t ) ⊕ χ [ t ,t ) ( t ) ⊕ ... we have ∀ x ∈ f ( u ) , x ( t ) = x ( t ) · χ ( t ,t + d r ) ( t ) ⊕ χ [ t + d r , ∞ ) ( t ) , ∀ x ∈ f ( u ) , x ( t ) = x ( t ) · χ ( t ,t + d r ) ( t ) ⊕ χ [ t + d r ,t ] ( t ) ⊕ x ( t ) · χ ( t ,t + d f ) ( t ) , ∀ x ∈ f ( u ) , x ( t ) = x ( t ) · χ ( t ,t + d r ) ( t ) ⊕ χ [ t + d r ,t ] ( t ) ⊕ x ( t ) · χ ( t ,t + d f ) ( t ) ⊕⊕ x ( t ) · χ ( t ,t + d r ) ( t ) ⊕ χ [ t + d r , ∞ ) ( t ) , ∀ x ∈ f ( u ) , x ( t ) = x ( t ) · χ ( t ,t + d r ) ( t ) ⊕ χ [ t + d r ,t ] ( t ) ⊕ x ( t ) · χ ( t ,t + d f ) ( t ) ⊕⊕ x ( t ) · χ ( t ,t + d r ) ( t ) ⊕ χ [ t + d r ,t ] ( t ) ⊕ x ( t ) · χ ( t ,t + d f ) ( t ) ... We conclude that u is a fundamental mode of f : ∀ x ∈ f ( u ) , x | ( −∞ ,t ) = 0 and x | [ t , ∞ ) = 1 ,u | ( −∞ ,t ) = u | ( −∞ ,t ) , u | ( −∞ ,t ) = u | ( −∞ ,t ) , u | ( −∞ ,t ) = u | ( −∞ ,t ) , ... ∀ x ∈ f ( u ) , x | [ t , ∞ ) = 0 , ∀ x ∈ f ( u ) , x | [ t , ∞ ) = 1 , ∀ x ∈ f ( u ) , x | [ t , ∞ ) = 0 , ... Accessibility vs fundamental mode
Theorem 9.1
Let be the non-anticipatory system (Definition 5.1) f : U → P ∗ ( S ( n ) ) , U ∈ P ∗ ( S ( m ) ) and we suppose that the following requirements arefulfilled: a ) for any ( t k ) ∈ Seq and any ( u k ) ∈ U, we have u · χ ( −∞ ,t ) ⊕ u · χ [ t ,t ) ⊕ u · χ [ t ,t ) ⊕ ... ∈ U ; b ) f has race-free initial states and bounded initial time, i.e. ∀ u ∈ U, ∃ µ ∈ B n , ∃ t ∈ R , ∀ x ∈ f ( u ) , x | ( −∞ ,t ) = µ ; c ) any vector from B n is the common final value of the states under an inputhaving arbitrary initial segment ∀ µ ∈ B n , ∀ u ∈ U, ∀ t ∈ R , ∃ v ∈ U, ∃ t ′ > t,u | ( −∞ ,t ) = v | ( −∞ ,t ) and ∀ y ∈ f ( v ) , y | [ t ′ , ∞ ) = µ. Then there is some µ ∈ B n such that for any sequence µ k ∈ B n , k ≥ ofbinary vectors, there are the sequences ( t k ) ∈ Seq, u k ∈ U, k ∈ N and an input e u ∈ U such that ∀ x ∈ f ( u ) , x | ( −∞ ,t ) = µ and x | [ t , ∞ ) = µ , e u | ( −∞ ,t ) = u | ( −∞ ,t ) , e u | ( −∞ ,t ) = u | ( −∞ ,t ) , e u | ( −∞ ,t ) = u | ( −∞ ,t ) , ... ∀ x ∈ f ( u ) , x | [ t , ∞ ) = µ , ∀ x ∈ f ( u ) , x | [ t , ∞ ) = µ , ∀ x ∈ f ( u ) , x | [ t , ∞ ) = µ , ... Proof.
Let v ∈ U be an arbitrary input. From b) we get the existence of µ ∈ B n and t ∈ R depending on v , such that ∀ x ∈ f ( v ) , x | ( −∞ ,t ) = µ . (17)Let us fix the sequence µ k ∈ B n , k ≥ δ > . At thismoment the property c) implies the existence of u ∈ U and t > t + δ suchthat v | ( −∞ ,t ) = u | ( −∞ ,t ) and ∀ x ∈ f ( u ) , x | [ t , ∞ ) = µ , of u ∈ U and t > t + δ such that u | ( −∞ ,t ) = u | ( −∞ ,t ) and ∀ x ∈ f ( u ) , x | [ t , ∞ ) = µ , of u ∈ U and t > t + δ such that u | ( −∞ ,t ) = u | ( −∞ ,t ) and ∀ x ∈ f ( u ) , x | [ t , ∞ ) = µ ,... The way that ( t k ) was constructed guarantees the fact that this sequencebelongs to Seq . Thus, by a), the input e u defined as e u = u · χ ( −∞ ,t ) ⊕ u · χ [ t ,t ) ⊕ u · χ [ t ,t ) ⊕ ... U . We have e u | ( −∞ ,t ) = u | ( −∞ ,t ) , e u | ( −∞ ,t ) = u | ( −∞ ,t ) , e u | ( −∞ ,t ) = u | ( −∞ ,t ) , ... Remark 9.2