The consistency, the composition and the causality of the asynchronous flows
JJournal of Progressive Research in Mathematics(JPRM)
ISSN: 2395-0218
SCITECH
Volume 3, Issue 2
RESEARCH ORGANISATION|
April 20, 2015 | Journal of Progressive Research in Mathematics
The Consistency, the Composition and the Causality of the Asynchronous Flows
Serban E. Vlad Oradea City Hall, p-ta Unirii, nr. 1, 410100, Oradea, Romania.
Abstract
Let nn }1,0{}1,0{: . The asynchronous flows are (discrete time and real time) functions that result by iterating the coordinates },...,1{, ni i independently on each other. The purpose of the paper is that of showing that the asynchronous flows fulfill the properties of consistency, composition and causality that define the dynamical systems. The origin of the problem consists in modelling the asynchronous circuits from the digital electrical engineering. Keywords consistency; composition; causality; asynchronous flow; asynchronous circuit. Introduction
The Boolean autonomous deterministic regular asynchronous systems have been defined by the author in 2007 and a study of such systems can be found in [12]. The concept has its origin in switching theory, the theory of modelling the asynchronous (or switching) circuits from the digital electrical engineering. The attribute Boolean vaguely refers to the Boole algebra with two elements; autonomous means that there is no input; determinism means the existence of a unique state function; and regular indicates the existence of a function ,}1,0{}1,0{: nn ),...,( n that ‘generates’ the system. Time is discrete: ,...}1,0,1{ , or continuous: R . The system, which is analogue to the (real, usual) dynamical systems, iterates (asynchronously) on each coordinate },...,1{ ni one of - i : we say that is computed, at that time instant, on that coordinate; - }1,0{),...,,...,(}1,0{ inin : we use to say that is not computed, at that time instant, on that coordinate. ournal of Progressive Research in Mathematics(JPRM) ISSN: 2395-0218
Figure 1: Asynchronous circuit
The flows are these that result by analogy with the dynamical systems. The ‘nice’ discrete time and real time functions that the (Boolean) asynchronous systems work with are called signals. The functions that show when and how the coordinates i are computed are called computation functions. In order to point out the source of inspiration, we give the example of the circuit from Figure 1, where }1,0{,...}1,0,1{:ˆ x is the signal representing the state of the system, and the initial state is )0,0( . The function that generates the system is }1,0{}1,0{: , ,}1,0{ ),()( . The evolution of the system is shown in its state diagram from Figure 2, where the arrows indicate the time increase Figure 2: The state diagram of the circuit from Figure 1 and we have underlined the coordinates i i that, by the computation of , change their value: ii )( . Let }1,0{,...}2,1,0{: be the computation function whose values ki show that i is computed at the time instant k if ki , respectively that it is not computed at the time instant k if ki , where i and ,...}2,1,0{ k . The uncertainty related with the modelled circuit, depending in general on the technology, the temperature, etc, manifests in the fact that the order and the time of computation of each coordinate function i are not known. The situation )0,0( , when no coordinate of is computed at the time instant , shows that the system remains in )0,0( . If the first coordinate of is computed at the time instant , i.e. )0,1( , then Figure 2 indicates the transfer from )0,0( in )0,1( . ournal of Progressive Research in Mathematics(JPRM) ISSN: 2395-0218 , i.e. )1,0( , and in this case the system transfers from )0,0( to )1,0( , where it remains indefinitely long for any values of ,...,, , since )1,0()1,0( . Such a signal x ˆ is called eventually constant and it corresponds to a stable system. The last possibility is given by )1,1( that indicates the transfer from )0,0( to )1,1( , as resulted by the simultaneous computation of )0,0( and )0,0( . If the system is in one of the points )1,1(),0,1( and the set }1,|{ k kk N is infinite, then it switches infinitely many times between )0,1( and )1,1( and this corresponds to an unstable system. The purpose of our paper is that of showing that the flows of these systems fulfill the properties of consistency, composition and causality that define the dynamical systems. Preliminaries. Signals
Notation 1
We denote by }1,0{ B the binary Boole algebra. Its laws are the usual ones: Table 1.
011 100 10,111 100 10,101 000 10,01 10 and they induce laws that are denoted with the same symbols on n n B . Definition 2
Both sets B and n B are organized as topological spaces by the discrete topology. Notation 3 ,...}1,0,1{ N is the notation of the discrete time set. Notation 4
We denote ...},,|){(ˆ kkkandjkkqeS jj NN }...,|){( abovefromunboundedtttandkttSeq kk NR . Notation 5 BR : A is the notation of the characteristic function of the set R A : , R t otherwiseAtift A ,0 ,,1)( . Definition 6
The discrete time signals are by definition the functions n x BN :ˆ . Their set is denoted with )( ˆ n S . The continuous time signals are the functions n x BR : of the form , R t ...)()(...)()()()( )1,[)1,0[0)0,( ttxttxttx ktktkttt (1) where n B and .)( Seqt k Their set is denoted by )( n S . Remark 7
The signals model the electrical signals of the circuits from the digital electrical engineering.
Remark 8
At Notation 4 and Definition 6 a convention of notation has occurred, namely a hat ^ is used to show that we have discrete time. The hat will make the difference between, for example, the notation of the discrete time signals ,...ˆ,ˆ yx and the notation of the real time signals ,..., yx Lemma 9
For any )( n Sx and any n txt BR )0(, exists with the property )0()(),,(,0 txxtt . (2) Proof.
We presume that tx , are arbitrary and fixed and that x is of the form (1), with n B and Seqt k )( . If tt , then any makes (2) be true with )0( tx ; and if k exists with ],( kk ttt , then any ),0( kk tt makes (2) be true with )()0( k txtx . Definition 10
The function n txt BR )0( is called the left limit function of x . ournal of Progressive Research in Mathematics(JPRM) ISSN: 2395-0218
Definition 11
The discrete time forgetful function )()(' ˆˆ:ˆ nnk SS is defined for any N ' k by )'(ˆ))(ˆ(ˆ,,ˆˆ ')( kkxkxkSx kn N (3) and the real time forgetful function )()(' : nnt SS is defined for R ' t in the following manner '.),0'( ,'),())((,, ')( tttx tttxtxtSx tn R (4) Computation functions
Definition 12
The discrete time computation functions are by definition the sequences n BN : . Their set is denoted by ' ˆ n . In general, we write k instead of .),( N kk The real time computation functions n BR : are by definition the functions of the form ...)(...)()()( }{}1{1}0{0 tttt ktktt (5) where Seqt k )( . Their set is denoted by ' n . Remark 13
The meaning of the computation functions '' ,ˆ nn , subject to the additional property of progressiveness that will be stated later, is that of showing when –in discrete time and in real time- and how the Boolean functions nn BB : are computed. Lemma 14
For any ' n and any R t , we have )0,...,0()(),,(,0 tt . (6) Proof.
Analogue with the proof of Lemma 9.
Definition 15
The discrete time ''' ˆˆ:ˆ nnk , N ' k and the continuous time ''' : nnt forgetful function , R ' t , are defined by: ,,ˆ ' N k n '' ))(ˆ( kkkk (7) and ,, ' R t n )()())(( ),'[' ttt tt . (8) Remark 16
Definition 15, equation (8) was given by analogy with Definition 11, equation (4), taking into account (6): , R t )()('),0,...,0( ,'),('),0'( ,'),())(( ),'[' tttttttttt tttt tt indeed. Progressiveness
Definition 17
The discrete time computation function ' ˆ n is called progressive if infiniteiskksettheni ki }1,|{},,...,1{ N . (9) The set of the discrete time progressive computation functions is denoted by n ˆ . The real time computation function ' n is called progressive if abovefromunboundedis tttsettheni i }1)(,|{},,...,1{ R (10) is true. The set of the real time progressive computation functions is denoted by n . ournal of Progressive Research in Mathematics(JPRM) ISSN: 2395-0218
Theorem 18 a) Let the computation function ' ˆ n . The following equivalence holds: nn ˆ)(ˆˆ . b) The computation function ' n and R ' t are given. The following equivalence holds: ntn )( ' . Proof. a) For any },...,1{ ni , the sets }1,0|{ ki kk , }1,1|{ ki kk are simultaneously finite or infinite. b) We suppose that is of the form ...)()(...)()()()()( }{}1{1}0{0 ttttttt ktktt (11) with Seqt k )( . We denote with k the rank of the sequence )( k t that is defined by ...)()()()())(( }1'{1'}'{'' ttttt ktkktkt For any },...,1{ ni , the sets }1)(,'|{},1)(,0|{ kikkik tkkttkt are simultaneously bounded or unbounded from above. Remark 19
From Theorem 18 a) we get the following conclusion. For ' ˆ n , we have the equivalence nkn k ˆ)(ˆ,ˆ N . Flows
Definition 20
For the function nn BB : and n B , we define nn BB : by n B , ))(),...,(()( nnnn . Definition 21
Let ,..., , k , nk B k . We define the functions nnkk BB : iteratively by n B , ))(()( ...011...0 kkkk . Definition 22 a) The function nnn kk BNB ),(ˆ),,(ˆ defined by , N k ...0 kifkifk k is called (discrete time) evolution function , or ( state ) transition function , or next state function . n B is called state space (or phase space ), is called the initial ( value of the ) state and is the computation function . The value ),(ˆ)(ˆ kkx a is the state )(ˆ kx resulted at the time instant k from the initial (value of the) state under the (action of the) computation function . b) We define the function nnn tt BRB ),(),,( in the following way. Let , R t ...)(...)()()( }{}1{1}0{0 tttt ktktt (12) where n ˆ and Seqt k )( . Then ...)(),(ˆ...)()0,(ˆ)()1,(ˆ),( )1,[)1,0[)0,( tkttt ktktttt ournal of Progressive Research in Mathematics(JPRM) ISSN: 2395-0218 evolution function , or ( state ) transition function , or next state function . n B is the state space , is the initial ( value of the ) state and is the computation function . The value ),()( ttx is the state resulted at the time instant t from the initial (value of the) state under the (action of the) computation function . Definition 23 a) We fix n B and n ˆ in the argument of the discrete time evolution function. The signal )( ˆ),(ˆ n S is called (discrete time) flow ( through , under ) and, more general, if previously ' ˆ n , then ),(ˆ is called semi-flow . b) We fix n B and n in the argument of the real time evolution function. The signal )( ),( n S is called (real time) flow ( through , under ). More general, if previously ' n , then ),( is called semi-flow . Remark 24
The function applied to the argument is computed on all its coordinates: ))(),...,(()( n . The function applied to computes those coordinates i of for which i and it does not compute those coordinates i for which i : },...,1{ ni , .0, ,1),()( ii iii if if Unlike the usual computations from the dynamical systems theory that happen synchronously on all the coordinates: ),( ),)((),)(( … here things happen on some coordinates only, as shown in Definitions 20, 21, 22. The asynchronous flows represent a generalization of the computations from the dynamical systems theory, since the constant sequence NB k nk ,)1,...,1( belongs to n ˆ and it gives for any n B , that ),()( ),)(()( ),)(()( … Remark 25
We give the meaning of progressiveness: n ˆ , n show that ),(),,(ˆ compute each coordinate ni i ,1, infinitely many times as k . In electrical engineering, this corresponds to the so called unbounded delay model of computation of the Boolean functions , stating basically that each coordinate i of is computed independently on the other coordinates, in finite time. Remark 26
In the following we shall always suppose that the progressiveness requirement on , is fulfilled, thus we shall work with flows. Consistency, composition and causality
Remark 27
The properties stated in Theorems 28, 29, 30 and 31 to follow are the adaptation to the present context of the properties of consistency, composition and causality of the transition function that are contained in the definition of a dynamical system from [9], page 11. At the same page, the authors show that the words ‘dynamical’, ‘non-anticipatory’ and ‘causal’ have approximately the same meaning, making us conclude that the property of causality to be introduced may be also called non-anticipation. We must add here the remark that in the cited work the systems had an input, unlike here where it is convenient to omit this aspect, and consequently there causality referred to the input, unlike here where it refers to the computation function. The input controls the state and the computation function shows when and how the state is computed. We suppose in this section that a function nn BB : is given, together with nn ˆ, B and n . The relation between and is given by (12), where Seqt k )( . Theorem 28 (Consistency) )1,(ˆ , (13) )0,( t . (14) ournal of Progressive Research in Mathematics(JPRM) ISSN: 2395-0218
Proof. a) This follows from Definition 22. b) Definition 22 shows that we have ),(, ttt , wherefrom (14) follows. Theorem 29 (Composition) a) ,,' NN kk )),1',(ˆ(ˆ)))(,(ˆ(ˆ )('ˆ' kkk kk . (15) b) ,,' RR tt )),0',(()))(,(( )('' ttt tt . (16) Proof.
Let us notice first of all that ntnknn )(,ˆ)(ˆ,ˆ '' result from Theorem 18 and Remark 19, wherefrom the right members of equations (15), (16) make sense. a) We have the following possibilities. Case N kk ,0' arbitrary, when )),1,(ˆ(ˆ),(ˆ),(ˆ)))(,(ˆ(ˆ )(0ˆ)(0ˆ0 kkkk . Case kk )1),1',(ˆ(ˆ)1',(ˆ)1))(,(ˆ(ˆ )('ˆ' kk kk . Case N kk ,1' arbitrary, for which )()',(ˆ)))(,(ˆ(ˆ '...1''...0' kkkkk kkk )),((ˆ))(( k kkkkkkkkk )),1',(ˆ(ˆ)),((ˆ )('ˆ1'...0)('ˆ kkk kkk . b) Equation (12) shows that we can put ),( under the form ...)()(...)()()(),( )1,[...0)1,0[0)0,( tttt ktktkttt (17) We take an arbitrary R ' t and we have the following possibilities. Case ' tt In this situation ),())(( ' tt t ,)0',( t thus )),0',((),()))(,(( )('' tttt tt . Case ],(', kk tttk N In this case we infer ...)()())(( }2{2}1{1' ttt ktkktkt ),()0',( ...0 k t ...)()()()()))(,(( )2,1[1...0)1,(...0' ttt ktktkkktkt ...)())(()()( )2,1[...01)1,(...0 tt ktktkkktk )),(( ...0...}2{2}1{1 t kktkktk ournal of Progressive Research in Mathematics(JPRM) ISSN: 2395-0218 )),0',(()),(( )('...0)(' ttt tkt . Theorem 30 (Composition) a) For arbitrary N ' k we can write: ,' kk )1'),',(ˆ(ˆ),(ˆ )(1'ˆ kkkk k . (18) b) R ' t we have: ,' tt ).),',((),( ),'( ttt t (19) Proof. a) We make the substitution pkk ' , where N p and we prove (18) by induction on p . For p , (18) becomes ),1),',(ˆ(ˆ)',(ˆ )(1'ˆ kk k obvious. We suppose that )1),',(ˆ(ˆ)',(ˆ )(1'ˆ pkpk k (20) is true and we infer that ))',(ˆ()1',(ˆ pkpk pk ))1),',(ˆ(ˆ( ,...',...,2',1'1')20( pk pkkkpk )))',(ˆ(( '...2'1'1' k pkkkpk ))',(ˆ( k pkpkkk )),',(ˆ(ˆ)),',(ˆ(ˆ )(1'ˆ,...1',',...,2',1' pkpk kpkpkkk . b) Indeed, we shall suppose in the following that (12) is true. In the case ' tt , we have ),()()( ),'( ttt t )',( t and (19) is true under the form ).,(),(,' tttt In the case ,),,[' N kttt kk ...)()()()( }2{2}1{1),'( tttt ktkktkt ),()',( ...0 k t ),( t is given by (17) and )),(()),',(( ...0...}2{2}1{1),'( ttt kktkktkt ...)()()()( )2,1[1...0)1,(...0 tt ktktkktk For ' tt , (19) is true. Theorem 31 (Causality) For any N k and any n ˆ, with '' },,...,1{' kk kk , (21) we have ),(ˆ),(ˆ kk . (22) b) Let R ' t and n ', with the property that )(')(,' tttt . (23) Then ournal of Progressive Research in Mathematics(JPRM) ISSN: 2395-0218 )',()',( ' tt . (24) Proof. a) We infer ),(ˆ)()(),(ˆ ...0...0 kk kk . (25) b) If R ' t is such that )0,...,0()(')(,' tttt , (26) then )',()',( ' tt . Let N k be arbitrary and fixed. We suppose that n ˆ, and Seqtt jj )(),( ' exist such that ,},,...,0{' ' '' kk ttkk ...)()()(...)()( }2{2}1{1}{}0{0 ttttt ktkktkktkt ...)()()(...)()(' }' 2{2}' 1{1}{}0{0 ttttt ktkktkktkt )0,...,0( and ),[),[' ' 11 kkkk ttttt hold. We get )',()()',( '...0 tt k . (27) Conclusion
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