finite automata over continuous time -...

Universidade Tecnica de Lisboa

Instituto Superior Tecnico

Finite automata overcontinuous time

Alexandre Paulo Lourenco Francisco

Applied Mathematics and Computation

Diploma Thesis

Supervisor:

Prof. Jose Felix Costa

June 2002

Acknowledgements

I would like to thank to my supervisor, Professor Jose Felix Costa, for hisguidance during the elaboration of this dissertation. His support and motivationhave contributed decisively to my future as a scientist and mathematician.

I am indebted to Dr. Alexander Rabinovich and Professor Boris Trakhten-brot for their suggestions concerning this dissertation. Specially, I would like tothank to Professor Boris Trakhtenbrot for helping me with some bibliographyand Dr. Alexander Rabinovich for his critical apreciation of chapter 7.

I would like to thank Professors Amilcar Sernadas and Cristina Sernadasfor their help and suggestions. Specially, I want to thank Professor AmilcarSernadas, as President of the Center for Logic and Computation, for invitingme to give a seminar about the topic of this dissertation.

Thanks also to all people in the Computer Science Section of the Departmentof Mathematics of IST for their support, specially to Dr. Miguel Dionısio andDr. Paula Gouveia for their patience along the preparation of this dissertation.

I wish to thank to my colleagues Ricardo Silva, Pedro Adao, and Luıs Russofor their friendship.

Thanks also to Joao Rasga, Hugo Lourenco, and Dr. Isabel Nunes for pro-viding me some references I needed.

I am also indebted to my family for their support. I would like to thankCatia Vaz for her patience and love during the preparation of this dissertationand for helping me with some details in the text.

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Contents

List of symbols 3

1 Introduction 4

2 Postulates 72.1 Postulates of Automata Theory . . . . . . . . . . . . . . . . . . . 7

2.1.1 Nature of Time . . . . . . . . . . . . . . . . . . . . . . . . 72.1.2 Finiteness Postulates . . . . . . . . . . . . . . . . . . . . . 82.1.3 Input-Output Behavior . . . . . . . . . . . . . . . . . . . 8

2.2 Non-Zeno Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3 Speed Independence and Stability 153.1 Speed Independent Operators over

Right Open Signals . . . . . . . . . . . . . . . . . . . . . . . . . . 183.2 Speed Independent Operators over

Non-Zeno Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4 Examples of Operators 28

5 Properties of Operators 435.1 Closure Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 435.2 Properties of Finite Memory Retrospective Operators . . . . . . 46

5.2.1 Finite Memory Signals . . . . . . . . . . . . . . . . . . . . 465.2.2 State Function . . . . . . . . . . . . . . . . . . . . . . . . 48

6 Continuous-time Automata 546.1 Finite State Transducers over

Non-Zeno Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . 546.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

7 Circuits of operators 647.1 Primitives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 647.2 Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 667.3 Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 717.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

8 Final Remarks 76

References 79

1

Index 81

A Notes about Partial Order Theory 83

2

List of symbols

T+ Time set[a,b[ Time intervalsΣ Set of states of a channelSig(Σ) Set of signalssuf(x, t) Suffix of a signal xxct; z Concatenation of x and zRes(F, ~x, t) Residual of FnZSig(Σ) Set of non-Zeno signals over ΣConsta Constant signal equal to aρ Order preserving bijectionx, y, x1, x2, ... Signalsα, β, α1, α2, ... ω-strings⊥ Undefinedout~G

Output functionstate ~G

State functionQ Set of states of a transducer[X ; OP ] Function algebraOFMR Set of finite memory retrospective operatorsLLima Left limit operatorLJV n

k,a Last jump value operatorRec(G)~x Feedback operator∨ Least upper bound∧ Greatest lower bound

3

Chapter 1

Introduction

Nowadays control systems are commonly found in many devices such as indus-trial robots and airplanes. The study of these devices and their interaction withtheir environment leads us to the mathematical theory of control, which dealswith the analysis and design of control systems.

As we can see in [Son90], there have been two main lines of work in controltheory. One take us over the models in mathematics and physics and over theirpossible optimization with respect to a particular behavior. The other mainline is based on the constraints of a specific object and the goal is to correct thedeviations of such object from a desired behavior.

So and specially in what concerns the second main line, we need to studysystems which involve interacting networks of digital and continuous systems,i.e., hybrid systems. These incorporate both discrete and continuous dynamicsin which the continuous aspects may require incursions into calculus and differ-ential equations. As we know, differential equations have nothing in commonwith existing and well understood tools of automata theory and logic.

The approximation between automata theory and continuous systems lead usto extensions of the basic finite automata paradigm. A first extension arise fromthe idea of interaction with environment seen as an oracle. A second one focuseson the use of continuous time instead of discrete time and ignores interactionwith the oracle. As is put in [RT98], it is believed that these two orthogonalextensions may facilitate a structured formalization of hybrid systems and alucid adaptation of basic automata theory to hybrid systems.

In this work we will be around the second extension and we will follow ideasdiscussed by B. A. Trakhtenbrot and A. Rabinovich, see e.g. [Rab97, RT98,Tra98, Tra99]. Their work include many definitions and formalizations withrespect to automata over continuous time, namely with respect to lift conceptsof the classical automata theory from discrete to continuous time; many of theresults achieved by them will be discussed in this dissertation. We note also thatthis work is specially related to speed independent properties which rely on theorder of real numbers, metric aspects which deal with the distance between realnumbers are not considered because, as we will see, operators which rely onmetric of reals have uncountable memory.

R. Alur and D. L. Dill (see, e.g., [AD94]) have proposed timed automata tomodel the behavior of real-time systems and in their approach metric propertiesof the reals are taken into account. In fact the work of R. Alur and D. L.

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Dill deals with timed automata, finite automata which have a finite numberof clocks associated, that accept words in which real-valued time of occurrenceis associated with each symbol. I.e., a timed automaton accepts words as aclassical automaton with a major difference: the time between the occurrenceof two symbols is not necessarily constant and may be a real number. Note that,in the approach took by B. A. Trakhtenbrot and A. Rabinovich, signals overreal-time are considered in place of words and automata compute operators onsignals in place of accept words.

This dissertation is organized as follows. In chapter 2 we will discuss somedefinitions and concepts of classic automata theory and we will define in anaxiomatic manner the behavior of finite state automata operating in contin-uous time. As is done in [Rab97], many postulates of automata theory willbe analyzed and concepts of retrospective operators and finite memory oper-ators will be studied with some improvements. Namely and with respect to[Rab97, RT98, PRT01], it will be formalized in detail the concept of time set,the extension to n-ary operators will be considered and proofs of propositions2.2.5 and 2.2.7 will be given too.

Stability and speed independence will be object of study in chapter 3. Withthe lift to continuous time comes up some properties of signals invisible at dis-crete time, for example majority of signals will be sensible to expansion andcompression of time space. Concepts such as stable operator and speed inde-pendent operator found, e.g., in [Rab97, PRT01] will be given and some gen-eralizations will be done with respect to n-ary operators. We also prove someresults about these properties, namely the characterization of speed indepen-dent operators over non-Zeno signals and over right open signals indicated in[Rab97] will be generalized and complete proofs of propositions 3.1.3 and 3.2.3will be provided.

Chapter 4 gathers many examples found in [Rab97, RT98, PRT01] whichones we have studied here in detail. They will illustrate many of concepts andproperties introduced in chapters 2 and 3.

In chapter 5 we will study closure properties of operators on signals andsome properties of finite memory retrospective operators. These have beenintroduced in [Rab97] and we will contribute with some improvements, namelythe generalization to n-ary operators and complete proofs of propositions 5.1.1,5.1.2 and 5.2.8 will be given.

The representation of finite memory operators found in [Rab97] is discussedin chapter 6 where we generalize the notion of finite state transducer to n-aryoperators, i.e., finite state transducers with multiple input channels are given.We also give illustrative examples of automata to finite memory retrospectiveoperators already introduced in chapter 4, these examples will clarify some ideasabout the relation between the states of a transducer and the residuals of anoperator.

Our main contribution to this theory is given in chapter 7. A physical device,in which complex transformations are implemented, is usually an appropriatecombination of elementary parts that interact as desired. These idea conductsus to the concept of circuit which appears many times in literature (see, e.g.,[KT65]) and which permits, for example, describe a given automaton as thecombination of elementary automata.

We introduce circuits of finite memory retrospective operators over signals,i.e., we choose a set of elementary finite memory operators and we study how to

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obtain all finite memory retrospective operators by constructing circuits withthe elementary operators. In order to perform this study, we use the notion offunction algebra in [Clo99] and we obtain an algebra of finite memory retrospec-tive operators. The equivalence between this algebra of operators and the set offinite memory retrospective operators is stated in propositions 7.3.1 and 7.2.15.We note that the proof of proposition 7.3.1 is constructive and so, making useof the elementary operators and operations provided by this algebra, we canconstruct circuits of operators.

Chapter 7 includes also examples of circuits for the finite memory retrospec-tive operators given in chapter 4.

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Chapter 2

Postulates

In this chapter we pretend to lift the basic concepts of classical automata theoryfrom discrete to continuous time. For that purpose we will follow the main ideasfound in [Rab97, PRT01, RT98].

Automata Theory is commonly introduced as a study of sets of strings ac-cepted by finite machines, see e.g. [HMU01]. However as is stated in [Tra99],the operators realized by these machines are more basic than the sets acceptedby them. This is in accordance with the belief that in Automata Theory as wellas in Computability Theory operators are more fundamental than sets and thispoint of view was followed consistently in [KT65].

Therefore our interest is the study of Automata Theory considering operatorsrealized by some finite machines.

Before going on we observe that a machine is considered as a closed box withinput and output channels. The user interacts with the machine through theinput channels and gets the answers from the output channels.

With respect to [Rab97, PRT01, RT98] we provide the following extensionsand more deep explanations: definition of time set, extension to n-ary operatorsand proofs of propositions 2.2.5 and 2.2.7.

2.1 Postulates of Automata Theory

The aim of this section is to define in an axiomatic way the behavior of finitestate devices operating in continuous time. For decades, most of the followingconcepts have been employed for the behavior of finite state devices operating indiscrete time. However, concerning continuous time, formalisms and notationsare many times obscured by imprecise definitions.

Taking advantage of ideas presented in [Rab97], we will examine some pos-tulates commonly followed in automata theory.

2.1.1 Nature of Time

Looking at discrete time systems, the underlying time set is discrete and weusually think that it is the set of integers Z. When we consider a continuoustime system, each instant is a real number and therefore we think in the time

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set as the set of real numbers R. For both cases there are common assumptionsand we consider the following postulates.

Postulate 2.1.1 (Linear Time) The set of moments of time is a linearly or-dered set.

Postulate 2.1.2 (Discrete Time) The set of moments of time is the set ofnatural numbers where 0 is the beginning of time.

For current work we will consider the next postulate.

Postulate 2.1.3 (Continuous Time) The set of moments of time is repre-sented by the non-negative real numbers where 0 is the beginning of time.

To treat both continuous and discrete time cases simultaneously, we use thenotation followed for example in [Son90]. So we define time set.

Definition 2.1.4 A time set T is a subgroup of (R, +) with addition and theusual order relation <. For any such set, T + is the set of non-negative elements{t ∈ T : t ≥ 0}.

Whenever the time set T is understood from the context, the intervals arerestricted1 to T and we have:

[a, b[ = {t ∈ T : a ≤ t < b}.

2.1.2 Finiteness Postulates

For our devices we adopt the following postulates too.

Postulate 2.1.5 (Finiteness of the Channels’ number) A machine has afinite number of input and output channels.

Postulate 2.1.6 (Finiteness of Channels’ states) At a given instant of timea channel can be in one among a finite number of possible states.

If Σ is the set of states, then an element of Σ is a possible state of a Σ-channel.

2.1.3 Input-Output Behavior

Given a channel c, the input to a machine through c is the state of that channelat each moment of time. We define the sequence of states of a channel alongtime as follows.

Definition 2.1.7 A signal over a channel is a function

s : T+ −→ Σ

where T is the time set and Σ is the set of the channel’s states.

1If T = Z and k > 0, then [m, m + k[ = {m, m + 1, ...,m + k − 1}.

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Let Σ be the set of channel’s states for some channel c, we denote the set ofsignals defined through c as Sig(Σ).

The following postulates are relative to the acceptable behavior of a machine.

Postulate 2.1.8 (Deterministic Behavior) The output signals are completelydetermined by input signals.

Postulate 2.1.9 (Causal Behavior) The output at a moment t does not de-pend on future inputs.

Postulate 2.1.10 (Strong Causal Behavior) The output at a moment t onlydepends on the past inputs.

Before continuing we observe that we did some generalizations with respectto [Rab97]. In [Rab97] only unary operators have been considered, but we knowthat a machine may have many input channels and the output through a givenchannel may depend on multiple input channels. So we will consider n-aryoperators from signals to signals.

The following definition formalizes the concepts found in the last three pos-tulates.

Definition 2.1.11 ((Strong) Retrospective Operator) Let F be an oper-ator from signals to signals,

F : Sig(Σ)n −→ Sig(Σ).

• F is retrospective with respect to the i-th argument if for any ~x, ~y ∈Sig(Σ)n such that xj = yj for i 6= j and t ∈ T+ the following conditionholds: If xi and yi coincide in the interval [0, t] then F~x and F~y coincidein the interval [0, t].

• F is strong retrospective with respect to the i-th argument if for any~x, ~y ∈ Sig(Σ)n such that xj = yj for i 6= j and t ∈ T+ the followingcondition holds: If xi and yi coincide in the interval [0, t[ then F~x and F~ycoincide in the interval [0, t].

Given a set S ⊂ {1, ..., n} of components, an operator F is (strong) retrospectivewith respect to S when it is (strong) retrospective with respect to i for any i ∈ S.An operator F is (strong) retrospective when it verifies the above conditions forall components i.

As is indicated in [Rab97], the postulates 2.1.9 and 2.1.10 imply that theinput-output behavior of a machine is a retrospective or strong retrospectiveoperator.

The last postulate is a key postulate of finite automata theory [Rab97].

Postulate 2.1.12 (Finite Memory) A machine can distinguish by its presentand future behavior between only a finite number of classes of possible signalshistories.

Through the rest of this section we will try to formalize this postulate. In firstplace we define history of a signal, prefix and suffix of a signal and concatenationof signals.

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Definition 2.1.13 A t-history over Σ is an operator from the interval [0, t] toΣ. A t-history h is a t-history of a signal x if h(τ) = x(τ) for τ ≤ t.

Definition 2.1.14 The restriction2 of x ∈ Sig(Σ) to the interval [0, t[ is calledt-prefix of x. The suffix of x at t, suf(x, t), is the signal y ∈ Sig(Σ) defined as

y(t′) = x(t + t′).

Definition 2.1.15 Let x, z ∈ Sig(Σ). The concatenation of t-prefix of x and z(notation xct; z) is defined as:

(xct; z)(τ) =

{

x(τ) if τ < tz(τ − t) if τ ≥ t

Suppose that we have two copies, M1 and M2, of the a machine M . Letx1, x2 ∈ Sig(Σ) be two signals and h1, h2 be t-histories of x1, x2, respectively.Therefore, we say that h1, h2 are indistinguishable if at time t and after bothmachines M1, M2 produce the same output. We will say that a machine, oran operator over signals, has finite memory when given an instant t and somet-history it will assume a behavior among a finite number of possibilities.

In order to formalize this idea we introduce the concept of residual. As isdiscussed in [PRT01], given an ω-operator F , an ω-operator G is the residualof F with respect to a string u of length k if y = G(x) and z = F (ux) implies∀ τ ≥ 0, y(τ) = z(τ + k). The extension of this notion to the case of signalsis straightforward, given an operator F on signals, a signal operator G is theresidual of F with respect to a t-history h of a signal x′ if y = G(x) andz = F (x′ct; x) implies ∀ τ ≥ 0, y(τ) = z(τ + t). Clearly, G is a residual of asignal operator F if there are a signal x and an instant t such that G is theresidual of F with respect to the t-history h of x, i.e., with respect to x and t.

The following definition generalizes the notion of residual to n-ary operatorsand some notation is introduced.

Definition 2.1.16 (Residual [Rab97]) Let F : Sig(Σ)n −→ Sig(Σ) be anoperator on signals, ~x ∈ Sig(Σ)n a signal and t ∈ T+ a time point. The residualof F with respect to ~x and t, Res(F, ~x, t), is the operator:

λz1...zn.λt′.F ((x1ct; z1), ..., (xnct; zn))(t + t′).

We note that if F is a retrospective operator, then Res(F, ~x, t) maps signal ~zon z′ iff F maps (x1ct; z1), ..., (xnct; zn) on yct; z′ for some y ∈ Sig(Σ).

Definition 2.1.17 (Finite Memory [Rab97]) An operator F is a finite mem-ory operator if it has finitely many distinct residuals, i.e., the set

{Res(F, ~x, t) : ~x ∈ Sig(Σ)n, t ∈ T+}

is finite.

So the postulates 2.1.8, 2.1.9, 2.1.10 and 2.1.12 are summarized as:

2As is done in [Rab97] we will use xct, x]t, xbt and x[t for the restriction of x to [0, t[,[0, t], ]t, +∞[ and [t,+∞[, respectively.

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Postulate 2.1.18 (Input-output) The input-output behavior of a machine isa finite memory retrospective operator.

As we have done in definition 2.1.17, we may define also countable memory.

Definition 2.1.19 (Countable Memory) An operator F is a countable mem-ory operator if it has countable many distinct residuals, i.e., the set

{Res(F, ~x, t) : ~x ∈ Sig(Σ)n, t ∈ T+}

is countable.

2.2 Non-Zeno Signals

As is stated in [Rab97], the piecewise constant signals are physically more real-istic than general signals. So we define next the concept of non-Zeno signal.

Definition 2.2.1 (Non-Zeno Signal) A signal x is non-Zeno or piecewiseconstant if there exists an unbounded increasing ω-sequence t0 = 0 < t1 <... < tn < ... such that x is constant in all sub-intervals ]ti, ti+1[. nZSig(Σ) willdenote the set of non-Zeno signals over the alphabet Σ.

Figure 2.1: A non-Zeno signal.

However, we can get signals physically more realistic than non-Zeno signalsif we impose the condition of non-zero duration.

Definition 2.2.2 (Non-zero Duration) A signal satisfies the requirement ofnon-zero duration if it does not have instantaneous jumps, i.e, if for every t ∈ T +,x is constant in an interval with non empty interior that contains t.

Unfortunately and as is observed in example 2.2.3, the set of signals satisfyingthe above requirement is not closed under boolean operations.

Example 2.2.3 Let x and y be the following signals:

x(t) =

{

a if 0 ≤ t < 1b otherwise

y(t) =

{

a if 0 ≤ t ≤ 1b otherwise

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When we test their equality we obtain the following non-Zeno signal:

eq(t) =

{

False if t = 1True otherwise

with an instantaneous jump at t = 1.

But we can make more restrictions and get the class of right open signals.

Definition 2.2.4 (Right Open Signal) A signal x is right open if for everyt ∈ T+ there exists t′ ∈ T+ such that t′ > t and x is constant in [t, t′[.

Figure 2.2: A right open signal.

It is easy to check that both the set of non-Zeno signals and the set of right-open signals are closed under suffix and concatenation. Constant signals areobviously included in these sets.

Proposition 2.2.5 The set of non-Zeno signals and the set of right open signalsare closed under suffix and concatenation.

Proof: Let t ∈ T+. If x is a non-Zeno signal, by definition 2.2.1, there existsan unbounded increasing ω-sequence t0 = 0 < t1 < ... < tn < ... such that x isconstant in all sub-intervals ]ti, ti+1[. Therefore, since tk ≤ t < tk+1 for some k,we know that suf(x, t) is constant in all sub-intervals ]sj , sj+1[ for s0 = t andsj = tk+j for j ≥ 1, i.e., suf(x, t) is a non-Zeno signal by definition 2.2.1.

Let y be another non-Zeno signal, by definition 2.2.1 there exists an un-bounded increasing ω-sequence s0 = 0 < s1 < ... < sn < ... such that y isconstant in all sub-intervals ]si, si+1[. So xct; y will be constant in all sub-intervals obtained from the ω-sequence t0 = 0 < t1 < ... < tk ≤ t < s1 + t <... < sn + t < ..., i.e., it is a non-Zeno signal by definition 2.2.1.

Let x be a right open signal and t ∈ T +, by definition 2.2.4 we know thatfor every τ ∈ T + there exists t′ ∈ T+ such that t′ > t + τ and x is constantin [t + τ, t′[. Therefore, for every τ ∈ T + there exists t′′ such that suf(x, t) isconstant in [τ, t′′[, i.e., such that x is constant in [t + τ, t + t′′[. Take t′′ = t′ − t.By definition 2.2.4, suf(x, t) is a right open signal.

Let y be another right open signal and t, τ ∈ T +, we want to prove thatthere exists t′ ∈ T+ such that xct; y is constant in [τ, t′[. The case τ ≥ t isclear because y is a right open signal. If τ < t and since x is a right opensignal, there exists t′′ such that x is constant in [τ, t′′[. Then, xct; y is constant

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in [τ, min(t′′, t)[ and we choose t′ = min(t′′, t), i.e., xct; y is a right open signalby definition 2.2.4. �

In order to study what happen to signals and operators on signals when westretch time, we give in the following definition the concept of order preservingbijection.

Definition 2.2.6 (Order Preserving Bijection) Let ρ : T −→ T be a bi-jective function. Then ρ is an order preserving bijection if it also verifies:

• ρ(0) = 0;

• if t < t′, then ρ(t) < ρ(t′).

Clearly, we may restrict an order preserving bijection ρ : T −→ T to T + andwe get an order preserving bijection over T +. We note also that, in the discretetime case, the identity is the unique order preserving bijection.

Let us suppose that ρ is an order preserving bijection. If C is the class ofnon-Zeno signals or the class of right open signals, then we prove that x ◦ ρ ∈ Cfor any x ∈ C.

Proposition 2.2.7 Let C be the set of non-Zeno signals or the set of right opensignals. If ρ is an order preserving bijection and x ∈ C, then x ◦ ρ ∈ C.

Proof: Let t ∈ T+. If x is a non-Zeno signal, by definition 2.2.1, there existsan unbounded increasing ω-sequence t0 = 0 < t1 < ... < tn < ... such that x isconstant in all sub-intervals ]ti, ti+1[. Therefore, since ρ is an order preservingbijection, there are t′0, ..., t

′n, ... ∈ T+ such that ρ(t′i) = ti and t′0 = 0 < t′1 <

... < t′n < ... and we also know that x◦ρ is constant in all sub-intervals ]t′i, t′i+1[,

i.e, x ◦ ρ is a non-Zeno signal by definition 2.2.1.Let x be a right open signal and t ∈ T +, by definition 2.2.4 we know that

for ρ(t) ∈ T+ there exists t′ ∈ T+ such that t′ > ρ(t) and x is constant in[ρ(t), t′[. Since ρ is an order preserving bijection, there exists t′′ ∈ T+ such thatρ(t′′) = t′ and therefore x is constant in [ρ(τ), ρ(t′′)[. Then x ◦ ρ is constant in[τ, t′′[, i.e., by definition 2.2.4, x ◦ ρ is a right open signal. �

By previous propositions3 and observations we know that the only propersubsets C of non-Zeno signals which verify the following conditions:

• C is closed under suffix, i.e., if x ∈ C, then suf(x, t) ∈ C for any t ∈ T +.

• C is closed under concatenation, i.e., if x, y ∈ C, then xct; y ∈ C for anyt ∈ T+.

• if ρ is an order preserving bijection and x ∈ C, then x ◦ ρ ∈ C.

• C contains all constant signals.

are the set of non-Zeno signals, the set of right open signals, the set of non-Zenosignals that have finitely many changes and the set of right open signals thathave finitely many changes.

3The proof is easily generalized for these cases.

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Results of this chapter

2.2.5 The set of non-Zeno signals and the set of right open signals are closedunder suffix and concatenation.

2.2.7 Let C be the set of non-Zeno signals or the set of right open signals. If ρis an order preserving bijection and x ∈ C, then x ◦ ρ ∈ C.

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Chapter 3

Speed Independence and

Stability

The aim of this chapter is to study speed independent and stable operators,namely we will give a characterization of these operators whenever they aredefined over right open signals and non-Zeno signals. Beforehand we give somenecessary definitions.

Definition 3.0.1 A signal x is constant at t ∈ T + if there are t1, t2 ∈ T+ suchthat t1 < t < t2 and x is constant in the interval ]t1, t2[.

In the discrete case, T = Z, it is clear that every signal is constant at everymoment t.

Definition 3.0.2 (Left limit) Let x be a signal, x has left limit c at t ∈ T +

if there exists t′ ∈ T+ such that t′ < t and x(τ) = c for τ ∈ ]t′, t[.

Definition 3.0.3 (Right limit) Let x be a signal, x has right limit c at t ∈ T +

if there exists t′ ∈ T+ such that t < t′ and x(τ) = c for τ ∈ ]t, t′[.

Definition 3.0.4 Let x be a signal, x is continuous from the left (right) att ∈ T+ if the left (respectively right) limit of x at t is equal to x(t). The signalx is continuous at t ∈ T + if it is continuous from the left and from the right att.

Note that in discrete time every signal is continuous for a given instant t.As we can see, the conditions in definitions 3.0.2 and 3.0.3 are always verifiedwith an arbitrary c because the sets ]t′, t[ and ]t, t′[ may be empty. Therefore,by definition 3.0.4, it follows that every signal is continuous.

Definition 3.0.5 (Stability [Rab97]) A total operator F from signals to sig-nals is stable if for every instant t > 0 and signal ~x ∈ Sig(Σ)n the followingimplication holds: xi is constant at t for all i implies F~x is constant at t.

It is clear that in the discrete case every operator is stable since every signalis constant at every instant as we observed above.

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Next proposition, stated but not proved in [Rab97], states an interestingproperty of stable operators and follows straightforward from what we haveseen in previous definition.

Proposition 3.0.6 A stable operator maps non-Zeno signals to non-Zeno sig-nals.

Proof: Let F : Sig(Σ) −→ Sig(Σ′) be a stable operator and ~x ∈ nZSig(Σ)n.By definition 2.2.1, there exists an unbounded increasing ω-sequence t0 = 0 <t1 < ... < tn < ... such that ~x is constant in all sub-intervals ]ti, ti+1[ and, sinceF is stable, F (~x) is also constant in all sub-intervals ]ti, ti+1[. Therefore, bydefinition 2.2.1, F (~x) is a non-Zeno signal. �

We also have an interesting property that relates countable memory oper-ators that map non-Zeno signals into non-Zeno signals and stability. Theseproperty was conjectured in [Rab97] for arbitrary countable memory operators,however it was not proved. (Countable memory was defined in 2.1.19.)

Proposition 3.0.7 Every countable memory operator that maps non-Zeno sig-nals to non-Zeno signals is stable.

Proof: Let F be a countable memory operator that maps non-Zeno signals intonon-Zeno signals and suppose that F is not stable. Then, by definition 3.0.5,there exists a non-Zeno signal ~x and an instant t ∈ T + such that xi is constant att, for every i ∈ {1, ..., n}, and F~x is not constant at t, i.e., for every ε > 0, thereexists τ ∈ [t−ε, t+ε] such that F~xτ 6= F~xt. Since ~x is constant at t, there existst1 < t and t2 > t such that ~x is constant in [t1, t2], i.e., ~x(t′) = (a1, ..., an), fort′ ∈ [t1, t2]. We know that F~x is a non-Zeno signal, therefore it will be constantin [t′1, t[ and ]t, t′2] for some t1 < t′1 < t and t < t′2 < t2 such that t− t′1 = t′2 − t.Also F~xτ 6= F~xt, for τ ∈ [t′1, t[, or F~xτ 6= F~xt for τ ∈ ]t, t′2].

tttt t1 2’’ ’ ’’

Figure 3.1: F (x)

Suppose that F~xτ 6= F~xt for τ ∈ [t′1, t[ (see figure 3.2) and let t′, t′′ ∈ [t′1, t[such that t′ < t′′, F1 be the residual of F with respect to ~x and t′ and F2 bethe residual of F with respect to ~x and t′′, then by definition 2.1.16 we have

F1(Consta1, ..., Constan

)(t − t′′) = F (~x)(t′ + t − t′′)

6= F (~x)(t)

= F (~x)(t′′ + t − t′′)

= F2(Consta1, ..., Constan

)(t − t′′).

16

tttt t1 2’’ ’ ’’ τ

Figure 3.2: F (x)

Suppose now that F~xτ 6= F~xt, for τ ∈ ]t, t′2], and that F~xτ ′ = F~xt, forτ ′ ∈ [t′1, t[ (see figure 3.2). Let t′, t′′ ∈ [t′1, t[ such that t′ < t′′, F1 be the residualof F with respect to ~x and t′ and F2 be the residual of F with respect to ~x andt′′. By definition 2.1.16 and for τ ∈ ]t, t′2] such that τ − t < t′′ − t′ we have

F1(Consta1, ..., Constan

)(τ − t′′) = F (~x)(t′ + τ − t′′)

= F (~x)(τ − (t′′ − t′))

6= F (~x)(τ)

= F (~x)(t′′ + τ − t′′)

= F2(Consta1, ..., Constan

)(τ − t′′).

By the above, given any t′ ∈ [t′1, t[, we know that the residual of F withrespect to ~x and t′ is different from the residual of F with respect to ~x and t′′,for all t′′ ∈ ]t′1, t[ such that t′ 6= t′′, because t′ > t′′ or t′ < t′′. Then, for eacht′ ∈ [t′1, t[ we get a different residual of F with respect to ~x and t′, obtaining anuncountable number of residuals of F , i.e., F cannot have countable memory.Hence, if F has countable memory, it is stable. �

We will now study another property of operators over signals, speed inde-pendence. We will say that a given operator has this property when it is notsensible to stretching of time.

Definition 3.0.8 (Speed Independence) An operator F from signals to sig-nals is speed independent if for every order preserving bijection ρ on time:

∀ ~x, F ((x1 ◦ ρ), ..., (xn ◦ ρ)) = F (~x) ◦ ρ.

The following proposition relates stability and speed independence.

Proposition 3.0.9 If F : Sig(Σ)n −→ Sig(Σ) is speed independent then F isstable.

Proof 1: Assume that ~x ∈ Sig(Σ) is constant at t ∈ T +. Then there existsτ1, τ2 ∈ T+ such that ~x is constant in ]τ1, τ2[ and τ1 < t < τ2. Let t1 bean arbitrary point in ]τ1, τ2[. Clearly there exists an order preserving bijectionρ1 : ]τ1, τ2[ −→ ]τ1, τ2[ such that ρ1(t) = t1. Let ρ be the bijection on T + definedas

1This proof is a slate corrected version of the proof of proposition 2 in [Rab97].

17

ρ(τ) =

{

ρ1(τ) if τ ∈ ]τ1, τ2[τ otherwise

It is clear that ρ is an order preserving bijection and that xi ◦ ρ = xi fori = 1, ..., n. Therefore,

F~x)(t1) = F (~x)(ρ ◦ ρ−1(t1))

= F ((x1 ◦ ρ), ..., (xn ◦ ρ))(ρ−1(t1))

= F (~x)(t).

Therefore, F~x is constant in ]τ1, τ2[ and F is stable. �

3.1 Speed Independent Operators over

Right Open Signals

Let x be a right open signal, by definition 2.2.4 we can see that x is definedunivocally by an ω-sequence α = < ai : i ∈ N > over Σ and an unboundedincreasing ω-sequence τ = < ti : i ∈ N > such that ti ∈ T+, t0 = 0 and

∀ i, ∀ t ∈ [ti, ti+1[, x(t) = ai.

Suppose now that ~x ∈ Sig(Σ)n is a right open signal, i.e., xi is a right opensignal, for i = 1, ..., n. By the above, each xi is characterized by ω-sequencesαi and τi. Let τ be an unbounded increasing ω-sequence such that τi ⊆ τ andlet α′

i be the ω-sequence obtained from αi by inserting the value of xi in αi ateach new instant of τ . It is clear that α′

i, τ characterize xi and we note that α′i

may have repetitions. Thus, we can obtain an unbounded increasing ω-sequenceτ = < ti : i ∈ N > such that τj ⊆ τ for j = 1, ..., n, ti ∈ T+, t0 = 0 and

∀ j, ∀ i, ∀ t ∈ [ti, ti+1[, xj(t) = aji ,

where α = < (a1i , ..., a

ni ) : i ∈ N > is an ω-sequence over Σn such that < aj

i :i ∈ N > and τ characterize xj , for j = 1, ..., n.

When the above conditions hold we say that the pair α, τ characterizes ~x or~x is characterized by α, τ .

Notation 3.1.1 An unbounded increasing sequence t0 < t1 < ... such thatti ∈ T+ and t0 = 0 is called time scale. Throughout this section τ , τ ′ will denotetime scales and α, β will denote ω-sequences over an alphabet Σn. (Σn)ω willdenote the set of all ω-sequences over the alphabet Σn.

Given an order preserving bijection ρ : T + → T+ and a time scale τ , westate that if τ ′

i = ρ(τi), for all i ∈ N, then τ ′ is a time scale. Moreover, α, τcharacterizes ~x if and only if α, τ ′ characterizes ~x◦ρ = (x1◦ρ, ..., xn◦ρ). Clearly,for every time scales τ and τ ′, there exists an order preserving bijection ρ suchthat α, τ characterizes x if and only if α, τ ′ characterizes ~x ◦ ρ.

It is also clear that, if ~x is characterized by α, τ and ~x is not constant att, then t appears in τ and if τ contains all points at which x is not constantthen there exists α such that α, τ characterizes ~x. So, if F is a stable operatorfrom right open signals to right open signals and α, τ characterizes ~x, then thereexists β such that β, τ characterizes F~x.

18

Assume that F is a speed independent operator from right open signals toright open signals, by proposition 3.0.9 we know that F is stable. Suppose thatα, τ characterizes ~x and let β be such that β, τ characterizes y = F (~x), whichexists as we saw in the previous paragraph. Since F is speed independent, itfollows that for any τ ′ and for the ~x′ characterized by α, τ ′ the signal F~x′ ischaracterized by β, τ ′.

Therefore, we can associate with every speed independent operator F anoperator G from ω-strings over Σn to ω-strings over Σ such that, for all α andτ , if α, τ characterizes ~x, then G(α), τ characterizes F~x. In this case G is knownas a discrete characterization of F .

However not every G on ω-strings characterizes an operator on right opensignals. Indeed, if G characterizes an operator, then, whenever α, τ and α′, τ ′

characterize the same signal, Gα, τ and Gα′, τ ′ should also characterize thesame signal. Many distinct α, τ may characterize the same signal. For example,assume that α = < a0, ..., ai, ai+1, ... > and τ = < t0, ..., ti, ti+1, ... >. Lett ∈ ]ti, ti+1[ and let α′ and τ ′ be defined as < a0, ..., ai, ai, ai+1, ... > and <t0, ..., ti, t, ti+1, ... > respectively. Then α, τ characterize ~x if and only if α′, τ ′

characterize ~x. Therefore, if G characterizes an operator on right open signalsit should satisfy the SI 2 condition.

Definition 3.1.2 (SI Condition [Rab97]) An operator G on ω-strings sat-isfies the SI condition if for any < a0, ...ai, ai+1, ... > and < b0, ...bi, bi+1, ... >with ai ∈ Σn and bi ∈ Σ:

G(< a0, ..., ai, ai+1, ... >) = < b0, ..., bi, bi+1, ... >if and only if

G(< a0, ..., ai, ai, ai+1, ... >) = < b0, ..., bi, bi, bi+1, ... >.

The following proposition resume some relevant observations and the givenproof is a completion of the proof found in [Rab97].

Proposition 3.1.3 .

1. Every speed independent operator F that maps right open signals into rightopen signals is characterized by a function G on ω-strings that satisfies theSI condition.

2. Every function G on ω-strings that satisfies the SI condition characterizesa speed independent operator F that maps right open signals into rightopen signals.

3. If G, a function on ω-strings, characterizes F , a speed independent oper-ator that maps right open signals into right open signals, then

(a) G is retrospective if and only if F is retrospective.

(b) G and F have the same number of distinct residuals.

(c) G has finite memory if and only if F has finite memory.

Proof: 1. Let F be a speed independent operator that maps right open signalsinto right open signals. We define G : (Σn)ω −→ Σω as follows:

2Speed Independence

19

G < (a1i , ..., a

ni ) : i ∈ N > = < bi : i ∈ N >,

where < bi : i ∈ N >, τ characterizes F~x, for some time scale τ and for ~xcharacterized by < (a1

i , ..., ani ) : i ∈ N >, τ . G is clearly well defined, i.e., given

an ω-sequence < (a1i , ..., a

ni ) : i ∈ N >, τ , the ω-sequence < bi : i ∈ N > is

univocally determined because < (a1i , ..., a

ni ) : i ∈ N >, τ is a right open signal

and, since F is speed independent, < bi : i ∈ N > will not depend on τ (we maychoose any time scale). As we have seen in this section, it is clear that G is acharacterization of F by definition of G.

Let < a0, ..., ai, ai+1, ... > ∈ (Σn)ω and < b0, ..., bi, bi+1, ... > ∈ Σω. If

< a0, ..., ai, ai+1, ... >, < t0, ..., ti, ti+1, ... >

characterizes a right open signal x, being τ = < t0, ..., ti, ti+1, ... > a time scale,then

< a0, ..., ai, ai, ai+1, ... >, < t0, ..., ti, t, ti+1, ... >,

with ti < t < ti+1, will characterize also x. As F is an operator, F~x is univocallydetermined and we have that

G < a0, ..., ai, ai+1, ... > and G < a0, ..., ai, ai, ai+1, ... >

will both characterize the right open signal F~x. Therefore, if

G < a0, ..., ai, ai+1, ... > = < b0, ..., bi, bi+1, ... >,

G < a0, ..., ai, ai, ai+1, ... > must be < b0, ..., bi, bi, bi+1, ... > otherwise the se-quences will not characterize the same signal for the chosen time scales. Theconverse implication in definition 3.1.2 follows similarly. Thus, we conclude thatG verifies the SI condition.

2. Let G be a function from ω-strings over Σn to ω-strings over Σ whichsatisfies the SI condition. We will consider only ω-strings < ai : i ∈ N > ∈ (Σn)ω

such that

∀ i, ai 6= ai+1

or (3.1)

∃ k, ∀ i < k, ai 6= ai+1 and ∃ a ∈ Σn, ∀ i ≥ k, ai = a.

As G verifies the SI condition, its values on ω-strings that verify the condition3.1 determine completely its values on any ω-string. Let ~x be a right open signaland ξ the set of instants in which ~x is not continuous, i.e., in which xi is notcontinuous for some 1 ≤ i ≤ n. We have two possible cases:

1. ξ = < ti : i ∈ N >, i.e., ξ is a time scale;

2. ξ = < t1, ..., tk >.

In first case we may consider the ω-string α = < (x1(ti), ..., xn(ti)) : i ∈ N >which clearly characterizes ~x with the time scale τ = ξ and verifies the condition3.1. In second case we extend ξ to the time scale τ = < t0, ..., tk, tk+1, tk+2, ... >and we consider the ω-string α = < (x1(t), ..., xn(t)) : t ∈ τ >. It is clear thatin this case α also verifies the condition 3.1 and that it also characterizes x withτ as time scale. Let y be the unique right open signal characterized by Gα, τand we define F as follows:

20

F~x = y

As we have seen, if ρ is an order preserving bijection, then < ρ(t) : t ∈ τ > isalso a time scale and by definition of F it follows that it is speed independent andwe will show now that G characterizes F . Assuming that α′, τ ′ characterize ~x,it does not satisfy 3.1 necessarily but we clearly may obtain α, τ from this whichcharacterizes also ~x and verifies 3.1. Gα, τ will characterize F~x by definition of Fand, as G verifies the SI condition, Gα′, τ ′ will also characterize F~x. Therefore,G characterizes F .

3. Suppose that G characterizes F . Let α, α′ ∈ (Σn)ω such that α(i) = α′(i),for i ≤ k and τ, τ ′ be two time scales such that τ(i) = τ ′(i), for i ≤ k + 1,therefore ~x characterized by α, τ and ~x′ characterized by α′, τ ′ are right opensignals and they will verify xi(t) = x′

i(t), for 0 ≤ t ≤ τ(k + 1) and for all i. If Fis retrospective by definition 2.1.11,

F (~x)(t) = F (~x′)(t), for 0 ≤ t ≤ τ(k + 1),

and, since G is a characterization of F , we have

(Gα)(i) = (Gα′)(i), for i ≤ k,

i.e., G is retrospective.Conversely, let ~x and ~x′ be two right open signals such that xi(t

′) = x′i(t

′),for t′ ≤ t with t ∈ T+ and for all i. If G is retrospective and these signals arecharacterized by α, τ and α′, τ ′, then α(i) = α′(i), for i ≤ k, and τ(i) = τ ′(i),for i < k, with k such that tk−1 is the last instant before t in which ~x and ~x′

change and we have

(Gα)(i) = (Gα′)(i), for i ≤ k.

Since G is a characterization of F , Gα, τ characterizes F~x and Gα′, τ ′ charac-terizes F~x′ and by the previous observations we get

F (~x)(t′) = F (~x′)(t′), for t′ ≤ t,

i.e., F is retrospective by definition 2.1.11. Therefore, F is retrospective if andonly if G is retrospective.

As we saw in 1 and 2, exists an univocally correspondence between speed in-dependent operators on right open signals and functions on ω-strings that verifythe SI condition. Therefore, since the residuals of F still are speed independentand the residuals of G still verify the SI condition, G and F will have the samenumber of distinct residuals. Clearly, by definition 2.1.17, G has finite memoryif and only if F has finite memory. �

Since every retrospective function on ω-strings has at most countable mem-ory and, as we saw in last proposition, all speed independent retrospectiveoperators on right open signals are characterized by retrospective functions onω-strings, we obtain the following corollary.

Corollary 3.1.4 Every speed independent retrospective operator on right opensignals has at most countable memory.

21

3.2 Speed Independent Operators over

Non-Zeno Signals

Now we provide a similar description for speed independent operators over non-Zeno signals.

A non-Zeno signal x over an alphabet Σ is said to be characterized by α, α′, τif α = < ai : i ∈ N > and α′ = < a′

i : i ∈ N > are ω-strings over Σn,τ = < ti : i ∈ N > is a time scale and x(ti) = ai and x(t) = a′

i for everyi ∈ N and every t ∈ ]ti, ti+1[. We observe that for every non-Zeno signal x thereexists a triple α, α′, τ that characterizes x and that every α, α′, τ characterizesa non-Zeno signal by definition 2.2.1.

Suppose now that ~x ∈ Sig(Σ)n is a non-Zeno signal, i.e., xi is a non-Zenosignal, for i = 1, ..., n. By the above, each xi is characterized by ω-sequencesαi, α′

i and τi. Let τ be an unbounded increasing ω-sequence such that τi ⊆ τand let βi and β′

i be the ω-sequences obtained from αi and α′i by inserting the

necessary values of xi in αi and α′i for each new instant of τ . It is clear that

β, β′i, τ characterize xi and we note that βi and β′

i may have repetitions. Thus,we can obtain an unbounded increasing ω-sequence τ = < ti : i ∈ N > suchthat τj ⊆ τ for j = 1, ..., n, ti ∈ T+, t0 = 0 and

∀ j, ∀i, xj(ti) = aji and ∀ t ∈ ]ti, ti+1[, xj(t) = a′j

i ,

where β = < (a1i , ..., a

ni ) : i ∈ N > and β′ = < (a′1

i , ..., a′ni ) : i ∈ N > are

ω-sequences over Σn such that < aji : i ∈ N >, < a′j

i : i ∈ N >, τ characterizexj , for j = 1, ..., n.

When the above conditions hold we say that the pair α, τ characterizes ~x or~x is characterized by α, τ .

An operator F from non-Zeno signals over Σ1 to non-Zeno signals over Σ2

is said to be characterized by a function G : (Σn1 )ω × (Σn

1 )ω −→ Σω2 × Σω

2 ifwhenever α, α′, τ characterizes ~x then G(α, α′), τ characterize F~x.

As we will see, every speed independent operator is characterized by a func-tion on ω-strings but not every function G : (Σn

1 )ω×(Σn1 )ω −→ Σω

2 ×Σω2 charac-

terizes a speed independent operator. Let α = < a0, ..., ai, ai+1, ... > and α′ = <a′0, ..., a

′i, a

′i+1, ... > with ai, a

′i ∈ Σn

1 be ω-strings and τ = < t0, ...ti, ti+1, ... > bea time scale. Assume that β, β′ are obtained from α, α′ by inserting a′

i after thei position, i.e., β = < a0, ..., ai, a

′i, ai+1, ... > and β′ = < a′

0, ..., a′i, a

′i, a

′i+1, ... >,

and that τ ′ is obtained from τ by inserting any t ∈ ]τi, τi+1[ after the i position,i.e., τ ′ = < t0, ..., ti, t, ti+1, ... >. Then α, α′, τ characterize ~x if and only ifβ, β′, τ ′ characterize ~x.

Hence, if G characterizes a speed independent operator it should satisfy thefollowing condition.

Definition 3.2.1 (Generalized SI Condition [Rab97]) Let α and β be ω-strings and let i ∈ N. Let β and β′ be obtained from α and α′ by inserting a′

i

after i as above. Similarly, let ζ and ζ ′ be obtained from ξ and ξ′ by insertingx′

i after i. Then:

G(α, α′) = (ξ, ξ′)if and only if

G(β, β′) = (ζ, ζ ′).

22

Assume that α, α′, τ characterizes ~x1 and that β, β′, τ characterizes ~x2. Then~x1 is equal to ~x2 in [0, t] if either t ∈]ti, ti+1[ and aj = bj and a′

j = b′j forj ∈ 0, ..., i or t = ti and aj = bj for j ∈ {0, ..., i} and a′

j = b′j for j ∈ {0, ..., i−1}.Hence, if G characterizes a retrospective operator F , then G should satisfy thefollowing condition.

Definition 3.2.2 (Generalized Retrospective Condition [Rab97]) Let Gbe a function on (Σn

1 )ω × (Σn1 )ω and let

α = < ai : i ∈ N >,

α′ = < a′i : i ∈ N >,

β = < bi : i ∈ N >,

β′ = < b′i : i ∈ N >,

ξ = < xi : i ∈ N >,

ξ′ = < x′i : i ∈ N >,

ζ = < zi : i ∈ N >,

ζ ′ = < z′i : i ∈ N >

be ω-strings. Then G satisfies the generalized retrospective condition if

1. G is retrospective;

2. if G(α, α′) = (ξ, ξ′), G(β, β′) = (ζ, ζ ′), aj = bj for j ∈ {0, ..., i} anda′

j = b′j for j ∈ {0, ..., i− 1}, then xj = zj for j ∈ {0, ..., i} and x′j = z′j for

j ∈ {0, ..., i− 1}.

The following proposition resume some relevant observations and it wasstated (but not proved) in [Rab97].

Proposition 3.2.3 ([Rab97]) .

1. Every speed independent operator F over non-Zeno signals is character-ized by a function G on (Σn

1 )ω × (Σn1 )ω that satisfies the generalized SI

condition.

2. Every function G on (Σn1 )ω × (Σn

1 )ω that satisfies the generalized SI condi-tion characterizes a speed independent operator F over non-Zeno signals.

3. If G, a function on (Σn1 )ω × (Σn

1 )ω, characterizes F , a speed independentoperator over non-Zeno signals, then:

(a) F is retrospective is and only if G satisfies the generalized retrospec-tive condition;

(b) F has finite memory if and only if G has finite memory;

(c) F has countable memory if and only if G has countable memory.

Proof: 1. Let F be a speed independent operator that maps non-Zeno signalsinto non-Zeno signals. We define G : (Σn

1 )ω × (Σn1 )ω −→ Σω

2 × Σω2 as follows:

G(α, α′) = (β, β′),

23

where α, α′, τ characterizes a signal ~x and β, β′, τ characterizes F~x, for sometime scale τ . G is clearly well defined, i.e, given ω-sequences α, α′ and τ , theω-sequences β and β′ are univocally determined because α, α′, τ is a non-Zenosignal and, since F is speed independent, β and β′ will not depend on τ . Aswe have seen in this section, it is clear that G is a characterization of F bydefinition of G.

Let

< a0, ...ai, ai+1, ... >, < a′0, ...a

′i, a

′i+1, ... > ∈ (Σn

1 )ω

and

< b0, ...bi, bi+1, ... >, < b′0, ...b′i, b

′i+1, ... > ∈ Σω

2 .

If < a0, ...ai, ai+1, ... >, < a′0, ...a

′i, a

′i+1, ... >, < t0, ..., ti, ti+1, ... > characterizes

a non-Zeno signal x, being τ = < t0, ..., ti, ti+1, ... > a time scale, then

< a0, ...ai, a′i, ai+1, ... >, < a′

0, ...a′i, a

′i, a

′i+1, ... >, < t0, ..., ti, t, ti + 1, ... >,

being ti < t < ti+1, will characterize also x. Since F is an operator, F~x isunivocally determined and we have that

G(< a0, ...ai, ai+1, ... >, < a′0, ...a

′i, a

′i+1, ... >)

andG(< a0, ...ai, a

′i, ai+1, ... >, < a′

0, ...a′i, a

′i, a

′i+1, ... >)

will both characterize the non-Zeno signal F~x. Therefore, if

G(< a0, ...ai, ai+1, ... >, < a′0, ...a

′i, a

′i+1, ... >)

=(< b0, ...bi, bi+1, ... >, < b′0, ...b

′i, b

′i+1, ... >),

we must have

G(< a0, ...ai, a′i, ai+1, ... >, < a′

0, ...a′i, a

′i, a

′i+1, ... >)

=< b0, ...bi, b

′i, bi+1, ... >, < b′0, ...b

′i, b

′i, b

′i+1, ... >

otherwise they will not characterize the same signal for the chosen time scales.The converse implication in definition 3.2.2 follows similarly. Thus, we concludethat G verifies the generalized SI condition.

2. Let G : (Σn1 )ω × (Σn

1 )ω −→ Σω2 × Σω

2 be a function over ω-strings whichsatisfies the generalized SI condition. We will consider only ω-strings

(< ai : i ∈ N >, < a′i : i ∈ N >) ∈ (Σn)ω × (Σn)ω

such that

∀ i, (ai, a′i) 6= (ai+1, a

′i+1)

or (3.2)

∃ k, ∀ i < k, (ai, a′i) 6= (ai+1, a

′i+1)

and ∃ a, a′ ∈ Σn, ∀ i ≥ k, (ai, a′i) = (a, a′).

As G verifies the generalized SI condition, its values on ω-strings that verifythe condition 3.2 determine completely its values on any ω-string. Let ~x benon-Zeno signal and ξ the set of instants in which ~x is not continuous, i.e., inwhich xi is not continuous for some 1 ≤ i ≤ n. We have two possible cases:

24

1. ξ = < ti : i ∈ N >, i.e., ξ is a time scale;

2. ξ = < t1, ..., tk >.

In first case we may consider the ω-strings

α = < (x1(ti), ..., xn(ti)) : i ∈ N >α′ = < (x1(t), ..., xn(t)) : i ∈ N and ti < t < ti+1 >

which clearly characterizes ~x with the time scale τ = ξ and verifies the condition3.2. In second case we extend ξ to the time scale τ = < t0, ..., tk, tk+1, tk+2, ... >and we consider the ω-strings

α = < (x1(t), ..., xn(t)) : t ∈ τ >α′ = < (x1(t), ..., xn(t)) : ti ∈ τ and ti < t < ti+1 >.

It is clear that in this case α, α′ also verifies the condition 3.2 and that it alsocharacterizes x with τ as time scale. Let y be the unique right open signalcharacterized by G(α, α′), τ and we define F as follows:

F~x = y.

As we have seen, if ρ is an order preserving bijection, < ρ(t) : t ∈ τ > is alsoa time scale and by definition of F it follows that it is speed independent andwe will show now that G characterizes F . Assumes that β, β ′, τ ′ characterizes~x, it does not satisfy 3.2 necessarily but we clearly may obtain α, α′, τ from thiswhich characterizes also ~x and verifies 3.2. G(α, α′), τ will characterize F~x bydefinition of F and, as G verifies the generalized SI condition, G(β, β ′), τ ′ willalso characterize F~x. Therefore, G characterizes F .

3. Suppose that G characterizes F . Let α, α′, β, β′ ∈ (Σn)ω such thatα(i) = β(i), for i ≤ k, and α′(i) = β′(i), for i < k, and τ, τ ′ be two time scalessuch that τ(i) = τ ′(i), for i ≤ k, therefore ~x characterized by α, α′, τ and ~x′

characterized by β, β′, τ ′ are non-Zeno signals and they will verify xi(t) = x′i(t),

for 0 ≤ t ≤ τ(k) and for all i. If F is retrospective by definition 2.1.11,

F (~x)(t) = F (~x′)(t), for 0 ≤ t ≤ τ(k)

and as G is a characterization of F , for ξ, ξ′ and ζ, ζ ′ such that G(α, α′) = (ξ, ξ′)and G(β, β′) = (ζ, ζ ′), we have

ξ(i) = ζ(i), for i ≤ k, and ξ′(i) = ζ ′(i), for i < k,

i.e., G is retrospective.Let ~x and ~x′ be two non-Zeno signals such that xi(t

′) = x′i(t

′) for t′ ≤ t witht ∈ T+ and all i. If G is retrospective and these signals are characterized byα, α′, τ and β, β′, τ ′, then α(i) = β(i), for i ≤ k, and α′(i) = β′(i), for i < k,with k such that tk ≥ t, and so, for ξ, ξ′ and ζ, ζ ′ such that G(α, α′) = (ξ, ξ′)and G(β, β′) = (ζ, ζ ′), we have

ξ(i) = ζ(i), for i ≤ k, and ξ′(i) = ζ ′(i), for i < k.

Since G is a characterization of F , G(α, α′), τ characterizes F~x and G(β, β′), τ ′

characterizes F~x′ and by the previous observations we get that

F (~x)(t′) = F (~x′)(t′), for t′ ≤ t,

25

i.e., F is retrospective by definition 2.1.11. Therefore, F is retrospective if andonly if G is retrospective.

As we saw in 1 and 2, there exists an univocally correspondence betweenspeed independent operators on non-Zeno signals and functions on ω-stringsthat verify the generalized SI condition. Therefore, since the residuals of Fstill be speed independent and the residuals of G still verify the generalized SIcondition, G and F will have the same number of distinct residuals. Clearly, bydefinition 2.1.17, G has finite memory if and only if F has finite memory. �

Since every retrospective function on ω-strings has at most countable mem-ory and, as we saw in the last proposition, all speed independent retrospectiveoperators on non-Zeno signals are characterized by retrospective functions onω-strings, we obtain the following corollary.

Corollary 3.2.4 Every speed independent retrospective operator on non-Zenosignals has at most countable memory.

26


3.0.6 A stable operator maps non-Zeno signals to non-Zeno signals.

3.0.7 Every operator that maps non-Zeno signals to non-Zeno signals with count-able memory is stable.

3.0.9 If F : Sig(Σ)n −→ Sig(Σ) is speed independent then F is stable.

3.1.3 1. Every speed independent operator F that maps right open signalsinto right open signals is characterized by a function G on ω-stringsthat satisfies the SI condition.

2. Every function G on ω-strings that satisfies the SI condition charac-terizes a speed independent operator F that maps right open signalsinto right open signals.

3. If G, a function on ω-strings, characterizes F , a speed independentoperator that maps right open signals into right open signals, then

(a) G is retrospective if and only if F is retrospective.

(b) G and F have the same number of distinct residuals.

(c) G has finite memory if and only if F has finite memory.

3.1.4 Every speed independent retrospective operator on right open signals hasat most countable memory.

3.2.3 1. Every speed independent operator F over non-Zeno signals is charac-terized by a function G on (Σn

1 )ω×(Σn1 )ω that satisfies the generalized

SI condition.

2. Every function G on (Σn1 )ω × (Σn

1 )ω that satisfies the generalized SIcondition characterizes a speed independent operator F over non-Zeno signals.

3. If G, a function on (Σn1 )ω × (Σn

1 )ω, characterizes F , a speed indepen-dent operator over non-Zeno signals, then:

(a) F is retrospective is and only if G satisfies the generalized retro-spective condition;

(b) F has finite memory if and only if G has finite memory;

(c) F has countable memory if and only if G has countable memory.

3.2.4 Every speed independent retrospective operator on non-Zeno signals hasat most countable memory.

27

Chapter 4

Examples of Operators

In this chapter we gather a considerable number of examples of operators onsignals1 found in [Rab97, PRT01]. However discussed there, these examples werenot studied in detail and some mistakes have been committed. We will studyall examples carefully, each one will be classified according to the propertiesstudied in previous chapters, namely we will verify if each one is:

1. speed independent;

2. stable;

3. mapping non-Zeno signals into non-Zeno signals;

4. strong retrospective;

5. retrospective;

6. finite memory;

7. countable memory.

We note that in these examples the time set T will be considered as R, i.e.,we will study operators that receive real-time signals.

Example 4.0.1 (Signal Jumpa→b [Rab97]) Jumpa→b is the signal, i.e, the0-ary operator on signals, defined as follows:

Jumpa→b(t) =

{

a if t = 0b if t > 0.

1. Let ρ : T+ −→ T+ be an order preserving bijection, therefore:

Jumpa→b(t) = Jumpa→b(ρ(t)),

for every t ∈ T +, i.e, Jumpa→b is speed independent by definition 3.0.8.2. By 1 and by proposition 3.0.9, we know that Jumpa→b is stable.3. Since Jumpa→b is a 0-ary operator on signals, we only need to verify if

it is a non-Zeno signal, which is true by definition 2.2.1.

1Not necessarily non-Zeno signals.

28

4. We know that Jumpa→b has no arguments, therefore it is strong retro-spective by definition 2.1.11.

5. Jumpa→b is retrospective by 4.6. Accordingly with definition 2.1.16, Jumpa→b has the following residuals:

{suf(Jumpa→b, t) : t ∈ T+} = {Jumpa→b, Constb}.

By definition 2.1.17, Jumpa→b has finite memory.7. By 6, it has countable memory.

Example 4.0.2 (Signal Rational [Rab97]) Rational is the signal, i.e, the 0-ary operator on signals, defined as follows:

Rational(t) =

{

True if t is a rational numberFalse otherwise.

1. Let ρ : T+ −→ T+ be an order preserving bijection such that

ρ(1) =√

2,

therefore:

Rational(1) = True 6= False = Rational(√

2) = Rational(ρ(1)),

i.e.,

Rational 6= Rational ◦ ρ.

So, Rational is not speed independent by definition 3.0.8.2. Rational is not constant for any t > 0, therefore it is not stable by

definition 3.0.5.3. Since Rational is a 0-ary operator on signals, we only need to verify if it

is a non-Zeno signal, which is false by definition 2.2.1 and because Rational isnot constant for any t > 0.

4. By definition 2.1.11, Rational is clearly a strong retrospective operator.5. Rational is retrospective by 4.6. Since Rational is a signal, the set of residuals is the set of the possible

suffixes. Let q, q′ ∈ T+ be irrationals and r ∈ T + be a rational such that r > q,q′ > q and q′ − q is not rational, therefore

Rational(q) = False,

Rational(q′) = False,

Rational(r) = True

and we will see that suf(Rational, q) and suf(Rational, q′) are different forr − q. Clearly:

suf(Rational, q)(r − q) = Rational(r − q + q) = True,

however,

suf(Rational, q′)(r − q) = Rational(r − q + q′) = False,

29

where r − q + q′ is irrational because q′ − q is irrational. Therefore, since theset of irrationals is uncountable, the set of irrationals q, q′ such that q′ − q isirrational is also uncountable and then the set of suffixes is uncountable, i.e.,Rational has not finite memory by definition 2.1.17.

7. As we have seen in 6, the set of prefixes is uncountable and then Rationalhas not countable memory.

Example 4.0.3 (Operator ∃ [Rab97, PRT01]) The unary operator ∃ overboolean signals is defined as follows:

∃(x)(t) =

{

True if there exists t′ such that x(t′) = TrueFalse otherwise.

1. Let ρ : T+ −→ T+ be an order preserving bijection. Thus, if x(t) = Truefor some t ∈ T+, there exists t′ such that ρ(t′) = t and therefore (x ◦ ρ)(t′) =True. Then

∃(x ◦ ρ)(t) = ∃(x)(ρ(t)),

for every t ∈ T +. If x(t) = False, for every t ∈ T +, then

∃(x ◦ ρ)(t) = False = ∃(x)(ρ(t)),

for every t ∈ T +. Therefore, ∃ is speed independent by definition 3.0.8.2. By 1 and by proposition 3.0.9, we know that ∃ is stable.3. By 2 and by proposition 3.0.6, we know that ∃ maps non-Zeno signals

into non-Zeno signals.4. Looking at definition of ∃, we see that the value at t may depend on t′

with t′ > t. Therefore ∃ is not strong retrospective by definition 2.1.11.5. By what we have seen in 4, ∃ can not be retrospective.6. Accordingly with definition 2.1.16, if x is a boolean signal and t ∈ T +,

the residual of ∃ with respect to x and t is one of the following operators:

• if x(t′) = False for t′ such that 0 ≤ t′ < t,

λzλτ. ∃(x)(τ);

• if exists t′ such that 0 ≤ t′ < t and x(t′) = True,

λzλτ. T rue.

Therefore, ∃ has finite memory by definition 2.1.17.7. By 6, it has countable memory.

Example 4.0.4 (Operator LeftCont [Rab97]) LeftCont is an unary oper-ator on signals defined as follows:

LeftCont(x)(t) =

{

True if x is left continuous at tFalse otherwise.

30

1. Let ρ : T+ −→ T+ be an order preserving bijection,

LeftCont(x ◦ ρ)(t) =

{

True if x ◦ ρ is left continuous at tFalse otherwise

=

{

True if x is left continuous at ρ(t)False otherwise

= LeftCont(x)(ρ(t))

= (LeftCont(x) ◦ ρ)(t).

Therefore, LeftCont is speed independent by definition 3.0.8.2. By 1 and by proposition 3.0.9, we know that LeftCont is stable.3. By 2 and by proposition 3.0.6, we know that LeftCont maps non-Zeno

signals into non-Zeno signals.4. Given a signal x, we need to know its value at t in order to decide about

the left continuity of x at t. Therefore, LeftCont is not strong retrospective bydefinition 2.1.11.

5. As we see in 4, we need to know the value of x at t and, clearly, its valuebefore t. Thus, since we do not need future values, LeftCont is retrospectiveby definition 2.1.11.

6. Accordingly with definition 2.1.16, if x is a signal and t ∈ T +, the residualof LeftCont with respect to x and t is one of the following operators:

• if t = 0,

λzλτ. LeftCont(z)(τ);

• if t > 0 and ∃ t′ < t, ∀ u ∈ [t′, t[, x(u) = a,

λzλτ.

True if τ = 0 and z(0) = aFalse if τ = 0 and z(0) 6= aLeftCont(z)(τ) if τ > 0;

• if t > 0 and no ∃ t′ < t, ∀ u ∈ [t′, t[, x(u) = a,

λzλτ.

{

False if τ = 0LeftCont(z)(τ) if τ > 0.

Since x is defined over a finite alphabet, LeftCont has a finite number of resid-uals, i.e., it has finite memory by definition 2.1.17.

7. By 6, it has countable memory.

Example 4.0.5 (Operator Cont [Rab97]) The unary operator Cont on sig-nals is defined as follows:

Cont(x)(t) =

{

True if x is continuous at tFalse otherwise.


Cont(x ◦ ρ)(t) =

{

True if x ◦ ρ is continuous at tFalse otherwise

=

{

True if x is continuous at ρ(t)False otherwise

= Cont(x)(ρ(t))

= (Cont(x) ◦ ρ)(t).

31

Therefore, Cont is speed independent by definition 3.0.8.2. By 1 and by proposition 3.0.9, we know that Cont is stable.3. By 2 and by proposition 3.0.6, we know that Cont maps non-Zeno signals

into non-Zeno signals.4. Given a signal x, we need to know its value at t in order to decide about

the continuity of x at t. Therefore, Cont is not strong retrospective by definition2.1.11.

5. As we see in 4, we need to know the value of x at t and, clearly, its valuebefore and after t. Thus, since we need future values, Cont is not retrospectiveby definition 2.1.11.

6. Accordingly with definition 2.1.16, if x is a signal and t ∈ T +, the residualof Cont with respect to x and t is one of the following operators:

• if t = 0,

λzλτ. Cont(z)(τ);

• if t > 0 and ∃ t′ < t, ∀ u ∈ [t′, t[, x(u) = a,

λzλτ.

True if τ = 0, a = z(0) andz is right continuous at 0

False if τ = 0 and(a 6= z(0) or z is not right continuous at 0)

Cont(z)(τ) if τ > 0;


λzλτ.

{

False if τ = 0Cont(z)(τ) if τ > 0.

Since x is defined over a finite alphabet, Cont has a finite number of residuals,i.e., Cont has finite memory by definition 2.1.17.

7. By 6, Cont has countable memory.

Example 4.0.6 (Operator LLim [Rab97]) The unary operator LLim on sig-nals is defined as follows:

LLim(x)(t) =

{

a if ∃ t′ < t, ∀ u ∈ [t′, t[, x(u) = a⊥ otherwise


LLim(x ◦ ρ)(t) =

{

a if ∃ t′ < t, ∀ u ∈ [t′, t[, x(ρ(u)) = a⊥ otherwise

=

a if ∃ t′, ρ(t′) < ρ(t)and ∀ v ∈ [ρ(t′), ρ(t)[, x(v) = a

⊥ otherwise

=

{

a if ∃ t′′ < ρ(t), ∀ v ∈ [t′′, ρ(t)[, x(v) = a⊥ otherwise

= LLim(x)(ρ(t))

= (LLim(x) ◦ ρ)(t).

32

Therefore, LLim is speed independent by definition 3.0.8.2. By 1 and by proposition 3.0.9, we know that LLim is stable.3. By 2 and by proposition 3.0.6, we know that LLim maps non-Zeno signals

into non-Zeno signals.4. Given a signal x and an instant t ∈ T +, by the definition of LLim, we

conclude that its value at t depends only on instants before t. Therefore, LLimis strong retrospective by definition 2.1.11.

5. By 4 and by definition 2.1.11, LLim is retrospective.6. Accordingly with definition 2.1.16, if x is a signal and t ∈ T +, the residual

of LLim with respect to x and t is one of the following operators:

• if t = 0,

λzλτ. LLim(z)(τ);

• if t > 0 and ∃ t′ < t, ∀ u ∈ [t′, t[, x(u) = a,

λzλτ.

{

a if τ = 0LLim(z)(τ) if τ > 0;


λzλτ.

{

⊥ if τ = 0LLim(z)(τ) if τ > 0.

Since x is defined over a finite alphabet, LLim has a finite number of residuals,i.e., it has finite memory by definition 2.1.17.

7. By 6, it has countable memory.

Example 4.0.7 (Operator RLim [Rab97]) The operator RLim on signalsis defined as follows:

RLim(x)(t) =

{

a if ∃ t′ > t, ∀ u ∈ ]t, t′], x(u) = a⊥ otherwise.


RLim(x ◦ ρ)(t) =

{

a if ∃ t′ > t, ∀ u ∈ ]t, t′], x(ρ(u)) = a⊥ otherwise

=

a if ∃ t′, ρ(t′) > ρ(t)and ∀ v ∈ ]ρ(t), ρ(t′)], x(v) = a

⊥ otherwise

=

{

a if ∃ t′′ > ρ(t), ∀ v ∈ ]ρ(t), t′′], x(v) = a⊥ otherwise

= RLim(x)(ρ(t))

= (RLim(x) ◦ ρ)(t).

Therefore, RLim is speed independent by definition 3.0.8.2. By 1 and by proposition 3.0.9, we know that RLim is stable.3. By 2 and by proposition 3.0.6, we know that RLim maps non-Zeno signals

into non-Zeno signals.

33

4. Given a signal x and an instant t ∈ T +, by the definition of RLim, weconclude that its value at t depends on instants after t. Therefore, RLim is notstrong retrospective by definition 2.1.11.

5. Given a signal x and an instant t ∈ T +, by the definition of RLim, weconclude that its value at t depends on instants after t. Therefore, RLim is notretrospective by definition 2.1.11.

6. Accordingly with definition 2.1.16, if x is a signal and t ∈ T +, the residualof RLim with respect to x and t is itself. Therefore RLim has a finite numberof residuals, i.e., RLim has finite memory by definition 2.1.17.

7. By 6, RLim has countable memory.

Example 4.0.8 (Pointwise extension [Rab97, PRT01]) Given a functiong : Σ1× ...×Σn −→ Σ, its pointwise extension G is the n-ary operator on signalsdefined as follows:

G(x1, ..., xn)(t) = g(x1(t), ..., xn(t)).


G(x1 ◦ ρ, ..., xn ◦ ρ)(t) = g(x1(ρ(t)), ..., xn(ρ(t)))

= G(x1, ..., xn)(ρ(t))

= (G(x1, ..., xn) ◦ ρ)(t).

Therefore, G is speed independent by definition 3.0.8.2. By 1 and by proposition 3.0.9, we know that G is stable.3. By 2 and by proposition 3.0.6, we know that G maps non-Zeno signals

into non-Zeno signals.4. Given a signal ~x and an instant t ∈ T +, by the definition of G, we

conclude that its value at t depends on the value of ~x at t. Therefore, G is notstrong retrospective by definition 2.1.11.

5. Given a signal ~x and an instant t ∈ T +, by the definition of G, weconclude that its value at t depends only on the value of ~x at t. Therefore, G isretrospective by definition 2.1.11.

6. Accordingly with definition 2.1.16, if ~x is a signal and t ∈ T +, theresidual of G with respect to ~x and t is itself. Therefore G has a finite numberof residuals, i.e., G has finite memory by definition 2.1.17.

7. By 6, G has countable memory.

Example 4.0.9 (Operator Prime [Rab97]) The unary operator Prime onsignals is defined as:

Prime(x)(t) =

{

True if x changes a prime number of times in [0, t[⊥ otherwise.

34

Prime(x ◦ ρ)(t) =

True if x ◦ ρ changes a primenumber of times in [0, t[

False otherwise

=

True if x changes a primenumber of times in [0, ρ(t)[

False otherwise

= Prime(x)(ρ(t))

= (Prime(x) ◦ ρ)(t).

Therefore, Prime is speed independent by definition 3.0.8.2. By 1 and by proposition 3.0.9, we know that Prime is stable.3. By 2 and by proposition 3.0.6, we know that Prime maps non-Zeno

signals into non-Zeno signals.4. Given a signal x and an instant t ∈ T +, by the definition of Prime, we

conclude that its value at t depends only on instants before t. Therefore, Primeis strong retrospective by definition 2.1.11.

5. By 4 and by definition 2.1.11, Prime is retrospective.6. Accordingly with definition 2.1.16, if x is a signal and t ∈ T +, the residual

of Prime with respect to x and t is one of the following operators:

• if t = 0,

λzλτ. P rime(z)(τ);

• if t > 0 and n is the number of changes of x in [0, t[,

λzλτ.

True if xct; z changes at t and∃ p prime, z changes p − n − 1 times in [0, t′[

True if xct; z does not change at t and∃ p prime, z changes p − n times in [0, t′[

False otherwise.

Since N0 is countable, Prime has not a finite number of residuals, i.e., Primehas not finite memory by definition 2.1.17.

7. By 6, Prime has countable memory since N0 is countable.

Example 4.0.10 (Operator T imer [Rab97, PRT01]) The unary operatorT imer on signals is defined as follows:

T imer(x)(t) =

True if ∃ τ < t such thatx is constant in [τ, t[ and t − τ ≥ 1

False otherwise.

1. Let ρ : T+ −→ T+ be an order preserving bijection and x a signal suchthat

ρ(1) = 12 and x(t) =

{

0 if 0 ≤ t < 12

1 otherwise.

35

Therefore,

T imer(x ◦ ρ)(1) = True

6= False

= T imer(x)(1

2)

= (T imer(x) ◦ ρ)(1),

i.e., T imer is not speed independent by definition 3.0.8.2. Clearly

T imer(Consta)(t) = False,

for every t < 1, and

T imer(Consta)(t) = True,

for every t ≥ 1, therefore it is not constant at t = 1 while Consta is constant att = 1. By definition 3.0.5, we know that T imer is not stable.

3. T imer change at most twice in any interval of length one. Therefore,T imer maps non-Zeno signals into non-Zeno signals.

4. Given a signal x and an instant t ∈ T +, by the definition of T imer, weconclude that its value at t depends only on instants before t. Therefore, T imeris strong retrospective by definition 2.1.11.

5. By 4 ad by definition 2.1.11, T imer is retrospective.6. Accordingly with definition 2.1.16, if x is a signal and t ∈ T + such that

x is constant in [0, 1[ and t < 1, the residual of T imer with respect to x and tis one of the following operators:

• if t = 0,

λzλτ. T imer(z)(τ);

• if t > 0 and with a = x(0),

λzλτ.

{

True if z(t′) = a for t′ ∈ [0, τ [ and τ ≥ 1 − tT imer(z)(τ) otherwise.

As the number of such operators is uncountable (for each t ∈ [0, 1[, there existsa different operator), it follows that T imer has not a finite number of residuals,i.e., T imer has not finite memory by definition 2.1.17.

7. By 6, T imer also has not countable memory.

Example 4.0.11 (Operator Delaya [Rab97, PRT01]) The unary operatorDelaya on signals is defined as follows:

Delaya(x)(t) =

{

a if t < 1x(t − 1) otherwise.

1. Let a, b ∈ Σ such that a 6= b and let ρ : T + −→ T+ be an order preservingbijection such that

ρ( 12 ) = 1.

36

Therefore,

Delaya(Constb ◦ ρ)(1

2) = a

6= b

= Delaya(Constb)(1)

= (Delaya(Constb) ◦ ρ)(1

2),

i.e., Delaya is not speed independent by definition 3.0.8.2. Clearly Delaya(Constb) is not constant at t = 1 where Constb is the

signal constant everywhere. Therefore, Delaya is not stable by definition 3.0.5.3. If x is a non-Zeno signal, it is also clear that Delaya(x) is a non-Zeno

signal. Thus, Delaya maps non-Zeno signals into non-Zeno signals.4. Given a signal x and an instant t ∈ T +, by the definition of Delaya,

we conclude that its value at t depends only on instants before t. Therefore,Delaya is strong retrospective by definition 2.1.11.

5. By 4 and by definition 2.1.11, Delaya is retrospective.6. Accordingly with definition 2.1.16, if x is a signal and t ∈ T + such that

t < 1, the residual of Delaya with respect to x and t is one of the followingoperators:

• if t = 0,

λzλτ. Delaya(z)(τ);

• if t > 0,

λzλτ.

{

a if τ < 1 − txct; z(τ − 1 + t) otherwise.

As the number of such operators is uncountable (for each t ∈ [0, 1[, there existsa different operator), it follows that Delaya has not a finite number of residuals,i.e., Delaya has not finite memory by definition 2.1.17.

7. By 6, Delaya also has not countable memory.

Example 4.0.12 (Operator PT imer [PRT01]) The unary operator PT imeron signals is defined as:

PT imer(x)(t) =

False if t < 1changes value if x has changed value at t − k

and x was constant in ]t − k, t[for some k ∈ N s.t. k ≤ t.


ρ( 12 ) = 1.

37

Therefore,

PT imer(Jumpb→c ◦ ρ)(1

2) = False

6= True

= PT imer(Jumpb→c)(1)

= (PT imer(Jumpb→c) ◦ ρ)(1

2),

i.e., PT imer is not speed independent by definition 3.0.8.2. Clearly PT imer(Jumpb→c) is not constant at t = 1 where Jumpb→c is a

signal constant for t > 0. Therefore, PT imer is not stable by definition 3.0.5.3. By its definition, it is clear that PT imer maps non-Zeno signals into

non-Zeno signals.4. Given a signal x and an instant t ∈ T +, by the definition of PT imer,

we conclude that its value at t depends only on instants before t. Therefore,PT imer is strong retrospective by definition 2.1.11.

5. By 4 and by definition 2.1.11, PT imer is retrospective.6. Accordingly with definition 2.1.16, if x is a signal and t ∈ T + such that

t < 1, the residual of PT imer with respect to x and t is one of the followingoperators:

• if t = 0,

λzλτ. PT imer(z)(τ);

• if t > 0,

λzλτ.

a if τ < 1 − tchanges value if xct; z has changed value at τ − k

and xct; z was constant in ]τ − k, τ [for some k ∈ N s.t. k ≤ τ .

As the number of such operators is uncountable (for each t ∈ [0, 1[, there exists adifferent operator), it follows that PT imer has not a finite number of residuals,i.e., PT imer has not finite memory by definition 2.1.17.

7. By 6, PT imer also has not countable memory.

Example 4.0.13 (Operator Filtera [PRT01]) The unary operator Filtera

on signals is defined as follows:

Filtera(x)(t) =

a if t < 1x(t − 1) if t ≥ 1 and x is constant in [t − 1, t[does not change otherwise.


ρ( 12 ) = 1.

38

Therefore,

Filtera(Constb ◦ ρ)(1

2) = a

6= b

= Filtera(Constb)(1)

= (Filtera(Constb) ◦ ρ)(1

2),

i.e., Filtera is not speed independent by definition 3.0.8.2. Clearly Filtera(Constb) is not constant at t = 1 where Constb is the

signal constant everywhere. Therefore, Filtera is not stable by definition 3.0.5.3. By its definition, it is clear that Filtera maps non-Zeno signals into

non-Zeno signals.4. Given a signal x and an instant t ∈ T +, by the definition of Filtera,

we conclude that its value at t depends only on instants before t. Therefore,Filtera is strong retrospective by definition 2.1.11.

5. By 4 and by definition 2.1.11, Filtera is retrospective.6. Accordingly with definition 2.1.16, if x is a signal and t ∈ T + such that

t < 1, the residual of Filtera with respect to x and t is one of the followingoperators:

• if t = 0,

λzλτ. F iltera(z)(τ);

• if t > 0,

λzλτ.

a if τ < 1 − txct; z(τ − 1 + t) if τ ≥ 1 − t and

xct; z is constant in [τ − 1, τ [does not change otherwise.

As the number of such operators is uncountable (for each t ∈ [0, 1[, there existsa different operator), it follows that Filtera has not a finite number of residuals,i.e., Filtera has not finite memory by definition 2.1.17.

7. By 6, Filtera also has not countable memory.

Example 4.0.14 (Operator Lasta [Rab97]) The unary operator Lasta onsignals is defined as:

Lasta(x)(t) =

b if ∃ τ1, τ2, τ1 < τ2 < t and ∀ τ ∈ ]τ1, τ2[, x(τ) = band x changes at every point in ]τ2, t[

a otherwise.

39

Lasta(x ◦ ρ)(t) =

b if ∃ τ1, τ2, τ1 < τ2 < t and∀ τ ∈ ]τ1, τ2[, (x ◦ ρ)(τ) = band x ◦ ρ changes at every point in ]τ2, t[

a otherwise

=

b if ∃ τ ′1, τ

′2, τ ′

1 < τ ′2 < ρ(t) and

∀ τ ′ ∈ ]τ ′1, τ

′2[, x(ρ(τ ′)) = b

and x changes at every point in ]ρ(τ ′2), ρ(t)[

a otherwise

= Lasta(ρ(t))

= (Lasta(x) ◦ ρ)(t).

Therefore, Lasta is speed independent by definition 3.0.8.2. By 1 and by proposition 3.0.9, we know that Lasta is stable.3. By 2 and by proposition 3.0.6, we know that Lasta maps non-Zeno signals

into non-Zeno signals.4. Given a signal x and an instant t ∈ T +, by the definition of Lasta, we

conclude that its value at t depends only on instants before t. Therefore, Lasta

is strong retrospective by definition 2.1.11.5. By 4 and by definition 2.1.11, Lasta is retrospective.6. Accordingly with definition 2.1.16, if x is a signal and t ∈ T +, the residual

of Lasta with respect to x and t is one of the following operators:

• if t = 0,

λzλτ. Lasta(z)(τ);

• if t > 0 and x(τ) = b for τ ∈ ]t′, t[ with t′ < t,

λzλτ.

b if ∃ τ2, 0 ≤ τ2 < τand ∀ τ ′ ∈ [0, τ2[, z(τ ′) = band z changes at every point in ]τ2, τ [

Lasta(z)(τ) otherwise;

• if t > 0 and x(τ) = b for τ ∈ ]t′, t′′[ and x changes at every point in ]t′′, t[with t′ < t′′ < t,

λzλτ.

{

b if z changes at every point in [0, τ [Lasta(z)(τ) otherwise.

Therefore, Lasta has a finite number of residuals, i.e., Lasta has finite memoryby definition 2.1.17.

7. By 6, Lasta has countable memory.

Example 4.0.15 (Operator F [Rab97]) The unary operator F on signals isdefined as follows:

F (x)(t) =

True if x changes a finite number of timesin [0, t[ or t is rational.

False otherwise.

40


ρ(1) =√

2,

therefore:

F (Rational)(1) = True

6= False

= F (Rational)(√

2)

= F (Rational)(ρ(1)),

i.e.,

F (Rational) ◦ ρ 6= F (Rational ◦ ρ).

Thus, F is not speed independent by definition 3.0.8.2. Let x be a signal such that x = Rationalc1; ConstFalse. x is constant at

t = 2, however F (x)(2) = True and therefore F is not constant at 2 because, ift is irrational, F (x)(t) = False. Clearly, F is not stable by definition 3.0.5.

3. Let x be a non-Zeno signal, it is clear that x changes a finite numberof times in [0, t[ for any t. Therefore, F (x) = ConstTrue which is a non-Zenosignal, i.e, F maps non-Zeno signals into non-Zeno signals.

4. Given a signal x and an instant t ∈ T +, by the definition of F , weconclude that its value at t depends only on instants before t. Therefore, F isstrong retrospective by definition 2.1.11.

5. By 4 and by definition 2.1.11, F is retrospective.6. Accordingly with definition 2.1.16, if x is a signal and t ∈ T + such that x

does not change a finite number of times in [0, t[, F will have has residual withrespect to x and t the following operator:

• λzλτ.

{

True if τ − t is rationalFalse otherwise.

Therefore, since the set of irrationals is uncountable and for t, t′ irrationalsRes(F, x, t) 6= Res(F, x, t′), F has not a finite number of residuals, i.e., F hasnot finite memory by definition 2.1.17.

7. By 6, F has not also countable memory.

Example 4.0.16 (Operator G [Rab97]) The unary operator G on signals isdefined as:

G(x)(t) =

True if there is an irrational t0 ≤ t such thatx is constant in [0, t0[ and x(t0) 6= x(0)

False otherwise.


ρ(1) =√

2,

and let x be a signal such that x(√

2) = 1 and x(t) = 0 for t 6=√

2, therefore:

G(x ◦ ρ)(1) = False

6= True

= (G(x) ◦ ρ)(1),

i.e.,

41

G(x) ◦ ρ 6= G(x ◦ ρ).

Thus, G is not speed independent by definition 3.0.8.2. Let x be a signal and t ∈ T + such that x is constant at t, i.e., exists

t1, t2 ∈ T+ such that x is constant in [t1, t2] and t1 < t < t2. If exists t0 < t suchthat t0 is irrational, x is constant in [0, t0[ and x(t0) 6= x(0), then G(x)(τ) =True, for every τ ∈ [t1, t2]. Otherwise, G(x)(τ) = False, for every τ ∈ [t1, t2].Therefore, G(x) is constant at t, i.e., G is stable by definition 3.0.5.

3. By 2 and by proposition 3.0.6, we know that G maps non-Zeno signalsinto non-Zeno signals.

4. Given a signal x and an instant t ∈ T +, by the definition of G, weconclude that its value at t may depend from t. Therefore, G is not strongretrospective by definition 2.1.11.

5. Given a signal x and an instant t ∈ T +, by the definition of G, weconclude that its value at t may depend only on t and on instants before t.Therefore, G is retrospective by definition 2.1.11.

6. Accordingly with definition 2.1.16, if x is a signal and t ∈ T +, G willhave as residual with respect to x and t one of the following operators:

• if t = 0,

λzλτ. G(z)(τ);

• if there exists an irrational 0 < t0 < t such that x is constant in [0, t0[ andx(t0) 6= x(0),

λzλτ. T rue;

• if x is not constant at 0 ≤ t0 < t with t0 rational,

λzλτ. False;

• if x(τ) = a for τ ∈ [0, t[ and 0 < t is irrational,

λzλτ.

{

True if z(0) 6= aG(z)(τ) otherwise;

• if x(τ) = a for τ ∈ [0, t[ and 0 < t is rational,

λzλτ.

True if there is an irrational t0 ≤ τ such thatz(t′) = a for t′ ∈ [0, t0[ and z(t0) 6= a

False otherwise.

Therefore, as the set Σ is finite, G has a finite number of residuals, i.e., G hasfinite memory by definition 2.1.17.

7. By 6, G has also countable memory.

42

Chapter 5

Properties of Operators

5.1 Closure Properties

In this section we will prove some propositions found in [Rab97] on some closureproperties of operators on signals. Namely, we provide the complete proofsof propositions 5.1.1, 5.1.2, 5.2.4 and 5.2.8, which were partially given or notprovided in the preceding paper [Rab97]. We also emphasize some other resultswhich have been proved in [Rab97] and which ones will be important for ourstudy.

Proposition 5.1.1 ([Rab97]) The following sets of operators on signals areclosed under taking residual:

1. retrospective operators;

2. strong retrospective operators;

3. stable operators;

4. speed independent operators;

5. finite memory retrospective operators.

Proof: Assume that G is the residual of an operator F , i.e., G is the residual ofF with respect to some ~x ∈ Sig(Σ)n and some t ∈ T+. From definition 2.1.16we know that:

G(~z)(t′) = F ((x1ct; z1), ..., (xnct; zn))(t + t′),

for ~z ∈ Sig(Σ)n.(1) Suppose that F is retrospective and ~z, ~w ∈ Sig(Σ)n such that

zi(t′) = wi(t

′),

for t′ ∈ [0, τ ] and i = 1, ..., n. Then

(xict; zi)(τ) = (xict; wi)(t + t′)

with t′ ∈ [−t, τ ] and i = 1, ..., n. Since F is a retrospective operator, we have

F ((x1ct; z1), ..., (xnct; zn))(t + t′) = F ((x1ct; w1), ..., (xnct; wn))(t + t′),

43

for t′ ∈ [−t, τ ] and i = 1, ..., n. Hence for t′ ∈ [0, τ ], we have

G(~z)(t′) = G(~w)(t′),

i.e, by definition 2.1.11, G is retrospective.(2) Similar to (1).(3) Suppose that F is a stable operator and take ~z ∈ Sig(Σ)n such that ~z isconstant at instant τ . Then (xict; zi) is constant at instant t + τ . Since Fis stable, we have that F ((x1ct; z1), ..., (xnct; zn)) is constant at instant t + τ .Hence G(~z) is constant at instant τ , i.e, by definition 3.0.5, G is stable.(4) Suppose that F is a speed independent operator and that ρ is an orderpreserving bijection. Defining ρ′ as

ρ′(τ) =

{

τ if τ < tt + ρ(τ − t) if τ ≥ t

we have that ρ′ is an order preserving bijection too. From the definition of ρ′

we observe that

(xict; zi) ◦ ρ′ = (xict; (zi ◦ ρ)),

for i = 1, ..., n. Therefore, since F is speed independent, we have that

G(z1 ◦ ρ, ..., zn ◦ ρ)(t′) = F ((x1ct; (z1 ◦ ρ)), ..., (xnct; (zn ◦ ρ))(t + t′)

= F ((x1ct; z1) ◦ ρ′, ..., (xnct; zn) ◦ ρ′)(t + t′)

= F ((x1ct; z1), ..., (xnct; zn))(ρ′(t + t′))

= F ((x1ct; z1), ..., (xnct; zn))(t + ρ(t′))

= G(z1, ..., zn)(ρ(t′)).

Hence G(~z ◦ ρ) = G(~z) ◦ ρ, i.e., G is speed independent.(5) Suppose that F is a finite memory retrospective operator and G is a residualof F as before. Let H be the residual of G with respect to ~y and τ , then

H(~z)(t′) = G((y1cτ ; z1), ..., (yncτ ; zn))(τ + t′)

= F ((x1ct; (y1cτ ; z1)), ..., (xnct; (yncτ ; zn)))(t + τ + t′)

= F (((x1ct; y1)ct+τ ; z1), ..., ((xnct; yn)ct+τ ; zn))((t + τ) + t′).

Thus, H is a residual of F with respect to ((x1ct; y1), ..., (xnct; yn)) and t+τ . Weknow that F has a finite number of distinct residuals and that G is a residual ofF . Hence, G has a finite number of distinct residuals and so is a finite memoryretrospective1 operator. �

Proposition 5.1.2 ([Rab97]) The following sets of operators from signals tosignals are closed under composition:




1Retrospective by 1.

44



Proof:(1) Let Fi : Sig(Σ)n −→ Sig(Σ), for i = 1, ..., m, and G : Sig(Σ)m −→ Sig(Σ)be retrospective operators. Suppose ~x, ~x′ ∈ Sig(Σ)n such that xj(τ) = x′

j(τ),for j = 1, ..., n and τ ∈ [0, t]. If Fi is retrospective, for every i, then Fi(~x)(τ) =Fi(~x

′)(τ), for every i and τ ∈ [0, t]. If G is retrospective, then by the abovestatements and definition 2.1.11 we know that

G(F1(~x), ..., Fn(~x))(τ) = G(F1(~x′), ..., Fn(~x′))(τ)),

for τ ∈ [0, t], i.e., the composition of G with F1, ..., Fm is a retrospective opera-tor.(2) Similar to (1).(3) Let Fi : Sig(Σ)n −→ Sig(Σ), for i = 1, ..., m, and G : Sig(Σ)m −→ Sig(Σ)be stable operators. Suppose ~x ∈ Sig(Σ)n such that xj is constant at t withj = 1, ..., n. If Fi is stable, for every i, then Fi(~x) is constant at t for every i(by definition 3.0.5). If G is stable, then by the above statements we know that

G(F1(~x), ..., Fn(~x))

is constant at t, i.e., by definition 3.0.5, G ◦ (F1, ..., Fm) is a stable operator.(4) Let Fi : Sig(Σ)n −→ Sig(Σ), for i = 1, ..., m, and G : Sig(Σ)m −→ Sig(Σ)be speed independent operators. Suppose that ρ is an order preserving bijection.If Fi is speed independent, for every i, then

Fi(~x ◦ ρ) = Fi(~x) ◦ ρ,

for every i and ~x ∈ Sig(Σ)n. If G is speed independent, then

(G(F1(~x), ..., Fm(~x))) ◦ ρ = G(F1(~x) ◦ ρ, ..., Fm(~x) ◦ ρ)

= G(F1(~x ◦ ρ), ..., Fm(~x ◦ ρ)),

for every ~x ∈ Sig(Σ)n, i.e., the composition of G with F1, ..., Fm is a speedindependent operator.(5) Let Fi : Sig(Σ)n −→ Sig(Σ), for i = 1, ..., m, and G : Sig(Σ)m −→ Sig(Σ)be finite memory retrospective operators. Suppose that F ′

i is the residual ofFi with respect to ~x and t, for every i, and that G′ is the residual of G withrespect to (F1(~x), ..., Fm(~x)) and t. Then by definition 2.1.16 the residual R ofthe composition of G with F1, ..., Fm with respect to ~x and t is

R(~z)(t′) =

= G(F1((x1ct; z1), ..., (xnct; zn)), ..., Fm((x1ct; z1), ..., (xnct; zn)))(t + t′)

= G((F1(~x)ct; F ′1(~z)), ..., (Fm(~x)ct; F ′

m(~z)))(t + t′)

= G′(F ′1(~z), ..., F ′

m(~z))(t′)

with ~z ∈ Sig(Σ)n. Thus, R is equal to the composition of G′ with F ′1, ..., F

′m and

as G and Fi, for i = 1, .., m, have finitely many residuals by definition 2.1.17,the composition of G with F1, ..., Fm have finitely many residuals too. In fact,if G has k residuals and each Fi has ni residuals, the composition of G with

45

F1, ..., Fm will have k×n1× ...×nm or less residuals because this is the numberof possible compositions of G′ with F ′

1, ..., F′m. Therefore, by definition 2.1.17

and by (1), the composition of G with F1, ..., Fm is a finite memory retrospectiveoperator. �

Analyzing the above proofs we can check that they hold for the set of oper-ators over any set C of signals which is closed under concatenation, suffix andorder preserving bijections. In particular the next proposition holds:

Proposition 5.1.3 ([Rab97]) Let C be a set of signals which is closed underconcatenation, suffix and the order preserving bijections. The following sets ofoperators over C signals are closed under composition and under residual:






Proof: Similar to the previous proofs of this section. �

5.2 Properties of Finite Memory Retrospective

Operators

To study the properties of finite memory retrospective operators we will dealwith general signals and we will follow the main ideas in [Rab97].

5.2.1 Finite Memory Signals

The following proposition is proved2 in [Rab97] and is a key technical propositionneeded in our study.

Proposition 5.2.1 ([Rab97]) A general signal x is finite memory if and onlyif x is constant on the positive reals.

Such signals are of the form:

Definition 5.2.2 (Jumpa→b) For some a, b ∈ Σ we define Jumpa→b ∈ Sig(Σ)as follows:

Jumpa→b(τ) =

{

a if τ = 0b otherwise.

As was stated in proposition 5.2.1, the finite memory signals are very simple.We have seen before that a signal is a 0-ary operator on signals, may we concludethe same about the simplicity of general operators on signals? Looking at theexamples given in chapter 4 we say no. In fact is enough to consider the pointwise

2This is a long proof.

46

extension of a function g : Σn −→ Σ, which has finite memory as we have seenin example 4.0.8.

We will state now some consequences of proposition 5.2.1. The followingtwo propositions were proved in [Rab97], however we reproduce their proofs inorder to emphasize the importance of proposition 5.2.1 and because some minorcorrections are needed.

Proposition 5.2.3 ([Rab97]) If F : Sig(Σ)n −→ Sig(Σ) is a finite memoryretrospective operator, then

F (Jumpa1→b1 , ..., Jumpan→bn) = Jumpc→d,

for some c and d in Σ.

Proof: F is a finite memory retrospective operator. For every i, Jumpai→bi

is a finite memory signal by proposition 5.2.1. Since signals are 0-ary oper-ators on signals, Jumpai→bi

is a finite memory 0-ary operator. Therefore,F (Jumpa1→b1 , ..., Jumpan→bn

) is a finite memory 0-ary operator by proposi-tion 5.1.2, i.e., F (Jumpa1→b1 , ..., Jumpan→bn

) is a finite memory signal and weconclude by proposition 5.2.1 that

F (Jumpa1→b1 , ..., Jumpan→bn) = Jumpc→d,

for some c and d in Σ. �

Proposition 5.2.4 ([Rab97]) Every finite memory retrospective operator isstable.

Proof: Assume that F : Sig(Σ)n −→ Sig(Σ) is a finite memory retrospectiveoperator. We pretend to show that if ~x ∈ Sig(Σ)n is constant at t > 0 thenF (~x) is constant at t. If ~x is constant at t > 0, then there exists ε > 0 such that

xi(τ) = xi(t) = bi, for τ ∈ [t − ε, t + ε].

Let G = Res(F, ~x, t − ε), from the previous and because F is retrospective itfollows that

G(Constb1 , ..., Constbn)(τ) = F (~x)(t − ε + τ),

where τ ∈ [0, 2ε] and Constbiis the signal which is constant (and equal to bi)

everywhere, for every i. So, F (~x) is constant at t iff G(Constb1 , ..., Constbn)

is constant at ε. By proposition 5.1.1, since G is a residual of F , G is a fi-nite memory retrospective operator. From proposition 5.2.1 we may infer thatG(Constb1 , ..., Constbn

) is constant on the positive reals and therefore it is con-stant at ε, i.e., F (~x) is constant at t. �

From this last proposition and from proposition 3.0.6, we get the followingcorollary.

Corollary 5.2.5 A finite memory retrospective operator maps non-Zeno signalsinto non-Zeno signals.

Let F be an operator over general signals, we denote the restriction of F tonon-Zeno signals by Rest(F ) which is defined as

47

Rest(F ) = λ~x ∈ nZSig(Σ)n. F~x.

As result of corollary 5.2.5, the following proposition holds and is proved in[Rab97].

Proposition 5.2.6 ([Rab97]) If F is a finite memory retrospective operatoron signals then Rest(F ) is an operator from non-Zeno signals to non-Zenosignals. Moreover, Rest(F ) is a finite memory retrospective operator over thenon-Zeno signals.

5.2.2 State Function

We pretend in this subsection generalize to n-ary case the definitions and resultspresented in [Rab97] about the state function.

Definition 5.2.7 (State Operator [Rab97]) Let G0 be a finite memory ret-

rospective operator from Sig(Σ)n to Sig(Σ′) and let ~G =< G0, G1, ..., Gk > bea sequence of all its residuals. It is clear that any residual of Gi is a residual ofG0. Define functions

out~G: Σn × {0, .., k} −→ Σ′

state ~G: Sig(Σ)n −→ Sig({0, ..., k} → {0, ..., k})

as follows:

out~G(a1, ..., an, i) = Gi(Consta1

, ..., Constan)(0),

state ~G(~x)(t)i = j, if Gj = Res(Gi, ~x, t)

The following result holds and, however stated, its proof is not provided in[Rab97].

Proposition 5.2.8 (Properties of the State Operator [Rab97]) .

1. state ~G(~x)(0) = id, the identity permutation.

2. state ~Gis a strong retrospective operator.

3. state ~G((x1ct1 ; z1), ..., (xnct1 ; zn))(t1 + t2) =

= (state ~G(~z)(t2)) ◦ (state ~G

(~x)(t1))

4. G(~x)(t) = out ~G(~x(t), state ~G

(~x)(t)0)

Proof:1. By definition 2.1.16, for every Gi and ~x ∈ Sig(Σ)n, we have that:

Res(Gi, ~x, 0) = λz1...zn.λt′.Gi((x1c0; z1), ..., (xnc0; zn))(0 + t′) = Gi.

Therefore state ~G(~x)(0)i = i, for every i, i.e., state ~G

(~x)(0) is the identity per-mutation.2. Again, by definition 2.1.16, we know that if ~x, ~y ∈ Sig(Σ)n coincide in theinterval [0, t[, then Res(Gi, ~x, τ) = Res(Gi, ~y, τ), for every τ ∈ [0, t] and i, i.e.,state ~G

(~x)(τ) = state ~G(~y)(τ) for every τ ∈ [0, t]. Therefore, by definition 2.1.11,

state ~Gis a strong retrospective operator.

48

3. Assume that state ~G((x1ct1 ; z1), ..., (xnct1 ; zn))(t1 + t2)i = j for some i and j,

i.e., Gj = Res(Gi, ((x1ct1 ; z1), ..., (xnct1 ; zn)), t1 + t2). By definition 2.1.16, wehave the following:

Gj = Res(Gi, ((x1ct1 ; z1), ..., (xnct1 ; zn)), t1 + t2)

= λ~wλt′. Gi(((x1ct1 ; z1)ct1+t2 ; w1), ..., ((xnct1 ; zn)ct1+t2 ; wn))(t1 + t2 + t′)

= λ~wλt′. Gi((x1ct1 ; (z1ct2 ; w1)), ..., (xnct1 ; (znct2 ; wn)))(t1 + t2 + t′)

= λ~wλt′. Gk((z1ct2 ; w1), ..., (znct2 ; wn))(t1 + t2 + t′),

where Gk = Res(Gi, ~x, t1) and more, Gj = Res(Gk, ~z, t2). Therefore thereexists k such that

state ~G(~x)(t1)i = k

state ~G(~z)(t2)k = j,

i.e.,

state ~G((x1ct1 ; z1), ..., (xnct1 ; zn))(t1 + t2)

=(state ~G

(~z)(t2)) ◦ (state ~G(~x)(t1)).

4. By definition 5.2.7 we verify that

out~G(~x(t), state ~G

(~x)(t)0) = out ~G(~x(t), i),

where i is such that Gi = Res(G0, ~x, t). Therefore

out~G(~x(t), i) = Gi(Constx1(t), ..., Constxn(t))

= G0((x1ct; Constx1(t)), ..., (xnct; Constxn(t))).

Since G0 is retrospective, we have

out~G(~x(t), state ~G

(~x)(t)0) = G0(~x)(t). �

The above proposition implies the following one which proof in [Rab97] canbe generalized straightforward with respect to n-ary operators on signals.

Proposition 5.2.9 ([Rab97]) state ~Gis a finite memory strong retrospective

operator on signals. Moreover, there exists

δ~G: Σ2n −→ ({0, ..., k} → {0, ..., k})

such that:

1. δ~G(a1, ..., an, b1, ..., bn) = state ~G

(Jumpa1→b1 , ..., Jumpan→bn)t,

for every t > 0;

2. δ~G(a1, ..., an, b1, ..., bn) = δ~G

(b1, ..., bn, b1, ..., bn) ◦ δ~G(a1, ..., an, b1, ..., bn),

for any a1, ..., an, b1, ..., bn ∈ Σ.

49

We will now study the same concepts over non-Zeno signals. Let G0 be afinite memory retrospective operator from non-Zeno signals over Σ to non-Zenosignals over Σ′. As we did before, let ~G = < G0, ..., Gk > be the sequence of allits residuals. Thus, defining state ~G

like in definition 5.2.7, we state the followingtheorem which proof may be found in [Rab97] and generalized straightforwardtoo.

Theorem 5.2.10 ([Rab97]) The operator state ~Gmaps non-Zeno signals to

non-Zeno signals. Moreover, propositions 5.2.8 and 5.2.9 hold whenever allnotions are relativized to non-Zeno signals. In particular, there exist

δ~G: Σ2n −→ ({0, ..., k} → {0, ..., k})

and

out~G: Σn × {0, .., k} −→ Σ′

such that:

1. δ~G(a1, ..., an, b1, ..., bn) = δ~G

(b1, ..., bn, b1, ..., bn) ◦ δ~G(a1, ..., an, b1, ..., bn);


(~x)(t)0);

3. state ~Gis a strong retrospective operator from non-Zeno signals over Σ to

non-Zeno signals over ({0, ..., k} → {0, ..., k}).

4. state ~G(~x)(0) = id, the identity permutation.;

5. state ~G(Jumpa1→b1 , ..., Jumpan→bn

)t = δ~G(a1, ..., an, b1, ..., bn);

6. state ~G((Jumpa1→b1ct′ ; x1), ..., (Jumpan→bn

ct′ ; x1))(t + t′) == (state ~G

(~x)t) ◦ δ ~G(a1, ..., an, b1, ..., bn).

Motivated by the above theorem we get the following definition.

Definition 5.2.11 (Definability [Rab97]) Let Σ and Q be finite sets and letδ : Σ2n −→ (Q → Q). An operator F from non-Zeno signals over Σ to non-Zenosignals over Q → Q is definable by δ if it satisfies the following conditions:

1. F is a strong retrospective operator;

2. F (~x)(0) = idQ;

3. F (Jumpa1→b1 , ..., Jumpan→bn)(t) = δ(a1, ..., an, b1, ..., bn), for every t > 0

and a1, ..., an, b1, ..., bn ∈ Σ;

4. F ((Jumpa1→b1ct′ ; x1), ..., (Jumpan→bnct′ ; xn))(t + t′) =

= (F (~x)t) ◦ δ ~G(a1, ..., an, b1, ..., bn).

We conclude this chapter with two propositions which will be useful onnext chapter. Both them have been proved in [Rab97] and their proofs can bestraightforward generalized.

Proposition 5.2.12 ([Rab97]) Let δ be a function in Σ2n −→ (Q → Q).Then there exists at most one operator definable by δ.

50

Proposition 5.2.13 ([Rab97]) If for every a1, ..., an, b1, ..., bn ∈ Σ:

δ~G(a1, ..., an, b1, ..., bn) =

= δ~G(b1, ..., bn, b1, ..., bn) ◦ δ~G

(a1, ..., an, b1, ..., bn),

then there exists a finite memory speed independent operator definable by δ.

The above propositions imply the following corollary.

Corollary 5.2.14 ([Rab97]) If for every a1, ..., an, b1, ..., bn ∈ Σ:

δ~G(a1, ..., an, b1, ..., bn) =

= δ~G(b1, ..., bn, b1, ..., bn) ◦ δ~G

(a1, ..., an, b1, ..., bn),

then there exists a unique operator definable by δ. Moreover, the operator de-finable by δ is finite memory strong retrospective and speed independent.

51


5.1.1 The following sets of operators on signals are closed under taking residual:






5.1.2 The following sets of operators from signals to signals are closed undercomposition:






5.1.3 Let C be a set of signals which is closed under concatenation, suffix andthe order preserving bijections. The following sets of operators over Csignals are closed under composition and under residual:






5.2.1 A general signal x is finite memory if and only if x is constant on thepositive reals.

5.2.3 If F : Sig(Σ)n −→ Sig(Σ) is a finite memory retrospective operator thenF (Jumpa1→b1 , ..., Jumpan→bn

) = Jumpc→d for some c and d in Σ.

5.2.4 Every finite memory retrospective operator is stable.

5.2.5 A finite memory retrospective operator maps non-Zeno signals to non-Zeno signals.

5.2.6 If F is a finite memory retrospective operator on signals then Rest(F ) is anoperator from non-Zeno signals to non-Zeno signals. Moreover, Rest(F )is a finite memory retrospective operator over the non-Zeno signals.

5.2.8 1. state ~G(~x)(0) = id, the identity permutation.

2. state ~Gis a strong retrospective operator.

3. state ~G((x1ct1 ; z1), ..., (xnct1 ; zn))(t1 + t2) =

= (state ~G(~z)(t2)) ◦ (state ~G

(~x)(t1))


(~x)(t)0)

52

5.2.9 state ~Gis a finite memory strong retrospective operator on signals. More-

over, there exists

δ~G: Σ2n −→ ({0, ..., k} → {0, ..., k})

such that:

1. δ~G(a1, ..., an, b1, ..., bn) = state ~G

(Jumpa1→b1 , ..., Jumpan→bn)t,

for every t > 0;

2. δ~G(a1, ..., an, b1, ..., bn) =

= δ~G(b1, ..., bn, b1, ..., bn) ◦ δ~G

(a1, ..., an, b1, ..., bn),for any a1, ..., an, b1, ..., bn ∈ Σ.

5.2.10 The operator state ~Gmaps non-Zeno signals to non-Zeno signals. More-

over, propositions 5.2.8 and 5.2.9 hold whenever all notions are relativizedto non-Zeno signals. In particular, there exist

δ~G: Σ2n −→ ({0, ..., k} → {0, ..., k})

and

out~G: Σn × {0, .., k} −→ Σ′

such that:

1. δ~G(a1, ..., an, b1, ..., bn) =

= δ~G(b1, ..., bn, b1, ..., bn) ◦ δ~G

(a1, ..., an, b1, ..., bn);


(~x)(t)0);

3. state ~Gis a strong retrospective operator from non-Zeno signals over

Σ to non-Zeno signals over ({0, ..., k} → {0, ..., k}).4. state ~G

(~x)(0) = id, the identity permutation.;

5. state ~G(Jumpa1→b1 , ..., Jumpan→bn

)t = δ~G(a1, ..., an, b1, ..., bn);

6. state ~G((Jumpa1→b1ct′ ; x1), ..., (Jumpan→bn

ct′ ; xn))(t + t′) == (state ~G

(~x)t) ◦ δ ~G(a1, ..., an, b1, ..., bn).

5.2.13 If for every a1, ..., an, b1, ..., bn ∈ Σ:

δ~G(a1, ..., an, b1, ..., bn) =

= δ~G(b1, ..., bn, b1, ..., bn) ◦ δ~G

(a1, ..., an, b1, ..., bn),

then there exists a finite memory speed independent operator definable byδ.

5.2.14 If for every a1, ..., an, b1, ..., bn ∈ Σ:

δ~G(a1, ..., an, b1, ..., bn) =

= δ~G(b1, ..., bn, b1, ..., bn) ◦ δ~G

(a1, ..., an, b1, ..., bn),

then there exists a unique operator definable by δ. Moreover, the operatordefinable by δ is finite memory strong retrospective and speed indepen-dent.

53

Chapter 6

Continuous-time Automata

The finite state transducer will be defined in this chapter and for that theresults studied in the last chapter will be very helpful. Our main objective is toshow that every finite state transducer computes a finite memory retrospectiveoperator over non-Zeno signals and that the inverse also holds.

In the first section we give also a result found in [Rab97] that implies theimpossibility of have a finite description of finite memory retrospective opera-tors over general signals. We conclude this chapter by given some examples oftransducers.

6.1 Finite State Transducers over

Non-Zeno Signals

As we have done in the last chapter with respect to the operators over signals,we will consider transducers with multiple input channels. We define now afinite state transducer.

Definition 6.1.1 (Finite State Transducer [Rab97]) A finite state trans-ducer A over non-Zeno signals is a tuple < Q, q0, Σin, Σout, n, out, δ > such that

• Q is a finite non empty set of states,

• q0 ∈ Q is the initial state,

• Σin1 is the input alphabet,

• Σout is the output alphabet,

• n is the arity of A,

• out : Q × Σnin −→ Σout is the output function and

• δ : Σ2nin −→ (Q → Q) is the transition function which verifies

δ(b1, ..., bn, b1, ..., bn) ◦ δ(a1, ..., an, b1, ..., bn) = δ(a1, ..., an, b1, ..., bn).

1We assume that all input signals are defined over the same alphabet, however multipleinput alphabets may be considered without lost of generality.

54

Note that this definition translates the ideas introduced in the last chapter.Each one of the states corresponds to a residual and each transition correspondsto a jump from a1, ..., an to b1, ..., bn. The condition for δ comes straightfor-wardly from the results obtained before.

We will use a graphical representation for transducers. As we may see infigure 6.1, the states will be represented by nodes and the transitions we will berepresented by labeled arcs. The output function can be represented by labelingthe nodes as follows:

q has label < (c11, ..., cn1)/d1, ..., (c1k , ..., cnk)/dk >if and only if

out(q, c1i, ..., cni) = di for every i.

The initial state will be indicated with an arrow as is done in figure 6.1.A transducer receives a signal ~x ∈ nZSig(Σin) as input and produces as

output a signal y ∈ nZSig(Σout). If ~x(t) = (a1, ..., an), for t ∈ T+, theny(t) = b where b is given accordingly with the label of the current node, i.e.,accordingly with the out function in each state. A transition between statesoccurs accordingly with the jump of ~x at t, if ~x produces a jump from (a1, ..., an)to (b1, ..., bn) at t, then occurs a transition from the current state to other one,where the transition arc is labeled by < (a1, ..., an), (b1, ..., bn) >.

The operator computable by a transducer is obtained as follows.

Definition 6.1.2 (Operator Computable by a Transducer [Rab97]) .Let A =< Q, q0, Σin, Σout, n, out, δ > be a transducer. Note that by proposi-tion 5.2.13 there exists a unique operator F δ definable by δ. The operator FA

computable by A is defined as out(F δ~xtq0, ~xt).

Making use of the results achieved in the last chapter, we will give now acharacterization of finite memory retrospective operators and we will state thatthey are speed independent. The following two theorems have been stated andpartially proved in [Rab97], complete proofs will be provided by us.

Theorem 6.1.3 An operator over non-Zeno signals is a finite memory retro-spective operator if and only if it is computable by a transducer.

Proof: Let A =< Q, q0, Σin, Σout, n, out, δ > be a transducer. As we haveseen in previous chapter, namely in corollary 5.2.14 , F δ is a finite memoryretrospective operator. Therefore, the operator FA is a finite memory becauseit is defined as the composition of the pointwise operator out2 and F δ. �

Theorem 6.1.4 Every finite memory retrospective operator over non-Zeno sig-nals is speed independent.

Proof: By corollary 5.2.14, the operator F δ is speed independent and thenthe operator computable by a transducer is also speed independent. Thereforethe theorem 6.1.3 implies that every finite memory retrospective operator overnon-Zeno signals is speed independent. �

In [Rab97] is provided a description of the operator computable by a trans-ducer in terms of ω-languages and proved that the set of finite memory speed

2Clearly a pointwise operator is retrospective because it only depends on current instantand it is finite memory because it only have one residual, itself (chapter 4).

55

independent retrospective operators can not be representable by finite means.We will not prove this result, we only state a further result that implies it,proved in [Rab97].

Theorem 6.1.5 ([Rab97]) The set of finite memory speed independent retro-spective operator (over general signals) is at least uncountable.

6.2 Examples

In this section we will provide three examples of transducers for operators thatwe already give as example or that we will use in next chapter. In the construc-tion of such transducers, we will make use of the properties shown in chapter 4for each case.

Example 6.2.1 The unary operator

LLim0 : nZSig({0, 1}) −→ nZSig({0, 1})

is defined as follows:

LLim0(x)(t) =

{

0 if t = 0b if t > 0 and b ∈ {0, 1} is the left limit of x at t.

As we will see in chapter 7, this operator have two residuals:

LLim0(z)(τ) =

{

0 if τ = 0b if τ > 0 and b ∈ {0, 1} is the left limit of z at τ ,

LLim1(z)(τ) =

{

1 if τ = 0b if τ > 0 and b ∈ {0, 1} is the left limit of z at τ .

Thus, we construct the transducer ALLim0=< Q, q0, Σin, Σout, n, out, δ >.

• Since there are two residuals, we need two states, i.e, Q = {q0, q1}. Eachstate is associated to a residual as follows:

q0 99K LLim0 and q1 99K LLim1.

• The initial state is the state associated to the operator LLim0, i.e., q0.

• Σin = {0, 1}.

• Σout = {0, 1}.

• Since LLim0 is an unary operator, we have n = 1.

• We obtain the function out as follows:

out(q0, 0) = LLim0(Const0)(0) = 0out(q0, 1) = LLim0(Const1)(0) = 0out(q1, 0) = LLim1(Const0)(0) = 1out(q1, 1) = LLim1(Const1)(0) = 1

• Since Σin has two symbols, we know that there exist four different jumps:

56

State Transition δ outq0 0, 0 q0 0 7→ 0q0 0, 1 q1 0 7→ 0q0 1, 0 q0 1 7→ 0q0 1, 1 q1 1 7→ 0q1 0, 0 q0 0 7→ 1q1 0, 1 q1 0 7→ 1q1 1, 0 q0 1 7→ 1q1 1, 1 q1 1 7→ 1

Table 6.1: δ and out functions in ALLim0.

Jump0→0, Jump0→1, Jump1→0 and Jump1→1,

i.e., there exist four possible transitions at each state. Therefore, thefunction δ is defined as follows:

δ(0, 0)q0 = q0 because Res(LLim0, Jump0→0, t) = LLim0,δ(0, 0)q1 = q0 because Res(LLim1, Jump0→0, t) = LLim0,δ(0, 1)q0 = q1 because Res(LLim0, Jump0→1, t) = LLim1,δ(0, 1)q1 = q1 because Res(LLim1, Jump0→1, t) = LLim1,δ(1, 0)q0 = q0 because Res(LLim0, Jump1→0, t) = LLim0,δ(1, 0)q1 = q0 because Res(LLim1, Jump1→0, t) = LLim0,δ(1, 1)q0 = q1 because Res(LLim0, Jump1→1, t) = LLim1,δ(1, 1)q1 = q1 because Res(LLim1, Jump1→1, t) = LLim1,

for t > 0.

In figure 6.1 we give the transducer ALLim0.

<0,1><1,1>

<0,0><1,0>

<1,1><0,1>

<0,0><1,0>

<0/0,1/0>q0

<0/1,1/1>q1

Figure 6.1: Transducer ALLim0.

57

LeftCont : nZSig({0, 1}) −→ nZSig({True, False})


LeftCont(x)(t) =

{


As we have seen in chapter 4 and since we are dealing with non-Zeno signals,LeftCont has three residuals:

LeftCont(z)(τ) =

{

True if z is continuous at τFalse otherwise.

G0(z)(τ) =

True if τ = 0 and z(0) = 0False if τ = 0 and z(0) 6= 0LeftCont(z)(τ) if τ > 0

G1(z)(τ) =

True if τ = 0 and z(0) = 1False if τ = 0 and z(0) 6= 1LeftCont(z)(τ) if τ > 0

Thus, we construct the transducer ALeftCont =< Q, q0, Σin, Σout, n, out, δ >.

• Since there are three residuals, we need three states, i.e, Q = {q0, q1, q2}.Each state is associated to a residual as follows:

q0 99K LeftCont, q1 99K G0 and q2 99K G1.

• The initial state is the state associated to the operator LeftCont, i.e., q0.

• Σin = {0, 1}.

• Σout = {True, False}.

• Since LeftCont is an unary operator, we have n = 1.

State Transition δ outq0 0, 0 q1 0 7→ Falseq0 0, 1 q2 0 7→ Falseq0 1, 0 q1 1 7→ Falseq0 1, 1 q2 1 7→ Falseq1 0, 0 q1 0 7→ Trueq1 0, 1 q2 0 7→ Trueq1 1, 0 q1 1 7→ Falseq1 1, 1 q2 1 7→ Falseq2 0, 0 q1 0 7→ Falseq2 0, 1 q2 0 7→ Falseq2 1, 0 q1 1 7→ Trueq2 1, 1 q2 1 7→ True

Table 6.2: δ and out functions in ALeftCont.

58

out(q0, 0) = LeftCont(Const0)(0) = Falseout(q0, 1) = LeftCont(Const1)(0) = False

out(q1, 0) = G0(Const0)(0) = Trueout(q1, 1) = G0(Const1)(0) = Falseout(q2, 0) = G1(Const0)(0) = Falseout(q2, 1) = G1(Const1)(0) = True

<0,1><1,1>

<0,0><1,0>

<1,1><0,1>

<0,0><1,0>

<0/T,1/F>q1

<0/F,1/T>q2

<0/F,1/F>q0

<0,0><1,0>

<0,1><1,1>

Figure 6.2: Transducer ALeftCont.




δ(0, 0)q0 = q1 because Res(LeftCont, Jump0→0, t) = G0,δ(0, 0)q1 = q1 because Res(G0, Jump0→0, t) = G0,δ(0, 0)q2 = q1 because Res(G1, Jump0→0, t) = G0,




59

for t > 0.

In figure 6.2 we give the transducer ALeftCont.


LJV 11,0 : nZSig({0, 1})n −→ nZSig({0, 1})


LJV 11,0(x)(t) =

0 if t = 0x(τ) if t > 0 and ∃ τ, x is constant in ]τ, t[

and x is not continuous at τ.

As we will see in chapter 7, this operator have four residuals:

F 11,0,0(z)(τ) = LJV 1

1,0((Jump0→0c1; z))(τ + 1),

F 11,0,1(z)(τ) = LJV 1

1,0((Jump0→1c1; z))(τ + 1),

F 11,1,0(z)(τ) = LJV 1

1,1((Jump1→0c1; z))(τ + 1),

F 11,1,1(z)(τ) = LJV 1

1,1((Jump1→1c1; z))(τ + 1).

Thus, we construct the transducer ALJV 11,0

=< Q, q0, Σin, Σout, n, out, δ >.

• Since there are four residuals, we need four states, i.e, Q = {q0, q1, q2, q3}.Each state is associated to a residual as follows:

q0 99K F 11,0,0, q1 99K F 1

1,0,1, q2 99K F 11,1,0 and q3 99K F 1

1,1,1.

• The initial state is the state associated to the operator LJV 11,0 (= F 1

1,0,0),i.e., q0.

State Transition δ outq0 0, 0 q0 0 7→ 0q0 0, 1 q1 0 7→ 0q0 1, 0 q2 1 7→ 0q0 1, 1 q3 1 7→ 0q1 0, 0 q0 0 7→ 0q1 0, 1 q1 0 7→ 0q1 1, 0 q2 1 7→ 0q1 1, 1 q1 1 7→ 0q2 0, 0 q2 1 7→ 1q2 0, 1 q1 1 7→ 1q2 1, 0 q2 0 7→ 1q2 1, 1 q3 0 7→ 1q3 0, 0 q0 1 7→ 1q3 0, 1 q1 1 7→ 1q3 1, 0 q2 0 7→ 1q3 1, 1 q3 0 7→ 1

Table 6.3: δ and out functions in ALJV 11,0

.

60

• Σin = {0, 1}.

• Σout = {0, 1}.

• Since LJV 11,0 is an unary operator, we have n = 1.


out(q0, 0) = F 11,0,0(Const0)(0) = 0

out(q0, 1) = F 11,0,0(Const1)(0) = 0

out(q1, 0) = F 11,0,1(Const0)(0) = 0

out(q1, 1) = F 11,0,1(Const1)(0) = 0

out(q2, 0) = F 11,1,0(Const0)(0) = 1

out(q2, 1) = F 11,1,0(Const1)(0) = 1

out(q3, 0) = F 11,1,1(Const0)(0) = 1

out(q3, 1) = F 11,1,1(Const1)(0) = 1

<0,0>

<1,0>

<0,0>

<1,0>

<0/0,1/0>q0

<0/1,1/1>q2

<0,1><1,1> <1,1>

<0,0><1,0>

<0,1>

<1,1>

<0,1>

<1,0> <1,1><0,0> <0,1>

<0/0,1/0>q1

<0/1,1/1>q3

Figure 6.3: Transducer ALJV 11,0

.




61

δ(0, 0)q0 = q0 because Res(F 11,0,0, Jump0→0, t) = F 1

1,0,0,δ(0, 0)q1 = q0 because Res(F 1

1,0,1, Jump0→0, t) = F 11,0,0,


1,1,0,δ(0, 0)q3 = q0 because Res(F 1

1,1,1, Jump0→0, t) = F 11,0,0,


1,0,1,δ(0, 1)q1 = q1 because Res(F 1

1,0,1, Jump0→1, t) = F 11,0,1,


1,0,1,δ(0, 1)q3 = q1 because Res(F 1

1,1,1, Jump0→1, t) = F 11,0,1,


1,1,0,δ(1, 0)q1 = q2 because Res(F 1

1,0,1, Jump1→0, t) = F 11,1,0,


1,1,0,δ(1, 0)q3 = q2 because Res(F 1

1,1,1, Jump1→0, t) = F 11,1,0,


1,1,1,δ(1, 1)q1 = q1 because Res(F 1

1,0,1, Jump1→1, t) = F 11,0,1,


1,1,1,δ(1, 1)q3 = q3 because Res(F 1

1,1,1, Jump1→1, t) = F 11,1,1,

for t > 0.

In figure 6.3 we give the transducer ALJV 11,0

.

62


6.1.3 An operator over non-Zeno signals is a finite memory retrospective oper-ator if and only if it is computable by a transducer.

6.1.4 Every finite memory retrospective operator over non-Zeno signals is speedindependent.

6.1.5 The set of finite memory speed independent retrospective operator (overgeneral signals) is at least uncountable.

63

Chapter 7

Circuits of operators

In this chapter we will study the possibility of characterizing the class of finitememory retrospective operators over non-Zeno signals by a function algebra[Clo99]. Circuits of finite memory retrospective operators will be consideredand our main results will concern the construction of such circuits from a finitenumber of primitives.

The concept of function algebra is introduced1 in [Clo99] and is clarified inthe following definition.

Definition 7.0.1 (Function Algebra [Clo99]) An operation maps functionsinto functions. If X is a set of functions and OP is a set of operations, then[X ; OP ] denotes the smallest set of functions containing X and is closed underthe operations of OP . The set [X ; OP ] is called a function algebra.

The set of functions that we pretend to characterize is the set of finite mem-ory retrospective operators over non-Zeno signals, OFMR. In the next sectionwe will specify the set X of primitive operators and verify that in fact X is a setof finite memory retrospective operators. The set of operations will be given inthe second section and some properties involving them will be stated. The mainresult will arise in the third section, namely will be proved that the functionalgebra specified coincide precisely with the class OFMR.

7.1 Primitives

Intuitively we may consider some operators that seem to be most relevant orbasic. Given a finite memory retrospective operator F over non-Zeno signals,its value at an instant t depends on the past values of the signal ~x given asargument. Thus, in order to compute the value of F at t, it will be useful toknow the behavior of ~x before t, namely the last jump of ~x before t and the leftlimit of ~x at t.

Following these intuitions, we define now two primitive operators which areneeded to get the main characterization results.

1However introduced before, the concept of function algebra was used for the first time tocharacterize a complexity classes in [Clo99].

64

Definition 7.1.1 (Left Limit LLima) Let Σ be a finite set and a ∈ Σ. Wedefine the unary operator

LLima : nZSig(Σ) −→ nZSig(Σ)

as follows:

LLima(x)(t) =

{

a if t = 0b if t > 0 and b is the left limit of x at t.

Definition 7.1.2 (Last Jump Value LJV nk,a) Let Σ be a finite set, a ∈ Σ

and n, k ∈ N such that k ≤ n. We define the n-ary operator

LJV nk,a : nZSig(Σ)n −→ nZSig(Σ)

as follows:

LJV nk,a(~x)(t) =

a if t = 0xk(τ) if t > 0 and τ < t such that ~x is constant in ]τ, t[

and ~x is not continuous at τ .

Since the left limit is undefined at the instant 0 and there is no jumpsbefore that, we have introduced default values at t = 0 in the above definitions.Therefore, we grant that LLima and LJV n

k,a are well defined at t = 0.

Example 7.1.3 Some applications of the above operators:

LLima(Constbc1; Jumpc→d)(0) = a

LLima(Constbc1; Jumpc→d)(1) = b

LLima(Constbc1; Jumpc→d)(2) = d

LJV 21,a((Constbc1; Jumpc→d), (Jumpb→ec2; Jumpc→d))(0) = a

LJV 22,a((Constbc1; Jumpc→d), (Jumpb→ec2; Jumpc→d))(0) = a

LJV 21,a((Constbc1; Jumpc→d), (Jumpb→ec2; Jumpc→d))(2) = c

LJV 21,a((Constbc1; Jumpc→d), (Jumpb→ec2; Jumpc→d))(3) = c

LJV 22,a((Constbc1; Jumpc→d), (Jumpb→ec2; Jumpc→d))(3) = b

These operators are evidently strong retrospective operators. We shall seethat they are finite memory operators.

Proposition 7.1.4 LLima is a finite memory operator.

Proof: Let x ∈ Sig(Σ) and t ∈ T +, by definition 2.1.16 we know that theresidual G of LLima with respect to x and t is defined as follows:

G(z)(t′) = LLima(xct; z)(t + t′)

= LLimb(z)(t′)

where b is the left limit of x at t whenever t > 0, otherwise b = a. Thus, the setof residuals of LLima is given by

{LLima : a ∈ Σ}.

65

Since Σ is a finite set, the set of residuals is finite and LLima is a finite memoryoperator. �

Proposition 7.1.5 LJV nk,a is a finite memory operator.

Proof: Let ~x ∈ Sig(Σ)n and t ∈ T+, by definition 2.1.16 we know that theresidual G of LJV n

k,a with respect to ~x and t is given by

G(~z)(t′) = LJV nk,a((x1ct; z1), ..., (xnct; zn))(t + t′)

= F nk,b,ξ(~z)(t′)

where b = LJV nk,a(~x)(t), ξ =< c1, ..., cn >=< LLima(xi)(t) : i = 1, ..., n > and

F nk,b,ξ is defined as follows:

F nk,b,ξ(~z)(t′) = LJV n

k,b((Jumpb→c1c1; z1), ..., (Jumpb→cn

c1; zn))(t′ + 1)

Therefore the set of residuals of LJV nk,a is given by

{F nk,b,ξ : b ∈ Σ and ξ ∈ Σn}.

Since Σ is a finite set, the set of residuals is finite and LJV nk,a is a finite memory

operator. �

Definition 7.1.6 (Pointwise Extension) If f : Σn −→ Σ′ is a functionwhere Σ and Σ′ are both finite non-empty sets, then we define the operatorPf : nZSig(Σ)n −→ nZSig(Σ′) as the pointwise extension of f as follows:

Pf (~x)(t) = f(~x(t))

Since the value of these operators only depends on the current instant, wesee that these operators are retrospective and that have finite memory.

Definition 7.1.7 (Set X [Σ1, ..., Σm] of Primitives) Given Σ1, ..., Σm finitenon-empty sets, the set X [Σ1, ..., Σm] of primitives is defined as follows:

X [Σ1, ..., Σm] = {LLima : a ∈ Σi, i = 1, ..., m}∪ {LJV n

k,a : a ∈ Σi, n, k ∈ N s.t. k ≤ n, i = 1, ..., m}∪ {Pf : n ∈ N, f : Σn

i −→ Σ′j , i, j = 1, ..., m}

7.2 Operations

We will discuss now which operators are needed in the set OP referred in defi-nition 7.0.1.

In definition 5.1.2, we stated that the set of finite memory retrospective oper-ators is closed under composition and naturally we will consider the composition◦ as one of the available operations.

Definition 7.2.1 (Composition) Let G : nZSig(Σ′)m −→ nZSig(Σ′′) bea m-ary finite memory retrospective operator and let Fi : nZSig(Σ)n −→nZSig(Σ′) be also a n-ary finite memory operator, for i = 1, ..., m. We de-fine the composition of G with F1, ..., Fm,

66

◦(G, F1, .., Fm) : nZSig(Σ)n −→ nZSig(Σ′′),

as follows:

◦(G, F1, ..., Fm)(~x) = G(F1(~x), ..., Fm(~x)).

The state function studied in section 5.2.2 give us some ideas about anotheroperation. As we have seen, sometimes we need to know previous values ofoperator to get its current value, so we define next the operation Rec.

Definition 7.2.2 (Rec) Let G : Sig(Σ)n+1 −→ Sig(Σ) be a finite memoryretrospective operator that verify the following conditions:

• is strong retrospective with respect to the last argument;

• given x1, ..., xn, y, z ∈ Sig(Σ), t > 0 and t′ ∈ [0, t[ such that xi is constantin ]t′, t[ for all i and xj is not continuous at t′ for some j, if y(τ) = z(τ)for τ ∈ [0, t′[ then, G(x1, ..., xn, y)(t) = G(x1, ..., xn, z)(t).

For an arbitrary a ∈ Σ, the operation Rec is defined as follows:

Rec(G)~xt =

{

G(x1, ..., xn, consta)t if t = 0

G(x1, ..., xn, (Rec(G)(~x)ct′ ; consta))t if t > 0

and clearly Rec(G) : Sig(Σ)n −→ Sig(Σ).

We will prove that the operators obtained with Rec, are well defined andare finite memory retrospective operators.

Given the operator H~x : Sig(Σ) −→ Sig(Σ) such that H~x(y) = G(~x, y) for~x ∈ Sig(Σ) and making use of the ideas in chapter 3, we will study the fixpointsof H~x as fixpoints of a function over ω-strings. We observe that Rec(G)(~x) is afixpoint of H~x.

Proposition 7.2.3 Let G be in the conditions of definition 7.2.2 and let ~x ∈Sig(Σ)n. The operator H~x, such that H~x(y) = G(~x, y) for y ∈ Sig(Σ), ischaracterized by a function K~x : Σω × Σω −→ Σω × Σω with the property:

if αi = βi and α′i = β′

i, for i < n, then K(α, α′)i = K(β, β′)i, for i ≤ n.

Proof: If G is a finite memory operator, then G is speed independent as wehave seen in proposition 6.1.4. Thus H~x is a finite memory operator and byproposition 3.2.3 there exists a function K~x : Σω × Σω −→ Σω × Σω thatcharacterizes H~x and verifies the generalized SI condition in definition 3.2.1.

In order to prove the above property let α, α′, β, β′ ∈ Σω such that αi = βi

and α′i = β′

i for i < n. Suppose also that y is the non-Zeno signal characterizedby α, α′, τ and z is the non-Zeno signal characterized by β, β′, τ where the timescale τ = < ti : i ∈ N0 > satisfies:

if t ≤ t′, then t ∈ τ iff ∃ i, xi is not constant at t,

for t′ ∈ T+ such that ~x is not constant at t′ and such that ~x is constant at τ forτ > t′ 2. Thus we verify that y(t) = z(t) for t ∈ [0, tn−1[ and then we concludeby the given properties of G in definition 7.2.2 that

2Clearly, such t′ may not exist. This condition express the minimality of the time scale τ

needed to characterize the non-Zeno signal ~x.

67

H~x(y)(t) = H~x(z)(t)

with t ∈ [0, tn[. Therefore K~x(α, α′)i = K~x(β, β′)i for i ≤ n because K~x is acharacterization of H~x(y). �

In appendix A we provide a few notes about partial orders that will be usefulto prove the following results. Before that and since we will work with prefixes,we recall the concept of prefix.

Definition 7.2.4 Let α ∈ Σω be a ω-string, we define the set of its prefixes as:

Pα = {u ∈ Σ∗ : ∃ β ∈ Σω, α = uβ}

Definition 7.2.5 (Set Closed for Prefixes) A set X ⊆ Σ∗ is closed underprefixes whenever it satisfies the following condition:

∀ u ∈ X, ∀ v ∈ Σ∗, if u = sv, then s ∈ X .

Let X ⊆ Σ∗ be a set closed under prefixes such that there exists u ∈ X withlength n, for all n ∈ N. Thus, it is clear that there exists at least one α ∈ Σω

such that Pα ⊆ X . Moreover, if for all u ∈ X there exists only one a ∈ Σ suchthat ua ∈ Xv, then X = Pα for some α ∈ Σω and we say that X characterizesunivocally3 α.

Proposition 7.2.6 Let P = {X ⊆ Σ∗ : X is closed for prefixes}, then is a complete lattice with ≤ defined as:

(X, X ′) ≤ (Y, Y ′) iff X ⊆ Y and X ′ ⊆ Y ′.

Proof: Let (X, X ′), (Y, Y ′), (Z, Z ′) ∈ P × P . Clearly:

- (X, X ′) ≤ (X, X ′);

- if (X, X ′) ≤ (Y, Y ′) and (Y, Y ′) ≤ (X, X ′), then (X, X ′) = (Y, Y ′);

- if (X, X ′) ≤ (Y, Y ′) and (Y, Y ′) ≤ (Z, Z ′), then (X, X ′) ≤ (Z, Z ′).

Therefore is a partial order.Let S ⊆ P×P and (Y, Y ′) ∈ P×P such that, for all (X, X ′) ∈ S, (X, X ′) ≤

(Y, Y ′). Then ∨S = (⋃

(X,X′)∈ SX,

⋃

(X,X′)∈ SX ′) ≤ (Y, Y ′), i.e, ∨S is the

least upper bound of S.Suppose now that (Z, Z ′) ∈ P×P is such that, for all (X, X ′) ∈ S, (Z, Z ′) ≤

(X, X ′). Then (Z, Z ′) ≤ ∧S = (⋂

(X,X′)∈ SX,

⋂

(X,X′)∈ SX ′), i.e, ∧S is the

greatest lower bound of S.Clearly ∨S ∈ P ×P and ∧S ∈ P ×P , therefore is a complete

lattice. �

Through this section we will work with the complete lattice .Given a subset S ⊆ P ×P , the least upper bound is easily given by

∨S = (⋃

(X,X′)∈ SX,

⋃

(X,X′)∈ SX ′)

as we already see in proof of proposition 7.2.6 and the greatest lower bound isobtained simply by

3When we do not consider sets closed under prefixes, a ω-string α may be characterizedby more sets than Pα, namely by any infinite subset of Pα.

68

∧S = (⋂

(X,X′)∈ SX,

⋂

(X,X′)∈ SX ′).

Thus, in , we have ⊥ = ({ε}, {ε}).

Definition 7.2.7 Let K~x : Σω × Σω −→ Σω × Σω be the function obtainedin proposition 7.2.3. Given a1a2...an, a′

1a′2...a

′n ∈ Σ∗, for n ∈ N, we define the

function K ′~x as follows:

K ′~x(a1a2...an, a′

1a′2...a

′n) = (b1b2...bn+1, b

′1b

′2...b

′n+1)

iff∀ α, α′ ∈ Σω, ∃ β, β′ ∈ Σω,

K~x(a1a2...anα, a′1a

′2...a

′nα′) = (b1b2...bn+1β, b′1b

′2...b

′n+1β

′).

Let (X, X ′) ∈ P × P , we extend K ′~x to P × P as follows:

K ′~x(X, X ′) = {K ′

~x(a1...an, a′1...a

′n) : a1...an ∈ X, a′

1...a′n ∈ X ′} ∪ {(ε, ε)}.

Proposition 7.2.8 K ′~x is well defined.

Proof: Let a1a2...an, a′1a

′2...a

′n ∈ Σ∗, for n ∈ N. Suppose that

K ′~x(a1a2...an, a′

1a′2...a

′n) = (b1b2...bn+1, b

′1b

′2...b

′n+1),

by definition 7.2.7, we have

∀ α, α′ ∈ Σω, ∃ β, β′ ∈ Σω,K~x(a1a2...anα, a′

1a′2...a

′nα′) = (b1b2...bn+1β, b′1b

′2...b

′n+1β

′)

and by proposition 7.2.3 we conclude that (b1b2...bn+1, b′1b

′2...b

′n+1) is univocally

determined. Thus, the extension of K ′~x to P × P is well defined. �

Proposition 7.2.9 Let K ′~x be the extension obtained in definition 7.2.7. This

function is defined from P × P to P × P and is continuous.

Proof: Let (X, X ′) ∈ P × P . By the property of K ′~x in proposition 7.2.3, we

know that if a1a2...an ∈ X , a′1a

′2...a

′n ∈ X ′ and

K ′~x(a1a2...an, a′

1a′2...a

′n) = (b1b2...bnbn+1, b

′1b

′2...b

′nb′n+1),

then, for every k ≤ n,

K ′~x(a1a2...ak, a′

1a′2...a

′k) = (b1b2...bkbk+1, b

′1b

′2...b

′kb′k+1).

Thus, since {(ε, ε)} ∈ K ′~x(X, X ′) by definition 7.2.7, K ′

~x(X, X ′) is closed underprefixes, i.e., K ′

~x(X, X ′) ∈ P × P .Let S ⊆ P × P be a directed subset. If (X, X ′) ∈ S, then K ′

~x(X, X ′) ≤K(∨S) where ∨S = (

⋃

(X,X′)∈ SX,

⋃

(X,X′)∈ SX ′). Clearly

∨K ′~x(S) ≤ K(∨S).

Suppose that (b1b2...bnbn+1, b′1b

′2...b

′nb′n+1) ∈ ∨K ′

~x(S), for some n ∈ N0, thenthere exists (a1a2...an, a′

1a′2...a

′n) ∈ ∨S such that

K ′~x(a1a2...an, a′

1a′2...a

′n) = (b1b2...bnbn+1, b

′1b

′2...b

′nb′n+1).

Thus, there exist (X1, X′1), (X2, X

′2) ∈ S such that

69

a1a2...an ∈ X1 and a′1a

′2...a

′n ∈ X2.

Because S is directed, we know that there exists (Y, Y ′) ∈ S such that

(X1, X′1) ≤ (Y, Y ′) and (X2, X

′2) ≤ (Y, Y ′)

and clearly

(b1b2...bnbn+1, b′1b

′2...b

′nb′n+1) ∈ K ′

~x(Y, Y ′).

As (Y, Y ′) ≤ ∨S, we have that (b1b2...bnbn+1, b′1b

′2...b

′nb′n+1) ∈ K ′

~x(∨S), i.e.

K ′~x(∨S) ≤ ∨K(S).

Therefore K ′~x(∨S) = ∨K ′

~x(S), K ′~x is continuous. �

Since we have a continuous function, it is now possible find its fixpoint. Inthe following three propositions we will find that and verify that Rec(G) is welldefined as promised.

Proposition 7.2.10 If K ′~x is the extension obtained in definition 7.2.7, then

K ′~x has a least fixpoint (P, P ′) and

(P, P ′) =∨{K ′n

~x ({ε}, {ε}) : n ∈ N0}.

Proof: Directly from propositions 7.2.9 and A.0.8. �

Since for all u ∈ P there exists only one a ∈ Σ such that ua ∈ P , we concludethat P characterizes an ω-string ξ. Similarly, we conclude that P ′ characterizesan ω-string ξ′ and the following result holds.

Proposition 7.2.11 Let ξ, ξ′ be the ω-strings characterized by the sets P, P ′ ofproposition 7.2.10, ~x = (x1, ..., xn) ∈ Sig(Σ)n and τ be a time scale such that

if t ≤ t′, then t ∈ τ iff ∃ i, xi is not constant at t,

for t′ ∈ T+ such that ~x is not constant at t′ and such that ~x is constant at τ forτ > t′. Then the non-Zeno signal characterized by ξ, ξ ′, τ is a fixpoint of H~x.

Proof: Since ξ, ξ′ are characterized by P, P ′ respectively and P, P ′ is a fixpointof K ′

~x, we have that K~x(ξ, ξ′) = (ξ, ξ′). This function K ′~x is a characterization

of H~x as we have seen in proposition 7.2.3 and then, if ξ, ξ′, τ characterizes z,K ′

~x(ξ, ξ′), τ characterizes H~x(z). But K ′~x(ξ, ξ′) = (ξ, ξ′), therefore H~x(z) = z,

i.e., z is a fixpoint of H~x. �

Proposition 7.2.12 Rec(G) is well defined.

Proof: Given ~x ∈ Sig(Σ)n and if we look at the definition 7.2.2, we verifythat the operator Rec constructs the least fix point found in proposition 7.2.11.Therefore it is well defined since that there exists only one least fixpoint whenwe fix the scale τ as we did in proposition 7.2.11. �

About retrospectivity and memory finiteness of Rec(G) we leave the follow-ing result.

Proposition 7.2.13 Let G be in the conditions of definition 7.2.2, then Rec(G)is a finite memory retrospective operator.

70

Proof: Looking at definition 7.2.2 we can see that G is retrospective and there-fore Rec(G) is obviously retrospective. If F = Rec(G) and t = 0, then we getfrom definition 2.1.16:

Res(F, ~x, t)(~z)(τ)

= F (~x)(t + τ)

= G(x1, ..., xn, Consta)(τ).

If t > 0, then we have:

Res(F, ~x, t)(~z)(τ)

= F ((x1ct; z1), ..., (xnct; zn))(t + τ)

= G((x1ct; z1), ..., (xnct; zn), (F ((x1ct; z1), ..., (xnct; zn))ct′ ; Consta))(t + τ)

= G((x1ct; z1), ..., (xnct; zn), (((F~x)ct; Res(F, ~x, t)~z)ct′ ; Consta))(t + τ)

= G((x1ct; z1), ..., (xnct; zn), ((F~x)ct; ((Res(F, ~x, t)~z)ct′−t; Consta)))(t + τ)

= Res(G, (x1, ..., xn, F~x), t)(z1, ..., zn, (Res(F, ~x, t)~z)ct′−t; Consta))(τ),

where t′ < t + τ , i.e, t′ − t < τ .As Res(G, (x1, ..., xn, F~x), t) is a finite memory retrospective operator, by

proposition 7.2.12, the number of residuals of F and G is the same, i.e., F is afinite memory operator. �

We may now give the complete definition of [X [Σ1, ..., Σm], OP ].

Definition 7.2.14 Given Σ1, ..., Σm finite non-empty sets, the function algebra[X [Σ1, ..., Σm]; OP ] is defined as follows:

OP = {◦, Rec},X [Σ1, ..., Σm] = {LLima : a ∈ Σi, i = 1, ..., m}

∪ {LJV nk,a : a ∈ Σi, n, k ∈ N s.t. k ≤ n, i = 1, ..., m}

∪ {Pf : n ∈ N, f : Σni −→ Σ′

j , i, j = 1, ..., m}

The following important result holds.

Theorem 7.2.15 If F is in [X [Σ1, ..., Σm]; OP ], then F is a finite memoryretrospective operator.

Proof: Directly from the above propositions. �

7.3 Circuits

If we consider the set of circuits obtained from the function algebra defined inthe previous section, we know that all representable operators are finite memoryretrospective operators by theorem 7.2.15. Now we will prove that all of themcan be represented by these circuits.

Before that we note that we will use the operation Rec with ~x such that eachxi may be defined over different alphabets. If we look at the above section, weeasily verify that the results continue to be true, in fact they are independentof the alphabets.

71

Theorem 7.3.1 If G : Sig(Σ)n −→ Sig(Σ) is a finite memory retrospective op-erator, then there is a circuit obtained from the function algebra [X [Σ1, ..., Σm]; OP ]that represents G.

Proof: Given a finite memory retrospective operator G : Sig(Σ)n −→ Sig(Σ),let

{G0, ..., Gk}with G0 = G be the set of its residuals. We define now the function

g : Σ2n × {0, ..., k} −→ {0, ..., k}as follows:

g(b1, ..., bn, a1, ..., an, p) = qiff

Res(Gp, Jumpa1→b1 , ..., Jumpan→bn, t) = Gq

where t > 0, ~a = (a1, ..., an) and ~b = (b1, ..., bn).

x1

xn

LJV

Pg

Ph G(x1,..,xn)

n1,d1

LJVnn,dn

LJVn+1n+1,0

LLim0

LLime1

LLimen

Figure 7.1: Construction schema for theorem 7.3.1.

Making use of the function algebra [X [Σ1, ..., Σm]; OP ], we can define

Jg(~x, r) = Pg(LLime1(x1), ..., LLimen

(xn),

LJV n1,d1

(~x), ..., LJV nn,dn

(~x),

LJV n+1n+1,0(~x, LLim0(r)))

with ~e and ~d such that g(d1, ..., dn, e1, ..., en, 0) = 0 4. Looking at the definitions7.1.1 and 7.1.2, we verify that Jg is in the conditions for the application ofoperator Rec.

4To guarantee this requisite we introduce if needed new symbols in the domain of g, thatwill not affect the result.

72

Given ~x ∈ Sig(Σ)n, we have

Rec(Jg)(~x)(t) = Pg(LLime1(x1), ..., LLimen

(xn),

LJV n1,d1

(~x), ..., LJV nn,dn

(~x),

LJV n+1n+1,0(~x, LLim0(Rec(Jg)~x)))(t)

= g(LLime1(x1)(t), ..., LLimen

(xn)(t),

LJV n1,d1

(~x)(t), ..., LJV nn,dn

(~x)(t),

LJV n+1n+1,0(~x, LLim0(Rec(Jg)(~x)))(t))

If t′ ∈ [0, t[ is such that xi is constant in ]t′, t[ for all i and xj is not continuousat t′ for some j, for each i

xi(t) = xict′ ; JumpLJV ni,di

(~x)→LLimei(xi)(t)

and GLJVn+1

n+1,0(~x,LLim0(Rec(J)(~x)))(t) is the residual of G with respect to ~x and t′.

By definition of g:

GRec(J)(~x)(t) = Res(G, ~x, t)

Defining h : Σn × {0, ..., k} −→ Σ as

h(~a, j) = b iff Gj(Consta1, ..., Constan

)(0)

we get G(~x)(t) = Ph(x1, ..., xn, Rec(Jg)(~x))(t). This is the circuit of G which isexemplified in figure 7.1. �

Theorems 7.2.15 and 7.3.1 taken together state that the function algebra[X [Σ1, ..., Σm]; OP ] coincide precisely with the set of finite memory retrospectiveoperators, i.e.,

[X [Σ1, ..., Σm]; OP ] = OFMR.

7.4 Examples

In this section we will construct circuits for three finite memory retrospectiveoperators, some of them already studied in chapter 4. We will use the functionalgebra [X [Σ1, ..., Σn], OP ], however the general construction schema providedin theorem 7.3.1 will not be followed5.

Example 7.4.1 Constb ∈ nZSig(Σ)

Constb(t) = b

LJV11,0 Ph

Figure 7.2: Constb circuit.

Constb is obviously a 0-ary finite memory retrospective operator with onlyone residual, Constb. Therefore it is possible to represent this operator by thecircuit of figure 7.2, where h : {0} −→ Σ is such that h(0) = b.

5The circuits obtained by the general construction schema become more complex.

73

Example 7.4.2 Jumpa→b ∈ nZSig(Σ)

Jumpa→b(t) =

{

a if t = 0b if t > 0

As we have seen in chapter 4, this signal is a 0-ary finite memory retrospectiveoperator with two residuals:

Jumpa→b, for t = 0;Constb, for t > 0.

In order to get a circuit to Jumpa→b is enough compose the circuit of figure7.2 with LLima as we did in figure 7.3.

LJV11,0 Ph LLima

Figure 7.3: Jumpa→b circuit.

Example 7.4.3 LeftCont : nZSig(Σ) −→ nZSig({True, False})

LeftCont(x)(t) =

{


Clearly we can get this operator as follows:

LeftCont(x) = P=(x, LLimw(x))

where w is a symbol such that w /∈ Σ and P= is the pointwise extension of

=: (Σ ∪ {w})2 −→ {True, False}

defined as

= (a, b) =

{

True if a = bFalse otherwise.

LLimw

P=

Figure 7.4: LeftCont circuit.

Therefore, in figure 7.4, we give a circuit to LeftCont.

74


7.1.4 LLima is a finite memory operator.

7.1.5 LJV nk,a is a finite memory operator.

7.2.15 If F is in [X [Σ1, ..., Σm]; OP ], then F is a finite memory retrospectiveoperator.

7.3.1 If F : Sig(Σ)n −→ Sig(Σ) is a finite memory retrospective operator, thenthere is a circuit obtained from the function algebra [X [Σ1, ..., Σm]; OP ]that represents F .

75

Chapter 8

Final Remarks

The behavior of finite devices with multiple input channels and operating incontinuous time has been formalized in chapter 2, in fact we have studied thebehavior of such devices operating in a time set T , where T is a subgroup of R.This formalization was performed by looking at the classical theory of automata,i.e., looking at discrete time devices. Many postulates of automata theory wereanalyzed, the concept of time set was formalized in detail and the proofs ofpropositions 2.2.5 and 2.2.7 were provided, these are our main contributions inthis chapter 2 with respect to [Rab97, RT98, PRT01].

A characterization of speed independent operators over non-Zeno signalsand over right open signals was given in chapter 3, together with also somerelated concepts. With the lift to continuous time comes up some properties ofsignals invisible at discrete time, for example majority of signals will be sensibleto expansion and compression of time. In this chapter our contributions withrespect to [Rab97] are the complete proofs of the propositions 3.1.3 and 3.2.3.

In chapter 4, many examples were provided and studied in detail and manyproperties have been clarified.

In chapter 5, we studied closure properties of operators over signals and someproperties of finite memory retrospective operators, which have permitted inchapter 6 to introduce the concept of a transducer and to define a representationof finite memory retrospective operators. With respect to [Rab97], the completeproofs of propositions 5.1.1, 5.1.2 and 5.2.8 were provided.

The representation of finite memory operators found in [Rab97] was dis-cussed in chapter 6. Our contribution concerns transducers for n-ary operatorsand illustrative examples of automata, which construction was discussed in de-tail.

We note that the contents of chapters 5 and 6 relay in the characterizationgiven in chapter 3.

Circuits of operators were introduced in chapter 7. The concept of functionalgebra in [Clo99] was used in order to obtain an algebra of finite memoryretrospective operators. Our main contribution is the proof of the equivalencebetween this algebra of operators and the set of finite memory retrospectiveoperators.

It is important to note that we did not use time delay operators, which arecommonly used in the classical theory of circuits. In order to know the valuesof signals at previous intants we used the LLim operator and the LJV operator

76

Finite Memory

Speed-Independent

Stable

Countable Memory

Figure 8.1: Properties of the retrospective operators that map non-Zeno signalson non-Zeno signals.

and with these operators we have proved that it is possible construct circuitsfor any finite memory retrospective operator.

The chapter 7 includes also examples of circuits for some operators. Howevernot used for these examples, the general construction schema provided in theproof of theorem 7.3.1 permits the construction of circuits for any finite memoryretrospective operator using the function algebra [X [Σ1, ..., Σn], OP ].

We recall two main properties showed within the dissertation.In first place it follows straightforward that finite memory strong retrospec-

tive operators map non-Zeno signals into left open signals. Since they have finitememory, they are stable and speed independent and, if a non-Zeno signal is notconstant at t, the output at t depends only on previous instants and thereforeit must be equal to previous outputs.

A second property that follows is the fact that countable memory operatorsmay not be representable by a transducer with a countable number of states be-cause the set {0, ..., n, ...} → {0, ..., n, ...} in definition 5.2.7 have an uncountablenumber of elements1.

In figure 8.1, we summarize the inclusions stated among the properties ofretrospective operators on non-Zeno signals. We have stated in theorem 6.1.4the inclusion Finite Memory ⊂ Speed-Independent, which is proper giventhe example 4.0.9. The inclusion Speed-Independent ⊂ Countable Memorywas proved in corollary 3.2.4. In proposition 3.0.7 we proved the inclusionCountable Memory ⊂ Stable for operators that map non-Zeno signals into non-Zeno signals, therefore this inclusion is true for retrospective operators with thisproperty. We also proved in proposition 6.1.4 that the finite memory retrospec-tive operators are stable and in proposition 3.0.9 that the speed independentretrospective operators are stable, however it is now clear that these resultsfollow immediately from the above inclusions.

The research around automata over continuous time, started in [Tra98] and[Rab97], was expanded in chapter 7, where we succeed to develop a theory ofcircuits of retrospective operators to fully characterize the class of finite memory

1As we know, the set of functions from N to N is uncountable.

77

retrospective operators.However many other open problems are to be solved and the field to be

enlarged towards a theory on foundations of hybrid systems. Boaz Trakhtenbrotgave us a copy of his personal notes, where many interesting problems weredesigned, e.g., the problem of oracles (the relativization problem), a theory ofreducibilities between operators has been considered and sketched, a theory ofreliable feedback has been developed in some depth, etc. We thus hope thatsuch a fundational seminal work prosper in the near future.

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Bibliography

[AD94] R. Alur and D. L. Dill. A Theory of Timed Automata. TheoreticalComputer Science, 126:183–235, 1994.

[Clo99] P. Clote. Computational models and function algebras. In E. R.Griffor, editor, Handbook of Computability Theory, pages 589–681.Elsevier, 1999.

[HMU01] J. E. Hopcroft, R. Motwani, and J. D. Ullman. Introduction to Au-tomata Theory, Languages, and Computation. Addison Wesley, 2edition, 2001.

[KT65] N. E. Kobrinskii and B. A. Trakthtenbrot. Introduction to the The-ory of Finite Automata. North-Holland Publishing Company Ams-terdam, 1965.

[Llo87] J. W. Lloyd. Fundations of Logic Programming. Symbolic Com-putation - Artificial Intelligence. Springer-Verlag, second, extendededition, April 1987.

[LSFV96] N. Lynch, R. Seagala, and H. B. Weiberg F. Vaandreger. Hybrid I/Oautomata. In R. Alur, T. A. Henzinger, and E. D. Sontag, editors,Lecture Notes in Computer Science - Hybrid Systems III, volume1066 of Lecture Notes in Computer Science, pages 496–510. Springer-Verlag, 1996.

[PRT01] D. Pardo, A. Rabinovich, and B. A. Trakhtenbrot. Synchronous Cir-cuits over Continuous Time: Feedback Reliability and Completeness.Technical report, School of Computer Science, Tel Aviv University,Ramat Aviv, Tel Aviv 69978, Israel, 2001.

[Rab97] A. Rabinovich. Automata over Continuous Time. To appear in The-oretical Computer Science in print, 1997.

[Rab98] A. Rabinovich. On Translations of Temporal Logic of Actionsinto Monadic Second Order Logic. Theoretical Computer Science,193:197–214, 1998.

[RT98] A. Rabinovich and B. A. Trakhtenbrot. From Finite Automata to-ward Hybrid Systems (Extended Abstract). In FCT: Fundamentals(or Foundations) of Computation Theory, 1998.

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[Son90] E. D. Sontag. Mathematical Control Theory — Deterministic FiniteDimensional Systems, volume 6 of Texts in Applied Mathematics.Springer-Verlag, June 1990.

[Tra98] B. Trakhtenbrot. Automata and Hybrid Systems. Technical Report153, Uppsala University, February 1998.

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Index

Jumpa→b, 28, 46LJV n

k,a, 65LLima, 65OP , 71Rec(G)~x, 67Res(F, ~x, t), 10Sig(Σ), 9T+, 8[X ; OP ], 64[X [Σ1, ..., Σm]; OP ], 71, 73[a, b[, 8Σ, 8OFMR, 64, 73∧, 83X [Σ1, ..., Σm], 66∨, 83nZSig(Σ), 11suf(x, t), 10xct; z, 10

boundgreatest lower, 83least upper, 83lower, 83upper, 83

channel, 7input, 8output, 8

circuits, 71closure properties, 43complete lattice, 83continuous mapping, 84continuous signal, 15

definability, 50directed set, 83

finite state transducer, 54function

function algebra, 64state, 48

hybrid systems, 4

least fixpoint, 84left limit, 15

machine, 7behavior, 9

memorycountable, 11finite, 9, 10

non-Zeno signal, 11

operatorcomputable by a transducer, 55countable memory, 11finite memory, 10, 11residual, 10retrospective, 9speed independent, 17stability, 15strong retrospective, 9

order preserving bijection, 13

partial order, 83pointwise extension, 66

residual, 10retrospective operator, 9

strong, 9strong retrospective, 9

right limit, 15right open signal, 12

SI condition, 19generalized, 23generalized retrospective, 23

signal, 8concatenation, 10constant at t, 15continuous, 15finite memory, 46

81

history, 9left limit, 15non-Zeno, 11non-zero duration, 11piecewise constant, 11prefix, 10restriction, 10right limit, 15right open, 12suffix, 10

speed independence, 17stability, 15

time, 8continuous time, 8discrete time, 8linear time, 8time set, 8

transducer, 54

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Appendix A

Notes about Partial Order

Theory

In this appendix we provide some definitions and results about partial ordersthat will be important for our goal in chapter 7.

Definition A.0.1 (Partial Order) A relation R on a set S is a partial orderif the following conditions are satisfied:

(a) xRx, for all x ∈ S.

(b) xRy and yRx imply x = y, for all x, y ∈ S.

(c) xRy and yRz imply xRz, for all x, y, z ∈ S.

Definition A.0.2 (Upper and Lower Bounds) Let S be a set equiped witha partial order ≤. Then a ∈ S is an upper bound of a subset X of S if x ≤ a,for all x ∈ X . Similarly, b ∈ S is a lower bound of X if b ≤ x, for all x ∈ X .

Definition A.0.3 (Least Upper and Greatest Lower Bounds) .Let S be a set equiped with a partial order ≤. Then a ∈ S is the least upperbound of a subset X of S if a is an upper bound of X and, for all upper boundsa′ of X , we have a ≤ a′. Similarly, b ∈ S is the greatest lower bound of a subsetX of S if b is a lower bound of X and, for all lower bounds b′ of X , we haveb′ ≤ b.

Let L = (S,≤) be a partial order. As usual, for X ⊂ L, we will write ∨LXfor the least upper bound and ∧LX for the greatest lower bound. ⊥L will denotethe least upper bound of L when it exists.

Definition A.0.4 (Complete Lattice) A partially ordered set L is a com-plete lattice if ∨LX and ∧LX exist for every subset X of L.

Definition A.0.5 (Directed Set) Let L be a complete lattice and X ⊆ L.We say X is directed if every finite subset of X has an upper bound in X .

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Definition A.0.6 (Continuous Mapping) Let L = (S,≤) be a complete lat-tice and T : S −→ S be a map. We say that T is continuous if T (∨LX) =∨LT (X), for every directed subset X of L.

Definition A.0.7 (Least (Greatest) Fixpoint) Let L be a complete latticeand T : L −→ L be a mapping. We say a ∈ L is the least fixpoint of T if a is afixpoint (that is, T(a)=a) and for all fixpoints b of T , we have a ≤ b. Similarly,we define greatest fixpoint.

The following result have an important role in our study and is due to Kleene,which is supported by an earlier result due to Knaster and Tarski.

Proposition A.0.8 Let L = (S,≤) be a complete lattice and T : S −→ S be acontinuous map. Then lfp(T ) = ∨L{T n(⊥L) : n ∈ N0}.

If L is a complete lattice and T : S −→ S is a continuous map, then weknow that it is monotonic, i.e, T (X) ≤ T (Y ) whenever X ≤ Y . Therefore{T n(⊥) : n ∈ N0} is a directed set. In practice when we want to obtain a leastfix point is enough to compute the ∨L{T n(⊥L) : n ∈ N0}.

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