PLC-Automaton Composition · tomata, described by H. Dierks, the problem of the composition of PLC automata is presented. Two methods for performing this composition are described,

PLC-Automaton Composition

Honours Project: 1999

Andre Renaud Dr. Padmanabhan Krishnan 1

1Supervisor

Abstract

Based on a discussion of timed automata and a subset of these called PLC au-tomata, described by H. Dierks, the problem of the composition of PLC automatais presented. Two methods for performing this composition are described, syn-chronous and asynchronous.

The synchronous method requires that if and only if both initial automataaccept a timed word, then it will appear in the resultant automaton. However,we prove that PLC automata are not closed under this form of composition.From this we introduce the asynchronous composition, which requires synchro-nization on the symbol element of the timed word alone. Four possible methodsof asynchronous composition are described, and their effects on the ‘essential’language, the language containing only words with no ignored symbols, arediscussed. A few elementary results are also presented.

Contents

1 Introduction 31.1 Report Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Preliminaries 52.1 Timed Automata . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.1 Automata Run . . . . . . . . . . . . . . . . . . . . . . . . 52.1.2 Accepting Runs . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 PLC Automata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2.1 Graphical Notation . . . . . . . . . . . . . . . . . . . . . . 82.2.2 Acceptance Criteria . . . . . . . . . . . . . . . . . . . . . . 92.2.3 Conversion to Timed Automata . . . . . . . . . . . . . . . 9

3 Composition: Definition and Examples 113.1 Synchronous Composition . . . . . . . . . . . . . . . . . . . . . . 11

3.1.1 Augmentation . . . . . . . . . . . . . . . . . . . . . . . . . 113.1.2 Synchronous Composition of Timed Automata . . . . . . . 12

3.2 Asynchronous Method . . . . . . . . . . . . . . . . . . . . . . . . 133.2.1 Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.3.1 Initial Automata . . . . . . . . . . . . . . . . . . . . . . . 143.3.2 Composite Automata . . . . . . . . . . . . . . . . . . . . . 14

4 Languages 174.1 Essential Language . . . . . . . . . . . . . . . . . . . . . . . . . . 174.2 Synchronous Composition Closure . . . . . . . . . . . . . . . . . 174.3 Comparison of languages . . . . . . . . . . . . . . . . . . . . . . 19

4.3.1 Minimum, ∩ . . . . . . . . . . . . . . . . . . . . . . . . . 194.3.2 Maximum, ∪ . . . . . . . . . . . . . . . . . . . . . . . . . 214.3.3 Minimum, ∪ . . . . . . . . . . . . . . . . . . . . . . . . . 224.3.4 Maximum, ∩ . . . . . . . . . . . . . . . . . . . . . . . . . 244.3.5 Language Conclusions . . . . . . . . . . . . . . . . . . . . 25

5 Conclusion 265.1 Further Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

1

Chapter 1

Introduction

Real time systems are currently used in a wide variety of situations. Most con-sumer devices are required to have real time response to input, and many crit-ical systems are also required to have deterministic response times to avoidfailure. When designing these systems a simple modelling technique is highlyuseful. Such a modelling system was described by Rajeev Alur and David Dill[AD94], in which they discuss an extension to standard automata which al-lows restricting transitions based on the values of multiple timers. Howeverthese timed automata proved to be too general, and so a simplified model, PLCautomata, was defined by Henning Dierks [Die97]. These PLC automata area simplification of timed automata, allowing only one timer, and limiting thetimer restrictions to a form of delay for symbols.

When dealing with these systems, a large problem is that of complexity. Acomplex structure may be difficult to build monolithically, and once built it maycontain errors due to its innate complexity. One standard method of dealingwith this complexity is composition, combining various sub-parts which con-tain individual desirable properties into the larger whole, which will hopefullyexhibit the required properties, without introducing any additional problems.

We will consider two distinct methods for composing PLC automata. In Sec-tion 3.1 a method for synchronous composition will be discussed. An asyn-chronous method, with four different variants will be considered in Section 3.2.

Synchronous Method

For standard automata we are normally interested in the intersection composi-tion. This is where all symbols occur simultaneously within both sub-automata.However with timed automata, there are two separate elements to each transi-tion, a symbol and a time. Thus we can have compositional methods which aresynchronized on either one or both of these elements. The synchronous compo-sition method will attempt to synchronize on both of these, thus requiring theresultant timed word to be accepted by both sub-automata.

Asynchronous Method

Conversely to the synchronous method, which attempted to use both elementsof a timed word, the asynchronous method will attempt to use only the sym-

3

PLC Automata Composition Introduction

bol, not the timing information. Thus the resultant automaton will acceptedthe intersection of the untimed languages of the original automata. We willconsider the properties of the timing constraints associated with the compositeautomaton. The asynchronous method is a structure based method, in whichwe maintain a PLC structure, and attempt to create logical values fitting withthe original system.

1.1 Report Structure

Chapter 2 will present an introduction to the foundation work in this area,precisely defining both real time automata and PLC automata, along with theirbehaviour. We will discuss both timed and PLC automata, and demonstrate amethod for converting from PLC automata to timed automata. This chapter willbe used as the basis for the additional research conducted for this report.

In Chapter 3 methods for composing automata in both the synchronous andasynchronous methods are discussed. We will also present an example of eachof the four different asynchronous techniques, and an explanation of how thiscomposite result relates to the original automata.

In Chapter 4 a simplified version of the languages used in the composition,called the ‘essential’ language is described. Using this language we will discussthe properties satisfied by the various methods of composition techniques.

Finally, Chapter 5 presents out conclusions, and describes some ideas forfurther development.

Acknowledgements

I would like to thank Dr. Padmanabhan Krishnan for assisting and guiding mewith this research, as well as Jane McKenzie for proof reading this paper manytimes.

4

Chapter 2

Preliminaries

2.1 Timed Automata

An introduction to timed automata [AD94], describes them as an extension tofinite automata. Formally an automaton is a tuple A = (Σ, S, S0, C,E) where:

• Σ is a finite alphabet,

• S is a finite set of states,

• S0 ⊆ S is a set of initial states,

• C is a finite set of clocks,

• E ⊆ S×S×Σ×2C×Φ(C) gives the set of transitions. An edge (s, s′, α, λ, δ)represents a transition from state s to state s′ on input symbol α. The setλ ⊆ C gives the clocks to be reset with this transition, and δ is a clockconstraint over C.

The transitions within timed automata require the satisfaction of clock con-straints. We will define a set Φ(X), to be a set of clock constraints δ. Theseconstraints are defined inductively by:

δ := x ≤ c|c ≤ x|¬δ|δ1 ∧ δ2

where x is a clock in X, and c is a constant in Q.However to keep the notation simple and consistent, the following notation,

which is similar to standard automata notation, will be used; S will be calledQ, and S0, which typically contains only one element, will be represented byπ0/q0. An example of a timed automaton is displayed in Figure 2.1. This showsa two state automaton with one timer, which accepts strings of alternating a’sand b’s. Each ‘a’ must be no more than two time units after the preceding ‘b’,and each ‘b’ must be at least one time unit after the preceding ‘a’. The clock Xis reset on both transitions.

2.1.1 Automata Run

In order to define a run, a few auxiliary concepts are required.

5

PLC Automata Composition Preliminaries

// ?>=<89:;q0

a,X<2

X

;;?>=<89:;q1

b,X>1

X

Figure 2.1: Example Timed Automaton

A clock interpretation v for a set C of clocks, assigns a real value to eachclock; that is, it is a mapping from C to R. We say that a clock interpretation vfor C satisfies a clock constraint δ over C if and only if δ evaluates to true usingthe values given by v. We will define two notational conveniences. [C ← R]specifies the clock mapping. vi = [λ 7→ r](X) will be used to assign all clocksc ∈ λ the value r, after which the equation X will be evaluated, and its resultassigned to vi.

A time sequence τ = τ1τ2τ3 · · · is an infinite sequence of time values τi ∈ Rwith τi > 0, satisfying the following constraints:

Monotonicity: τ increases strictly monotonically; that is τi < τi+1 for all i ≥ 1.

Progress: For every t ∈ R, there is some i ≥ 1 such that τi > t.

A timed word over an alphabet Σ is a pair (σ, τ), where σ = σ1σ2 . . . is an infiniteword over Σ and τ is a time sequence. A timed word is viewed as an input to anautomaton, it presents the symbol σi at time τi.

We can now define an automata run. A run over a timed word, (q, v), of atimed automaton is an infinite sequence of the form

r : 〈q0, v0〉σ17−→τ1〈q1, v1〉

σ27−→τ2〈q2, v2〉

σ37−→τ3· · ·

with qi ∈ Q and vi ∈ [C ← R], for all i ≥ 0, satisfying:

• Initiation: q0 = π0, and v0(x) = 0 for all x ∈ C

• Consecution: for all i≥1, there is an edge inE of the form 〈qi−1, qi, σi, λi, σi〉such that (vi−1 + τi− τi−1) satisfies σi and vi = [λi 7→ 0](vi−1 + τi− τi−1).

2.1.2 Accepting Runs

We will add the notion of acceptance to our timed automata through the useof Buchi, [Tho90], acceptance criteria. Buchi acceptance criteria add an extraelement to the timed automaton called F . This is a set of accepting states. A runr over a Buchi automaton is accepted if and only if inf(r) ∩ F 6= 0, where inf(r)is defined as all states visited infinitely often during the run r. Graphically wewill denote these states as a double circle, as in Figure 2.2

This acceptance criteria implies that a run is accepted if and only if it visitsone or more of the accepting states infinitely often. So in the example in Figure2.2 a run must contain an infinite number of b’s to be accepted. It may or maynot contain an infinite number of a’s.

6

Preliminaries PLC Automata Composition

// ?>=<89:;q0

a,X<1,X

a,X>1,X33 ?>=<89:;/.-,()*+q1

b,true,

b,X<1,ss

Figure 2.2: Example Acceptance Automaton

2.2 PLC Automata

PLC automata, as described by Henning Dierks [Die97], are a model de-signed to express the behaviour of polling systems. They are a subclassof timed automata, being restricted to only one clock, and certain types ofclock restrictions. Formally they are an automaton defined as a tuple, A =(Q,Σ, δ, π0, ε, St, Se,Ω, ω), where

• Q is a nonempty, finite set of states.

• Σ is a nonempty, finite set of inputs.

• δ is a transition function of type Q× Σ→ Q

• π0 ∈ Q is the initial state.

• ε > 0 is the upper bound for a cycle.

• St is a function of type Q→ R≥0.

• Se is a function of type Q → P(Σ)\0, assigning to each state a set ofdelayed inputs that cause no change of the state during the first time unitsthe automaton stays in this state.

• Ω is a nonempty, finite set of outputs.

• ω is a function of type Q→ Ω (output function).

The St function assigns to each state π a delay time. This delay time is usedto determine the duration for which a specified set of inputs (defined in Se) areignored. Given St(qi) = 3.5, any symbol ∈ Se(qi) arriving whilst the automatonis in state qi, and has been there for < 3.5 time units, will be ignored. This willnot reset the clock, so if α ∈ Se(qi) arrived at time 3.4, and was ignored, then asecond α one time unit later will be visible.

Within the concepts discussed in this paper we will ignore the ε value, whichrelates to the time it takes to poll the inputs, as well as both ω and Ω, as we arenot interested in the output functionality. This simplification does not removethe ability of our restricted PLC automata to model the polling property of sys-tems.

It should also be noted that St is not a partial function, that is, every statehas some transition on every input. We will relax this slightly and only requirethat each state qi ∈ Q has a transition on every symbol in Se(qi), that is, everysymbol that can be ignored at a state for a period must eventually have the pos-sibility of a transition from that state. PLC automata also have a deterministicrequirement, and are unable to cope with λ (null) transitions.

7


2.2.1 Graphical Notation

We will use the following graphical notation to represent a particular state of aPLC automaton: [

π

D, s1 . . . sn

]

where:

• π ∈ Q is a particular state.

• D = St(π), the delay for the current state.

• s1 . . . sn = Se(π), the set of inputs which can be ignored within thetiming constraints.

An example of a PLC automaton using this graphical notation is given inFigure 2.3. Informally, state q0 is sensitive to a and b after 5 time units, while itis always sensitive to c. However state q2 is always sensitive to a and b, but issensitive to c only after 4 units of time.

//[

q0

5, a, b

]b

11

a &&

[q1

3, a

]aqq

b

[q2

4, c

]

c

LL

a,b

@@

Figure 2.3: Example PLC Automaton

Sample Run

Using the automaton shown in Figure 2.3 we will now describe a sample run,demonstrating how the timing constraints work within PLC automata.

(a, 4) 7→ (a, 6) 7→ (b, 8) 7→ (a, 9) 7→ (a, 12) . . .

Figure 2.4: Sample Run of PLC Automaton

In the sample run shown in Figure 2.4, we start in state q0. The ‘a’ at time4 will be ignored, as it falls before the settling time of 5 for the state q0. The‘a’ at time 6 will now be acknowledged, and the transition to state q2 is made.The relative clock is now 0, as it is reset on every transition, and thus the ‘b’ attime 8 has a relative time for state q2 of 2. Since ‘b’ is not in the ignored set forstate q2, the transition is made to state q1. Again the relative clock is now 0, sothe transition on ‘a’ at time 9 is inactive, as it has a relative timing of 1, whichis less than the delay time of state q1, 3. The ‘a’ at time 12 has a relative clockvaluation of 4, and is thus active, taking us back to state q1.

8

Preliminaries PLC Automata Composition

2.2.2 Acceptance Criteria

For demonstrative purposes, we will define a simple acceptance criterion, whichwill allow us to display acceptable and unacceptable runs of an automaton. Wewill first define the timed word. Consider tw, a timed word, such that

tw : ω → (Σ× R+)

Given an automata A, a run of A over tw is

ρ : ω → Q such that ρ0 = π0, ρitw1(i)7−→tw2(i)

ρi+1

These timed steps are defined as

qa7−→tq′

iff

• St(q) ≥ t, a ∈ Se(q), q = q′ =⇒ a ignored or

• St(a) < t,∃q a→ q′ ∈ δ =⇒ delay expired or

• a /∈ Se(q),∃qa→ q′ =⇒ “urgent” action

2.2.3 Conversion to Timed Automata

We would like to be able to use many of the properties of timed automata withinour PLC automata; as such we will define a method for converting between thetwo, [DFMV98]. This conversion is fairly straightforward.

Given a PLC-Automaton A1, such that, A1 = (Q1,Σ1, δ1, π01 , St1 , Se1), theequivalent timed A2 = (Q2,Σ2, q02 , C2, E2) can be formed as follows:

• Σ2 = Σ1

• Q2 = Q1

• p02 = π01

• C2 = X, a new clock

• ∀qi ∈ Q1,∀αi ∈ Se1(qi), (qi, qi, αi, X ≤ St1(qi), ∅) ∈ E2

• ∀qi ∈ Q1,

∀αi s.t δ1(qi, αi)→ qj ,

(qi, qj , αi, X > St1(qi), X) ∈ E2 αi ∈ Se1(qi)(qi, qj , αi, true, X) ∈ E2 αi /∈ Se1(qi)

The final rule above shows the creation of transitions from the delayed andnon-delayed symbols. If a state has a delay time of St1(qi), then the transitionrequires the clock X to be greater than the delay on any symbols in Se1(qi).However if the symbol is not in the set, then it can be instantly moved on withno clock restriction.

An example of this conversion is shown in Figure 2.5. Figure 2.5(a) showsthe original timed automaton. Figure 2.5(b) shows the subsequent timed au-tomata generated from the method shown above. The self loop on q0 corre-sponds to the ignored a, while the self loop on q1 corresponds to the ignoredb.

9


//[

q0

3, a

]a,b

11

[q1

4, b

]a,bqq

(a) PLC Automaton

// ?>=<89:;q0

a,X≤3,

a,X>3,X

;;b,true,X

22 ?>=<89:;q1

b,X≤4,

b,X>4,X

a,true,Xrr

(b) Timed Automaton

Figure 2.5: Example Conversion from a PLC Automaton to a Timed Automaton

10

Chapter 3

Composition: Definition andExamples

In this chapter we will examine various methods of combining PLC automata.The first, and simplest, is the synchronous product, over perhaps different al-phabets.

3.1 Synchronous Composition

We will define the synchronous composition as the intersection of two automata.The method we will use for this composition will require us to convert the PLC-Automata models into standard Timed Automata. We will then synchronouslycompose these timed automata, [AD94]. Finally we will attempt to take the re-sultant composite timed automaton, and convert it back into a PLC automaton.

3.1.1 Augmentation

When composing two PLC automata we will require that Σ1 = Σ2. If this is notthe case, we can augment the alphabets of the individual automaton using thefollowing method.

Augment both automata with the symbols which are unique to the other.

∀α, α ∈ Σ1, α /∈ Σ2 =⇒ ∀qi ∈ Q2, (qi × α→ qi) ∈ δ′

2

Similarly, the reverse:

∀α, α ∈ Σ2, α /∈ Σ1 =⇒ ∀qi ∈ Q1, (qi × α→ qi) ∈ δ′

1

The result is that when one automaton makes a move about which the otherhas no knowledge, i.e.: it does not recognise the symbol, the move does notinvolve a state change or timer reset. This allows the automaton to see as many‘unknown’ symbols as we like, without affecting its interaction with the knownalphabet.

11

PLC Automata Composition Composition: Definition and Examples

3.1.2 Synchronous Composition of Timed Automata

We would like to be able to compose two PLC automata in a synchronous fash-ion to produce a new automaton with desirable properties, such as word accep-tance or language intersection.

Given the two automata described in Figure 3.1, their equivalent timed au-tomata, Figure 3.2, can be determined using the method described above.

//[

q0

10, a, b

]b

a11

[q1

6, c

]cqq

(a) Automaton 1

//[

p0

7, a

]a

11

[p1

8, b

]bqq

c

(b) Automaton 2

Figure 3.1: Initial PLC-Automata

// ?>=<89:;q0

(a,b),X≤10,∅

(b),X>10,X

WW(a),X>10,X

22 ?>=<89:;q1

(c),X≤6,∅

(c),X>6,Xrr

(a) Timed Automaton 1

// ?>=<89:;p0

(a),X≤7,∅

(a),X>7,X

22 ?>=<89:;p1

(b),X≤8,∅

(c),true,X

WW

(b),X>8,Xrr

(b) Timed Automaton 2

Figure 3.2: Equivalent Timed Automata

The synchronous composition of these two automata, presented as a timedautomata, is shown in Figure 3.3. Converting this back to a PLC automatonnow requires each transition to be considered individually. Transitions of theform ‘α,X ⊕ n ∧ Y ⊕ n, S’, and ‘α,Z ⊕ n, S’, where ⊕ ranges over >,≤, andS ranges over X,Y , ∅, are easily translated back to a PLC automaton. Thisis because the final PLC automaton can have only one clock, and transitionsof these forms are easily converted into transitions of only one clock, suitablefor PLC automaton. However transitions which involve the reseting of only oneclock, or have restrictions on both clocks which are not identical, cannot be con-verted into restrictions on only one clock. Thus the composite result cannot beconverted back from a timed automaton to a PLC automaton. Because it is nowimpossible to successfully compose two PLC automata in a synchronous fashion,we will look at some of the alternative methods for asynchronous composition.From this we will accept Lemma 1.

Lemma 1 (Synchronous Composition) There exist PLC automata A1 and A2

such that there is no PLC automaton that accepts L(A1) ∩ L(A2)

Proof: Examples in Figures 3.1–3.3, and above discussion.

12

Composition: Definition and Examples PLC Automata Composition

GFED@ABCq0,p0

a,X≤7,∅

a,X>10∧Y >10,X,Y --

a,Y >7∧X≤10,X,Y

GFED@ABCq1,p1

c,X≤6,Y

c,X>6,X,Y

GFED@ABCq0,p1

b,X≤8,∅

WW

b,X≤10∧Y >8,Y

LL

b,X>10∧Y >8,X,Y

@@

Figure 3.3: Composite Timed Automata

3.2 Asynchronous Method

As the previous examples showed, it does not appear possible to compose PLCautomata in a synchronous manner. Thus the construction of the language inter-section is not possible. As an alternative we will look at methods which requirethe retainment of the PLC structure, and observe the effects they have on thecombined language.

In this section we will look into methods for structure based asynchronouscomposition, demonstrating these methods with a brief example.

Given any two automata, A1 =(Q1,Σ1, St1 , Se1), and A2 =(Q2,Σ2, St2 , Se2),whose states are disjoint, i.e.: Q1 ∩ Q2 = ∅, the composition, to obtain A3 =(Q3,Σ3, St3 , Se3), is as follows.

3.2.1 Structure

The states of the new automaton, A3, will be the product of the states of thetwo initial automata, A1 and A2. Thus Q3 = Q1 ×Q2.

(q1,q2)a // (q′1,q′2)

iff q1a // q′1 and q2

a // q′2

St and Se composition

There are two different ways of composing each of the St and Se arrays.For Se, either the intersection or union is used.

∀q = (qi, qj) ∈ Q3, St3(q) = St1(qi)⊕ St2(qj)

where ⊕ ranges over ∪, ∩.For St, either the min or max is used.

∀q = (qi, qj) ∈ Q3, Se3(q) = Se1(qi)⊕ Se2(qj)

13


where ⊕ ranges over min, max.It should be noted that the transition structure does not change between the

different methods of composition, only the internal sensors and input sets ofeach node changes as the ‘untimed’ languages intersect.

3.3 Example

In this section we will look at a sample composition, showing each of the fourproposed methods.

3.3.1 Initial Automata

Consider the two automata shown in Figure 3.4. There is no need to augmenteither of these, as they share the same alphabet, a, b, c.

//[

q0

10, a, b

]b

a11

[q1

6, c

]cqq

(a) Automaton 1

//[

p0

7, a

]a

11

[p1

8, b

]bqq

c

(b) Automaton 2

Figure 3.4: Initial Automata

3.3.2 Composite Automata

Minimum, ∪

The automaton displayed in Figure 3.5 is the result of the minimum and ∪composition of the two initial automata. The result here appears to take the‘best’ sensor (i.e.: the one with the lowest value), and apply it to all of thepotentially unreliable inputs.

//[q0, p0

7, a, b

]a

11

[q1, p1

6, b, c

]

c

[q0, p1

8, a, b

]b

RR

Figure 3.5: Minimum, ∪ Composition

14

Composition: Definition and Examples PLC Automata Composition

Maximum, ∪

This composition, shown in Figure 3.6, takes the ‘strongest’ sensor in everycase. It selects the maximum value, thus the slowest sensor, most reliable to allunreliable inputs, and it takes the union of the sets of symbols, thus requiringthe largest set of symbols to be restricted. This could be viewed as the mostconservative composition; for any given transition it selects the slower/morerestrictive option.

//[

q0, p0

10, a, b

]a

11

[q1, p1

8, b, c

]

c

[q0, p1

10, a, b

]b

RR

Figure 3.6: Maximum, ∪ Composition

Minimum, ∩

This composition, shown in Figure 3.7, is, not surprisingly, the opposite of themaximum, union composition. For each composite node, the fastest sensor ischosen, and it is applied to the smallest set of symbols possible (i.e.: the mini-mum). The result of this is an automaton which delays for as little as possible.If the maximum, union composition was conservative, then this would be themost permissive, containing the fewest restraints. For example in the (q1, p1)state, the fastest sensor for b is used; that from q1, where b is assumed to beinstantly stable (i.e.: time delay of 0). Similarly for c, the sensor in p1 is used,which gives c a delay of 0. The result of this is the empty set, as both b and chave delays of 0, and as such do not appear in the delay set.

//[q0, p0

7, a

]a

11

[q1, p1

6,

]

c

[q0, p1

8, b

]b

RR

Figure 3.7: Minimum, ∩ Composition

15


Maximum, ∩

Shown in Figure 3.8, this composition is similar to the first, minimum, union.It selects the slowest sensor, but only attaches it to the smallest number of ele-ments. The result is an automaton which is slow on symbols that were commonto previous timing constraints, but ignores symbols that were from a single au-tomaton only.

//[q0, p0

10, a

]a

11

[q1, p1

8,

]

c

[q0, p1

10, b

]b

RR

Figure 3.8: Maximum, ∩ Composition

Thus concludes the definition of the structure based composition of PLCautomata. While they do not exactly correspond to the standard product con-struction, there are a number of similarities. In the next chapter we study thelanguage behaviour of the composite automata.

16

Chapter 4

Languages

4.1 Essential Language

For any PLC-Automata, A, we define the language of words it can accept asL(A). Within this language L(A), if every word is collapsed, by removing allignored transitions and merging identical collapsed words, we get a languageL(A)0, which contains all the minimal runs of the automata. We will call thisthe essential language. This essential language allows for easier manipulationand comparison, and will be used for the following language comparisons.

Using the sample run previously shown in Figure 2.4, we can demonstratethe effect of employing only the essential language. Initially we had a transitionon ‘a’ at time 4. This was ignored and therefore it will not appear in the es-sential version of this run. Removing the ignored symbols, the sample run nowbecomes that shown in Figure 4.1.

(a, 6) 7→ (b, 8) 7→ (a, 12) . . .

Figure 4.1: Sample Run for Essential Language

It should be noted that for every word in the standard language of an au-tomaton, there is a word in the essential language which is equivalent, exceptthat all ignored symbols have been removed. Thus there is an N : 1 relationshipbetween the two sets.

4.2 Synchronous Composition Closure

As shown in Section 3.1.2, we are unable to build a new PLC automaton whichaccepts the intersection of the languages of the two initial automata. However ifwe consider the essential language, rather than the standard language, a slightlymodified construction can now be attempted.

In the modified construction transitions will no longer be allowed to occurduring periods of ignored time for the PLC states. This will not have any in-fluence on the essential language, as these transitions would have been invalid

17

PLC Automata Composition Languages

anyway. We now only produce transitions of symbols which are non-ignored.This will result in transitions only being bounded by > as a timing constraint,since ≤ was used when symbols occurred before their delay time, which is nowimpossible.

[q0

5, a

]a

b,c

11

[q1

7, b, c

]b,c

aqq

(a) Automaton 1

[p0

3, a, b, c

]b,c

a11

[p1

4, a

]b,c

aqq

(b) Automaton 2

Figure 4.2: Initial PLC Automata

?>=<89:;q0

(a),X>5,X

(b,c),true,X

22 ?>=<89:;q1

(b,c),X>7,X

a,true,Xrr

(a) Automaton 1

?>=<89:;p0

(b,c),Y >3,Y

a,Y >3,Y

22 ?>=<89:;p1

(b,c),true,Y

a,Y >4,Y rr

(b) Automaton 2

Figure 4.3: Resultant Timed Automata

In the resultant timed automata, shown in Figure 4.3, we can see that theconversion has added no additional transitions, unlike the case for the originalconversion. These extra transitions were all due to symbols arriving before theirdelay time had elapsed, which no longer occurs.

(q1,p1)

(b,c),X>7,X,Y

a,X>5∧Y >3,X,Y // (q0,p0)(b,c),Y >3

X,Y //

a,X>5∧Y >3,X,Y

(q1,p0)

(b,c),X>7∧Y >3,X,Y

a,Y >3,X,Y

zzzzzzzzzzzzzzzzzzzzzzzzzz

(q0,p1)

(b,c),true,X,Y

aaDDDDDDDDDDDDDDDDDDDDDDDDDD

a,X>5∧Y >4,X,Y

RR

Figure 4.4: Synchronous Composite Automaton

The final composite automaton, shown in Figure 4.4, with each state hav-ing only one transition on any given symbol, is now much simpler than thecomposite result obtained by the general method. We would like to be able toconvert this back to a PLC automaton. However determining the timing con-straints and delayed symbols is non-trivial. From most states there is a choice

18

Languages PLC Automata Composition

between either of the delayed times of the two original combining states. Thiswould lead us to choose either the minimum or maximum of these two values,but the composition would then tend to the asynchronous method. This wouldalso mean one transition was either over, or under-compensated. For exam-ple in Figure 4.4, state (q0, p0) has two transitions, (b, c), Y > 3, X,Y , anda,X > 5 ∧ Y > 3, X,Y . If we choose the maximum of the two timing con-straints, 5, the (b, c) transition is too heavily restricted. However, if we choose3 the a transition will be too permissive. Determining the delayed symbols issimpler however. If a transition from a state, qi, has a timing restriction of any-thing other than true, then the symbol(s) of that transition must appear in thedelayed set, Se(qi). If the transition has a restriction of true, then that symbolwill not appear in the set. This set construction turns out to be union, becauseif a symbol was delayed in its initial automaton, it must be delayed in the com-posite result. Thus every delayed symbol from the initial states must be delayedin the composite state.

From this we can conclude that the synchronous essential language compo-sition is of little use. It is restricted to the essential language, and can thus notbe generalized, so its ability to model standard PLC automata is diminished. Itis also highly similar to two of the asynchronous methods (min,max,∪), andas these are more general methods, which still work on the entire language,they are more useful compositions.

4.3 Comparison of languages

For the purposes of this report, we are interested in comparing the resultantlanguages from the asynchronous composition methods discussed earlier, withthe intersection of the two languages from the original automata used in thecomposition.

4.3.1 Minimum, ∩We are interested in the status of the following class of conjectures.

Conjecture 1 (Minimum, ∩ Comparison)

L0min,∩ L

0∩ where could range over =,⊂,⊆,⊃,⊇

[qi

ni, Si

]α //

q′in′i, S

′i

, i = 1, 2

Figure 4.5: General PLC Automaton

In comparing the language of the intersection, which as shown in Section3.1 cannot be represented as a PLC automaton, with the language of the asyn-chronous composition, discussed in Section 3.2, consider the general PLC tran-sition, as shown in Figure 4.5. Both will be converted into standard timedautomata and the languages compared. When performing this conversion, thethree scenarios that must be investigated are:

19


• α ∈ S1 ∧ α ∈ S2

• α ∈ S1 ∧ α /∈ S2

• α /∈ S1 ∧ α /∈ S2

We will thus construct the two composite transitions for the three abovescenarios. These are displayed in Figures 4.6, 4.7 and 4.8 respectively.

(q1,q2)α,Z>min(n1,n2)

Z// (q′1,q′2)

(a) Asynchronous

(q1,q2)α,X>n1∧Y >n2

X,Y // (q′1,q′2)

(b) Synchronous

Figure 4.6: α ∈ S1 ∧ α ∈ S2

From the first of these, α ∈ S1 ∧ α ∈ S2, shown in Figure 4.6, we can seethat there are no transitions accepted by the synchronous result, that are notaccepted by the asynchronous one. For example, given n1 = 5 ∧ n2 = 6, thecomposite conditions become Z > 5 and X > 5 ∧ Y > 6 respectively. A timestep of (α, 5.1) would now be permitted by the first transition, but not by thesecond.

(q1,q2)α,true

Z// (q′1,q′2)

(a) Asynchronous

(q1,q2)α,X>n1∧true

X,Y // (q′1,q′2)

(b) Synchronous

Figure 4.7: α ∈ S1 ∧ α 6∈ S2

Similarly to the first minimum, ∩, scenario, we can see that in the behaviourshown in Figure 4.7, the asynchronous transition will be enabled on all clockvaluations on α, as its clock constraint is true. However, there is still a constrainton the synchronous composition, X > n1. Thus the synchronous method mustbe delayed by a minimum of n1, whereas the asynchronous transition is alwaysenabled.

(q1,q2)α,true

Z// (q′1,q′2)

(a) Asynchronous

(q1,q2)α,true

X,Y // (q′1,q′2)

(b) Synchronous

Figure 4.8: α 6∈ S1 ∧ α 6∈ S2

In the final of the three minimum, ∩, scenarios, shown in Figure 4.8 we findboth automata accepting all timing conditions of α. When the semantics of thisfinal composition are considered this is not surprising, as this is the case whenα is not in either of the restricted sets, and is thus completely unrestricted.

20


From the three scenarios for minimum, ∩, we can see that overall, the asyn-chronous composition would always accept at least all of the words accepted bythe synchronous composition. Further to this we can see that in the first twocases, Figures 4.6 and 4.7, there are additional words accepted by the asyn-chronous method, as the timing constraints on the synchronous transitions aremore strict. From this Conjecture 1 can be completed.

Lemma 2 (Minimum, ∩, Language Comparison)

L0min,∩ ⊃ L0

∩

Proof: Let (α0, t0)(α1, t1) . . . (αn, tn) . . . ∈ L0∩ be the given word.

∴ There is a run of A1 ‖ A2, the synchronous composition, over it. Byobservation, and the constructions in Figures 4.6 – 4.8, this run can beconverted to a run of A1 ∩min,∩ A2 over the given word.It should be noted that this conversion only works because in PLC au-tomata, every non-ignored transition resets the clock. Thus when contem-plating the subsequent transition we know that the clock is zero, and itcan be examined independently of previous results.

4.3.2 Maximum, ∪Using the same technique as above, the state of the following conjecture will beinvestigated:

Conjecture 2 (Maximum, ∪ Comparison)

L0max,∩ L


We will again consider the three case scenarios described in Section 4.3.1.

(q1,q2)α,Z>max(n1,n2)

Z// (q′1,q′2)

(a) Asynchronous

(q1,q2)α,X>n1∧Y >n2

X,Y // (q′1,q′2)

(b) Synchronous

Figure 4.9: α ∈ S1 ∧ α ∈ S2

For the first case, where α ∈ S1 ∧ α ∈ S2, the composite transitions areas described in Figure 4.9. This differs from that described in the minimum,∩, composition, in that the restrictions on both transitions are now equivalent.The asynchronous restriction, Z > max(n1, n2), and the synchronous, X >n1∧Y > n2, both require that the larger of the two timing constraints, n1, n2,is satisfied before the transition is active.

The second scenario, α ∈ S1 ∧ S 6∈ S2, shown in Figure 4.10, is a slightlydifferent case to those seen before. In this case, we have the most restrictive con-ditions on the asynchronous transition, Z > max(n1, n2), whilst the restrictionon the synchronous transition, X > n1, would appear to be weaker. Howeverthere are now two possibilities:

21



Z// (q′1,q′2)

(a) Asynchronous


X,Y // (q′1,q′2)

(b) Synchronous

Figure 4.10: α ∈ S1 ∧ α 6∈ S2

• n1 < n2 – The synchronous composition can now accept additional timesteps, any of whose clock valuations are between (n1, n2].

• n1 ≥ n2 – The transitions are now equivalent, as max(n1, n2) = n2. Thereare no additional time steps accepted by either transition.

(q1,q2)α,true

Z// (q′1,q′2)

(a) Asynchronous

(q1,q2)α,true

X,Y // (q′1,q′2)

(b) Synchronous

Figure 4.11: α 6∈ S1 ∧ α 6∈ S2

In the final of the maximum, ∪, composition scenarios, shown in Figure4.11, we have the trivial case, where α is not an ignored symbol. Since both ofthe restrictions are true, there is no difference in the activation properties of thetransition; both accept the same timing constraints.

From the three cases of the maximum, ∪, we can see that in the first andthird case, the restrictions were equivalent between asynchronous and syn-chronous. However, in the second case, α ∈ S1 ∧α 6∈ S2, there was a possibilityfor additional transitions to be active, as long as n1 < n2. Thus, in general,Conjecture 2 can be adapted into the Lemma below.

Lemma 3 (Maximum, ∪, Language Comparison)

L0max,∪ ⊂ L0

∩

Proof: As shown in the above constructions (Figures 4.9 – 4.11)

4.3.3 Minimum, ∪Similarly to above we will now examine the essential language for the mini-mum, ∪, composition.

Conjecture 3 (Minimum, ∪ Comparison)

L0min,∪ L


The first of the three scenarios for minimum, ∪, is shown in Figure 4.12. Itcan be seen that it is identical to the first scenario for the minimum, ∩, as thedetermining factor here is the minimum function. The set combining element

22



Z// (q′1,q′2)

(a) Asynchronous

(q1,q2)α,X>n1∧Y >n2

X,Y // (q′1,q′2)

(b) Synchronous

Figure 4.12: α ∈ S1 ∧ α ∈ S2

(∪, ∩) only affects the status of the second scenario. We can see here that, asin the minimum, ∩, example, there are additional time steps accepted by theasynchronous composition; any steps whose clock valuation occurs betweenmin(n1, n2) and max(n1, n2).


Z// (q′1,q′2)

(a) Asynchronous


X,Y // (q′1,q′2)

(b) Synchronous

Figure 4.13: α ∈ S1 ∧ α 6∈ S2

In the second scenario, α ∈ S1 ∧ α 6∈ S2, shown in Figure 4.13, the situationis similar to the second scenario for maximum, ∪. However, we now have theminimum, not maximum, and thus the timed step behaviour is reversed. Thereare now two possible cases:

• n1 ≤ n2 – The synchronous transition has an identical constraint to theasynchronous, and the timed step acceptance is equivalent.

• n1 > n2 – The synchronous transition is more restrictive than the asyn-chronous, and, as such, timed steps whose clock valuation falls between(n1, n2) will now be active on the asynchronous, but not on the syn-chronous.

(q1,q2)α,true

Z// (q′1,q′2)

(a) Asynchronous

(q1,q2)α,true

X,Y // (q′1,q′2)

(b) Synchronous

Figure 4.14: α 6∈ S1 ∧ α 6∈ S2

In the third scenario, displayed in Figure 4.14, we again have the trivial case,in which α is a non-ignored symbol, and, as such, the timed step acceptance isequivalent among the two transitions.

For the minimum, ∪, composition, we now have two scenarios in which it ispossible for the asynchronous composition to allow additional timed steps. Inone of these two, the case in the second scenario in which n1 was the smallestclock delay, there is the possibility for the timed step acceptance to be identical.

23


In the final scenario, we have the trivial case in which timed step acceptance isagain identical. Thus we can complete Conjecture 3 to that displayed in Lemma4.

Lemma 4 (Minimum, ∪, Language Comparison)

L0min,∪ ⊃ L0

∩

Proof: As shown in the above constructions (Figures 4.12 – 4.14)

4.3.4 Maximum, ∩For the final compositional method, maximum, ∩, we consider the situationwhere the slowest sensor is chosen, but only on those inputs that are commonto both automata. This could be the case where maximum ‘safety’ was required,but non-ignored symbols were assumed to be perfectly read by the sensor.

Similarly to the other language comparisons we will be considering the sta-tus of the following class of conjecture:

Conjecture 4 (Maximum, ∩, Comparison)

L0max,∩ L



Z// (q′1,q′2)

(a) Asynchronous

(q1,q2)α,X>n1∧Y >n2

X,Y // (q′1,q′2)

(b) Synchronous

Figure 4.15: α ∈ S1 ∧ α ∈ S2

In the first scenario, α ∈ S1 ∧ α ∈ S2, shown in Figure 4.15 the case isidentical to that of maximum, ∪. Both transitions are active on the same timingconstraints, requiring timed steps to have a clock valuation of greater than themaximum of n1 and n2.

(q1,q2)α,true

Z// (q′1,q′2)

(a) Asynchronous


X,Y // (q′1,q′2)

(b) Synchronous

Figure 4.16: α ∈ S1 ∧ α 6∈ S2

In α ∈ S1 ∧ α 6∈ S2, the second scenario, shown in Figure 4.16, we can seethat since the asynchronous transition has no restriction, whilst the synchronouscomposition has a restriction, X > n1, there are thus timed steps which areactive under the asynchronous composition, but not under the synchronous.All timed steps whose clock valuations are less than n1 will behave differentlyunder the two methods.

24


(q1,q2)α,true

Z// (q′1,q′2)

(a) Asynchronous

(q1,q2)α,true

X,Y // (q′1,q′2)

(b) Synchronous

Figure 4.17: α 6∈ S1 ∧ α 6∈ S2

As with the other methods, the third scenario, where α 6∈ S1 ∧ α 6∈ S2, istrivial. We can thus see that the timed words active on both transitions wouldbe identical.

For the maximum, ∩, compositional language, there is only one scenario inwhich there is a difference in the timed word activation, similarly with min-imum, ∩. In the second scenario, the asynchronous transition has no timingconstraint, and, as such, the overall language acceptance was more lenient forthe asynchronous automaton. We can complete the earlier conjecture to thatshown in Lemma 5.

Lemma 5 (Maximum, ∩, Language Comparison)

L0max,∩ ⊃ L0

∩

Proof: As shown in above constructions (Figures 4.15 – 4.17)

4.3.5 Language Conclusions

It is surprising to note that there is only one situation where the asynchronouslanguage is a subset of the intersection language, maximum, ∪. This is the mostrestrictive situation, where the most symbols are ignored for the longest lengthof time. A method for synchronous composition was considered but was nothighly applicable, and should not be used. This was due to both its restrictionto the essential language, and its inability to accurately distinguish the timingvalues, requiring the choice of either the minimum or maximum clock value oneach composite state.

The significance of these results is described in the next chapter and fur-ther work is proposed to complete the language theory relative associated withpolling systems.

25

Chapter 5

Conclusion

Synchronous composition was initially examined as a model for strict languageintersection. However, due to the limitations on the clock constraints within PLCautomata, it was shown that PLC automata are not closed under this operation.This led to the development of a structure based asynchronous method, whichwe examined from a purely intuitive level in Section 3.3. Their similarity instructure, and differences in timing constraints were demonstrated.

For simplicity we defined the essential language, to allow for easier manip-ulation of the languages involved. Although this did remove some of the gener-ality of the compositional methods, the essential language for PLC automata isvery similar to the general languages for standard timed automata. With stan-dard timed automata, symbols which occur outside of their timing restrictionscause the automata to crash. However PLC automata allow these symbols, butsimply ignore them. The essential language restricts the PLC automata to ac-cepting only words not containing these ignored symbols. In this model the PLCautomata now react in a similar manner to the standard timed automata.

The impact of the essential language on the synchronous composition is thatalthough it is now possible to build a synchronous result, the composition isreduced to a highly restricted version of one of the asynchronous methods.

We showed how the various asynchronous languages compared with theessential version of the intersection language of the initial automata. It wasinteresting to note that three of the methods resulted in a language which wasa superset of the intersection language. However the maximum, ∪, methodaccepted a subset of the intersection language. This can be attributed to itsuse of the most restrictive conditions in both timing constraints and symbolrestrictions. Although this method was identical to the synchronous method inall but one scenario, this was one of the more common scenarios. In this, onestate ignores a symbol, but the delay for the second state is larger. Thus in thecomposite result the symbol is ignored for longer than in either of the initialstates.

5.1 Further Work

A very preliminary analysis of a method for complementing PLC automata wasmade. However it presents several issues, and thus remains incomplete. The

26

Conclusion PLC Automata Composition

method used was a conversion from PLC Automata to Event-Recording TimedAutomata [AFH94], which have been proven to be closed under complementa-tion. We would then complement the resultant event-recording automaton, andattempt to convert it back to PLC automaton. The conversion from PLC–ERA,however was non trivial. This appears to result in an infinite state expansion,because every action, ignored or not, must reset clocks in an ERA. The con-version thus has a problem with ignored symbols, and attempts to count thenumber of occurrences of an ignored symbol until it is finally past the clockrestriction. However, since there can be an arbitrary number of these ignoredoccurrences, it is impossible to build a finite automaton to model the result.It should be noted that these problems do not occur when using the essentiallanguage, as keeping track of ignored symbols is no longer necessary.

27

Bibliography

[AD94] R. Alur and D. Dill. A Theory of Timed Automata. Theoretical Com-puter Science, 126:183–235, 1994.

[AFH94] R. Alur, L. Fix, and T. A. Henzinger. A determinizable class of timedautomata. In Proc. Sixth International Conference on Computer AidedVerification, LNCS818, pages 1–13. Springer Verlag, 1994.

[DFMV98] H. Dierks, A. Fehnker, A. Mader, and F. W. Vaandrager. Operationaland Logical Semantics for Polling Real-Time Systems. In A. P. Ravnand H. Rischel, editors, Proceedings of the Symposium on FormalTechniques in Real-Time and Fault-Tolerant Systems, volume LNCS1486, pages 29–40, Lyngby, Denmark, 1998. Springer Verlag.

[Die97] H. Dierks. PLC-Automata: A new class of Implementable Real-Time Automata. In M. Bertran and T. Rus, editors, Transforma-tion Based Reactive Systems Development: ARTS’97, volume 1231 ofLNCS, pages 126–137. Springer-Verlag, 1997.

[Tho90] W. Thomas. Automata on infinite objects. In J. van Leeuwen, ed-itor, Handbook of Theoretical Computer Science: Formal Models andSemantics, pages 133–192. Elsevier, 1990.

28

Documents

PLC-Automaton Composition · tomata, described by H. Dierks, the problem of the composition of PLC automata is presented. Two methods for performing this composition are described,