Identifying Equivalence of DEVSs: A Language Approach Moon Ho Hwang Simulation and Control Center, Cubictek, Co., Ltd., 684-1, Deung Chon-Dong, Kang Seo-Ku, Seoul, South Korea e-mail: [email protected] Keywords: DEVS, Logical Analysis, Untimed Event Language, Equivalence

Abstract Logical Analysis functionality is required in the system design and implementation process even if we can simulate behavior of the target system using a system theory approach. This paper presents a way of analysis for a discrete event system specification (DEVS) models with a language approach. Firstly, we are clarifying the language that is a set of event sequences of input events, output events, and/or the internal event. Secondly, we are presenting a identifying way of equivalence states of DEVS in terms of languages traced by the DEVS model. From the testing way of equivalent states, we extend a way of identifying equivalent DEVSs. This procedure can be expected to apply the logical analysis step in the whole design processes.

1

Introduction

For designing a complex and huge discrete event dynamic system(DEDS), DEVS(discrete event system specification)[3] has been recognized one of promising formalisms [2]. As the same as in others, the design processes using DEVS can be divided into modeling, verification, and implementation [1]. In terms of time, verification step can be partitioned into two approaches: timed and untimed. As timed verification approaches, [5] introduced a symbolic DEVS and proposed the solving method satisfying a group of time constraints using the simplex method; [6] proposed a verifying framework using dual specification approach in which DEVS is employed for the operational specification and temporal logic is used for the assertional specification, as the result, they provided a methodology identifying if a DEVS satisfies temporal requirements. As untimed verification methods, [4] proposed an error diagnosis, which was extended from Reiter’s logic; [1] defined the communicating DEVS(C-DEVS), and an equivalence of states and two C-DEVS, and

showed the minimization algorithm of C-DEVS. Similar to [1], we would like to address equivalences associated with DEVS. However, the main different point from [1] is that we employee a language theory so we are defining a language accepted and generated by DEVS first. From that language, this paper defines equivalence of states and equivalence of two DEVSs. Therefore we can answer to following questions: 1. What kinds of events sequences can be generated by DEVS model? 2. Can two states p and q of a DEVS be considered equivalent? 3. Can two DEVS models D1 and D2 be considered equivalent? The language approaches have been proven to be useful for analysis of DEDS. Automata-based language theory has been employed to design a compiler [7] and it has been also applied to supervisory control theory of DEDS [8]. However, the ordinary automata does not have any system concept so that there is no clear idea for interaction with models. For this reason, this paper starts from DEVS formalism that has been based on a system theory so it has input and output functionality, moreover, it has also hierarchical modeling capability. First of all, it needs to define the system behavior of DEVS, thus this paper employs an event languages concept, similar to automata language, from which logical analysis of systems can be extended. We organized this paper as follows. Section 2 presents a brief review of general language. In section 3, we review DEVS and in section 4, trajectory functions and languages associated with DEVS are introduced. Based on the DEVS language, equivalence of two states and equivalence of two DEVSs are identified in Section 5 and Section 6, respectively. Section

7 summarizes this paper and presents some further researches.

2

Alphabet, String, and Language

In this section we shall introduce the general language. For more information of language based on automata, the reader can refer [7]

2.1

Alphabet and String

An alphabet is a finite, nonempty set of events. This paper assumes a event set is a finite, nonempty set so an alphabet means just a event set. For example, A = {0, 1} is the binary alphabet and nonevent is denoted by ø such that for arbitrary event set A, A ∩ {ø} = ∅. A string is a finite sequence of events chosen from some alphabet. For example, 0001 is a string from the binary alphabet A={0,1}. The length of string s, denoted |s| is the number of events in the string. For instance, |0001|=4 and |ø|=0. If A is an alphabet, we define Ak to be the set of strings of length k, each of whose symbols is in A. For example, A0 = {ø}, regardless of what alphabet A is. If A ={0,1}, A2 ={00,01,10,11}, The set of all strings over an alphabet A is conventionally denoted A∗ and the * operator is called the Kleene-closure. And its meaning is A∗ = A0 ∪ A1 ∪ A2 ∪ · · · . Let x and y be strings. The x · y denotes the concatenation of x and y, that is, x = a1 ·a2 . . .·ai and y = b1 · b2 . . . · bj , then x · y = a1 · a2 . . . · ai · b1 · b2 . . . · bj . For any string x, the equations ø · x = x · ø = x hold. That is, ø is the identity for concatenation. And if there is no confusion, we can omit ‘ ·’. For example x · y = xy.

2.2

Language

If A is an alphabet, and L ⊆ A∗ , then L is a language over A. For example, the language of all strings consisting of n 0’s followed by n 1’s, for some n ≥ 0 : {ø, 01, 0011, 000111, ...}. There are some notes: A∗ is a language for any alphabet A. ∅, the empty language, is a language over any alphabet A. {ø}, the language consisting of only the empty string, is also a language over any alphabet A. ∅ = {ø}. The usual set operations, such as union, intersection, difference, and complement with respect to A∗ , are applicable to languages since languages are sets. In addition, we will also defined concatenation of language from string concatenation. Let LA and LB be languages LA ⊆ A∗ and LB ⊆ B ∗ , respectively, then we can define concatenation as LA LB = {sa sb ∈ (A ∪ B)∗ |∀sa ∈ LA , ∀sb ∈ LB }. (1)

We call it Kleene-closure that L∗ = {ø} ∪ L ∪ LL ∪ LLL ∪ · · · . And we can define equivalence of two languages using equivalence of two sets as follows. LA = LB iff ∀a ∈ LA , a ∈ LB , ∀b ∈ LB , b ∈ LA . (2) For example, let LA = {a, ab, ac} and LB = {b, dc}. Then LA LB = {ab, abb, acb, adc, abdc, acdc} and LA = LB because a ∈ LA but a ∈ LB .

3

Review of DEVS

This section introduces a DEVS formalism. Basically DEVS used in this paper is the same as DEVS of [3] with respect to their semantics. However, for easiness of explanation, we added some features.

3.1

Atomic DEVS

For definition of language(we will mention it Section 4.2), we add two features associated with the state set: one is the initial state and the other is the final acceptable state set. An atomic DEVS is define as follows: Definition 3.1 (Atomic DEVS) An atomic DEVS is a 8-tuple, D =< X, Y, S, s0 , SF , ta , δ, λ > where, • X is the input events set. • Y is the output events set. • S is the sequential state set. • s0 is the initial state. • SF ∈ S is the finally acceptable state set whose elements can be considered task completion states. is the time advance function where • ta : S → +∞ 0 +∞ denotes non-negative real numbers. Here, we 0 can define the total state such that Q = {(s, σ)|s ∈ S, 0 ≤ σ ≤ ta (s)} where r is the remaining time in a state s ∈ S. Let tL be the last event time and t be the current time, then the remaining time σ is equal to tL + ta (s) − t. • δ : S × (X ∪ {@}) → S is the transition function where @ ∈ X denotes the internal event or scheduled event occurring when its remaining time reaches 0 without any interruption of x ∈ X. • λ : S × {@} → Y ∪ {ø} is the output function. This output function outputs an event y ∈ Y ∪ {ø} at every scheduled time.

3.2

Coupled DEVS

To make an event flow between DEVSs possible, the way of coupling them are required. To meet this requirement, following coupled DEVS is used. Definition 3.2 (Coupled DEVS)

ˆ : S × (X ∪ {@})∗ → (X ∪ 3. The total trajectory Γ {@} ∪ Y )∗ is defined as

An coupled DEVS is a 7-tuple, N =< X, Y, M, EIC, IT, EOC, select > where • X is the input event set. • Y is the output event set. • M is the sub-component model set where model can be either atomic
DEVS or coupled DEVS. • EIC ⊆ X → m∈M m.X is the external input coupling relation.

• IT ⊆ m∈M m.Y → m∈M m.X is the internal coupling relation.
• EO ⊆ m∈M m.Y → Y is the external output coupling relation. • select : 2M → M is a tie-breaking function which is used to resolve the situation that there are lots of sub-models whose the next event times are identical. The coupled model provides the hierarchical and modular modelling functionality. Theoretically, the coupled DEVS can be considered an atomic DEVS because it is proven that DEVS is closed under coupling operation [3]. Therefore, we would like to pay our attention to just atomic DEVS rather than both of atomic DEVS and coupled DEVS forwards in this paper.

4

 ˆ a) = ø Γ(s, ˆ b), c) ˆ b)cλ(δ(s, Γ(s,

4.2

if a = ø if a = bc

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Languages Generated by DEVS

Based on the trajectories of Section 4.1, we can define some languages associated with DEVS. Suppose that D is a DEVS D =< X, Y, S, s0 , SF , ta , δ, λ >. The transition language, denoted by LX@ (D) is defined as ˆ 0 , a) ∈ SF } LX@ (D) = {a ⊆ (X ∪ {@})∗ | δ(s

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The output language, denoted by LY (D), and is defined as ˆ 0 , a) = b} LY (D) = {b | ∀a ∈ LX@ (D), Λ(s

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The total language or language tracing all kinds of input events X, the internal event @, and output events Y , denoted by LX@Y (D), and is defined as ˆ 0 , a) = b} LX@Y (D) = {b | ∀a ∈ LX@ (D), Γ(s

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Behavior of DEVS based-on its Language

4.1

Trajectories of DEVS

For description of trajectories we are handling here is a sequence of events in Section 2.1. Suppose D is an atomic DEVS D =< X, Y, S, s0 , SF , ta , δ, λ >, s ∈ S and a, b, c ⊆ (X ∪ {@})∗ . Then we can define the following trajectories: 1. The transition trajectory δˆ : S × (X ∪ {@})∗ → S, is defined as  s if a = c ∧ ¬δ(s, c)!    δ(s, c) if a = c ∧ δ(s, c)! ˆ a) = δ(s, ˆ ˆ b), c)!  δ(s, b) if a = bc ∧ ¬δ(δ(s,    ˆ ˆ b), c)! δ(δ(s, b), c) if a = bc ∧ δ(δ(s, (3) where δ(s, c)! denotes that δ(s, c) is defined while ¬δ(s, c)! denotes that δ(s, c) is not defined. ˆ : S × (X ∪ {@})∗ → Y ∗ 2. The output trajectory Λ is defined as  if a = ø or a ∈ X ˆ a) = ø Λ(s, ˆ b), c) if a = bc ˆ b)λ(δ(s, Λ(s, (4)

Example 4.1 (Simple Traffic Light) Consider a traffic light system of Figure 1(a) that has a red(R) light and a green(G) light. Initially, the green light is on and a few seconds later, its green becomes off and its red is turned on. After a few seconds, the red becomes off and the green is turned on. If someone pushes its button, the circulation is performed without spending the circulation time. We model the system as shown in 1(b) using DEVS. And its language LX@Y = {ø, @G : 0, @G : 0@R : 1, @G : 0x, @G : 0x@G : 0, ...}

5

Equivalence of States

As we can see Example 4.2, the language generated by DEVS has a infinite size of event strings. Thus it is not practical to identifying if two language generated by DEVSs are equivalent using the Equation (2.2) of Section 2.2. For solving this problem, this section provides the key idea for identifying if two DEVSs are equivalent. Suppose D is a DEVS and p and q are states of it. If from p and q, D generates the identical LX@Y languages then we can say that p and q are LX@Y equivalent. Following are the formal definition of it.

2 3 4 5 6 7 8 9 10 11 }

add (p, q) to U P ; ∀(p, q) ∈ U P , if( λ(p, @) = λ(q, @) or p ∈ SF , q ∈ SF or p ∈ SF , q ∈ SF ) move (p, q) from U P to DP ; ∀(r, s) ∈ DP , ∀(p, q) ∈ IP (r, s), if( (p, q) ∈ U P ) move (p, q) from U P to DP ; return U P ;

Theorem 5.1 (Undistinguishable = Equivalent) Let D be a DEVS D =< X, Y, S, s0 , SF , ta , δ, λ > and p, q ∈ S. Then states p and q are LX@Y equivalent, that is, LX@Y (D(p)) = LX@Y (D(q)) if (p, q) ∈ U P of Finding UP.

Figure 1: Traffic Light Definition 5.1 (LX@Y Equivalent states) Let D be a DEVS D =< X, Y, S, s0 , SF , ta , δ, λ >. For p, q ∈ S, two states p and q is said to be LX@Y equivalent if LX@Y (Dp ) = LX@Y (Dq ) where LX@Y (Dp )(LX@Y (Dq )) denotes LX@Y of D with its initial state is p(q) instead of s0 . If two states are not LX@Y equivalent, then we say they are distinguishable. That is, state p is distinguishable from state q if (1) λ(p, @) = λ(q, @) or (2) there exists an event string u ⊆ (X ∪ {@})∗ such that ˆ u) and δ(q, ˆ u) is in SF and the other exactly one of δ(p, is not in SF . To find states those are LX@Y equivalent, we make our best efforts to find pairs of state that are distinguishable. The algorithm, which we refer to as the Finding UP: D → 2S×S where D is a DEVS and S is its state set. StatePairSet Finding UP(D) { // D =< X, Y, S, s0 , SF , ta , δ, λ > // U P : StatePairSet;(undistinguished) // DP : StatePairSet;(distinguished) // IP : S × S → 2S×S : input state pair set // IP (r, s):={(p, q)|∀x ∈ X ∪ {@}, // δ(r, x):=p, δ(s, x):=q} 1 ∀p, q ∈ S,

PROOF: By testing of lines 7-10 of Finding UP, U P guarantees that every pair (p, q) ∈ U P has no transition making (δ(p, x), δ(q, x)) enter into DP . That means from (p, q) can remain in U P forever. Therefore, every pair (p, q) ∈ U P , ∀u ⊆ (X ∪ {@})∗ , ˆ u) ∈ SF . So following is true: ˆ u) ∈ SF iff δ(q, δ(p, LX@ (D(p)) = LX@ (D(q)).

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ˆ u) = ø = Γ(q, ˆ u) reSuppose u = ø, then Γ(p, gardless (p, q) is in U P or not. For a (p, q) ∈ U P , suppose u ∈ LX@ (D(p))(= LX@ (D(q))) and u = vx where v ⊆ (X ∪ {@})∗ , x ∈ X ∪ {@} ∪ {ø} and let ˆ v) = r and δ(q, ˆ v) = s. Since there is no path from δ(p, (p, q) ∈ U P to DP so (r, s) must be in U P . Moreover, by line 4 and 5 of Finding UP, λ(r, @) = λ(s, @) ˆ u) = Γ(p, ˆ v)xλ(r, x) = for (r, s) ∈ U P . Thus Γ(p, ˆ v)xλ(s, x) = Γ(q, ˆ u). Therefore, Γ(q, LX@Y (D(p)) = LX@Y (D(q)).

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Example 5.1 (Equivalent States) For modelling of Example 4.2, suppose that someone builds a DEVS model as shown in Figure 2. Let’s test there are equivalent state pairs by using the algorithm Finding UP. First, for difference test in the lines 4-6, the distinguishable pair set(DP ) is calculated like DP={ (GE1,GX1), (GE1,RS1), (GE1,RE1), (GE1,RX1), (GE1,GS1), (GE1,GX2), (GE1,RS2), (GE1,RE2), (GE1,RX2), (GE1,GS2), (GX1,RS1), (GX1,RE1), (GX1,RX1), (GX1,GS1), (GX1,GE2), (GX1,RS2), (GX1,RE2), (GX1,RX2), (GX1,GS2), (RS1,RE1), (RS1,RX1), (RS1,GS1), (RS1,GE2), (RS1,GX2),

D1 and D2 is undistinguishable. If they are undistinguishable than they are equivalent. That means if LX@Y (D(s10 )) = LX@Y (D(s20 )) and it is the same as LX@Y (D1 ) = LX@Y (D2 ) because LX@Y (D1 ) = LX@Y (D(s10 )) and LX@Y (D2 ) = LX@Y (D(s20 )). Following algorithm is testing if two DEVS D1 and D2 are LX@Y equivalent assuming X1 = X2 = X, Y1 = Y2 = Y, and S1 ∩ S2 = ∅.

Figure 2: Another DEVS for Traffic Light (RS1,RE2), (RS1,RX2), (RS1,GS2), (RE1,RX1), (RE1,GS1), (RE1,GE2), (RE1,GX2), (RE1,RS2), (RE1,RX2), (RE1,GS2), (RX1,GS1), (RX1,GE2), (RX1,GX2), (RX1,RS2), (RX1,RE2), (RX1,GS2), (GS1,GE2), (GS1,GX2), (GS1,RS2), (GS1,RE2), (GS1,RX2), (GE2,GX2), (GE2,RS2), (GE2,RE2), (GE2,RX2), (GE2,GS2), (GX2,RS2), (GX2,RE2), (GX2,RX2), (GX2,GS2), (RS2,RE2), (RS2,RX2), (RS2,GS2), (RE2,RX2), (RE2,GS2), (RX2,GS2) }. So UP={ (GE1,GE2), (GX1,GX2) (RS1,RS2), (RE1,RE2), (RX1,RX2), (GS1,GS2) }. For testing of lines 7-10 of the algorithm, check (GE1, GX1) ∈ DP . IP(GE1,GX1)={(GE1,GS2)}. However, (GE1,GS2) is already in DP but not in UP, so there is no pairs in IP(GE1,GX1) which should be moved from U P to DP . In addition to (GE1,GX1) case, since there is no pairs (p,q) such that (r, s) ∈ DP, (p, q) ∈ IP (r, s) ∩ U P so no pairs should be moved from U P to DP . That means the pairs ∈ U P such as (GE1,GE2), (GX1,GX2) (RS1,RS2), (RE1,RE2), (RX1,RX2), and (GS1,GS2), are equivalent in terms of LX@Y .

6

Equivalence of DEVSs

Based-on their languages, we can define the equivalency between two DEVSs as follows. Definition 6.1 (LX@Y Equivalent DEVSs) Two DEVS D1 and D2 are said to be LX@Y equivalent if LX@Y (D1 ) = LX@Y (D2 ). To compare the two languages generated by two DEVS, we can use the Finding UP algorithm. Imagine one DEVS D whose states are the union of the states of D1 and D2 . After using the Finding UP algorithm, test if the start states of the two original

Bool Is LX@Y Equivalent(D1 , D2 ) { // D1 =< X1 , Y1 , S1 , s10 , S1F , ta1 , δ1 , λ1 > // D2 =< X2 , Y2 , S2 , s20 , S2F , ta2 , δ2 , λ2 > // D =< X, Y, S, s0 , SF , ta , δ, λ > // EP : StatePairSet; 1 S = S1 ∪ S 2 ; 2 s0 = s10 or s20 ; 3 SF = S1F ∪ S2F ; 4 ta (s) = ta1 (s), if s ∈ S1 5 = ta2 (s), if s ∈ S2 6 δ(s, x) = δ1 (s, x), if s ∈ S1 ,x ∈ X ∪ {@} 7 = δ2 (s, x), if s ∈ S2 ,x ∈ X ∪ {@} 8 λ(s, @) = λ1 (s, @), if s ∈ S1 9 = λ2 (s, @), if s ∈ S2 10 EP =Finding UP(D); 11 return ( (s10 , s20 ) ∈ EP ); } Example 6.1 (Equivalency of two DEVSs) Let the DEVS of Figure 1(b) be D1 and the DEVS of Figure 2 be D2 . Let’s test if they are equivAs mentioned alent in terms of their LX@Y s. in the Is LX@Y Equivalent algorithm, first this is merging two DEVS into one so merging state set S = {GE1,GX1,RS1,RE1,RX1,GS1,GE2, GX2,RS2,RE2,RX2,GS2,GE,GX,RS,RE,RX,GS }. s0 = GE1 and SF = {GE1,GE2,GE}. After testing λ differences of Finding UP algorithm(lines 4-6), we can first achieve the undistinguishable pair set U P = { (GE1,GE2),(GE1,GE), (GE2,GE) (GX1,GX2), (GX1,GX), (GX2,GX), (RS1,RS2), (RS1,RS), (RS2,RS), (RE1,RE2), (RE1,RE), (RE2,RE), (RX1,RX2), (RX1,RX), (RX2,RX), (GS1,GS2), (GS1,GS), (GS2,GS) }. And the rest all pairs have been achieved as the distinguishable pair set(DP). For testing of lines 7-10 of the algorithm, let’s check (GE1, GX1) ∈ DP . IP (GE1, GX1) = {(GE1, GS2)}. However, (GE1, GS2) is already in DP but not in U P , so there is no pairs ∈ IP (GE1, GX1) which should be moved from U P to DP . In addition to (GE1,GX1) case, since there is no pairs (p, q) such that {(p, q) ∈ U P ∧(p, q) ∈ IP (r, s)∧(r, s) ∈ DP so no pairs should be moved from U P to DP . That means

the pairs ∈ U P such as (GE1,GE2),(GE1,GE), (GE2,GE) (GX1,GX2), (GX1,GX), (GX2,GX), (RS1,RS2), (RS1,RS), (RS2,RS), (RE1,RE2), (RE1,RE), (RE2,RE), (RX1,RX2), (RX1,RX), (RX2,RX), (GS1,GS2), (GS1,GS), and (GS2,GS) are equivalent in terms of LX@Y . As a result, since the pair (s10 , s20 ) = (GE1, GE) is equivalent so LX@Y (D1) = LX@Y (D2) and we can say that D1 and D2 are equivalent in terms of their LX@Y s.

7

Summary and Further Researches

This paper introduced a logical analysis way based on language theory. We defined languages of input, internal, and output events generated by DEVS. Based on the language, we could define a equivalence of states of a DEVS and it could be extended to identify a equivalence of two DEVSs. The result of this paper can be extended to develop a way of minimizing states of DEVS that can guarantee the equivalent language. That will help us to analysis of any huge and complex models. And the language describing not only events order but exact occurring time of them can be employed by introducing the event segment mentioned in [3]. Now the author have been researching the timed languages generated by an output augmented DEVS and sooner or later he will publish the result in his web page is http://hmh2002.hihome.com.

References [1] Kim, T.G., S.M. Cho and W.B. Lee, 2001, “DEVS Framework for Systems Development: Unified Specification for Logical Analysis, Performance Evaluation and Implementation,” Discrete Event Modeling And Simulation Technologies(H.S.Sarjoughian and F.E. Cellier Editors), Springer-Verlag, New York, Inc., 131-166 [2] Hom Y.C., 1993, “Forward to the Special Issue,” Discrete Event Dynamic Systems: Theory and Applications, 3, 111 [3] Zeigler, B.P.; H. Praehofer; T.G. Kim, 2000,Theory of Modeling and Simulation, (2nd Edition): Integrating Discrete Event and Continuous Complex Dynamic Systems, Academic Press [4] Zeigler,B.P., 1992, “System Formulation of a Theory of Diagnosis from First Principles,” IEEE Transactions on Reliability, 41, no. 1(March):3848

[5] Zeigler, B.P. and S. Chi,1992, “Symbolic Discrete Event System Specification,” IEEE Transactions on Systems, Man, and Cybernetics, 22, no. 6,(Nov./Dec.):1428-1443 [6] Hong, G.P.and T.G. Kim,1996, “A Framework for Verifying Discrete Devent Models Within a DEVSBased System Development Methodology,” Transactions of The Society for Computer Simulation, 13, no. 1: 19-34 [7] Hopcroft J.H.; R. Motwani,; J.D. Ullman, 2001, Introduction to Automata Theory, Languages, and Computation, (2nd Edition), Pearson Education [8] Lin, F and W.M. Wonham, 1990, “Decentralized control and coordination of discrete event systems with partial observation,” IEEE Transactions on Automatic Control, 35, (Dec) : 1330-1337

Author Biographies MOON HO HWANG is a Senior Researcher of Simulation and Control Center(SCC) at Cubictek, Co., Ltd, Seoul, South Korea. He received a BS from Hong-Ik University, Korea, in 1990, and MS and Ph.D. from the Korea Advanced Institute of Science and Technology (KAIST) in 1992 and 1999, respectively, all in industrial engineering. He have developed several simulation softwares since 1998 as the Chief of SCC at Cubictek. His research interests include discrete-event system modelling, analysis, and simulation, visual interactive simulation. He is a member of SCS, ACM, and IEEE. His e-mail address is [email protected] and his web page is http://hmh2002.hihome.com.