On-Line Identification of Discrete Event Systems by ... - Poliba

On-Line Identification of Discrete Event Systems by Interpreted Petri Nets Mariagrazia Dotoli, Member, IEEE, Maria Pia Fanti, Senior, IEEE, Agostino Marcello Mangini

Abstract—The paper proposes an on-line identification strategy for Discrete Event Systems (DES). The identifier stores a sequence of events and the corresponding output symbols. Moreover, by solving an integer linear programming problem, an identification procedure synthesizes an Interpreted Petri Net (IPN) modeling the DES. More precisely, we assume that the fixed numbers of places are given and that a finite sequence of transitions and the corresponding markings are completely or partially known. Moreover, the identification algorithm working in real-time identifies the IPN assuming the DES dynamics deterministic, i.e., the event occurrence from a given state yields only one new state. Keywords: Discrete event systems, Petri nets, identification, integer linear programming.

T

I. INTRODUCTION

HE term

identification is an expression for the problem of “determining the input-output relationships of a black box by experimental means” [17]. In particular, we consider the identification problem in the context of Discrete Event Systems (DES). The DES behavior can be described by a language that specifies all the admissible sequences of events that the DES is capable of “processing” or “generating” [3]. The problem of identifying a language and an automaton from finite data has been explored by Gold [8] that shows that the problem of determining a finite automaton accepting positive samples of a regular language is NP complete. Moreover, Angluin proposes in [1] a learning algorithm that constructs a finite automaton accepting a given regular language. Starting from these pioneer contributions, authors have shown different applications of DES identification, e.g. the synthesis of supervisors [10] and the solution of diagnostic problems [16]. Petri Nets (PN) are a standard model for the study of DES [5] and in [10] an algorithm is presented for the construction of a free-labeled PN model from the knowledge of a finite set of its firing sequence. An analogous problem is solved by Bourdeaud’huy and Yim [2] that propose a method to synthesize a PN on the basis of a set of structural constraints: the PN obtained by the solution are used to specify the system Manuscript received March 22, 2006. M. Dotoli is with Dipartimento di Elettrotecnica ed Elettronica Politecnico di Bari, Italy (e-mail: [email protected]). M. P. Fanti is with Dipartimento di Elettrotecnica ed Elettronica, Politecnico di Bari, Via Re David 200, 70125 Bari, Italy (corresponding author: phone: +39 080 5963643; fax: +39 080 5963410; e-mail: [email protected]). A. M. Mangini is with Dipartimento di Elettrotecnica ed Elettronica, Politecnico di Bari, Italy (e-mail: [email protected]).

controller. Moreover, Giua and Seatzu [6] propose a linear algebraic approach for the identification of a PN from the knowledge of the DES language, i.e., the finite set of strings generated by the set of transitions. The authors solve the identification problem by Integer Linear Programming (ILP) that is applied to λ -free labeled PN [13]. This paper deals with the identification problem of DES using an on-line identification strategy that records a sequence of events and the resulting output response sequence. A similar approach has been applied by Chung et al. [4] that propose an on-line modeling refinement able to update incrementally the observed sample path. Moreover, Meda et al. [11] use an incremental synthesis approach to identify an Interpreted Petri Net (IPN) [14] modeling a DES on the basis of the partial knowledge of input and output symbols. IPN are an extension of PN that allows representing the output signals generated when a marking is reached and associating labels with transitions and events. The identification approach here proposed records on-line a word of events generated by the DES starting from an initial state. In addition, the system dynamics is supposed deterministic, i.e., the event occurrence from a given state yields only one new state. However, a kind of non-determinism is here associated to events that are indistinguishable because two transitions may share the same label [7]. Moreover, an algorithm stores in real time the occurred event and the system output. On the basis of the observed behavior, a set of constraints and a metric, i.e., an indicator of the IPN size [15], are defined. Consequently, an ILP problem is established in order to minimize the selected metric. Hence, the algorithm refines recursively the IPN modeling the DES using the on-line solution of the ILP problem. The paper is organized as follows. Section II summarizes some basic definitions and notations on PN and IPN, Section III defines the DES identification problem and proposes a solution technique with an on-line identification algorithm. Moreover, Section IV presents two examples showing how the methodology works and Section V draws the conclusions. II. BASIC DEFINITIONS ON PN AND IPN The state space of DES satisfies the two properties that it is a discrete set and the associated state transition mechanism is event-driven. PN are a widely used tool for the description of the structure and the dynamics of DES [12]. This section provides some basic definitions on PN and IPN used in this paper [3, 13, 14].

A. Petri Nets Definition 1: A PN is a bipartite digraph described by the four-tuple N=(P, T, Pre, Post) where P is a set of places with cardinality m, T is a set of transitions with cardinality n, Pre: P×T→Ν and Post: P×T→Ν are the pre- and post-incidence matrices respectively, which specify the arcs connecting places and transitions. More precisely, for each p∈P and t∈T element Pre(p,t) (Post(p,t)) is equal to a natural number indicating the arc multiplicity if an arc going from p to t (from t to p) exists, and it equals 0 otherwise. Note that Ν is the set of non-negative integers. The m×n incidence matrix of the net is defined as follows: C(p,t)=Post(p,t)-Pre(p,t).

(1)

For pre- and post-sets we use the dot notation, e.g., •p={t∈T: Pre(p,t)>0}. Moreover, a transition t∈T is called source transition if it has no input arcs, i.e. it holds •t= ∅ . The state of a PN is given by its current marking, which is a mapping M: P→Ν, assigning to each place of the net a non-negative number of tokens. A PN system 〈 N , M 0 〉 is a net N with an initial marking M0. A transition tj∈T is enabled at a marking M if and only if (iff) for each p∈•tj, it holds: M(p)≥Pre(p,tj)

(2)

and we write M[tj> to denote that tj∈T is enabled at a marking M. When fired, tj produces a new marking M’, denoted by M[tj>M’ that is computed by the PN state equation: G M’=M+C t j , (3) G where t j is the n-dimensional firing vector corresponding to

the j-th canonical basis vector. Let σ = tα1 , tα 2 ,..., tαk be a sequence of transitions (or firing sequence) and let its length k be indicated by | σ | and given by the number of transitions that σ contains. If a transition t∈T appears in the sequence σ , we write t∈ σ . In addition, we denote by | σ |t the number of occurrences of transition t in σ . Moreover, we use the notation M[ σ >M’ to denote that the sequence of enabled transitions σ may fire at M yielding M’. The firing vector of G σ is denoted by σ . A marking M is said reachable from 〈 N , M 0 〉 iff there exists a firing sequence σ such that M0[ σ >M. The set of all markings reachable from M0 defines the reachability set of 〈 N , M 0 〉 and is denoted by R(N,M0). B. Interpreted Petri Nets A language may be a formal way describing the behavior of a DES. The event set E is viewed as an alphabet and L⊂E* is the set of all words (sequence of events) generated by the system. Moreover, if a PN N=(P, T, Pre, Post) is used to model the DES, the system events are associated with transitions. In addition, state information is described by the place markings capturing the conditions that govern the DES.

However, each transition of T does not necessarily correspond to a distinct event from the set E, and vice-versa. Moreover, the system state could be partially observed from the system output. Hence, this leads to the introduction of IPN [13, 14]. Definition 2: An IPN is the eight-tuple Q=(P, T, Pre, Post, E, Y, λ , Ψ ) where: N=(P, T, Pre, Post) is a PN; E={e} is the event set of cardinality ν for transition labeling; Y={y} is the set collecting the output symbols y associated with places; λ : T→E is the transition labeling function that assigns to each transition t∈T a symbol e∈E. The labeling function is λ -free according to [13, 5], i.e., the same label e∈E may be associated to more than one transition while no transition may be labeled with the empty string ε . Moreover, Ψ : R 〈 N , M 0 〉 →{Y}q is an output function and q is the number of available outputs associated with places. An IPN system 〈Q, M 0 〉 is an IPN Q with initial marking M0. Remark 1. We assume that the output set of symbols is Y=Ν and the output function Ψ is a q×n matrix Ψ = {Ψ (i, j )} where the i-th row vector Ψ (i, ⋅) is the transpose of the elemental vector corresponding to the j-th canonical vector. Consequently, Ψ imposes a correspondence between the i-th output symbol and the j-th place by the relation y= Ψ M, where y is the measured marking vector of dimension q. Definition 3: Given an IPN Q, a place pi ∈P is said measurable if ∃ j∈{1,…,q} such that Ψ (i,j)=1. We denote Pm the set of measurable places and Pnm the set of non measurable places. Now, we recall a definition about determinism [7] as a behavioral property depending on the reachable set. Definition 4. The IPN system 〈Q, M 0 〉 with labeling function λ is said deterministic if ∀ M∈R(N,M0) and ∀ t1,t2∈T: t1≠t2, λ (t1)= λ (t2), M[t1> → ¬M[t2>, i.e., any two transitions with the same associated symbol can not have the same input places. The set of all firing sequences in 〈Q, M 0 〉 is denoted by L(Q,M0)={ σ ∈T*  M0[ σ >} and for each σ ∈L(Q,M0) with σ = tα1 , tα 2 ,..., tαk the evolution of the net is the following: M0 [tα1 〉 M α1 [tα 2 〉...[tα k 〉 M αk .

(4)

We denote by w the word of events associated to the sequence σ with w= λ (σ ) , using the extended form of the transition labeling function λ : T* → E* in the usual manner. Moreover, the empty string ε is associated with the word of

null length. The language generated by the IPN system 〈Q, M 0 〉 is:

events and the corresponding sequence of output vectors yαi for i=1,…,k, that are collected in the (q×k) matrix Y with

(5)

α α α Y(.,i)= yαi . Denoting by M0 [t β 1 〉 M α1 [t β 2 〉...[tβ k 〉 M αk the

This section defines the DES Identification Problem (IP) and proposes a technique to obtain a deterministic IPN modeling the DES. Let us introduce the following PN set:

evolution of the net corresponding to the sequence σ , each Y (., j )  marking is M α j =  nm  for j=1,…,k, where M αnm j  Mα j    indicates the marking of the non measurable places. Hence, we define a set of linear algebraic constraints and a metric to minimize establishing an ILP problem. The ILP solution

E

L (Q,M0)={w∈E*  λ (σ ) = w , σ ∈T*, M0[ σ >}. III. THE DES IDENTIFICATION PROBLEM

D={N=(P, T, Pre, Post) : Pre∈Ν m×n , Post∈Ν m×n }. (6) The DES identification problem is stated as follows. Identification Problem (IP): Let us consider a DES and let E be the set of events of the DES of cardinality v. Moreover, let w∈E* be a sequence of events generated by the DES, Y be the set of output symbols and y∈Ν q be the output vector of dimension q . Chosen a set of places P of cardinality m≥q, a set of transitions T of cardinality n≥v and an output function Ψ , the IP consists in determining an IPN system 〈Q, M 0 〉 deterministic over a labeling function λ , with Q=(P, T, Pre, Post, E, Y, λ , Ψ ), N∈D and M0∈Ν w∈LE(Q,M0). w = eα1 ,..., eαk

DEDS

m

such that

yα1 ,..., yαk

IPN System

Fig. 1 The scheme of the identification technique.

A. The Proposed Identification Technique A schematic representation of the proposed identification technique to solve the defined IP is shown in Fig. 1. We consider a DES whose dynamics is driven by the event set E and provides the output vector y∈Ν q . On the basis of the available knowledge of the system, we choose a set of places P of cardinality m≥q and an output function Ψ , imposing the relation between places and output symbols. We present the identification procedure in two steps. In the first step we assume that the labeling function λ and the complete set T of transitions are known. Hence, we denote by Te={ti ∈T | λ(ti )=e}={ t1et2e ...t ze } the set of transitions e

associated with the same event e and ze=|Te|. Consequently, if we observe a sequence of events α

1

2

α

w = eα1 eα 2 ...eα k = λ(tβ 1 )λ(tβ 2 )...λ(tβ k ) , then we infer α α

α

σ = tβ 1 tβ 2 ...tβ k with tβαii ∈ T 1

2

k

k

eαi

. Moreover, assuming the

DES in an unknown initial state, we record the sequence w of

2

k

determines the PN N∈D and the vector M0∈Ν m so that the IPN system 〈Q, M 0 〉 such that w∈LE(Q,M0) is singled out. In the second step of the identification procedure we propose a recursive algorithm that, starting from the first observed event, can build progressively the set of associated transitions and the labeling function λ , under the hypothesis of system determinism. A this point it is possible to introduce the following theorem. Theorem 1: An IPN system 〈Q, M 0 〉 deterministic over the labeling function λ is a solution of the IP, with N∈D and M0∈Ν m , if it satisfies the following set of linear algebraic constraints:

ξ ( w, Y, λ,n) =

Identifier Algorithm

α

1

(7)

 Pre , Post ∈ N mxn  M0 ∈ N m  G G G  Post T ⋅ 1mx1 + PreT ⋅ 1mx1 ≥ 1nx1 G G G   Post ⋅ 1nx1 + Pre ⋅ 1nx1 ≥ 1mx1  G =  M - Pre tGα1 ≥ 0 0 m×1 β1  ∀tα i : λ(tαi ) = e with e ∈ w and i ≠ 1 Pre tGα i ≤ M αi αi ai-1 βi βi  βi Gα Gα  1 1  M 0 + Post tβ1 − Pre tβ1 = Mα 1  α G G ∀tβ i : λ(tαβ i ) = eα i with eαi ∈ w and i ≠ 1 Post tβαi − Pre tβα i = Mα i - Mα i-1 i i i  i

G G where 1mxn ( 0mxn ) is a matrix of dimensions m × n with each element being 1 (0).

Proof: The first two constraints of the set ξ (w,Y, λ ,n) are obvious. Since we have to impose that ∀ t∈T ∃ p∈P such that either t∈p• or t∈•p, it holds G G G (8) Post T ⋅ 1mx1 + PreT ⋅ 1mx1 ≥ 1nx1 . Analogously, the following constraints imposes that ∀ p∈P ∃ t∈T such that either p∈t• or p∈t•: G G G (9) Post ⋅ 1nx1 + Pre ⋅ 1nx1 ≥ 1mx1 .

By the definition of IP and the assumption of deterministic IPN, if the sequence α

α

α

1

2

k

w = eα1 eα 2 ...eα k = λ(tβ 1 )λ(tβ 2 )...λ(tβ k ) is observed, then

we

α α

α

α

σ = tβ 1 tβ 2 ...tβ k

infer

1

α

2

k

and

α

M0 [t β 1 〉 Mα1 [t β 2 〉...[tβ k 〉 Mα k . Consequently, by condition 1

k

2

(2) the following constraints are derived: G Gα M 0 - Pre t β 1 ≥ 0m×1 1

(10)

and α

α

i

i

∀tβ i : λ(tβ i ) = eα i with eα i ∈ w and i ≠ 1 Gα Pre tβ i ≤ M ai -1

(11)

i

Moreover, by the state equation (3), it must hold: Gα Gα M0 + Post t β 1 − Pre t β 1 = M a1 1

1

(12)

and α

α

i

i

∀tβ i : λ(tβ i ) = eαi with eαi ∈ w and i ≠ 1 Gα Gα Post tβ i − Pre tβ i = Mα i - Mα i -1 i

.

i

(13) We remark that in constraints (11) and (13) for each Y (., j )  marking M α j =  nm  vectors Y(.,j) with j=1,…,k  Mα j    represent known terms. In general, there is not only one IPN satisfying the constraint set ξ (w,Y, λ ,n). Hence, to select an IPN Q and an initial marking M0 , we introduce a metric as a linear m

function φ: D× Ν → Ν [15] and we choose the PN N∈D and the marking M0 ∈Ν

m

that satisfy the constraint set and

minimize a metric φ . Proposition 1: A solution of the IP is obtained solving the following ILP problem: G G G min φ ( Pre , Post , M 0 ) = 11xm ⋅ ( Pre + Post ) ⋅ 1nx1 + 11xm ⋅ M 0

On the other hand, if N has to be restricted, then we have to add the following constraints: ( Pre + Post ) ≤ 1mxn .

Moreover, if ∀ p∈P ∃ t∈T it holds p∈t•, the following constraint has to be verified: G G Pre ⋅ 1nx1 ≥ 1mx1 . (17) Analogously, if ∀ t∈T ∃ p∈P such that t∈p• then it must hold: G G Post T ⋅ 1mx1 ≥ 1nx1 . (18) To evaluate the computational complexity of the optimization problem defined by (7), we recall that the primary determinants of the computational difficulty of an ILP problem are the number of integer variables and the special structure of the problem [9]. It is easy to infer that in the worst case the unknowns are θ =m(2n+1+|w|) in number. B. The Algorithm Implementing the Identification Technique We introduce the following Identification Algorithm (IA) that is applied on-line to recursively obtain the IPN modeling the DES with language L. The procedure refines the IPN at each event occurrence obtaining an IPN that progressively approaches the IPN such that LE(Q,M0)=L. Identification Algorithm (IA) G 1. w := ε , σ := ε , i:=1, Y := 0q×k For j = 1,...,ν

G z j := 0 , K j := 0 q×k endfor 2. Wait until a new event eαi ∈ E and the corresponding

output vector yαi ∈ Ν q are observed; j := α i α 3. If i=1 then zj:=zj+1; call t β 1 the transition such that 1 α

α

eα1 = λ (tβ 1 ) ; t:= t β 1 ; Y (⋅,1) := yα1 ; γ := j 1

1

endif Goto 5 4. If i > 1 then α i 4.1. If zj=0 then zj:=zj+1; call t α such that eαi = λ (t β i ) ; βi i i t:= t α ; K j (⋅,1) := yαi − Y (⋅, i − 1) and Y (⋅, i ) := yαi β i

s.t. ξ (w,Y, λ ,n).

(14)

Moreover, some constraints can be added to the above ILP problem if additional structural properties are imposed. For example, if we force the IPN not to contain any source transition, then we impose: G G PreT ⋅ 1mx1 ≥ 1nx1 . (15)

(16)

endif 4.2. If z j > 0 then if j = γ then θ :=2 else θ := 1 endif flag:=0

{

for h ∈ θ ,..., z j

}

two cases are considered.

K j (⋅, h) = yαi − Y (⋅, i − 1)

if

flag:=1 endif endfor 4.2.1. If

then

h := h

and

α 4.1. Case z j =0. Transition t β i ∈ T is associated to event i

eαi by function λ . Hence, the first column of K j

stores yαi − Y (⋅, i − 1) . Then the current output yαi is flag=0

zj:=zj+1,

then

K j (⋅, z j ) := yαi − Y (⋅, i − 1) ,

call

α

tz i j

the

α α transition such that eαi = λ (t β i ) ; t:= t z i ; i

j

Y (⋅, i ) := yαi αi

4.2.2. else t:= t h and Y (⋅, i ) := yαi endif endif endif 5. w := weαi , σ := σ t , µ := 0 for j = 1,...ν if z j > 1 then µ := µ + z j − 1 endif endfor n := ν + µ

stored in the i-th column of Y. Subsequently, the IA goes to step 5. 4.2. Case z j >0. First, the IA checks whether the current event, which was already observed in the sequence, is equal to the first one (i.e., eαi = e j = eγ ) or not (i.e. ej≠eγ). Hence, the IA checks whether it holds K j (., h ) = yαi − Y (⋅, i − 1) for some h ∈{2,…, z j } in the first case or for some h ∈{1,…, z j } in the second case. If this is the case, then the transition associated to eαi was previously detected and defined by function

λ , otherwise a novel transition has to be associated to the event. Hence, two sub-cases are distinguished. 4.2.1. The transition to associate to the already occurred event eαi is different from those previously defined. Hence, z j is incremented,

solve min φ ( Pre , Post , M 0 ) s.t. ξ ( w, λ , Y , n) C := Post − Pre 6. flag:=0

{

for h ∈ 2,..., zγ

}

if K γ (⋅, h) = yα1 − M 0 , then h := h and flag:=1 endif endfor α α if flag=1 and C (⋅,1) = C (⋅, th 1 ) then C (⋅,η ) := C (⋅, th 1 ) ;

C :=[C (⋅,1: η − 1) C (⋅,η + 1: n)] endif 7. i := i + 1 and goto 2.

C. Discussion on the IA We now discuss and clarify the steps of the above algorithm for DES on-line identification. 1. The variables used in the IA are initialized. 2. The j-th event in the DES input sequence and the DES outputs are recorded. α

3. Transition t β 1 ∈ T is associated to the first observed event 1

yαi − Y (⋅, i − 1) is stored in the z j -th column of K j and eαi is associated to the new transition α

t z i ∈ T by function λ . The IA goes to step 5. j

4.2.2. The transition to associate to eαi was already α

defined by λ and is t h i . The IA goes to step 5. 5. The event sequence w and the transition sequence σ are updated by the new event and associated transition, respectively. Moreover, the number of columns n of the incidence matrix C of the IPN is determined. Hence, the ILP problem is solved and matrix C and the initial marking M0 identify the IPN. α α 6. Since the first transition t β 1 = t1 1 associated to the 1 sequence is a priori assumed different from the other transitions associated to eα1 , the IA checks whether after

the ILP problem solution there exists an h ∈{2,…, z j } such that it holds K γ (⋅, h ) = yα1 − M 0 and, if this is the α

case, if it holds C (⋅,1) = C (⋅, th 1 ) , as well. If both these α

α1 h

eα1 by function λ . Moreover, the first column of Y stores

conditions hold it is inferred that t1 1 = t

output yα1 and γ stores the index of the first event.

matrix C is modified by eliminating the column

4. A transition is associated to an event eαi subsequent to the first one in the observed string, i.e. with i>1. The IA checks whether such an event occurred previously in the sequence (i.e. z j >0) or not (i.e. z j =0). Accordingly, the following

α

and hence

corresponding to t h 1 , otherwise C is unchanged. 7. Finally, the IA returns to recording the events.

IV. EXAMPLES Example 1: Let us consider a DES with E = {e1 , e2 } . Suppose that sequence w = eα1 eα 2 eα3 eα 4 eα5 = e1e2e1e2 e1 is recorded with yα1 =[1 4 1 0 0]T, yα2 =[1 1 1 1 0]T, yα3 =[0 5 1 1 0]T, yα4 =[0 4 0 1 1]T, yα5 =[1 4 1 0 0]T.

Assuming m=5 and Ψ = I m , it holds M ai = yαi for i=1,…,5. Imposing (7) with (17) and (18) leads to the IPN in Fig. 2. In particular, we obtain n=5 and T={ t11, t12 , t13 , t12 , t22 }

{

where T e1 = t11 , t12 , t13

}

{

, T e2 = t12 , t22

}

,

eα1 = λ (t11 ) ,

eα 2 = λ (t12 ) , eα3 = λ (t12 ) , eα 4 = λ (t22 ) and eα5 = λ (t13 ) .

p1

t12

t12

3

p2

p4

t31

the DES states. Assuming that the IPN is deterministic, i.e., each event occurrence from a given state yields only one new state even if two or more transitions can be labeled with the same symbol, an ILP problem is established to select an IPN exhibiting minimal dimensions. Hence, an on-line algorithm is proposed to obtain the IPN modeling the DES with the recursive solution of the defined ILP problem at each novel event occurrence. Moreover, some examples show the efficiency and the simplicity of the proposed method. Obviously, imposing suited structural constraints, different solutions of the ILP problem can be obtained. Moreover, the procedure allows us to provide a solution to the IP even if the knowledge of the system is very poor and on the basis of only one observation of the system behavior, i.e. the observed transition sequence. Future research deals with characterizing the DES language in relationship with the identified IPN language. REFERENCES

4

[1]

4

t22

t11

p5

[2]

p3

[3]

Fig. 2. The IPN identified in Example 1. [4]

t12

t12

p1

p4

3

p2

t31

[6]

4 4

t11

[5]

t22

p5

p3

[7] [8]

Fig. 3. The IPN identified in Example 2.

[9]

Example 2. Let us now consider the DES of the previous example and assume that Ψ ≠ I m . More precisely, the set of

[10]

measurable and non measurable places are Pm= { p1 , p2 , p3 }

[11]

and Pnm= { p4 , p5 } , respectively. Moreover, we assume that w = eα1 eα 2 eα3 eα 4 eα5 = e1e2 e1e2 e1 and the corresponding outputs are yα1 =[1 4 1]T, yα2 =[1 1 1]T, yα3 =[0 5 1]T, T

[12] [13]

T

yα4 =[0 4 0] , yα5 =[1 4 1] . Solving (7) subject to (17) and

[14]

(18) we identify the IPN in Fig. 3. Comparing Figs. 3 and 2, we remark that in the former one the poorer information on outputs yαi leads to a different IPN able to model the

[15]

observed DES behavior. [16]

V. CONCLUSION The paper addresses the identification problem of DES by a real time procedure that synthesizes an IPN model. The identifier stores the observed sequence of DES events and the corresponding outputs that can provide a partial knowledge of

[17]

Angluin D., “Learning regular sets from queries and counterexamples”, Information and Computation, Vol. 75, pp. 87-106, 1987. Bordeaud’Huy T., Yim P., “Synthèse de rèseaux de Petri à partir d’exigences”, Proceedings of the 5th French Conference on Modeling and Simulation, MOSIM’04, Nantes, 8 pages, 2004 (in French). Cassandras C.G., Lafortune S., An Introduction to Discrete Event Systems, Kluwer Academic Publisher, Boston, MA, 1999. Chung S.L., Li C.L., Wu J.C., Wang S.T., “Online modeling refinement for discrete event systems”, Proceedings of the IEEE Conference on Systems, Man and Cybernetics, Vol. 3, pp. 2739-2744, 2003. Gaubert S., Giua A., “Petri net languages and infinite subsets of Ν m”, Journal of Computer and System Sciences 59, pp. 373-391, 1999. Giua A., Seatzu, C., “Identification of free-labeled Petri nets via integer programming”, Proceedings of the 44th IEEE Conference on Decision and Control, pp. 7639-7644, 2005. Giua A., Corona D. and Seatzu C., “State estimation of λ-free labeled Petri nets with contact-free nondeterministic transitions”, Journal of Discrete Event Dynamic Systems, Vol. 15, no. 1, pp. 85-108, 2005. Gold E.M., “Complexity of automaton identification from given data”, Information and Control, Vol. 37, pp. 117-135, 1980. Hillier F.S., Lieberman G.J., Introduction to Operations Research, Mc Graw Hill, New York, NY, USA, 2005. Hiraishi K., “Synthesis of supervisor for discrete event systems allowing concurrent behavior”, IEEE Proceedings, Vol. 5, pp. 13-20, 1999. Meda-Campaña M.E., López-Mellado E., “Incremental synthesis of Petri net models for identification of discrete event systems”, Proceedings of the 41st IEEE Conference on Decision and Control, Vol. 1, pp. 805-810, 2002. Murata T., “Petri nets: properties, analysis and applications”, IEEE Proceedings, Vol. 77, no. 4, pp. 541-580, 1989. Peterson J.L., Petri Net Theory and the Modeling of Systems, Prentice Hall, Englewood Cliffs, NJ, USA, 1981. Ramirez-Trevino A., Riviera-Rangel I., López-Mellado E., “Observability of discrete event systems modeled by interpreted Petri nets”, IEEE Transactions on Robotics and Automation, Vol. 19, no. 4, pp. 557- 565, 2003. Sreenivas R.S., “On minimal representations of Petri net languages”, Proceedings of the 6th International Workshop on Discrete Event Systems, pp. 237-242, 2002. Supavatanakul P., Falkenberg C., Lunze J., “Identification of timed discrete-event models for diagnosis”, Proceedings of the 14th International Workshop on Principles of Diagnosis, Washington D.C., USA, pp. 193-198, 2003. Zadeh L.A., “On the identification problem”, IRE Transactions on Circuit Theory, Vol. 3, pp. 277-281, 1956.