Chapter 22The Synthesis Problem of ConcurrentSystems Speci�ed by Dynamic InformationSystemsZbigniew SurajInstitute of MathematicsPedagogical UniversityRejtana 16A, 35-310 Rzesz�ow, Polande-mail: zsuraj@univ.rzeszow.plAbstract.We discuss the synthesis problem of concurrent systems from observa-tions or speci�cation encoded in data table (information system) [Pawlak,1991].In the paper we �rst introduce a new notion of a so-called dynamic informationsystem, and then we apply this notion as a tool for speci�cation of concurrentsystems behaviour [Pawlak,1992], [Pawlak,1997]. Finally, we present two meth-ods of construction from any dynamic information systemDS with its underlyingsystem S, and transition system TS describing the behaviour of DS, a concur-rent model in the form of an elementary net system N [Thiagarajan,1987] withthe following property: a given transition system TS is isomorphic to the transi-tion system associated with the constructed net system N . In the �rst methodwe assume that the data table representing a given dynamic information systemDS contains the whole knowledge about the observed or speci�ed behaviourof the system. For this setting, we adopt a method of construction a solutionof the synthesis problem of concurrent system models suggested by [Desel andReisig,1996]. A solution of the synthesis problem is any net which is constructedusing the concept of regions of transition systems, introduced in [Ehrenfeuchtand Rozenberg,1990]. The second method presented in the paper is based onapproach that a given data table consists of only partial knowledge about thesystem behaviour. Thus, we at �rst compute an extension DS0 of the dynamicinformation system DS, i.e. the system in which the set of all global states ofDS0 is consistent with all rules true in the underlying information system S ofDS, and the set of all global states of DS0 represents the largest extension of Sconsistent with the knowledge represented by S. Next, for �nding a solution ofthe synthesis problem considered here we use the �rst method. This approach isbased on rough set theory [Pawlak,1991] and Boolean reasoning [Brown,1990].We have implemented program on IBM PC generating a net model from a dy-namic information system.In our approach we also use a modi�cation of the process independence def-inition presented in [Pawlak,1992]. This paper is an attempt to present a newapproach to concurrency based on the rough set philosophy.

We illustrate our ideas by an intuitive example of tra�c signal control [Paw-lak,1997].We assume that the reader is familiar with the basic ideas of concurrentsystems [Milner,1989], Petri nets [Murata,1989], [Reisig,1985] and informationsystems [Pawlak,1991].Our results seem to have some signi�cance for methods of explanation of thesystem behaviour. Besides, the proposed approach can be seen as basis for acertain class of control system design [Pawlak,1997], and it could be also usedfor software speci�cation [Hurley,1983].Key words: information systems, rough sets, concurrent systems, Petri nets.1 IntroductionThe synthesis problem of concurrent systems is the problem of synthesizing aconcurrent system model from observations or speci�cation of certain processes.This problem has been discussed for various formalisms, among others: parallelprograms [Lengauer and Hehner,1982], COSY-expressions [Janicki,1985], Petrinets [Krieg,1977], [Ehrenfeucht and Rozenberg,1990], [Nielsen, Rozenberg, andThiagarajan,1992], [Mukund,1992], [Bernadinello,1993], [Desel and Reisig,1996].In the paper we consider the synthesis problem of concurrent systems speci-�ed by a so-called dynamic information systems and denoted by DS. The syn-thesis problem informally can be formulated as follows.Synthesis problem. Let A = fa1; :::; amg be a non-empty �nite set ofprocesses. With every process a 2 A we associate a �nite set Va of its localstates. We assume that the behaviour of such a process system is presented by adesigner in a form of two integrated subtables denoted by S and TS, respectively.Each row in the �rst subtable includes the record of local states of processes fromA, and each record is labelled by an element from the set U of global states of thesystem, whereas the second subtable represents a transition system. Columns ofthe second subtable are labeled by events, rows, analogously as for the underlyingsystem, by objects of interest and entries of the subtable for a given row (state)are follower states of that state. The �rst row in the �rst subtable represents theinitial state of a given transition system.The problem is: For a given dynamic information system DS with its transi-tion system TS, �nd a concurrent model in the form of an elementary net systemN [Thiagarajan,1987] with the property: the transition system TS is isomorphicto the transition system associated with the constructed elementary net systemN . Two approaches here are possible. In the �rst case we assume that the ta-ble representing a given dynamic information system contains all possible statecombinations, i.e. the table contains the whole knowledge about the observedbehaviour of the system. In the second one only a part of possible observationsis contained in the table, i.e. they contain partial knowledge about the systembehaviour only. In the paper we discuss both approaches.

Some relationships of information systems and rough set theory with thesynthesis problem have been recently discussed in [Pawlak,1992], [Pawlak,1997].Our considerations are based on the notion of processes independence. Weapply the de�nition of the total independence of processes which is a modi�cationof the independence de�nition used in [Pawlak,1992]. The main idea of the totalindependence of two sets B and C of processes can be explained as follows: twosets B and C of processes are totally independent in a given information systemS if and only if in S the set of local states of processes from B (from C) does notuniquely determine the set of local states of processes from C (from B). Thisproperty can be formulated by applying the partial dependency and rule notions[Pawlak,1991]. The total independency of processes allows us to obtain our mainresult, i.e. a method for constructing from a given dynamic information systemDS its concurrent model in the form of an elementary net system N with thefollowing property: a given transition system TS is isomorphic to the transitionsystem associated with the constructed net system N . The set of all global statesof DS is consistent with all rules true in the underlying information system Sof DS. The set of all global states of DS represents the largest extension of Sconsistent with the knowledge represented by S.Our method for constructing a Petri net model consists of two phases. In the�rst phase, all dependencies between processes in the system are extracted fromthe given set of global states, the extension of the system is computed and, ifnecessary, a modi�cation of the given transition system is done. In the secondphase, an elementary net system corresponding to the computed extension ofthe given dynamic information system is built by employing a method solvingthe synthesis problem of Petri nets presented in [Desel and Reisig,1996].This paper is an attempt to present a new approach to concurrency basedon the rough set philosophy.A designer of concurrent systems can draw Petri nets directly from a speci�-cation in a natural language. We propose a method which allows automatically togenerate an appropriate Petri net from a speci�cation given by a dynamic infor-mation system and/or rules. This kind of speci�cation can be more convenientfor the designers of concurrent systems than drawing directly nets especiallywhen they are large. The designer of concurrent systems applying our method isconcentrated on a speci�cation of local processes dependencies in global states.These dependencies are represented by an information system [Pawlak,1991],[Pawlak and Skowron,1993], [Skowron,1993a,b], [Skowron and Suraj,1993b,c,d].The computing process of the solution is iterative. In a successive step the con-structed so far net is automatically redesigned when some new dependencies arediscovered and added to a speci�cation. The nets produced automatically byapplication of our method can be simpli�ed by an application of some reduc-tion procedures. This problem is out of scope of this paper. We expect that ourmethod can be applied as a convenient tool for the synthesis of larger systems[Baar,Cohen, and Feigenbaum,1989], [Shapiro and Eckroth,1987].We illustrate our ideas by an example of tra�c signal control [Pawlak,1997].The idea of concurrent system representation by information systems is dueto Professor Z. Pawlak [1992].

It is still worth to mention that discovering relations between observed datais the main objective of the machine discovery area (cf. [ _Zytkow,1991]). Our mainresult can be interpreted as a construction method of all global states consistentwith knowledge represented by the underlying system S of DS (i.e. with allrules true in S). For example, checking if a given global state is consistent withS is equivalent to checking if this state is reachable from the initial state of thenet system N representing DS. It seems that our approach can be applied forsynthesis and analysis of knowledge structure by means of its concurrent models.We assume that the reader is familiar with the basic ideas of concurrentsystems [Milner,1989], Petri nets [Murata,1989], [Reisig,1985] and informationsystems [Pawlak,1991].The text is organized as follows. In Section 2 we recall some basic notions ofrough set theory [Pawlak,1991]. Section 3 describes how to compute a concurrentdata models from information systems. The relationships between dependenciesin information systems and partially (totally) independent sets of processes arediscussed in Subsection 3.1. In Subsection 3.2 we explain the role of reductsas maximal partially independent sets of processes. In particular, we show thatmethods for reducts computing can be applied for computing maximal partiallyindependent sets of processes. Subsection 3.3 deals with maximal totally inde-pendent sets of processes. Section 4 contains a method for generating rules inminimal form, i.e. with a minimal number of descriptors on its the left hand side.The method is based on the idea of Boolean reasoning [Brown,1990] applied todiscernibility matrices de�ned in [Skowron and Rauszer,1992] and modi�ed herefor our purposes. This section realizes the �rst step in the construction of a con-current model of knowledge embedded in a given information system. Section5 and 6 contain basic de�nitions and notation from transition systems and nettheory. In section 7 we de�ne the notion of a dynamic information system andwe state the synthesis problem formally. Section 8 contains the solution of thesynthesis problem based on synthesis of rules describing transition relation ofa given dynamic information system. In the conclusions we suggest some direc-tions for further research related to the representation of information systemsby concurrent models.2 Preliminaries of Rough Set TheoryIn this section we recall basic notions of rough set theory. Among them arethose of information systems, indiscernibility relations, discernibility matrices,functions, reducts and rules.2.1 Information SystemsInformation systems (sometimes called data tables, attribute-value systems, con-dition-action tables, knowledge representation systems etc.) are used for repre-senting knowledge. The notion of an information system presented here is dueto Z. Pawlak and was investigated by several authors (see e.g. the bibliography

in [Pawlak,1991]). Among research topics related to information systems are:rough set theory, problems of knowledge representation, problems of knowledgereduction, dependencies in knowledge bases. Rough sets have been introduced[Pawlak,1991] as a tool to deal with inexact, uncertain or vague knowledge inarti�cial intelligence applications.This subsection contains basic notions related to information systems thatwill be necessary in order to understand our results.An information system is a pair S = (U;A); where U - is a non-empty, �niteset called the universe, A - is a non-empty, �nite set of attributes, i.e. a : U ! Vafor a 2 A; where Va is called the value set of a.Elements of U are called objects and interpreted as e.g. cases, states, pa-tients, observations. Attributes are interpreted as features, variables, processes,characteristic conditions etc.In the paper attributes are meant to denote the processes of the system, thevalues of attributes are understood as local states of processes and objects areinterpreted as global states of the system.The set V = Sa2A Va is said to be the domain of A:For S = (U;A); a system S0 = (U 0; A0) such that U � U 0; A0 = fa0 :a 2 Ag; a0(u) = a(u) for u 2 U and Va = Va0 for a 2 A will be called aU 0�extension of S (or an extension of S, in short). S is then called a restrictionof S0. If S = (U;A) then S0 = (U;B) such that A � B will be referred to as aB � extension of S:Example 1 [Pawlak,1997]. Let us consider an information system S = (U;A)such that U = fu1; u2; u3g; A = fa; b; cg and the values of the attributes arede�ned as in Table 1. U=A a b cu1 1 1 0u2 0 2 0u3 0 0 2Table 1. An example of an information systemThis information system we can treat as a speci�cation of system behaviourconcerning distributed tra�c signals control presented in Figure 1.

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Figure 1. T-intersectionIn this case we assume that attributes a; b; and c denote the tra�c signals,objects labeled by u1; u2; u3 denote the possible states of the observed system,whereas entries of the table 0, 1 and 2 denote colours of the tra�c lights, red,green and green arrow, respectively.In a given information system, in general, we are not able to distinguish allsingle objects (using attributes of the system). Namely, di�erent objects can havethe same values on considered attributes. Hence, any set of attributes dividesthe universe U into some classes which establish a partition [Pawlak,1991] of theset of all objects U . It is de�ned in the following way.Let S = (U;A) be an information system. With any subset of attributesB � A we associate a binary relation ind(B); called an indiscernibility relation,which is de�ned by: ind(B) = f(u; u0) 2 U � U for every a 2 B; a(u) = a(u0)g:Notice that ind(B) is an equivalence relation and ind(B) = Ta 2 B ind(a),where ind(a) means ind(fag):If u ind(B) u0; then we say that the objects u and u0 are indiscernible withrespect to attributes from B. In other words, we cannot distinguish u from u0 interms of attributes in B.Any information system S = (U;A) determines an information functionInfA : U ! P (A� V )de�ned by InfA(u) = f(a; a(u)) : a 2 Ag; whereV = Sa 2 AVa and P (X) denotes the powerset of X . The set fInfA(u) : u 2Ug is denoted by INF(S).Hence, u ind(A) u0 if and only if InfA(u) = InfA(u0) .The values of an information function will be sometimes represented by vec-tors of the form (v1; :::; vm); vi 2 Va; for i = 1; :::;m, where m = card(A). Suchvectors are called information vectors (over V and A ).

Let S = (U;A) be an information system, where A = fa1; :::; amg. Pairs (a; v)with a 2 A; v 2 V are called descriptors. Instead of (a; v) we also write a = vor av.The set of terms over A and V is the least set containing descriptors (overA and V ) and closed with respect to the classical propositional connectives: :(negation), _ (disjunction), and ^ (conjunction), i.e. if �; � 0 are terms over Aand V then :�; (� _ � 0); (� ^ � 0) are terms over A and V .The meaning k � kS (or in short k � k) of a term � in S is de�ned inductivelyas follows: k (a; v) k= fu 2 U : a(u) = vg for a 2 A and v 2 Va;k � _ � 0 k=k � k [ k � 0 k;k � ^ � 0 k=k � k \ k � 0 k;k :� k= U� k � k :Two terms � and � 0 are equivalent, � , � 0, if and only if k � k=k � 0 k. Inparticular we have: :(a = v), Wfa = v0 : v0 6= v and v0 2 Vag.2.2 Rules in Information SystemsRules express some of the relationships between values of the attributes describedin the information systems. This subsection contains the de�nition of rules aswell as other related concepts.Let S = (U;A) be an information system and let B � A. For every a 2=B) wede�ne a function dBa : U ! P (Va) such thatdBa (u) = fv 2 Va : there exists u0 2 U u0 ind(B) u and a(u0) = vg;where P (Va) denotes the powerset of Va.Hence, dBa (u) is the set of all the values of the attribute a on objects indis-cernible with u by attributes from B. If the set dBa (u) has only one element, thismeans that the value a(u) is uniquely de�ned by the values of attributes fromB on u.Let S = (U;A) be an information system and let B;C � A. We say that theset C depends on B in S in degree k (0 � k � 1); symbolically B�!S,kC; if k =card(POSB(C))card(U) , where POSB(C) is the B-positive region of C in S [Pawlak,1991].If k = 1 we write B�!S C instead of B�!S,kC: In this case B�!S C means thatind(B) � ind(C). If the right hand side of a dependency consists of one attributeonly, we say the dependency is elementary.It is easy to see that a simple property given below is true.Proposition 1. Let S=(U,A) be an information system and let B;C;D � A.If B�!S C and B�!S D then B�!S C [D.A rule over A and V is any expression of the following form:

(1) ai1 = vi1 _ ::: _ air = vir ) ap = vpwhere ap; aij 2 A; vp; vij 2 Vaij for j = 1; :::; r.A rule of the form (1) is called trivial if ap = vp appears also on the lefthand side of the rule. The rule (1) is true in S (or in short: is true) if; 6=k ai1 = vi1 ^ ::: ^ air = vir k�k ap = vp kThe fact that the rule (1) is true in S is denoted in the following way:(2) ai1 = vi1 ^ ::: ^ air = vir=)S ap = vp:In the case (2) we also shall say that the values (local states) vi1 ; :::; vir ofprocesses ai1 ; :::; air can coexist in S.By D(S) we denote the set of all rules true in S.Let R � D(S). An information vector v = (v1; :::;vm) is consistent with Rif and only if for any rule ai1 = vi1 ^ ::: ^ air = vir=)S ap = vp in R if vij = vijfor j = 1; :::; r then vp = vp. The set of all information vectors consistent withR is denoted by CON(R).Let S0 = (U 0; A0) be a U 0-extension of S = (U;A). We say that S0 is aconsistent extension of S if and only if D(S) � D(S0). S0 is a maximal consistentextension of S if and only if S0 is a consistent extension of S and any consistentextension S00 of S is a restriction of S.We apply here the Boolean reasoning approach to the rule generation [Skow-ron,1993a].The Boolean reasoning approach [Brown,1990], due to G. Boole, is a generalproblem solving method consisting of the following steps: (i) construction of aBoolean function corresponding to a given problem; (ii) computation of primeimplicants of the Boolean function; (iii) interpretation of prime implicants lead-ing to the solution of the problem.It turns out that this method can be also applied to the generation of ruleswith certainty coe�cients [Skowron,1993b]. Using this approach one can alsogenerate the rule sets being outputs from some algorithms known in machinelearning, like AQ-algorithms [Michalski,Carbonell,and Mitchell,1983], [Skowronand Stepaniuk,1994].2.3 Reduction of AttributesLet S = (U;A) be an information system. Any minimal subset B � A such thatind(B) = ind(A) is called a reduct in the information system S [Pawlak,1991].The set of all reducts in S is denoted by RED(S).Now we recall two basic notions, namely those of discernibility matrix anddiscernibility function [Skowron and Rauszer,1992]], which will help to computeminimal forms of rules with respect to the number of attributes on the left handside of the rules.Let S = (U;A) be an information system, and let us assume that U =fu1; :::; ung, and A = fa1; :::; amg. By M(S) we denote an n � n matrix (cij),

called the discernibility matrix of S, such that cij = fa 2 A : a(ui) 6= a(uj)g fori; j = 1; :::; n.Intuitively an entry cij consists of all the attributes which discern objectsui and uj . Since M(S) is symmetric and cii = ; for i = 1; :::; n;M(S) can berepresented using only elements in the lower triangular part of M(S), i.e. for1 � j < i � n.With every discernibility matrix M(S) we can uniquely associate a discerni-bility function fM(S), de�ned in the following way:A discernibility function fM(S) for an information system S is a Booleanfunction of m propositional variables a�1; :::; a�m (where ai 2 A for i = 1; :::;m)de�ned as the conjunction of all expressions Wc�ij , where Wc�ij is the disjunctionof all elements of c�ij = fa� : a 2 cijg; where 1 � j < i � n and cij 6= ;. In thesequel we write a instead of a�.Proposition 2 gives an important property which enables us to compute allreducts of S.Proposition 2. [Skowron and Rauszer,1992]. Let S =(U;A) be an infor-mation system, and let fM(S) be a discernibility function for S. Then the setof all prime implicants [Wegener,1987] of the function fM(S) determines the setRED(S) of all reducts of S, i.e. ai1 ^ ::: ^ aik is a prime implicant of fM(S) ifand only if fai1 ; :::; aikg 2RED(S).In the following propositions [Pawlak,1991] the important relationships be-tween the reducts and the dependencies are given.Proposition 3. Let S=(U;A)be an information system and let B 2RED(S).If A�B 6= ; then B�!S A�B.Proposition 4. If B�!S C then B�!S C 0, for every ; 6= C 0 � C. In particular,B�!S C implies B�!S fag; for every a 2 C.Proposition 5. Let B 2RED(S): Then attributes in the reduct B are pair-wise independent, i.e. neither fag �!S fa0g nor fa0g �!S fag holds, for any a,a0 2 B; a 6= a0.Below we present a procedure for computing reducts [Skowron and Rau-szer,1992].PROCEDURE for computing RED(S):Input: An information system S.Output: The set of all reducts in S.Step 1. Compute the discernibility matrix for the system S.Step 2. Compute the discernibility function fM(S) associated with the dis-cernibility matrix M(S).Step 3. Compute the minimal disjunctive normal form of the discernibilityfunction fM(S) (The normal form of the function yields all the reducts).

One can show that the problem of �nding a minimal (with respect to car-dinality) reduct is NP-hard [Skowron and Rauszer,1992]. In general the num-ber of reducts of a given information system can be exponential with respectto the number of attributes (i.e. any information system S has at most mover [m=2] reducts, where m=card(A)). Nevertheless, existing procedures forreduct computation are e�cient in many applications and for more complexcases one can apply some e�cient heuristics (see e.g. [Bazan,Skowron, andSynak,1994b], [Nguyen and Skowron,1995], [Skowron,1995], [Skowron,Polkowski,and Komorowski,1996], [Nguyen,1997]).Example2: Applying the above procedure for the information system S fromExample 1, we obtain the following discernibility matrixM(S) presented in Table2 and discernibility function fM(S) presented below:U u1 u2 u3u1u2 a; bu3 a; b; c b; cTable 2. The discernibility matrix M(S) for the information system Sfrom Example 1fM(S)(a; b; c) = (a _ b) ^ (a _ b _ c) ^ (b _ c):We consider non-empty entries of the table (see Table 2), i.e. a; b; b; c anda; b; c; next a; b; c are treated as Boolean variables and the disjunctions a_b; b_cand a_ b_ c are constructed from these entries; �nally, we take the conjuction ofall the computed disjunctions to obtain the discernibility function correspondingto M(S).After reduction (using the absorption laws) we get the following minimaldisjunctive normal form of the discernibility function fM(S)(a; b; c) = (a^ c)_ b.There are two reducts: R1 = fa; cg and R2 = fbg of the system. ThusRED(S) = fR1; R2g.Example 3 illustrates how to �nd all dependencies among attributes usingPropositions 3 and 4.Example 3: Let us consider again the information system S from Example1. By Proposition 3 we have for the system S the dependencies:fa; cg �!S fbg and fbg �!S fa; cg:Next, by Proposition 4 we get the following elementary dependencies:fa; cg�!S fbg; fbg�!S fag; fbg�!S fcg:

3 Computing Concurrent Data Models from InformationSystemsWe base our considerations about independency of processes on the notions ofdependency and partial dependency of sets of attributes in an information systemS. The set of attributes C depends in S on the set of attributes B in S if onecan compute the values of attributes from C knowing the values of attributesfrom B. The set of attributes C depends in S partially in degree k (0 � k < 1)on the set of attributes B in S if the B-positive region of C in S consists of k %of global states in S.A set of processes B � A in a given information system S = (U;A) is calledpartially independent in S if there is no partition of B into sets C and D suchthat D is dependent on C in S. We show that maximal partially independent setsin S are exactly reducts in S. In this way we have a method for computing max-imal partially independent sets (in S) based on methods of reducts computing[Skowron and Rauszer,1992].We say that a set B � A is a totally independent set of processes in S=(U;A)if there is no partition of B into C and D such that D depends on C in S in thedegree 0 < k � 1.In the following we show a method for computing maximal totally indepen-dent sets of processes in S = (U;A). These are all totally independent maximalsubsets of reducts in S.3.1 Dependencies in Information System and Independence ofProcessesIn this section we present two basic notions related to independency of processes.Let S = (U;A) be an information system (of processes) and let ; 6= B � A.The set B of processes is called totally independent in S if and only if card(B) = 1or there is no partition of B into C;D such that C �!S;k D, where k > 0.Let S = (U;A) be an information system (of processes) and let ; 6= B �A. The set B of processes is called partially independent in S if and only ifcard(B) = 1 or there is no partition of B into C;D such that C �!S D.One can prove from the above de�nitions the following properties.Proposition 6. If B is a totally independent set of processes in S and ; 6=B0 � B then B0 is also totally independent set of processes in S.Proposition 7. B is a totally independent set of processes in S if and onlyif card(B)=1 or B � fag �!S,0 fag for any a 2 B.Proposition 8. B is a partially independent set of processes in S if and onlyif card(B)=1 or B consists of B-indispensable [Pawlak,1991] attributes in S only.

3.2 Reducts as Maximal Partially Independent Sets of ProcessesWe have the following relationship between the partially independent sets ofprocesses and reducts:Proposition 9. B is a maximal partially independent set of processes in Sif and only if B 2RED(S); where RED(S) denotes the set of all reducts in S.In order to compute the partially independent parts of a given informa-tion system, �rst we have to execute the presented above procedure generatingreducts (see Section 2).3.3 Maximal Totally Independent Sets of ProcessesIn the previous section we have discussed the problem of construction of thefamily of partially independent sets of processes and a relationship betweenthese sets and reducts. Now we are interested in a construction of all maximaltotally independent sets of processes.From the de�nition of totally independent sets of processes in a given infor-mation system S it follows that for an arbitrary totally independent set B inS there is a reduct C 2 RED(S) such that B � C. Hence to �nd all maximaltotally independent sets of processes it is enough to �nd for every C 2RED(S)all maximal independent subsets of C.To �nd all maximal totally independent sets of processes in S = (U;A) it isenough to perform the following steps:Step 1: T:= RED(S); I:= ffa1g; :::; famgg;Step 2: if (T is empty) then goto Step 4else beginCHOOSE A SET B 2 T;T:= T�fBgend;Step 3: if card(B) � 1 then goto Step 2;L := 0;for every a 2 B doif B � fag�!S,kfag for some k > 0 then T := T [ fB � faggelse L := L+ 1;if L = card(B) then I := I [ fBg;goto Step 2;Step 4. The maximal sets in I (with respect to the inclusion �) are maximaltotally independent sets in S.Let OPT(S) be the set of all rules of the form (1) ai1 = vi1 ^ ::: ^ air =vir=)S a = v; with the left hand side in minimal form (see Section 4). If is inthe form (1) then by L( ) we denote the set fai1 ; :::; airg. It is easy to see thatone can take in the �rst line of Step 1 the instruction T := fL( ) : 2OPT(S)ginstead of T:= RED(S). In this way we obtain more e�cient version of the

presented method. The time and space complexity of the discussed problemis, in general, exponential because of the complexity of RED(S) computing.Nevertheless, existing procedures and heuristics help us to compute all maximalindependent sets for many practical applications.At the end let us note the following characterization of reducts being maximaltotally independent set of processes:Proposition 10. Let S = (U;A) be an information system and let C 2RED(S)with card(C) > 1. C is a maximal totally independent set of processes in S ifand only if for every u 2 U and a 2 C card(dCa (u)) > 1.4 Minimal Rules in Information SystemsIn this section we present a method for generating the minimal form of rules(i.e. rules with a minimal number of descriptors on the left hand side).Let S = (U;A [ fa�g) be an information system and a� 2=A. We are lookingfor all minimal rules in S of the form: ai1 = vi1 ^ ::: ^ air = vir=)S a = v, wherea 2 A [ fa�g; v 2 Va; aij 2 A and vij 2 Vaij for j = 1; :::; r.The above rules express functional dependencies between the values of theattributes of S. These rules are computed from systems of the form S0 = (U;B[fag) where B � A and a 2 A�B or a = a�.First, for every v 2 Va; ul 2 U such that dBa (ul) = fvg a modi�cationM(S0; a; v; ul) of the discernibility matrix is computed from M(S0).ByM(S0; a; v; ul) = (c�ij) (orM , in short) we denote the matrix obtained fromM(S0) in the following way:IF i 6= l THEN c�ij = ;;IF clj = ; and dBa (uj) 6= fvg THEN c�lj = clj \ B ELSE c�lj = ;.Next, we compute the discernibility function fM and the prime implicants[Wegener,1987] of fM taking into account the non-empty entries of the matrixM (when all entries c�ij are empty we assume fM to be always true).Finally, every prime implicant ai1 ^ ::: ^ air of fM determines a rule ai1 =vi1 ^ ::: ^ air = vir=)S a = v, where aij (ul) = vij for j = 1; :::; r, a(ul) = v.Let S = (U;A) be an information system. In the following we shall apply theabove method for every R 2RED(S). First we construct all rules correspondingto nontrivial dependencies between the values of attributes from R and A � Rand next all rules corresponding to nontrivial dependencies between the valuesof attributes within a reduct R. These two steps are realized as follows.(i) For every reduct R 2RED(S), R � A and for every a 2 A�R we considerthe system S0 = (U;R [ fag). For every v 2 Va; ul 2 U such that dRa (ul) =fvg we construct the discernibility matrixM(S0; a; v; ul), next the discernibilityfunction fM and the set of all rules corresponding to prime implicants of fM .(ii) For every reduct R 2RED(S) with card(R) > 1 and for every a 2 R weconsider the system S00 = (U;B[fag), where B = R�fag. For every v 2 Va; ul 2U such that dBa (ul) = fvg we construct the discernibility matrixM(S00; a; v; ul),

then the discernibility function fM and the set of all rules corresponding to primeimplicants of fM .The set of all rules constructed in this way for a given R 2RED(S) is denotedby OPT(S;R).We put OPT(S) = Sf OPT(S;R) : R 2RED(S)g.Let us observe that if ai1 = vi1 ^ ::: ^ air = vir=)S ap = vp is a rule fromOPT(S), then U\ k ai1 = vi1 ^ ::: ^ air = vir kS 6= ;.Proposition 11 [Pawlak,1992]. Let S=(U;A) be an information system,R 2RED(S), and R � A. Let fM(S0) be a relative discernibility function forthe system S0 = (U;R [ fa�g) where a� 2 A� R . Then all prime implicants ofthe function fM(S0) correspond to all fa�g - reducts of S'.Now we are ready to present a very simple procedure for computing an ex-tension S0 of a given information system S: Let OPT(S) be the set of all rulesconstructed as described above.PROCEDURE for computing an extension S0 of S:Input: An information system S = (U;A) and the set OPT(S) of rules.Output: An extension S0 of S.Step 1. Compute all admissible global states of S, i.e. the cartesian productof the value sets for all attributes a from A.Step 2. Verify using the set OPT(S) of rules which admissible global statesof S are consistent with rules true in S.The next example illustrates how to �nd all nontrivial dependencies betweenthe values of attributes in a given information system. At the end of example wegive information about an extension of the information system.Example 4: Let us consider the information system S from Example 1 andthe discernibility function for S presented in Table 2. We compute the set of rulescorresponding to nontrivial dependencies between the values of attributes fromthe reduct R1 of S with b (i.e. those outside of this reduct) as well as the set ofrules corresponding to nontrivial dependencies between the values of attributeswithin the reduct of that system. In both cases we apply the method presentedabove.Let us start by computing the rules corresponding to nontrivial dependenciesbetween the values of attributes from the reduct R1 = fa; cg of S with b.We have the following subsystem S1 = (U;B [ fbg), where B = R1, fromwhich we compute the rules mentioned above:

U=B a c b dBbu1 1 0 1 f 1 gu2 0 0 2 f 2 gu3 0 2 0 f 0 gTable 3. The subsystem S1 = (U;B [ fbg with the function dBb , where B = fa; cgIn the table the values of the function dBb are also given. The discernibilitymatrix M (S1; b; v; ul) where v 2 Vb, ul 2 U , l = 1; 2; 3, obtained from M(S1)in the above way is presented in Table 4.U u1 u2 u3u1 a a; cu2 a cu3 a; c cTable 4. The discernibility matrixM(S1; b; v; ul) for the matrix M(S1)The discernibility functions corresponding to the values of the function dBbare the following:Case 1. For dBb (u1) = f1g : a ^ (a _ c) = a .We consider non-empty entries of the column labelled by u1 (see Table 4),i.e. a and a; c; next a; c are treated as Boolean variables and the disjunctions aand a_ c are constructed from these entries; �nally, we take the conjuction of allthe computed disjunctions to obtain the discernibility function corresponding toM(S1; b; v; ul).Case 2. For dBb (u2) = f2g : a ^ c.Case 3. For dBb (u3) = f0g : (a _ c) ^ c = c.Hence we obtain the following rules: a1=)S b1; a0 ^ c0=)S b2; c2=)S b0.Now we compute the rules corresponding to all nontrivial dependencies be-tween the values of attributes within the reduct R1.We have the following two subsystems (U;C [fcg), (U;D[ fag) of S, whereC = fag, and D = fcg, from which we compute the rules mentioned above:

U=C a c dCcu1 1 0 f0gu2 0 0 f0; 2gu3 0 2 f0; 2gTable 5. The subsystem (U;C [ fcg) with the function dCc , where C = fagU=D c a dDau1 0 1 f0; 1gu2 0 0 f0; 1gu3 2 0 f0gTable 6. The subsystem (U;D [ fag) with the function dDa , where D = fcgIn the tables the values of the functions dCc and dDa are also given.The discernibility functions corresponding to the values of these functionsare the following:Table 5. For dCc (u1) = f0g: a.Table 6. For dDa (u3) = f0g: c.Hence we obtain the following rules:From Table 5: a1=)S c0.From Table 6: c2=)S a0.Finally, the set of rules corresponding to all nontrivial dependencies betweenthe values of attributes within the reduct R1 has the form: a1=)S c0, c2=)S a0.Eventually, we obtain the set OPT(S;R1) of rules corresponding to all non-trivial dependencies for the reduct R1 in the considered information system S:a1=)S b1, a0 ^ c0=)S b2, c2=)S b0, a1=)S c0, c2=)S a0.In a similar way one can compute the set OPT(S;R2) of rules correspondingto all nontrivial dependencies for the reduct R2 in the system S. This set consistsof one kind of rules, i.e. the rules corresponding to all nontrivial dependenciesbetween the values of attributes from R2 with a; c of the form: b1=)S a1, b0 _b2=)S a0, b1 _ b2=)S c0, b0=)S c2, whereas the second set of rules corresponding toall nontrivial dependencies between the values of attributes within the reductR2 is empty, because this reduct has only one element.The set OPT(S) of all rules constructed in this way for the informationsystem S of Example 1 is the union of sets OPT(S;R1) and OPT(S;R2).It is easy to verify that in this case the extension S0 of the system S computedby using our procedure presented above is the same as the original one.Remark 1. The above rules explain behaviour of the system from Figure 1.Remark 2. Our approach to rule generation is based on procedures for thecomputation of reduct sets. It is known that in general the reduct set can be of

exponential complexity with respect to the number of attributes. Nevertheless,there are several methodologies allowing to deal with this problem in practicalapplications. Among them are the feature extraction techniques or clusteringmethods known in pattern recognition [Nadler and Smith,1993] and machinelearning [Michalski, Carbonell, and Mitchell,1983], allowing to reduce the num-ber of attributes or objects so that the rules can be e�ciently generated fromthem. Another approach is suggested in [Bazan Skowron, and Synak,1994a]. Itleads to the computation of only so called the most stable reducts from thereduct set in a sampling process of a given decision table (i.e. a special case ofan information system, see [Pawlak,1991]). The rules are produced from thesestable reducts only. This last technique can be treated as relevant feature ex-traction from a given set of features. The result of the above techniques appliedto a given information system is estimated as successful if rules can be e�cientlygenerated from the resulting compressed information system by the Boolean rea-soning method and if the quality of the classi�cation of unseen objects by theserules is su�ciently high. We assume that the information systems which createinputs for our procedures satisfy those conditions.5 Transition systemsTransition systems create a simple and powerful formalism for explaining theoperational behaviour of models of concurrency. This section contains basic no-tions and notations connected with transition systems that will be necessary forunderstanding of our main result.A transition system is a quadruple TS = (S;E; T; s0), where S is a non-empty set of states, E is a set of events, T � S�E�S is the transition relation,s0 2 S is the initial state.A transition system can be pictorially represented as a rooted edge-labelleddirected graph. Its nodes and its directed arcs represent states and state transi-tion, respectively. As di�erent state transitions may be caused by equal events,di�erent arcs may be labelled by equal symbols. If (s; e; s0) 2 T then a transitionsystem TS can go from s to s0 as a result of the event e occurring at s.Example 5. In Figure 2 a transition system is shown, where the initial stateis indicated by an extra arrow without source and label.An isomorphism between transition systems is de�ned in the following way:Let TS = (S;E; T; s0) and TS0 = (S0; E0; T 0; s00) be two transition systems. Abijection f : S ! S0 is an isomorphism from TS to TS0 (denoted f : TS ! TS0)if and only if the following two conditions are satis�ed:(i) f(s0) = s00(ii) (s; e; s0) 2 T if and only if (f(s); e; f(s0)) 2 T 0 .Two transition systems TS and TS0 are called isomorphic (denoted TS 'TS0) if and only if there exists an isomorphism f : TS ! TS0.It is worth to observe that we demand that the set of events of E from atransition system TS coincides with the set of events of E0 from TS0.

m s1 ms2ms3 ������z AAAAAUx� y?

Figure 2. An example of a transition systemLet TS = (S;E; T; s0) be a transition system. We say that the event e hasconcession in the state s (is enabled at s) if there exists a state s0 such that(s; e; s0) 2 T .The notion of regions, introduced in [Ehrenfeucht and Rozenberg,1990] isimportant for this paper.Let TS = (S;E; T; s0) be a transition system. A set R of states of TS is aregion of TS if and only if for equally labelled arcs (s; e; s0) and (s1; e; s01) holds:if s 2 R and s0 2=R then s1 2 R and s01 2=R, andif s 2=R and s0 2 R then s1 2=R and s01 2 R.; and S are called trivial regions of TS. By RTS we denote the set of allnon-trivial regions of TS.Let TS = (S;E; T; s0) be a transition system.For e 2 E,�e = fR 2 RTS : there exists (s; e; s0) 2 T s 2 R and s0 2=Rgis called the pre� region of e,e� = fR 2 RTS : there exists (s; e; s0) 2 T s 2=R and s0 2 Rgis called the post� region of e.Example6: For the transition system shown in Figure 2, X = fs1g; Y = fs2gand Z = fs3g are regions, and �x = fXg; y� = fZg.6 Elementary net systemsIn this section we recall basic notions connected with the basic system model ofnet theory, called elementary net system [Thiagarajan,1987].In net theory, models of concurrent systems are based on objects called netswhich specify the local states and local transitions and the relationships betweenthem.A triple N =(S; T; F ) is called a net if and only if(i) S and T are disjoint sets (the elements of S are called S � elements, theelements of T are called T � elements).(ii) F � (S � T ) [ (T � S) is a binary relation, called the ow relation.(iii) For each x 2 S [ T there exists y 2 S [ T such that (x; y) 2 F or(y; x) 2 F .

In the following the S-elements will be called conditions and the T -elementswill be called events. Moreover, we use B to denote the set of conditions andE to denote the set of events; consequently a net will be denoted as the triple(B;E; F ).Let N = (B;E; F ) be a net. For x 2 B [ E;� x = fy : (y; x) 2 Fg is calledthe preset of x, x� = fy : (x; y) 2 Fg is called the postset of x.The element x 2 B [ E is called isolated if and only if �x [ x� = ;.It is worth to observe that the condition (iii) in the net de�nition states thatwe do not permit isolated elements in considered nets.The net N = (B;E; F ) is called simple if and only if distinct elements do nothave the same pre- and postset, i.e. for each x 2 B [ E the following conditionis satis�ed:if �x =� y and x� = y� then x = y.A quadruple N= (B;E; F; c0) is called an elementary net system if and onlyif (i) N = (B;E; F ) is a simple net without isolated elements, called the un-derlying net of N and denoted by NN ,(ii) c0 � B is the initial state.In diagrams the conditions will be drawn as circles, the events as boxes andelements of the ow relations as directed arcs. The initial state will be indicatedby marking (with small black dots) the elements of the initial state.Example7: An elementary net system shown in Figure 3 has three conditionsX, Y, Z, and three events x, y, z. Its initial state is fXg. The preset of x is equalto fXg, and the postset of y is fZg.m xmym̀bcz 6Z

- X ?Y?��� �

Figure 3. An elementary net systemFrom now on we will often refer to elementary net systems as just net systems.The dynamics of a net system are straightforward. The states of a net systemconsists of a set of conditions that hold concurrently. The system can go froma state to a state through the occurrence of an event. An event can occur ata case if and only if all its pre-conditions (i.e. conditions in its preset) holdand none of its post-conditions (i.e. conditions in its postset) hold at the state.When an event occurs then all its pre-conditions cease to hold and all its post-conditions begin to hold. Formally, the dynamics of a net system is described bythe so-called the transition relation of that net system.

Let N = (B;E; F ) be a net. Then trN � P (B)�E � P (B) is the transitionrelation of N de�ned as follows: (c; e; c0) 2 trN if and only if c � c0 =�e andc0 � c = e�.Let N= (B;E; F; c0) be a net system.(i) CN is the state space ofN and it is the smalest subset of P (B) containingc0 which satis�es the condition: if (c; e; c0) 2 trNN and c 2 CN then c0 2 CN .(ii) trN is the transition relation ofN and it is trNN restricted to CN�E�CN .(iii) EN is the set of active events of N and it is the subset of E given byEN = fe : there exists (c; e; c0) 2 trNg.It is possible to associate a transition system with a net system to explainits operational behaviour.Let N= (B;E; F; c0) be a net system. Then the transition system TSN =(CN ; EN ; trN ; c0) is called the transition system associated with N .A transition system TS is an abstract transition system if and only if thereexists a net system N such that TS ' TSN .Example 8: The state space of the net system presented in Figure 3 isffXg; fY g; fZgg. It is easy to verify that the transition system associated withthe net system of Figure 3 is isomorphic with the transition system shown inFigure 2.7 Dynamic Information SystemsNow we introduce the notion of a dynamic information system which plays acentral role in this paper.A dynamic information system is a quintuple DS = (U;A;E; T; u0) where(i) S =(U;A) is an information system called the underlying system of DS,(ii) TS =(U;E; T; u0) is a transition system.Dynamic information systems will be presented in the form of two integratedsubtables. The �rst subtable represents the underlying system, wheras the secondone the transition system. Columns of the second subtable are labeled by events,rows, analogously as for the underlying system, by objects of interest and entriesof the subtable for a given row (state) are follower states of that state. The �rstrow in the �rst subtable represents the initial state of a given transition system.We will both subtables have the same number of rows, but the number of columnsis di�erent.Example9: In Table 7 is shown an example of a dynamic information systemDS = (U;A;E; T; u0) such that its underlying system is represented by Table 1,whereas the transition system is represented by the graph in Figure 2. In thiscase the initial state of the system is represented by u1. We show also that, forinstance in the state u2 the event y has concesion and when it occurs then a newstate u3 of DS appears.

U=A a b c U=E x y zu1 1 1 0 u2u2 0 2 0 u3u3 0 0 2 u1Table 7. A dynamic information systemNow we are ready to formulate the synthesis problem of concurrent systemsspeci�ed by dynamic information systems.The synthesis problem:Let DS = (U;A;E; T; u0) be a dynamic information system. Is a given tran-sition system TS = (U;E; T; u0) an abstract transition system? If yes, constructa net system N satisfying TS ' TSN .8 The solution of the synthesis problemIn this section we present a solution of the synthesis problem stated in this paper.8.1 The �rst approachA solution method of the problem is based on the approach proposed in [De-sel and Reisig,1996]. Now we describe shortly their approach connected with aprocedure to decide whether or not a given transition system TS is an abstracttransition system. In the positive case, the procedure provides a net systemwhose transition system is isomorphic to TS.Since every condition corresponds to a region and every region generates apotential condition we can construct a net system from a transition system,using only generated conditions.Let DS = (U;A;E; T; u0) be a dynamic information system, let TS =(U;E; T; u0) be the transition system of DS, and let m be a set of regions of TS.Then the m-generated net system is NTSm = (m;E; F; c0) where for each regionR 2 m and each event e 2 E the following conditions are satis�ed:(i) (R; e) 2 F if and only if R 2�e,(ii) (e;R) 2 F if and only if R 2 e�,(iii)R 2 c0 if and only if u0 2 R.Example 10: The transition system from Example 5 with the regions X, Y,Z of Example 6 generates the net system shown in Figure 3.We can now formulate the synthesis problem in the following way: Givena transition system TS, construct the net system generated by the regions ofTS. If the transition system associated with this net system is isomorphic toTS, then the net system is a basic solution to the synthesis problem and the

procedure is �nished. In the oposite case, there exists no a net system whichcorresponds to TS and so TS is no abstract transition system. This fact followsfrom the followingTheorem [Desel and Reisig,1996]. A transition system TS is an abstracttransition system if and only if TS ' TSNTSm , where m denotes the set of allregions of TS.Example 11: The transition system TS from Example 5 is an abstract tran-sition system. The transition system associated with the net system from Figure3 is shown in Figure 4. It is isomorphic to TS.f Z; S g f Y; S g

f X;S g S = fs1; s2; s3g�������z AAAAAAUx� yFigure 4. The transition system associated with the net system from Figure 3Remark 3. To decide if two graphs are isomorphic is in general a nontrivialproblem. Fortunately, the procedure proposed above, decides this problem veryeasily since there exists at most one isomorphism transforming a given transitionsystem TS onto a transition system associated with a net system generated bythe regions of TS. It follows from the following proposition, which is reformulatedto our formalism:Proposition 12 [Desel and Reisig,1996]. Let DS = (U;A;E; T; u0) be adynamic information system, let TS = (U;E; T; u0) be its transition system, andlet m denotes the set of all regions of TS. Then there is exactly one isomorphismf from TS to TSNTSm , where NTSm denotes the m-generated net system which isde�ned as follows: f(s) = fR 2 m : s 2 Rg.8.2 The second approachNow we describe shortly a solution of the synthesis problem stated in the paperbased on the second approach, i.e. we assume that a given data table DS consistsof only partial knowledge about the system behaviour. Thus, we at �rst computean extension DS0 of the data table DS, i.e. the system in which the set of globalstates of DS0 is consistent with all rules true in the underlying informationsystem S of DS as well as the set of global states of DS0 represents the largestextension of S consistent with the knowledge represented by S. Next, for �ndinga solution of the synthesis problem in the form of a net system we use the

method described in the previous section. The idea of our method is presentedby example and a very simple procedure given below.At �rst, we give one more de�nition from rough set theory.A decision table is any information system of the form S =(U;A[fdg), whered 2=A is a distinguished attribute called decision. The elements of A are calledconditional attributes (conditions).Example 12. Let us consider an example of a decision table S = (U;A[ fdg)de�ned by the data table presented in Table 8.In the example we have U = fu1; u2; :::; u9g, A = fa; b; c; a0; b0; c0g. The de-cision is denoted by d. The possible values of attributes (conditions and thedecision) from A [ fdg are equal to 0, 1 or 2. This data table has been con-structed on the basis of the dynamic information system DS = (U;A;E; T; u0)from Example 9. Table 8 contains all possible pairs of global states from theunderlying system of DS. The value of decision d is equal to 1 if and only ifthere exists an event e 2 E such that (u; e; u0) 2 T . Thus, this decision table wecan treat as a description of the characteristic function of the transition relationT . For the decision table S we obtain the following discernibility matrix M(S)presented in Table 9.U=A a b c a0 b0 c0 du1 1 1 0 0 2 0 1u2 1 1 0 1 1 0 0u3 1 1 0 0 0 2 0u4 0 2 0 0 0 2 1u5 0 2 0 0 2 0 0u6 0 2 0 1 1 0 0u7 0 0 2 1 1 0 1u8 0 0 2 0 0 2 0u9 0 0 2 0 2 0 0Table 8. An example of a decision table

U u1 u2 u3 u4 u5 u6 u7 u8 u9u1u2 a0; b0;du3 b0; c0; d a0; b0;c0u4 a; b; b0 a; b; a0; a; b; dc0 b0; c0; du5 a; b; a; b; a0; a; b; b0; c0; dd b0 b0; c0u6 a; b; a0; b0; a; b a; b; a0; b0; a0; b0d a0; b0; c0 c0; du7 a; b; c; a0; a; b; c; d a; b; c; a0; b; c; a0; b; c; a0; b; c; db0 b0; c0; d b0; c0 b0; du8 a; b; c; b0; a; b; c; a; b; c b; c; d b; c; b0; b; c; a0; a0; b0;c0; d a0; b0; c0 c0 b0; c0 c0; du9 a; b; c; a; b; c; a; b; c b; c; b0; b; c b; c; a0; a0; b0; b0; c0d b0; c0 b0; c0 c0; d b0 dTable 9. The discernibility matrix M(S) for the decision table SU=A a b c a0 b0 c0 d dAdu1 1 1 0 0 2 0 1 f 1 gu2 1 1 0 1 1 0 0 f 0 gu3 1 1 0 0 0 2 0 f 0 gu4 0 2 0 0 0 2 1 f 1 gu5 0 2 0 0 2 0 0 f 0 gu6 0 2 0 1 1 0 0 f 0 gu7 0 0 2 1 1 0 1 f 1 gu8 0 0 2 0 0 2 0 f 0 gu9 0 0 2 0 2 0 0 f 0 gTable 10. The decision table S with the function dAdNow we compute the set of rules corresponding to nontrivial dependenciesbetween the values of conditions and the decision values. In this case we alsoapply the method for generating the minimal form of rules presented in theSection 4. Let us start by computing the decision rules corresponding to theconditions A = fa; b; c; a0; b0; c0g and the decision d. We have the decision table

S = (U;A [ fdg) from which we compute the decision rules mentioned below.In the table the values of the function dAd are also given. The discernibilitymatrix M(S; d; v; ul) where v 2 Vd, ul 2 U; l = 1; 2; :::; 9, obtained from M(S)in the above way is presented in Table 11.U u1 u2 u3 u4 u5 u6 u7 u8 u9u1u2 a0; b0u3 b0; c0u4 a; b; a0; a; bb0; c0u5 a; b b0; c0u6 a; b; a0; b0 a0; b0;c0u7 a; b; c a; b; c; a0; b; c; a0; b; cb0; c0 b0u8 a; b; c; b0; b; c a0; b0;c0 c0u9 a; b; c; b; c; b0; a0; b0;c0Table 11. The discernibility matrixM(S; d; v; ul) for the M(S)The discernibility functions corresponding to the values of the function dAdafter reduction (using the absorption laws) are the following:Case 1. For dAd (u1) = f1g : a ^ a0 ^ c0 _ b ^ a0 ^ c0 _ a ^ b0 _ b ^ b0 .Case 2. For dAd (u2) = f0g : a ^ a0 _ b ^ a0 _ c ^ a0 _ a ^ b0 _ b ^ b0 _ c ^ b0 .Case 3. For dAd (u3) = f0g : a ^ b0 _ a ^ c0 _ b ^ b0 _ b ^ c0 .Case 4. For dAd (u4) = f1g : b ^ c0 _ b ^ b0 _ a ^ c ^ b0 _ a ^ c ^ c0 .Case 5. For dAd (u5) = f0g : a ^ b0 _ b ^ b0 _ b ^ c0 _ a ^ c ^ c0 _ a ^ a0 ^ c0 .Case 6. For dAd (u6) = f0g : b ^ a0 _ b^ b0 _ b ^ c0 _ c ^ b0 _ c^ a0 _ a ^ c ^ c0 .Case 7. For dAd (u7) = f1g : b ^ a0 _ b ^ b0 _ c ^ a0 _ c ^ b0 .Case 8. For dAd (u8) = f0g : b ^ a0 _ b ^ b0 _ b ^ c0 _ c ^ a0 _ c ^ b0 _ c ^ c0 .Case 9. For dAd (u9) = f0g : b^ a0 _ c^ a0 _ a^ b0 _ a^ a0 ^ c0 _ b^ b0 _ c^ b0 .Hence we obtain the following decision rules:a1 ^ a00 ^ c00 _ b1 ^ a00 ^ c00 _ a1 ^ b02 _ b1 ^ b02=)S d1;b2 ^ c02 _ b2 ^ b00 _ a0 ^ c0 ^ b00 _ a0 ^ c0 ^ c02=)S d1;

b0 ^ a01 _ b0 ^ b01 _ c2 ^ a01 _ c2 ^ b01=)S d1;a1 ^ a01 _ b1 ^ a01 _ c0 ^ a01 _ a1 ^ b01 _ b1 ^ b01 _ c0 ^ b01=)S d0;a1 ^ b00 _ a1 ^ c02 _ b1 ^ b00 _ b1 ^ c02=)S d0;a0 ^ b02 _ b2 ^ b02 _ b2 ^ c00 _ a0 ^ c0 ^ c00 _ a0 ^ a00 ^ c00=)S d0;b2 ^ a01 _ b2 ^ b01 _ b2 ^ c00 _ c0 ^ b01 _ c0 ^ a01 _ a0 ^ c0 ^ c00=)S d0;b0 ^ a00 _ b0 ^ b00 _ b0 ^ c02 _ c2 ^ a00 _ c2 ^ b00 _ c2 ^ c02=)S d0;b0 ^ a00 _ c2 ^ a00 _ a0 ^ b02 _ a0 ^ a00 ^ c00 _ b0 ^ b02 _ c2 ^ b02=)S d0;These decision rules allow us to verify which global states of the dynamicinformation system DS from Example 9 are in the transition relation T of DS.Let DS =(U;A;E; T; u0) be a dynamic information system and S =(U;A)its underlying system. Sometimes, it is possible that an extension of the under-lying system S of DS contains new global states consistent with the knowledgerepresented by S, i.e. with the all rules from the set OPT(S). The extension ofthe system S we can obtain applying the procedure for computing an extensionS0 of S described in Section 4. Thus, the method of �nding the decision rules ina given dynamic information system presented in the above example allows usto extend the transition relation T of DS to a new transition relation T 0. In con-sequence, we obtain a new dynamic information system DS0 =(U 0; A;E0; T 0; u0)called an extension of the dynamic information system DS, where S0 =(U 0; A) isan extension of S; E0, is a set of events, E � E0, and T 0 is the extension of thetransition relation T; T 0 � U 0 � E0 � U 0. Further, for constructing from a dy-namic information system DS0 with its transition system TS0 =(U 0; E0; T 0; u0)describing the behaviour of DS0 a concurrent model in the form of an elementarynet system we can proceed analogously to the method presented in Subsection8.1.Now we are ready to present a very simple procedure for computing an exten-sion DS0 =(U 0; A;E0; T 0; u0) of a given dynamic information system DS =(U;A;E; T; u0).PROCEDURE for computing an extension DS0 of DS:Input: A dynamic information system DS = (U;A;E; T; u0) with its under-lying system S =(U;A).Output: An extension DS0 of the system DS.Step 1. Construct the decision table S0 =(U 0; A [ fdg) with the function dAdin the way described in Section 4.Step 2. Compute the discernibility matrix M(S0).Step 3. Compute the discernibility matrixM(S0; d; v; ul) where v 2 Vd; ul 2U 0; l = 1; 2; :::;card(U 0) for the M(S0).Step 4. Compute the discernibility functions corresponding to the values ofthe function dAd in the way described in Section 4.Step 5. Compute the decision rules true in S0, i.e. the setD0(S0) of rules corre-sponding to nontrivial functional dependencies between the values of conditionsand the decision values from the decision table S0.

Step 6. Compute an extension S00 =(U 0; A) of the underlying system S ofDS using procedure described in Section 4.Step 7. Compute an extension T 0 of the transition relation T using the deci-sion rules obtained in Step 5 in the followig way:(i) construct all possible pairs of global states of S, i.e. a set U � U ,(ii) verify using the set of decision rules obtained in Step 5 which pairs ofglobal states of S are consistent with these rules, i.e. execute instructions1. T 0 := ;; E0 := ;.2. For every pair (u; u0) 2 U � U doif (u; u0) 2 U �U and an information vector v corresponding to a pair (u; u0)is consistent with D0(S0) then add (u; e; u0) to T 0 and e to E0.Step 8. Construct the extension DS0 =(U 0; A;E0; T 0; u0).It is easy to verify that the extension DS0 of the dynamic information systemDS from Example 9 computed by using our procedure presented above is thesame as the system DS (see Example 4 and Example 12). Thus, the net systemfor the extension DS0 is identical as for the system DS (see Figure 3).9 ConclusionsWe have formulated a method of the synthesis problem of concurrent systemsspeci�ed by dynamic information systems. Our solution is based on a construc-tion of a solution of the synthesis problem of Petri nets discussed in [Desel andReisig,1996]. We have proposed a solution of the synthesis problem of a net sys-tem from a dynamic information system. It is also possible to solve this problemfor �nite place/transition Petri nets, since �nite self-loop-free place/transitionnets are equivalent to vector addition systems, introduced by Karp and Miller[1969]. The solution of our problem for place/transitions Petri nets is also simpleto obtain.The paper is concerned with some approach to concurrency based on roughset theory. Petri nets have been chosen as a model for concurrency. The ap-plication of Petri nets to represent a given information system and a modi�edde�nition of these systems enables:- to discover in a simple way new dependencies between local states of pro-cesses being in the system,- to represent in an elegant and visual way the dependencies between localstates of processes in the system,- to observe concurrent and sequential subsystems of the system.On the basis of Petri net approach it was possible to understand better thestructure and dynamics of a given information system.Moreover, to some extent, it is a matter of taste which of the modellingmethod of concurrent systems to use. Drawing Petri nets by hand one can pro-duce very compact solutions for problems solved rather by small nets. For largemodels some automatic methods could be accepted even if the produced by them

nets are not so compact or small. Comparing the presented examples it is pos-sible to see that our method can also produce solutions close to those obtainedby designers.The method presented in the paper allows to generate automatically from anarbitrary dynamic information system its concurrent model in the form of a netsystem. We have implemented a program on IBM PC generating a net modelof the system speci�ed by a dynamic information system. The resulting net canbe analyzed by the PN-tools system for computer aided design and analysis ofconcurrent models [Suraj,1995].It seems for us that the presented in the paper results as well as the furtherinvestigations of relationships between Petri net theory and rough set theory willstimulate the theoretical research related to them and new practical applicationsof both of them, e.g. in the area of knowledge discovery systems, control sys-tem design, decomposition of information systems as well as for real-time stateidenti�cation.Moreover, we would like to investigate to what extent our method could beapplied for automatic synthesis of parallel programs from examples [Shapiro andEckroth,1987], [Smith,1984].Acknowledgement. I am grateful to Professor A. Skowron for stimulatingdiscussions and interesting suggestions about this work. This work was partiallysupported by the grant #8T 11C 01011 from the State Committee for Scienti�cResearch (KBN) in Poland and by the ESPRIT project 20288 CRIT-2.References1. Baar, A., Cohen, P.R,. Feigenbaum, E.A.: The handbook of arti�cial intelligence4 Addison Wesley (1989)2. Bazan, J., Skowron, A., Synak, P.: Dynamic reducts as a tool for extracting lawsfrom decision tables. In: Z. W. Ras, M. Zemankova (eds.), Proceedings of theEighth Symposium on Methodologies for Intelligent Systems, Charlotte, NC, Oc-tober 16-19, Lecture Notes in Arti�cial Intelligence 869, Springer-Verlag (1994)346{3553. Bazan, J., Skowron, A., Synak, P.: Discovery of decision rules from experimentaldata. In: T.Y. Lin (ed.), Proc. of the Third International Workshop on Rough Setsand Soft Computing, San Jose CA, November 10-12 (1994) 526{5334. Bernadinello, L.: Synthesis of net systems. In: Proc. of the Application and Theoryof Petri Nets. Lecture Notes in Comput. Sci. 691, Springer-Verlag, Berlin (1993)89{1055. Brown E.M.: Boolean reasoning. Kluwer Academic Publishers, Dordrecht (1990)6. Desel, J., Reisig, W.: The synthesis problem of Petri nets. Acta Inf. 33/4 (1996)297{3157. Ehrenfeucht, A., Rozenberg, G.: Partial 2-structures Part II. State space of con-current systems. Acta Inf. 27 (1990) 348{3688. Hack, M.: Decidability questions for Petri nets. Ph.D thesis. Department of Elec-trical Engineering, Massachusetts Institute of Technology, Cambridge MA (1975)9. Hurley, R.B.: Decision tables in software engineering. Van Nostrad Reinhold Com-pany, New York (1983)

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