Fundamenta Informaticae 3,4(1996)353-376 IOS Press
353
Discovery of Concurrent Data Models from Experimental Tables: A Rough Set Approach Zbigniew Suraj
Institute of Mathematics Pedagogical University Rzeszow, Poland email:
[email protected]
Abstract. The main objective of machine discovery is the determination of relations
between data and of data models. In the paper we describe a method for discovery of data models represented by concurrent systems from experimental tables. The basic step consists in a determination of rules which yield a decomposition of experimental data tables; the components are then used to de ne fragments of the global system corresponding to a table. The method has been applied to automatic data models discovery from experimental tables with Petri nets as models for concurrency.
Keywords: data mining, system decomposition, rough sets, concurrent models, laws discovery from experimental data
1. Introduction Machine discovery has elaborated methods for the discovery of relations among observed data (cf. [42]) and of data models. Recent research shows that these methods can be applied among others in knowledge discovery in databases [20], scienti c discovery [22], concept discovery [14], automated data analysis [8] and discovery in mathematics [6], [1]. Large databases can be a source of useful knowledge. Yet this knowledge is implicit in the data. It must be mined and expressed in a concise, useful form (e.g. in the form of rules, conceptual hierarchies, statistical patterns, equations, and the like). Automation of knowledge discovery is important because databases are growing in a size and a number, and the standard data analysis techniques are not designed for exploration of huge hypotheses spaces. Decomposition is the breakdown of a complex system into smaller, relatively independent subsystems. It is the main tool available to simplify the construction of complex man-made systems. System decomposition problems are an essential part of the system analysis and design process. However, despite the importance of system decomposition, there is no general approach for accomplishing it. Rather, decomposition relies on an analyst's experience and expertise. The system decomposition methods are applied in many areas (see e.g. in the eld of system engineering and large-scale systems [37], [21], [16], [7], [23], in the logic synthesis [36], in Petri net theory [4]. This work has been supported by the grant #8T11C01011 from the State Committee for Scienti c Research in Poland
The aim of this paper is to present an approach to the decomposition of information systems. In the paper we represent experimental data by information systems. Our approach can be applied to the discovery of data models in the form of concurrent systems. Decomposition of large experimental data tables can be treated as one of the fundamental tools in data mining. It is usually imposed by the high computational complexity of the search for relations between data on one hand and/or the structure of the process of data models discovery on the other. Our approach is based on rough set theory [17] and Boolean reasoning [5]. It consists of three levels. First we show how experimental data tables are represented by information systems [17]. Next we discuss how any information system S can be decomposed (with respect to any of its reduct) into components linked by some connections which allow to preserve some constraints. Any component represents in a sense the strongest functional module of the system. The connections between components represent constraints which must be satis ed when these functional modules coexist in the system. The components together with the connections de ne a so called covering of S . Finally, we use the coverings of the information system S to construct its concurrent model in the form of a marked Petri net (NS ; MS ) [30] with the following property: the reachability set R(NS ; MS ) is in one-toone correspondence with the set of all global states consistent with all rules true (valid) in S . The idea of concurrent system speci cation by information systems is due to Z. Pawlak [18]. The behaviour of the constructed concurrent systems is consistent with data tables from which they are extracted; their properties (like their invariants) can be considered as higher level laws of experimental data. From these invariants some new forms of laws can be deduced to express e.g. relationships between di erent components of the system. In the paper we investigate decomposition problems which can now be roughly de ned as follows:
Component Extraction Problem:
Input: An information system Output: All components of S .
S.
Covering Problem:
Input: An information S . Output: The set of all coverings
of S . Our approach can be applied for automatic feature extraction and for control design of systems represented by experimental data tables. The text is organized as follows. Section 2 deals with some basic de nitions concerning information systems and rough set theory. Section 3 contains a method for generating the minimal form of rules with respect to the number of attributes on the left hand side of the rules. The method is based on the idea of discernibility matrix de ned in [26] and modi ed here for our purposes. In Section 4 we present basic concepts and notation related to the decomposition of information systems as well as a method for constructing components and coverings of a given information system with respect to its reducts. Section 5 contains basic de nitions and notation from Petri net theory. In Section 6 we present a method for constructing a concurrent representation (in the form of a Petri net) of an information system. In the last section we give some comments related to the decomposition problem of information systems.
2. Preliminaries of Rough Set Theory In this section we recall basic notions of rough set theory. Among them are those of information systems, indiscernibility relations, discernibility matrices, functions, reducts and rules.
2.1. Information Systems
Information systems (sometimes called data tables, attribute-value systems, condition-action tables, knowledge representation systems etc.) are used for representing knowledge. The notion of an information system presented here is due to Z. Pawlak and was investigated by several authors (see e.g. the bibliography in [17]). Among research topics related to information systems are: rough set theory, problems of knowledge representation, problems of knowledge reduction, dependencies in knowledge bases. Rough sets have been introduced [17] as a tool to deal with inexact, uncertain or vague knowledge in arti cial intelligence applications. This subsection contains basic notions related to information systems that will be necessary for understanding our results. An information system is a pair S = (U; A), where U { is a non-empty, nite set called the universe, A { is a non-empty, nite set of attributes, i.e. a : U !Va for 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, patients, observations. Attributes are interpreted as features, variables, processes, characteristic conditions etc. S The set V = Va is said to be the domain of A. a2A If S = (U; A) then S 0 = (U 0 ; A0) such that U U 0 , A0 = fa0 : a 2 Ag, a0 (u) = a(u) for u 2 U and Va = Va for a 2 A will be called a U 0 -extension of S (or an extension of S , in short). S is also called a restriction of S 0. If S = (U; A) then S 0 = (U; B ) such that A B will be referred to as a B ? extension of S . Example 2.1. [18]. Let us consider an information system S = (U; A) such that U = fu ; u ; u ; u ; u g, A = fa; b; c; d; eg and the values of the attributes are de ned as in Table 1. 0
1
2
3
4
5
U=A u u u u u 1 2 3 4 5
a 1 0 2 0 1
b 0 0 0 0 1
c 2 1 2 2 2
d 1 2 1 2 1
e 0 1 0 2 0
Table 1 An example of an information system In a given information system, in general, we are not able to distinguish all single objects (using attributes of the system). Namely, di erent objects can have the same values on considered attributes. Hence, any set of attributes divides the universe U into some classes which establish a partition [17] of the set of all objects U . It is de ned in the following way. Let S = (U; A) be an information system. With any subset of attributes B 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. T ind(a), where ind(a) Notice that ind(B ) is an equivalence relation and ind(B ) = a2B means ind(fag). If u ind(B )u0, then we say that the objects u and u0 are indiscernible with respect to attributes from B . In other words, we cannot distinguish u from u0 in terms of attributes in B .
Any information system S = (U; A) determines an information function InfA : U !P (AV ) S de ned by InfA(u) = f(a; a(u)) : a 2 Ag, where V = Va and P (X ) denotes the powerset a2A of X . The set fInfA(u) : u 2 U g 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 vectors of the form (v ; . . .; vm), vi 2 Vai , for i = 1; . . .; m, where m = card(A). Such vectors are called information vectors (over V and A). Let S = (U; A) be an information system, where A = fa ; . . .; amg. Pairs (a; v) with a 2 A, v 2 V are called descriptors. Instead of (a; v) we also write a = v or av . The set of terms over A and V is the least set containing descriptors (over A and V ) and closed with respect to the classical propositional connectives: :(negation), _(disjunction), and ^(conjunction), i.e. if ; 0 are terms over A and V then : ; ( _ 0); ( ^ 0 ) are terms over A and V . The meaning jj jjS (or in short jj jj) of a term in S is de ned inductively as follows: jj(a; v)jj = fu 2 U : a(u) = vg for a 2 A and v 2 Va; jj _ 0 jj = jj jj [ jj 0jj; jj ^ 0 jj = jj jj \ jj 0jj; jj: jj = U ? jj jj Two terms Wand 0 are equivalent, , 0, if and only if jj jj = jj 0jj. In particular we have: :(a = v), fa = v0 : v0 6= v and v0 2 Va g. 1
1
2.2. Rules in Information Systems
Rules express some of the relationships between values of the attributes described in the information systems. This subsection contains the de nition of rules as well as other related concepts. Let S = (U; A) be an information system and let B A. For every a 2= B we de ne a function dBa : U !P (Va) such that dBa (u) = fv 2 Va : 9u0 2 U [u0ind(B )u and a(u0) = v]g where P (Va) denotes the powerset of Va. Hence, dBa (u) is the set of all the values of the attribute a on objects indiscernible with u by attributes from B . If the set dBa (u) has only one element, this means that the value a(u) is uniquely de ned by the values of attributes from B on u. Let S = (U; A) be an information system and let B; C A. We say that the set C depends on B in S in degree k(0 k 1), symbolically B ! C , if k = POSUB C , where S;k POSB (C ) is the B-positive region of C in S [17]. If k = 1 we write B ! C instead of B S;k ! C . In this case B !S C means that ind(B ) S ind(C ). If the right hand side of a dependency consists of one attribute only, we say the dependency is elementary. It is easy to see that a simple property given below is true. Proposition 2.1. Let S = (U; A) be an information system and let B; C; D A. If B !S C and B ! D then B ! C [ D. S S card(
card(
(
)
))
A rule over A and V is any expression of the following form: (1) ai1 = vi1 ^ . . . ^ air = vir )ap = vp where 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 left hand side of the rule. The rule (1) is true in S (or in short: is true) if ; 6= kai1 = vi1 ^ . . . ^ air = vir k kap = vpk : The fact that the rule (1) is true in S is denoted in the following way: a = vp : (2) ai1 = vi1 ^ . . . ^ air = vir ) S p By D(S ) we denote the set of all rules true in S . Let R D(S ). An information vector v = (v ; . . .; vm ) is consistent with R if and only if for any rule ai1 = vi1 ^ . . . ^ air = vir )ap = vp in R if vi = vi for j = 1; . . .; r then vp = vp . The set of all information vectors consistent with R is denoted by CON(R). Let S 0 = (U 0 ; A0) be a U 0-extension of S = (U; A). We say that S 0 is a consistent extension of S if and only if D(S ) D(S 0). S 0 is a maximal consistent extension of S if and only if S 0 is a consistent extension of S and any consistent extension S 00 of S is a restriction of S 0. We apply here the Boolean reasoning approach to the rule generation [24]. The Boolean reasoning approach [5], due to G. Boole, is a general problem solving method consisting of the following steps: (i) construction of a Boolean function corresponding to a given problem; (ii) computation of prime implicants of the Boolean function; (iii) interpretation of prime implicants leading to the solution for the problem. It turns out that this method can be also applied to the generation of rules with certainty coecients [25]. Using this approach one can also generate the rule sets being outputs from some algorithms known in machine learning, like AQ-algorithms [10], [27]. 1
j
j
2.3. Reduction of Attributes
Let S = (U; A) be an information system. Any minimal subset B A such that ind(B ) = ind(A) is called a reduct in the information system S [17]. The set of all reducts in S is denoted by RED(S ). Now we recall two basic notions, namely those of discernibility matrix and discernibility function [26], which will help to compute minimal forms of rules with respect to the number of attributes on the left hand side of the rules. Let S = (U; A) be an information system, and let us assume that U = fu ; . . .; ung, and A = fa ; . . .; amg. By M (S ) we denote an nn matrix (cij ), called the discernibility matrix of S , such that cij = fa 2 A : a(ui) 6= a(uj )g for i; j = 1; . . .; n. Intuitively an entry cij consists of all the attributes which discern objects ui and uj . Since M (S ) is symmetric and cii = ; for i = 1; . . .; n, M (S ) can be represented using only elements in the lower triangular part of M (S ), i.e. for 1 j < i n. With every discernibility matrix M (S ) we can uniquely associate a discernibility function fM S , de ned in the following way: A discernibility function fM S for an information system S is a Boolean function of m propositional variables a; . . .; aWm (where ai 2 A for i = 1; . . .; m) de ned as the conjunction W of all expressions cij , where cij is the disjunction of all elements of cij = fa : a 2 cij g, where 1 j < i n and cij 6= ;. In the sequel we write a instead of a . Proposition 2.2. gives an important property which enables us to compute all reducts of S . 1
1
( )
( )
1
Proposition 2.2. [26]. Let S = (U; A) be an information system, and let fM S be a discernibility function for S . Then the set of all prime implicants [38] of the function fM S determines the set RED(S) of all reducts of S , i.e. ai ^ . . . ^ aik is a prime implicant of fM S if and only if fai ; . . .; aik g 2 RED(S ). ( )
( )
( )
1
1
In the next propositions [17] the important relationships between the reducts and the dependencies are given. Proposition 2.3. Let S =(U; A) be an information system and let B 2 RED(S ). If A?B 6= ; then B ! A ? B. S
Proposition 2.4. If B !S C then B !S C 0, for every ; 6= C 0 C . In particular, B !S C implies B ! fag, for every a 2 C . S Proposition 2.5. Let B 2 RED(0 S ). Then all attributes in the reduct B are pairwise 0 independent, i.e. neither fag ! f a g nor f a g ! fag holds, for any a; a0 2 B , a 6= a0. S S
Below we present a procedure for computing reducts [26]. PROCEDURE for computing RED(S ): Step 1. Compute the discernibility matrix M (S ) for the system S . Step 2. Compute the discernibility function fM S associated with the discernibility matrix M (S ). Step 3. Compute the minimal disjunctive normal form of the discernibility function 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 cardinality) reduct is NP-hard [26]. In general the number of reducts of a given information system can be exponential with respect to the number of attributes (i.e. any information system S has at most m over [m/2] reducts, where m = card(A)). Nevertheless, existing procedures for reduct computation are ecient in many applications and for more complex cases one can apply some ecient heuristics (see e.g. [3]). Example 2.2. Applying the above procedure for the information system S from Example 2.1., we obtain the following discernibility matrix M (S ) presented in Table 2 and discernibility function presented below: ( )
( )
U u u u u u
1 2 3 4 5
u
1
a,c,d,e a a,d,e b
u
2
a,c,d,e c,e a,b,c,d,e
u
3
a,d,e a,b
u
4
u
5
a,b,d,e
Table 2 The discernibility matrix M (S ) for the information system S from Example 2.1
fM S (a; b; c; d; e) = a ^ b ^ (a _ b) ^ (c _ e) ^ (a _ d _ e) ^ (a _ b _ d _ e)^ ^(a _ c _ d _ e) ^ (a _ b _ c _ d _ e) : We consider non-empty entries of the table (see Table 2), i.e. a; b; a, b; c, e; a, d, e; a, b, d, e; a, c, d, e and a, b, c, d, e; next a, b, c, d, e are treated as Boolean variables and the ( )
disjunctions a; b; a _ b; c _ e; a _ d _ e; a _ b _ d _ e; a _ c _ d _ e and a _ b _ c _ d _ e are constructed from these entries; nally, we take the conjuction of all the computed disjunctions to obtain the discernibility function corresponding to M (S ). After simpli cation (using the absorption laws) we get the following minimal disjunctive normal form of the discernibility function fM S (a; b; c; d; e) = a ^ b ^ (c _ e) = (a ^ b ^ c) _ (a ^ b ^ e) : There are two reducts: R = fa; b; cg and R = fa; b; eg of the system. Thus RED(S ) = fR ; R g. Example 2.3. illustrates how to nd all dependencies among attributes using Propositions 2.3. and 2.4. Example 2.3. Let us consider again the information system S from Example 2.1. By Proposition 2.3. we have for the system S the dependencies: fa; b; cg !S fd; eg and fa; b; eg !S fc; dg : ( )
1
1
2
2
Next, by Proposition 2.4. we get the following elementary dependencies: fa; b; cg !S fdg; fa; b; cg !S feg; fa; b; eg !S fcg; fa; b; eg !S fdg :
3. Minimal Rules in Information Systems In 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 looking for all a = v, where a 2 A [ fag, minimal rules in S of the form: ai1 = vi1 ^ . . . ^ air = vir ) S v 2 Va ; aij 2 A and vij 2 Vaij for j = 1; . . .; r. The above rules express functional dependencies between the values of the attributes of S . These rules are computed from systems of the form S 0 = (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 cation M (S 0; a; v; ul) of the discernibility matrix is computed from M (S 0). By M (S 0; a; v; ul) = (cij ) (or M , in short) we denote the matrix obtained from M (S 0 ) in the following way: IF i 6= l THEN cij = ;; IF clj 6= ; and dBa (uj ) 6= fvg THEN clj = clj \ B ELSE clj = ;. Next, we compute the discernibility function fM and the prime implicants [38] of fM taking into account the non-empty entries of the matrix M (when all entries cij are empty we assume fM to be always true). Finally, every prime implicant ai1 ^ . . . ^ air of fM determines a rule a=v; ai1 = vi1 ^ . . . ^ air = vir ) S 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 the above method for every R 2 RED(S ). First we construct all rules corresponding to nontrivial dependencies between the values of attributes from R and A ? R and next all rules corresponding to nontrivial dependencies between the values of attributes within a reduct R. These two steps are realized as follows.
(i) For every reduct R 2 RED(S ), R A and for every a 2 A ? R we consider the system S 0 = (U; R [ fag). For every v 2 Va, ul 2 U such that dRa(ul ) = fvg we construct the discernibility matrix M (S 0; a; v; ul), next the discernibility function fM and the set of all rules corresponding to prime implicants of fM . (ii) For every reduct R 2 RED(S ) with card(R) > 1 and for every a 2 R we consider the system S 00 = (U; B [ fag), where B = R ? fag. For every v 2 Va ; ul 2 U such that dBa (ul) = fvg we construct the discernibility matrix M (S 00; a; v; ul), then the discernibility function fM and the set of all rules corresponding to prime implicants of fM . The set of all rules constructed in this way for a given R 2 RED(S ) is denoted by OPT(S; R). S We put OPT(S ) = fOPT(S; R) : R 2 RED(S )g. a = vp is a rule from OPT(S ), then Let us observe that if ai1 = vi1 ^ . . . ^ air = vir ) S p U \ kai1 = vi1 ^ . . . ^ air = vir kS 6= ;. Proposition 3.1. [18]. Let S = (U; A) be an information system, R 2 RED(S ), and R A. Let fM S be a relative discernibility function for the system S 0 = (U; R [ fa g) where a 2 A ? R. Then all prime implicants of the function fM S correspond to all fag { reducts of S 0. The next example illustrates how to nd all nontrivial dependencies between the values of attributes in a given information system. Example 3.1. Let us consider the information system S from Example 2.1. and the discernibility function for S presented in Table 2. We compute the set of rules corresponding to nontrivial dependencies between the values of attributes from the reduct R of S with c; d (i.e. those outside of this reduct) as well as the set of rules corresponding to nontrivial dependencies between the values of attributes within the reduct of that system. In both cases we apply the method presented above. Let us start by computing the rules corresponding to nontrivial dependencies between the values of attributes from the reduct R = fa; b; eg of S with c; d. We have the following two subsystems S = (U; B [ fcg) and S = (U; B [ fdg) of S , where B = R , from which we compute the rules mentioned above: (
0
)
(
0
)
2
2
1
2
2
U=B u u u u u 1 2 3 4 5
a 1 0 2 0 1
b 0 0 0 0 1
e 0 1 0 2 0
c 2 1 2 2 2
dBc f2g f1g f2g f2g f2g
Table 3 The subsystem S = (U; B [ fcg) with the function dBc , where B = fa; b; eg 1
In the table the values of the function dBc are also given. The discernibility matrix M (S ; c, v; ul) where v 2 Vc, ul 2 U , l = 1; 2; 3; 4; 5; obtained from M (S ) in the above way is presented in Table 4. The discernibility functions corresponding to the values of the function dBc are the following: Case 1. For dBc (u ) = f2g : a _ e. Case 2. For dBc (u ) = f1g : (a _ e) ^ (a _ e) ^ e ^ (a _ b _ e) = e. 1
1
1
2
U u u u u u
u
u a,e
1
2
1
a,e
2
a,e e a,b,e
3 4 5
u
3
u
u
a,e
e
a,b,e
4
5
Table 4 The discernibility matrix M (S ; c; v; ul) for the matrix M (S ) 1
1
We consider non-empty entries of the column labelled by u (see Table 4), i.e. a; e; a; e; e and a; b; e; next a; b; e are treated as Boolean variables and the disjunctions a _ e, a _ e, e and are constructed from these entries; nally, we take the conjuction of all the computed disjunctions to obtain the discernibility function corresponding to M (S1 ; c ; v ; ul ). Case 3. For dBc (u ) = f2g : a _ e. Case 4. For dBc (u ) = f2g : e. Case 5. For dBc (u ) = f2g : a _ b _ e. Hence we obtain the following rules: a _ a _ b _ e _ e ) c,e ) c. S S In a similar way we compute the rules for the subsystem S = (U; B [ fdg) where B = fa; b; eg. 2
3
4
5
1
2
1
0
2
2
1
1
2
U=B u u u u u
a 1 0 2 0 1
1 2 3 4 5
b 0 0 0 0 1
e 0 1 0 2 0
d 1 2 1 2 1
dBd f1g f2g f1g f2g f1g
Table 5 The subsystem S = (U; B [ fdg) with the function dBd , where B = fa; b; eg 2
In the table the values of the function dBd are also given. The discernibility functions corresponding to the values of these functions are the following: Case 1. For dBd (u ) = f1g : (a _ e) ^ (a _ e) = a _ e Case 2. For dBd (u ) = f2g : (a _ e) ^ (a _ e) ^ (a _ b _ e) = a _ e Case 3. For dBd (u ) = f1g : (a _ e) ^ (a _ e) = a _ e Case 4. For dBd (u ) = f2g : (a _ e) ^ (a _ e) ^ (a _ b _ e) = a _ e Case 5. For dBd (u ) = f1g : (a _ b _ e) ^ (a _ b _ e) = a _ b _ e Hence we get the following rules: a _ a _ b _ e ) d , a _e _e ) d. S S Finally, the set of rules corresponding to all nontrivial dependencies between the values of attributes from R with c; d has the following form: e ) c, S 1
2
3
4
5
1
2
1
0
1
2
0
1
1
2
2
1
a _a _b _e _e ) c ; a _a _b _e ) d ; a _e _e ) d : S S S 1
2
1
0
2
2
1
2
1
0
1
0
1
2
2
Now we compute the rules corresponding to all nontrivial dependencies between the values of attributes within the reduct R . We have the following three subsystems (U; C [ feg), (U; D [ fbg), (U; E [ fag) of S , where C = fa; bg, D = fa; eg, and E = fb; eg, from which we compute the rules mentioned above: 2
U=C u u u u u 1 2 3 4 5
a 1 0 2 0 1
b
0 0 0 0 1
e
0 1 0 2 0
dCe
f0g
f1; 2g f0g
f1; 2g f0g
Table 6 The subsystem (U; C [ feg) with the function dCe , where C = fa; bg U=D u u u u u 1 2 3 4 5
a 1 0 2 0 1
e
0 1 0 2 0
b
0 0 0 0 1
dDb f0; 1g f0g f0g f0g
f0; 1g
Table 7 The subsystem (U; D [ fbg) with the function dDb , where D = fa; eg U=E u u u u u 1 2 3 4 5
b
0 0 0 0 1
e
0 1 0 2 0
a 1 0 2 0 1
dEa f1; 2g f0g
f1; 2g f0g f1g
Table 8 The subsystem (U; E [ fag) with the function dEa , where E = fb; eg In the tables the values of the functions dCe , dDb , and dEa are also given. The discernibility functions corresponding to the values of these functions are the following: Table 6. Case 1. For dCe (u ) = f0g : a ^ a = a. Case 2. For dCe (u ) = f0g : a ^ a = a. Case 3. For dCe (u ) = f0g : (a _ b) ^ (a _ b) = a _ b. Table 7. Case 1. For dDb (u ) = f0g : (a _ e) ^ (a _ e) = a _ e. Case 2. For dDb (u ) = f0g : a ^ a = a. Case 3. For dDb (u ) = f0g : (a _ e) ^ (a _ e) = a _ e. Table 8. Case 1. For dEa (u ) = f0g : e ^ e(b _ e) = e. Case 2. For dEa (u ) = f0g : e ^ e ^ (b _ e) = e. Case 3. For dEa (u ) = f1g : b ^ (b _ e) ^ b ^ (b _ e) = b. Hence we obtain the following rules: 1
3
5
2
3
4
2
4
5
From Table 6: a _ a _ b ) e. S From Table 7: a _ a _ e _ e ) b. S From Table 8: e _ e ) a ;b ) a. S S Finally, the set of rules corresponding to all nontrivial dependencies between the values of attributes within the reduct R has the form: a _a _b ) e ; a _a _e _e ) b ; e _e ) a;b ) a: S S S S 1
2
1
0
2
1
1
2
0
0
2
0
1
1
2
1
2
1
0
0
2
1
2
0
1
2
0
1
1
Eventually, we obtain the set OPT(S; R ) of rules corresponding to all nontrivial dependencies for the reduct R in the considered information system S : a _a _b _e _e ) c,e ) c , a _a _b _e ) d , a _e _e ) d , a _a _b ) e, S S S S S a _a _e _e ) b , e _e ) a,b ) a. S S S In a similar way one can compute the set OPT(S; R ) of rules corresponding to all nontrivial dependencies for the reduct R in the system S . This set consists of two kinds of rules. The rst kind consists of the rules corresponding to all nontrivial dependencies between the values of attributes from R with d; e of the form: a _ a ) d,a ) d,c ) e, S S S a _a ) e , a ^c ) e , while the second { of the rules corresponding to all nontrivial S S dependencies between the values of attributes within the reduct R of the following form: a _a _b ) c,c ) a,b ) a , a _a _c ) b. S S S S The set OPT(S ) of all rules constructed in this way for the information system S of Example 2.1. is the union of sets OPT(S; R ) and OPT(S; R ). Remark 3.1. Our approach to rule generation is based on procedures for the computation 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 practical applications. Among them are the feature extraction techniques or clustering methods known in pattern recognition [15] and machine learning [10], allowing to reduce the number of attributes or objects so that the rules can be eciently generated from them. Another approach is suggested in [2]. It leads to the computation of only so called the most stable reducts from the reduct set in a sampling process of a given decision table (i.e. a special case of an information system, see [17]). The rules are produced from these stable reducts only. This last technique can be treated as relevant feature extraction from a given set of features. The result of the above techniques applied to a given decision table is estimated as successful if rules can be eciently generated from the resulting compressed decision table by the Boolean reasoning method and if the quality of the classi cation of unseen objects by these rules is suciently high. We assume that the information systems which create inputs for our procedures satisfy those conditions. 2
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4. Decomposition of Information Systems
We present in this section concepts and notation related to the decomposition of information systems as well as a method for constructing components and coverings of a given information system with respect to its reducts. Let S = (U; A) be an information system. An information system S is said to be covered with constraints C (or C-covered, in short) by information systems S = (U ; A ); . . .; Sk = (Uk ; Ak ), if INF(S 0) = fInfA1 (u ) [ . . . [ InfAk (uk ) : InfA1 (u ) [ . . . [ InfAk (uk ) 2 CON(C ) and ui 2 Ui for i = 1; . . .; kg, where S 0 is a maximal consistent extension of S and C is a set of rules. 1
1
1
1
1
The pair (fS ; . . .; Sk g; C ) is called a C { covering of S (or a covering of S , in short). The sets S ; . . .; Sk are its components and C is the set of constraints (connections). Example 4.1. Let us consider the information system S from Example 2.1. It is easy to see that the information systems S = (U ; A ), S = (U ; A ), S = (U ; A ), and S = (U ; A ) represented by Table 9, 10, 11 and 12, respectively, and the set of constraints C containing rules: a _ a ) e , e _e ) a , a _a _e _e ) b ,b ) a , and b ) e yield a C {covering S S S S S of S . 1
1
1
1
2
0
1
2
0
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1
1
2
2
1
2
U =A u u u 1
2
0
1
a
1
2 3
3
1
1
2
c
2
d
2 1 2
1 2 4
3
1 2 3 4
0
0 1 2 2
0 1
Table 11 The information system S 4
4
b
3
5
4
4
e
1 2 2
1
U =A u u u u
4
1
Table 10 The information system S U =A u u
3
1 2 1
Table 9 The information system S U =A u u u
3
d
1 0 2
1
2
a 1 0 2 1
3
b
0 0 0 1
Table 12 The information system S
4
From the de nition of information system covering follows the obvious proposition presented below. Proposition 4.1. Every information system has at least one covering. If S = (U; A) then the system S = (U 0 ; A0) such that U 0 U , A0 = fa0 : a 2 B Ag, a0(u) = a(u) for u 2 U 0 and Va = Va for a 2 A is said to be a subsystem of S . Example 4.2. Every information system in Example 4.1. is a subsystem of the system S from Example 2.1. 0
Let S = (U; A) be an information system and let R 2 RED(S ). An information system S 0 = (U 0 ; A0) is a normal component of S (with respect to R) if and only if the following conditions are satis ed: (i) S 0 is a subsystem of S , (ii) A0 = B [ C where B is a minimal (with respect to ) subset of R such that B ) fag S for some a 2 A ? R, and C is the set of all attributes a with the above property. The set of all normal components of S (with respect to R) is denoted by COMPR (S ). Remark 4.1. From condition (ii) we have that the sets B and C establish a partition of A0. Example 4.3. The subsystems S , S , S are normal components of S (with respect to the reduct R ) from Example 2.1, but the subsystem S is not. A more detailed explanation of this fact is included in Example 4.4. The next proposition is a direct consequence of the de nition of a normal component of an information system and Proposition 2.3. Proposition 4.2. Every information system has at least one normal component (with respect to any of its reduct). Let S 0 2 COMPR (S ) and S 0 = (U 0 ; BS [ CS ). By XR we denote the set of all attributes which simultaneously occur in normal components of S (with respect to R) and in the reduct S R, i.e. XR = BS . S2 RS Let XR be a set de ned for S and R as above. We say that a subsystem S 0 = (U 0 ; A0) of S is to be a degenerated component of S (with respect to R) if and only if A0 = fag for some a 2 R ? XR . We denote this fact by fag ! ; (the empty set). S In the sequel a component (with respect to a reduct) will be assumed to be either a normal component or a degenerated component (with respect to the reduct). Proposition 4.3. Let S = (U; A) be an information system and let R be its reduct. Then the information system S consists of card(R ? XR ) degenerated components (with respect to R). Let S = (U; A) be an information system, R 2 RED(S ). We say that S is R-decomposable into components or that S is C-coverable by components (with respect to R) if and only if there exist components S = (U ; B [ C ); . . .; Sk = (Uk ; Bk [ Ck ) of S (with respect to R) with a set of constraints C such that B [ . . . [ Bk = R and C [ . . . [ Ck = A ? R, yielding a C -covering of S . The set of constraints (connections) includes: (i) rules corresponding to nontrivial dependencies between the values of attributes in Bi (i = 1; . . .; k) called internal linkings (the internal connections) within the component Si of S, (ii) rules corresponding to nontrivial dependencies between the values of attributes in a set Bi(i = 1; . . .; k) and those in the set A ? Ai , where Ai = Bi [ Ci called external linkings (the external connections) with the outside of Si . From the above de nition and from Proposition 2.3. follow the theorem and the proposition presented below. Theorem 4.1. Every information system is C-coverable by components (with respect to any its reduct), where C is the set of all internal and external linkings of S. 1
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COMP ( )
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We obtained a constructive method of the information system (data table) decomposition into functional modules interconnected by external linkings. One can observe a similarity of our data models to those used in general system theory and control design. Proposition 4.4. Let R be a reduct of an information system S . Then S has at least one C -covering by components (with respect to R), where C is the set of all internal and external linkings of S . We denote by COVERR (S ) the family of all C -coverings of S (with respect to R), where C is the set of all internal and external linkings of S .
4.1. Procedures for Computing Components and Coverings
Now we are ready to present a method for computing of the components of a given information system (with respect to its a reduct). All normal components of a given information system S = (U; A) (with respect to a reduct R 2 RED(S )) can be obtained by the following procedure: PROCEDURE for computing COMPR (S ): Input: An information system S = (U; A), a reduct R 2 RED(S ). Output: Components of S (with respect to R), i.e. the set COMPR (S ). Step 1. Compute all dependencies of the form: R ! fag, for any a 2 A ? R. S Step 2. Compute the discernibility function fM S for each subsystem S 0 = (U; R [ fag) of S with a 2 A ? R. In this step we compute the so called fag - reducts of R, for a 2 A ? R [17] (see also Proposition 3.1.). Step 3. For all dependencies of the form B ! fa g; . . .; B !S faik g, where B is any subset S i1 of R obtained in Step 2, construct a dependency B ! C , where C = fai1 g [ . . . [ faik g. The S set C is the maximal (with respect to ) subset in A ? R such that the dependency B ! C S 00 0 is true. Now the subsystem S = (U ; B [ C ) of S de nes a normal component of S (with respect to R). The correctness of this method follows from Proposition 2.1., Proposition 3.1. and from the de nition of components (with respect to a reduct). One can see that the time and space complexity of the discussed problem is, in general, exponential because of the complexity of RED(S ) computing. Example 4.4. Let us perform the procedure for the computation COMPR2 (S ) for the information system S of Example 2.1. and its reduct R . Step 1. The following elementary dependencies are valid in the system S for the reduct R : fa; b; eg ! fcg, fa; b; eg !S fdg (see Example 2.3.). S Step 2. We compute the minimal subsets of R on which the sets fcg and fdg depend, i.e. we compute the relative reducts (cf. [18]) of the left hand sides of the above dependencies. To reduce the rst elementary dependency we consider the information system S = (U; B [fcg) with B = fa; b; eg. Hence, fM S1 (a; b; e) = e. Thus fa; b; eg ! fcg can be simpli ed to S feg !S fcg. We reduce the second dependency in a similar way. As a consequence, fa; b; eg ! fdg S can be reduced either to fag ! fdg or feg !S fdg. Eventually, we get the following minimal S dependencies (i.e. dependencies with a minimal number of attributes on the left hand side) with respect to R in the information system S : fag ! fdg; feg !S fcg; feg !S fdg. This S completes Step 2 of the above procedure. (
0
)
2
2
2
1
(
2
)
Step 3. For dependencies feg ! fcg and feg !S fdg we construct a new dependency S feg !S fc; dg. Now we have fag !S fdg and feg !S fc; dg. They de ne two normal components S1 = (U1 ; A1) and S2 = (U2 ; A2) of the system S , where: A1 = B1 [ C1; B1 = fag; C1 = fdg; A2 = B2 [ C2; B2 = feg; C2 = fc; dg : Since XR2 = fa; eg, we have R2 ? XR2 = fbg. This means that fbg ! ; is true in S . S
Hence S has the degenerated component S = (U ; A ) of the form: A = B [ C ; B = fbg; C = ; : Eventually, the system is decomposed into three components (with respect to R ). They are shown in Table 9, 10 and 11, respectively (see Example 4.1.). There are no the internal linkings in components of the system (with respect to the reduct R ), since each component of S contains only one attribute from R . However, the components are connected by external linkings of the form: 1. For S and S : a _ a ) c ; a _a ) e ; e _e ) a. S S S 2. For S and S : a _ a ) b;b ) a ;b ) d. S S S 3. For S and S : e _ e ) b,b ) e,b ) c ;b ) d. S S S S It is easy to see that the rules: a _ a ) c ,b ) d ,b ) c can be omitted, since they follow S S S from nontrivial dependencies between the values of attributes within the reduct R as well as the dependencies between the values of attributes from R with c and d. In this case the set of constraints C consists of the following rules: a _ a ) e , e _e ) a , a _a _e _e ) b, S S S b) a,b ) e (see Example 4.1.). S S In a similar way we compute the components of S (with respect to the reduct R = fa; b; cg0 ). After0 appropriate calculations we obtain three components S 0 , S 0 , and S 0 such that S and S are the same as the components S and S of S (with respect to R ), and the component S 0 has the form: S 0 = (U 0 ; B 0 [ C 0 ), B 0 = fa; cg, C 0 = feg. The component S 0 of S is normal and it is shown in Table 13. There are only some internal linkings in the component B 0 = fa; cg of the system S (with respect to R ) represented by rules corresponding to nontrivial dependencies between the values of attributes within the set B 0 of the form: a _ a ) c,c ) a . Besides, the S S components of the system S are connected by external linkings of the form: a _ a ) e, S a _a ) b,c ) b,b ) a,b ) c. S S S S Eventually, in the considered case the set of constraints C 0 consists of the following rules: a _a ) c,c ) a , a _a ) e , a _a ) b,c ) b,b ) a,b ) c. S S S S S S S 3
3
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To compute a covering of an information system by its components (with respect to a reduct) it is sucient to perform the following procedure. PROCEDURE for computing COVERR (S ): Input: An information system S = (U; A), a reduct R 2 RED(S ). Output: The covering family of S , i.e. COVERR (S ). Step 1. Compute all normal and degenerated components of S (with respect to R). Step 2. Compute the set C of all external and internal linkings of S . Step 3. Choose those combinations of components which together with C yield a C covering by components of S (with respect to R). This step is to be performed as long as new solutions are obtained.
U 0 =A0 u u u u
a
2
2
c
1 0 2 0
1 2 3 4
e
2 1 2 2
0 1 0 2
Table 13 The component S 0 = (U 0 ; B 0 [ C 0 ) of the system S (with respect to R ) from Example 2.1, where B 0 = fa; cg, C 0 = feg 2
2
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2
Example 4.5. The information system S of Example 2.1. has one0 covering (fS ; S ; S g; C ) (with respect to the reduct R ) as well as one covering (fS ; S ; S g; C 0) (with respect to 0 1
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3
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3
the reduct R ), where S ; S ; S , and S denote components of S (with respect to these reducts), and C , C 0 are the sets of constraints determined for these coverings, respectively (see Example 4.4.). Now we show an example of a construction of a given information system from its components. Example 4.6. Let us consider the information system S from Example 2.1., its components S ; S ; S (with respect to the reduct R ), and the set of constraints C computed in Example 4.4. This system we reconstruct from its components and the set of constraints in two steps. First, we compute the subsystem S 0 = fS ; S g of S presented in Table 14. 1
1
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U 0 =A0 u u u u
2
a
2
c
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1 2 3 4
d
2 1 2 2
e
1 2 1 2
0 1 0 2
Table 14 The subsystem S 0 of the system S from Example 2.1 Next, we construct the system S 00 = fS 0; S g shown in Table 15. 3
U 00=A00 u u u u u u u u 2
1 2 3 4 5 6 7 8
a 1 0 2 0 1 0 2 0
b
0 0 0 0 1 1 1 1
c
2 1 2 2 2 1 2 2
d 1 2 1 2 1 2 1 2
e
0 1 0 2 0 1 0 2
Table 15 The subsystem S 00 of the system S from Example 2.1
It is worth to observe that in this case we have U U 00 . Additional states from U 00 ? U are objects with indexes from 6 up to 8. It is easy to see that the rules from C eliminate the states from u up to u . One can see that the remaining states are consistent with the above rules and the resulting system S 00 is the same as the original system S . 6
8
5. Petri Nets
Petri nets are useful for describing and analyzing the structure of systems and information
ow in them. In this paper Petri nets are used as a tool for representing and analyzing the knowledge represented by an information system. After modelling an information system by a Petri net, many desirable properties of the system can be revealed by analyzing properties of the constructed Petri net (see e.g. [35]). At rst, we recall some basic concepts from Petri net theory. A Petri net contains two types of nodes, circles P (places) and bars T (transitions). The relationship between the nodes is de ned by two sets of relations; de nes the relationship between places and transitions, and de nes the relationship between transitions and places. The relations between nodes are represented by directed arcs. A Petri net N is de ned as a quadruple N = (P; T; ; ). Such Petri nets are called ordinary. In the paper we only use the ordinary Petri nets. A marking m of a Petri net is an assignment of black dots (tokens) to the places of the net for specifying the state of the system. The number of tokens in a place pi is denoted by mi and then m = (m ; . . .; ml ), where l is the total number of places of the net. The initial distribution of tokens among the places is called the initial marking and is denoted by M . A Petri net N with a marking M is called a marked Petri net and it is denoted by (N; M ). In the paper we only use nets in that all markings are binary, i.e. m(p) 2 f0; 1g for any place p. Input and output places of a transition are those which are initial nodes of an incoming or terminal nodes of an outgoing arc of the transition, respectively. In a similar way we de ne input and output transitions of a place. The dynamic behaviour of the system is represented by the ring of the corresponding transition, and the evolution of the system is represented by a ring sequence of transitions. We assume that nets constructed in the paper act according to the following transition ( ring) rules: (1) A transition t is enabled if and only if each input place p of t is marked by one token. (2) A transition can re only if it is enabled. (3) When a transition t res, a token is removed from each input place p of t, and t adds a token to each output place p0 of t. A marking m0 is said to be reachable from a marking M if there exists a sequence of rings that transforms M to m0. The set of all possible markings reachable from M in a net N is called the M-reachability set of N , and it is denoted by R(N; M ). Let N = (P; T; ; ) be an ordinary Petri net. 1. A Petri net N is called coverable by the nets N =(P ; T ; ; ); . . .; Nk =(Pk ; Tk ; k ; k ) if and only if P = P [ . . . [ Pk ; T = T [ . . . [ Tk ; = [ . . . [ k ; = [ . . . [ k : The set fN ; . . .; Nk g is called a covering of a net N . 2. A Petri net N 0 = (P 0; T 0; 0; 0) we call a subnet of if all places and all transitions of N 0 belong to N , i.e. if P 0 P , T 0 T , and if in a net N 0 there are precisely those arcs of a net N which connect the nodes included in a net N 0 , i.e. if 0 [ 0 = ( [ ) \ ((P 0T 0) [ (T 0P 0)) : 1
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3. A subnet N 0 = (P 0; T 0; 0; 0) of N is called a component, if a set of all input transitions for places from P 0 is equal to a set of all output transitions for places from P 0. 4. A Petri net N is said to be covered by components N ; . . .; Nk (or decomposable on components N ; . . .; Nk ), if these components consists of a covering of N . From these de nitions follow that a component N 0 of a net N is determined unambiguously by a set of its places, i.e. a set P 0. Besides, a component N 0 of N is a strongly connected subnet of N . Notice that for any net N = (P; T; ; ) a component, de ned by a set of places P 0, where ; 6= P 0 P , is N (P 0) = (P 0; T 0; 0; 0), where T 0 consists of all input and output transitions for places from P 0 and 0 [ 0 = ( [ ) \ ((P 0T 0) [ (T 0P 0)) : Example 5.1. A Petri net from Figure 2 is covered by components: N (fa ; a ; a g), N (fb ; b g), and N (fe ; e ; e g), whereas a net shown in Figure 3 has, for instance, the following components: N 0(fa ; a ; a ; d ; d g), N 0(fb ; b g), and N 0 (fe ; e ; e ; c ; c ; d ; d ; d g). All components of these nets are also their the strongly connected subnets. For more detailed information about Petri nets we refer the reader to [13]. 1
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6. How to Compute Concurrent Data Models from Information Systems?
We present a method for constructing a marked Petri net (NS ; MS ) for an arbitrary information system S such that the reachability set R(NS ; MS ) represents the set of all global states consistent with a given information system S . That method consists of two steps. First, the rules corresponding to two kinds of dependencies are generated. The rst kind consists of the dependencies between the values of attributes within reducts, the second - of the dependencies between the values of attributes not in reducts and those within reducts. Next, the rules so obtained (and represented in minimal form with respect to the number of descriptors on the left hand side) [24] are implemented by means of a Petri net.
6.1. Initial Transformations of Rules
Now we present a method for transforming rules representing a given information system into a Petri net (cf. [30]). First initial transformations of rules are performed. There are two rules (see Figure 1). A rule A Petri net p )S q p ? mR mq p ^ q )S r p ? m q R rm
m
Figure 1 Illustration of two initial transformation rules, where p, q, r are descriptors in S
6.2. Transformation of Rules into Petri Nets
In this subsection we illustrate on examples a method for transforming rules representing an information system (with respect to any reduct of a given system) into a Petri net. This method consists of three levels: 1. A net representing all attributes in a reduct of a given information system is constructed. 2. The net obtained in the rst step is extended by adding the elements (arcs and transitions) of the net induced by the rules determined by: all nontrivial dependencies between the values of attributes not in a reduct and those within a reduct of the information system, dependencies between the values of attributes within a reduct of the information system. 3. We add to the net obtained so far the subnets corresponding to situations when between some values of attributes (but not all values) there are no dependencies represented by the information system. This method is repeated for all reducts of the given information system. Finally, the obtained nets are merged. Such an approach makes the appropriate construction of a net much more readable. Moreover, one can compare better our approach with that presented in [18]. For more detailed information about this transformation method see [30]. In the examples which follows we only illustrate some steps of the above transformation method. These examples use the reduct R of the information system S of Example 2.1. Example 6.1. Let us consider again the information system S from Example 2.1. The attributes a; b; e 2 R are represented by the nets shown in Figure 2. 2
2
# ? pa m
?pm b
0
' ?me
0
0
? ? ma ? "?ma
? ?m b
1
? ?m `p e ? &?me
1
1
2
2
Figure 2 Nets representing attributes a; b; e 2 R
2
Example 6.2. Consider the rules from Example 3.1. for the system S of Example 2.1. of
the form:
a ) d ; a _a ) d;e ) d ;e _e ) d ; e _e ) c;e ) c : S S S S S S These rules correspond to the dependencies between attributes within the reduct R of S and those outside of the reduct. A net representation of the above rules obtained by an application of our construction (after some simpli cations consisting in the deletion of super uous arcs) is illustrated in Figure 3. The initial markings of the nets presented in Figures 2 and 3 corresponds to the second row of Table 1. It is worth to observe in Figure 3 that the components N 0 (fa ; a ; a ; d ; d g) and 0 N (fe ; e ; e ; c ; c ; d ; d g) have the common places d and d . These places are drawn separately for readibility reasons. 0
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The construction method shortly described above has the following properties [30]: Theorem 6.1. Let S be an information system, and let (NS ; MS ) be a marked Petri net representing a system S . Then INF(NS ; MS ) = CON(D(S )), where INF(NS ; MS ) denotes the set fv(M ) : M 2 R(NS ; MS )g and R(NS ; MS ) is the reachability set of NS from MS . Let S = (U; A) be an information system, U 0 U be a set of the same cardinality as INF(NS ; MS ) and let f be a bijection between U 0 and INF (NS ; MS ) such that f (u) = (a (u); . . .; am(u)) for u 2 U 0 . We assume also A = fa ; . . .; amg and A0 = fa0i : ai 2 Ag. By S 0 = (U 0 ; A0) we denote U 0 -extension of S such that a0i(u) = (f (u))i for u 2 U 0 and i = 1; . . .; m. Theorem 6.2. Let S be an information system, S 0 its U 0 -extension constructed as above. Then S 0 is the largest consistent extension of S . From Theorem 6.2. and the de nition of C -covering of a given information system we obtain the theorem establishing an important property of the decomposition of information systems proposed in the paper. Corollary 6.1. Let S 0 be a maximal consistent extension of an information system S constructed by the method presented in the paper. Then COVERR (S ) = COVERR (S 0), for any reduct R of S . 1
1
-- d ?p @@ ?`p ?e mc Rm m+m pd b m a `m 0
1
2
@@R ?? ? ? ?? ? eZZ ?Qs md b m ? d pm= p?m ~ pmc a mQ ? ?? ? a ? m ?me 0
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Figure 3 A net representation of rules from Example 6.2 In Figure 4 and Figure 5 we show nets representing the external linkings between components S , S and S , S , respectively, computed in Example 4.4. These nets also include arcs which guarantee the correctness of the construction. 1
3
2
3
a- ?PiPPPP ? p PPPqP pb ? 1 ? ?b a ?)
?
a ? 0
1
0
1
2
Figure 4 External linkings between components S and S One can see that the rechability sets of these nets are consistent with all rules valid in the information system from Example 2.1. In the Petri net (NS ; MS ) constructed for a given information system S one can distinguish components and connections between them. The Petri net (NS ; MS ) can be decomposed with respect to any C -covering in COVERR (S ), where R is any reduct of S and C is the set of all internal and external linkings of S . 1
3
-
mi : ?me b p? 9 PPq ? ?I p?me b ?m R? ?me 0
0
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1
2
Figure 5 External linkings between components S and S 2
3
Remark 6.1. A net representation of components S , S , and S of the system of Example 1
4.4. is shown in Figure 3.
3
2
7. Conclusions
The decomposition method has been implemented in C ++ and preliminary tests are promising. Our method can be applied for automatic feature extraction. The properties of the constructed concurrent systems (e.g. their invariants) can be interpreted as higher level laws of experimental data. New features can be also obtained by performing for a given decision table S = (U; A [ fdg) the following steps: Step 1. Extract from S a subtable Si corresponding to the decision i, for any i 2 Vd , i.e. Si = (Ui; Ai), where Ui = fu 2 U : d(u) = ig, Ai = fai : a 2 Ag, and ai(u) = a(u) for u 2 Ui . Step 2. Compute the components of Si for any i 2 Vd . Step 3. For a new object u compute the values of components de ned on information included in InfA(u) and check in what cases the computed values of components are matching InfA(u) (i.e. they are included in InfA(u)). For any i 2 Vd compute the ratio ni (u) of the number of components matching InfA(u) to all components of Si. The simplest strategy classi es u to the decision class i , where ni0 (u) = maxi ni(u). We also study some applications of our method in control design from experimental data tables. The application of Petri nets to representing a given information system enable us: to represent in an elegant and visual way the dependencies between components in the system, and their dynamic interactions, to observe concurrent and sequential subsystems (components) of the system. On the basis of Petri net approach it was possible to understand better the structure of those rules which are true in a given information system. Acknowledgements. I am grateful to Professor A. Skowron for stimulating discussions and interesting suggestions about this work. I am also would like to thank J. Wegrzyn for his helpful comments and high quality programming. 0
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