S In a practical diapason of values m and N rank r turns out to be small. This allows defining all the implicative max laws using a check for intervals «forbiddances» of a rank that is not more than r. The disjunctive union of all max the found forbidden intervals as conjunctions of combinations of informative features of DMO forms an analytic (predicate) description of the forbidden area, corresponding BtkZ.

Definition 3. The algorithmic procedure INDS INDS(tk20;AZ;tk2BM)= tk20 tk2BM, (12) AZ implementing inductive derivation of non-odd BtkZ= tk2 BM in a form of a set of simple prohibitions from the learning knowledge quantum tk using the algorithm АZ, is called an operator of inductive derivation of 2 implicative tk-knowledge (INDS-operator) [Sirodzha, 1992].

Algorithm АZ Input: TED in a form of quantum tk of size mn threshold M* =10-2, maximal rank r = 3.

2 0, S max Output: minimized BtkZ = tk2 BM as a system of simple forbidden quanta, i.e. that do not result one from another.

Steps:

1. according to r patterns of features prohibitions combinations are formed. For r = 3 there are 8 patterns:

max max <000>, <001>, <010>, <011>, <100>, <101>, <110>, <111>. Forbidden combinations are searched between domains components, but not inside a domain.

2. In the cycle in tk all the combinations of features values are taken as doubles, and then as triples, etc. till 2 r. The non-found in tk pattern combinations are added to tk2 B.

max 2 3. The formed quantum of prohibitions tk2 B is minimized in BtkZ = tk2 BM using operators of gluing, merging and compression.

Let’s assume that in the result of step 2 in the algorithm AZ we got a quantum tk2 B. DMO is characterized by three features x, x, x.

1 2 XII-th International Conference "Knowledge - Dialogue - Solution" x1 x2 x 01 -:- 1:- 1 - 01-:0-:-1 -, where «-» defines «it is indifferent if it is 0 or 1».

3.2. Deductive derivation operator of decisions from implicative tk-knowledge.

It is necessary to solve the task of building the algorithm AL, implementing deductive operating process to ( j search the needed decisions being correspondent with the logical consequence tk Y, tk1Y, tk0ik) :

DED DED DED ( j tk2 BM tk Y, tk BM tk1 Y, tk BM tk0ik), (13) 2 2 AL1 AL3 ALwhere tk BM is a known database of implicative tk-knowledge.

( j The searched sequences tk Y, tk1Y tk0ik) (13) represent the different-level tk-knowledge, characterizing the decisions being made in basic tasks В and С according to the observed results.

t t Let a base of implicative tk-knowledge tk2 BM is given and a quantum tk1Y of knowledge about the observed DMO of a data domain being investigated. The algorithm АL to evaluate the possible condition of DMO according to quanta of observations tk1Y, based on a known BtkZ, is a implementation of deductive derivation for the searched decision according to the scheme (13). Let’s note that under the possible condition of DMO we understand a class or pattern and the DMO is concerned to it while solving the В t-task or a category (value) of prognosis connected with DMO if we solve the С t-task.

Algorithm АL Input: tk-knowledge BtkZ= tk2 BM and observations tk1Y for DMO.

* Output: deductively derived tk-knowledge tk2 Y from BtkZ about the possible condition of DMO, according to the observations tk1Y.

Steps:

1. To make a substitution of quantum values tk1 Y in BtkZ= tk2 BM in this way: to delete columns in a matrix quantum tk BM, meeting the features of the observed quantum tk1 Y.

Decision Making 2. To delete the rows, which are orthogonal to the observation tk1 Y row, from the formed minor (respectively to * the known features; ‘orthogonal’ means those having opposite in the meaning). In such a way we get tk2 Y.

* * 3. To invert the received quantum and consider it to be the result tk2 = tk2 Y.

( j The algorithm is analogical for deriving the logical sequences tk1Y, tk0ik) [Sirodzha, 2002].

Let’s assume the DMO is characterized with 4 features (x,x,x,x ), and the BtkZ has been inductively received in 1 2 3 a form of:

x1 x2 x x 01- : -1: -1- - - : 01- : 0- : -1- -- :1- -tk2 BM = :1- : - - -0 :1 - -- : 0 1- - ::1- ::- - -0: --0 - -1 0 - - : -0 : - -1- : -There is also a quantum to observe the DMO tk1Y =[001:10:0100:--]. It is required to define the possible value of the non-measured feature x. According to the algorithm steps we get:

001 : 10 : 0100 : -- tk1Y = [ ] 01- : -1 : -1-- : 01 01- : -1 : -1-- : tk2 BM = 01- : 0- : -1-- : 1- 01- : 0- : -1-- : 1- -10 : 1- : ---0 : 1- -10 : 1- : ---0 : 1- --- : 1- : ---0 : 0- --- : 1- : ---0 : 0- 1-- : -0 : --1- : -0 1-- : -0 : --1- : - 0-- : -0 : --1- : -1 0-- : -0 : --1- : - After applying algorithm AL steps 1,2 a quantum [--- : 1- : ---0 : 0-] is left. After the inversion (step 3 of the ( j algorithm AL) tk0ik) =[1], i.e. the 4th feature (the 4th domain corresponds to it) takes the first value. Analogically the tasks to prognosis the several features values are being solved. In such a way the В t-, С t-tasks have been solved with a help of the algorithm AL.

Conclusion Operating derivation of identification and prognostic decisions using tRAKZ-method suppose such a sequence of operating transformations of different-level tk-knowledge: using induction operator according to the given table of empirical data (TED) as learning tk-knowledge the database of authentic knowledge quanta (BtkZ) is synthesized. Then using deduction operator according to the observed (input) tk-knowledge of DMO, the searched identification or prognostic decisions are derived on the basis of BtkZ in a form of resulting tkkhowledge.

Operating method of decision derivation is based on the computer manipulation of vector-matrix structures (unlike the existing methods), that allows to abbreviate the time for BtkZ synthesis as a conclusive rule and to increase the efficiency of computer decision-making.

Bibliography [Sirodzha, 2002] Sirodzha, I.B. Quantovye modeli I metody iskusstvennogo intellekta dlya prinyatiya reshenij I upravleniya.

(Quantum models and methods of artificial intelligence for decision-making and management). Naukova dumka. – Kyiv:

2002. – 420 pp.

[Sirodzha,1992] Sirodzha, I.B. Matematicheskoe I programmnoe obespechenie intellektualnykh compiuternykh sistem.

(Mathematical provision and programming software of intellectual computer systems.) – Kharkiv: KhAI,1992.

Authors' Information Liudmyla Molodykh – a post-graduate student of Computer System Software Department, National Aerospace University named after N.I. Zhuckovsky "Kharkov Aviation Institute"; room 518, Impulse Building, Chkalova st., 17, Kharkiv, Ukraine, 61070; e-mail: molodykh@onet.com.ua; flamelia@mail.ru Igor B. Sirodzha – Professor, Doctor of Technical Sciences, Head of Computer System Software Department, National Aerospace University named after N.I. Zhuckovsky "Kharkov Aviation Institute"; room 414, Impulse Building, Chkalova st., 17, Kharkiv, Ukraine, 61070.

XII-th International Conference "Knowledge - Dialogue - Solution" CONSTRUCTING OF A CONSENSUS OF SEVERAL EXPERTS STATEMENTS Gennadiy Lbov, Maxim Gerasimov Abstract: Let be a population of elements or objects concerned by the problem of recognition. By assumption, some experts give probabilistic predictions of unknown belonging classes of objects a, being already aware of their description X (a). In this paper, we present a method of aggregating sets of individual statements into a collective one using distances / similarities between multidimensional sets in heterogeneous feature space.

Keywords: pattern recognition, distance between experts statements, consensus.

Introduction We assume that X (a) = (X1(a),..., X (a),..., Xn(a)), where the set X may simultaneously contain j qualitative and quantitative features X, j = 1,n. Let Dj be the domain of the feature X, j = 1,n. The j j n feature space is given by the product set D = Dj. In this paper, we consider statements Si, i = 1,M ;

j=represented as sentences of type “if X (a) Ei, then the object a belongs to the -th pattern with probability n i i pi ”, where {1,...,k}, Ei = Ei, Ei Dj, Ei = [, ] if X is a quantitative feature, Ei is a j j j j j j j j=finite subset of feature values if X is a nominal feature. By assumption, each statement Si has its own weight j wi. Such a value is like a measure of “assurance”.

Without loss of generality, we can limit our discussion to the case of two classes, k = 2.

Distances between Multidimensional Sets In the works [1, 2] we proposed a method to measure the distances between sets (e.g., E1 and E2 ) in heterogeneous feature space. Consider some modification of this method. By definition, put n n (E1, E2) = k (E1, E2) or (E1, E2) = k ( (E1, E2))2, j j j j j j j j j=1 j=n where 0 k 1, k =1.

j j j=| E1E2 | j j Values (E1, E2 ) are given by: (E1, E2 ) = if X is a nominal feature, j j j j j j j | Dj | 12 2 rj + | E1E2 | 1 + 1 + j j 12 j j j j (E1, E2) = if X is a quantitative feature, where rj = -.

j j j j | Dj | 2 It can be proved that the triangle inequality is fulfilled if and only if 0 1 2.

The proposed measure satisfies the requirements of distance there may be.

Consider the set (1) = {S(1),..., S(m }, where S(u is a statement concerned to the first pattern class, 1) 1) u = 1,m1. Let Eu be the relative sets to statements S(u, Eu D, u = 1,m1. By analogy, determine 1) ~ (2) = {S(2),..., S(m }, S(v, Ev as before, but for the second class.

2) 2) The work was supported by the RFBR under Grant N04-01-00858.

Decision Making m1 m2 ~v j By definition, put k =, where = (Eu, E ), j = 1,n.

j j j j j n u=1 v=i i=Consensus We first treat single expert’s statements concerned to a certain pattern class: let be a set of such statements, = {S1,...,Sm}, Ei be the relative set to a statement Si,i = 1,m.

n 1 1 2 1 2 1 Denote by Ei i2 := Ei Ei = (Ei Ei ), where Ei Ei is the Cartesian join of feature values j j j j j=1 Ei and Ei for feature X and is defined as follows.

j j j 1 2 1 2 1 When X is a nominal feature, Ei Ei is the union: Ei Ei = Ei Ei.

j j j j j j j 1 2 1 2 1 When X is a quantitative feature, Ei Ei is a minimal closed interval such that Ei Ei Ei Ei.

j j j j j j j 1 1 1 Denote by ri i2 := d(Ei i2, Ei Ei ).

k | E' | j j The value d(E, F) is defined as follows: d(E, F) = max min, where E' is any subset such E'E\F j |E' ||E | j j diam(E) that its projection on subspace of quantitative features is a convex set.

u By definition, put I1 = {{1},...,{m}}, …, Iq = {{i1,...,iq}| ri iv < u,v =1,q}, where is a threshold decided by the user, q = 2,Q ; Q m.

Take any set Jq = {i1,...,iq}of indices such that Jq Iq and Jq Jq+1 Jq+1 Iq+1.

J q q Now, we can aggregate the statements Si, …, Si into the statement S :

J q q q S = “if X (a) EJ, then the object a belongs to the -th pattern with probability pJ ”, where q ciJ wi pi iJq q q q q q EJ = Ei... Ei, pJ =, ciJ = 1- (Ei, EJ ).

q ciJ wi iJq q ciJ wi iJq J q q 1- Jq By definition, put to the statement S the weight wJ = d(E, Ei ).

iJq q ciJ iJq J q The procedure of forming a consensus of single expert’s statements consists in aggregating into statements S for all Jq under previous conditions, q = 1,Q.

After coordinating each expert’s statements separately, we can construct an agreement of several independent q q experts for each pattern class. The procedure is as above, except the weights: wJ = ciJ wi.

iJq Solution of Disagreements After constructing of a consensus for each pattern, we must make decision rule in the case of contradictory statements. Take any sets E(u and E(v such that E(u E(v = Euv, where the set E(u corresponds to 1) 2) 1) 2) ) a statement S(u from the experts agreement concerned to the -th pattern class, = 1,2.

) i * * * Consider the sets I(uv ={i | (S ( ) ) and ((Ei, Euv ) < )}, where is a threshold, 0 < < 1.

} XII-th International Conference "Knowledge - Dialogue - Solution" (1- (E(i, Euv ))wi pi ) iI(uv) * By definition, put p(uv =. Denote by := arg max( p(uv ).

) ) i (1- (E{ ), Euv ))wi uv iI{ ) Thus, we can make decision statement:

* Suv = ” if X (a) Euv, then the object a belongs to the -th pattern with probability p(uv ” * ) (1- (E(i, Euv ))wi - (1- (E(i, Euv ))wi 1) 2) iI(uv) iI(uv) 1 with the weight wuv =.

(1- (E(i, Euv )) * iI(uv* ) ) Bibliography [1] G.S.Lbov, M.K.Gerasimov. Determining of distance between logical statements in forecasting problems. In: Artificial Intelligence, 2’2004 [in Russian]. Institute of Artificial Intelligence, Ukraine.

[2] G.S.Lbov, V.B.Berikov. Decision functions stability in pattern recognition and heterogeneous data analysis [in Russian].

Institute of Mathematics, Novosibirsk, 2005.

Authors' Information Gennadiy Lbov – Institute of Mathematics, SB RAS, Koptyug St., bl.4, Novosibirsk, Russia;

e-mail: lbov@math.nsc.ru Maxim Gerasimov – Institute of Mathematics, SB RAS, Koptyug St., bl.4, Novosibirsk, Russia, e-mail: max_post@mail.ru ANALYSIS AND COORDINATION OF EXPERT STATEMENTS IN THE PROBLEMS OF INTELLECTUAL INFORMATION SEARCHGennadiy Lbov, Nikolai Dolozov, Pavel Maslov Abstract: The paper is devoted to the matter of information presented in a natural language search. The method using the statements agreement process is added to the known existing system. It allows the formation of an ordered list of answers to the inquiry in the form of quotations from the documents.

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