OPERATING MODEL OF KNOWLEDGE QUANTUM ENGINEERING FOR DECISION-MAKING IN CONDITIONS OF INDETERMINACY Liudmyla Molodykh, Igor Sirodzha Abstract: The operating model of knowledge quantum engineering for identification and prognostic decisionmaking in conditions of -indeterminacy is suggested in the article. The synthesized operating model solves three basic tasks: А to formalize tk-knowledge; В to recognize (identify) objects according to observed t-task t-task results; С to extrapolate (prognosticate) the observed results. Operating derivation of identification and t-task prognostic decisions using authentic different-level algorithmic knowledge quantum (using tRAKZ-method) assumes synthesis of authentic knowledge quantum database (BtkZ) using induction operator as a system of implicative laws, and then using deduction operator according to the observed tk-knowledge and BtkZ a derivation of identification or prognostic decisions in a form of new tk-knowledge.

ACM Classification Keywords: I.2.3 Deduction and Theorem Proving; I.2.4 Knowledge Representation Formalisms and Methods; I.2.5 Programming Languages and Software Introduction Knowledge-oriented modelling of human being’s intellectual skills to make decisions in conditions of indeterminacy to recognize patterns and prognostic situations for artificial intelligence systems (AIS) is being developed in the article. Operating model for knowledge quantum engineering for decisions derivation in conditions of indeterminacy, which is based on using the method of authentic different-level algorithmic knowledge quantum or portions (tRAKZ-method) is suggested. The existing artificial neural networks (ANN) and knowledge engineering methods, based on frame, production and other knowledge models, are not effective enough because of the imperfection of representation ways and computer knowledge manipulation. Unlike these approaches the suggested model has a form of strictly formalized knowledge quantum, different in the level of complexity (tk-knowledge). Such tk-knowledge as substantial algorithmic structures of authentic data allow computer manipulation of knowledge using an finite predicates algebra and vector-matrix operators, and also inductive synthesis of knowledge quantum database (BtkZ) while teaching computer using selective plot examples of situations from the concrete data domain.

XII-th International Conference "Knowledge - Dialogue - Solution" 1. Target Setting The model-based process of human’s classification and prognostic decision-making in conditions of indeterminacy is always aimed (motivated by a target criterion) at the decision-making object (DMO), which can be described with a set of characteristics (features), measured in different scales and allowing logical representation. Target features are also contained in this set. Their values determine the class and pattern of the considered DMO. To identify the class (pattern) of DMO, i.e. to make a classification decision, means to define a value of the target feature according to the observed initial characteristics, relying on the knowledge quantum database (BtkZ), represented by classification law systems. Analogically to make a prognostic decision it is necessary to have a prognostic BtkZ, allowing to define the value of the target prognostic feature on the segment t+t, according to the situation on the time segment t.

The discussed -indeterminacy is characterized by such limitations:

data about DMO are of different type (i.e. measured in quantitative as well as in qualitative scales) and can be reached in incomplete volumes and from different sources (experts, technical documentation, reference books, instruments measurements etc.);

the target criteria are given implicitly, it is unknown which ones, in what quantity and how to select informative features of DMO according to targets of decision-making;

the rules of making classification and prognostic decisions are unknown, and also the inductive principles of their building by teaching on selective experimental data are unknown too;

the sought rules of decision-making are impossible to be defined by regular calculus of approximations directly, but it is possible to create knowledge engineering tools to model and imitate intellectual human’s skills to find solutions, relying on intuition and knowledge database.

In -indeterminacy the authentic k-knowledge (tk-knowledge) are used.

The main task of this article is to create a method of synthesis for operating model in knowledge quantum engineering to derive classification and prognostic decisions in conditions of -indeterminacy. In general this task is deducted to solving three basic tasks [Sirodzha, 2002]:

1. А t-task for formalization of tk-knowledge;

2. В t-task for object recognition (identification) according to observation results;

3. С t-task for extrapolation (prognostic) of observation results.

In the А t-task it is required to define the terms “tk-knowledge” and “tRAKZ-models” formally in conditions of indeterminacy, to describe their algorithmic design using quantum structuring of different-type data about DMO considering its semantics in a concrete data domain.

А t-task is described formally using the multiple four:

At = S, Kt,,Qt (1) t and consists in building the class M of substantial algorithmic structures and operating tools for manipulating t them on a character language S from a set of letters, numbers, special symbols and algorithmic operations of algorithm theory on the basis of using rules for constructing t-quantum to terminal t-quantum from K with a t t help of finite set Q of semantic codes. Under semantic code tks Qt (s=0,1,2,…) we assume symbols, coding t t-quantum, which corresponds the form and content of authentic knowledge of level s.

The В t-task is to synthesize recognizing tRAKZ-models and algorithms to manipulate tk-knowledge to define values of target characteristic for the recognized DMO, i.e. its identification with the given reliability according to the external observations, relying on the preliminary cumulated BtkZ.

The С t-task is to synthesize prognostic tRAKZ-models and algorithms for manipulation tk-knowledge to predict with the given reliability of DMO permanent characteristics values according to the measured values of the observed characteristics, relying on the preliminary built BtkZ.

To solve B t- and C t-tasks it is required:

Decision Making 1) to synthesize the induction operator INDS(tk20;AZ;tk2BM) for inductive derivation of the sought BtkZ from a set of selected teaching tk-knowledge, where in brackets the parameters of INDS operator are shown: tk20 - teaching selective tk-knowledge of the 2nd level; AZ – operating algorithm of inductive derivation for BtkZ as new knowledge; tk2BM - minimized BtkZ in the form of a matrix tquantum of the 2nd level as a system of implicative laws.

2) to synthesize the deduction operator DED(tk ;tk Y ;AL;tk R) for deductive derivation of the sought 2 0 1 s decision as a new tk-knowledge of the level s (s=1,2) tk R in observations tk1Y for DMO, relaying s on BtkZ= tk BM, where AL is a deduction algorithm.

2. Algorithmic Formalization and Vector-Matrix Representation of tk-knowledge (A t-task) The general structure of t-quantum of knowledge (tk-knowledge) has two compounds: semantic and informational to represent a knowledge portion about DMO conditions in semantic, informational and algorithmic aspects at the same time. It is supposed that a portion (quantum) of knowledge about the DMO condition describes some authentic quantum event (QE) in a production form “message - consequence” according to the scheme (2) IF (logical combination of messages e), THEN (consequence C), i j (2) i =1, k; j = 1, h.

Semantic compound of t-quantum in a form of special structure of data represents meaning information about this QE, showing the scales for measuring the DMO features, semantic code and quantum purpose as knowledge model about facts or laws. Semantic code from the set Q has a symbolic form tk Y, k is a quantum t s symbol; s {0,1,2…} is a level, Y is a name and {p, tr, b, t…} – quantum status (precondition, target, basic, terminal).

Information compound describes different-type features (characteristics) of DMO in a sectioned (domain) vector-matrix form, suitable to manipulate tk-knowledge and logical derivation using computer algebra. In a substantial and formal representation the domains d meet non-target (precondition) and target features of DMO, j they are called active and are separated by a symbol “ : ”. Binary components of active domains i d j j correspond to the features values. All the active domains define the QE logics, as far as a postulate is taken about the fact that active domains are connected with a conjunction (“:” is a strap “”), the compounds in domains – with a disjunction (“,” is a strap “”), and precondition domains to a target – with an implication («») in a form of (2). The logics of QE can be described in sentential formulas of propositional logic or in finite predicates, where the arguments are components of ij domains.

The main idea of strictly formalization is in axiomatic building of tRAKZ-model on the basis of postulating the three terminal quanta tk y, tk a, tk b and using operators of superposition (-operator) known in the theory 1 т 0 т 1 т of algorithm, a string concatenation (CON•-operator) and a column concatenation (CON[•]-operator).

The generalized terminal quantum tk y represents a vector of domains, corresponding to different-type 1 т features x, …,x DME with values (in the domain components) from the finite sets Xj, (j=1,2,…, n):

1 n n 1 ={1,...,1 },..., n ={1,...,n },.The generalized quantum tk y has a form:

1 т 1 r1 rn x x x 1 2 n 2 n (3) tk1yT = [ d : d : …: d ] = [1,…,1 : 1,…,2 : …: 1,…,n ], 1 2 n 1 r1 r2 rn where tk1 QT ; name yT Sv.

The generalized terminal selecting quantum tk a is described with a selection function V of the argument 0 т k(t) k from t-consequence of numbers or symbols:

( tk a =[Vk ) (1,…,k,…, )=k ], 0 т (4) ( where tk0 Q; name a,Vk ) S;

;

The generalized terminal characteristic quantum tk b is described with a characteristic function of a set Y 1 т Yj j for admissible values ij j of the j feature x :

XII-th International Conference "Knowledge - Dialogue - Solution" j 1, if k Yj, j tk b = [Y (k )] = k = (1,2,...,rj).

1 т (5) 0, if j j Yj, k Definition 1. The different-level algorithmic structures, being received from terminal quantum tk y (3), tk a (4) 1 т 0 т and tk b (5) with a help of finite number of applying -operator, CON•-operator and CON[•]-operator, are 1 т called different-level algorithmic tk-knowledge or tRAKZ-models of knowledge in conditions of indeterminacy, which form a class of authentic tRAKZ-models M.

t B(3) In Fig.1 a quantum area of tRAKZ-model of DMO is shown, being described by three features: x with r = 1 t 2 values from X1 ={1,1 } ; x with r = 4 values from X2 = {1,2,3,2} and x with r = 3 values from the 2 2 3 1 2 2 set X3 ={1,3,3}.

X 2 Vector domains are separated with a semicolon «:» and meet the different-type tk A features of DMO, and components of domains X – for the features values so that i component 1 3 of j domain should contain «1», if we observe i value of j feature, otherwise i component equals to «0». If every domain of a quantum of the 1st level contains strictly only one «1», tk C tk B it is called an element one, otherwise it is called – an interval vector quantum. The points А and В of the area B(3) are X t Fig.1. Area B(3) of tRAKZ-model t responsible for element vector tk-knowledge tk А and tk В:

1 x1 x x 2 tk1A = :0010:010, tk1B = [10:0100:010], (6) The interval С B(3) corresponds with an authentic interval vector quantum of the 1st level t x1 x2 x (7) tk1C = [11: 0110 : 010], which can be represented by a matrix t-quantum of the 2nd level tk С, containing the joint 4 element vector tquantum of the 1st level:

x1 x2 x 01: 0010 : 10 : 0010 : tk2C = (8) 01: 0100 : 10 : 0100 : Besides, is t-quantum tk1C (7) represents a conjunct, an elementary conjunction corresponds to it:

( ( (x1 {11),(1)})(x2 {(2),32)})(x3 {(3)}) (9) 2 2 The elementary conjunction (9) can be represented as a predicate equation:

( ( ((x1 = 11))(x1 = (1)))((x2 = (2))(x2 = 32))) (x3 = (3))= 1 (10) 2 2 So, the class М of tRAKZ-models represents a set of uniform quantum tools for describing implicative laws, and t also different facts to represent them in the three equivalent forms: multiple (points, intervals of area В );

Decision Making 3. Inductive Search and Deductive Derivation of Solutions as tk-knowledge Under the facts we understand the measured DMO features of different type and their logical combinations, and also any observed events and situations, having relation to DMO and being represented by knowledge quantum of different levels, i.e. by tRAKZ-models. The tables of empirical data (TED) Т (m,n) are typical examples of о real facts.

Under the laws (DMO are subordinated to them) we consider implicative (forbidden) logical connections between features of DMO, they are rather stable to be defined while analyzing a limited TED Т (m,n).

о Definition 2. A stable connection between r characteristics of DMO from the general number of n, (rn), expressing inadmissibility of at least one combination of their values on a set of tk-knowledge, is called an implicative law or a prohibition of r rank.

In tRAKZ-method of decision-making the inductive derivation of tk-knowledge is used to build a general “world model” in a form of BtkZ as a range of implicative laws being found by learning tk-knowledge, represented in a form of TED.

The deductive derivation of tk-knowledge is necessary to receive partials conclusions for the observed facts, basing on the BtkZ.

3.1. Inductive derivation operator of implicative BtkZ (INDS-operator) The existence of implicative law as some forbidden knowledge quantum of s-level tksY from T, according to r TED Т (m,N), (s=1,2), is defined by the evaluation of its certainty, satisfying the inequality N!2r(1-m) (2r -1)m M {m,N,r}= M* (11) S S r!(N - r)! where the given possible limit value (threshold) of M* [Sirodzha,1992] evaluation.

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