Some definitions : < O, k > is the model of the subject domain, where

1. Method of Limiting Generalizations for Solving Logical and Computing TasksYuriy ProkopchukInstitute of Technical Mechanics of NASU & NSAU,Ukrainian State University of Chemical Engineeringitk3@ukr.net

2. Суть доклада состоит в представлении доступного и эффективного метода решения задач реальной сложности The basic contents of report will consist in representation of an accessible and effective method for solving of real complexity tasks. Метод ориентирован на задачи обработки данных, принятия решений и управления в слабо формализованных предметных областях The method is focused on tasks of data processing, decision-making and management in poorly formalized subject domains.

3. Some definitions: <O, k>is themodel of the subject domain, where Ois themodel of ontology, kis themodel of knowledge. A(<O, k>)isthe model of reality.

4. Some definitions: • Represent the model of knowledgek in a developed view as follows: • k = {f/: k1  k2} Pk, • where • f/is mapping realizing mathematical models; • are the distinct mechanisms of realization of mapping; • k1are the input data of the task (description of information environment and job); • k2are the output data of the task; • Pkare the rules of composition of tasks schemas

5. Specifications of tasks for some classes of knowledge models /Tare theresults of tests; d/Dare theconclusions, diagnoses; h/Hare theprediction hypotheses; r/Rare thecontrol programs; T, D, H, Rare the sorts or thedomains The choice of domains determines thereality description generality level.

6. The term “domain” has been borrowed from databases, but in the context of this work its treatment is much wider, namely: • a domain contains all constructions (linguistic, mathematical and other ones) that make it possible to form the result of a test or conclusion; • a domain is certain to involve a semantic situational metric The domain T is not necessarily a discrete value set; for example, T may be a lexical tree. Thus, one can speak only of the deducibility of the result of a test on the basis of the constructions of the domain T.

7. F1 = {f/: {/T}1{/T}2}is the class of models for computing knowledge; F2 = {f/: {/T} d/D}is the class of models of diagnosticknowledge; F3 = {f/: {/T}d/D}is the class of knowledge models describing the domain of prohibitions; F4 = {f/: {/T}, {d/D}{h/H}}is the class of models for predictionknowledge; F5 = {f/: {/T}, {d/D}, {h/H}{r/R}}is the class of knowledge models for optimization of control;

8. F6 = {f/: {/T}{/T}’}is the class of knowledge models for description of the structure and the dynamics of complicated systems represented as collection of causal and consequent relations. The general knowledge model k includes all the above-mentioned classes of models, namely: F1F2 F3 F4 F5 F6k. The closure of the set of data mappingF+/Pk is built by means of the rules of composition Pkin solving a specific task

9. The above schemes of the knowledge model classes F1 – F6 illustrate the formal logic levelof knowledge representation. • Original structures of knowledge representation and ontologies at the procedural level are given in Refs. (by the example of clinical medicine). • The basis for the structures islexical trees

10. Examples with complete information R+ = {1, …, m}is the sample of examples with complete information Example:knowledge bases for hospital systems, telemedicine systems and learning systems Electronic patient or pupil/student records are used as a priori information. {E-Cards} = {1, …, m} k Ontology: • LPL = Limited Professional Language (set of lexical trees); • Bank of models of tests; • Bank of models ofconclusions, diagnoses; • ets.

11. Condition of separability Suppose that there is a finite set of elementary tests{} when any situation of reality  is uniquely re-established from R+ by values of tests {}. Assume that one of the tests takes values from finite and alternative sets D = {d1,…, dn}. Denote that test by d. Introduce the condition of separability of real situations based on sets of tests {/T}\d and some transitive metric : {}, {}’ where {}{/T}\d and , ’  R+:  = ({},d),’ = ’({}’, d’) the following condition should be met: ({}, {}’)= 0 d = d’.

12. 1st task. Assume that a representative sample of real situations R+ with complete information is given at a particular level of abstraction (the level is determined by domains). Assume that the metric is given in such a manner that the condition of separability is performed on the set R+. It is required to build a minimal remainder–free model of knowledge on sets R+ from the point of view of an efficiency function : “the classification of the conclusions from D”

13. In Ref., algorithms for solving the task for different classes of models of knowledge (in the context of a fixed combination of domains) are given. Below are some examples of these classes: KI = {{/T} d}  {{/T}1…{/T}m d} (d1… dn,  d1 …dn-1 dn). KII = {{/TX} d} {{/TX}1…{/TX}m d} (d1… dn,  d1 …dn-1 dn). KIII = j=1,m(pj({/TX}j)=t  d)  {j=1,m pj({/TX}j)=f d} (d1… dn,  d1 …dn-1 dn).

14. As the mappings {{/T} d}, we will consider all irredundant mappings, i.e. mappings whose left parts are minimal combinations of test results which are sufficient to draw a conclusion from the available data (the example set R+). For each conclusion dj D there exists a minimum set of irredundant mappings which in the aggregate cover all the examples from R+(dj).

15. Note If the sample of examples R+ at the given level of abstraction is not representative, one cannot use the class KI(KII) of models of knowledge because in this case the mappings {{/T}1…{/T}m d} are incorrect. The representativeness of sample is judged from tests that form the true minimal model of knowledge at the given level of abstraction (if any).

16. For the left-hand sides of any of the minimum sets of irredundant mappings covering R+(dj), the following notation will be used: • (dj) = l{/T}jl, (l =1,…,Lj.) Let us call the numbers Ljconclusion indices. Abasic model of knowledgek0 k0 = j=1,nl=1,Lj({/T}jldj |{/T}jl(dj)) {d1… dn = true}. is defined. Clearly, the basic model of knowledge is not unique because the (dj)’s are formed in a non-unique manner. The construction of a basic model of knowledge does not imply the representativeness of the sample R+; however, it implies the separability condition.

17. Oriented Graph of Domains(2nd task) Domains can represent a distinct level of generality. Consider the examples. Assume that T1 – T4 are distinct domains for description of the human temperature: T1 = [34, 42] degrees; T2 = {[34, 35], (35, 36.5), [36.5, 36.8], [36.9, 37.4], [37.5, 40]}; T3 = [decreased; normal; elevated; high] temperature; T4 = [normal; abnormal] temperature. The above-mentioned groups of domains have the desired property that if the value of the test is given on one domain, values of the test may be determined on domains with a greater number by using the fixed (single) rules of recalculation.

18. In other words, by using the domains cited an improper order can be given by the criterion of generality (the relation of domination), namely: T1  T2  T3  T4. The rules by which the values from one domain are translated into another may be specified in different ways, for example, on the basis of fuzzy-set theory or using neural networks. By way of example, below are the simplest rules: T2.{[34, 35], (35, 36.5)}  T3.{decreased temperature}; T2.{[36.5, 36.8]}  T3.{normal temperature}; T2.{[36.9, 37.4]}  T3.{elevated temperature}; T2.{[37.5, 40]}  T3.{high temperature}; T3.{normal temperature} T4.{normal temperature}; T3.{decreased; elevated; high} temperatureT4.{abnormal temperature}.

19. If we replace the sign ‘’ with the implication sign ‘’, then for the relation of domination we will obtain an oriented graph of domains with a single root node, which symbolizes the objective level (the minimum level of generality). Examples: T1  T2  T3  T4. The domain graph for the test “Age”:

20. Complete set of descriptions of reality The oriented graphs of all test domains are part of the nonprimitive ontology of the subject domain. An oriented graph of domainscan be determined for each test (2nd task). One can set some graphs for any test. Any path on the graph implies a possibility of a unique recalculation of values from one domain to other one. In searching through all possible combinations of domains for diverse tests we derive acomplete set of descriptions of realitywith a variety of levels.

21. Critical, subcritical and postcritical descriptions We name suchdescriptions which cannot be generalizedby one test without breaking the condition of separability ascriticalones. Descriptions that can be generalizedfrom at least one test without violating the separability condition will be termedsubcritical. Descriptions that violate the separability conditionwill be termedpostcritical. Aset of optimal models of knowledgefor all descriptions (subritical, critical and postcritical) forms acomplete model of multilevel description of reality. We name the model which allows of solving a target taskfor any presented situationof reality astrueone.

22. Как правило, достаточно хранить только истинные оптимальные модели знаний для критических описаний As a rule, it is enough to store only true optimal models of knowledge for critical descriptions

23. The Method of Limiting Generalizations: 1. Themaximum branched graph of domains (or some graphs with different domination relation realization mechanisms) is built for each test involved in description of the task.Experts in the subject domain play a large role in the construction of graphs. 2. An optimal model of knowledge is built for each combination of domains defining the level of generality of description. A set of all optimal models of knowledge defines thecomplete model of a multilevel description of reality. 3. In searching the solution for a new situation a given situation is generalized at most to one of descriptions including a true model of knowledge (it is desirable to generalize to a critical description). The solution is situated at a new level of description (3rd task). If the solution is not available, it is necessary to correct models of knowledge.

24. Example: Tests: 1– temperature; 2 – Age; d –conclusion. Domains for description of the Human Temperature: T1 = [34, 42] degrees; T2 = {[34, 35], (35, 36.5), [36.5, 36.8], [36.9, 37.4], [37.5, 40]}; T3 = [decreased temperature; normal temperature; elevated temperature; high temperature]; T4 = [normal temperature; abnormal temperature]. The domain graph:T1  T2  T3  T4 Domains for description of the Age: В1 = [0…120]; В2 = {молодой, средних лет, пожилой, старческий} = {young, middle-aged, elderly, old}. The domain graph: B1  B2

25. Example (Ru):

26. Example (En):

27. An optimal model of knowledge «T3 – B2» - critical (true)KII k= {1/T3{decreased; normal} t  DS1; 1/T3{elevated; high} t  DS2} k= {1/T3{пониженная; N}  DS1; 1/T3{повышенная; высокая}  DS2}

28. Example (Ru):

29. Example (En):

30. Example: Oriented Graphs of Domains T1  T2  T3 T3-4 T4.

31. 1st Situation (Ru):

32. 1st Situation (En):

33. 2nd Situation (Ru):

34. 2nd Situation (En):

35. References [1] Prokopchuk Yu. Intellectual Medical Systems: Formal and Logic Level. - Dnepropetrovsk: ITM NASU & NSAU, 2007.-259 p. (In Russian) [2] Alpatov A., Prokopchuk Yu., Yudenko O., Khoroshilov S. Information Technologies for Education and Public Health. - Dnepropetrovsk: ITM NASU & NSAU, 2008.-287 p. (In Russian) [3] Alpatov A.P., Prokopchuk Yu. A., Kostra V. Hospital Information Systems. - Dnepropetrovsk: Ukrainian State University of Chemical Engineering, 2005.-257 p. (In Russian)