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

Method of Limiting Generalizations for Solving Logical and Computing

Tasks

Yuriy Prokopchuk

Institute of Technical Mechanics of NASU & NSAU ,Ukrainian State University of Chemical Engineering

[email protected]

Суть доклада состоит в представлении доступного и эффективного метода решения задач реальной

сложностиThe 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.

Some definitions:

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

O is the model of ontology, k is the model of knowledge.

A(<O, k>) is the model of reality.

Some definitions:

Represent the model of knowledge k 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;k1 are the input data of the task (description of

information environment and job);k2 are the output data of the task;Pk are the rules of composition of tasks schemas

Specifications of tasks for some classes of knowledge models

/T are the results of tests;

d/D are the conclusions, diagnoses;

h/H are the prediction hypotheses;

r/R are the control programs;

T, D, H, R are the sorts or the domains

The choice of domains determines the reality description generality level.

The term “domain” has been borrowed from databases, but in the context of this work its treatment is much wider, namely:

1) a domain contains all constructions (linguistic, mathematical and other ones) that make it possible to form the result of a test or conclusion;

2) 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.

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 diagnostic knowledge;

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 prediction knowledge;

F5 = {f/: {/T}, {d/D}, {h/H} {r/R}} is the class of knowledge models for optimization of control;

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 mapping F+/Pk is built

by means of the rules of composition Pk in

solving a specific task

• The above schemes of the knowledge model classes F1 – F6 illustrate the formal logic level of

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 is lexical trees

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} kOntology:• LPL = Limited Professional Language (set of lexical trees);• Bank of models of tests;• Bank of models of conclusions, diagnoses;• ets.

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’.

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”

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 = {{/TX} d}{{/TX}1 …{/TX}m d}

(d1… dn, d1 … dn-1 dn).

KIII = j=1,m (pj({/TX}j)=t d)

{j=1,m pj({/TX}j)=f d}

(d1… dn, d1 … dn-1 dn).

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).

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).

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 Lj conclusion indices.

A basic model of knowledge k0

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.

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.

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}.

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” :

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 domains can 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 a complete set of descriptions of reality with a variety of levels.

Critical, subcritical and postcritical descriptions We name such descriptions which cannot be generalized

by one test without breaking the condition of separability as critical ones.

Descriptions that can be generalized from at least one test without violating the separability condition will be termed subcritical.

Descriptions that violate the separability condition will be termed postcritical.

A set of optimal models of knowledge for all descriptions (subritical, critical and postcritical) forms a complete model of multilevel description of reality.

We name the model which allows of solving a target task for any presented situation of reality as true one.

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

The Method of Limiting Generalizations:

1. The maximum 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 the complete 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.

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

Example (Ru):Форма «T1-B1» - a priori data «T3 – B1» - subcritical (true) KI-III

1/T1 2/B1 d 1/T3 2/B1 d 36.0 12 Пониженная 12 36.7 87

DS1 Нормальная 87 DS1

37.2 32 Повышенная 32 39.0 50

DS2 Высокая 50 DS2

«T3 – B2» - critical (true) KI-III «T4 – B2» - postcritical

1/T3 2/B2 d 1/T4 2/B2 d Пониженная Молодой Ненорм. Молодой Нормальная Старческ.

DS1 Нормальная Старческ. DS1

Повышенная Молодой Ненорм. Молодой Высокая Сред. лет

DS2 Ненорм. Сред. лет DS2

«T2 – B2» - subcritical «T4 – B1» - critical

1/T2 2/B2 d 1/T4 2/B1 d (35, 36.5) Молодой Ненорм. 12 [36.5, 36.8] Старческ.

DS1 Нормальная 87 DS1

[36.9, 37.4] Молодой Ненорм. 32 [37.5, 40] Сред. лет

DS2 Ненорм. 50 DS2

Гипотеза: 1/T3 – представительный тест

Example (En):«T1-B1» - a priori data «T3 – B1» - subcritical (true) KI-III

1/T1 2/B1 d 1/T3 2/B1 d 36.0 12 decreased t 12 36.7 87

DS1 normal t 87 DS1

37.2 32 elevated t 32 39.0 50

DS2 high t 50 DS2

«T3 – B2» - critical (true) KI-III «T4 – B2» - postcritical

1/T3 2/B2 d 1/T4 2/B2 d decreased t young abnormal t young normal t old

DS1 normal t old DS1

elevated t young abnormal t young high t middle-aged

DS2 abnormal t middle-aged DS2

«T2 – B2» - subcritical «T4 – B1» - critical

1/T2 2/B2 d 1/T4 2/B1 d (35, 36.5) young abnormal t 12 [36.5, 36.8] old

DS1 normal t 87 DS1

[36.9, 37.4] young abnormal t 32 [37.5, 40] middle-aged

DS2 abnormal t 50 DS2

Hypothesis: 1/T3 is representative

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}

Example (Ru):

Добавим новые домены: T3-4 = {пониженная; нормальная; повышенная}

В3 = {юный, молодой, немолодой}.

В4 = {молодой, немолодой}

Новое критическое описание «T3-4 – B4»

«T3-4 – B4» - Critical (true)

1/T3-4 2/B4 d

Пониженная Молодой Нормальная Немолодой

DS1

Повышенная Молодой Повышенная Немолодой

DS2

Гипотеза: 1/T3-4 – представительный тест

Example (En):T3 = [decreased; normal; elevated; high] temperature; T4 = [normal temperature; abnormal temperature].

T3-4 = {пониженная; нормальная; повышенная} = {decreased; normal; elevated } temperature (t)

В3 = {юный, молодой, немолодой} = {adolescent, young, aged}.

В4 = {молодой, немолодой} = {young, aged}

New critical description «T3-4 – B4»

«T3-4 – B4» - Critical (true)

1/T3-4 2/B4 d

decreased t young normal t aged

DS1

elevated t young elevated t aged

DS2

Hypothesis: 1/T3-4 is representative

Example:

T1 T2 T3 T3-4 T4.

Oriented Graphs of Domains

1st Situation (Ru):

Новая ситуация {/T}P = {1/T1 = 39С; 2/B1 = 5 лет}

«Гипотеза +» означает справедливость гипотезы о представительности тестов 1/T3, 1/T3-4.

Результаты классификации

Описание Гипотеза + Гипотеза - Т1 – В1 Отказ Отказ Т3 – В1 DS2 Отказ Т2 – В2 DS2 Отказ Т3 – В2 DS2 Отказ Т4 – В2 артефакт артефакт Т4 – В1 Отказ Отказ Т4 – В1-2 DS1 DS1 Т3-4 – В2 DS2 DS2 Т4 – В3 (critical) DS1 DS1 Т3-4 – В4 (critical, true) DS2 DS2

Если гипотеза верна, то решением задачи является заключение DS2. С точки зрения методологии интересным является факт отказа в классификации на уровне Т1 – В1. Противоречивость классификации на высоких уровнях общности означает необходимость пересмотра моделей знаний для описаний Т4 – В1-2 и Т4 – В3 с учетом нового наблюдения

1st Situation (En):

New situation: {/T}P = {1/T1 = 39С; 2/B1 = 5}

The “hypothesis+” means that the validity of the tests represenativeness hypothesis: 1/T3, 1/T3-4.

Results of classification Description Hypothesis + Hypothesis -

Т1 – В1 Refusal Refusal

Т3 – В1 DS2 Refusal



Т4 – В2 Artifact Artifact Т4 – В1 Refusal Refusal Т4 – В1-2 DS1 DS1 Т3-4 – В2 DS2 DS2 Т4 – В3 (critical) DS1 DS1 Т3-4 – В4 (critical, true) DS2 DS2

If the hypothesis is valid, the solution is the conclusion DS2.

2nd Situation (Ru): Новая ситуация: {/T}P = {1/T3-4 = «Пониженная»; 2/B2 = «Пожилой»}

«Гипотеза +» означает справедливость гипотезы о представительности тестов 1/T3-4. Результаты классификации

Описание Гипотеза + Гипотеза - Т3-4 – В2 DS1 Отказ Т4 – В3 (critical) DS2 DS2 Т3-4 – В4 (critical, true) DS1 Отказ

Если гипотеза ВЕРНА, то решением задачи является заключение DS1. Если гипотеза неверна, то решением задачи будет объявлено, скорее

всего, заключение DS2. Однако поскольку представительной выборки нет, необходимо в дальнейшем перестроить все модели с учетом нового наблюдения.

2nd Situation (En): New situation: {/T}P = {1/T3-4 = «decreased»; 2/B2 = «elderly»}

The “hypothesis+” means that the validity of the tests represenativeness hypothesis: 1/T3-4. Results of classification

Description Hypothesis + Hypothesis - Т3-4 – В2 DS1 Refusal Т4 – В3 (critical) DS2 DS2 Т3-4 – В4 (critical, true) DS1 Refusal

If the hypothesis is valid, the solution is the conclusion DS1. If the hypothesis is not valid, it is the conclusion DS2 that will most probably be declared the solution.

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)