CC BY
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выделяем сильно связный блок [1], преобразуя схему, полученную декомпозицией, к блочно-ациклическому виду[ 1].
Оценим сложность попарного соединения и полного соединения всех отношений БД. Предположим, что в БД входит k отношений, каждое из которых содержит n кортежей. Тогда количество соединений Ef при полном соединении определяется формулой: Ef = nk, при попарном соединении количество соединений Ep вычисляется по формуле k (k-1)
E p = £ n2 = Mk-1) n2
: pn3 , где p =
1
k (k -1) 2
Если при оценке вычислительной сложности брать во внимание не только количество отношений, участвующих в соединении, то возможно, что сильно связный блок несколько увеличит вычислительную сложность, хотя в общем он дает значительное преимущество перед полным соединением цикли -ческой схемы.
4. Заключение
Рассмотренные задачи поддержки целостности данных выполняются на этапе как проектирования, так и ведения БД. Распределенное ведение БД требует проверки уникальности ключа во всех локальных отношениях, где существует ключевой атрибут, а именно проверки однозначности ключа на схеме слабоуниверсального отношения.
Литература: 1. ДедиковЭ.А.., БусликН.Н., ТанянскийС.С. Использование обобщенных операторов реляционной алгебры при ведении крупномасштабных баз данных. 1996. 8с. Деп. в ГНТБ Украины 16.04.96, N 957-Ук96. 2. Мейер Д. Теория реляционных баз данных. М.: Мир, 1987. 608 с. 3. Цикритзис Д, Лоховски Ф. Модели данных. М.: Финансы и статистика, 1985. 344 с.
Поступила в редколлегию 22.02.98 Танянский Сергей Станиславовоич, канд. техн. наук, доцент кафедры информационных систем и технологий в деятельности МВД Университета внутренних дел. Научные интересы: логическое проектирование и поддержка баз данных. Адрес: 310170, Украина, Харьков, ул. Блюхера, 22, кв. 159, тел. 65-47-52, 50-33-17.
Кобзев Игорь Владимирович, канд. техн. наук, старший преподаватель кафедры информационных систем и технологий в деятельности МВД Университета внутренних дел. Научные интересы: модели данных, распределенные неоднородные базы данных. Адрес: 310166, Украина, Харьков, ул. Новгородская, 44, кв. 19, тел. 30-71-75, 50-33-17.
Яковлева Ирина Александровна, канд. техн. наук, доцент кафедры информатики Университета внутренних дел. Научные интересы: структуры данных, универсальные алгебры. Адрес: 310058, Украина, Харьков, ул. Ромена Роллана, 7, кв. 40, тел. 43-75-14, 50-31-88.
УДК 519.713
EXPERT SYSTEMS USING FUZZY LOGIC
KRIVULIA G. F., RAMI J. MATARNEH
We propose offer some methods for human knowledge representation and making inferences in rule based expert systems, in terms of the theory of approximate reasoning developed by Zadeh, we extend our results on the rule based expert systems with a quantifier of the first and the second kind, these rules which satisfaction of sertain conditions in the antecedent portion.
1. Introduction
The essence of the fuzzy logic is that it underlies the modes approximately rather than exactly [1], derived from the fact that most modes or behavior of human reasoning are approximate in nature. The concept, which plays the main role in this logic, is the possibility distribution, which suggests that, ifwe have X, where Xis a collection of objects taking a value in a discourse U, then Пх is the fuzzy set of all possible values ofX, in symbolic notation Vu є U, Пх (u): U > [0,1], which means the possibility that X may take u as its value.
In general, the problem database in the expert systems consists of an attribute, an object and a value, so in the expert systems the rule is checked, if it is satisfied, then its value is added to the ED (explanatory database), taking into consideration that the rule considered to be satisfied, if the information in the database satisfies the antecedent
portion of the rule. The procedure is repeated until no more information can be added to the ED, after this procedure we apply other methods to obtain an approximate (acceptable for us) result.
2. Principal modes of fuzzy logic
It will be very helpful to take a short look at some of the principal modes of reasoning in the fuzzy logic [1].
1. Categorical representation: A is B and A is Cimplies that A is B and C which gives us A is (BaC), then A= MIN(B, C).
2. Syllogistic reasoning: Most A is B and Most B is C implies that Most2 A is B and C, with taking most as a quantifier named Q, then Q2 A is B and C, in symbolic form, where ''and'' stands for intersection, then Q2 A is (B a C), which gives Q2 A is Min (B,C), where Q2 is evaluated as an arithmetic value using the fuzzy arithmetic (the evaluation of Q will be discussed in section 4).
3. Dispositional reasoning: usually A is not B and usually A is not C, implies (2 usually -1) A is not B and not C, in symbols (2 usually-1) A is—(B) and — (C), then (2 usually-1) A is Min (—B a —C), which is equal to (2 usually -1) A is Min(—B,— C), usually evaluated as an arithmetic value, ''and'' — is a subtraction operator in the fuzzy arithmetic.
4. Qualitative reasoning: A is a if B is C and A is p if B is not C, implies that A is (a or p) if B is C (or not C), then A is (CaDaC v not CaDaP) is D, (here D<C and p>D), then A is Max[Min(C,D,C), Min(C,D, p)] is D which is equal to A is D if B is D.
For example: volume is small, if pressure is high and volume is large, if pressure is low, then volume is medium, if pressure is medium.
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3. Rule representation
Based upon the idea of the fuzzy sets introduced by Zadeh, if we have a large set X and have a fuzzy subset A of X, where Лє X, then for all x є X , and the membership grade for each x, is a value or element in the interval [0,1]. Maxim is tall. This proposition can take the form: G is A, where A is a fuzzy set, and G is a (fuzzy) variable, which can take A (the elements of A) as its value, then nG(x) = pA(x), which induces the possibility distribution of nG over A.
For such a simple rule G is A, we can represent its meaning using the test score method [1], that is, we check all possible values, which can be determined, and then make a comparison between them to take a suitable meaning or a value using min or max according to our choice, which depends on the real facts (the value is always determined approximately). Briefly, we can take a score Q, where Q satisfies the degree to which the constrain C is satisfied, then aggregate these scores to determine the compatibility of the proposition compared with our database or ED (the meaning of the proposition not by the overall test of Q but by the procedure, which leads to it).
For example: A is B and A is C and A is D then A is B and C and D. To get a value for such a proposition (to represent its meaning), we will search our database to get Q1, Q2, ..., Qn , where n is the number of all satisfied conditions C1, C2, ... Ci. To apply this procedure firstly, we are looking for all satisfied B, where B is a fuzzy value and take Ci , for all Ci > 0 , and then give a value to Qn, which means that we will have Qn=p(Ci) to take the better possible value for all Q, this leads us to aggregate all Q (to take the maximum value of Q): 5n = Q1vQ2vQ3 v ...
,VQn, sn Max (Qn Q2, ^ ... , Qn).
The same method used to get A is C and A is D. To obtain 52,52, Sn from the query A is (B and C and D), we see that the fuzzy variables were combined by the AND operation, for this reason the aggregation of 5s will be made by the Min operation: 5= 5пл5пл5пл5п; 5= Min (51, 52,53, 5n).
What we should take in consideration is that any modifier applied with the constraints Ci will effect the score Q: * not C = 1-Q; * very C = Q2 ; * more or less C = C2; also in the same manner for the rules pertaining to the composition: * C1 and C2 = Q^ Q2; * C1 or C2 = Qiv Q2; * if C1, then C2 =1 л (1 - Qi + Q2 ).
Example: assume we have the following data query: A is approximately equal to B and larger than C (note: approximately equal, larger than are constrains): A= {1, 2, 3, 5, 7, 9}; B=8, C= 4.
Firstly, find the membership grade for all x in A, according to A is approximately equal B, which is mean: 5i = pi(xi), then (51.. 5n) =(0, 0.1, 0.3, 0.6, 0.8, 0.8); Q1 = Max (0, 0.1, 0.3, 0.6, 0.8, 0.8) = 0.8. Secondly, find A is larger than Cusing the previous steps 5i = pi(xi), then (51.. 5n) = (0,0,0, 0.2, 0.6, 0.9); Q2 = Max (0,0,0,
0.2, 0.6, 0.9) = 0.9. Finally, get Q = Q1 л Q2 with taking л as Min, we will have Q = Min (Q1, Q2); Q = Min(
0.8, 0.9) = 0.8. This leads us to the conclusion that the rule A is approximately equal to B and larger than C equal to Q, then A is approximately equal to B and larger than C = 0.8.
But in the rule based expert systems, which have more complex structure than the previous rule, such rules, in general, take the form: ifG is A, then S is B, where G and S are (fuzzy) variables, A and B are fuzzy sets, to take the possibility distribution of G and S over AxB this implies the following: П(х1!... m, y1,..., yn) = A © B, that is, the
possibility distribution of G and S is the bounded sum of A and B, using the definition of the bounded sum [2]; A © B = Min(1, pA(x) + pB(y) ), so the possibility distribution of G and S over AxB can be the following: n(x1,... xn, y1,..., yn) = A © B = Min(1, pa(x) + рб(у)) which is equal to
nG|s = Min(1, 1-pa(x) + №(y)). (1)
According to the rule of modens pollens [2] for the rules of the form A ^ B it gives us the following derivation: A ^ B = —A v B; — A=1-A, then—pA= 1-pA (v treated as MAX), by substituting the value of—A and the meaning of v, this result in A ^ B = Max (1-pa(x), рб(у)), then
nG|s = Max (1-pa(x), рб(у)), (2)
one of the previous rules (1) or (2) can be used to represent our knowledge.
4. Rule representation with more than one antecedent and consequent
In the previous section we represented the rule with one antecedent, but in the rules with more than one antecedent such as: if G1 is A1 and G2 is A2 and G3 is A3, then S is B, we note that the antecedent G1 is A1 and G2 is A2 and G3 is A3 consists of more than one condition and is connected with the AND operator, to take the possibility distribution of the antecedent ( of all Gs over all fuzzy sets A's) we will have П G1, G2, G3,. Gn (x 1x x2x x3x xn), which is equal to the cartesian product [2]. Rewrite this equation by mean of the cartesian product definition, then П G1, G2, G3,... Gn (x1x x2x x3x xn) = Mini[Ai(xi)], which in other words means that G1 is
A1 and G2 is A2 and G3 is A3 = Mini[Ai(xi)]. (3)
Substitute (3) in (1) this will produce the following:
П S1| G1, G2, G3,... Gn(x1x x2x xnx y1)=Min(1,1- Mini[Ai(xi)] +B(y)). The rules with more than one antecedent and more than one consequent, such as: if G1 is A1 and G2 is A2 and Gn is An, then S1 is B1 or S2 is B2 or Sn is Bn in the same manner will take the form П S1, S2, S3 ...j Sn| G1, G2, G3,... Gn (x1x x2x xnx y1x y2x Уп) = Min (1, 1-Mini[Ai(xi)]+ Maxi[Bi(yi)]).
The use of Min and Max depends on the connection operator between the conditions in the antecedent portion of the rule or the variables in the consequent portion, that is, if we change the AND operation to OR in the antecedent portion of the rule ifG1 is A1 or G2 is A2 or Gn is An then S1 is B1 or S2 is B2 or Sn is Bn, then the representation of this rule will be П S1, S2, S3...j Sn| G1, G2, G3,... Gn (x1x x2x xnx y1x y2x yn) ’= ’ Min (1, ’ 1-Maxi[Ai(xi)]+ Maxi[Bi(yi)]).
5. Rule representing with quantifier
In some cases one may stipulate the terms whose grade of membership falls under a specific grade or in an interval of grades. To represent the meaning of such rule we must find an approximated value for such quantifier to get a complete meaning for that proposition, in such a case the cardinality of the fuzzy set is used [1].
Let A be a fuzzy set of U, then the cardinality of A is the summation of all its items’ membership, in other word card A= em(ui). A in this case is the fuzzy set of all possible values without a quantifier, and let B is the fuzzy set which has all elements which satisfy the condition or the quantifiers B= QA. We can conclude then that B<A and B є A. From this point we can represent the meaning of the rule using the cardinality of the two fuzzy sets A and B, that is Q is equal to the cardinality of A intersected
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by the cardinality of B, the result is divided by the cardinality of B; in a symbolic representation
q _ ECount(A n B)
Q _ ECount(B) ’
Q _ A (Ui) л2Ь В (Ui)
2 hВ (Ui)
so, if we have the rule most A is B, rewrite this rule in other form and it will be QA is B. To represent the meaning of this rule, we take the cardinality of A ( the summation of its membership grades hA(ui) ), and the cardinality of B (the summation of its membership grades pB(ui)). By substituting in the above formula we will get:
QA is B ^ QA is B ^
ZpA (ui) Л^М- B (ui) 2pB (ui) ’
Min(ZpA (u), ZpB (u)) ZhB(u)
But for rules with a complex structure such as: if Q [G1 is A1 or G2 is A2 or Gn is An ], then S is B the representation of the quantifiers is obtained by other effective method. This method is based on taking possible values for the quantifier [ 1], these values are used to get the value of the antecedent portion Maxi[Ai(xi)] in the base rule П S1| G1, G2, G3,... Gn, fax x2X xnx yO = Min (1, 1- Mini[Ai(xi)]+B(yj). The procedure of this method is illustrated in the following:
1. Find all grade of membership of all items in all fuzzy sets in the antecedent portion pAi(xi).
2. Take a pointer called (for example) Ki, where i is the number of all variables in the antecedent portion. That is, if we have the following rule: if Q [ G1 is A1 or G2 is A2 or G3 is A3 ], then S is B, then i will take the values 1,3 (K1, K2, K3), Ki will take 0 or 1 depending on the following: Ki = 1, if at least one of the conditions in the antecedent is true, else Ki = 0; K2 = 1, ifat least two of the conditions in the antecedent are true, else K2 = 0; Kn = 1, if all conditions in the antecedent are true, else Kn= 0.
3. Define the quantifier, in another words, take a number n of items with an equal period in the interval [0, 1] and give every item a quantification value, this value being increased depending on the items, that is, the largest item will take the largest quantification value, such as: if we take 1/2, and 3/2, then Q(1/2) < Q(3/2).
4. Find Maxi[Ai(xi) using the formula: Maxi[Q(i) л Ki] for the quantifiers of the second type, at least 7, about 9 .... etc. and Maxi[Q(i/n) л Ki] at least half, most all, most, .... etc., respectively.
5. Substitute the resulting value from the fourth step in the base formula to find the value of the fuzzy rule.
Example: take the rule: if most of G1 is A1 and G2 is A2 and G3 is A3, then S is B. To give a value for the fuzzy variable S, assume A1, A2, A3 with the data:
A1 = {1/a, 0/b}, A2 = {1/c, 0/d}, A3 = {1/e, 0/f}, B = {1/g, 0/h}; (the grade of the membership of all elements is defined that is, items with 1 membership are available and 0 not available). Define the quantifier Q(0)=0, Q(1/
3)= 1/4, Q(2/3) = 1/2, Q(1)=1 ( note, the period between all quantifiers is equal to 1/3 and Q(0)<Q(1/3)<Q(3/
2)<Q(1)). Take the points a, c, f. Find Ki for (a, c, f) , get K1= 1, because at least a is true (available) in (a, c, f); K2 = 1, because at least a and c are true (available) in (a, c, f); K3 = 0, because at least a and c are true (available)
in (a, c, f) but f is not available. The following table shows the combinations of the variables and the corresponding value Kn for each combination, and the formulation of Mini[Ai(xi)], which is as H in the table. Now determine Maxi[Q(i/n) л Ki] for i = 1 to 3 (the number of the conditions): Max1 = (1/4 л 1) = 1/4, Max2 = (1/2 л 1) = 1/2, Max3 = (1/1 л 0) = 0, then Maxi[Q(i/n) л Ki] = Max[1/4, 1/2, 0] = 1/2.
Table
x1 x2 x3 Afa) Afa) Afa) k1 k2 k3 H
a c e 1 1 0 1 1 0 1/2
a c f 1 1 1 1 1 1 1
a d e 1 0 0 1 0 0 1
a d f 1 0 1 1 1 0 1/2
b c e 0 1 0 1 0 0 0
b c f 0 1 1 1 1 0 1/2
b d e 0 0 0 0 0 0 0
b d f 0 0 1 1 0 0 0
Substituting this result in the base rule,
Min (1, 1- Mini[Ai(xi)]+B(y)), for B(g) we get Min (1, 1 - x + 1) = 1 for B(h), we get Min (1, 1 - x +0) = x ,then S=(1/g, x/h).
6. Certainty qualifications
Ifwe have a fuzzy set A, then the statement A ispossible, it means that nx(u) =1 and pA(u) is always less than or equal to nx(u). On the other hand, when we assume that A is certain, this is mean that nx(u) =0, because any value outside A is impossible, and pA(u) is always larger than or equal to nx(u). We can conclude that, if A is certain, then any value outside A is impossible. Note that the principle depends on taking the complement of the possibility meaning to represent the certainty meaning. In the same manner, if we take A1, A2, ..., An with pA1(u), hA2(u),..., pAn(u), assume that A is possible, then the maximum of pA1 (u), pA2(u),..., pAn(u) is always less than or equal to nx(u). Ifwe take A is certain, then the minimum of pA1(u), pA2(u),..., pAn(u) is always larger than or equal to 7Tx(u).
Now let us take the previous assumption with an extra condition, such as A is possible with degree a. This means that nx(u) is always larger than or equal to a, and the minimum of pA(u) and a is always less than or equal to nx(u) [1-5]. But ifwe take that A is certain with degree a, to represent this rule remember what we said before, that this principle depends on taking the complement to 1 of the membership degree, which leads us to conclude that nx(u) is always less than or equal to the maximum of pA(u) and 1-a.
Ifwe now take the general case, where A is quantified and fuzzy, then A can be represented in terms of a collection of ordinary subsets, namely the a-cut relating to the definition of Zadeh [2] of the a-cut: Vu є U, sup ae[oj1] min(pAa(u), a), then if A is possible to a degree a, this means that u є U, sup aє[0д] min(pAa(u), a) is always less than or equal to nx(u), take the complement to 1 to represent A is certain to a degree a,: 1 - pA(u) = inf ає[0д] min(pE1-a(u),1- a); changing E into A we get hA(u) = inf ає [0,1] min(pA1-a(u),1- a), which means that nx(u) is always less than or equal to pA(u) and less than or equal to maximum of pr1-a(u) and 1- a).
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The general conclusion of this situation: if A is possible and certain, by substituting in the formula of A is possible and A is certain we will get: pA(u) = nx(u) for all u in U. Indeed, when A is possible, it may be modelled as Vu є U, pA(u)< nx(u), and when A is an ordinary subset, then nx(u) =1, while the possibility distribution of a non-fuzzy variable according to the definition ofZadeh: n(A) = sup u є Anx(u), then n(A)=1, only when the statement x is in A is consistent with the variable information described by nx.
To specify a fuzzy rule relate a variable x in U to a variable y in V. For the rules of the first kind, possibility rules, this kind of rules take the form: the more x is A, the more possible y in B; if we translate this rule into the form: Vu, ifx =u, B is range for x in at least pA(u)-possible, then we will get the following: u є U, V v є V, mm(pA(u), №(v)) < %y(v, u).
For the second kind of rules, certainty rules, which take the form: the more x is A, the more certainty y in B, translate this statement into the form Vu, if x =u, B is a range for x in at least pA(u)-certain, then we will get the form: V u є U, V v є V, %y(v, u) < max(1-pA(u), pb(v)). In particular, when A is an ordinary subset, we know that, if x in A, then B is a certain and possible range for y, this yields: u є A, %y(v, u) = pb(v)); V u g A, ny (v, u) unspecified. In the general case, when the person is uncertain as to the value he provides to the system [1-2], thus he provides the following information: Vis A with certainty a, the question is, how we can make an inference from this rule to get a meaning for the variable B in the rule: Vis A (with certainty a) ^ V is B?As a conclusion, if we have a rule of the form: Vis A with certainty a, it can be translated into the equivalent
rule: V is B, where for any x єХ: B(x)=(arA(x))+(1-a). Take into consideration that for Vis A with certainty 1, we get V is A, and for Vis A with certainty 0, we get Vis X.
7. Conclusion
The aim of the fuzzy logic is to provide an easy way to deal with the systems which are full of uncertainty. In such systems or environments the fuzzy logic is considered to be very effective when the conclusions need not to be precise, but acceptable for a degree of certainty.
References: 1. Ronald R Yager, Lotfi Zadeh //An introduction to fuzzy logic applications in intelligent systems. Fuzzy sets, fuzzy systems, intelligent control. ISBN 0-79239191. 1993. 299 p. 2. Zimmerman //Fuzzy set theory and its applications. Second edition, fuzzy sets, operation research. ISBN 0-7923-9075-X. 1993. 399 p. 3. WolfgangH. Janko, Marc Roubens and Zimmerman //Progress in fuzzy sets and systems. ISBN 0-7923-0730-5. 1992. 188 p. 4. Bart Kosko // Neural networks and fuzzy systems, a dynamical systems approach to machine intelligence. ISBN 0-13-612334-1. 1993. 449 p.
5. Dubis D. and Prade H. // Fuzzy sets in approximate reasoning. Part 2: logical approaches. Fuzzy sets and systems. 1990. 298 p.
Paper submission 12.01.98
G. F. Krivulia, Head of Automation design computing technique department. Adress: 14, Lenin Avenue, Kharkov, 310726, Ukraine. Tel. 409326. Hobby: travel and fishing.
Rami J. Matarneh, postgraduate student of automation design computing technique department. Adress: 14, Lenin Avenue, Kharkov, 310726, Ukraine. Tel. 409326. Hobby: sports and reading. Scientific interests: AI and Expert systems.
УДК 519.7
ОТОБРАЖЕНИЯ КАК ОБЪЕКТЫ ФОРМУЛЬНОГО ОПИСАНИЯ
ДУДАРЬ З.В, САМУЙЛИКИ.Г., ШАБАНОВ-КУШНАРЕНКО Ю.П.
Рассмотрены связи и переходы между бинарными отношениями, предикатами и отображениями, соответствующими друг другу. Предложен способ формульного представления отображений, основанный на использовании языка алгебры предикатов. Найден общий вид однозначных отображений. Результаты обобщены на случай произвольной арности.
В [1] был рассмотрен вопрос о формульном описании отношений. В предлагаемой статье продолжена разработка той же темы, но понятие отношения анализируется теперь с несколько иной точки зрения. На практике с отношениями во многих случаях обращаются как с функциями. Об этом свидетельствует частое употребление терминов «частичная функция» и «многозначная функция». Строго говоря, такая терминология неверна, поскольку противоречит определению функции. Как известно, функцией называется отношение, обладающее свойствами всюду определенности и однозначности. То, что фактически подразумевается под словами «частичная функция» или «многозначная функция» — это, на самом деле, не функция, а отношение. Понятие отношения не совпадает с понятием функции, так как не любое отношение является функцией. И все же следует признать, что в стремлении многих
авторов рассматривать отношения как функции есть нечто такое, что заслуживает внимательного изучения.
Сказанное поясним примером. Любой текст можно понять только в том случае, если каждое слово в нем обладает вполне определенным, единственным значением. При выполнении указанного требования значения слов в тексте будут функцией этих слов, взятых вместе с окружающими их контекстами. Но если слово выделено из контекста и предъявлено без него, то его значение может стать неоднозначным. Например, значения слова «коса» в словосочетаниях «песчаная коса», «острая коса» и «русая коса» различны, но они однозначно определены в каждом из этих контекстов. Если же слово «коса» взять отдельно от контекста, оно будет иметь несколько различных значений. Факты такого рода приводят к выводу, что связь между отдельно взятыми словами и их значениями — это не функция, а отношение. Однако если использовать для характеристики связи между словом х и его значением у некоторое отношение xFy, то придется рассматривать сигналы х и у как равноправные. Если F — отношение, а не функция, бессмысленно спрашивать, какой из сигналов (х или у ) входной, а какой — выходной. И тем не менее, каждый человек, как носитель языка, ясно чувствует, что слово первично, а его значение — вторично. Это явствует даже из анализа словоупотребления: мы говорим «значение слова» , но нельзя сказать «слово значения». Именно это чувство заставляет исследователя языка рассматривать отношение xFy как нечто «функциеподобное» и писать y=F(x), считая слово х аргументом, а его смысл у — значением некоторого преобразования F, соответствующего отношению F.
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