Fuzzy Models of Language Structures
Fransisco Criado Torralba, Tamaz Gachechiladze, Hamlet Meladze, and Guram Tsertsvadze
Abstract—Statistical distribution of language structures reflect important regularities controlling informational and psycho-phys-iological processes, which accompany the generation of verbal lan-guage or printed texts. In this paper, fuzzy quantitative models of language statistics are constructed. The suggested models are based on the assumption about a super-position of two kinds of un-certainties: probabilistic and possibilistic. The realization of this super-position in statistical distributions is achieved by the split-ting procedure of the probability measure. In this way, the fuzzy versions of generalized binomial, Fucks’, and Zipf–Mandelbrot’s distributions are constructed describing the probabilistic and pos-sibilistic organization of language at any level: morphological, syn-tactic, or phonological.
Index Terms—Fuzzy sets, linguistic modeling, membership func-tions, probability theory.
I. SETSPLITTING
L
ET be a finite set and any subset, . Considera correspondence , where is the
indi-cator of subset , and
(1)
In this paper, it is supposed that is the support of the mappings
and [i.e., , ]. More
precisely, is the support of and determines the support
of : , that is if is
a subnormal fuzzy subset ( , ),
. In other cases . In all instances
determines the support of .
Notice that “ ” does not denote especially the functional correspondence but only that any pair is called a
splitting of a crisp set if and
equa-tion (1) holds.
According to [1], the splitting components and are fuzzy subsets of . Call the dual subset with respect to . The splitting procedure of some subsets induces corresponding splitting of the union and intersection of these two subsets.
Let and and the results of
their splitting. How can be split when and are given? This splitting must satisfy the natural requirement
(2)
Manuscript received February 6, 2001; revised October 30, 2001 and Feb-ruary 26, 2002.
F. Criado Torralba is with the Department of Statistics, Málaga University, Málaga E-29071, Spain (e-mail: [email protected]).
T. Gachechiladze, H. Meladze, and G. Tsertsvadze are with the Department of Applied Mathematics and Computer Science, Tbilisi State University, Tbilisi 380028, Georgia.
Publisher Item Identifier 10.1109/TFUZZ.2002.800655.
One has
Taking the previous into account, there are two direct possi-bilities of grouping the terms
1)
i.e., the grouping corresponds to the splitting of , when
(3)
This splitting has the same character as in.1 Indeed, consider and first split as
Now, split as
This leads to (3). Evidently, the splitting order is not essential. For this reason, [2], (3), and in are called “sequential splitting”
2)
if ,
if
1Throughout the whole paper, one must always keep in mind that the tilde (or double tilde) in notations of intersection and union of crisp subsetsA and
B stands for splitting induced by splitting of the corresponding components A
andB.
i.e., this grouping corresponds to following splitting of the in-tersection
(4)
Such splitting in [2] is called “simultaneous.”
In the case of the union fulfillment of the natural re-quirement
is demanded, and operating as in the case of intersection one can easily obtain
(sequential splitting) (5)
(simultaneous splitting) (6)
Now consider the Cartesian product. Let two universal sets and be given, and , . If their splittings are given, then according to
one can write
(sequential splitting) (7)
(simultaneous splitting) (8)
Finally, some simple relations connected with the dual subset are listed, which can be proved directly from the following def-initions:
1) involution ;
2) connection with Zadeh’s complement ,
where is the usual complement of in the universal set;
3) duality laws for intersection and union (true for both kinds of splitting)
(9)
(10)
which turn into the De Morgan’s laws for fuzzy subsets when and , i.e., when supports of both and mappings coincide with the universal set.
II. THELATTICE OFSPLIT ELEMENTS OFORDINARY INDICATORS’ BOOLEANLATTICE
First, let some definitions be considered.
Definition 1: A set , partially ordered, is called a lattice if any two elements and have a infimum (intersection ) and a supremum (union ). Brouwer and Heiting have introduced a generalization of the Boolean Algebra (see [4]).
Definition 2: The lattice is called a Brouwer’s lattice if for any given element of this lattice and the set of all such that have a greatest element,2 called relative pseudocomplement of in . Relative pseudocomplement of in , is called pseudocomplement of and is denoted by
.
Consider the Boolean lattice with natural
or-dering (if and for , , then
). The set of all split elements of this lattice with natural
ordering is a lattice.
Theorem 1: is a Brouwer’s lattice.
The direct demonstration of this theorem [i.e., the demon-stration that for any two elements and the set of
all such that3 has the greatest element
called the relative pseudocomplement of in ] can be done according to [4]. It is easy to see that
(11)
where is a pseudocomplement of and, as a
function of , represents the indicator of the usual complement of the set in . Next, the following theorem is easy to demon-strate.
The theorem shown below is well known, but here it is pre-sented in terms of fuzzy subsets.
Theorem 2: The following statements hold in the lattice :
If then
(12)
The two theorems previously considered, as will be seen fur-ther on, are important when calculating probabilities of some random events.
III. THESPLITTING OF ASET
The splitting of a set, which corresponds to the indicator split-ting, as seen, is represented as
(13)
Here, is the operation of set synthesis.
The term “splitting” of corresponds to the splitting procedure of the classical indicator resulting in the pair
2Greatest element of the setX is an element b 2 X such that x b for all
x 2 X.
. The term “split element” means the first element of a pair , i.e., . All results can be formulated in terms of the splitting procedure. All formulas in this paper must be considered as rules for operating with first component of dual pairs in practical use of the theory of fuzzy sets and can be founded by the splitting procedure.
On the basis of (13) one can obtain a more general expression that, obviously, will make sense under the condition
that , or . One can also obtain the existence
conditions for expressions , etc.
Considering that such a condition holds for the aforemen-tioned expressions, one can easily prove that
(14)
For example, to prove the last two formulas, one can write
Let it be assumed that in these formulas the following rela-tions hold:
(15)
which are evident because of (4), (6), and (13).
In the lattice of split subsets, almost all Boolean lattice rules hold.
1) reflexivity: ;
2) antisymmetry: , ;
3) transitivity: , ;
4) idempotency: and ;
5) commutativity: and ;
6) associativity: and
;
7) distributivity: and
;
8) the annihilation laws: and
;
9) involution law for fuzzy complement: ;
10) identity laws: , and ,
;
11) order inversion laws: and
;
12) De Morgan’s laws: and
.
In connection with the introduced notion of dual subsets, one can prove the following laws:
13) involution law for the dual subset:
14) duality laws for the union and intersection of split sub-sets:
For example, to prove law 13), one can write
On the other hand, according to (12)
Comparing these expressions, one obtains the required proof. Now, to prove the second law (14) one has
Notice that in lattice laws of contradiction and tertium non datur do not hold.
In Section I, some examples of indicator splitting were con-sidered. Let other examples be concon-sidered.
Splitting the set difference . Let be the universal set,
. The equality holds. If one splits
subsets and , then the splitting of this equality, according to (4), will be
(16)
For splitting the symmetric difference , one has
(17)
On the other hand
Thus, for the split symmetric difference, one also has the for-mula
According to (17)
(18)
Actually, taking into account law 7), one has
Equations (17) and (18) can be rewritten as
(19)
Let be the universal set and . By the
equality
the splitting of , according to the aforementioned example, leads to the equality
(20)
Under the condition that is a narrowing of on
cor-responding , i.e., .
Let be the universal set again but now .
It is widely known that
The splitting of leads to the formula
(21)
Under the condition that .
Splitting of the element of the universal set (fuzzy point)
(22)
where
(23)
Theorem 3: Let be the universal set, and a corresponding split point, then the splitting of the universal set determined by the splitting of the point will be the relative
pseudocomplement of in :
(24)
Proof:
IV. DUAL ELEMENT AND FUZZINESS (QUALITATIVECONSIDERATION)
As illustrated before, the dual element plays an important role in describing split subset lattices. Now, the role of the dual ele-ment in understanding fuzziness will be considered.
There is an important difference between usual and fuzzy sub-sets: The usual subset (set) can be represented as an aggregate of real objects when only the real measured potential possibility of aggregate formation corresponds to fuzzy subsets. Fuzzy subset is a medium of formation for real aggregate. It is important to notice that the term “medium of formation” is borrowed from [5] to underline the following circumstance: any sequence of research outcomes is a result of acts of free decision-making by the subject (observer), any concrete sequence is a crisp finite subset of some universum, but the fuzzy subset is analogous of Weil’s continuum.
In the lattice of fuzzy subsets, a dual element is defined by splitting procedure [2], [3]. Its sense can be explained as fol-lows: The value of the membership function is a degree of concordance of an element with the concept represented by ; the value has the same sense with respect to the concept represented by , which in the pair with ,
[6], then more fuzzy is the statement “elements of posses property ( ).” Below, a qualitative description of fuzziness is considered analogously with [6], but with the following dif-ference: in [6], the fuzziness is characterized by the relation between and Zadeh’s negation . In the present case, the less rigid relation between and , which in the authors’ opinion underlines the fact that fuzziness is an intrinsic property of and is independent of pseudocomplement, is assumed as a basis. The basis for considering the relation is a relation in distributive lattice, “ is between and , ” [6].
Definition 3: Let and (distributive lattice). is
not less fuzzy than , if and
are in between and . Here means sequential splitting of the intersection of and [3]. So that
(25)
Theorem 4: Relation is reflexive and transitive on , i.e.,
and and
It can be seen that on is not antisymmetric and, hence, it is not a partial order.
Theorem 5: Relation on is such that
1) and ;
2) .
On the lattice define such a relation such that
if or or . is an equivalence
relation. Each equivalence class consists of a fuzzy subset and corresponding dual. If , then the equivalence class consists of only one element.
Check that is really an equivalence relation. The reflexivity
and symmetry are obvious. Let and and let
from the first follows ; if the second means that
then in this case ; and if then
. The other cases are checked analogously. Thus,
and or or . The
transitivity is thus proved.
The subset consisting of any fuzzy subset and its corre-sponding dual subset is called the dual pair. According to Theorem 5, if one component of the dual pair is more fuzzy than any component of the other pair, then any component of the first pair is more fuzzy than any component of the second pair. So it is reasonable to introduce the notion of fuzziness of the dual pair. Definition 4: Let be a set of dual pairs. Define on a
relation such that if for takes place , then
one can say that the dual pair is no less fuzzy than the dual pair .
Theorem 6: The relation on the set of dual pairs is a partial-order relation.
Proof:
1) Reflexivity: . Consider the component , as
then .
2) Transitivity: and .
means that for all and all .
means that for all and all .
According to Theorem 4 and
, , , consequently .
3) Antisymmetry: and . One
has if and , then .
if and , then .
Ac-cording to Theorem 5, and
or and ;
besides, it must be , i.e., or .
Consequently . The dual pair corresponds to the splitting procedure. From the aforementioned considera-tion, one can speak about a rise in fuzziness at splitting (fuzziness of splitting). Relation defines a partial order in the set of split subsets.
The properties of fuzzy subsets studied in previous sections enables one to consider problems of fuzzy subset theory in terms of the splitting procedure.4 However, the aim here is that of considering fuzzy quantitative models of language structures.
Remark 1: The term “splitting” of corresponds to the splitting procedure of the classical indicator, which results in the pair ( , ). The term “split element” refers to the component of a pair ( , ), for instance . All results can be formulated in terms of the splitting procedure. All formulas in this paper must be consider as rules for dual pair operations with respect to the practical use of the theory of fuzzy sets and can be supported by the splitting procedure.
V. PROBABILITYMEASURESPLITTING
Let be a given probability space. The probability of the event is calculated from the formula
(26)
According to the splitting procedure of the set , this formula can be rewritten in the following form:
(27)
where is a -measurable membership function (the corre-sponding subset is a fuzzy random event). Define and
as follows:
and
(28)
the probability of fuzzy event and the probability of dual fuzzy event , correspondingly. Call the representation
(29)
the procedure of probability measure splitting.
4This idea, put forward by V. Kreinovich, is very elegant, especially for considering fuzziness quantitatively, this fundamental quality was presented in terms of the dual pair( ~X; ~X ). The authors would like to thank the referee for his interesting suggestions.
The main property of additivity
can be split in the following way. The left-hand side is
Because are not intersected, then
(29 )
and the right-hand side
(29 )
Finally, the property of additivity for split events is written in the following way:
or
(30)
Thus, the denumerable additivity property also holds for fuzzy subsets. Similarly
(31)
Actually, if , then also, ,
. From these relations, it follows that for noninter-secting subsets, the law (14) can be represented as
(14 )
In general, for any finite
(32)
From , where is an arbitrary subset, it
immediately follows that:
(33)
Furthermore, since , it is evident that
(34)
From the properties of the Lebesgue–Stieltjes integral the properties of the split probability measures are as follows:
1) monotony: ;
2) continuity with respect to monotonic sequences:
(35)
3) strong additivity:
4) -semiadditivity:
(36)
Measure 3) follows from the equality:
and measure 4) from the inequality
(37)
A. Conditional Probability (Nonsplit Conditions)
The condition expressed by the phrase “for a given event ” means that the initial probability space is
re-placed by the probability space , is a
condi-tional probability measure. As it is known, the condicondi-tional prob-ability of some event is the conditional mathematical expec-tation of indicator
(38)
In [7], this quantity is interpreted as the value of function
More generally, if is the denumerable partition of and is the minimal -field induced by this partition, then the -mea-surable function
(40)
is a value of the conditional probability for a given -field . The procedure of splitting indicator leads to the notion of the conditional probability of fuzzy event for a given (nonsplit)
-field
(41)
and
(42)
The splitting rule is
(43)
Formula (40) defines the conditional expectation for any non-null crisp event . Actually, may be represented as a union over a subclass of :
and, according to (38), one may write
(44)
One can see that if is known, then can be evalu-ated.
Let be a narrowing of on , which is determined by the formula
Then, the right-hand side of (38) can be represented as
(45)
The left-hand side is equal to . The authors obtained the descriptive definition of conditional probability [7]. The conditional expectation for a given (conditional probability of for a given ) is the -measurable function whose indefinite integral with respect to is a narrowing on
of the definite integral of with respect to
(46)
It is easy to see that
1) ;
2) if , or is a -measurable, then a.s.
Note that (46) also makes sense for nondenumerable partitions [7].
B. Conditional Probability (Split Conditions)
The constructive definition of is such that the direct application of the splitting procedure for obtaining conditional probability in the case of a fuzzy condition is impossible. How-ever, as it will been seen below, one can obtain a formula similar to (38) in the case of fuzzy condition (split condition). For this purpose, when splitting the corresponding measure is essential to retain some features of this formula. Proceeding from the no-tion of the mathematical expectano-tion of a random event indicator for a given function [7], if for such a function one takes a func-tion corresponding to fuzzy condifunc-tion (membership funcfunc-tion of fuzzy condition) and performs the convenient splitting, a rea-sonable measure that has almost all basic properties of ordinary conditional probability can be obtained.
Let induces the denumerable partition of ; ( , , ). In this case
(47)
Thus, for a function of conditional probability in the case of fuzzy condition one can take the expression
(48)
where the numbers
A similar expression is obtained for . It is clear that
and
(50)
Now consider any measurable indicator . If for any natural one defines the function
then the sequence tends to . One
has
(51)
and
It is clear that
(52)
These definitions make sense because of the generalized Radon–Nikodym’s theorem [7]. Comparing the aforementioned formulas, one can conclude that can be considered as the conditional probability function in the case of the measurable fuzzy condition. All considered formulas were related with measures of crisp events in the case of the fuzzy condition. If one performs the splitting of indicator on the right-hand side of basic formula (51), then one obtains the definition of
(53)
where [see (3)].
The considered version of conditional probability in the case of the fuzzy condition almost surely has all properties of ordi-nary probabilities except the condition , which must be replaced by
a.s. (54)
One has
a.s. a.s.
a.s. (55)
Formulas (54) and (55) are evident. For example, in the case of (54) one has (taking into account the absolute continuity of measures and with respect to measure )
a.s.
VI. SPLITTINGSHANNON’SENTROPY
Let set be split point by point. In this case, Shannon’s entropy of probability distribution , turns into , where is the membership function of a fuzzy point , . If the branching property of func-tion [8] is used, then
(56)
where5
(57)
and
(58)
is Zadeh’s entropy [9], i.e., an entropy of fuzzy set with respect to probability distribution
(59)
Function is actually a Kullback directed
diver-gence , [10] and
(60)
is a weighted nonprobabilistic entropy of [11].
It can be seen that (56) is no more than Hiroto’s measure of uncertainty [12]. Notice that if the branching property is used in some other way, then (56) can be rewritten in the following form:
(61)
where
In addition, functions and are
connected by Jumarie’s entropy [13]:
(62)
VII. FUZZYDISTRIBUTIONS
A. Binomial Distribution With Fuzzy Elementary Events
Let be the space of elementary events. One can obtain the fuzzy elementary events by splitting usual events and . For membership functions, one can write
(63)
where , .
According to (28), the probability of fuzzy elementary events is
(64)
where and are the probabilities of the corresponding crisp events.
Now it is easy to write the split binomial distribution corre-sponding to fuzzy elementary events. Only two variants will be considered: completely simultaneous and completely sequen-tial. The intermediate cases are not of any interest and for this reason they will not be considered here.
For the completely simultaneous case, the split binomial dis-tribution is
(65)
where is the fuzzy Bernoulli event. The normalization factor is
For the completely sequential case, one gets
(66)
and
The important characteristic of split Bernoulli probability (65) is the composition law; in the simultaneous case
(67)
and in the sequential case
(68)
and
As well as the characteristic of binomial probabilities in the case of fuzzy elementary events, one may consider the known property of exponential distribution; in the simultaneous case
(69)
and in the sequential case
where
B. The Binomial Distribution With Fuzzy Number of Successes
Let set be considered. The fuzzy
quantity “approximately from ” is defined as the fuzzy subset of . Therefore, the corresponding distribution is
(71)
where is the membership function of fuzzy number “ap-proximate from .”
This distribution is also called the binomial distribution be-cause it is characterized by the above composition law and the property of exponential distribution.
C. Fuzzy Upper Binomial Distribution
The consideration of the usual upper binomial distribution is based on the model of superposition of two events. The Bernoulli event and the emergence of the total amount of failures characterized by a priori probability .
If is the probability of elementary success, and are values of membership functions corresponding to
compli-cated events when distinguishing the events of a Bernoulli and non-Bernoulli origin, then the universal set , which is the composition , is split in the fol-lowing way:
The corresponding membership function
[condition is obviously satisfied].
The probability measure corresponding to fuzzy upper bino-mial distribution is
(72)
where
The Poisson limit
(73)
where and are connected by the relation
From a practical viewpoint, what is interesting is the expres-sion of the sum over all values of and
(74)
where
Taking into account the relation between , , and , then
D. Negative Binomial Distribution With Fuzzy Elementary Events [10]
Let the sequence of Bernoulli trials with probability of fuzzy
success be considered ( ),
probability of usual Bernoulli elementary event, de-notes the probability that the th success takes place at th trial, provided trials are continued up to th success. Accepting a splitting scheme that is used for binomial distribution with fuzzy elementary events, one can write
(75)
then the aforementioned formula can be written in the following form:
(76)
Define negative binomial distribution with fuzzy elementary
events, but fixed real number and as sequence
(77)
where
Note that if , or , then (77) reduces
to usual negative binomial distribution.
Fuzzy Fucks’ Distribution: As in the case of “upper Bernoulli” distribution, all variants of Fucks’ distributions [15] are based on the assumption that Fucks’ event is a superposition of Bernoulli and deterministic events
(78)
where is deterministic (certainly successes in trials) and is a Bernoulli event [ successes in
random events].
There are many variants of Fucks’ event splitting. Only some of them are considered in this paper.
1) The deterministic event is nonfuzzy, but Bernoulli ele-mentary events are fuzzy. In this case
The corresponding probability measure is
(79)
(for simultaneous splitting) with
and
(80)
(for sequential splitting) with
Here is connected to linguistic spectrum [15].
2) events are split ) and Bernoulli
events are crisp:
Evidently
(81)
where is the membership function of fuzzy set and
(82)
3) In the case when both deterministic and Bernoulli events are split, one must discriminate clearly the simultaneous and successive or sequential splitting of Fucks’ event. In the last case, it is easy to obtain the final result. Consid-eration of the two aforesaid cases allows one to write
(83)
consequently
(84)
(simultaneous splitting of Bernoulli event) and
(85)
(completely sequential splitting of Bernoulli event). When Fucks’ event is split simultaneously the author’s reasoning is as follows: is a realized chain of distributed successes and failures, a chain that is a con-catenation of two others: Deterministic in which there are only successes and Bernoulli sequence of length containing successes. Therefore, simultaneous split-ting must take place according to the rule
Consequently
(87)
The considered fuzzy Fucks’ distributions play a leading part in constructing fuzzy quantitative microlin-guistical models of language.
4) Some language structures are often described by gener-alized Fucks’ distributions when there are two kinds of successes with probabilities and . In this case
(88)
If one is only interested in one kind of success, then
(89)
Corresponding Poisson limit ( , and
) is
(90)
where
(91)
is Euler integral, and , an
incom-plete gamma function. Taking into account the relation between incomplete gamma-function and -distribution one finally obtains
(92)
where is -distribution with
degrees of freedom. Distribution (92) is called the “ -distribution with approximately degrees of freedom.”
Fuzzy Zipf–Mandelbrot Distribution: It is well known that the Mandelbrot’s theory of recurrent coding constitutes the basis of statistical macrolinguistics. If the vocabulary of volume is divided into classes according to informational cost [16] of words of a given class, then the probability of the word of th class can be expressed as
(93)
where , , do not depend on the cost and is a informa-tional cost of th class.
Let three cases of splitting be considered.
1) The set of classes . In
this case
(94)
2) The set of informational costs
. For is a function of , according to the principle of generalization [18] one obtains
(95)
3) When the number of classes is fuzzy number , by analogy with binomial distribution with fuzzy number of trials, one can write
(96)
The aforementioned formulas must be applied to the whole language as a formation medium, while the clas-sical one must be applied to individual texts.
VIII. GRAPHEMESDISTRIBUTIONMODELINGBYWORDS IN THESPANISHLANGUAGE
In this section, the description of the research method of probability–possibility organization of graphemes distribution by words is suggested. The aim of this investigation is not to construct a final quantitative model of the word formation process, but rather to illustrate the possibilities of the suggested model.
Description of the Model
Consider -seat carteges of three types of symbols conven-tionally called empty symbols, zero and unit elements. Unit el-ements are obtained directly from experience and the real struc-tures of natural language, which are formed by picked up lan-guage elements, are estimated using the said elements.
According to the suggested model, the process of any an-alyzed structure formation is considered as the super-position of purely random (probabilistic) and possibilistic (fuzzy) pro-cesses, i.e., as the composition of two bodies of evidence: Prob-abilistic (dissonant) and possibilistic (consonant).
The structure of the probabilistic body of evidence corre-sponds to Bernoulli events where is the cartege length, the whole number of unit elements, the number of a priori fixed (determined) unit elements (d.u.e.). As to the structure of the possibilistic body of evidence, it consists of consonant
(focal) events to which the
pos-sibility distribution corresponds; here is
the event “no d.u.e.,” “one d.u.e.,” , “ d.u.e.” Probabilities of focal events are expressed by the following possibility distribution:
(97) The compositional rule is described by the following rela-tions:
(98)
(99)
where is a combined event whose probability is defined by (99).
Notice that can be considered as the mathematical
expectation of the random variable
de-fined on the constant body of evidence.
The generating function corresponding to (99) is
(100)
where is the generating function of the Bernoulli distri-bution.
Moments of the random variable defined on the com-bined body are
(101)
where by convention
when
when
For example
(102)
(103)
Here denotes mathematical expectation.
In the case of practical calculations, the corresponding Poisson limit
(104)
is used instead of (99). The generating function is
(105)
The meaning of constant is defined by the relation
(106)
i.e., is the difference between common and focal mathematical expectations.
Formula (99) describes a class of carteges that can be called “nonuncomponent.” All seats in these carteges are filled with elements (no empty symbols). Such a class of carteges can be used as a model if the frequency of the zero event can be repre-sented by number . Experience shows that for any grapheme this condition is not valid. One must use the more general model based on distribution (92).
This distribution differs from (99) by supplementary factors and turns into (99) when .
It is easy to see that the corresponding generating function is
(107)
and
(108)
where is the mean over the focal probabilities.
Let it be supposed that distribution (92) must be considered as a model grapheme distribution by words. The division of all words of natural language into fuzzy classes corresponds to focal events. Each of them corresponds to specific features of word formation of a given class.
TABLE I
RESULTS OFDIFFERENTFREQUENCIES FORGRAPHEME“a”
Illustrative Example
The considered model will be used for investigating grapheme distribution in Spanish words. The consideration of all graphemes, the comparative analysis of their distributions and models and informational analysis will be given in a separate paper.
Let “a” be considered as an example. The initial material was obtained from the Diccionario manual Español–Ruso [17], with about 6000 dictionary units. The observation results of different frequencies for grapheme “a” are given in Table I. The vertical entries correspond to the number of meetings of graphemes of a word, the horizontal ones to the different initial graphemes of a word.
TABLE II
RESULTSOBTAINEDFROM(110)
As it will be shown below, a good possibility distribution is the following one:
(109)
Using (108) and the observation results, one gets
which for defining gives
From this equation, .
Focal probabilities 0, 3951, and 0,
6049.
The data obtained gives the following combined distribution for grapheme “a” over words
(110)
Results of calculations are given in Table II. For comparing the mean of observations, results over all initial graphemes are given in the same table.
Fuzzy subset of the number of determined graphemes
(111)
can be interpreted as “approximately 0,” i.e., “almost all” graphemes “a” are random.
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Fransisco Criado Torralba was born in 1945. He received the Ph.D. degree from Malaga University, Malaga, Spain, in 1979.
Currently, he lectures on statistics and operational research at Malaga University. He has published a large number of papers on mathematical systems modeling, fuzzy systems, game theory and control theory, as well as having participated actively in several EU research programs.
Tamaz Gachechiladze was born in 1929. He is an Assistant Professor with the Department of Stochastic Processes Theory, Tbilisi State Univer-sity, Georgia. His current research interests include mathematical cybernetics and informatics.
Hamlet Meladze was born in 1939.
He is currently Chair of Computer Mathematical Providing and Information Technologies, and Full Professor in the Department of Applied Mathematics and Computer Sciences, Tbilisi State University, Georgia. His current research interests include mathematical cybernetics and informatics.
Guram Tsertsvadze was born in 1933.
He is Chair of Mathematical Cybernetics and Informatics, and Full Professor in the Department of Applied Mathematics and Computer Sciences, Tbilisi State University, Georgia. His current research interests include mathematical cybernetics and informatics.
