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LINGUISTIC ALGEBRA AND SIMILAR-TO-BRAIN COMPUTER
BONDARENKO M.F.,
DUDAR Z.K
Karkov National University of Radio Electronics dudar@kture.kharkov.ua
Abstract. The attempt to identify some natural language mechanisms in the form of mathematical structure called linguistic algebra is made. There are two tiers in it- the semantic and syntactic. The first tier is represented by one type of predicate algebra, the second - by predicate operation algebra. The method ofexperimental validation of algebra-logic language models (on the Russian language example) is considered. The method of a formula record of natural language word combinations and sentences sense is processed.
Introduction
In everywhere used machines of consecutive operation only a small number of decision elements works anytime, all the rest are inactive. Program control principle opposes simultaneous work of all computer elements. Nature has created humanbrain, which is a “computer” in a way; it works on different and more progressive principles. Brain is a “machine” of parallel operation. Although nerve cells act million times slower than computer electronic elements, brain acts quicker and is able to solve more complex tasks than the most powerful modern computers due to parallel principle of operation. At the same time the way of communication between a man and a computer is changing fundamentally. If modern (consecutive) computers “live on” programs then the parallel ones like people “live on” knowledge. The communicationbetween themwill be performed with the help of sentence (statements) exchange. These machines will operate guided by knowledge but not by programs.
Natural language is a complex research object. In the given article the hypothesis is used as a mathematical analysis of language mechanism: natural language is so called linguistic algebra.
Let us characterize some basis algebra concepts. Any elements formula record system of any set A is called algebra A over A. The set A is called algebra A carrier. Any algebra A over A is characterized by its basis operations used as its elements translator, by its basis elements chosen from its carrierA. The set of all algebra A basis operations is called operation basis, the set of all basis elements is called algebra A element basis. Operations basis and elements basis form algebra A basis.
Any record, which denotes any superposition of this algebra basis operations applied to its basis elements is called algebra formula. Every algebra formula stands for some element of its carrier and can be used as its name. Algebra A is called full if each element of its carrier can be stated as some algebra A formula. The algebra basis is called full if this algebra is full. It is called irreducible if an elimination of any operation or element from basis is making it incomplete. Any two algebra formulas expressing same element of its carrier are called identical. Any record indicating any pair of algebra A identical formulas is called algebra A identity. If algebras A and B are given over a same carrierA, then identical formulas of different
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algebras can be considered. The algebra A identity system is called full if the fact of any two algebra A formulas identity or non-identity can be derived from it. The system of algebra identities is called irreducible if none of its identities can be logically derived from the collection of others. Any record indicating any identity family of this algebra is called algebra identity scheme.
Many various consequences, which allow experimental check, can be derived from hypothesis that natural language is algebra. Thus, each sentence and text made from this sentence express some thought, so thoughts should be considered as elements of linguistic algebra carrier, and corresponding sentences (texts) - as describing formulas. The sentences and texts should be built the way formulas were built. The sentences expressing one idea should be considered as identical formulas of linguistic algebra.
Thoughts are international; each can be expressed using any language. Different languages should be considered as different linguistic algebras given over one carrier - the set ofvarious thoughts operated by people. Sentences in different languages expressing one idea are identical formulas. Texts translationfrom one language to another should be considered as a transition of one linguistic algebra formulas to its identical formulas of another given over the same carrier.
By checking consequences derived from this hypothesis and other assumptions in a linguistic experiment the initial aspects (axioms) of language theory set up this way may be confirmed or disproved. If a lot of various axiom consequences may be obtained without language and speech facts contradiction then this will be the confirmation of this theory. If some of formulated hypothesis are not true or are partly confirmed then they may be replaced by more complete ones based on negative results.
1. Thoughts as predicates
Let us consider mathematical nature of thoughts expressed by sentences and texts, that is the sense of any natural language sentence (text). If thoughts are predicates then linear algebra should be considered as predicate algebra. This hypothesis is motivated by such a fact that all thoughts (ideas and concepts) are expressed by predicates in mathematics. If natural language is also a mathematical obj ect (we have presumed that it is an algebra) then the situation here is the same.
Any function P(x1, x2,..., xm)=x, which represents set Um into set S={0, 1} is called predicate of dimension m defined at set U. The set of all predicates P: Um^S is denoted by P symbol.
Values of predicate P independent variables xi (i= 1, m) is
called objects, the variables are called object ones. The set U is called object universal set, the set Um is called object space of dimension m. The set V={x1, x2,..., xm} is called universe set of object variables. The sets U and V can be chosen arbitrarily. The elements of set S are called logical. The element 0 stands forfalse; the element 1 stands for true. The dependent variable x of predicate P is called true, its values are called true ones. Any algebra, which medium is a set P of all predicates defined on object universe set U, is called predicate algebra on P.
In order to convince that some predicate really serves as sentence content, it is enough to consider what a content of
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some formula is. The content of the formula is the function, which it expresses. But if a sentence is a formula, then its content should also be any function. If a sentence is used for any quite certain situation characteristic, then it will be either true, or false. If the sentence is considered out of any situation, then the true or false question does not arise. Similarly, while arguments values are not substituted in the formula, the function concrete meaning expressed by this formula cannot be considered. The true value of each sentence is unequivocally defined by situation, which it is referred to. Similarly, any formula value is unequivocally defined by a set of all argument values included in it.
Thus, each sentence expresses some function with binary value, in other words, it sets some predicate P(x)=x. Independent variable x of this function is variable situation, dependent variable is represented by true variable x. After substitution instead of variable x of a concrete constant situation x=a the given sentences become true (x=1) or false (x=0) depending on whether this sentence corresponds with a situation a, which it is referred to. The variable situation x should represent a set x=(xj, x2,..., xm) of object variables x1, x2,..., xm. Any constant situation x=a should be a set a=(a1, a2,..., am) of any objects x1=a1, x2=a2,..., xm=am.
So, each sentence should express some predicate P(x1, x2,..., xm)=x, which represents some dependence of a true variable x on object variables x1, x2,..., xm.. However, if addressing to the concrete natural language sentences, there are no any object variables. The explanation is the following. Unlike mathematical formula, natural language sentence expresses not function P(x1, x2,..., xm) itself, but only its name P. Every time a man by transforming a sentence into an appropriate idea, completes the sentence up to a predicate, adding to it a missing object variable (as to a predicate name). Only after that the sentence becomes accessible to understanding. And vise versa, by transforming some idea into a sentence a man excludes object variables from it, reproducing only a name of an idea but not the idea itself.
The sentence completed by object variables is called a statement. Any statement expressing some predicate is called this way in mathematical logic.
The sentence used as a name of a predicate, becomes ambiguous, if it is considered out of a context and without object variables. Such ambiguities are eliminated completely by object variables introduction. This fact evidently demonstrates the necessity of sentence completing with object variables for their unequivocal understanding. At transition of sentences to the appropriate statements, the intuition of Russian native speaker can be relied upon. However formal performance of such a transition is possible, which should be carried out only mechanically on the basis of the text and context analysis without sense reference. This task is rather complex, howeverthe automation ofprocess ofnatural language texts understanding is impossible without it.
There is a following discrepancy of the accepted initial theoretical circuit and actual situation. The algebraic approach to natural language requires all object variables x1, x2,..., xm of predicate algebra to present in each predicate P(x1, x2,..., xm). In fact it is not so. However the same discrepancy is observed in mathematics. In all used in practice formula there is only a small part of variables of that algebra, which language they are written in. In mathematics this discrepancy is overcome by insignificant variable introduction.
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Argument xi (i = 1,m) of predicate P(x1, x2,..., xi,..., xm) is called insignificant, if at any xj, x2,..., xi-1, xi’, xi’’, xi+1,..., Xm e U P(xj, X2,...,XM, Xi’, Xi+j,..., Xm)=P(Xj, X2,..., Xi_j, X’’, xi+1,..., xm). According to this definition, the predicate P value does not depend on value of insignificant argument xi at any other fixed variable values. If there is no variable xi in the predicate P formula, it is the evidence of its insignificance. Insignificant variables can be omitted in the list of arguments ofpredicate P(x1, x2,..., xm). For example, predicate P(x1, x2,..., xm) with the only significant variables x2 and x4, could be written down as P(x2, x4). If there is an insignificant argument in the predicate formula, then it could be transformed in such a way that this argument will disappear. The predicate, at which all but one arguments are insignificant, is called unary, all but two is binary, all but three - ternary, all but n (n < m)
- n-ary. The number n is called a predicate arity. Number m is a predicate dimension, which should be distinguished. At m= 1 the predicate is called single, at m=2 - double, at any m
- m-placed.
Every man in the speech practice uses significant number of object variables. Therefore, the number m, which describes object variables quantity in set V of linguistic algebra, is rather big. The additional research is necessary for its concrete size estimation. It is formally considered that each sentence, which is included in such texts structure, has all these arguments. However only a few object variables (usually not more than ten) will be significant in some statements.
2. Natural language as Boolean algebra
The sentences are built from separate words. Therefore, the fact that words are basic elements in linguistic algebra is natural to assume. As any elements of a linguistic algebra carrier are predicates, then words considered as basic elements should also be predicates. From the substantial point of view any linguistic algebra elements are ideas, and ideas are expressed by the sentences. Hence, the separate words are to be considered as sentences.
The object name as name P of a predicate P (x), and variable x - as its argument is used as an example. The values of variable x are objects showed to the examinee. Set of all possible objects, which the examinee is capable to react on, serves as objects U universe set. Thus, each word P can be understood as a name of some predicate P(x), given on set of every possible objects U. This predicate is real and is quite defined, as it can be reproduced by any man in practice by answering a question: “whether the subject x is right for a concept expressed by a predicate P”. It also concerns adjectives, verbs, quantitative numerals, adverbs etc. Prepositions are also can be considered as binary predicates, but not unary ones. We assume that any word (except for small quantity of words expressing predicates operation such as “He”, «h», «huh») may be presented similarly as some predicate. The special research in word sense analysis is necessary for the substantiation.
Let us consider the question of linguistic algebra basic operations. The negation can be formed from any sentence by adding particle “He” or the expression “uo^ho, hto”. The negation can affect separate words and word combinations. It is naturally to assume that from the algebraic point of view the negation of the sentence P is Boolean operation of predicate P(x) negation expressing the content. Designating negation operation by a word He, in general case we have
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He(P(Xi, X2,..., Xm))=(HeP)(Xi, X2,..., Xm). (1)
Here P(x1, x2,..., xm) is any statement; x1, x2,..., xm are its obj ect variables; P is a sentence appropriate to this statement; He (P) is a sentence received from the sentence P by negation operation He use.
Similarly conjunctions “h” and “nun” are considered which can help to connect any sentences, and as a result receive new sentences. It is naturally to assume that the words h and h™ correspond with double operations of conjunction and disjunction working on the statements P(x1, x2,..., xn) and Q(x1, x2,..., xn), which express sense of sentences P and Q. W e can write down:
(P(x1, x2,..., xm))H(Q(x1, x2,..., xm))=(PuQ)(x1, x2,..., xm); (2)
(P(x1, x2,. .., xm))H^H(Q(x1, x2,..., xm))=(PU™Q)(x1, x2,..., xm),
(3)
where P and Q are initial sentences; PhQ and Ph™Q are sentences received as a result of connection of the initial sentences by conjunctions «n» and «huh».
As a result of natural language algebraization the questions being rather difficult for the traditional language and speech analysis can be answered.
Let us introduce concept of Boolean predicate algebra. Let P(x1, x2,..., xm) and Q(xb x2,..., xm) are predicates on U. Such operations on predicates, which for any xb x2,..., xmeU are defined by equities:
(-P)(x1, x2,..., xm) =^(P(X1, X2,..., Xm));
(PaQ)(x1, x2,..., xm) =P(x1 ,x2,..., xm)aQ(x1 , x2,..., xm);
(PvQ)(xb x2,..., xm)=P(x1, x2,..., xm)vQ(x1, x2,..., xm)
are called negation P, conjunction PvQ, and disjunction PaQ of predicates P and Q.
These operation equalities -., a and v on predicates are reduced to operations -., a and v on their values, i.e. to operations on logic elements 0 and 1. The latter is called negation, disjunction and conjunction of logic elements and is defined as follows: -.0=1, 1=0; 0a0=0a1= 1 a0=0, 1 a1=1; 0v0=0,0 v1=1 v0=1 v1= 1. Any predicate algebra with operation basis consisted of predicates negation, conjunction and disjunction is called Boolean predicate algebra.
The negation, conjunction and disjunction operations are called boolean. Boolean operations can be defined abstractly (i.e. system of properties) with the help of Boolean algebra concept. Boolean algebra is any set M along with given single operation and double operations a and v. By definition these operations have the following properties [1 ]: for any x, y, z e M xax=x, xvx=x; XAy=yAX, xvy=yvx; (xAy)Az=XA(yAz), (xvy)vz=xv(yvz); (xvy)Az=(xAz)v(yAz), (xAy)vz=(xvz)A(yvz); xv(yA—y) =x, XA(yv—y) =x; —1(—x) =x; -i(xvy)=-xA-y, -.(xAy)=-xv-y These properties are called Boolean algebra axioms. At given M boolean operation are defined uniquely in abstract sense (i.e. to within set M elements designations). It is important to note that the specified system of axioms is superfluous. All axioms but one are paired in it. In any pair one of axioms can be excluded (either all left, or all right) without a detriment to system completeness. Thus, seven axioms are enough for Boolean algebra exhaustive characteristic.
We introduce one more hypothesis: the linguistic algebra is Boolean algebra. Operations on words and word combinations, sentences and texts expressed by words He, h and ii.™ are represented as Boolean operations. These words for specified operations expression are not always used. Two sentences, on which the operation and works, can be connected with a coma or a full stop. The connective words “however”, “nevertheless” etc can be used instead of conjunction «h». But words “He”, “h” and “nun” not always express operations of negation, conjunction and disjunction. The particle “ He” may mean a statement but not a negation, and, the conjunction “h” can express not operation of conjunction but the sense of word “gouro”. The conjunction «huh» can be used in dividing sense “ nun - nun “. As the disjunction the conjunction «huh» is used in uniting sense “nun TaK^e”.
Generalizing, it is possible to say that in natural language the texts and the appropriate meaning are not connected mutually. The meaning of the same text may vary depending on a obj ect variables choice and a context. The same meaning can be expressed by various texts. The meaning are to be studied and formally described regardless of the text, and texts - regardless of its meaning. Moreover, the connection between texts and their meaning can be formally described. The meaning can be written down in statement language, texts are easier to express directly in their natural language form. Apparently, special mathematical language is required for the connection between the texts and meaning description.
There are cases, when the words “He”, “n” and “nun” express boolean operation on texts, but they are used differently or with additional sense. For example, the conjunction “n”, used as a conjunction, may also express events contraposition or two events sequence in time described by these statements.
3. Sentences as formulas
The fact that natural language sentences are to be predicate algebra formulas follows from accepted above hypothesis. Let us compare this conclusion with language facts. Any formula has a certain structure, which can be expressed by some pattern. For example, let us consider formula of Boolean functions X1X2 v X3X4 algebra. It can be expressed graphically by the scheme shown on picture.
The circles with Boolean operations -., a and v stand for formula translator. The scheme synthesizes formula X1X2 v X3X4 from its arguments X1, X2, X3, X4.
In grammar the syntactical hypotaxis trees are used for a sentence structure pictorial presentation [3]. Words in a sentence are connected in pairs by arc. The result of such a connection is a word combination. The word, which the arc comes from, is the main one and the word, which the arc comes to, is called the dependent one. The word, which no arc comes to, is called the root. A man, who understands a sentence meaning, is supposed to build its syntactical
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hypotaxis tree. At the heart of this ability there is a feeling that each pair of dependent words in the sentence construction process appears in a head right after the main one.
The tree structure of connections between words and sentences is the evidence that along with linear word order there are directional connections between words. The arcs do not intersect there, and there is no root under any arc. Such trees along with corresponding sentences are called descriptive. Unlike descriptive sentences non-descriptive are interpreted as unnatural.
Words serve as formula argument X1 +Xn values. The circles (marked with numbers) depict words and word combination translators. They carry out words and word combination joining operations. The scheme synthesizes a sentence from separate words. So, words are transformed into word combinations at block transition. At the scheme output the finished sentence is received. The numbered blocks show the operation consequence carried out by scheme translators.
This consequence is defined by the following algorithm. Words in a sentence from left to right are checked and all possible operations on every step are carried out. The fact whether a word is connected with a following one is determined. The coherent and finished word consequence (word combination) should be a result of every operation realization The sentences are formed by operation situated on sentence root level performed by a predicate (if there is no predicate then a root becomes subject). The first word in each pair is considered the main regardless of its position concerning the dependent word. Native-speaker intuition is used for algorithm realization. Additional research for this algorithmbringing to computer realization is necessary.
The sentence formula scheme is built according to its syntactical hypotaxis tree, so, the return from the scheme to the tree can easily be done. The scheme contains: blocks, which synthesize text from its separate elements: poles, where sentences and word combinations appear; order of sentence formula synthesis by scheme blocks. The sentence formula can be built on the scheme basis.
Numbers serve as operation names, brackets show realization order and operation to words application consequence, word forms are formula argument values. At transition to formula arguments from its values the words are replaces by variables
X
The example of formula of predicate operation algebra, which expresses syntactical structure of some sentence:
((X12(X21(X3)))4(X43(X5)))10(X69((X75X8)S
((X96Xio))7Xn)))),
where X1^X10 are predicate variables, numbers 1a10 are predicate operations.
Let us consider that natural language is two-tiered. The first tier is described by predicate algebra, the second - by predicate operation algebra. The sentence semantics, i.e. its content, is formally described on predicate algebra language, the syntax, i.e. the sentence construction, - on language of predicate operation algebra.
Let us give the formal definition of predicate operation algebra. Let Uis a universe set of objects; x1, x2,xm -are object variables; P is a set of all predicates P(x1, x2, xm) on
object space Um. The set P is called the universe set of predicates. Variables X1, X2..., Xk, defined on set P, are called predicate variables. Their values are predicates given on Um. The set Pk is called predicate space of dimension k over object space Um. The elements of set Pk (k-component set of predicates ) are called predicate vectors. The predicate space is two-tiered construction: there are objects on its first tier, there are predicates on the second tier. Any function F(Xj, X2, ..., Xk)=Y, describing set Pk into set P, is called predicate operation. Let us generate the set R of all predicate operations. Any algebra given of carrier R is called predicate operation algebra over R.
Let F(Xj, X2, ..., Xk)=is a predicate operation, describing a set Pk into set P. HereX\,X2,..., Xkare predicate variables, acting as operation F argument; Y is a predicate variable, which is a value of operation F, which values are defined by the pattern
(-F)(X, X2, ..., Xk)=—F(X\, X2, ..., Xk)
for any X1t X2, ..., XkeP is called the negation —.F= F of predicate operation F.
Let F and G are predicate operations, describing Pk into P. The predicate operation, which values are defined by the pattern
(FvG)(X1, X2,..., Xk)=F(X1, X2,..., Xt)vG(X1, X2,..., Xk)
for anyX1, X2, ..., XkeM is called disjunction FvG of predicate operations F and G. The predicate operation, which values are defined by the pattern
(FaG)(X1, X2,..., Xk)=F(X1, X2,..., Xk)AG(X1, X2,..., Xk)
for any X1, X2, ..., Xke.M is called conjunction G ofpredicate operations F and G. In the last three equities there are operations -., v, a on predicate operations to the left of equality sign; the signs -., v, a to the right stand for operations on predicates. Any predicate operation algebra with operation basis consisted of negation, conjunction, and disjunction is called Boolean predicate operation algebra.
4. Word combinations as formulas
Any sentence expresses some idea, which can be described by definite polyadic predicate P(x1, x2,..., xn). Mechanism of sentence creation is presented as following. Firstly, sentence variables x1, x2,..., xn are connected by predicate, which realizes predicate S(x1, x2,..., xn), where n is an quantity of object variables at verb S. By expanding the sentence with some word combinations, which answer questions according to object variables x1, x2,..., xn, gradual approach of a text sense to required is achieved.
Similarly, sense narrowing (specification) and bringing the meaning, which is expressed by a man, is achieved by sentence broadening. After adding every following word to the sentence the content always narrows. If operated backwards by cutting words off a finished sentence, the chain of sentences with broadening meaning inserted into each other is obtained.
It means that a word adding is expressed by conjunction. Indeed, if predicates P and Q are known to be in relation P c Q, then there is always such a predicate R, that P=QR. Thus, word Q or word combination R added to the sentence is carried out as conjunctive factor. On the contrary, the sentence P is obtained as a result of realization of word or
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word combination R adding to sentence Q operation QR=P, which meet the requirement P c Q.
Let T\(xiu , Xj2l Xis l ), T2(Xm , Xh2 Xis22 Tr(Xilr ,
Xi2r,..., Xisrr ) are predicates expressed by words (or word combinations) , which are added to its predicate.
Here r is a quantity of all words (or word combinations), added to sentence predicate. Some of predicate S(x\, x2,..., xn) arguments of predicate S serve as every predicate Tj arguments
(j= 1, r). The symbol Sj designates the number of significant
variables of predicate Tj. Then sentence predicate is expressed as follows:
P(X\, X2,..., X„)=S(X1, X2,..., Xn)T\(Xin , Xi21 ,..., Xis11 )a
AT2(Xi\2 , Xi22 >...> ^2 )...Tr(X\r > Xi2r v- XSrr ^ (4)
The equity (4) shows that sentence P content is formed from predicate S contentby its limitationby word combinations T\, T2,..,Tr content. By splitting the idea P on parts S, T\, T2,..., Tr according to formula (4) the speaker is performing conjunctive decomposition of predicate P. The listener A is performing predicate P composition from predicates S, T\, T2,..., Tr, with conjunction operation.
The predicate P representation in P=Q\ aQ2a...aQ[, where Q\, Q2,..., Qi are some predicates is called predicate P conjunctive decomposition; l is the predicate number received at decomposition.
Let us consider concrete ways of word combination building from separate words. The subject T connects with predicate S by the pattern
P(x)=T(x)aS(x) (5)
as a result the sentence P is obtained.
The sequence of adjective T\ and noun T2 is realized by the same pattern:
T(x)=T\(x)aT2(x). (6)
As a result the word combination T is obtained
The noun T2 control of cardinal numeral T\ may be realized by the pattern:
T(x)=T\(x)av Oy(yeX3 T2(y)). (7)
Conclusion
The text meaning can be represented as a formula of predicate algebra, and its syntactical structure - as formula of predicate operation algebra. The research of various types of sentence object variables identification (for example, temporal and special character) and methods of practical language use study are considered to be perspective.
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