The paper proposes a method for clustering predicates of arbitrary dimensionality. To this end, a theorem of the general form of first-order predicate is formulated, which justifies the firstorder two-layer decomposition method of predicate, resulting in a predicate defined on a set of significantly smaller power than the original. This allows for the identification of conditions for the most effective identification of processes of human intellectual activity using the generalized comparator identification decomposition method of predicate is developed, which is based on the concept of the general form of second-order predicate. Based on the combination of the concepts of two-layer decomposition of first and second-order predicate, a method of three-layer decomposition of predicate is developed. The resulting method of multilayer decomposition of predicate is suitable for building electronic circuits that implement arbitrary relationships. The predicate scheme is widely parallelized, resulting in its high performance.
decomposition Text classification, predicate algebra, equivalence predicate, general form of predicate, text isomorphism, decomposition, comparator identification, three-layer
In recent years, great attention has been paid to the development of new methods for parallel information processing. Practically all existing approaches [1-3] are based on the concepts of decomposition and composition in one way or another - be it programming, databases, neural networks, and so on. Therefore, the development of new theoretical methods of decomposition is a highly promising and interesting direction in algebraic logic. This work proposes a new method of multi-layer decomposition of predicates, which will be obtained from the generalized properties of comparator identification.
In predicate algebra, comparator identification is widely used, which is a type of indirect identification [4, 5]. A comparator K is a device with m inputs y1, y2,…, ym and one output t, where t0, 1 is the binary reaction of the comparator. The comparator determines whether its input signals y1, y2,…, ym are in a given relation K or not. Comparator identification is successfully applied in solving many artificial intelligence problems. In [5], its capabilities for alternative evaluation models in decision-making systems are shown. In [6], interesting results were obtained for the theory of color vision using the comparison method. This method is designed for investigating objects with input signals that are inaccessible for direct measurement. The subject of such identification is often the human intelligence. EMAIL: (G.
Proniuk);
Geseleva);
2023 Copyright for this paper by its authors.
In predicate algebra, during the comparator identification process, the identifiable object P implements a predicate Р (x1, x2, …, xm) = K(f1(x1), f2(x2), …, fm(xm)), referred to as the predicate of object P. The simplest task of comparator identification is to mathematically describe the output signals у1, у2, …, уm of processes f1, f2, …, fm and the processes themselves based on the given comparator and the known properties of object P. The comparator's behavior implements a predicate K (y1, y2,…, ym)=t, corresponding to the relation K. The identifiable processes f1, f2,…, fm. are connected to the inputs of the comparator by their outputs. Here, x1A1, x2A2,…, xmAm are input signals of processes, while y1В1, y2В2,…, ym Вm are their output signals. A1, A2,…, Am are sets of input signals of processes, and В1, В2,…, Вm are sets of output signals of processes.
Let us consider the main concepts that will help us generalize the method of comparator identification to a wider class of predicates.
Let E (x, y) be a predicate defined on the Cartesian product of a non-empty set M. Then the predicate E is called [7–9]: • reflexive if it satisfies the condition xМ E (x, х); • symmetric if it satisfies the condition ∀х, у ∈ М (Е(х, у) ⊃ Е(у, х)); • transitive if ∀x,y,z∈x, y, z M , from хЕу and уЕz result хЕz.
Any reflexive, symmetric, and transitive predicate is called an equivalence predicate [1, 7].
Let N be a non-empty set, f – be a surjective function mapping set M to set N and let D – be the equality predicate defined on N x N with the condition: х, уN D (х, у) = аN хауа.
It is known [8, 9] that any predicate E on M x M expressed for any x, yM as
( , ) = ( ( ), ( )), (1) is an equivalence predicate. The function f is called the characteristic function of the equivalence predicate.
In predicate algebra, any equivalence predicates, and only they, can be represented in the general form (1) with a suitable choice of the set N and function f. From a mathematical point of view, this result is trivial, but it is very important for the theory of comparator identification because it indicates the necessary and sufficient features that can always be used to determine whether an object implementing the predicate E can be identified by the comparator method.
If a system implements the predicate t=E (x, y), and this predicate satisfies the conditions of reflexivity, symmetry, and transitivity, then it can be identified using the comparator method. However, if at least one of these three conditions is not satisfied, then the comparator method is not applicable for such an object. These results of comparator identification can be applied to any physical objects that satisfy the aforementioned conditions.
Above it has been shown that a pair (N, f), where f: M→N, determines a unique equivalence predicate E (x, y) =D(f(x), f(y)) on the set M M. However, does every equivalence predicate E uniquely determine the pair (N, f)? It turns out, no. There exist different pairs (N, f) and (N’, f’) that define the same equivalence predicate E. The property formulated below specifies the necessary and sufficient condition under which two pairs (N, f) and (N', f') determine the same equivalence predicate E.
Statement 1. In order for two pairs (N, f) and (N', f') to define the same equivalence predicate E on the Cartesian product of the set M, it is necessary and sufficient for there to exist a bijection T between the domain of N and the range of N' such that for all xM f’(x)=T(f(x)).
From Statement 1 it follows that if the equivalence predicate E (x, y) can be represented as (1) for any x, уM, then it can also be represented as
Е( , ) = ( ( ( )), ( ( ))), (2) where T is an arbitrarily chosen bijection.
It follows from statement 1 that if the predicate E is represented by two different methods Е (x, y)= =D(f(x), f(y))=D(f’(x), f’(y)), then there always exists a bijection T that links the functions f and f' by the dependence f’(x)=T(f(x)), which holds for any x ∈ M. Therefore, it is impossible to specify a unique characteristic function f for the equivalence predicate E.
Thus, if some function f is found that mathematically describes the identification object, then an entire family of other functions can also claim to describe this object. In other words, the output signals of the identification object through comparator identification allow for various options for mathematical descriptions. Such multiplicity of object representation may indicate the incompleteness of its description by the comparator identification method and, consequently, the disadvantage of this method compared to classical direct identification methods. In fact, the degree of completeness of object description in these two identification methods is absolutely the same. The fact is that in direct identification, the object description is obtained only by virtue of the fact that the method of describing its output signals was chosen before the identification process began. In the case of comparator identification, however, the method of describing the output signals is chosen in the identification process itself, and this is precisely what leads to multiple descriptions of the object.
It is known that the comparator identification method describes an object up to isomorphism [4, 9]. Essentially, this means that comparator identification, just like direct identification, provides a unique description of the object up to notations.
To solve problems of comparator identification, an important question is the issue of isomorphism of equivalence characteristic functions.
Predicates Р and Р on А В и А В, are called weakly isomorphic (or simply isomorphic) if there exist bijective functions : A→A and : В→В, such that for all xА and yВ the equality is satisfied:
P (x , y) = P ((x ), (y)). (3)
We also say that the predicate P(x, y) is isomorphic to the predicate P. The bijections and that satisfy condition (3) are called left and right isomorphisms of the predicates P and Р.
The predicates Р (x, y) and Р(x, y) on sets А В and А В are called strongly isomorphic if there exists a bijection : AB→AB, such that for all xА and yВ, the equation is satisfied
P (x , y) =P ((x ), (y)). (4)
We will also say that predicate Р is -isomorphic to predicate Р. A bijection that satisfies condition (4) is called an isomorphism of predicates Р and Р.
The concepts of weak and strong isomorphisms of predicates play an important role in the theory of comparator identification. The point is that the choice of designations for the signals of the identified system is within the power of the researcher and is determined by the unit’s system adopted by him. If two researchers studying the behavior of the same system use different designations for its input signals, they will obtain different predicates for it. If all input signals of the same system under study are recorded by each researcher in a single (but their own) units’ system, then the predicates obtained by them will be strongly isomorphic, and if they are recorded in different systems, the predicates will be weakly isomorphic. In this case, it is said that the studied systems are identified up to designations (common or separate). In the case of strong isomorphism of predicates, it is said that the identified systems coincide up to designations in a single unit’s system. In the case of weak isomorphism of predicates, it is said that the identified systems coincide up to designations in different units’ systems.
Statement 2. If АВ= and АВ=, then weakly isomorphic predicates Р and Р, defined on А В and А В, will also be strongly isomorphic.
In essence, this property means the following: two different descriptions of the same identifiable system P (x, y), whose input signals are defined on non-intersecting domains, always coincide up to strong isomorphism, i.e., they coincide up to the designation of input signals x and y of the system Р described in a unified notation system.
Statement 3. If the equivalence predicates Е and Е on sets А А and А А weakly isomorphic, then they are also strongly isomorphic.
The substantive content of property 3 means that the signals x and y of the equivalence predicate E (x, y) cannot be described in different notation systems, but only in the same one. To describe the system E (x, y) with an equivalence model, a researcher must express its output signals x and y in a single notation system.
Statement 4. Let E be an equivalence predicate on А А and f: A→B be its characteristic function. Then, equivalence relation E is isomorphic to equality relation D on В В if and only if f is injective.
This statement determines in which cases a human perceives complete information about objects and in which cases not. Information is not lost when a person's sense organs assign a subjective image (regardless of what it is) to each object. However, if the number of images is less than the number of perceived objects, some information about the objects is lost. For example, the human eye loses some information about light when perceiving light radiation. This is proven by the existence of different types of light radiation that appear as the same color to the eye. For instance, there is a mix of red and green monochromatic radiation that appears the same color as yellow monochromatic radiation.
Let us now turn to the study of isomorphism of characteristic functions of equivalences. Suppose we have a signal transformer that implements a function у=f(х), which maps set A to set B. By renaming its input and output signals x and y using bijective maps : A→B and : A→B, we obtain x=(x), у=(y). As a result, the same signal transformer is now described by a different function у=f (x), which maps set A' to set B'. Using the inverse function, denoted by -1, we express the function f' in terms of f: f (x) = (f ( − 1(x)). Similarly, the function f is expressed in terms of f: ( ) = − 1( ( ( )), (5) where − 1 is the inverse function of the bijection .
Let Е and Е be equivalences on А А and А А; D and D be equality predicates on В В and В В.
Statement 5. If the predicate E is -isomorphic to the predicate E', then there exists a bijection : A →B, such that the function f (, )-isomorphic to the function f, and the predicate D is - isomorphic to the predicate D.
Statement 6. If the function f (, )-isomorphic to the function f, then the predicate E is - isomorphic to the predicate Е, and the predicate D is -isomorphic to the predicate D.
It directly follows from statements 5 and 6 that the following property holds.
Statement 7. For the equivalence E to be φ-isomorphic to the equivalence Е, it is necessary and sufficient for the function f to be (, )-isomorphic to the function f.
The substance of statements (5)–(7) means that the behavior of E (x, y) = D(f(x), f(y)) of the identified system E, is fully determined (i.e., up to notation) by the action of the identified object f and vice versa. In addition, the action of the zero organ D (u, v) is fully determined both by the behavior of the system E and the action of the object f. All of the above indicates that the comparator method is an effective means of identifying the object f, the internal state u=f(x) of the system E, and the zero organ D (u, v).
There is some inequality between the external behavior E of the test and the corresponding internal information process f since strong isomorphism of the predicates E and Е corresponds to weak isomorphism of the functions f and f. The following statement establishes a condition under which the predicate E and the function f become equal in this sense.
Statement 8. In order for the -isomorphism of any equivalences E on А А and Е on А А to be equivalent to the -isomorphism of their characteristic functions f: A→B, f: A→B, it is necessary and sufficient for the sets А and В, А and В to be disjoint.
Statement 8 states that if it is required that objects and their images can be measured in the same system of physical units and always obtain the system's action in the form of an equivalence predicate, it is necessary to ensure that the set of all analyzed objects and the set of their images do not intersect. For example, when creating an artificial color vision system, colors as physical objects should be represented not by light emissions, but by some physical processes, such as magnetic fields.
Thus, a comparator identification method is described for mathematically describing subjective phenomena. The behavior predicate Р of the subject allows determining the set of images or thoughts, as well as the intellectual functions of the human (perception, understanding, recognition) uniquely up to isomorphism. Human behavior in many cases allows description using an equivalence predicate. The question arises: is there another general form of a binary predicate, and what is it? Undoubtedly, there must be some general expression that gives some binary predicate. To answer this question, consider the following concepts.
Let us consider an arbitrary binary predicate P defined on А1 А2, and seek a representation for it, in which the comparison of the values of the two corresponding functions f1 and f2 is carried out using a simple predicate, in some sense. This predicate should replace the equality predicate in formula (1). In order to obtain the required form of the predicate, let us first consider several important concepts.
Statement 9. On accompanying equivalences. For any predicate P defined on the Cartesian product АВ, the predicates ЕL on АА and ЕR on ВВ of the form ( 1, 2) = ( ( 1, ) ( 2, )), (6) ( 1, 2) = ( ( , 1) ( , 2)) (7) are equivalences.
Statement 10. Generalization of the theorem on accompanying equivalences for a predicate of arbitrarity. For any predicate Р (х1, х2, … хn) on A1A2…A, the predicates Ei on AiAi ( = 1, ) of the form ( ′ , ′′ ) = ∀ 1 ∈ 1 ∀ 2 ∈ 2 … ∀ −1 ∈ −1∀ +1 ∈ +1 … ∀ ∈ (8) P( 1, 2, …, −1, ′ , +1, …, ) P( 1, 2, …, −1, ′ ′ , +1, …, ) are equivalences.
The predicates ЕL and ЕR, defined by expressions (6) and (7), are called accompanying equivalences (left and right) of the predicate P. The predicate Ei, defined by expression (8), is called the i-th accompanying equivalence of the predicate P.
Let Е and E1 be equivalences on АА. We will say that the equivalence E is embedded in the equivalence Е1 and write ЕЕ1, if for any х, уА from Е (х, у)=1 then Е1(х, у)=1. If ЕЕ1 and Е Е1, then we will write Е <Е1 and say that the equivalence E is strictly embedded in the equivalence E1. If Е <Е1, we will say that the partition R corresponding to the equivalence E is finer than the partition R1, corresponding to the equivalence Е1. We will also say that the partition R1 is coarser than the partition R. If ЕЕ1, then we will say that the partition R is finer than or equal to the partition R1. If ЕЕ1, then the partition R corresponding to the equivalence E is called a sub-partition of the partition R1, corresponding to the equivalence Е1. It is easy to see that the embedding relation defined on the set of equivalence predicates is reflexive, transitive, and antisymmetric, i.e., it is a partial order relation.
Theorem 1. On the general form of a binary predicate of the 1-st order. Let P be a predicate on А1 А2; ЕL and ЕR be its accompanying equivalences on А1 А1 and А2 А2, respectively; Е1 on А1 А1 and Е2 on А2 А2 be equivalences that satisfy the conditions Е1ЕL, Е2ЕR; f1: A1→B1 and f2: A2 →B2 be characteristic functions of the equivalences Е1 and Е2 respectively. Then there exists a unique predicate L on В1 В2, such that for any хА1 and уА2
( , y) = L( 1( ), 2( )). (9)
Expression (9) represents the general form of a binary predicate Р. Surjections f1 and f2 are called the characteristic functions (left and right) of predicate Р. Predicate L is called the image of predicate P under equivalences Е1 and Е2. Equivalences Е1 and Е2 can be taken as accompanying equivalences ЕL and ЕR, in which case predicate L takes on the simplest form and is called the absolute image of predicate P.
Below, we describe a method for finding the image of a predicate. Given Р, f1 and f2, predicate L is found using the formula:
( , ) = Р( 1−1( ), 2−1( )), (10) where f1-1 and f2-1 are inverse mappings of surjections f1 and f2.
Theorem 1 can be extended to the case of an arbitrary n-ary predicate Р(х1, х2, ..., хn). In this case, the theorem can be formulated as follows.
Statement 11. Let P be a predicate on на А1 А2…Аn, Еic be its accompanying equivalence relation on Ai ( = 1, ), Еi be an equivalence relation satisfying the condition Еi Еic, fi: Ai → Bi be the characteristic function of equivalence relation Еi. Then there exists a unique predicate L on В1 В2…Bn such that for any х1А1, х2А2,…, хnАn
( 1, 2, … , ) = ( 1( 1), 2( 2), … , ( )). (11) Expression (11) represents the general form of an n-ary predicate P.
By representing the predicate in its general first-order form (9, 10), we have achieved that the maximum amount of information carried by predicate P about the relationships between the elements of sets А1 and А2, has been transferred to functions f1 and f2, while the comparator bears minimal burden.
The following property states that by this comparator identification method, п objects f1, f2, ..., fn. can be identified exhaustively (i.e., up to notations). This means that even in the most general case, the depth of analysis of objects using the comparator identification method is not inferior to that of direct identification method.
Statement 12. Let the predicate Р(х1, х2,..., хn) be defined on the set А1 А2 ... Аn, the predicate Р'(х'1, х'2,..., х'n) on the set А'1 А'2 ... А'n, and let the predicates L(v1, v2, ..., vn) and L'(v'1, v'2, ..., v'n) be defined on the sets В1 В2 ... Вn, В'1 В'2 ... В'n respectively, where L is the image of predicate P under equivalences Е1 Е2 ... Еn, and L' is the image of predicate P' under equivalences Е'1 Е'2 ... Е'n. Suppose that the predicates P and Р' are (1, 2,..., n)-isomorphic, and that the predicates Еi and Е'i are i-isomorphic, where i: Ai →A'i, i=1, . Then there exist bijections i: Ai→A'i, i=1, , such that the predicates L and L' (1, 2, ..., n) are isomorphic.
Identification of human intellectual activity using this scheme opens the way to a mathematical description and artificial reproduction of such important aspects of the mind for machine intelligence as perception, understanding, recognition, and awareness. Undoubtedly, subjective states in the human brain are implemented in some, as yet poorly understood, material structures and processes. Clearly, direct identification methods are unacceptable in this case, since images of situations and meanings of texts, being subjective states of a human being, are inherently inaccessible to direct physical measurement.
A first-order two-layer decomposition of predicate is called its decomposition into characteristic functions and images according to its general first-order form. The most important case for practice is the decomposition using accompanying equivalences. There is also a case of using equivalences nested in accompanying equivalences.
It was previously mentioned that any equivalence predicate can be represented in the form (1), where f: А→В is a surjection, B is the set of images of the objects in set А; u= f(x) is the image of object x. The question arises: what form will the predicate E (type of predicate) take if an arbitrary mapping is used instead of a surjection f? To answer this question, the general form of the predicate needs to be slightly modified. The following statement provides the required modified form of the predicate E.
Statement 13. On the variant of the general form of the equivalence predicate. Let F (x, u) be a predicate on A B, corresponding to the surjection f: A → B, f(x)=u. Then the predicate E, whose values for any x, y А are expressed as
( , ) = В ( ( , ) ( , )), (12) is an equivalence relation on A.
And vice versa: For any equivalence predicate E on A, there exist a set B and a well-defined, oneto-one, and surjective predicate F on А В, such that for any x, y А equality (12) holds.
The predicate F (x, u) is called the characteristic predicate of the equivalence. It uniquely determines the characteristic function f of the equivalence. It is important to have a method for constructing the characteristic predicate for any equivalence.
Let us ask the following question: What kind of predicate E will we obtain if, in equality (12) that characterizes the general form of predicate E, we take an arbitrary predicate F(x, u) on A B instead of a surjective, one-to-one, and well-defined predicate F? That is, instead of the surjection f: A → B, we take an arbitrary mapping f(x)=u, acting from A to B. To answer this question, we will consider some concepts.
The predicate E on A А is called quasi-reflexive if it satisfies the condition ∀ ∈ ((∃ ∈ ( ( , ) ∨ ( , ))) ⊃ ( , ). (13)
This property implies that the predicate E is reflexive, but not on the entire set A, but on some of its subset А, defined by the formula ′( ) = ∃ ∈ ( , ).
On the domain АА the predicate E is reflexive, but outside of it, i.e., for any хА or уА, this predicate becomes zero Е (х, у)0.
A reflexive and symmetric predicate is called tolerant. A quasi-reflexive and symmetric predicate are called quasi-tolerant. After the natural restriction of the domain of the predicate E from A to АА, the quasi-reflexive predicate E on A А becomes a reflexive predicate E on АА.
Statement 14. On the general form of a tolerant predicate. Let E be a predicate on BB. Then E is tolerant if and only if there exist a set A and a predicate F on BA, such that a) for any x, yB
( , ) = ( ( , ) ( , )); (14) b) for any xB, the condition uA F (x, u) is satisfied.
The expression of tolerance E given by formula (14) is referred to as its general form, and the predicate F is the characteristic predicate of tolerance. The mapping f corresponding to predicate F is called the characteristic mapping of tolerance.
If all restrictions are removed from the predicate F, then the following theorem on the general form of quasi-tolerance predicate [10] holds, according to which the predicate E on BB is a quasi-tolerance if and only if there exists a set A and a predicate F on BA, such that equality (14) holds for any x, yB.
If we replace the surjection f with an arbitrary (in general, partial and multivalued) mapping in the equivalence scheme E (x, y) =D(f(x), f(y)), we obtain the quasitolerant predicate E (x, y) = u, u B (F (x, u) F (x, u) D(u, u)). The equality predicate D (u, u)=1, if at least one of the values of f(x)=f(y) coincides, and F is a mapping defined everywhere. If we change the surjection f: А → В in the scheme to the function f: А → В, then nothing will change – we will get any equivalence on the left. Thus, quasitolerance is the most general case of a symmetric predicate.
In conclusion, it should be noted that the general form for the most general case of a symmetric predicate, i.e., the quasitolerant predicate, has been obtained. If we remove the last restriction - the symmetry of the predicate E – the following theorem will be valid.
Theorem 2. On the general form of a binary predicate of the 2-nd order. For any binary predicate E on A1 А2, there exist a set B and predicates F1 on А1B and F2 on А2B such that for any x1А1, x2А2, the following equality holds:
( 1, 2) = В ( 1( 1, ) 2( 2, )). (15)
The formula (15), which represents the general form of a 2nd-order predicate Е (x1, x2), can be expressed differently as:
( 1, 2) = В ( 1( 1, ) 2( 2, )) = (ℎ1( 1), ℎ2( 2)), (16) where DВ is a symmetric and reflexive predicate, which we define as follows
1, 2 В ( 1, 2) ( В 1( ) 2( )). (17) It should be noted that in formula (16), h1 and h2 are not functions, as in the general form (9) of a 1st-order binary predicate, but rather mappings, i.e., objects of a more general nature than functions. The predicate F1, which appears in expression (15), is called the left characteristic predicate of the predicate E, and F2 is the right characteristic predicate.
Statement 15. Generalization of the general view theorem of the 2-nd order into n-ar predicates. For any predicate E on А1 А2…Аn, there exist a set B and predicates Fi on АiB ( = 1, ) such that for any х1А1, х2А2, …, хnАn the equality holds: ( 1, 2, … , ) = В ( 1( 1, ) 2( 2, ) … ( , )), (18) where characteristic predicates Fi on АiB ( = 1, ) of the predicate E on А1 А2…Аn can be found using the formula: ( , ) = ∃ 1 ∈ 1∃ 2 ∈ 2 … ∃ −1 ∈ −1∃ +1 ∈ +1 … ∃ ∈ (19)
S( 1, 2, … , , ), where S is a function that assigns different names u to all sets (x1, x2, …, хn), for which Е (x1, x2, …, хn) =1; В is the set of all such names.
Expression (18) represents the general form of the 2nd kind predicate Е (x1, x2, …, хn) on А1 А2…Аn.
In turn, the 2nd-order two-layer decomposition of a predicate is a representation of the predicate in its general 2nd-order form using formula (16). Thus, the representation of any predicate in its general 2nd-order form has been considered. In this form of predicate representation, a certain classifying function appears that assigns names to all sets of variables. This property is very useful in describing the structures of many information objects (such as databases, microchip design) [11, 12].
In the previous sections, two types of two-layer decomposition of predicates were considered – the first and second orders. Their combination results in a three-layer decomposition of the predicate, which completes the construction of the method of multi-layer decomposition of predicates. The first order of decomposition transforms the predicate E into a construction E(x, y) = L(f1(x), f2(y)), where f1 and f2 are functions, and L is a simpler predicate than E (defined on a set of smaller cardinality). The 2-nd order of decomposition transforms the predicate E into a construction of the form E(x, y) = DВ(h1(x), h2(y)), where h1 and h2 are mappings, i.e., objects of a more general nature than functions; DВ is a predicate defined by expression (17), the same for all predicates E, which in some sense is the simplest predicate.
By performing a 2nd-order two-layer decomposition of the predicate L, it can be represented as L(v, w) = DВ(h1(v), h2(w)). Here, the mappings h1 and h2 have a special form: h1(v) = g1-1(v), h2(w) = g2-1(w), where g1: R→В1 and g2: R→В2 are some functions, and h1: В1→R and h2: В2→R are mappings whose inverses are functions g1 and g2 , respectively.
Thus, a three-layer decomposition gives a representation of the predicate E as:
E( , ) = ( 1−1 ( 1( )), 2−1 ( 2( ))), where f1, f2, g1, g2 are some functions. Rewriting formula (20) in a different way gives a more compact form of the multilayer decomposition of the predicate: (20) Е( , ) = (p, q) = ⋁ = .
(21) ∈
We will interpret the obtained result in technical terms. The signal transformer E (Figure 1a) is transformed into a two-layer connection of blocks f1, f2, L with intermediate signals v and w (Figure 1b). The signal transformer L, in turn, is transformed into a two-layer connection of blocks DB, g1-1 и g2-1. As a result, we obtain a three-layer connection of blocks DB, g1-1, g2-1, f1 and f2 with intermediate signals v, w and p, q (Figure 1c). In it, the blocks that implement the functions f1 and g1, f2 and g2, are included in reverse order. Above, it was shown how the functions g1 and g2 are practically sought.
a) Figure 1: Signal transformation schemes b) c)
In this section, a method was developed for constructing schemes that implement arbitrary relations, and relations, as is known, represent a universal tool for modeling any objects and processes. It is important to note that the brain also implements relations, and no other neural structures have been found in the brain [1, 2]. It is natural to assume that the principle of brain operation is also based on a three-layer decomposition of predicates. The mathematical results of the work can be used in systems for automatic processing of textual information (effective support and implementation of databases, knowledge bases, expert systems, etc.), as well as in automated design of new information technologies.
It is well known that predicate logic is a natural and convenient tool for modeling natural language relations. This tool satisfies all the requirements imposed on language formalizations. Moreover, all types of language processing are reduced to solving algebraic equations with different input data. Predicate logic is highly formalized and well-studied. It is designed to describe a very limited part of semantics, the one that deals with the truth or falsity of statements. Nevertheless, its elements – logical connectives, quantifiers, and especially predicates - allow for a broader range of applications.
To enable a computer to understand natural language, it is not only necessary to break down the language into its basic elements and input this information into the computer, but also to create a complete system for natural language processing [13–15].
Special importance in word inflection is played by the endings of word forms (flexional morphemes or simply flexions). In language morphology, there exists a certain dependency (relationship) between flexion and the surrounding text. The task is to mathematically describe the existing dependency, i.e. formalize the concept of flexion. The text surrounding the ending is heterogeneous with respect to it. We will distinguish between the proximate text (bordering the ending directly in the word form) and the distant text (bordering the word form). According to the principle of unambiguousness, the ending always unambiguously depends on its meaning. This principle can be interpreted as a requirement for completeness of the set of features used to select the ending. We will call the set of features complete if it ensures unambiguousness of the selection of the corresponding flexion for any feature values. A set of features that satisfies the completeness requirement will be considered meaningful.
Let us describe the mathematical formulation of the task of flexional processing of complete nonpossessive adjectives in the Ukrainian language. In other words, the task is to formally describe the morphological predicate Р (X, Y, Z), which is the model of flexional processing of Ukrainian adjectives. Thus, it is necessary to form a three-letter ending Z=z1z2z3 of the word form X depending on the set of grammatical features Y. In the Ukrainian language, there are 24 endings of complete non-possessive adjectives. z1 є z2 - z3- = (a3 a4) t10.
Thus, formulas (23) – (27) form a model of the flexion of adjectives in the Ukrainian language. It is evident that processing such a system of equations is extremely difficult and it is necessary to further decompose the original morphological predicate P(X, Y, Z) as described above, i.e., to perform its binarization and then exclude uninformative variable pairs from consideration. Similarly, it is possible to describe the declension of all adjectives, nouns, pronouns, and numerals, as well as the conjugation of verbs. Naturally, formalizing the concept of flexion for each part of speech presents certain problems.
The results obtained in this work can find broad applications in various areas of human activity related to computer and information technologies. The most promising direction is the creation of a new generation computer based on the principles of parallel information processing [18]. Clearly, if various regularities of natural language are described using the two-layer decomposition method of 1st and 2nd order predicates and microprocessors are built on this basis, they will be able to perform the functions of certain structures of human intelligence that participate in the implementation of corresponding aspects of human language activity.
The scientific problem of developing algebraic methods for predicate decomposition for the formal analysis of information processes, particularly for the formal representation of natural language text semantics, has been solved in this work.
A series of theorems have been considered and proven, characterizing the method of comparator identification, which is a method suitable for studying and modeling subjective states of a person. By representing the predicate in its general form of the first kind (9), we have achieved that the maximum possible amount of information carried by the predicate R about the relationships between elements of the sets А1 and А2 has been transferred to the functions f1 and f2, while the comparator L bears a minimal load. This general form of the predicate is a new model of comparator identification. The relationship between the types of isomorphisms of the model of comparator identification (equivalence predicate) and the practical features of measuring input signals has been investigated, which allowed the theory of comparator identification to be developed for a wider class of objects.
We were also able to present the predicate in its general form of the 2nd kind, which gives the researcher even more opportunities for formalizing any relations in logic algebra. The combination of the two-layer decomposition of the 1st and 2nd order made it possible to obtain a three-layer decomposition of any predicate of any dimensionality. This representation of the predicate, and therefore the relation, allows for parallel processing of information, which significantly speeds up the process of formalization and brings it closer to the workings of the human brain.
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