American Journal of Technology and Applied Sciences ISSN (E): 2832-1766 Volume 36, May - 2025 P a g e | 64 www.americanjournal.org ADDITIONAL NEW MATHEMATICAL MODELS OF VERB PHRASEOLOGY FOR COMPUTER TRANSLATION FROM ENGLISH TO UZBEK Muftakh Khakimov 1, Vazira Bekova 1, Ziyoviddin Sirojiddinov 1, Akbar Omonov 2. 1National University of Uzbekistan of name Mirzo Ulugbek, University Street 4, 100104 Tashkent, Republic of Uzbekistan 2Tashkent Pediatric Medical Institute, Department of "Biophysics, Medical Informatics" vazira.bekova16@gmail.com orcid.org/ 0009000865181312. A B S T R A C T K E Y W O R D S It is well known that the process of any computer translation system involves encoding the meaning of the input text in a natural language and re-encoding this meaning in the target language while maintaining semantic consistency with the source text. One approach to achieving this goal is the formalization of the grammatical structures of the natural languages involved in translation systems. More precisely, grammatical structures are based on linguistic rules that determine word structures and their classification according to parts of speech. Analyzing word structures enables the construction of logically sound mathematical models. This article examines how verbs in English and Uzbek form syntactic relationships and how verb categories are classified during the computer translation process. The study analyzes the key aspects of a morpho-analyzer that ensures accurate and meaningful automatic translation. Additionally, it classifies verb-related words and verb-forming affixes in both languages, along with their patterns and mathematical models. Natural language, expandable input language, computer translation, database, English verb category, Uzbek verb category, additional new mathematical models, word weight. American Journal of Technology and Applied Sciences 36, May - 2025 P a g e | 65 www.americanjournal.org Introduction Each natural language (NL) is a complex system consisting of components that are neither mathematically structured nor formalized. However, through the processing of NL, it is possible to identify unstructured elements in the language and formalize them using a linear methodology. This process includes determining word structures, constructing logical-linguistic models based on word and sentence types, and developing mathematical models using a specialized meta-language. This methodology is referred to as the degree of formalization of a language. The degree of formalization, in turn, determines the level of semantic formalization of NL and the accuracy of the algorithm. A superficial understanding of NL formalization—i.e., perceiving a formalized language as an abstract, content-independent structure with a simple logical framework—leads to low efficiency in machine translation [1]. Formalization allows for the segmentation of a language into different components, the analysis of their interrelations, and the characterization of its semantic structure. There are numerous shortcomings in automatic translation systems (Google Translate, Microsoft Translator, DeepL Translator, Yandex Translate, Amazon Translate) when translating between English and Uzbek: Since Uzbek is an agglutinative language, word formation and the semantic structure of sentences depend on the affixes attached to words in each sentence. This difference between English and Uzbek has not been resolved in the automatic translation systems mentioned above. There is no tool for formalizing natural languages to address this issue. For high-quality translation between languages, particularly between English and Uzbek, natural languages have not been formalized, and their various models have not been developed.To address the shortcomings mentioned above, the first author of this paper has developed a specialized expandable input language to formalize natural languages [2, 3]. As a result of creating this expandable input language, linguistic and mathematical models have been developed for parts of speech and general sentence structures in English, Russian, and Uzbek [4, 5]. This study adopts the first approach, focusing on the efficiency of transferring words and sentences from language A to language B, classified as class 0 according to Chomsky's classification [6]. 1 Methods To date, various models for translation programs have been developed by mathematicians and linguists. However, due to the lack of high-quality text translation between English and Uzbek in both directions, this research aims to represent the linguistic capabilities of both languages using an expandable input language. This approach allows for the development of additional new mathematical models by analyzing the grammatical and morphological structures of natural languages belonging to different classes. The transformation from formalizing linguistic capabilities to modeling consists of the following stages: Analysis of parts of speech in natural languages: This stage involves studying the morphological, syntactic, and semantic aspects of word categories such as nouns, adjectives, verbs, pronouns, adverbs, and numerals in both languages. American Journal of Technology and Applied Sciences 36, May - 2025 P a g e | 66 www.americanjournal.org Formation of word chains: By identifying which prefixes, affixes, and suffixes are added to a root to form words in both languages, multiple word chains (A1, A2, ..., A3) are constructed based on word formation. Selection of logically and semantically valid chains: Only root and affix chains that are logically and semantically compatible in both languages are selected, leading to the construction of mathematical models. The functionality of words in natural languages (NL) is reflected in their polysemy. Each word has a specific meaning in a given context, although some words may cause exceptions due to their inherent polysemy. This phenomenon leads to two main approaches when creating mathematical models of natural language: Developing a unified system for linear processing of words and sentences. Considering each word and sentence as an independent structure and processing them accordingly. The following steps are also included in this research: Studying natural language (NL); Implementing the developed expandable input language; Developing a semantic database; Creating a bilingual database of terms and phrases specific to particular scientific fields; Modeling NL; Designing translation program algorithms and developing its software environment. 2.1 Mathematical model (Problem statement) Using an expandable input language is the most effective approach for developing a bilingual translation program for computer translation. Therefore, in this article, new terminal symbols related to the verb category of both languages have been introduced into the terminal symbols of the expandable input language, as presented in Table 1. Based on these terminal symbols, we construct new mathematical models for a computer translation program by analyzing and synthesizing the verb categories of both languages. American Journal of Technology and Applied Sciences 36, May - 2025 P a g e | 67 www.americanjournal.org New terminal symbols added to the expandable input language Table 1 Verb Types Terminal characters Verb Types Terminal characters Notional Verbs G Gerund G13 Action verbs G3 Modal verbs G14 Mental verbs G31 Present tense G15 Perception verbs G32 Future tense G16 Speech verbs G33 Past tense G17 Gesture verbs G34 Active voicei G18 visual perception verbs G35 Reciprocal voice G181 Stative verbs G4 Reciprocal voice G182 Auxiliary verbs G5 Causative voice G183 Defective verbs G6 Reflexive voice G184 Light verbs G7 Passive voice G185 Linking verbs G8 To be” verbs G (D1) G(D2) G (D3) Simple verbs G9 Transitive verbs G19 Compound verbs G10 Intransitive verbs G20 Paired (dual) verbs G11 Regular verbs G21 Infinitive G12 Irregular verbs G22 2.2. Initial data (Descriptions of datasets) Developing an algorithm for the verb category in computer translation software by analyzing the verb categories of two natural languages and incorporating additional new terminologies into the expandable input language. Creating additional new mathematical models for computer translation between English and Uzbek and vice versa using the expandable input language. 2.3 Computational algorithm (Solution method) Algorithm for Translating from English to Uzbek 1. A specialized word database consisting of multiple tables has been built for computer translation between the two languages. Using this specialized database, the following sequential steps are performed. 2. The input word in English is analyzed using data retrieved from the database. 3. The output word in Uzbek is synthesized based on the corresponding data in the database. 4. The input word in English is segmented into its root (K) and affixes, and new mathematical models are constructed accordingly. 5. The output word in Uzbek is synthesized using the newly constructed mathematical models. 6. Based on the newly constructed mathematical models for both languages, weight coefficients for words and affixes are calculated. 7. Translation is carried out using weight coefficients that are either equal or very close to each other in both languages. American Journal of Technology and Applied Sciences 36, May - 2025 P a g e | 68 www.americanjournal.org 8. If no exactly matching weight coefficients are found between the two languages, translation is performed by selecting words with the closest weight coefficients from the specialized database built for computer translation. 2 Results Based on the above-discussed points, we will conduct the following analysis for both languages. (G) (national verbs) — independent verbs — are categorized into action verbs and state verbs in both English and Uzbek, based on what they denote. In English, (G) verbs do not differentiate between action and state verbs in terms of lexical meaning. However, in Uzbek, (G) verbs are classified into six types of action verbs and three types of state verbs, depending on whether they indicate an action or a state in a sentence. This difference highlights the necessity of comparative analysis between the two languages. Although both natural languages have verbs as an independent part of speech, their grammatical and morphological structures differ due to belonging to different language families. Therefore, in this study, a new set of terminological symbols was introduced into the Expandable Input Language (EIL) developed by the first author of the paper, designed to formalize the operation of TT (Translation Tool) for computer translation. Based on these newly introduced terminological symbols, an algorithm was created for each verb category in both languages. In this paper, new mathematical models and numerical coefficients for notional verbs (independent verbs) and auxiliary verbs in both languages are introduced. The weight values of words belonging to each verb type are provided in the tables below. The (G) (notional verbs) in English and Uzbek are similar because, in both languages, (G) verbs perform the main semantic function in a sentence with their inherent meaning. In the field of Information Technology (IT), such verbs are used to describe processes like running programs, managing networks, compiling code, and processing data. The weight values of parts of speech in any natural language are assigned as defined in [7]. Words associated with noun (C) – 0.1; - Words associated with adjective (P) – 0.2; - Words associated with verb (G) – 0.3; - Words associated with adverb (N) – 0.4; - Words associated with pronoun (M) – 0.5; - Words associated with numeral (F) – 0.6. - Words associated with dependent closed word classes (U, D, Y, L) - 0.07. The numbering of word classes shown above helps to calculate the weight of the two languages. For example, tables below indicate that the weight of the declarative, interrogative and negative sentences in the English language, which include pronoun differentiates from those in Uzbek. The mathematical models of English interrogative and negative sentences, the weight of the word-forming affixes also differ from each other in both languages. (MM) - Mathematical model; V1 - word root weight; V3 - weight of affixes in a word. Signs mean: - ⊕ joining operation, ↓- operation of possible “connection” or “not connections” a component following it. $ selection operation, syntax is $[,<1/h>] [3]. American Journal of Technology and Applied Sciences 36, May - 2025 P a g e | 69 www.americanjournal.org In our previous studies on computer translation systems [8], we analyzed noun categories in both languages, developing new mathematical models and computing weight coefficients. In this paper, we conduct analysis and synthesis of the verb category (G) in English and Uzbek, deriving new mathematical models and calculating their weight coefficients for each verb type. For instance, the English verb translates to Uzbek as . The mathematical model for this verb in English is represented as: G = $[i,1-h]G[i] with a weight coefficient of 0.3. G The corresponding Uzbek translation of this English verb is G10, where: is the verb root (G) is the verbal noun-forming suffix To derive the mathematical model for the Uzbek translation, we first analyze the translated word based on [4], formulating its model as: (G, G (A1)) = $[i,1-h] G[i] ⊕↓$[j,1-h[]]G(A1[j]). This model helps the computer translation system process input text (EVX) and generate the translated output text (EVIX). The system first analyzes the input word using the mathematical model, then synthesizes the translation by comparing weight coefficients across both languages. (G, G (A1)) = $[i,1-h] G[i] ⊕↓$[j,1-h[]]G(A1[j]) For the translated Uzbek word G10: The root weight for is 0.3 The suffix weight coefficient for is 0.10221 The results are presented as an example in Table 2. New mathematical model and weight coefficient of the action verb of an independent verb in two languages Table 2 № Ingliz tilida (MM) V1 V3 O’zbek tilida (MM) V1 V3 1. store (G) 0.3 0 Saqla+moq G10 0.2 0.10221 The translations of English (G) verbs into Uzbek are presented in Table 3. For example, the English verb is a compound verb in Uzbek: Amal+ga osh+ir+moq = (amal) asos = K (C), ( -ga) jo’nalish kelisshigi = (X3), (osh) asos = K (C), (ir) G183 (A1) where: (Amal) is the root noun K (C) (-ga) is the dative case suffix (X3) (osh) is another root K (C) (-ir) is the causative verb-forming affix G183 (A1) (-moq) is the verbal noun-forming affix G3 (A1) When translating from English to Uzbek, the structure follows composition rules: (C+G) = G10 Thus, becomes a compound verb (G19) in Uzbek. The mathematical model for the English verb remains (G). However, when translated into Uzbek, it follows the formula: K (C) ⊕ X3 ⊕ K (C) ⊕ G183 (A1) ⊕ G (A1) = G10 A New mathematical model for computer translation was developed by analyzing and synthesizing translations. Weight coefficients for word categories and affixes were previously established by the first author in [4]. This paper utilizes [4] for computing weights across both languages. The English verb has: A weight coefficient of 0.3. Since the Uzbek translation is a compound verb, it contains two root words, requiring separate weight calculations: K (C) - 0.1 + K (C) - 0.1 = 0.2; For the suffixes: X3- 0.10201 + ir - 0.10217 + moq - 0.10221 = 0.3063 Thus, the computer translation system must automatically generate weight coefficients for words and affixes in both languages. Since English and Uzbek verbs are categorized into action and state verbs, weight coefficients for auxiliary verbs in both languages were calculated using the same methodology. To perform English-to-Uzbek translation, the proposed algorithm was extended to include (G3), (G4) and (G5) verb types. The final results are provided in the table below. American Journal of Technology and Applied Sciences 36, May - 2025 P a g e | 70 www.americanjournal.org Algorithmic mathematical models and weight coefficients of words belonging to the verb phrase class in two languages Table 3 № Ingliz tilida (MM) V1 V3 O’zbek tilida (MM) V1 V3 1. (G) store (G) 0.3 0 Saqlamoq G19 0.3 0.10221 2. (G) execute (G) 0.3 0 Amalga oshirmoq (G19) 0.2 0.3063 3. (G3) Install (G3) 0.3 0 O’rnat+moq (G3) 0.3 0.10221 4. G4 Exist (G4) 0.3 0 Mavjud bo’lmoq (G4) 0.6 0.10221 5. G5 I am (G51) 0.5 0.3 Men (G51) 0.5 0 6 G5 He is (G52) 0.5 0.3 U (G52) 0.5 0 7. G5 You (G53) 0.5 0.3 Siz (G53) 0.5 0 8. G5 has (G54) 0.3 0 bor (G54) 0.3 0 9. G5 have (G55) 0.3 0 bor (G55) 0.3 0 10. G5 do (G56) 0.3 0 qilmoq (G56) 0.3 0 11. G5 does (G57) 0.3 0 qilmoq (G57) 0.3 0 3 Discussion Unlike other Turkic languages, Uzbek is considered a low-resource language and has a highly agglutinative structure. A single word can form an entire sentence. There are insufficient rule-based machine translation resources for Uzbek. However, significant progress has been made in Turkic languages such as Turkish and Kazakh in this field. For example, sentiment analysis has been conducted in [10]. In the era of globalization, despite all challenges, Uzbek must become an active participant in the information community. Several studies have been conducted on Uzbek morphology and word stem identification, such as [11, 12]. There are very few studies on formalizing natural languages, but numerous papers focus on different aspects of Turkic languages. For example, sentiment analysis in Kazakh and Russian has been conducted in [13], and ontology- based sentiment analysis of Kazakh sentences has been performed. 4 Conclusion In this paper, new additional mathematical models have been developed for verbs using the Expandable Input Language. The significance of this study lies in achieving high translation accuracy in automatic translation between English and Uzbek. For future research, we plan to analyze the grammar and morphology of other parts of speech in English and Uzbek to develop additional mathematical models for each category. With these new mathematical models based on the Expandable Input Language, we can precisely develop the translation algorithm. These models enable the prediction of word and sentence alignment probabilities between English and Uzbek. From this perspective, developing new additional mathematical models is crucial for achieving high-quality machine translation between English and Uzbek. 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