Management and Future Technologies | Volume 2, Issue 3 | № 007

12-rasm. O‘rtacha migratsiya vaqti Shuningdek, barcha iteratsiyalar uchun seansni tiklash va uzilgan sessiyalar uchun taqsimlash funksiyasining grafigi tuzilgan. Bu taklif qilingan algoritmning ishlashiga to‘liq baho beradi. Xulosa Maqolada yuqori yuklamaga ega bo‘lgan tarmoqlarni qurish muammolari tahlili va ularni chekka periferik qurilmalarga yuklamalarni tushirish xizmatlari orqali hal qilish usullari keltirilgan. Tahlil asosida xizmatni optimal tushirish uchun tarmoq operatori va uchinchi tomon xizmat ko‘rsatuvchi provayderlar o‘rtasidagi o‘zaro munosabatlar modeli taklif etiladi. Tarmoq operatori va xizmat ko‘rsatuvchi provayder o‘rtasidagi o‘zaro aloqalarni hisobga olgan holda, chekka hisoblash qurilmalariga xizmatlarni optimal yuklash imkonini beruvchi yangi REx platformasi ishlab chiqildi. Kichik operator tarmog‘idagi ishtirokchilar o‘rtasidagi munosabatlar strukturasini modellashtirish platforma samaradorligini ijobiy baholashni ko‘rsatdi. Genetik algoritmga asoslangan trafikni bashoratlash so‘rovlar cho‘qqisiga chiqmasdan oldin uchinchi tomon xizmat ilovalarini tushirishga yordam berishi isbotlangan, bu ilovalarning ishlamay qolish ehtimolini kamaytiradi va transport tarmoqlarini o‘z vaqtida engillashtiradi. Adabiyotlar 1. Khakimov, A., Dynamic Algorithm for Building Future Networks Based onIntelligent Core Network / Khakimov, A., Muthanna, A., Elgendy, I. A., &Samouylov, K.// In International Conference on Distributed Computer andCommunication Networks Springer, Cham. (pp. 126136). 2020 2. Khakimov, A., Flexible architecture for deployment of edge computing applications. / Khakimov, A., Elgendy, I. A., Muthanna, A., Mokrov, E., Samouylov, K., Maleh, Y.,& Abd El-Latif, A. A.// Simulation Modelling Practice and Theory, 114, 102402. 2022 3. Атея А.А. Интеллектуальное ядро для сетей связи 5G и тактильного интернета на базе программно-конфигурируемых сетей / А.А. Атея, А.С. Мутханна, А.Е. Кучерявый // Электросвязь. – 2019. – № 3. – С. 34-40. 4. .S. Forge, K. Vu, Forming a 5G strategy for developing countries: A note for policy makers,/ S. Forge, K. Vu, // Telecommun. Policy 44 (7) (2020) 101975. 5. S. Dang, O., What should 6G be? / S. Dang, O. Amin, B. Shihada, M.-S. Alouini// Nat. Electron. 3 (1) (2020) 20–29. 6. 32.Parvez, I. A survey on low latency towards 5G: RAN, core network and caching solutions / 100

Parvez, I., Rahmati, A., Guvenc, I., Sarwat, A.I. and Dai, H. // IEEE Communications Surveys & Tutorials, 20(4), pp.3098-3130, 2018. 101

BLOKCHAIN TEXNOLOGIYASIDA FOYDALANILADIGAN KRIPTOGRAFIK XESH FUNKSIYALAR TAHLILI Abduraximov Baxtiyor; Allanov Orif; Turdibekov Baxtiyor Kiberxavfsizlik va kriminalistika kafedrasi, “Muhammad al-Xorazmiy nomidagi Toshkent axborot texnologiyalari universiteti” universiteti baxtiyorturdibekovo4@gmail.com Annotatsiya: Blokcheyn ma’lumotlarni ishonchli tarzda bog‘laydigan tartiblangan bloklar zanjirini saqlaydigan taqsimlangan ma’lumotlar bazasidir. Bog‘langan bloklar ro‘yxatini yaratish uchun xeshlash qo‘llaniladi. Ushbu ishda zamonaviy blokcheyn tarmoqlarida qo‘llanilishi mumkin bo‘lgan xeshlash algoritmlari ishlashining qiyosiy tahlili o‘tkaziladi. Xususan, xesh unumdorligi, bir bayt uchun aylanishlar soni (sikl/bayt), soniyada xeshlangan xabar miqdori (MB/s) va xesh tezligi (KHash/s) bo‘yicha solishtiriladi. Turli xil xeshlash algoritmlarini qiyosiy tahlil qilish markazlashtirilmagan blokcheyn tizimlarini qurish uchun eng munosib nomzodlarni tanlash imkonini beradi. Kalit so‘zlar: xeshlash, butunlik, kriptografik algoritm, blokcheyn, kriptovalyuta. Kirish Blokcheyn texnologiyasiga asoslangan zamonaviy markazlashmagan axborot tizimlari va tarmoqlari tobora keng tarqalmoqda va turli ilovalarda foydalanilmoqda. Shunday qilib, blokcheyn texnologiyalari kriptovalyutalarni qurishda, turli maqsadli va funktsional vazifalarning markazlashtirilmagan tizimlarini qurishda, masalan, asosiy sertifikatlashtirish markazlarini birlashtirgan, elektron identifikatsiyalash va elektron ovoz berishning taqsimlangan markazlashtirilmagan tarmoqlarini qurishda, aqlli shartnomalar deb ataladigan qo‘llabquvvatlanadigan axborot tizimlarini qurishda va boshqalarda foydalanilayabdi. Muayyan xesh funksiyasini tanlash blokcheyn tizimini loyihalashning asosiy vazifalaridan biridir. Quyida blokcheyn texnologiyasida foydalanish mumkin bo‘lgan xesh-funksiyalarning keng qamrovli tahlili keltiriladi. Blokchan texnologiyasi 1-rasmda markazlashtirilgan, markazlashtirilmagan, taqsimlangan tizimlarning o‘zaro farqi keltirilgan. Markazlashtirilgan Markazlashtirilmagan Taqsimlangan 1-rasm. Markazlashtirilgan, markazlashtirilmagan va taqsimlangan tizimlar. 102

1. Markazlashtirilgan. Kompaniyaning aktivlari, majburiyatlari, egalarining kapitali, daromadlari va xarajatlari bilan bog‘liq operatsiyalarni qayd etish uchun barcha hisoblarni o‘z ichiga oladi. Markaziy partiya ma’lumotlar ustidan nazorat qilish huquqiga ega. 2. Markazlashtirilmagan. Har bir tomon hisob yozuviga ega bo‘lib, bir tomonning ma’lumotlar ustidan vakolatini kamaytiradi va ma’lumotlarni kim va qachon o‘zgartirganligi uchun javobgarlikni ta’minlaydi. 3. Taqsimlangan. Har bir tugun har bir elementni qayta ishlaydi va tekshiradi, shu bilan har bir elementning yozuvini yaratadi va uning haqiqiyligi bo‘yicha konsensus yaratadi. Blokcheyn - bir-biriga bog‘langan bloklar zanjiridan iborat (2-rasm). Har bir tranzaksiya tizim ishtirokchilarining aksariyati tomonidan tekshiriladi. Har bir blok tranzaksiyaning har bir yozuvini o‘z ichiga olinadi. U P2P (Peer-to-Peer) protokoli, raqamli shifrlash texnologiyasi, konsensus mexanizmi, aqlli shartnoma va boshqa texnologiyalarni o‘zida birlashtiradi. 2-rasm. Blokchain tarkibidagi bloklar. Quyida blokcheyn tizimlarida qoʻllanilishi mumkin boʻlgan kriptografik xesh-funksiyalar samaradorligini tahlil qilish keltiriladi. Xalqaro miqyosda standartlashtirilgan eng keng tarqalgan va qo‘llaniladigan kriptografik xeshlash algoritmlari, shuningdek, standartlashtirilmagan bo‘lsada, blokcheyn texnologiyasi asosida qurilgan aksariyat zamonaviy markazlashtirilmagan tizimlarda qo‘llaniladigan algoritmlar ko‘rib chiqiladi[1]. CPU uchun xeshlash algoritmlarini o‘zaro taqqoslash Xususan, quyidagi kriptografik xeshlash funksiyalari o‘rganiladi: • ARGON kriptografik algoritmlar oilasi (ARGON2D va ARGON2I xesh funksiyalari), ba’zi kriptovalyutalarda, masalan, MMXVIda qo‘llaniladi; • BALLOON xeshlash algoritmi Deft kripto valyutasida qo‘llaniladi; • BLAKE kriptografik algoritmlar oilasi quyidagi xesh funksiyalardan iborat: BLAKE224, BLAKE256, BLAKE384, BLAKE512. Ushbu algoritmlar oilasi Monero, Blakecoin, Electroneum va boshqalar kabi kripto valyutalarida qo‘llaniladi. • BMW224, BMW256, BMW384, BMW512 kabi xesh-funksiyalardan tashkil topgan BMW Kriptografik Algoritmlar oilasi. Ushbu algoritmlar oilasi CLOAK, Dark, XST, MXT, ONION va boshqalar kabi kriptovalyutalarda qo‘llaniladi; • CUBEHASH algoritmlar oilasi: CUBEHASH224, CUBEHASH256, CUBEHASH384, CUBEHASH512. Bu xesh algoritmlari oilasi X17, X13, X11 va boshqalar kabi kriptovalyutalarda qo‘llaniladi; • DJB-2 xeshlash algoritmi. Bu xeshlash funksiyasi kriptografik emas, lekin u X17 kriptovalyutasida qo‘llaniladi; • ECHO224, ECHO256, ECHO384, ECHO512 kabi ECHO kriptografik algoritmlar oilasi. Ushbu algoritmlar DarkCoin, Navcoin, PinkCoin va boshqalar kabi kriptovalyutalarda qo‘llaniladi; 103

• ED2K xeshlash algoritmi, fayl almashish tarmoqlari va eDonkey2000 va Overnet fayl almashish dasturlarida qo‘llaniladi; • EDONR256 va EDONR512 kabi algoritmlarning EDONR oilasi; • DAGGER-HASHIMOTO xeshlash algoritmi va uni yanada takomillashtirilgan - ETHASH algoritmi. U Ethash algoritmi bilan almashtirilgan; • FUGUE kriptografik algoritmlar oilasi (FUGUE224, FUGUE256, FUGUE384, FUGUE512). Ushbu algoritmlar Navcoin, MinersCoin, KoboCoin va boshqalar kabi kriptovalyutalarda qo‘llaniladi[2]. • kriptografik xeshlash algoritmi GOST34.11-94-256; • GROESTL kriptografik algoritmlar oilasi (GROESTL224 xesh funksiyasi, GROESTL256; GROESTL384; GROESTL512), ba’zi blokcheyn tizimlarida, jumladan, Verge kriptovalyuta loyihasida qo‘llaniladi; • HAMSI224 xeshlash funksiyalaridan tashkil topgan HAMSI kriptografik algoritmlar oilasi, HAMSI256, HAMSI384, HAMSI512. Ushbu algoritmlar Firecoin, Orlycoin, EverGreenCoin va boshqalar kabi kriptovalyutalarda qo‘llaniladi; • Has160 xeshlash algoritmi; • J-H224, J-H256, J-H384, J-H512 kabi J-H xeshlash algoritmlari. Ular Orlycoin, EverGreenCoin, XSH, Dash va boshqalar kabi kriptovalyutalarda qo‘llaniladi; • SHA3 sifatida standartlashtirilgan KECCAK224, KECCAK256, KECCAK384, KECCAK512 kabi KECCAK xeshlash funksiyalari. Hozirgi kunda ushbu algoritmlar oilasi turli kriptovalyutalarda eng keng tarqalganlardan biri hisoblanadi. Ular Quark, METH, SMART, BIFI, GLN, TALK va boshqalar kabi kriptovalyutalarda qo‘llaniladi; • Kupyna xeshlash algoritmi (Kupyna256 va Kupyna512 xesh funksiyalari). • LOSELOSE xeshlash algoritmi. LOSELOSE xeshlash funksiyasi kriptografik emas, lekin u X17 kriptovalyuta algoritmida ishlatiladi; • LUFFA kriptografik algoritmlar oilasi (LUFFA224 xesh funksiyasi; LUFFA256; LUFFA384; LUFFA512). Ushbu xeshlash funksiyalari Dash, Navcoin, Halcyon, Orlycoin va boshqalar kabi kriptovalyutalarda qo‘llaniladi; • LYRA xeshlash algoritmlari: LYRA2RE, LYRA2REV2. Ushbu algoritmlar oilasi ORE, MONA, VTC, RUP va boshqalar kabi kriptovalyutalarda qo‘llaniladi; • xeshlash funksiyalarining MD oilasi (MD4 va MD5); • PANAMA256 kriptografik xeshlash algoritmi; • PROGPOW xeshlash algoritmi. ProgPoW algoritmidagi kriptovalyuta Bitcoin Interest (BCI) hisoblanadi; • ko‘plab zamonaviy kriptovalyutalarda qo‘llaniladigan RIPEMD160 kriptografik xeshlash algoritmi; • SCRYPT kriptografik xeshlash algoritmi. Scrypt ko‘plab kriptovalyutalarda tekshirish algoritmi sifatida ishlatiladi. Algoritm birinchi marta Tenebrix uchun amalga oshirilgan va Litecoin va Dogecoin uchun asos bo‘ldi. • SHA1 kriptografik xeshlash algoritmi; • SHA2 kriptografik algoritmlar oilasi (SHA2-256 va SHA2-512 xesh funksiyalari), Peercoin, Bitcoin, NXT, Namecoin va boshqalar kabi kriptovalyutalarda qo‘llaniladi; 104

• SHABAL256 va SHABAL512 kabi xeshlash funksiyalarining SHABAL oilasi. Ushbu kriptografik algoritm Firecoin, KoboCoin, Orlycoin va boshqalar kabi kriptovalyutalarda qo‘llaniladi; • xeshlash funksiyalarining SHAVITE oilasi (SHAVITE224, SHAVITE256; SHAVITE384; SHAVITE512). Bu algoritmlar Dash kabi kriptovalyutalarda qo‘llaniladi, Bu algoritmlar Dash, PURA, SNRG, EverGreenCoin va boshqalar kabi kriptovalyutalarda qo‘llaniladi; • SIMD224, SIMD256, SIMD384, SIMD512 kabi SIMD xeshlash funksiyalari. Ushbu algoritmlar Dash, EverGreenCoin, MLM, DeepOnion va boshqalar kabi kriptovalyutalarda qo‘llaniladi; • kriptografik algoritmlarning SKEIN oilasi (SKEIN224; SKEIN256; SKEIN384; SKEIN512). Ushbu algoritmlar Hshare, Pura, BitSend, KoboCoin va boshqalar kabi kriptovalyutalarda qo‘llaniladi: • SNEFRU256 xeshlash algoritmi; • STREEBOG xeshlash algoritmi (STREEBOG256 va STREEBOG512 variantlarida). Ushbu algoritm GOST R 34.11-2012 standartida tasvirlangan. Ochiq manba blokcheyn izzz.io va STREEBOG algoritmi bilan amalga oshirilgan kriptografik kutubxonalardan BitCoen, NWP Solution, Buzcoin, NS Platform va boshqalar kabi intellektual shartnomalarda foydalaniladi; • TIGER xeshlash algoritmi, Gnutella, Gnutella2, Direct Connect shuningdek, Phex, BearShare, LimeWire, Shareaza, DC++ va Valknut fayl almashish protokollarida ishlatiladi; • WHIRLPOOL xesh algoritmi, Halcyon, Orlycoin, KoboCoin, Verge va boshqalar kabi kriptovalyutalarda qo‘llaniladi; • “X” turkumidagi xeshlash algoritmlari (X11; X12; X13; X14). Algoritmlarning “X” oilasi xakerlik hujumlaridan yuqori darajada himoyalangan[3]. 1-jadval. 210 bayt matn uchun xeshlash algoritmlarining sinov natijalari. Algoritm ARGON2 D ARGON2 I BLAKE2 56 BLAKE512 BMW256 BMW512 CUBEHA SH-256 CUBEHA SH-512 DJB-2 ECHO- Mezonlar Sikl/bayt MB/s 0,00136 1904191,2 1 Mezonlar Sikl/bayt MB/s KHash/s LOSELO SE LUFFA256 LUFFA512 LYRA2R E 1,4*10-5 2,6*10+7 2,3*10+7 12,12 213,82 208,81 23,46 110,97 108,37 25,67 100,92 98,56 MD5 4,91 527,45 515,09 0,23 806,6 787,69 11,85 218,04 212,93 Algoritm KHash/s 0,001329 1905170,8 0,00136 0,001328 9,8 264,93 258,72 6,74 385,36 376,33 5,53 465,62 454,71 3,05 851,12 831,17 20,13 128,75 125,74 20,22 128,2 125,2 SCRYPT 722,61 1,8 1,75 1,5*10-5 24,05 2,3*10+7 107,86 2,2*10+7 105,33 SHA1 SHA2-256 7,81 13,46 331,83 192,54 324,05 188,03 PANAMA -256 RIPEMD160 105

256 ECHO512 45,03 57,55 56,21 3,32 780,19 761,9 3,69 699,52 683,12 1,86 1387,01 1354,5 17,58 147,46 144 35,86 72,31 70,62 36,76 70,44 68,79 20,86 124,28 121,37 33,12 85,24 83,25 29,49 87,94 85,88 87,49 29,31 28,62 HAS160 5,03 514,51 502,45 JH-256 29,69 87,48 85,45 JH-512 30,35 85,6 83,59 9,64 268,87 262,56 17,41 149,28 145,79 ED2K EDONR256 EDONR512 FUGUE256 FUGUE512 GOST34. 11-94 GROEST L-256 GROEST L-512 HAMSI256 HAMSI512 KECCA K-256 KECCA K-512 SHA2-512 7,94 327,37 319,7 6,15 420,61 410,75 6,31 416,6 406,83 14,33 178,63 174,45 23,77 106,31 103,47 SIMD-256 19,32 134,62 131,47 SIMD-512 23,88 108,45 105,91 4,02 643,69 628,61 4,02 645,67 630,54 89,21 29,02 28,34 30,4 85,4 83,4 31,59 82,2 80, 28 3,51 739,48 722,14 15,57 166,6 162,69 X11 40,95 63,76 62,26 X14 55,84 46,4 45,31 SHABAL256 SHABAL512 SHAVITE -256 SHAVITE -512 SKEIN256 SKEIN512 SNEFRU256 STREEB OG-256 STREEB OG-512 TIGER WHIRE POOL 1-jadvalda Intel Core i9-7980 2.60 GHz 64-bitli hisoblash platformalari uchun xeshlash algoritmlarining sinov natijalari keltirilgan. Natijalar uchta mezon bo‘yicha olingan: • bir bayt uchun hisoblash tizimining aylanishlar soni (Sikl/bayt); • har soniyadagi xabar hajmi (MB/s); • har soniyada hosil qilingan xesh-kodlar soni (KHash/s). Qiyosiy tahlil davomida HashCat dasturidan foydalanildi. Hashcat hozirgi kunda eng tezkor parolni tiklash vositalaridan biridir. Dastur 2015 yilgacha xususiy bo‘lgan, ammo hozirda bepul dasturiy ta’minot sifatida chiqarildi. Linux, macOS va Windows uchun versiyalari mavjud va ular CPU yoki GPU asosida ishlaydi[4]. HashCat dasturi yordamida quyidagi buyruq bilan tezlikni tekshirish bajariladi: hashcat64.exe -m1400 –b Hashcat barcha turdagi xeshlarni samarali sindirish uchun har xil turdagi hujumlardan foydalanishni nazarda tutadi. Bu Maska hujumi, Lug‘at hujumi, Permutatsiya hujumi, Jadvalni qidirish hujumi (faqat CPUda) va boshqalar kabi hujumlar bo‘lishi mumkin[5]. 106

Taqqoslash sinovlari quyidagi xesh algoritmlari va Geforce GTX1050ti 4GB GPU da bitta OpenCL hisoblash yadrosi uchun amalga oshiriladi. Sinov uchun tanlab olingan xesh algoritmlar quyidagilar: GOST 34.311, STRIBOG256, STRIBOG512, KECCAK256, KECCAK512, SHA2 256, SHA2 512, RIPEMD160, Blake2b, Whirlpool. 3-rasm. Hashcatda bitta OpenCL hisoblash yadrosi uchun xeshlash tezligini hisoblash. GPU uchun xeshlash algoritmlari tadqiqot sinovlari HashCat dasturidan foydalangan holda xeshlash algoritmlarining unumdorligi natijalari 2jadvalda keltirilgan. 2-jadval. Grafik video kartaning bitta OpenCL hisoblash yadrosiga to'g'ri keladigan xesh tezligini (KHash/s). Geforce GTX1050ti 4GB (KHash/s) GOST 34.311 STRIBOG256 STRIBOG512 KECCAK256 KECCAK25 6 10889,65 2268,9 2261,5 43400 43733,3 SHA2 256 SHA2 512 RIPEMD 160 Blake2b Whirlpool 148266,6 49550 230516,6 97516,6 10795,5 107

250000 230516 200000 148266 150000 97516 100000 43400 43733 50000 10889 2268 49550 10795 2261 0 GOST 34.311 STRIBOG256 STRIBOG512 KECCAK256 KECCAK512 SHA2 256 SHA2 512 Whirlpool RIPEMD160 Blake2b 4-rasm. OpenCL hisoblash yadrosiga to‘g‘ri keladigan xesh tezligini diagrammasi. 2-jadvalda qo‘llanilgan Geforce GTX1050ti 4GB GPU da bitta OpenCL hisoblash yadrosi uchun turli xil algoritmlar uchun xeshlash tezligi ko‘rsatilgan. 4-rasmda 2-jadvaldagi ma’lumotlar tasvirlangan. Diagrammadan eng tezkor algoritm RIPEMD160 ekanligini ko‘rish mumkin. Ikkinchi va uchinchi o‘rinlarda mos ravishda SHA2 va Blake2b algoritmlari joylashgan. Xulosa Yaxshi xesh funksiyasi ikkita xususiyatni qondirishi kerak: tez hisoblash va to‘qnashuvlar sonini minimallashtirish. Xeshlash blokcheyn tizimining xavfsizligi va ishonchliligini ta’minlaydi, uni to‘qnashuvlar va mumkin bo‘lgan xakerlik hujumlaridan himoya qiladi. Hozirgi vaqtda juda ko‘p xesh funksiyalari mavjud. Haqiqiy muammo blokcheyn tizimi uchun xesh funksiyasini tanlashdir. Xususan, ko‘pchilik taqsimlangan tarmoq loyihalari nisbatan yuqori tezlikda ishlashga ega ishonchli va vaqt sinovidan o‘tgan kriptografik xeshlash algoritmlaridan masalan, KECCAK, SHA2, RIPEMD160 lardan foydalanishadi. Ammo so‘nggi yillarda MD4, EDONR-256, EDONR512, ED2K yoki hatto eng oddiy DJB-2 va LOSELOSE xesh funktsiyalari qo‘llanila boshlandi, balki ular tezroq bo‘lishi mumkin, ammo qaytarilmaslik xususiyatlariga nisbatan ba’zi zaifliklarga ega. Muayyan ko‘rsatkichlarni hisoblashning soddaligi tufayli, qaytarib bo‘lmaydigan xususiyatlarning buzilishini e’tiborsiz qoldirib bo‘lmaydi, ayniqsa ular blokcheyn tarmoqlarining asosiy afzalliklariga asoslangan bo‘lsa 1. 2. Adabiyotlar R.Naveenkumar Raman “The Persistence of Blockchain Technology using Digital Signature and Hash Functions” International Journal of All Research Education and Scientific Methods (IJARESM), ISSN: 2455-6211 Volume 8, Issue 11, November-2020. E. Swathi, G. Vivek, and G. S. Rani, “Role of Hash Function in Cryptography”, in National Conference on Computer Security, Image Processing, Graphics, Mobility andAnalytics, India,2016, pp.10-13. 108

3. 4. 5. A. Kuznetsov, I. Oleshko, V. Tymchenko, K. Lisitsky, M. Rodinko, A. Kolhatin, “Performance Analysis of Cryptographic Hash Functions Suitable for Use in Blockchain”, International Journal of Computer Network and Information Security (IJCNIS), Vol.13, No.2, pp.1-15, 2021. DOI: 10.5815/ijcnis.2021.02.01 B. Seok, J. Park, J.H. Park. A Lightweight Hash-Based Blockchain Architecture for Industrial IoT / Byoungjin Seok, Jinseong Park, Jong Hyuk Park// Appl. Sci. 2019, 9, 3740; doi:10.3390/app9183740. Hany F. Atlam, Ahmed Alenezi, Madini O. Alassafi, Gary B. Wills, "Blockchain with Internet of Things: Benefits, Challenges, and Future Directions", International Journal of Intelligent Systems and Applications, Vol.10, No.6, pp.40-48, 2018. 109

DEVELOPMENT OF A LEMMATIZATION ALGORITHM: INTERPRETING OPEN COMPOUND WORDS Maksud Sharipov; Ogabek Sobirov Urgench State University named after Abu Rayhan Biruni maqsbek72@gmail.com, sobirovogabek0409@gmail.com Abstract: Open compound words pose significant challenges in the lemmatization process for morphologically rich and low-resource languages such as Uzbek, where multi-word expressions often carry unified meanings but appear as separate tokens. This research aims to address this issue by developing a lemmatization algorithm tailored specifically for Uzbek open compound words, which are underrepresented in existing NLP resources. The method relies on a dictionary-based approach supported by a finite-state machine (FSM) to recognize and normalize compound expressions without requiring prior tokenization. The compiled lexicon includes over 900 compound units across various parts of speech, including verbs, adverbs, pronouns, interjections, and auxiliary verb constructions. Evaluation of the algorithm on structured textual data demonstrates its ability to accurately identify compound forms and map them to their base lemmas with high precision. The approach does not require large annotated corpora or complex machine learning models, making it transparent, reproducible, and adaptable for resource-scarce languages. These findings highlight the potential of rule-based lexical methods combined with finite-state modeling in improving lemmatization performance and suggest promising directions for further development of NLP tools for the Uzbek language and other agglutinative languages with similar structural characteristics. Keywords: Natural Language Processing (NLP), compound words, morphological analysis, text normalization, lemma, lemmatization. Introduction Natural Language Processing (NLP) has witnessed remarkable advancements over the past decades, fundamentally transforming the way humans interact with machines. One of the key components of many NLP systems is lemmatization, the process of reducing inflected or derived words to their base or dictionary form, known as the lemma. For instance, the lemma of the words “baxt” - (happiness), “baxtda” - (in happiness), “baxtdan” - (from happiness), “baxtga” - (to happiness), “baxtning” - (of happiness) is “baxt” - (happiness). Accurate lemmatization is vital for a wide range of language technologies, including search engines, text summarization, information retrieval, and machine translation. However, the lemmatization of open compound words—multiword expressions that are syntactically independent but semantically unified—remains a significant challenge, especially in agglutinative and morphologically rich languages. In English, compound words are commonly categorized into three types: open compounds (e.g., ice cream), closed compounds (e.g., firefighter), and hyphenated compounds (e.g., up-to-date). These structures are generally well-documented in English lexicons and can be handled by established rules or dictionary-based methods in NLP systems. In contrast, the Uzbek language recognizes two primary forms of compound words: merged compounds (e.g., “balandparvoz” - high-flying) and syntactic compounds comprised of two or more separate words (e.g., “sotib olmoq” – to buy, “qaror qabul 110

qilmoq” - to make a decision). The latter, often referred to as “open compounds” present unique challenges for automatic lemmatization, as they are not explicitly marked and require contextual or syntactic cues for correct interpretation. Although significant work has been done in the field of NLP for many languages, Turkic languages, which are morphologically rich and low-resource, particularly Uzbek, have not seen sufficient work specifically in the area of lemmatization. However, it is worth reviewing the work that has been carried out in the Uzbek language segment within the scope of NLP. This article [1] discusses the pros and cons of lemma, stemming, and tokenization techniques in the field of corpus linguistics. In this article [2], it is argued that the lemmatization process is complicated due to the morphology of the Uzbek language, the richness of agglutinative aspects, lexical ambiguity, and the methods of lemmatization of word forms with different structures are presented. A variety of machine learning algorithms were employed in this approach, including Naive Bayes, K-Nearest Neighbors (KNN), Support Vector Machine (SVM), Decision Tree, and Random Forest [3]. This article [4] discusses the development of a linguistic resource for the lemmatization of Uzbek language verbs. It also examines the processes involved in gathering this resource and its importance for Uzbek verb forms. In this work [5], authors describe how they address the difficulties posed by harmonizing disparate lemmatization standards in a project that seeks to use the Linked Data paradigm to connect Latin language resources, in particular, the study discusses how a lemmatized and syntactically annotated corpus, a word formation lexicon, and its lexical foundation were added to the Knowledge Base. This work [6] introduces a novel hybrid technique to lemmatization of Turkish inflected words by utilizing graph models based on morphological similarity, which is gaining popularity lately. To do this, a unique Turkish similarity function is created to link the related word forms. A Turkish lemmatization dataset that has been double-checked is used to train and evaluate the suggested model. They evaluated how well various lemmatization techniques performed while retrieving information from a collection of Turkish texts. Additionally, they compared one fixed length truncation strategy and three distinct lemmatizers across a collection of Turkish texts. The first one is based on finite state technology, another one is dictionary-based, the third one is a simple dictionary-based top-down parser and the last one is truncation of words at fixed length [7]. In the lexicon and rule-based lemmatization method for Somali language, lexicons and derived term dictionaries were formed and enriched with rules, tested on different documents and achieved the best results of 95.87% [8]. In the work done for Bangla language, we can see that they also developed a lemmatization algorithm based on the dictionary and partially using rules and achieved 96.36%, developed a program based on the algorithm and made it open source together with the dictionary [9]. Looking at the results of the work done for Urdu language, we can also see the work done on the dictionary, they used the dictionary not only for lemmatization but also for tagging the part-of-speech (POS) and achieved the best 76.44% [10]. This study aims to fill this gap by proposing a lemmatization algorithm specifically designed to interpret and process open compound words. By doing so, we contribute to more accurate text analysis in languages where open compounds play a central role in communication. Methodology This study introduces a dictionary-based lemmatization algorithm developed to handle tokenized compound expressions in the Uzbek language. Rather than operating at the raw text level, 111

this approach assumes that the input has already been tokenized. The algorithm focuses on identifying multi-token compound expressions and converting them to their correct lemma forms using an exact-match lexicon. The methodology is designed to be transparent, rule-driven, and highly reproducible, particularly for morphologically rich and low-resource languages like Uzbek. The core linguistic resources consist of curated lexicons grouped by syntactic category. These include 383 verb compound expressions (e.g., ado et, hosil ber, toza qil), 71 adverbial compounds (e.g., bir dam, bir kam, bir payt), 26 compound pronouns (e.g., har bir, har kim, har narsa), and 12 interjections (e.g., assalomu alaykum, bor bo‘ling, omon bo‘ling). A key part of the dataset is the inclusion of 485 Auxiliary Verb Phrases (Ko‘makchi Fe’lli So‘z Qo‘shilmalari – KFSQ), such as “buzil ket”, “chiz tur”, and “ishla yur”, which are frequent in colloquial and literary Uzbek and require special handling due to their syntactic structure and meaning [Table 1]. Table 1. Types and frequencies of compound words in Uzbek. Compound word type Number Example Compound verbs 383 Ado etmoq (to carry out) Compound adverbs 71 Bir safar (one time) Compound pronouns 26 Ana shu (thet one) Interjections 12 Xush kelibsiz (welcome) Auxiliary Verb Phrases (Ko‘makchi 485 Olib chiqmoq (to take out) Fe’lli So‘z Qo‘shilmalari – KFSQ) Given the discrete and deterministic nature of this problem — matching fixed-length token sequences to known entries in a lexicon — the most appropriate mathematical model for this task is the Finite-State Machine (FSM). Unlike sequence-to-sequence neural models or statistical vectorization methods like TF-IDF, FSMs offer high efficiency, interpretability, and minimal data dependency, making them especially suitable for dictionary-based systems where the input space is predefined and rule-governed. FSMs can be used to scan token streams in a stateful manner, detecting the presence of compound patterns and handling variable-length matches (bigrams, trigrams, etc.) with minimal computational overhead [Figure 1]. Figure 1. FSM for compound n-gram detection in token sequences (2–4 grams). The algorithm processes each input sentence as a sequence of tokens and uses a sliding window to match n-grams (of size 2 to 4) against the compound lexicon. If a match is found, the matched phrase is replaced with its lemma and marked as complete. Overlapping matches are resolved based on maximal length preference, ensuring longer compounds are prioritized to avoid partial lemmatization. Algorithm (descriptive form) [Figure 2]: 1. Receive a tokenized sentence as input. 2. Begin scanning from the first token. 112

3. For each token position, check for n-gram matches (n = 4 down to 2) against the compound lexicon. 4. If a match is found, replace the entire n-gram with its corresponding lemma from the dictionary. 5. Skip ahead by n tokens and continue scanning. 6. If no match is found, retain the original token and move to the next position. 7. Repeat until the end of the token sequence is reached. 8. Return the updated sequence with compound expressions lemmatized. This algorithm is language-agnostic but is optimized through the use of a domain-specific lexicon. The data structure used for the compound dictionary is a Python hash map (i.e., dict) for O(1) average lookup time. The code supports integration with broader NLP pipelines and can be extended to include POS tagging, syntactic parsing, or neural reranking modules in future stages. All resources — including the compound lexicon, auxiliary verb phrase list, and implementation scripts — are planned to be released under an open-source license in a GitHub repository upon publication. The system has no external dependencies beyond standard Python libraries and is thus portable and reproducible across environments. This deterministic, lexicon-driven method not only ensures linguistic precision but also facilitates transparency, error analysis, and domain-specific customization. Figure 2. Lemmatization of compound words using sliding n-gram matching. Results 3.1. Lexicon Coverage and Matching Accuracy The compiled dictionary of compound expressions, consisting of 977 entries across verb phrases, adverbials, pronouns, interjections, and auxiliary verb phrases (KFSQ), was evaluated on a test corpus of 25,000 manually tokenized Uzbek sentences. The corpus was drawn from literary, 113

journalistic, and conversational sources to ensure variety. The dictionary-based FSM lemmatizer achieved a compound recognition rate of 92.6%, indicating that the majority of compound expressions in the test data were successfully identified and matched with entries in the lexicon. 3.2. Lemmatization Precision Among the correctly identified compounds, lemmatization precision was measured at 98.2%, with only a few errors caused by syntactic ambiguities or overlapping expressions. For example, the expression "chiqib ketdi" was sometimes misclassified when preceded by noise tokens or punctuation. However, the system consistently returned correct lemmas for standard expressions such as "sotib olibdi" → "sotib olmoq" and "qaror qabul qilganmish" → "qaror qabul qilmoq". 3.3. Error Analysis The 7.4% of unmatched compounds were mainly due to: • Variations in word order or intervening particles (e.g., “tezda sotib ol” vs. “sotib ol”), • Lexical innovations or informal constructs not present in the static lexicon, • Errors in tokenization (although not the focus of the algorithm). These limitations suggest potential extensions, such as fuzzy matching strategies, morphological normalization of intervening tokens, or dynamic expansion of the compound dictionary using corpus mining. 3.4. Processing Efficiency Thanks to the FSM-based implementation, the algorithm processed the full test corpus in less than 2.1 seconds, demonstrating suitability for real-time applications. Unlike neural models, which often require GPU acceleration and substantial memory, this approach maintained low computational overhead and high interpretability, crucial for resource-constrained NLP environments. 3.4. Example Outputs A few illustrative examples of input and output from the system are presented below: Input Tokens … sotib olgan mashinasi … … ular qaror qabul qilishgandi. Assalomu alaykum dedim … Lemma Output …, sotib olmoq, mashina, … …, u, qaror qabul qilmoq. assalomu alaykum, demoq, … The results confirm the effectiveness of using an FSM guided by a well-structured compound lexicon for high-precision lemmatization of open compounds in Uzbek. This approach not only provides reproducibility and transparency but also meets practical needs in tasks such as information retrieval, machine translation, and chatbot interaction in morphologically complex languages. Discussion The results obtained from the development and evaluation of a dictionary-based lemmatization algorithm for open compound words in the Uzbek language demonstrate the effectiveness of rule-governed symbolic approaches when sufficient lexical resources are available. The high recognition rate (92.6%) and lemmatization precision (98.2%) affirm the initial hypothesis that compound word lemmatization in morphologically rich, agglutinative languages such as Uzbek can be successfully implemented using a finite-state model guided by a manually curated lexicon. 114

These findings align with earlier research conducted for other morphologically complex languages such as Turkish, Somali, Bangla, and Urdu [6–10], where dictionary and rule-based techniques have yielded high accuracy. However, our work makes a novel contribution by focusing explicitly on open compound expressions, which remain under-addressed in many NLP pipelines for Turkic and low-resource languages. Prior works on Uzbek NLP have largely concentrated on single-word lemmatization or limited classes of derivational morphology [1–4], without dedicated treatment of syntactic compound units. This study addresses that gap by offering a systematic solution to a structurally challenging linguistic phenomenon. Furthermore, the use of a finite-state machine (FSM) as the core computational model proves to be a pragmatic design choice. While deep learning approaches such as sequence-to-sequence (seq2seq) models or transformer-based systems might offer robustness to noisy input and context variability, they typically require large annotated corpora and extensive training. In contrast, our approach ensures interpretability, scalability, and resource efficiency, making it well-suited for lowresource environments where computational power and training data are limited. In a broader context, accurate lemmatization of open compounds directly impacts the quality of higher-level NLP tasks. For instance, in information retrieval systems, the failure to recognize expressions like "sotib olmoq" as a single semantic unit can significantly degrade search relevance. Similarly, in machine translation, the misinterpretation of compound verbs or auxiliary phrases can lead to grammatical and semantic inaccuracies. In conclusion, this study not only confirms the viability of a structured, rule-based model for handling open compound lemmatization in Uzbek but also sets a foundation for further hybrid or adaptive systems. The approach is scalable, interpretable, and replicable, and has the potential to support the development of more advanced NLP applications in other under-resourced agglutinative languages as well. Conclusions This study presents a dedicated approach to the lemmatization of open compound words in the Uzbek language, a challenge that remains largely unresolved in the domain of Natural Language Processing (NLP) for low-resource and morphologically rich languages. By employing a dictionarybased method supported by a finite-state machine (FSM), we demonstrated that high-accuracy lemmatization can be achieved without the need for large annotated corpora or complex machine learning infrastructures. The proposed method successfully leverages a curated lexicon of compound expressions— including verb phrases, adverbial groups, pronouns, interjections, and auxiliary verb constructions—to perform accurate lemmatization at the token level. Experimental results affirm the robustness and scalability of this approach, particularly in structured or domain-specific texts where compound usage is consistent and well-defined. The significance of this work lies not only in its direct contribution to Uzbek NLP resources but also in its broader implications for processing compound structures in other agglutinative languages. Accurate compound lemmatization enhances downstream applications such as information retrieval, sentiment analysis, syntactic parsing, and machine translation, where semantic integrity across multi-word expressions is critical. 115

While the system achieves strong performance, it remains dependent on the completeness and maintenance of the underlying lexicon. As such, one of the key directions for future work includes the automated expansion of the compound word database using corpus-driven techniques, as well as the integration of statistical or neural methods to support more dynamic and adaptive compound recognition. In summary, this research provides a reproducible, transparent, and linguistically grounded method for lemmatizing Uzbek open compound words. It lays the groundwork for building more advanced tools that can process natural language in a nuanced and semantically faithful manner, thereby enriching the digital ecosystem for the Uzbek language and beyond. Literatures 1. Abdullayeva O. Korpus lingvistikasida lemmalashtirish, stemming va tokenlashtirish jarayoni. O’zMU xabarlari 2021; 1: 240–243. 2. Elov B, Khamroeva S, Khusainova Z. OʻZBEK TILIDAGI TURLI TUZILISHLI SOʻZLARNI LEMMALASH USULLARI. 2023; 13–19. 3. Sharipov M, Sobirov O, Matsalayev A. Creating a database of uzbek words and part-ofspeeches using machine learning methods, and lemmatization. DTAI – 2024: “Digital transformation and artificial intelligence: problems, innovations and trends” I international scientific - practical conference, 2024, 291–294. 4. Sobirov O, Sharipov M. Creating a linguistic supply for lemmatization of uzbek verbs. DTAI – 2024: “Digital transformation and artificial intelligence: problems, innovations and trends” I international scientific - practical conference 2024; 1: 187–189. 5. Mambrini F, Passarotti M. Harmonizing different lemmatization strategies for building a knowledge base of linguistic resources for Latin. Proceedings of the 13th Linguistic Annotation Workshop, 2019, 71–80. 6. Arslan E, Orhan U. Graph-based lemmatization of Turkish words by using morphological similarity. 2016 International Symposium on INnovations in Intelligent SysTems and Applications (INISTA), 2016, 1–5. 7. Ozturkmenoglu O, Alpkocak A. Comparison of different lemmatization approaches for information retrieval on Turkish text collection. 2012 International Symposium on Innovations in Intelligent Systems and Applications, 2012, 1–5. 8. Mohamed SA, Mohamed MA. Lexicon and Rule-based Word Lemmatization Approach for the Somali Language. 2023. 9. Afrin S, Chowdhury MdSM, Islam MdE et al. BanLemma: A Word Formation Dependent Rule and Dictionary Based Bangla Lemmatizer. 2023. 10. Shaukat S, Asad M, Akram A. Developing an Urdu Lemmatizer Using a Dictionary-Based Lookup Approach. Applied Sciences (Switzerland) 2023; 13. 116

DOIRANI GEOMETRIK SHAKILLAR BILAN TO‘LDIRISH YORDAMIDA OROL DENGIZINING FRAKTAL O‘LCHOVINI ANIQLASH Saidkulov Elyor Abdullayevich, Ibrohimova Zulayho Ergash qizi Muhammad al-xorazmiy nomidagi toshkent axborot texnologiyalari universiteti Samarqand filiali, elyorsamtiut9977@gmail.com Annotatsiya: Ushbu maqolada tabiatdagi fraktal shakllar, ularning xususiyatlari, shuningdek, fraktal o‘lchov tushunchasi haqida fikrlar berilgan. Fraktal geometrik shakllarning geofizik kelib chiqish hodisalari hamda murakkab shakllarni tavsiflash uchun potensial foydali vositalar taqdim etilgan. Fraktallarning geografik sohalarda ko‘plab qo‘llanilishiga qaramay, turli fraktal hisoblash algoritmlaridan kelib chiqadigan nomuvofiq natijalarni keltirib chiqaruvchi muammolari haqida fikrlar keltirilgan. Fraktal o‘lchov birinchi bo‘lib geometrik jihatdan murakkab shakllarni tavsiflovchi koyeffisiyent sifatida kiritilgan bo‘lib, ular uchun to‘liq chizilgan rasmdan ko‘ra detallar muhimroqdir. Kalit so‘zlar: Fraktal, fraktal o‘lchov, geometrik o’lchov, doirani qoplash, lokal-dispersiya usuli, Richardson effekti, masshtab, xarita. Kirish Fraktallarning ayrim asosiy xususiyatlarini va tabiiy geografiya sohasiga aloqadorligini bayon qilish kerak. Ma’lumki, fraktallar o‘ta murakkab shakllarga ega bo‘lib, ular fraktal gʻoyalarni va haqiqiy hayotiy dasturlarni tushunish uchun muhim bo‘lgan noyob obyektlardan tashkil topadi.Turli chegara hududlarini fraktal o‘lchov bilan tavsiflanishi mumkin. Biroq tasvir ma’lumotlari cheklangan bo‘lsa, chegaralarning fraktal o‘lchovlarini muntazam ravishda hisoblash birmuncha murakkab. Ammo chegara o‘lchovi ba’zan taqriban hisoblanadi. Ushbu ilmiy tadqiqot ishida fraktallar nazariyasiga asoslangan tadqiqotlar yordamida chegara o‘lchovlarini hisoblash uchun bir qator amaliy geometrik modellarni ishlab chiqishga qaratilgan. Bundan tashqari, chegara o‘lchovi va shakl indekslari hamda masshtablash o‘rtasidagi bog‘liqlik va farqlarni ko‘rish mumkun. Asosiy Qism Fraktal xususiyatli obyektlar fraktal o‘lchovga ega bo‘lib, ushbu tushunchaning asosiy va muhim yondashuvi geometrik o‘lchov munosabatlaridir. Yevklid fazosidagi geometrik o‘lchov munosabatlari o‘lchov izchilligi va prinsipidan kelib chiqadi. Fraktal o‘lchov (masalan, uzunlik) boshqa o‘lchovga (masalan, maydon) proportsional emas, ya’ni ular bir xil fazoviy o‘lchovga ega emas. O‘lchov bir xilligi printsipidan biz Yevklid geometrik o‘lchov munosabatlarini olishimiz mumkin, bu fraktal geometrik o‘lchov munosabatiga umumlashtirilishi mumkin. Fraktal o‘lchov munosabatlaridan fraktal o‘lchov va xaritadagi masshtab ko‘rsatkichlarini olish mumkin. Xaritadagi masshtablash munosabati umumlashtirilgan fraktal o‘lchov munosabati sifatida qaralishi mumkin. Turli xil geometrik o‘lchov munosabatlari orasida umumiylik mavjud bo‘lib, ikki o‘lchovli fazoga kiritilgan o‘ziga o‘xshash shakllarning chegaraviy o‘lchovini olish uchun foydalaniladi. Sirtlarning chegara uzunligini masshtablash parametri va chegara o‘lchovi fraktal o‘lchov munosabatlaridan olinishi mumkin. Agar yetarlicha ma’lumotlar mavjud bo‘lsa, turli fraktal parametrlarni hisoblash mumkin. Biror hudud chegarasini misol qilib oladigan bo‘lsak, katakchalar usulidan foydalangan 117

holda fazoviy tuzilmani o‘rganish va fraktal o‘lchov yordamida hududning kengayish darajasini aniqlash mumkin. Agar hududning chegara uzunligini aniqlash uchun yetarli ma’lumot mavjud bo‘lmasa, u holda chegara uzunligini fraktal o‘lchovini aniqlash orqali hisoblash mumkin. Yevklid fazosiga murojaat qilib, fraktal o‘lchovni hisoblashning bir qator formulalarini qurish mumkin. Bunda olingan hudud tasvirini oddiy uchburchak, kvadrat, muntazam olti burchak va doira shaklli qismlarga bo‘lib, fraktal o‘lchov aniqlanadi. Taqribiy geometrik modellarga asoslangan fraktal o‘lchov qiymatlari aslida fraktal ko‘rsatkichlar bo‘lib, tegishli ma’lumotlar yo‘qligi sharti bilan fraktal o‘lchovni almashtirish uchun ishlatilishi mumkin. Bunda notekis abyektning chegara chizig‘i yopiq prefraktal egri chiziq hisoblanadi. Prefraktal fraktalga o‘xshash obyekt bo‘lib, u haqiqiy fraktal emas. Haqiqiy fraktallarda chiziqning uzunligi cheksizdir. Agar shahar konverti kabi tartibsiz chegara haqiqiy fraktal egri chiziq bo‘lsa, fraktal ko‘rsatkichlar va shakl indekslari uchun oddiy formulani keltirib bo‘lmaydi. Prefraktallardan olingan aniqliklardan foydalanib, berilgan mos yozuvlar shaklidan chegara o‘lchovining bir qanch taxminiy geometrik modellarini topishimiz mumkin. Yo‘naltiruvchi shakllar ikki o‘lchovli Yevklid fazosida aniqlangan muntazam geometrik figuralarning ba’zi turlari quyidagi doira, muntazam uchburchak, kvadrat, muntazam olti burchakli va muntazam yulduzsimon shakillarda bo‘lishi mumkin. Uchburchakni Yevklid geometriyasining asosiy shakli deb hisoblash mumkin. Kvadratchalar, to‘rtburchaklar, trapezoidlar, doiralar, ellipslar va tartibsiz shakllarni o‘z ichiga olgan barcha geometrik shakllarni uchburchaklarga qisqartirish mumkin. Shunday qilib tartibsiz chegaralar uchun fraktal o‘lchovlarni aniqlashning birinchi formulasi muntazam uchburchaklardan foydalanish tartibi keltirilgan bo‘lib bunda tomon uzunligi r bo‘lgan teng tomonli uchburchak uchun A maydoni va perimetri P quyidagicha ifodalanishi mumkin. A= a sin( 3) 2 2 r (1) P = 3r (2) b c d 1-rasm. Fraktal o‘lchovlarni aniqlashda qo‘llanilgan to‘rtta (a) uchburchak, (b) kvadrat, (c) olti burchak va (d) yuldizsimon belgilar. Notekis chegaralarning fraktal o‘lchovining formulalarini ishlab chiqish uchun to‘rtta (a) uchburchak, (b) kvadrat, (c) olti burchak va (d) yuldizsimon belgilar olingan. Bu shakllardagi doiralar standart doiralar bo‘lib ulardan biz chegara o‘lchovining taxminiy formulalarini va shakl indekslari uchun qiymatlar olishimiz mumkin. Shunday qilib A maydonni va perimetri P o‘rtasidagi geometrik o‘lchov munosabatlarida birinchi va ikkinchi formulani birlashtirib r ni yo‘qotib olish mumkin. 118

12 2A r =( ) = P3 sin( 3) (3) Faraz qilaylik, uchta Yevklid tomoni uchta fractal chiziq bilan almashtirildi va bu chiziqlarning fraktal o‘lchovi D bo‘lsin. Bunday holda, muntazam shakl tartibsiz shaklga o‘zgaradi. Shunday qilib, Yevklid geometrik o‘lchovni aniqlovchi uchunchi formulaning fraktal geometrik o‘lchov munosabati bilan almashtirilishi kerak. 1 1 2A 2 P D sin( 3) = 3 (4) Bu yerda D chegara chizig‘ining fraktal o‘lchovini bildiradi. 4-formuladan foydalanib obyektning chegara chiziqlarining fraktal o‘lchovini aniqlash formulasini quyidagi tarzda yozish mumkin. P P P 2 ln 2 ln 3 = 3 3 D= 4A ln 2A 3 ln sin 3 (5) Agar olingan obektning shakli uchburchakka o‘xshash bo‘lsa chegara chizig‘ining fraktal o‘lchamini 5-formula bilan aniqlash mumkin. Ikkinchi formulani kvadrat asosida qurish mumkin. Kvadrat oddiy va muntazam bo‘lib, agar tomon uzunligi r ma’lum bo‘lsa, uning maydoni A va perimetri P ni hisoblash oson. Maydon va perimetr formulalari quydagi formulada berilgan. A = r2 (6) P = 4r (7) Oltinchi va yettinchi formulalarni birlashtirish natiyjasida A P perimetri oʻrtasidagi geometrik oʻlchov munosabati quyidagicha hosil qilinadi. 1 r = A2 = P 4 maydoni va (8) Agar kvadrat tomonlari D fraktal o‘lchovli fraktal chiziqlar bilan almashtirilsa, geometrik o‘lchov munosabati esa yuqoridagi 8-formula bilan almashtiriladi. 1 P D A = 4 1 2 (9) 10-formuladan fraktal o‘lchovni aniqlashda formulaga quyidagicha o‘zgartirishi kiritish mumkin. P 2ln 4 D= ln ( A) 119 (10)