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27.
BUL ALGEBRASI FUNKSIYALARI SISTEMASINI POST TEOREMASI
ASOSIDA TOʻLIQLIKKA TEKSHIRISH
Xolmatov Javlon Yusupovich
Oʻzbekiston Milliy universiteti Jizzax filiali
Xudoyshukurova Ruxsora Shuxrat qizi,
Ibadullayev Shohruh Sherali oʻgʻli
Oʻzbekiston Milliy universiteti Jizzax filiali talabalari
Annotatsiya:
Maqolada to`liq funksiyalar sistemasiga ta’rif va Post teoremasi
yordamida funksiyalar sistemasini to`liqlikka tekshirishning chinlik jadvali asosida
yaratilgan yangi algoritmlari qo`llanilgan. Murakkablik darajasi o`rtacha bo`lgan misol
shu algoritmlar asosida yechilgan va natijalari olingan.
Kalit soʻzlar:
Funksiyalar sistemasining to`liqligi, yopiq sinflar, Post teoremasi,
Post jadvali, P
0
- nol saqlovchi funksiyalar sinfi, P
1
- bir saqlovchi funksiyalar sinfi, M
- monoton funksiyalar sinfi, S - o’z-oziga ikki taraflama-dual funksiya sinfi, L - chiziqli
funksiyalar sinfi.
Kirish
Mantiq algebrasining
F=(f
1,
f
2,
...f
n
)
funksiyalar sistemasi berilgan bo‘lsin.
Agar mantiq algebrasining istalgan funksiyasini
F=(f
1,
f
2,
...f
n
)
sistemadagi
funksiyalar superpozitsiyasi orqali ifodalash mumkin bo‘lsa, u holda
F
sistema to‘liq
funksiyalar sistemasi deb ataladi.
Mantiq algebrasida hammasi bo‘lib beshta maksimal funksional yopiq sinf
mavjud. Bular quyidagilardir:
-nol saqlovchi,
-bir saqlovchi,
-monoton,
-
o`z-o`ziga ikki taraflama,
-chiziqli.
Post teoremasi yordamida funksiyalar sistemasi to`liqlikka tekshiriladi.
Asosiy qism
Amerikalik olim E.Post tomonidan funksiyalar sistemasi to‘liqligining yetarli va
zarur shartlari topilgan.
Post teoremasi.
F = {f
1
, f
2
, f
3
,.....,f
n
}
funksiyalar sistemasi to‘liq bo‘lishi uchun
bu sistemada
,
,
,
,
maksimal funksional yopiq sinflarning har biriga
kirmaydigan kamida bitta funksiya mavjud bo‘lishi yetarli va zarur (ya’ni
F = {f
1
,
f
2
, f
3
,.....,f
n
}
funksiyalar sistemasi faqat
,
,
,
,
maksimal funksional yopiq
sinflardan birortasining ham qism to‘plami bo‘lmaganda va faqat shundagina to‘liq
sistema bo‘ladi.
Post jadvali
0
P
1
P
M
S
L
0
P
1
P
M
S
L
0
P
1
P
M
S
L
67
F
P
0
P
1
M
L
S
f
1
...
...
...
...
...
...
f
n
Post jadvali asosida quyidagi misolnito`liqlikka tekshiramiz.
Misol:
𝐹 = {(𝑥 → 𝑦̅) ⋅ 𝑧; 𝑥̅ ∨ 𝑦 ↓ 𝑧
̅̅̅̅̅̅; (𝑥 → 𝑦̅)⨁𝑧; 𝑥̅⨁𝑦𝑧}
{
𝑓
1
= (𝑥 → 𝑦̅) ⋅ 𝑧
𝑓
2
= 𝑥̅ ∨ 𝑦 ↓ 𝑧
̅̅̅̅̅̅
𝑓
3
= (𝑥 → 𝑦̅)⨁𝑧
𝑓
4
= 𝑥̅⨁𝑦𝑧
Ushbu funksiyalarning chinlik jadvalini tuzamiz.
x
y
z
f
1
f
2
f
3
f
4
0
0
0
0
1
1
1
0
0
1
1
1
0
1
0
1
0
0
1
1
1
0
1
1
1
1
0
0
1
0
0
0
0
1
0
1
0
1
1
1
0
0
1
1
0
0
1
0
0
1
1
1
0
1
1
1
P
0
- Nol saqlovchilikka tekshirish:
Nol saqlovchilikga tekshirish uchun chinlik
jadvali qiymatlar satrining 1-chi satrdagi (x,y,z – o`zgaruvchilarning qiymatlari
yolg`on bo`lgan) qiymatda
f
funksiya 0 bo’lsa bu funksiya nol saqlovchi funksiya
bo`ladi.
Yoki, analitik usulda ham aniqlaymiz:
•
𝑓
1
(0,0,0) = (0 → 0̅) ⋅ 0 = 0
- nol saqlovchi.
•
𝑓
2
(0,0,0) = 0̅ ∨ 0 ↓ 0
̅̅̅̅̅̅̅ = 1
- nol saqlovchi emas
•
𝑓
3
(0,0,0) = (0 → 0̅)⨁0 = 1
- nol saqlovchi emas
•
𝑓
4
(0,0,0) = 0̅⨁0 ⋅ 0 = 1
- nol saqlovchu emas
P
1
- Bir saqlovchilikka tekshirish:
Berilgan chinlik jadvali qiymatlar satrining
so`nggi ustunidagi qiymatlarida (ya’ni o`zgaruvchilarning barchasi rost qiymat qabul
qilsa)
f
funksiyaning qiymati 1 bo’lsa, bu funksiya bir saqlovchi funksiya bo`ladi.
Analitik usulda:
•
𝑓
1
(1,1,1) = (1 → 1̅) ⋅ 1 = 0
- Bir saqlovchi emas;
•
𝑓
2
(0,0,0) = 1̅ ∨ 1 ↓ 1
̅̅̅̅̅̅̅ = 1
- Bir saqlovchi;
•
𝑓
3
(0,0,0) = (1 → 1̅)⨁1 = 1
- Bir saqlovchi;
•
𝑓
4
(0,0,0) = 1̅⨁1 ⋅ 1 = 1
- Bir saqlovchi.
M – Monotonlikga tekshirish:
chinlik jadvalidan ko’rinib turibdiki biz
ishloyatgan funksiyalar chinlik jadvalidagi qiymatlar satri yuqoridan pastga qarab
o’suvchi yoki o`zgarmas emas. Shu uchun funksiyalar monoton emas.
68
S – O`z-o`ziga ikki taraflamalikka tekshirish:
Funksiyalarni dualikka
tekshirishni chinlik jadval asosida qiymatlar satri n ta bo’lgan funksiya berilgan
bo’lsin. Bu satrlarda quyidagi
n
x
x
=
1
,
1
2
−
=
n
x
x
, ... ,
1
2
2
+
=
n
n
x
x
shartlar bajarilsa dual bo’ladi.
f
1
– Monoton emas, dual emas
f
3
– Monoton emas,dual emas
f
2
– Monoton emas, dual emas
f
4
– Monoton emas ,dual emas
Chiziqlilikga tekshirish:
chiziqlilikga tekshirish uchun bul funksiyasini
jegalkin ko’phadiga yoyamiz. Jegalkin ko’phadida ko’paytma amali qatnashmasa bu
funksiya chiziqli bo’ladi.
f
1
=
(
x
y
)z =(
x
y
)z=
z
→
y
x
=
(
)
z
xyz
z
xy
z
xy
+
=
+
=
1
chiziqli emas
f
2
=
x
v
z
y
(
)(
)
1
1
1
1
1
=
=
=
=
x
xz
xy
xyz
z
y
x
z
y
x
z
y
x
chiziqli emas
f
3
=(
y
x
→
)
z
=
( )
1
=
=
z
xy
z
xy
z
y
x
chiziqli emas
f
4
=
yz
x
=
1
yz
x
chiziqli emas
F
P
0
P
1
M
L
S
f
1
+
-
-
-
-
f
2
-
+
-
-
-
f
3
-
+
-
-
-
f
4
-
+
-
-
-
Post jadvalidan ko'rinib turibdiki, yuqorida keltirilgan funksiyalar sistemasi
to'liq, chunki har bir sistema uchun jadvalda bitta ustunda kamida bitta "-" ishtirok
etgan. Shuni ham ta’kidlash kerakki, har bir sistema uchun bu ustunlar har xil.
Post teoremasi orqali Bul funksiyalari sistemasini to`liqlikka tekshirishda yangi
usuldagi algoritmlarlardan, jumladan funksiyani L-chiziqlilikka tekshirish uchun bir
necha xil usullar mavjud biz yuqorida qo`llagan usul algoritm tuzish uchun va
keyinchalik dasturiy vositalar yordamida funksiyalar sistemasini to`liqlikka tekshirish
uchun minimal talab va maksimal natijadir.
Foydalanilgan adabiyotlar roʻyxati:
1.
H.T.To‘rayev, I.Azizov “MATEMATIK MANTIQ VA DISKRET
MATEMATIKA “ I jild.
2.
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BOYICHA DORILARNI QABUL QILGANLIK DARAJASINI ANIQLASH
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– №. 19. – С. 223-234.
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МНОГОФАЗНОЙ ФИЛЬТРАЦИИ В НЕФТЯНОМ ПЛАСТЕ ПРИ ЕГО
ЗАВОДНЕНИИ //RESEARCH AND EDUCATION. – 2022. – Т. 1. – №. 1. – С. 137-
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CLASSIFIER USING BIOMIMETIC PATTERN RECOGNITION
WITH
CONVOLUTIONAL NEURAL NETWORKS WITH A DYNAMIC THRESHOLD
METHOD FOR MOTION EXTRACTION USING EF SENSORS. International
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SHIFOKOR AXBOROT TIZIMINI YARATISH KONSEPSIYASI. Новости
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BILAN ISHLASH KOMPETENTSIYASINI SHAKILLANTIRISHDA DIDAKTIK
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THE ROLE OF ARTIFICIAL INTELLIGENCE IN ENHANCING THE
PERFORMANCE OF SHELL AND TUBE HEAT EXCHANGERS IN THE
CHEMICAL INDUSTRY
Davronbekov Abdurasul Abdumajidovich,
Abdunazzarov Asliddin Toxir ugli
Fergana Polytechnic Institute, Uzbekistan
Annotation:
Shell and tube heat exchangers are fundamental components in the
chemical industry, responsible for efficient heat transfer processes critical for various
manufacturing operations. As the chemical industry continues to evolve and strive for
increased efficiency and sustainability, the integration of artificial intelligence (AI)
technologies has emerged as a promising avenue to optimize the operation and
performance of these heat exchangers. This paper explores the current state of shell
and tube heat exchangers in the chemical industry and investigates the pivotal role that
AI plays in improving their efficiency, reliability, and overall effectiveness. We delve
into the applications of AI in the design, monitoring, and control of heat exchangers,
highlighting key benefits and challenges associated with its implementation.
Keywords:
Artificial Intelligence (AI), Shell and Tube Heat Exchangers,
Chemical Industry, Heat Exchanger Design, Heat Exchanger Optimization, Machine
Learning, Energy Efficiency, Monitoring and Control, Process Optimization, Data
Analytics, Predictive Maintenance, Computational Fluid Dynamics (CFD), Heat
Transfer Efficiency, Sustainability, Industrial Automation, Fault Detection and
Diagnosis, Thermal Performance, Materials Selection, Energy Savings, Operational
Efficiency
Introduction.
The chemical industry is characterized by its energy-intensive
processes, where heat exchangers play a vital role in heat recovery and temperature
control. Shell and tube heat exchangers are widely used due to their versatility, high
heat transfer efficiency, and robust design. However, traditional approaches to the
design and operation of these heat exchangers often rely on simplified models and
manual adjustments, leaving room for improvement in terms of energy efficiency,
sustainability, and cost-effectiveness.
Artificial intelligence, particularly machine learning and data analytics, has
gained significant attention in recent years for its potential to optimize various
industrial processes, including those involving heat exchangers. This paper explores
the applications of AI in enhancing the performance of shell and tube heat exchangers
in the chemical industry, addressing both theoretical and practical aspects.
