Abstract: Lectures on mathematical logic and theory of algorithms. 1

All universities Columbia University Novikontas Maritime College Khakass State University named after. N.F. Katanova Khakass Technical Institute (branch of Siberian Federal University) Caspian State University of Technology and Engineering named after. Yessenov Aktobe Regional State University named after. K. Zhubanov West Kazakhstan State Medical University named after. M. Ospanova Almaty Management University Almaty State College of Energy and Electronic Technologies Almaty Technological University Almaty University of Energy and Communications Kazakh Academy of Transport and Communications named after. M. Tynyshpayev Kazakh Head Architectural and Civil Engineering Academy Kazakh National Academy of Arts named after. T. Zhurgenova Kazakh National Agrarian University Kazakh National Medical University named after. S.D. Asfendiyarov Kazakh National Pedagogical University them. Abay Kazakh National Technical University named after. K. I. Satpayeva Kazakh National University named after. al-Farabi Kazakh University international relations and world languages ​​named after. Abylai Khan Kazakhstan Institute of Management, Economics and Forecasting Kazakh-British Technical University Kazakh-German University Kazakh-Russian Medical University International University of Information Technologies New Economic University named after. T. Ryskulova University international business University of Turan Donbass State Technical University Almetyevsk State oil institute Arzamas State Pedagogical Institute named after. A.P. Gaidar Arzamas Polytechnic Institute (branch of NSTU) Armavir State pedagogical academy Armavir Linguistic University Northern (Arctic) federal university them. M. V. Lomonosov Northern State Medical University Eurasian National University named after. L.N. Gumilyov Kazakh agrotechnical university them. S. Seifullina Kazakh Humanitarian-Law University Kazakh University of Technology and Business Astana Medical University Astrakhan State University of Architecture and Civil Engineering Astrakhan State Medical University Astrakhan State Technical University Azerbaijan Medical University Balakovo Institute of Engineering, Technology and Management Baranovichi State University Altai Academy Economics and Law Altai State Academy of Culture and Arts Altai State Agrarian University Altai State Medical University Altai State Pedagogical University Altai State Technical University named after. I.I.Polzunova Altai State University Altai branch of RANEPA(SibAGS AF) Altai Economics and Law Institute Technical School 103 Belotserkovsky National Agrarian University Belgorod State Agricultural Academy named after. V.Ya. Gorina Belgorodsky state institute arts and culture Belgorod State National research university Belgorod State Technological University named after. V.G. Shukhov Belgorod University of Cooperation, Economics and Law Belgorod Law Institute of the Ministry of Internal Affairs of Russia Berdyansk State Pedagogical University named after. Osipenko Berdyansk University of Management and Business Biysk Technological Institute (branch of ASTU named after Polzunov) Kyrgyz State Medical Academy named after. I.K. Akhunbaeva Kyrgyz State University of Construction, Transport and Architecture Kyrgyz National University named after. Zh. Balasagyn Kyrgyz-Russian Academy of Education Kyrgyz-Russian Slavic University named after. Yeltsin Amur State Medical Academy Amur State University Far Eastern State Agrarian University Boksitogorsk Institute (branch of Leningrad State University named after A.S. Pushkin) Bratsk State University Brest State Technical University Brest State University named after. A.S. Pushkin Bryansk State Engineering and Technology Academy Bryansk State Agrarian University Bryansk State Technical University Bryansk State University named after. Academician I.G. Petrovsky Bryansk Institute of Management and Business Bryansk Branch of RANEPA (ORAGS BF) Velikoluksk State Academy of Physical Culture and Sports Velikoluksk State Agricultural Academy Vinnitsa State Pedagogical University named after. M. Kotsyubinsky Vinnytsia National Agrarian University Vinnytsia National Medical University named after. N.I. Pirogova Vinnitsa National Technical University Vinnitsa Trade and Economic Institute (branch of KNTEU) Vinnitsa Financial and Economic University Vitebsk State Academy veterinary medicine Vitebsk State Medical University Vitebsk State Technological University Vitebsk State University named after. P. M. Masherova Vladivostok State University of Economics and Service Far Eastern State Technical Fisheries University Far Eastern State Technical University Far Eastern Federal University Maritime State University named after. Admiral G.I. Nevelskoy Pacific State Medical University Gorsky State Agrarian University North Caucasus Mining and Metallurgical Technological University (SKGMI) North Ossetian State Medical Academy North Ossetian State University named after. K. Khetagurova Vladimir State University named after. Stoletovs Vladimir branch RANEPA (RAGS VF) Volgograd State Academy of Physical Culture Volgograd State Agrarian University Volgograd State University of Architecture and Civil Engineering Volgograd State Institute of Arts and Culture Volgograd State Medical University Volgograd State Social Pedagogical University Volgograd State Technical University Volgograd State University Volgograd Institute of Business Volgograd branch of RANEPA ( VAGS) Volgodonsk Engineering and Technical Institute National Research Nuclear University MEPhI Volga Polytechnic Institute (branch of Volga State Technical University) Volkovysk Pedagogical College of the Grodno State University named after Y. Kupara Vologda State Dairy Academy named after. N.V. Vereshchagina Vologda State University Vologda Institute of Law and Economics of the Federal Penitentiary Service of Russia Pedagogical Institute of VoGU Voronezh State Forestry Academy Voronezh State Medical Academy named after. N.N. Burdenko Voronezh State Agrarian University named after. Emperor Peter I Voronezh State University of Architecture and Civil Engineering Voronezh State Institute of Physical Culture Voronezh State Medical University named after. N.N. Burdenko Voronezh State Pedagogical University Voronezh State Technical University Voronezh State University Voronezh State University of Engineering Technologies Voronezh Institute of the Ministry of Internal Affairs of the Russian Federation Voronezh Economic and Legal Institute Institute of Management, Marketing and Finance International Institute of Computer Technologies State Institute of Economics, Finance, Law and Technology Glazov State Pedagogical Institute . V.G. Korolenko Glukhov National Pedagogical University named after. A. Dovzhenko Belarusian State University of Transport Belarusian Trade and Economic University of Consumer Cooperation Gomel State Agrarian and Economic College Gomel State Medical University Gomel State Technical University them. BY. Sukhoi Gomel State University. Francysk Skaryna Belarusian State Agricultural Academy Gorlovka State Pedagogical Institute of Foreign Languages ​​DSPU Gorno-Altai State University Grodno State Medical University Grodno State University named after. Y. Kupala Chechen State University Dnepropetrovsk State financial Academy Dnepropetrovsk Medical Academy of the Ministry of Health of Ukraine Dnepropetrovsk State Agrarian-Economic University Dnepropetrovsk State University of Internal Affairs Dnepropetrovsk National University railway transport them. Academician V. Lazaryan Dnepropetrovsk National University named after. Olesya Gonchar Dnepropetrovsk University named after. A. Nobel National Metallurgical Academy of Ukraine National mining University Prydneprovsk State Academy of Civil Engineering and Architecture Ukrainian State Chemical-Technological University Moscow State University of Physics and Technology (MIPT) Academy civil protection Ministry of Emergency Situations of the DPR Donbass Law Academy Donetsk Institute Railway Transport Donetsk National Medical University named after. M. Gorky Donetsk National University Donetsk National University of Economics and Trade named after. M. Tugan-Baranovsky Donetsk Technical School of Industrial Automation Donetsk Law Institute of the Ministry of Internal Affairs of Ukraine Drogobych State Pedagogical University named after. I. Franko Tajik State Medical University named after. Abuali ibni Sino (Avicens) Tajik State Pedagogical University named after Sadriddin Aini Evpatoria Institute of Social Sciences (branch of KFU) Ekaterinburg State Theater Institute Institute of International Relations College of Railway Transport Russian State Vocational Pedagogical University Ural State Academy of Architecture and Art Ural State Conservatory named after. M.P. Mussorgsky Ural State Agrarian University Ural State Mining University Ural State Forestry University Ural State Medical University Ural State Pedagogical University Ural State University of Transport Ural State Economic University Ural State Law University Ural Institute business named after I. A. Ilyina Ural Institute of State Fire Service of the Ministry of Emergency Situations of Russia Ural Institute of Commerce and Law Ural Institute RANEPA (UrAGS) Ural Institute of Economics, Management and Law Ural Technical School of Automobile Transport and Service Ural Technical Institute of Communications and Informatics (branch of SibGUTI) Ural Federal University . B.N. Yeltsin "UPI" Ural Financial and Legal Institute Elabuga Institute of Kazan (Volga Region) Federal University (formerly EGPU) Yelets State University named after. I.A. Bunin Yerevan State University Zhytomyr State Technological University Zhytomyr State University named after. Ivana Franko Zhytomyr Institute of Nursing Zhytomyr National Agroecological University Zaporozhye Automotive Technical School Zaporizhzhya State Engineering Academy Zaporizhzhya State Medical University Zaporizhzhya Institute of Economics and Information Technologies Zaporizhzhya National Technical University Zaporizhzhya National University Institute of Arts and Information Technologies, Moscow branch Ivano-Frankivsk National Medical University Ivano-Frankivsk National Technical University of Oil and Gas Prykarpattia National University named after. V. Stefanika Ivanovo State Academy of Architecture and Civil Engineering Ivanovo State Medical Academy Ivanovo State Agricultural Academy Ivanovo State University Ivanovo State Chemical-Technological University Ivanovo State Energy University named after. IN AND. Lenin Textile Institute IvSPU Moscow Regional Institute of Management and Law Izhevsk State Medical Academy Izhevsk State Agricultural Academy Izhevsk State Technical University named after. M. T. Kalashnikova Kama Institute of Humanitarian and Engineering Technologies Udmurt State University Udmurt Republican Social Pedagogical College Izmail College of Mechanization and Electrification of Agriculture Baikal State University Irkutsk State Agrarian University named after. A.A. Ezhevsky Irkutsk State Linguistic University Irkutsk State Medical University Irkutsk State University Irkutsk State University of Transport Irkutsk National Research Technical University Pedagogical Institute (branch of ISU) Siberian Academy of Law, Economics and Management Institute of Law (branch of ISU) National University of State Tax Service of Ukraine Mari State University Interregional Open Social Institute Volga State Technological University Academy social education Institute of Social and Humanitarian Knowledge Institute of Economics and Finance KFU Institute of Economics, Management and Law Kazan State Academy of Veterinary Medicine named after. N.E. Bauman Kazan State Conservatory (Academy) named after. N. G. Zhiganova Kazan State Agrarian University Kazan State University of Architecture and Civil Engineering Kazan State Medical University Kazan State University of Culture and Arts Kazan State Energy University Kazan Cooperative Institute (branch of RUK) Kazan National Research Technical University named after. A. N. Tupolev Kazan National Research Technological University Kazan Federal University Volga Region State Academy of Physical Culture, Sports and Tourism Tatar State Humanitarian Pedagogical University TISBI University of Management Kalacheevsky Agrarian College Baltic State Academy of Fishing Fleet Baltic Information College Baltic Federal University named after. I. Kant Kaliningrad State Technical University St. Petersburg University of Service and Economics (Kaliningrad branch) Kaluga State University. K. E. Tsiolkovsky Kaluga branch of RANEPA Kamenets-Podolsk National University named after. I. Ogienko Podolsk State Agrarian-Technical University Kamyshin Technological Institute (branch of Volga State Technical University) Karaganda State Medical University Karaganda State Technical University Karaganda State University named after. E. A. Buketova Karaganda University Bolashak Karaganda Economic University Suleyman Demirel University Kemerovo State Medical University (formerly KemSMA) Kemerovo State Agricultural Institute Kemerovo State University Kemerovo State University of Culture and Arts Kemerovo Institute of Technology Food Industry Kuzbass State Technical University Kuzbass Institute of Economics and Law Kerch State Marine Technological University State University of Telecommunications State Economics and Technology University of Transport European University finance, information systems, management and business Kyiv State Academy water transport them. Konashevich-Sagaidachny Kiev Medical University UANM Kiev National Linguistic University Kiev National Trade and Economic University Kiev National University named after. T. Shevchenko Kiev National University of Culture and Arts Kiev National University of Construction and Architecture Kiev National University of Theater, Film and Television named after. I. K. Karpenko-Kary Kiev National University of Technology and Design Kiev National Economic University named after. V. Getman Kyiv Slavic University Kyiv University named after. B. Grinchenko Kiev University of Law of the National Academy of Sciences of Ukraine Kiev University of Tourism, Economics and Law International scientific and technical university them. Yu. Bugaya Interregional Academy of Personnel Management National Academy of Internal Affairs of Ukraine National Academy of Management Personnel of Culture and Arts National Academy of Statistics, Accounting and Audit National Academy of Management National Music Academy of Ukraine named after. P.I. Tchaikovsky National Aviation University National Medical University named after. A.A. Bogomolets National Pedagogical University named after. M.P. Dragomanova National Technical University of Ukraine "Kiev Polytechnic Institute" National Transport University National University "Kiev-Mohyla Academy" National University of Bioresources and Environmental Management National University of Food Technologies National University of Physical Education and Sports of Ukraine Open International University of Human Development Ukraine Ukrainian State University of Finance and international trade Samara State Agricultural Academy Volgo-Vyatka Institute (branch of Moscow State Law Academy) Vyatka State Agricultural Academy Vyatka State Humanitarian University Vyatka State University Vyatka Socio-Economic Institute Moscow Financial and Law University Kirov branch Kirovograd Flight Academy of the National Aviation University Kirovograd State Pedagogical University named after. V. Vinnichenko Kirovograd Institute of Regional Management and Economics Kirovograd National Technical University State Agrarian University of Moldova State University of Medicine and Pharmacology named after. Nicolae Testemitanu International Independent University of Moldova Kovrov State Technological Academy named after. V.A. Degtyareva Kolomna Institute branch of MSMU Moscow State Regional Social humanitarian Institute Amur Humanitarian-Pedagogical State University Komsomolsk-on-Amur State Technical University Konotop Institute SSU Financial and Technological Academy Kostanay State University named after. Akhmet Baitursynov Kostroma State Technological University Kostroma State University named after. ON THE. Nekrasova Donbass State Engineering Academy Donbass national academy construction and architecture Donetsk National Technical University Krasnoarmeysk Industrial Institute DonNTU Krasnodar State University of Culture and Arts Kuban State Agrarian University Kuban State Medical University Kuban State Technological University Kuban State University Kuban State University of Physical Culture, Sports and Tourism Kuban Socio-Economic Institute Modern Humanitarian Academy Humanitarian Institute of Siberian Federal University Engineering and Construction Institute of SFU Institute of Architecture and Design of SFU Institute of Mining, Geology and Geotechnology of SFU Institute of Natural and humanities SFU Institute of Engineering Physics and Radioelectronics SFU Institute of Space and Information Technologies SFU Institute of Oil and Gas SFU Institute of Pedagogy, Psychology and Sociology SFU Institute of Business Process Management and Economics SFU Institute of Philology and Language Communication SFU Institute of Fundamental Biology and Biotechnology SFU Institute of Non-Ferrous Metals and Materials Science Siberian Federal University Institute of Economics, Management and Environmental Management Siberian Federal University Krasnoyarsk State Academy of Music and Theater Krasnoyarsk State Academy of Architecture and Civil Engineering Siberian Federal University Krasnoyarsk State Agrarian University Krasnoyarsk State Medical University named after. V.F. Voino-Yasenetsky Krasnoyarsk State Pedagogical University named after. V.P. Astafiev Krasnoyarsk Institute of Railway Transport, branch of IrGUPS Polytechnic Institute Siberian Federal University Siberian State Technological University Siberian State University of Science and Technology. Academician M.F. Reshetnev Siberian Institute of Business, Management and Psychology Siberian Interregional Training Center Siberian Federal University Trade and Economics Institute SFU Institute of Law SFU Kremenchug National University named after. M. Ostrogradsky Krivoy Rog National University Krivoy Rog Economic Institute KNEU named after. V. Getman Aviation Technical College Kurgan State Agricultural Academy named after. T. S. Maltseva Kurgan State University Kursk State Agricultural Academy named after. Ave. I.I. Ivanova Kursk State Medical University Kursk Institute social education Regional Financial and Economic Institute Southwestern State University Tuva State University Lesosibirsk Pedagogical Institute(branch of Siberian Federal University) Lipetsk State Pedagogical University Lipetsk State Technical University Luga Institute (branch of Leningrad State University named after A.S. Pushkin) Lugansk State Academy of Culture and Arts Lugansk State Medical University Lugansk State University of Internal Affairs named after. E.A. Didorenko Lugansk State University named after. Vladimir Dahl Lugansk National Agrarian University Lugansk National University named after. Taras Shevchenko Eastern European National University named after. Lesya Ukrainka Lutsk National Technical University Lvov Commercial Academy Lvov National Academy of Arts Lvov State University of Internal Affairs Lvov State University of Physical Culture Lvov Institute of Economics and Tourism Lvov National Agrarian University Lvov National Medical University named after. D. Galitsky Lviv National University of Veterinary Medicine and Biotechnology named after. S.Z. Grzhitsky Lviv National University. I. Franko National University Lviv Polytechnic Russian Customs Academy North-Eastern State University Ingush State University Magnitogorsk State Technical University named after. G.I. Nosov Magnitogorsk Medical College named after. P.F. Nadezhdina Azov Maritime Institute Odessa National Maritime Academy Donetsk State University of Management Mariupol State University Priazov State Technical University Dagestan State Medical Academy Dagestan State Pedagogical University Dagestan State Technical University Dagestan State University Melitopol State Pedagogical University named after. B. Khmelnitsky Tauride State agrotechnological university Belarusian State Academy of Arts Belarusian State Academy of Music Belarusian State Academy of Communications Belarusian State Agrarian Technical University Belarusian State Medical University Belarusian State Pedagogical University named after. M. Tanka Belarusian State Technological University Belarusian State University Belarusian State University of Informatics and Radioelectronics Belarusian State University of Culture and Arts Belarusian State University of Physical Culture Belarusian State Economic University Belarusian National Technical University Institute of Information Technologies BSUIR Institute border service Republic of Belarus Institute of Modern Knowledge named after. A.M. Shirokov International State Ecological University named after. A. D. Sakharova International University MITSO Minsk State Higher Radio Engineering College Minsk State Polytechnic College Minsk Innovation University Minusinsk College of Culture and Art Mikhailovsky Technical School. A. Merzlova Belarusian-Russian University Mogilev State University named after. A. A. Kuleshova Mogilev State University of Food Mozyr State Pedagogical University named after. I.P. Shamyakin Academic International Institute Academic Law Institute Academy of the State Fire Service of the Ministry of Emergency Situations of Russia Academy of Standardization, Metrology and Certification Academy of Labor and Social Relations of the Federation of Independent Trade Unions of Russia Air Force Engineering Academy named after. Ave. N.E. Zhukovsky All-Russian Academy foreign trade Ministries economic development Russian Federation All-Russian State University of Cinematography named after. S.A. Gerasimov "VGIK" Higher Theater School (Institute) named after. M. S. Shchepkina GAPOU College of Entrepreneurship No. 11 State Academy of Slavic Culture State Classical Academy named after. Maimonides State academic university Humanities State Institute of Russian Language named after. A.S. Pushkin State University of Land Management State University of Management Humanitarian Institute of Television and Radio Broadcasting named after. M.A. Litovchina Institute of Humanitarian Education and Information Technologies Institute of Journalism and Literary Creativity Institute of International Law and Economics named after A.S. Griboyedov Institute of Postgraduate Professional Education FMBTS ( science Center) Institute of Market Economics, social policy and Law Institute of Textile and light industry MSUTU Institute of Tourism and Hospitality Institute of Management and Law Institute of Economics and Culture College of Urban Planning and Service No. 38 College of Multilevel Professional Education RANEPA Literary Institute named after. A.M. Gorky Medical Institute continuing education Medical College No. 1 International Academy of Business and Management International Institute of Economics and Law International Law Institute Moscow Academy of Astrology Moscow Academy of Entrepreneurship under the Government of Moscow Moscow Academy of Economics and Law Moscow State Academy of Veterinary Medicine and Biotechnology named after. K.I. Scriabin Moscow State Academy of Water Transport Moscow State Academy Municipal Services and Construction Moscow State Academy of Physical Culture Moscow State Conservatory named after. P.I. Tchaikovsky Moscow State Academy of Arts and Industry named after. S. G. Stroganova Moscow State Law Academy named after. O.E. Kutafina Moscow Humanitarian and Technical Academy Moscow Academy of Finance and Law Moscow Aviation Institute (National Research University) Moscow Automobile and Highway State Technical University Moscow Institute of Architecture and Construction Moscow architectural institute(state academy) Moscow Banking Institute Moscow Mining Institute (branch of NUST MISIS) Moscow City Pedagogical University Moscow City Psychological and Pedagogical University Moscow City University of Management of the Moscow Government Moscow State Agricultural Engineering University named after. V.P. Goryachkina Moscow State University for the Humanities and Economics Moscow State University for the Humanities. M.A. Sholokhov Moscow State industrial university Moscow State Institute of Tourism Industry named after. Yu.A. Senkevich Moscow State Institute of Radio Engineering, Electronics and Automation (Technical University) Moscow State Institute of Electronics and Mathematics (Technical University) Moscow State College of Information Technologies Moscow State Linguistic University Moscow State Mechanical Engineering University "MAMI" Moscow State Medical and Dental University. A.I. Evdokimova Moscow State regional university Moscow State Open University named after. V. S. Chernomyrdin Moscow State University of Civil Aviation Moscow State Technical University of Civil Aviation Moscow State Technical University named after. N.E. Bauman Moscow State Technological University "Stankin" Moscow State University of Geodesy and Cartography Moscow State University of Design and Technology Moscow State University. M.V. Lomonosov Moscow State University environmental engineering Moscow State University of International Relations of the Ministry of Foreign Affairs of Russia (MGIMO) Moscow State University of Printing Arts. I. Fedorova Moscow State University of Food Production Moscow State University of Instrument Engineering and Informatics Moscow State University applied biotechnology Moscow State University of Environmental Management Moscow State University of Transport Moscow State University of Technology and Management named after. K.G. Razumovsky Moscow State University of Fine Chemical Technologies named after. M.V. Lomonosov Moscow State University of Economics, Statistics and Informatics (MESI) Moscow Humanitarian-Economic Institute Moscow Humanitarian Institute. E.R. Dashkova Moscow Humanitarian University Moscow Institute of Public Administration and Law Moscow Institute of Entrepreneurship and Law Moscow Institute of Television and Radio Broadcasting "Ostankino" Moscow International University Moscow New Law Institute Moscow educational complex them. V. Talalikhina Moscow Pedagogical State University Moscow Psychological social university Moscow Socio-Economic Institute Moscow Technical University of Communications and Informatics Moscow Technological Institute "VTU" Moscow University. S.Yu. Witte (formerly Moscow Institute of Economics, Management and Law) Moscow University of the Ministry of Internal Affairs of the Russian Federation. V.Ya. Kikotya Moscow Financial and Industrial University Synergy Moscow Artistic - industrial institute Moscow Economic Institute Musical-Pedagogical State Institute named after. MM. Ippolitova-Ivanova National Institute Business National Research Technological University "MISiS" National Research University "Higher School of Economics" National Research University "MIET" National Research University "MPEI" National Research Nuclear University (MEPhI) Open University Israel in the CIS Pedagogical Institute of Physical Culture and Sports of the Moscow City Pedagogical University First Moscow State Medical University named after. THEM. Sechenov Polytechnic College named after P.A. Ovchinnikova Orthodox St. Tikhon's Humanitarian University Russian Academy of Music named after. Gnessins Russian Academy National economy and civil service under the President Russian Federation Russian International Academy of Tourism Russian Open Academy of Transport MIIT Russian State Agrarian University MSHA named after. Timiryazev Russian State Geological Prospecting University named after. S. Ordzhonikidze Russian State Humanitarian University Russian State Social University Russian State Technological University named after. K.E. Tsiolkovsky (MATI) Russian State Trade and Economic University Russian State University named after A.N. Kosygina Russian State University of Innovative Technologies and Entrepreneurship Russian State University of Oil and Gas named after. THEM. Gubkin Russian State University of Justice Russian State University of Tourism and Service Russian State University of Physical Culture, Sports, Youth and Tourism (GTSOLIFK) Russian National Research Medical University named after N. I. Pirogov Russian new university Peoples' Friendship University of Russia Russian University theatrical arts Russian Chemical-Technological University named after. DI. Mendeleev Russian Economic University. G.V. Plekhanov Capital Financial and Humanitarian Academy Theater Institute named after. B.V. Shchukin at the State Academic Theater them. E. Vakhtangov University of Russian Innovative Education University of the Russian Academy of Education Federal Institute advanced training and retraining Financial University under the Government of the Russian Federation School-studio (institute) named after. Vl. I. Nemirovich-Danchenko at the Moscow Art Theater. A. P. Chekhova Mukachevo State University International Institute of Business Education Murmansk State Humanitarian University Moscow State University of Forestry Moscow Altshul Cooperative College Russian University of Cooperation Kama State Engineering and Economic Academy Naberezhnye Chelny State Trade and Technological Institute Naberezhnye Chelny Institute KFU Naberezhnye Chelny Institute of Social and Pedagogical Technologies and Resources Kabardino-Balkarian State University named after. H. Berbekova Nanjing University of Science and Technology (Nanjing University of Science and Technology) Nezhin State University named after. N. Gogol Nemeshaevsky Agrotechnical College Nizhnevartovsk State University Nizhnekamsk Chemical-Technological Institute Kazan State Technological University Volga State Academy of Water Transport Nizhny Novgorod State Conservatory named after. M.I. Glinka Nizhny Novgorod State Agricultural Academy Nizhny Novgorod Law Academy Nizhny Novgorod State University of Architecture and Civil Engineering Nizhny Novgorod State Engineering and Economic University Nizhny Novgorod State Linguistic University named after. ON THE. Dobrolyubov Nizhny Novgorod State Pedagogical University named after. K. Minin Nizhny Novgorod State Technical University named after. R.E. Alekseev Nizhny Novgorod State University named after. N.I. Lobachevsky Nizhny Novgorod Institute management and business Nizhny Novgorod Institute of Management RANEPA (VVAGS) Privolzhsky Research Medical University (formerly NizhSMA) Nizhny Tagil State Social and Pedagogical Institute (branch of RGPPU) Nizhny Tagil Technological Institute (branch of UrFU) National University of Shipbuilding named after. adm. Makarov Nikolaev National Agrarian University Nikolaev National University named after. V.A. Sukhomlinsky Black Sea State University named after. Peter Mogila Novgorod State University named after. Yaroslav the Wise Novokuznetsk Institute (branch of Kemerovo State University) Siberian State Industrial University State Maritime University named after. Admiral F. F. Ushakov Institute of Catalysis named after. G.K. Boreskov Novosibirsk State Conservatory named after. M.I. Glinka Novosibirsk State Agrarian University Novosibirsk State University of Architecture and Civil Engineering Novosibirsk State Medical University Novosibirsk State Pedagogical University Novosibirsk State Technical University Novosibirsk State University Novosibirsk State University of Architecture, Design and Arts (formerly NGAHA) Novosibirsk State University of Economics and Management Novosibirsk Medical College Novosibirsk Law School institute (branch of TSU) Siberian Academy of Finance and Banking Siberian State University of Water Transport Siberian State University of Geosystems and Technologies Siberian State University of Transport Siberian State University of Telecommunications and Informatics Siberian Institute of Management RANEPA (SibAGS) Siberian University consumer cooperation South Russian State Technical University (Novocherkassk Polytechnic Institute) (SRSTU (NPI)) Obninsk Humanitarian Institute Obninsk Institute nuclear energy NRNU MEPhI National University Odessa Marine Academy(formerly ONMA) National University Odessa Law Academy Odessa State Academy of Construction and Architecture Odessa National Academy of Food Technologies Odessa National Academy of Communications named after. A.S. Popova Odessa State Agrarian University Odessa State Ecological University Odessa State Economic University Odessa Corporate Computer College Odessa National Medical University Odessa National maritime university Odessa National Polytechnic University Odessa National University named after. I.I. Mechnikov South Ukrainian National Pedagogical University named after. K.D. Ushinsky Ozyorsk Technological Institute Omsk Academy Ministry of Internal Affairs of Russia Omsk State Agrarian University named after. P. A. Stolypina Omsk State Institute of Service Omsk State Medical University Omsk State Pedagogical University Omsk State Technical University Omsk State University named after. F.M. Dostoevsky Omsk State University of Transport Omsk Economic Institute Omsk Law Institute Siberian State Automobile and Highway Academy Siberian State University of Physical Culture and Sports State University - educational, scientific and production complex (formerly Orel State Technical University) Medical Institute of Oryol State University Oryol State Institute of Arts and Culture Oryol State Institute of Economics and Trade Oryol branch RANEPA Orenburg State Agrarian University Orenburg State Institute of Management Orenburg State Medical University Orenburg State Pedagogical University Orenburg State University Orenburg Institute (branch of Moscow State Law Academy Kutafina) Orsk Humanitarian-Technological Institute (branch of OSU) Orsk Medical College GBPOU Ostashkov College Osh Technological University named after. acad. MM. Adysheva Innovative Eurasian University Pavlodar State Pedagogical University Pavlodar State University named after. S. Toraigyrov Pedagogical Institute named after. V. G. Belinsky Penza State University Penza State Agricultural Academy Penza State Technological University Penza State University Penza State University of Architecture and Construction Pereyaslav-Khmelnitsky State Pedagogical University named after. G.S. Skovoroda West Ural Institute of Economics and Law Perm State Academy of Art and Culture Perm State Agricultural Academy named after. D.N. Pryanishnikova Perm State Pharmaceutical Academy Perm State Humanitarian and Pedagogical University Perm State Medical University named after. ak. E.A. Wagner Perm State National Research University Perm Institute of Humanities and Technology Perm Institute Economics and Finance Perm National Research Polytechnic University Karelian State Pedagogical Academy Petrozavodsk State Conservatory named after. A.K. Glazunov Petrozavodsk State University North Kazakhstan State University named after. M. Kozybaeva Kamchatka State Technical University Pinsk State Vocational and Technical College of Mechanical Engineering Polesie State University Poltava State agricultural academy Poltava National Pedagogical University named after. V. G. Korolenko Poltava National Technical University. Yu. Kondratyuk Poltava University of Economics and Trade Ukrainian Medical Dental Academy Pskov Agrotechnical College Pskov State University Leningrad State University named after. A.S. Pushkin St. Petersburg State Agrarian University Pyatigorsk State Linguistic University Pyatigorsk State Technological University Pyatigorsk Medical-Pharmaceutical Institute (branch of Volga State Medical University) North Caucasus Institute RANEPA (SKAGS) Rezhevsk Polytechnic School International Economics and Humanities University named after. S. Demyanchuk National University water management and environmental management Rivne State Humanitarian University Academy of Architecture and Arts Southern Federal University Donskoy State Agrarian University Donskoy State Technical University Institute of Service and Tourism (branch of DSTU) Institute of Management, Business and Law Rostov State Conservatory. S. V. Rachmaninova Rostov State Medical University Rostov State University of Transport Rostov State Economic University "RINH" Rostov Institute for the Protection of Entrepreneurs Rostov Law Institute (branch of the RPA MU) Southern Federal University Rybinsk State Aviation Technical University named after. P. A. Solovyov Rybinsk River School named after. IN AND. Kalashnikov Rybnitsa Branch of the Transnistrian State University named after T.G. Shevchenko Ryazan State Agrotechnological University named after. P.A. Kostychev Ryazan State Medical University named after. acad. I.P. Pavlova Ryazan State Radio Engineering University Ryazan State University named after. S.A. Yesenina Medical University "REAVIZ" Volga Region State Social and Humanitarian Academy Volga Region State University of Telecommunications and Informatics Samara Academy state and municipal government Samara State Academy of Culture and Arts Samara Humanitarian Academy Samara State University of Architecture and Civil Engineering Samara State Medical University Samara State Technical University Samara State University of Transport Samara State Economic University Samara Institute- Higher School of Privatization and Entrepreneurship Samara National Research University named after. ak. S.P. Korolev (formerly SSAU, SamSU) Samarkand State medical school Academy of Russian Ballet named after. AND I. Vaganova Baltic Academy tourism and entrepreneurship Baltic State Technical University "VOENMEH" named after. D.F. 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Mathematical logic and theory of algorithms – Course of lectures
Introduction.

    1. Purpose.
This course serves to develop knowledge and skills that form the theoretical foundation necessary for setting and solving problems in the field of computer science, for a correct understanding of the limitations that arise when creating computational structures, algorithms and information processing programs.

New sections of the discrete mathematics course, although implemented in the form curricula and series of lectures, do not yet exist in the form of monographs, at least in Russian, since the course of discrete mathematics for technical universities is focused on old applied problems that engineers had to solve. In particular, in mathematical logic it was the minimization of logical circuits, which has lost its relevance today.

It is interesting to note that the theory of logic circuit synthesis, having gone through almost a complete “biological cycle” before the eyes of one generation of researchers, is a very instructive example of how industries are highly susceptible to obsolescence technical sciences, weakly related to fundamental science. 10 years ago everything technical magazines were filled with articles on minimization and synthesis of logic circuits. Most of the minimization methods developed by scientists are now forgotten and are not in demand in practice. And those ideas that were considered purely theoretical at that time found practical application in modern technology. For example, fuzzy logic, Petri nets and algorithm theory have stood the test of time and are widely used in various fields of cybernetics and programming, such as system Programming, computational complexity and artificial intelligence.

And the theory of algorithms became the central section of discrete mathematics. However, unlike most monographs in Russian, in the course of lectures these issues are presented as a means of solving practical, engineering problems.

As is known, after each decade, the component base of computers, OS, the means of access and the programs themselves are changing radically. However, the structures and algorithms underlying them remain unchanged for much longer. These foundations began to be laid thousands of years ago, when formal logic was developed and the first algorithms were developed.

Mathematical logic and the theory of algorithms traditionally belong to fundamental science and are considered to be of little practical relevance and difficult to understand. Indeed, when J. Bull created mathematical apparatus Boolean algebra, it took him a long time to find practical application, however, in the 20th century it was precisely this mathematical apparatus that made it possible to design all computer components. Consequently, the first of these prejudices is successfully refuted by the development of computer technology.

As for the prejudice about the difficulty of understanding this discipline, it largely stems from the fact that books on mathematical logic and the theory of algorithms were written by mathematicians for mathematicians.

Now, when the capabilities of computing technology have increased many times over, and there are much more personal computers themselves than there are people who know how to use them effectively, understanding what can and cannot be done with the help of modern computing technology is of exceptional importance.

Exactly general theory algorithms have shown that there are problems that are unsolvable no matter how powerful the computing power is, and its rapidly developing branch, the theory of computational complexity, gradually leads to the understanding that there are problems that can be solved, but are objectively complex, and their complexity may turn out to be in some sense absolute, those. practically inaccessible to modern computers.

This course set the following objectives:

1. Present all the issues under consideration as simply as possible, but not simpler than is required for a highly qualified specialist.

2. Practical problems of design and analysis of information systems are the starting point, and the formal apparatus is a means of systematically solving these problems. It is our deep conviction that a student is not a vessel that needs to be filled, but a torch that needs to be lit.

3. Each section of the course contains self-test questions. To complete this course, the student must answer all these questions.

As a result of mastering this course, the student, based on a clear understanding of the relevant theoretical sections, should be able to:

Implement the simplest type of logical transformation of information in an arbitrary basis of logical functions;

Highlight in evidentiary reasoning natural language logical structure, build formal proof schemes and check their correctness.

1.2 Logical representations
Logical representations - description of the system, process, phenomenon under study in the form of a set complex statements made up of simple (elementary) statements And logical connectives between them. Logical representations and their components are characterized by certain properties and a set of permissible transformations over them (operations, inference rules, etc.), implementing those developed in formal (mathematical) logic correct methods reasoning - the laws of logic.

Methods (rules) of formal presentation of statements, construction of new statements from existing ones using logically correct transformations, as well as methods (methods) of establishing the truth or falsity of statements are studied in mathematical logic. Modern mathematical logic includes two main sections: logic of statements and covering it predicate logic(Fig. 1.1), for the construction of which there are two approaches (languages), forming two variants of formal logic: algebra of logic And logical calculus. There is a one-to-one correspondence between the basic concepts of these languages ​​of formal logic. Their isomorphism is ultimately ensured by the unity of the underlying admissible transformations.

Rice. 1.1
The main objects of traditional branches of logic are statements.

Statement - declarative sentence (statement, judgment), o which it makes sense to say that it true or false. All scientific knowledge(laws and phenomena of physics, chemistry, biology, etc., mathematical theorems, etc.), events Everyday life, situations arising in economics and management processes are formulated in the form of statements. Imperative and interrogative sentences are not statements.

Examples of statements: “Twice two is four”, “We live in the 21st century”, “The ruble is the Russian currency”, “Alyosha is Oleg’s brother”, “The operations of union, intersection and addition are Boolean operations on sets”, “Man is mortal” , “Rearranging the places of the terms does not change the sum,” “Today is Monday,” “If it rains, you should take an umbrella.”

In order to further operate with these sentences as statements, we must know for each of them whether it is true or false, i.e. know them truth value (truth). Note that in some cases the truth or falsity of a statement depends on what specific reality (system, process, phenomenon) we are trying to describe with its help. In this case, the given statement is said to be true (or false) in a given interpretation (context). We further assume that the context is given and the statement has a certain truth value.

1.3 History of developed mathematical logic

Logic as a science was formed in the 4th century. BC. It was created by the Greek scientist Aristotle.

The word “logic” comes from the Greek “logos”, which on the one hand means “word” or “exposition”, and on the other, thinking. IN explanatory dictionary Ozhegova S.I. It is said: “Logic is the science of the laws of thinking and its forms.” In the 17th century German scientist Leibniz planned to create a new science, which would be “the art of calculating truth” . In this logic, according to Leibniz, each statement would have a corresponding symbol, and reasoning would have the form of calculations. This idea of ​​Leibniz, having not met the understanding of his contemporaries, was not spread or developed and remained a brilliant guess.

Only in the middle of the 19th century. Irish mathematician George Boole embodied Leibniz's idea. In 1854, he wrote the work “Investigation of the laws of thought,” which laid the foundations for the algebra of logic, in which laws similar to the laws of ordinary algebra apply, but the letters do not denote numbers , but statements. In the language of Boolean algebra, one can describe reasoning and “compute” its results. However, it does not cover all reasoning, but only a certain type of it. , Therefore, Boole algebra is considered a propositional calculus.

Boole's algebra of logic was the embryo of a new science - mathematical logic. In contrast, Aristotle's logic is called traditional formal logic. The name “mathematical logic” reflects two features of this science: firstly, mathematical logic is logic that uses the language and methods of mathematics; secondly, mathematical logic is brought to life by the needs of mathematics.

At the end of the 19th century. The set theory created by Georg Cantor seemed to be a reliable foundation for all mathematics, including mathematical logic, at least for propositional calculus (Boole algebra), because It turned out that Cantor algebra (set theory) is isomorphic to Boole algebra.

Mathematical logic itself became a branch of mathematics that at first seemed highly abstract and infinitely far from practical applications. However, this area did not remain the domain of “pure” mathematicians for long. At the beginning of the 20th century. (1910) Russian scientist Ehrenfest P.S. pointed out the possibility of using the apparatus of Boolean algebra in telephone communications to describe switching circuits. In 1938-1940, almost simultaneously, the works of the Soviet scientist V.I. Shestakov, the American scientist Shannon and the Japanese scientists Nakashima and Hakazawa appeared on the application of mathematical logic in digital technology. The first monograph devoted to the use of mathematical logic in the design of digital equipment was published in the USSR by the Soviet scientist M.A. Gavrilov. in 1950. The role of mathematical logic in the development of modern microprocessor technology is extremely important: it is used in the design of computer hardware, in the development of all programming languages ​​and in the design of discrete automation devices.

Scientists from different countries made a great contribution to the development of mathematical logic: professor of Kazan University Poretsky P.S., de-Morgan, Peirce, Turing, Kolmogorov A.N., Heidel K. and others.

1.4 Questions for self-test.

1. Formulate the objectives of the course

The book was written based on materials from lectures and seminars conducted by the authors for junior students of the Faculty of Mechanics and Mathematics of Moscow State University. It talks about the basic concepts of mathematical logic (propositional logic, first-order languages, expressibility, propositional calculus, decidable theories, the completeness theorem, principles of model theory). The presentation is intended for students mathematics schools, mathematics students and everyone interested mathematical logic. The book includes about 200 problems of varying difficulty.

Logic of statements.
Utterances and Operations
“If the number n is rational, then n - algebraic number. But it is not algebraic. This means p is not rational.” We do not have to know what the number n is, which numbers are called rational and which are algebraic, in order to recognize that this reasoning is correct in the sense that the conclusion actually follows from the two stated premises. This kind of situation - when a certain statement is true regardless of the meaning of the statements included in it - constitutes the subject of propositional logic.

This beginning (especially considering that the logic course was part of the program of the Faculty of Philosophy, where “dialectical logic” was also studied) is alarming, but in fact our considerations will have a completely precise mathematical nature, although we will start with informal motivations.

Table of contents
Preface
1. Propositional logic
1.1. Utterances and Operations
1.2. Complete systems bundles
1.3. Schemes of functional elements
2. Propositional calculus
2.1. Propositional Calculus (PC)
2.2. Second proof of the completeness theorem
2.3. Finding a counterexample and sequent calculus
2.4. Intuitionistic propositional logic
3. First order languages
3.1. Formulas and interpretations
3.2. Definition of truth
3.3. Expressible predicates
3.4. Expressibility in arithmetic
3.5. Inexpressible predicates: automorphisms
3.6. Elimination of quantifiers
3.7. Presburger Arithmetic
3.8. Tarski-Seidenberg theorem
3.9. Elementary equivalence
3.10. Ehrenfeucht's game
3.11. Power reduction
4. Predicate calculus
4.1. Generally valid formulas
4.2. Axioms and rules of inference
4.3. Correctness of predicate calculus
4.4. Conclusions in predicate calculus
4.5. Completeness of predicate calculus
4.6. Renaming Variables
4.7. Prefixed normal form
4.8. Herbrand's theorem
4.9. Skolemov functions
5. Theories and models
5.1. Axioms of equality
5.2. Power boost
5.3. Complete theories
5.4. Incomplete and undecidable theories
5.5. Diagrams and Extensions
5.6. Ultrafilters and compactness
5.7. Non-standard analysis
Literature
Subject index
Index of names.

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