1,095,253 research outputs found
Partial Correctness of a Power Algorithm
This work continues a formal verification of algorithms written in terms of simple-named complex-valued nominative data [6],[8],[15],[11],[12],[13]. In this paper we present a formalization in the Mizar system [3],[1] of the partial correctness of the algorithm: i := val.1 j := val.2 b := val.3 n := val.4 s := val.5 while (i n) i := i + j s := s * b return s computing the natural n power of given complex number b, where variables i, b, n, s are located as values of a V-valued Function, loc, as: loc/.1 = i, loc/.3 = b, loc/.4 = n and loc/.5 = s, and the constant 1 is located in the location loc/.2 = j (set V represents simple names of considered nominative data [17]).The validity of the algorithm is presented in terms of semantic Floyd-Hoare triples over such data [9]. Proofs of the correctness are based on an inference system for an extended Floyd-Hoare logic [2],[4] with partial pre- and post-conditions [14],[16],[7],[5].Institute of Informatics, University of BiaĆystok, PolandGrzegorz Bancerek, CzesĆaw ByliĆski, Adam Grabowski, Artur KorniĆowicz, Roman Matuszewski, Adam Naumowicz, and Karol PÄ
k. The role of the Mizar Mathematical Library for interactive proof development in Mizar. Journal of Automated Reasoning, 61(1):9â32, 2018. doi:10.1007/s10817-017-9440-6.R.W. Floyd. Assigning meanings to programs. Mathematical aspects of computer science, 19(19â32), 1967.Adam Grabowski, Artur KorniĆowicz, and Adam Naumowicz. Four decades of Mizar. Journal of Automated Reasoning, 55(3):191â198, 2015. doi:10.1007/s10817-015-9345-1.C.A.R. Hoare. An axiomatic basis for computer programming. Commun. ACM, 12(10): 576â580, 1969.Ievgen Ivanov and Mykola Nikitchenko. On the sequence rule for the Floyd-Hoare logic with partial pre- and post-conditions. In Proceedings of the 14th International Conference on ICT in Education, Research and Industrial Applications. Integration, Harmonization and Knowledge Transfer. Volume II: Workshops, Kyiv, Ukraine, May 14â17, 2018, volume 2104 of CEUR Workshop Proceedings, pages 716â724, 2018.Ievgen Ivanov, Mykola Nikitchenko, Andrii Kryvolap, and Artur KorniĆowicz. Simple-named complex-valued nominative data â definition and basic operations. Formalized Mathematics, 25(3):205â216, 2017. doi:10.1515/forma-2017-0020.Ievgen Ivanov, Artur KorniĆowicz, and Mykola Nikitchenko. Implementation of the composition-nominative approach to program formalization in Mizar. The Computer Science Journal of Moldova, 26(1):59â76, 2018.Ievgen Ivanov, Artur KorniĆowicz, and Mykola Nikitchenko. On an algorithmic algebra over simple-named complex-valued nominative data. Formalized Mathematics, 26(2):149â158, 2018. doi:10.2478/forma-2018-0012.Ievgen Ivanov, Artur KorniĆowicz, and Mykola Nikitchenko. An inference system of an extension of Floyd-Hoare logic for partial predicates. Formalized Mathematics, 26(2): 159â164, 2018. doi:10.2478/forma-2018-0013.Ievgen Ivanov, Artur KorniĆowicz, and Mykola Nikitchenko. Partial correctness of GCD algorithm. Formalized Mathematics, 26(2):165â173, 2018. doi:10.2478/forma-2018-0014.Ievgen Ivanov, Artur KorniĆowicz, and Mykola Nikitchenko. On algebras of algorithms and specifications over uninterpreted data. Formalized Mathematics, 26(2):141â147, 2018. doi:10.2478/forma-2018-0011.Artur Kornilowicz, Andrii Kryvolap, Mykola Nikitchenko, and Ievgen Ivanov. Formalization of the algebra of nominative data in Mizar. In Maria Ganzha, Leszek A. Maciaszek, and Marcin Paprzycki, editors, Proceedings of the 2017 Federated Conference on Computer Science and Information Systems, FedCSIS 2017, Prague, Czech Republic, September 3â6, 2017., pages 237â244, 2017. ISBN 978-83-946253-7-5. doi:10.15439/2017F301.Artur Kornilowicz, Andrii Kryvolap, Mykola Nikitchenko, and Ievgen Ivanov. Formalization of the nominative algorithmic algebra in Mizar. In Leszek Borzemski, Jerzy ĆwiÄ
tek, and Zofia Wilimowska, editors, Information Systems Architecture and Technology: Proceedings of 38th International Conference on Information Systems Architecture and Technology â ISAT 2017 â Part II, Szklarska PorÄba, Poland, September 17â19, 2017, volume 656 of Advances in Intelligent Systems and Computing, pages 176â186. Springer, 2017. ISBN 978-3-319-67228-1. doi:10.1007/978-3-319-67229-8_16.Artur KorniĆowicz, Andrii Kryvolap, Mykola Nikitchenko, and Ievgen Ivanov. An approach to formalization of an extension of Floyd-Hoare logic. In Vadim Ermolayev, Nick Bassiliades, Hans-Georg Fill, Vitaliy Yakovyna, Heinrich C. Mayr, Vyacheslav Kharchenko, Vladimir Peschanenko, Mariya Shyshkina, Mykola Nikitchenko, and Aleksander Spivakovsky, editors, Proceedings of the 13th International Conference on ICT in Education, Research and Industrial Applications. Integration, Harmonization and Knowledge Transfer, Kyiv, Ukraine, May 15â18, 2017, volume 1844 of CEUR Workshop Proceedings, pages 504â523. CEUR-WS.org, 2017.Artur KorniĆowicz, Ievgen Ivanov, and Mykola Nikitchenko. Kleene algebra of partial predicates. Formalized Mathematics, 26(1):11â20, 2018. doi:10.2478/forma-2018-0002.Andrii Kryvolap, Mykola Nikitchenko, and Wolfgang Schreiner. Extending Floyd-Hoare logic for partial pre- and postconditions. In Vadim Ermolayev, Heinrich C. Mayr, Mykola Nikitchenko, Aleksander Spivakovsky, and Grygoriy Zholtkevych, editors, Information and Communication Technologies in Education, Research, and Industrial Applications: 9th International Conference, ICTERI 2013, Kherson, Ukraine, June 19â22, 2013, Revised Selected Papers, pages 355â378. Springer International Publishing, 2013. ISBN 978-3-319-03998-5. doi:10.1007/978-3-319-03998-5_18.Volodymyr G. Skobelev, Mykola Nikitchenko, and Ievgen Ivanov. On algebraic properties of nominative data and functions. In Vadim Ermolayev, Heinrich C. Mayr, Mykola Nikitchenko, Aleksander Spivakovsky, and Grygoriy Zholtkevych, editors, Information and Communication Technologies in Education, Research, and Industrial Applications â 10th International Conference, ICTERI 2014, Kherson, Ukraine, June 9â12, 2014, Revised Selected Papers, volume 469 of Communications in Computer and Information Science, pages 117â138. Springer, 2014. ISBN 978-3-319-13205-1. doi:10.1007/978-3-319-13206-8_6.27218919
On Fuzzy Negations Generated by Fuzzy Implications
We continue in the Mizar system [2] the formalization of fuzzy implications according to the book of Baczynski and Jayaram âFuzzy Implicationsâ [1]. In this article we define fuzzy negations and show their connections with previously defined fuzzy implications [4] and [5] and triangular norms and conorms [6]. This can be seen as a step towards building a formal framework of fuzzy connectives [10]. We introduce formally Sugeno negation, boundary negations and show how these operators are pointwise ordered. This work is a continuation of the development of fuzzy sets [12], [3] in Mizar [7] started in [11] and partially described in [8]. This submission can be treated also as a part of a formal comparison of fuzzy and rough approaches to incomplete or uncertain information within the Mizar Mathematical Library [9].Institute of Informatics, University of BiaĆystok, PolandMichaĆ Baczynski and Balasubramaniam Jayaram. Fuzzy Implications. Springer Publishing Company, Incorporated, 2008. doi:10.1007/978-3-540-69082-5.Grzegorz Bancerek, CzesĆaw Bylinski, Adam Grabowski, Artur KorniĆowicz, Roman Matuszewski, Adam Naumowicz, Karol Pak, and Josef Urban. Mizar: State-of-the-art and beyond. In Manfred Kerber, Jacques Carette, Cezary Kaliszyk, Florian Rabe, and Volker Sorge, editors, Intelligent Computer Mathematics, volume 9150 of Lecture Notes in Computer Science, pages 261â279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1. doi:10.1007/978-3-319-20615-8_17.Didier Dubois and Henri Prade. Fuzzy Sets and Systems: Theory and Applications. Academic Press, New York, 1980.Adam Grabowski. Formal introduction to fuzzy implications. Formalized Mathematics, 25(3):241â248, 2017. doi:10.1515/forma-2017-0023.Adam Grabowski. Fundamental properties of fuzzy implications. Formalized Mathematics, 26(4):271â276, 2018. doi:10.2478/forma-2018-0023.Adam Grabowski. Basic formal properties of triangular norms and conorms. Formalized Mathematics, 25(2):93â100, 2017. doi:10.1515/forma-2017-0009Adam Grabowski. On the computer certification of fuzzy numbers. In M. Ganzha, L. Maciaszek, and M. Paprzycki, editors, 2013 Federated Conference on Computer Science and Information Systems (FedCSIS), Federated Conference on Computer Science and Information Systems, pages 51â54, 2013.Adam Grabowski and Takashi Mitsuishi. Extending Formal Fuzzy Sets with Triangular Norms and Conorms, volume 642: Advances in Intelligent Systems and Computing, pages 176â187. Springer International Publishing, Cham, 2018. doi:10.1007/978-3-319-66824-6_16.Adam Grabowski and Takashi Mitsuishi. Initial comparison of formal approaches to fuzzy and rough sets. In Leszek Rutkowski, Marcin Korytkowski, Rafal Scherer, Ryszard Tadeusiewicz, Lotfi A. Zadeh, and Jacek M. Zurada, editors, Artificial Intelligence and Soft Computing â 14th International Conference, ICAISC 2015, Zakopane, Poland, June 14-18, 2015, Proceedings, Part I, volume 9119 of Lecture Notes in Computer Science, pages 160â171. Springer, 2015. doi:10.1007/978-3-319-19324-3_15.Petr HĂĄjek. Metamathematics of Fuzzy Logic. Dordrecht: Kluwer, 1998.Takashi Mitsuishi, Noboru Endou, and Yasunari Shidama. The concept of fuzzy set and membership function and basic properties of fuzzy set operation. Formalized Mathematics, 9(2):351â356, 2001.Lotfi Zadeh. Fuzzy sets. Information and Control, 8(3):338â353, 1965. doi:10.1016/S0019-9958(65)90241-X.12112
Renamings and a Condition-free Formalization of Kroneckerâs Construction
In [7], [9], [10] we presented a formalization of Kroneckerâs construction of a field extension E for a field F in which a given polynomial p â F [X]\F has a root [5], [6], [3]. A drawback of our formalization was that it works only for polynomial-disjoint fields, that is for fields F with F â© F [X] = â
. The main purpose of Kroneckerâs construction is that by induction one gets a field extension of F in which p splits into linear factors. For our formalization this means that the constructed field extension E again has to be polynomial-disjoint.Institute of Informatics, University of Gdansk, PolandGrzegorz Bancerek, CzesĆaw ByliĆski, Adam Grabowski, Artur KorniĆowicz, Roman Matuszewski, Adam Naumowicz, Karol PÄ
k, and Josef Urban. Mizar: State-of-the-art and beyond. In Manfred Kerber, Jacques Carette, Cezary Kaliszyk, Florian Rabe, and Volker Sorge, editors, Intelligent Computer Mathematics, volume 9150 of Lecture Notes in Computer Science, pages 261â279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1. doi:10.1007/978-3-319-20615-8_17.Grzegorz Bancerek, CzesĆaw ByliĆski, Adam Grabowski, Artur KorniĆowicz, Roman Matuszewski, Adam Naumowicz, and Karol PÄ
k. The role of the Mizar Mathematical Library for interactive proof development in Mizar. Journal of Automated Reasoning, 61(1):9â32, 2018. doi:10.1007/s10817-017-9440-6.Nathan Jacobson. Basic Algebra I. Dover Books on Mathematics, 1985.Artur KorniĆowicz. Quotient rings. Formalized Mathematics, 13(4):573â576, 2005.Heinz LĂŒneburg. Gruppen, Ringe, Körper: Die grundlegenden Strukturen der Algebra. Oldenbourg Verlag, 1999.Knut Radbruch. Algebra I. Lecture Notes, University of Kaiserslautern, Germany, 1991.Christoph Schwarzweller. On roots of polynomials over F [X]/ ă pă. Formalized Mathematics, 27(2):93â100, 2019. doi:10.2478/forma-2019-0010.Christoph Schwarzweller. On monomorphisms and subfields. Formalized Mathematics, 27(2):133â137, 2019. doi:10.2478/forma-2019-0014.Christoph Schwarzweller. Field extensions and Kroneckerâs construction. Formalized Mathematics, 27(3):229â235, 2019. doi:10.2478/forma-2019-0022.Christoph Schwarzweller. Representation matters: An unexpected property of polynomial rings and its consequences for formalizing abstract field theory. In M. Ganzha, L. Maciaszek, and M. Paprzycki, editors, Proceedings of the 2018 Federated Conference on Computer Science and Information Systems, volume 15 of Annals of Computer Science and Information Systems, pages 67â72. IEEE, 2018. doi:10.15439/2018F88.28212913
Basic Formal Properties of Triangular Norms and Conorms
SummaryIn the article we present in the Mizar system [1], [8] the catalogue of triangular norms and conorms, used especially in the theory of fuzzy sets [13]. The name triangular emphasizes the fact that in the framework of probabilistic metric spaces they generalize triangle inequality [2].After defining corresponding Mizar mode using four attributes, we introduced the following t-norms: minimum t-norm minnorm (Def. 6),product t-norm prodnorm (Def. 8),Ćukasiewicz t-norm Lukasiewicz_norm (Def. 10),drastic t-norm drastic_norm (Def. 11),nilpotent minimum nilmin_norm (Def. 12),Hamacher product Hamacher_norm (Def. 13), and corresponding t-conorms: maximum t-conorm maxnorm (Def. 7),probabilistic sum probsum_conorm (Def. 9),bounded sum BoundedSum_conorm (Def. 19),drastic t-conorm drastic_conorm (Def. 14),nilpotent maximum nilmax_conorm (Def. 18),Hamacher t-conorm Hamacher_conorm (Def. 17). Their basic properties and duality are shown; we also proved the predicate of the ordering of norms [10], [9]. It was proven formally that drastic-norm is the pointwise smallest t-norm and minnorm is the pointwise largest t-norm (maxnorm is the pointwise smallest t-conorm and drastic-conorm is the pointwise largest t-conorm). This work is a continuation of the development of fuzzy sets in Mizar [6] started in [11] and [3]; it could be used to give a variety of more general operations on fuzzy sets. Our formalization is much closer to the set theory used within the Mizar Mathematical Library than the development of rough sets [4], the approach which was chosen allows however for merging both theories [5], [7].Institute of Informatics, University of BiaĆystok, PolandGrzegorz Bancerek, CzesĆaw ByliĆski, Adam Grabowski, Artur KorniĆowicz, Roman Matuszewski, Adam Naumowicz, Karol PÄ
k, and Josef Urban. Mizar: State-of-the-art and beyond. In Manfred Kerber, Jacques Carette, Cezary Kaliszyk, Florian Rabe, and Volker Sorge, editors, Intelligent Computer Mathematics, volume 9150 of Lecture Notes in Computer Science, pages 261â279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1. doi:10.1007/978-3-319-20615-817.Didier Dubois and Henri Prade. Fuzzy Sets and Systems: Theory and Applications. Academic Press, New York, 1980.Adam Grabowski. The formal construction of fuzzy numbers. Formalized Mathematics, 22(4):321â327, 2014. doi: 10.2478/forma-2014-0032.Adam Grabowski. On the computer-assisted reasoning about rough sets. In B. Dunin-KÈ©plicz, A. Jankowski, A. Skowron, and M. Szczuka, editors, International Workshop on Monitoring, Security, and Rescue Techniques in Multiagent Systems Location, volume 28 of Advances in Soft Computing, pages 215â226, Berlin, Heidelberg, 2005. Springer-Verlag. doi: 10.1007/3-540-32370-815.Adam Grabowski. Efficient rough set theory merging. Fundamenta Informaticae, 135(4): 371â385, 2014. doi: 10.3233/FI-2014-1129.Adam Grabowski. On the computer certification of fuzzy numbers. In M. Ganzha, L. Maciaszek, and M. Paprzycki, editors, 2013 Federated Conference on Computer Science and Information Systems (FedCSIS), Federated Conference on Computer Science and Information Systems, pages 51â54, 2013.Adam Grabowski and Takashi Mitsuishi. Initial comparison of formal approaches to fuzzy and rough sets. In Leszek Rutkowski, Marcin Korytkowski, Rafal Scherer, Ryszard Tadeusiewicz, Lotfi A. Zadeh, and Jacek M. Zurada, editors, Artificial Intelligence and Soft Computing - 14th International Conference, ICAISC 2015, Zakopane, Poland, June 14-18, 2015, Proceedings, Part I, volume 9119 of Lecture Notes in Computer Science, pages 160â171. Springer, 2015. doi: 10.1007/978-3-319-19324-315.Adam Grabowski, Artur KorniĆowicz, and Adam Naumowicz. Four decades of Mizar. Journal of Automated Reasoning, 55(3):191â198, 2015. doi: 10.1007/s10817-015-9345-1.Petr HĂĄjek. Metamathematics of Fuzzy Logic. Dordrecht: Kluwer, 1998.Erich Peter Klement, Radko Mesiar, and Endre Pap. Triangular Norms. Dordrecht: Kluwer, 2000.Takashi Mitsuishi, Noboru Endou, and Yasunari Shidama. The concept of fuzzy set and membership function and basic properties of fuzzy set operation. Formalized Mathematics, 9(2):351â356, 2001.Takashi Mitsuishi, Katsumi Wasaki, and Yasunari Shidama. Basic properties of fuzzy set operation and membership function. Formalized Mathematics, 9(2):357â362, 2001.Lotfi Zadeh. Fuzzy sets. Information and Control, 8(3):338â353, 1965.2529310
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