659,923 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
Partial Correctness of an Algorithm Computing Lucas Sequences
In this paper we define some properties about finite sequences and verify the partial correctness of an algorithm computing n-th element of Lucas sequence [23], [20] with given P and Q coefficients as well as two first elements (x and y). The algorithm is encoded in nominative data language [22] in the Mizar system [3], [1].
i := 0
s := x
b := y
c := x
while (i n)
c := s
s := b
ps := p*s
qc := q*c
b := ps â qc
i := i + j
return s
This paper continues verification of algorithms [10], [14], [12], [15], [13] written in terms of simple-named complex-valued nominative data [6], [8], [19], [11], [16], [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 [18], [21], [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.Adrian Jaszczak. Partial correctness of a power algorithm. Formalized Mathematics, 27 (2):189â195, 2019. doi:10.2478/forma-2019-0018.Adrian Jaszczak. General theory and tools for proving algorithms in nominative data systems. Formalized Mathematics, 28(4):269â278, 2020. doi:10.2478/forma-2020-0024.Adrian Jaszczak and Artur KorniĆowicz. Partial correctness of a factorial algorithm. Formalized Mathematics, 27(2):181â187, 2019. doi:10.2478/forma-2019-0017.Artur KorniĆowicz. Partial correctness of a Fibonacci algorithm. Formalized Mathematics, 28(2):187â196, 2020. doi:10.2478/forma-2020-0016.Artur KorniĆowicz, 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 KorniĆowicz, 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.Thomas Koshy. Fibonacci and Lucas Numbers with Applications, Volume 1. John Wiley & Sons, Inc., 2017. ISBN 978-1118742129. doi:10.1002/9781118742327.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.Steven Vajda. Fibonacci & Lucas Numbers, and the Golden Section: Theory and Applications. Dover Publications, 2007. ISBN 978-0486462769.28427928
General Theory and Tools for Proving Algorithms in Nominative Data Systems
In this paper we introduce some new definitions for sequences of operations and extract general theorems about properties of iterative algorithms encoded in nominative data language [20] in the Mizar system [3], [1] in order to simplify the process of proving algorithms in the future.
This paper continues verification of algorithms [10], [13], [12], [14] written in terms of simple-named complex-valued nominative data [6], [8], [18], [11], [15], [16].
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 postconditions [17], [19], [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.Adrian Jaszczak. Partial correctness of a power algorithm. Formalized Mathematics, 27 (2):189â195, 2019. doi:10.2478/forma-2019-0018.Adrian Jaszczak and Artur KorniĆowicz. Partial correctness of a factorial algorithm. Formalized Mathematics, 27(2):181â187, 2019. doi:10.2478/forma-2019-0017.Artur KorniĆowicz. Partial correctness of a Fibonacci algorithm. Formalized Mathematics, 28(2):187â196, 2020. doi:10.2478/forma-2020-0016.Artur KorniĆowicz, 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 KorniĆowicz, 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_6.28426927
The social impact and cultural issues affecting the e-learning performance in Libyan Higher Education institutes
This paper analyses the social impact and cultural issues which affect the e-learning performance in Libyan
Higher Education institutes (HEIs). It is described the development and implementation of e-learning systems in
various HEIs with the emphasis on the digital gap in Libya and barriers to successful e-learning implementation in
these institutions. Also the social impact of using e-learning packages and Internet by young people in Libya is
studied and a SWOT analysis of ICT and e-learning in Tripoli University is performed in order to improve the
effectiveness of the use of e-learning systems in Libyan HEIs
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