1,438 research outputs found
On an Algorithmic Algebra over Simple-Named Complex-Valued Nominative Data
This paper continues formalization in the Mizar system [2, 1] of basic notions of the composition-nominative approach to program semantics [14] which was started in [8, 12, 10].The composition-nominative approach studies mathematical models of computer programs and data on various levels of abstraction and generality and provides tools for reasoning about their properties. In particular, data in computer systems are modeled as nominative data [15]. Besides formalization of semantics of programs, certain elements of the composition-nominative approach were applied to abstract systems in a mathematical systems theory [4, 6, 7, 5, 3].In the paper we give a formal definition of the notions of a binominative function over given sets of names and values (i.e. a partial function which maps simple-named complex-valued nominative data to such data) and a nominative predicate (a partial predicate on simple-named complex-valued nominative data). The sets of such binominative functions and nominative predicates form the carrier of the generalized Glushkov algorithmic algebra for simple-named complex-valued nominative data [15]. This algebra can be used to formalize algorithms which operate on various data structures (such as multidimensional arrays, lists, etc.) and reason about their properties.In particular, we formalize the operations of this algebra which require a specification of a data domain and which include the existential quantifier, the assignment composition, the composition of superposition into a predicate, the composition of superposition into a binominative function, the name checking predicate. The details on formalization of nominative data and the operations of the algorithmic algebra over them are described in [11, 13, 9].Ievgen Ivanov - Taras Shevchenko National University, Kyiv, UkraineArtur KorniĆowicz - Institute of Informatics, University of BiaĆystok, PolandMykola Nikitchenko - Taras Shevchenko National University, Kyiv, UkraineGrzegorz 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.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.Ievgen Ivanov. On the underapproximation of reach sets of abstract continuous-time systems. In Erika ĂbrahĂĄm and Sergiy Bogomolov, editors, Proceedings 3rd International Workshop on Symbolic and Numerical Methods for Reachability Analysis, SNR@ETAPS 2017, Uppsala, Sweden, 22nd April 2017, volume 247 of EPTCS, pages 46â51, 2017. doi:10.4204/EPTCS.247.4.Ievgen Ivanov. On representations of abstract systems with partial inputs and outputs. In T. V. Gopal, Manindra Agrawal, Angsheng Li, and S. Barry Cooper, editors, Theory and Applications of Models of Computation â 11th Annual Conference, TAMC 2014, Chennai, India, April 11â13, 2014. Proceedings, volume 8402 of Lecture Notes in Computer Science, pages 104â123. Springer, 2014. ISBN 978-3-319-06088-0. doi:10.1007/978-3-319-06089-7_8.Ievgen Ivanov. On local characterization of global timed bisimulation for abstract continuous-time systems. In Ichiro Hasuo, editor, Coalgebraic Methods in Computer Science â 13th IFIP WG 1.3 International Workshop, CMCS 2016, Colocated with ETAPS 2016, Eindhoven, The Netherlands, April 2â3, 2016, Revised Selected Papers, volume 9608 of Lecture Notes in Computer Science, pages 216â234. Springer, 2016. ISBN 978-3-319-40369-4. doi:10.1007/978-3-319-40370-0_13.Ievgen Ivanov, Mykola Nikitchenko, and Uri Abraham. On a decidable formal theory for abstract continuous-time dynamical systems. In Vadim Ermolayev, Heinrich C. Mayr, Mykola Nikitchenko, Aleksander Spivakovsky, and Grygoriy Zholtkevych, editors, Information and Communication Technologies in Education, Research, and Industrial Applications: 10th International Conference, ICTERI 2014, Kherson, Ukraine, June 9â12, 2014, Revised Selected Papers, pages 78â99. Springer International Publishing, 2014. ISBN 978-3-319-13206-8. doi:10.1007/978-3-319-13206-8_4.Ievgen Ivanov, Mykola Nikitchenko, and Uri Abraham. Event-based proof of the mutual exclusion property of Petersonâs algorithm. Formalized Mathematics, 23(4):325â331, 2015. doi:10.1515/forma-2015-0026.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 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 KorniĆowicz, Ievgen Ivanov, and Mykola Nikitchenko. Kleene algebra of partial predicates. Formalized Mathematics, 26(1):11â20, 2018. doi:10.2478/forma-2018-0002.Artur KorniĆowicz, Andrii Kryvolap, Mykola Nikitchenko, and Ievgen Ivanov. Formalization of the nominative algorithmic algebra in Mizar. In Jerzy ĆwiÄ
tek, Leszek Borzemski, and Zofia Wilimowska, editors, Information Systems Architecture and Technology: Proceedings of 38th International Conference on Information Systems Architecture and Technology â ISAT 2017: Part II, pages 176â186. Springer International Publishing, 2018. ISBN 978-3-319-67229-8. doi:10.1007/978-3-319-67229-8_16.Nikolaj S. Nikitchenko. A composition nominative approach to program semantics. Technical Report IT-TR 1998-020, Department of Information Technology, Technical University of Denmark, 1998.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_6.26214915
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
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
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
Partial Correctness of a Fibonacci Algorithm
In this paper we introduce some notions to facilitate formulating and proving properties of iterative algorithms encoded in nominative data language [19] in the Mizar system [3], [1]. It is tested on verification of the partial correctness of an algorithm computing n-th Fibonacci number:
i := 0
s := 0
b := 1
c := 0
while (i n)
ââc := s
ââs := b
ââb := c + s
ââi := i + 1
return s
This paper continues verification of algorithms [10], [13], [12] written in terms of simple-named complex-valued nominative data [6], [8], [17], [11], [14], [15]. 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 [16], [18], [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 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_6.28218719
Developmental language disorder and universal grammar
L'étude de la Faculté des langues (FL), telle que définie par la grammaire
générative, a été principalement entreprise à travers l'examen des langues
adultes, l'acquisition de la langue premiĂšre, l'acquisition des langues secondes
et l'acquisition bilingue. Peu de travaux ont abordé la FL à partir d'une situation
d'acquisition atypique, communément appelée Trouble développemental du
langage (TDL). Cette thÚse est consacrée à l'étude de la façon dont FL est
affectée par cette condition malheureuse. Le TDL est manifesté par certains
jeunes enfants et adultes et peut ĂȘtre la cause de limitations importantes dans
le développement du langage. La production et la compréhension langagiÚres
de ce groupe d'enfants sont atypiques par rapport au comportement
linguistique d'autres enfants du mĂȘme Ăąge. Leur atypicitĂ© consiste en une
grammaire non-cible en ce qui concerne ce qui est autorisé et ce qui est interdit
dans la/les langue(s) à laquelle/auxquelles ils sont exposés. Les symptÎmes
les plus communs, d'un point de vue morpho-syntaxique, sont (a) l'omission de
morphÚmes et de mots, (b) les commissions, c'est-à -dire la présence
inadéquate de certains mots ou le remplacement inapproprié de morphÚmes
et (c) les redoublements, c'est-Ă -dire, l'apparition de mots ou de morphĂšmes
dans plus de positions que celles autorisées dans la langue cible. Ces
symptĂŽmes ont Ă©tĂ© pris comme lâindication que la FL est dĂ©ficiente. Le rĂ©sultat
de cette défaillance est une grammaire développée par les enfants ayant le
TDL qui est qualitativement différente de celle développée par leurs pairs
typiques. Cette thÚse examinera si la compétence linguistique sous-jacente
des enfants DLD est dĂ©terminĂ©e par les mĂȘmes traits, opĂ©rations et principes
qui régissent le langage naturel en général. Extraites de la littérature
expĂ©rimentale sur le TDL, les donnĂ©es pour lâanalyse incluent la
compréhension et la production par les enfants du TDL et concernent les
domaines nominal, temporel/verbal et propositionnel. Les propositionsiii
avancées pour rendre compte de ce disorder seront évaluées. Toutes
proposent explicitement ou implicitement que la grammaire universelle (GU),
c'est-à -dire l'ensemble des traits et opérations phonologiques, sémantiques et
syntaxiques qui sous-tendent FL, est dĂ©fectueuse: certains traits peuvent ĂȘtre
absents, ou des opĂ©rations peuvent ĂȘtre inactives ou fonctionner par
intermittence. Contrairement à ces propositions, l'hypothÚse défendue ici est
que la GU n'est pas affectée chez les enfants TDL. C'est-à -dire que malgré les
nombreuses différences entre le TDL et l'acquisition typique du langage, la GU
se rĂ©vĂšle ĂȘtre similaire Ă un certain niveau dans les deux situations
d'acquisition. Si la GU était altérée chez les enfants TDL, on s'attendrait à ce
que les enfants affectés par cette condition produisent des phrases
remarquablement différentes de celles produites par des enfants typiques.
Plusieurs études ont révélé que les enfants DLD et leurs pairs typiques peuvent
montrer des performances linguistiques similaires en termes de quantité et de
type d'erreurs. De plus, les données révÚlent que les énoncés TDL ne sont pas
toujours erronés; lorsque tous les éléments et les mécanismes linguistiques
sont présents, ils sont correctement utilisés. Ceci est considéré comme un
signe que les traits syntaxiques, bien qu'ils ne soient pas toujours réalisés
morpho-phonologiquement, sont présents dans les dérivations syntaxiques
des enfants TDL, et que les opérations syntaxiques Fusion et Accord sont
actives, tout comme dans les grammaires typiques. Enfin, l'analyse des
énoncés non-cibles par les enfants TDL met en évidence une grammaire
syntaxiquement normale et mĂȘme une ressemblance avec des langues
auxquelles ces enfants n'ont pas été exposés. La conclusion est que, malgré
la non-convergence entre le TDL et la langue cible, la GU dans cette situation
d'acquisition est intacte.The study of the Faculty of Language (FL), as defined by generative grammar, has been mainly undertaken through the examination of adult language, first language acquisition, second language acquisition and bilingual acquisition. Few works have approached the FL from an atypical acquisitional situation, standardly called Developmental Language Disorder (DLD). This dissertation is devoted to the study of how FL is affected by this unfortunate condition. DLD is displayed by some young children and adults and can be the cause of significant limitations in language development. The linguistic production and comprehension by this group of children is atypical compared to the linguistic behaviour of other children of the same age. Their atypicality consists in a non-target-like grammar with regard to both what is allowed and what is disallowed in the language(s) to which they are exposed. The most common symptoms, from a morpho-syntactic point of view, are (a) omission of morphemes and words, (b) commissions, i.e., the inadequate presence of certain words or the inappropriate replacement of morphemes and (c) doublings, i.e., the appearance of words or morphemes in more positions than are allowed in the target language. These symptoms have been taken to indicate that the FL is deficient. The result of this deficiency is a grammar developed by children with DLD that is qualitatively different from that developed by their typical peers. This dissertation will consider whether or not the underlying linguistic competence of children with DLD is determined by the same features, operations and principles that regulate natural language in general. Drawn from the experimental literature on DLD, the data for analysis include comprehension and production by children with DLD and concern the nominal, the temporal/verbal and the propositional domains. The proposals that have been put forth to account for this impairment will be evaluated. All of them explicitly or implicitly propose that Universal Grammar (UG), i.e., the set of phonological, semantic and syntactic features and operations that underlie FL, is faulty: Some features can be absent, or operations can be inactive or function intermittently. Contrary to these proposals, the hypothesis defended here is that UG is not affected in DLD children. That is to say, despite the many differences between DLD and typical language acquisition, UG is revealed to be similar at a certain level in both acquisitional situations. If UG were impaired in DLD, children affected by this condition would be expected to produce sentences remarkably different from those produced by typical children. Several studies have shown that children with DLD and their typical peers can display similar linguistic performance in terms of both quantity and type of errors. Moreover, the data reveal that DLD utterances are not always erroneous; when all linguistic elements and mechanisms are present, they are correctly used. This is taken as a sign that syntactic features, while not always realized morpho-phonologically, are present in DLD syntactic derivations, and that the syntactic operations Merge and Agree are active, just as in typical grammars. Finally, the analysis of non-target utterances by children with DLD evinces a syntactically normal grammar and even a resemblance with languages to which these children have not been exposed. The conclusion is that, despite the non-convergence of DLD and the target language, UG in this acquisitional situation is intact
- âŠ