Simultaneous Method for Solving Certain Systems of Matrix Equations with Two Unknowns
Quantitative bisimulations between weighted finite automata are defined as solutions of certain systems of matrix-vector inequalities and equations. In the context of fuzzy automata and max-plus automata, testing the existence of bisimulations and their computing are performed through a sequence of...
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2024-11-01
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| author | Predrag S. Stanimirović Miroslav Ćirić Spyridon D. Mourtas Gradimir V. Milovanović Milena J. Petrović |
| author_facet | Predrag S. Stanimirović Miroslav Ćirić Spyridon D. Mourtas Gradimir V. Milovanović Milena J. Petrović |
| author_sort | Predrag S. Stanimirović |
| collection | DOAJ |
| description | Quantitative bisimulations between weighted finite automata are defined as solutions of certain systems of matrix-vector inequalities and equations. In the context of fuzzy automata and max-plus automata, testing the existence of bisimulations and their computing are performed through a sequence of matrices that is built member by member, whereby the next member of the sequence is obtained by solving a particular system of linear matrix-vector inequalities and equations in which the previously computed member appears. By modifying the systems that define bisimulations, systems of matrix-vector inequalities and equations with <i>k</i> unknowns are obtained. Solutions of such systems, in the case of existence, witness to the existence of a certain type of partial equivalence, where it is not required that the word functions computed by two WFAs match on all input words, but only on all input words whose lengths do not exceed <i>k</i>. Solutions of these new systems represent finite sequences of matrices which, in the context of fuzzy automata and max-plus automata, are also computed sequentially, member by member. Here we deal with those systems in the context of WFAs over the field of real numbers and propose a different approach, where all members of the sequence are computed simultaneously. More precisely, we apply a simultaneous approach in solving the corresponding systems of matrix-vector equations with two unknowns. Zeroing neural network (ZNN) neuro-dynamical systems for approximating solutions of heterotypic bisimulations are proposed. Numerical simulations are performed for various random initial states and comparison with the <i>Matlab</i>, linear programming solver linprog, and the pseudoinverse solution generated by the standard function pinv is given. |
| format | Article |
| id | doaj-art-94c7a6ef2c9143a999a3945fa5bcf123 |
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| language | English |
| publishDate | 2024-11-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj-art-94c7a6ef2c9143a999a3945fa5bcf1232025-08-20T02:00:51ZengMDPI AGAxioms2075-16802024-11-01131283810.3390/axioms13120838Simultaneous Method for Solving Certain Systems of Matrix Equations with Two UnknownsPredrag S. Stanimirović0Miroslav Ćirić1Spyridon D. Mourtas2Gradimir V. Milovanović3Milena J. Petrović4Faculty of Sciences and Mathematics, University of Niš, Višegradska 33, 18108 Niš, SerbiaFaculty of Sciences and Mathematics, University of Niš, Višegradska 33, 18108 Niš, SerbiaLaboratory “Hybrid Methods of Modelling and Optimization in Complex Systems”, Siberian Federal University, Prosp. Svobodny 79, Krasnoyarsk 660041, RussiaFaculty of Sciences and Mathematics, University of Niš, Višegradska 33, 18108 Niš, SerbiaFaculty of Sciences and Mathematics, University of Priština in Kosovska Mitrovica, Lole Ribara 29, 38220 Kosovska Mitrovica, SerbiaQuantitative bisimulations between weighted finite automata are defined as solutions of certain systems of matrix-vector inequalities and equations. In the context of fuzzy automata and max-plus automata, testing the existence of bisimulations and their computing are performed through a sequence of matrices that is built member by member, whereby the next member of the sequence is obtained by solving a particular system of linear matrix-vector inequalities and equations in which the previously computed member appears. By modifying the systems that define bisimulations, systems of matrix-vector inequalities and equations with <i>k</i> unknowns are obtained. Solutions of such systems, in the case of existence, witness to the existence of a certain type of partial equivalence, where it is not required that the word functions computed by two WFAs match on all input words, but only on all input words whose lengths do not exceed <i>k</i>. Solutions of these new systems represent finite sequences of matrices which, in the context of fuzzy automata and max-plus automata, are also computed sequentially, member by member. Here we deal with those systems in the context of WFAs over the field of real numbers and propose a different approach, where all members of the sequence are computed simultaneously. More precisely, we apply a simultaneous approach in solving the corresponding systems of matrix-vector equations with two unknowns. Zeroing neural network (ZNN) neuro-dynamical systems for approximating solutions of heterotypic bisimulations are proposed. Numerical simulations are performed for various random initial states and comparison with the <i>Matlab</i>, linear programming solver linprog, and the pseudoinverse solution generated by the standard function pinv is given.https://www.mdpi.com/2075-1680/13/12/838weighted finite automatazhang neural networkbisimulationpseudoinverse |
| spellingShingle | Predrag S. Stanimirović Miroslav Ćirić Spyridon D. Mourtas Gradimir V. Milovanović Milena J. Petrović Simultaneous Method for Solving Certain Systems of Matrix Equations with Two Unknowns Axioms weighted finite automata zhang neural network bisimulation pseudoinverse |
| title | Simultaneous Method for Solving Certain Systems of Matrix Equations with Two Unknowns |
| title_full | Simultaneous Method for Solving Certain Systems of Matrix Equations with Two Unknowns |
| title_fullStr | Simultaneous Method for Solving Certain Systems of Matrix Equations with Two Unknowns |
| title_full_unstemmed | Simultaneous Method for Solving Certain Systems of Matrix Equations with Two Unknowns |
| title_short | Simultaneous Method for Solving Certain Systems of Matrix Equations with Two Unknowns |
| title_sort | simultaneous method for solving certain systems of matrix equations with two unknowns |
| topic | weighted finite automata zhang neural network bisimulation pseudoinverse |
| url | https://www.mdpi.com/2075-1680/13/12/838 |
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