Minimal Determinization Algorithm for Fuzzy Automata
The determinization of fuzzy automata is a well-studied problem in theoretical computer science celebrated for its practical applications. Indeed, in the fields of fuzzy discrete event systems, fault diagnosis, clinical monitoring, decision-making systems, and model checking, when a suitable model o...
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| Veröffentlicht in: | IEEE transactions on fuzzy systems Jg. 31; H. 11; S. 1 - 10 |
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IEEE
01.11.2023
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| Abstract | The determinization of fuzzy automata is a well-studied problem in theoretical computer science celebrated for its practical applications. Indeed, in the fields of fuzzy discrete event systems, fault diagnosis, clinical monitoring, decision-making systems, and model checking, when a suitable model of a fuzzy automaton is employed, it is desirable to find its language-equivalent deterministic version because of its computational efficiency. Although many methods have been developed to convert a fuzzy automaton to its language equivalent fuzzy deterministic finite automaton (FDfA), they can be applied only for fuzzy automata defined over specific underlying sets of truth values. For example, recently developed determinization methods employ the concept of maximal factorization, which can be defined only on non-locally finite lattices or the Boolean lattice. In addition, not all such determinization methods result in a minimal FDfA. On the other hand, even though such determinization methods have been developed for fuzzy automata over specific underlying structures, these methods cannot be generalized for fuzzy automata over locally finite lattices. This article focuses on filling this gap and develops a novel method for computing a minimal FDfA for a fuzzy automaton defined over a locally finite and divisible residuated lattice. Our method uses the new concept of a reduction graph that emerges from the strict order relation on the resulting fuzzy states, according to which we can construct all minimal FDfAs equivalent to a given fuzzy automaton. |
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| AbstractList | The determinization of fuzzy automata is a well-studied problem in theoretical computer science celebrated for its practical applications. Indeed, in the fields of fuzzy discrete event systems, fault diagnosis, clinical monitoring, decision-making systems, and model checking, when a suitable model of a fuzzy automaton is employed, it is desirable to find its language-equivalent deterministic version because of its computational efficiency. Although many methods have been developed to convert a fuzzy automaton to its language equivalent fuzzy deterministic finite automaton (FDfA), they can be applied only for fuzzy automata defined over specific underlying sets of truth values. For example, recently developed determinization methods employ the concept of maximal factorization, which can be defined only on non-locally finite lattices or the Boolean lattice. In addition, not all such determinization methods result in a minimal FDfA. On the other hand, even though such determinization methods have been developed for fuzzy automata over specific underlying structures, these methods cannot be generalized for fuzzy automata over locally finite lattices. This article focuses on filling this gap and develops a novel method for computing a minimal FDfA for a fuzzy automaton defined over a locally finite and divisible residuated lattice. Our method uses the new concept of a reduction graph that emerges from the strict order relation on the resulting fuzzy states, according to which we can construct all minimal FDfAs equivalent to a given fuzzy automaton. The determinization of fuzzy automata is a well-studied problem in theoretical computer science celebrated for its practical applications. Indeed, in the fields of fuzzy discrete event systems, fault diagnosis, clinical monitoring, decision-making systems, and model checking, when a suitable model of a fuzzy automaton is employed, it is desirable to find its language-equivalent deterministic version because of its computational efficiency. Although many methods have been developed to convert a fuzzy automaton to its language equivalent fuzzy deterministic finite automaton (FDfA), they can be applied only for fuzzy automata defined over specific underlying sets of truth values. For example, recently developed determinization methods employ the concept of maximal factorization, which can be defined only on nonlocally finite lattices or the Boolean lattice. In addition, not all such determinization methods result in a minimal FDfA. On the other hand, even though such determinization methods have been developed for fuzzy automata over specific underlying structures, these methods cannot be generalized for fuzzy automata over locally finite lattices. This article focuses on filling this gap and develops a novel method for computing a minimal FDfA for a fuzzy automaton defined over a locally finite and divisible residuated lattice. Our method uses the new concept of a reduction graph that emerges from the strict order relation on the resulting fuzzy states, according to which we can construct all minimal FDfAs equivalent to a given fuzzy automaton. |
| Author | Grau, Aitor G. de Mendivil Farina, Federico Stanimirovic, Stefan |
| Author_xml | – sequence: 1 givenname: Aitor G. de Mendivil orcidid: 0000-0001-8961-8736 surname: Grau fullname: Grau, Aitor G. de Mendivil organization: Departamento de Estadística, Informática y Matemáticas, Universidad Pública de Navarra, Pamplona, Spain – sequence: 2 givenname: Stefan orcidid: 0000-0003-3126-1265 surname: Stanimirovic fullname: Stanimirovic, Stefan organization: University of Niš, Faculty of Sciences and Mathematics, Višegradska 33, Niš, Serbia – sequence: 3 givenname: Federico surname: Farina fullname: Farina, Federico organization: Departamento de Estadística, Informática y Matemáticas, Universidad Pública de Navarra, Pamplona, Spain |
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| References_xml | – ident: ref22 doi: 10.1109/TFUZZ.2015.2404348 – volume: 249 start-page: 73 year: 2014 ident: ref21 article-title: Brzozowski type determinization for fuzzy automata publication-title: Fuzzy Sets Syst doi: 10.1016/j.fss.2014.02.021 – ident: ref35 doi: 10.1016/j.fss.2017.09.004 – ident: ref15 doi: 10.1360/02yf9038 – ident: ref37 doi: 10.1145/2631915 – year: 2012 ident: ref30 publication-title: Fuzzy Relational Systems Foundations and Principles – year: 2019 ident: ref25 article-title: Improved algorithms for determinization of fuzzy and weighted automata publication-title: Fac Sci Math – ident: ref13 doi: 10.1016/S0020-0255(02)00192-5 – volume: 181 start-page: 1358 year: 2011 ident: ref19 article-title: An improved algorithm for determinization of weighted and fuzzy automata publication-title: Inf Sci doi: 10.1016/j.ins.2010.12.008 – ident: ref6 doi: 10.1109/TFUZZ.2014.2310466 – ident: ref23 doi: 10.1016/j.ins.2014.06.033 – ident: ref14 doi: 10.1007/BF02713945 – ident: ref29 doi: 10.1016/j.fss.2019.03.009 – year: 2007 ident: ref1 publication-title: Introduction to Automata Theory – ident: ref2 doi: 10.1007/978-3-540-70844-5_17 – ident: ref17 doi: 10.1016/j.ins.2007.08.003 – ident: ref11 doi: 10.4236/jcc.2017.510003 – volume: 301 start-page: 79 year: 2015 ident: ref20 article-title: Further improvements of determinization methods for fuzzy finite automata publication-title: Fuzzy Sets Syst – ident: ref12 doi: 10.1016/j.fss.2015.12.011 – ident: ref33 doi: 10.1007/978-3-642-01492-5_12 – ident: ref10 doi: 10.1016/j.ins.2009.01.015 – ident: ref8 doi: 10.1109/TFUZZ.2020.2988848 – ident: ref18 doi: 10.1016/j.fss.2009.06.007 – ident: ref36 doi: 10.1016/j.fss.2015.09.012 – ident: ref26 doi: 10.1016/j.ins.2018.08.033 – volume: 17 start-page: 578 year: 2009 ident: ref5 article-title: Fuzzy discrete-event systems under fuzzy observability and a test algorithm publication-title: IEEE Trans Fuzzy Syst doi: 10.1109/TFUZZ.2008.924212 – ident: ref16 doi: 10.1016/j.fss.2005.04.004 – ident: ref32 doi: 10.1016/j.artint.2006.12.002 – ident: ref4 doi: 10.1109/TSMCB.2004.840457 – ident: ref28 doi: 10.1109/TFUZZ.2014.2315620 – ident: ref34 doi: 10.1016/j.ins.2009.11.030 – year: 2005 ident: ref31 article-title: Fuzzy equational logic publication-title: Studies in Fuzziness and Soft Computing – ident: ref7 doi: 10.1016/j.fss.2009.08.005 – year: 2002 ident: ref9 publication-title: Fuzzy Automata and Languages Theory and Applications – volume: 12 start-page: 529 year: 0 ident: ref3 article-title: Canonical regular expressions and minimal state graphs for definite events publication-title: Proc Symp Math Theory Automata MRI Symposia Ser – ident: ref24 doi: 10.1109/TFUZZ.2017.2775601 – ident: ref27 doi: 10.1016/j.fss.2019.07.006 |
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| SubjectTerms | Algorithms Automata Automata theory Brzozowski's procedure complete deterministic fuzzy automaton Discrete event systems Equivalence factorization of fuzzy states Fault diagnosis Fuzzy finite automata Fuzzy systems Lattices locally finite lattices Mathematical models minimal determinization method Model checking Monitoring Nickel |
| Title | Minimal Determinization Algorithm for Fuzzy Automata |
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