Abstract
Already in the seventies, strong results illustrating the intimate relationship between category theory and automata theory have been described and are still investigated. In this column, we provide a uniform presentation of the basic concepts that underlie minimization results in automata theory. We then use this knowledge for introducing a new model of automata that is an hybrid of deterministic finite automata and automata weighted over a field. These automata are very natural, and enjoy minimization result by design.
The presentation of this paper is indeed categorical in essence, but it assumes no prior knowledge from the reader. It is also non-conventional in that it is neither algebraic, nor co-algebraic oriented.
- Jiří Adámek, Filippo Bonchi, Mathias Hülsbusch, Barbara König, Stefan Milius, and Alexandra Silva. 2012. A Coalgebraic Perspective on Minimization and Determinization. In Proceedings of the 15th International Conference on Foundations of Software Science and Computational Structures (FOSSAC'12). Springer-Verlag, Berlin, Heidelberg, 58--73. Google ScholarDigital Library
- Jiří (ing) Adámek, Horst Herrlich, and George E. Strecker. 1990. Abstract and concrete categories : the joy of cats. Wiley, New York. http://opac.inria.fr/record=b1087598 A Wiley-Interscience publication. Google ScholarDigital Library
- Jiří (ing) Adámek and Věra Trnková. 1989. Automata and Algebras in Categories. Springer Netherlands, New York. http://www.springer.com/fr/book/9780792300106Google Scholar
- Michael A. Arbib and Ernest G. Manes. 1974a. Basic concepts of category theory applicable to computation and control. In Category Theory Applied to Computation and Control (Lecture Notes in Computer Science), Vol. 25. Springer, 1--34. Google ScholarDigital Library
- Michael A. Arbib and Ernest G. Manes. 1974b. A categorist's view of automata and systems. In Category Theory Applied to Computation and Control (Lecture Notes in Computer Science), Vol. 25. Springer, 51--64. Google ScholarDigital Library
- Michael A. Arbib and Ernest G. Manes. 1975. Adjoint machines, state-behavior machines, and duality. Journal of Pure and Applied Algebra 6, 3 (1975), 313 -- 344.Google ScholarCross Ref
- Michael A. Arbib and Ernest G. Manes. 1980. Machines in a category. Journal of Pure and Applied Algebra19 (1980), 9--20.Google Scholar
- E. S. Bainbridge. 1974. Adressed machines and duality. In Category Theory Applied to Computation and Control (Lecture Notes in Computer Science), Vol. 25. Springer, 93--98. Google ScholarDigital Library
- Nicolas Bedon. 1996. Finite automata and ordinals. (1996), 119--144. Google ScholarDigital Library
- Nicolas Bedon, Alexis Bès, Olivier Carton, and Chloe Rispal. 2010. Logic and Rational Languages of Words Indexed by Linear Orderings. Theory Comput. Syst. 46, 4 (2010), 737--760. Google ScholarDigital Library
- Nick Bezhanishvili, Clemens Kupke, and Prakash Panangaden. 2012. Minimization via Duality. Springer Berlin Heidelberg, Berlin, Heidelberg, 191--205.Google Scholar
- Mikołaj Bojańczyk. 2011. Data Monoids. In STACS 2011: 28th International Symposium on Theoretical Aspects of Computer Science (LIPIcs), Thomas Schwentick and Christoph Dürr (Eds.), Vol. 9. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 105--116.Google Scholar
- Mikołaj Bojańczyk. 2015. Recognisable Languages over Monads. In DLT (Lecture Notes in Computer Science), Vol. 9168. Springer, 1--13.Google Scholar
- Mikołaj Bojańczyk, Bartek Klin, and Slawomir Lasota. 2014. Automata theory in nominal sets. Logical Methods in Computer Science 10, 3 (2014).Google Scholar
- Mikołaj Bojańczyk and Igor Walukiewicz. 2008. Forest algebras. In Logic and Automata: History and Perspectives {in Honor of Wolfgang Thomas}. 107--132.Google Scholar
- Filippo Bonchi, Marcello Bonsangue, Michele Boreale, Jan Rutten, and Alexandra Silva. 2012. A coalgebraic perspective on linear weighted automata. Information and Computation 211 (2012), 77 -- 105. Google ScholarDigital Library
- Filippo Bonchi, Marcello M. Bonsangue, Helle Hvid Hansen, Prakash Panangaden, Jan J. M. M. Rutten, and Alexandra Silva. 2014. Algebra-coalgebra duality in Brzozowski's minimization algorithm. ACM Trans. Comput. Log. 15, 1 (2014), 3:1--3:29. Google ScholarDigital Library
- Filippo Bonchi, Marcello M. Bonsangue, Jan J. M. M. Rutten, and Alexandra Silva. 2012. Brzozowski's Algorithm (Co)Algebraically. In Logic and Program Semantics (Lecture Notes in Computer Science), Vol. 7230. Springer, 12--23. Google ScholarDigital Library
- Filippo Bonchi and Damien Pous. 2013. Checking NFA equivalence with bisimulations up to congruence. In POPL. ACM, 457--468. Google ScholarDigital Library
- Walter S. Brainerd. 1968. The Minimalization of Tree Automata. Information and Control 13, 5 (1968), 484--491.Google ScholarCross Ref
- Olivier Carton, Thomas Colcombet, and Gabriele Puppis. 2011. Regular Languages of Words over Countable Linear Orderings. In ICALP 2011 (2): Automata, Languages and Programming - 38th International Colloquium, Luca Aceto, Monika Henzinger, and Jiri Sgall (Eds.), Vol. 6756. 125--136. Google ScholarDigital Library
- Christian Choffrut. 1979. A Generalization of Ginsburg and Rose's Characterization of G-S-M Mappings. In Automata, Languages and Programming, 6th Colloquium, Graz, Austria, July 16--20, 1979, Proceedings (Lecture Notes in Computer Science), Hermann A. Maurer (Ed.), Vol. 71. Springer, 88--103. Google ScholarDigital Library
- Christian Choffrut. 2003. Minimizing subsequential transducers: a survey. Theor. Comput. Sci. 292, 1 (2003), 131--143. Google ScholarDigital Library
- Thomas Colcombet. 2009. The theory of stabilisation monoids and regular cost functions. In ICALP 2009 (2): Automata, Languages and Programming, 36th International Colloquium, Susanne Albers, Alberto Marchetti-Spaccamela, Yossi Matias, Sotiris E. Nikoletseas, and Wolfgang Thomas (Eds.), Vol. 5556. 139--150. Google ScholarDigital Library
- Thomas Colcombet. 2013. Regular cost functions, Part I: logic and algebra over words. 9, 3 (2013), 47.Google Scholar
- Thomas Colcombet and Sreejith A. V. 2015. Limited Set Quantifiers over Countable Linear Orders. In Automata, Languages, and Programming - 42nd International Colloquium, ICALP 2015 (Lecture Notes in Computer Science), Magnús M. Halldórsson, Kazuo Iwama, Naoki Kobayashi, and Bettina Speckmann (Eds.), Vol. 9135. Springer, 146--158. Google ScholarDigital Library
- Samuel Eilenberg. 1974. Automata, Languages, and Machines. Vol. Volume A. Academic Press, Inc., Orlando, FL, USA. Google ScholarDigital Library
- Samuel Eilenberg. 1976. Automata, Languages, and Machines. Vol. Volume B. Academic Press, Inc., Orlando, FL, USA. Google ScholarDigital Library
- Samuel Eilenberg and Jesse B. Wright. 1967. Automata in general algebras. Information and Control 11, 4 (1967), 452 -- 470.Google ScholarCross Ref
- Mai Gehrke, Serge Grigorieff, and Jean-Eric Pin. 2008. Duality and Equational Theory of Regular Languages. In ICALP (2) (Lecture Notes in Computer Science), Vol. 5126. Springer, 246--257. Google ScholarDigital Library
- Mai Gehrke, Serge Grigorieff, and Jean-Eric Pin. 2010. A Topological Approach to Recognition. In ICALP (2) (Lecture Notes in Computer Science), Vol. 6199. Springer, 151--162. Google ScholarDigital Library
- Mai Gehrke, Daniela Petrişan, and Luca Reggio. 2017. Quantifiers on languages and codensity monads. LICS (2017).Google Scholar
- J. A. Goguen. 1972. Minimal realization of machines in closed categories. Bull. Amer. Math. Soc. 78, 5 (09 1972), 777--783. http://projecteuclid.org/euclid.bams/1183533991Google Scholar
- Bart Jacobs and Jan Rutten. 1997. A Tutorial on (Co)Algebras and (Co)Induction. EATCS Bulletin 62 (1997), 62--222.Google Scholar
- R. E. Kalman. 1963. Mathematical Description of Linear Dynamical Systems. Journal of the Society for Industrial and Applied Mathematics Series A Control 1, 2 (Jan. 1963), 152--192.Google ScholarCross Ref
- Denis Kuperberg. 2011. Linear temporal logic for regular cost functions. In STACS 2011: 28th International Symposium on Theoretical Aspects of Computer Science (LIPIcs), Thomas Schwentick and Christoph Dürr (Eds.), Vol. 9. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 627--636.Google Scholar
- Saunders Mac Lane. 1978. Categories for the Working Mathematician. (1978). http://link.springer.com/book/10.1007/978-1-4757-4721-8Google Scholar
- Dominique Perrin and Jean-Éric Pin. 1995. Semigroups and automata on infinite words. In NATO Advanced Study Institute Semigroups, Formal Languages and Groups, J. Fountain (Ed.). Kluwer academic publishers, 49--72.Google Scholar
- Libor Polák. 2001. Syntactic Semiring of a Language. In Mathematical Foundations of Computer Science 2001, 26th International Symposium, MFCS 2001 Marianske Lazne, Czech Republic, August 27--31, 2001, Proceedings. 611--620. Google ScholarDigital Library
- Michael O. Rabin and Dana Scott. 1959. Finite automata and their decision problems. IBM J. Res. and Develop. 3 (April 1959), 114--125. Google ScholarDigital Library
- Jurriaan Rot. 2016. Coalgebraic Minimization of Automata by Initiality and Finality. Electronic Notes in Theoretical Computer Science 325 (2016), 253 -- 276.Google ScholarCross Ref
- J.J.M.M. Rutten. 2000. Universal coalgebra: a theory of systems. Theoretical Computer Science 249, 1 (2000), 3 -- 80. Google ScholarDigital Library
- M.P. Schützenberger. 1961. On the definition of a family of automata. Information and Control 4, 2 (1961), 245 -- 270.Google ScholarCross Ref
- Marcel-Paul Schützenberger. 1965. On finite monoids having only trivial subgroups. Information and Control 8 (1965), 190--194.Google ScholarCross Ref
- Alexandra Silva. 2015. A Short Introduction to the Coalgebraic Method. ACM SIGLOG News 2, 2 (April 2015), 16--27. Google ScholarDigital Library
Index Terms
- Automata and minimization
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