Abstract
A bialgebra is a structure which is simultaneously an algebra and a coalgebra, such that the algebraic and coalgebraic parts are “compatible”. In this paper, we apply the defining diagrams of algebras, coalgebras, and bialgebras to categories of semimodules and semimodule homomorphisms over a commutative semiring. We then show that formal language theory and the theory of bialgebras have essentially undergone “convergent evolution”. For example, formal languages correspond to elements of dual algebras of coalgebras, automata are “pointed representation objects” of algebras, automaton morphisms are instances of linear intertwiners, and a construction from the theory of bialgebras shows how to run two automata in parallel. We also show how to associate an automaton with an arbitrary algebra, which in the classical case yields the automaton whose states are formal languages and whose transitions are given by language differentiation.
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References
Adàmek, J., Trnkovà, V.: Automata and algebras in categories. Kluwer Academic Publishers, Dordrecht (1990)
Buchholz, P.: Bisimulation relations for weighted automata. Theoretical Computer Science 393, 109–123 (2008)
Crutchfield, J.P., Moore, C.: Quantum automata and quantum grammars. Theoretical Computer Science 237, 275–306 (2000)
Duchamp, G., Flouret, M., Laugerotte, È., Luque, J.-G.: Direct and dual laws for automata with multiplicities. Theoretical Computer Science 267, 105–120 (2001)
Duchamp, G., Tollu, C.: Sweedler’s duals and Schützenberger’s calculus. Arxiv Preprint. arXiv:0712.0125v2
Fitting, M.: Bisimulations and boolean vectors. Advances in Modal Logic 4, 97–125 (2003)
Golan, J.S.: Semirings and their applications. Kluwer Academic Publishers, Dordrecht (1999)
Grossman, R.L., Larson, R.G.: Bialgebras and realizations. In: Bergen, J., Catoiu, S., Chin, W. (eds.) Hopf Algebras, pp. 157–166. Marcel Dekker, Inc., New York (2004)
Grossman, R.L., Larson, R.G.: The realization of input-output maps using bialgebras. Forum Mathematicum 4, 109–121 (1992)
Katsov, Y.: Tensor products and injective envelopes of semimodules over additively regular semirings. Algebra Colloquium 4(2), 121–131 (1997)
Kozen, D.: Automata on guarded strings and applications. Matématica Contemporânea 24, 117–139 (2003)
Kozen, D.: A completeness theorem for Kleene algebras and the algebra of regular events. Infor. and Comput. 110(2), 366–390 (1994)
Lang, S.: Algebra: revised, 3rd edn. Springer, Heidelberg (2002)
Litvinov, G.L., Masloc, V.P., Shpiz, G.B.: Tensor products of idempotent semimodules. An algebraic approach. Mathematical Notes 65(4) (1999)
Majid, S.: Foundations of quantum group theory. Cambridge University Press, Cambridge (1995)
Rutten, J.J.M.M.: Universal coalgebra: a theory of systems. Theoretical Computer Science 249, 3–80 (2000)
Street, R.: Quantum groups: a path to current algebra. Cambridge University Press, Cambridge (2007)
Worthington, J.: Automatic proof generation in Kleene algebra. In: Berghammer, R., Möller, B., Struth, G. (eds.) RelMiCS/AKA 2008. LNCS, vol. 4988, pp. 382–396. Springer, Heidelberg (2008)
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Worthington, J. (2008). A Bialgebraic Approach to Automata and Formal Language Theory. In: Artemov, S., Nerode, A. (eds) Logical Foundations of Computer Science. LFCS 2009. Lecture Notes in Computer Science, vol 5407. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-92687-0_31
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DOI: https://doi.org/10.1007/978-3-540-92687-0_31
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