Abstract
The algebraic approach to formal language and automata theory is a continuation of the earliest traditions in these fields which had sought to represent languages, translations and other computations as expressions (e.g. regular expressions) in suitably-defined algebras; and grammars, automata and transitions as relational and equational systems over these algebras that have such expressions as their solutions.
As part of a larger programme to algebraize the classical results of formal language and automata theory, we have recast and generalized the Chomsky hierarchy as a complete lattice of dioid algebras. Here, we will formulate a general construction by ideals that yields a family of adjunctions between the members of this hierarchy.
In addition, we will briefly discuss the extension of the dioid hierarchy to semirings and power series algebras.
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References
Hopkins, M.W.: The Algebraic Approach I: The Algebraization of the Chomsky Hierarchy. RelMiCS 2008 (to be published, 2008)
Gruska, J.: A Characterization of Context-Free Languages. Journal of Computer and System Sciences 5, 353–364 (1971)
McWhirtier, I.P.: Substitution Expressions. Journal of Computer and System Sciences 5, 629–637 (1971)
Yntema, M.K.: Cap Expressions for Context-Free Languages. Information and Control 8, 311–318 (1971)
Ésik, Z., Leiss, H.: Algebraically complete semirings and Greibach normal form. Ann. Pure. Appl. Logic. 133, 173–203 (2005)
Ésik, Z., Kuich, W.: Rationally Additive Semirings. Journal of Computer Science 8, 173–183 (2002)
Berstel, J., Reutenauer, C.: Les Séries Rationelles et Leurs Langages. Masson (1984). English edition: Rational Series and Their Languages. Springer, Heidelberg (1988)
Kozen, D.: On Kleene Algebras and Closed Semirings. In: Rovan, B. (ed.) MFCS 1990. LNCS, vol. 452, pp. 26–47. Springer, Heidelberg (1990)
Conway, J.H.: Regular Algebra and Finite Machines. Chapman and Hall, London (1971)
MacLane, S.: Categories for the Working Mathematician. Springer, Heidelberg (1971)
Kuich, W., Salomaa, A.: Semirings, Automata and Languages. Springer, Berlin (1986)
Chomsky, N., Schützenberger, M.P.: The Algebraic Theory of Context-Free Languages. In: Braort, P., Hirschberg, D. (eds.) Computer Programming and Formal Systems, pp. 118–161. North-Holland, Amsterdam (1963)
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Hopkins, M. (2008). The Algebraic Approach II: Dioids, Quantales and Monads. In: Berghammer, R., Möller, B., Struth, G. (eds) Relations and Kleene Algebra in Computer Science. RelMiCS 2008. Lecture Notes in Computer Science, vol 4988. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-78913-0_14
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DOI: https://doi.org/10.1007/978-3-540-78913-0_14
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