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On characterizations of recursively enumerable languages. (English) Zbl 0686.68060

V. Geffert [Theor. Comput. Sci. 62, 235-249 (1988; Zbl 0664.68075)] has shown that each recursively enumerable language L over \(\Sigma\) can be expressed in the form \(L=\{h(x)^{-1}g(x)|\) x in \(\Delta^+\}\cap \Sigma^*\) where \(\Delta\) is an alphabet and g, h is a pair of morphisms. Our purpose is to give a simple proof for Geffert’s result and then sharpen it into the form where both of the morphisms are nonerasing. In our method we modify constructions used in a representation of recursively enumerable languages in terms of equality sets and in a characterization of simple transducers in terms of morphisms. As direct consequences, we get the undecidability of the Post correspondence problem and various representations of L. For instance, \(L=\rho (L_ 0)\cap \Sigma^*\) where \(L_ 0\) is a minimal linear language and \(\rho\) is the Dyck reduction \(a\bar a\to \epsilon\), \(A\bar A\to \epsilon\).
Reviewer: M.Latteux

MSC:

68Q45 Formal languages and automata

Citations:

Zbl 0664.68075
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[1] Baker, B.S., Book, R.V.: Reversal-bounded multipushdown machines. J. Comput. Syst. Sci.8, 315-332 (1974) · Zbl 0309.68043
[2] Book, R.V., Jantzen, M., Wrathall, C.: Monadic Thue systems. Theor. Comput. Sci.19, 231-251 (1982) · Zbl 0488.03020
[3] Culik, K. II: A purely homomorphic characterization of recursively enumerable sets. J. Assoc. Comput. Math.26, 345-350 (1979) · Zbl 0395.68076
[4] Engelfriet, J., Rozenberg, G.: Fixed point languages, equality languages, and representation of recursively enumerable languages. J. Assoc. Comput. Mach.27, 499-518 (1980) · Zbl 0475.68047
[5] Geffert, V.: A representation of recursively enumerable languages by two homomorphisms and a quotient. Theor. Comput. Sci.62, 235-249 (1988) · Zbl 0664.68075
[6] Geffert, V.: Context-free-like forms for the phrase-structure grammars. (Lect. Notes Comput. Sci., vol. 324, pp. 309-317) Berlin Heidelberg New York: Springer 1988 · Zbl 0652.68095
[7] Latteux, M., Leguy, J.: On composition of morphisms and inverse morphisms. (Lect. Notes Comput. Sci., vol. 154, pp. 420-432) Berlin Heidelberg New York: Springer 1983 · Zbl 0523.68067
[8] Latteux, M., Leguy, J., Ratoandromanana, B.: The family of one-counter languages is closed under quotient. Acta Inf.22, 579-588 (1985) · Zbl 0585.68069
[9] Latteux, M., Timmerman, E.: Bifaithful starry transductions. Inf. Process. Lett.28, 1-4 (1988) · Zbl 0666.68072
[10] Pin, J.E.: Sur le monoide syntactique deL * lorsqueL est un langage fini. Theor. Comput. Sci.7, 211-215 (1978) · Zbl 0388.20050
[11] Salomaa, A.: Formal languages. New York: Academic Press 1973 · Zbl 0262.68025
[12] Salomaa, A.: Jewels of Formal Language Theory. Rockville, Maryland: Computer Sience Press, 1981 · Zbl 0487.68064
[13] Savitch, W.J.: How to make arbitrary grammars look like context-free grammars. SIAM J. Comput.2, 174-182 (1973) · Zbl 0298.68057
[14] Turakainen, P.: On characterization of recursively enumerable languages in terms of linear languages and VW-grammars. Indagationes Math.40, 145-153 (1978) · Zbl 0369.68049
[15] Turakainen, P.: A machine-oriented approach to compositions of morphisms and inverse morphisms. Bull. EATCS20, 162-166 (1983)
[16] Turakainen, P.: Transducers and compositions of morphisms and inverse morphisms. In: Studies in honour of Arto Kustaa Salomaa on the occasion of his fiftieth birthday. Ann. Univ. Turku. Ser. A I186, 118-128 (1984) · Zbl 0541.68045
[17] Vitányi, P.M.B.: A note on DPDA transductions of {0,1}* and inverse DPDA transductions of the Dyck set. Int. J. Comput. Math.9, 131-137 (1981) · Zbl 0454.68098
[18] Vitányi, P.M.B., Savitch, W.J.: On inverse deterministic pushdown transductions. J. Comput. Syst. Sci.16, 423-444 (1978) · Zbl 0376.68051
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