no. 2
Decision problems among the main subfamilies of rational relations
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 40 (2006) no. 2, pp. 255-275.

We consider the four families of recognizable, synchronous, deterministic rational and rational subsets of a direct product of free monoids. They form a strict hierarchy and we investigate the following decision problem: given a relation in one of the families, does it belong to a smaller family? We settle the problem entirely when all monoids have a unique generator and fill some gaps in the general case. In particular, adapting a proof of Stearns, we show that it is recursively decidable whether or not a deterministic subset of an arbitrary number of free monoids is recognizable. Also we exhibit a single exponential algorithm for determining if a synchronous relation is recognizable.

DOI : 10.1051/ita:2006005
Classification : 3D05, 68Q45
Mots clés : multitape automata, Presburger arithmetics, decision problems 
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     title = {Decision problems among the main subfamilies of rational relations},
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Carton, Olivier; Choffrut, Christian; Grigorieff, Serge. Decision problems among the main subfamilies of rational relations. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 40 (2006) no. 2, pp. 255-275. doi : 10.1051/ita:2006005. http://www.numdam.org/articles/10.1051/ita:2006005/

[1] J. Berstel, Transductions and context-free languages. B.G. Teubner (1979). | MR | Zbl

[2] A. Bertoni and P. Massazza, On the inclusion problem for finitely ambiguous rational trace languages. RAIRO: Inform. Théor. Appl. 32 (1998) 79-98. | EuDML

[3] S. Eilenberg, Automata, Languages and Machines, Vol. A. Academic Press (1974). | MR | Zbl

[4] S. Eilenberg, C.C. Elgot and J.C. Shepherdson, Sets recognized by n-tape automata. J. Algebra 3 (1969) 447-464. | Zbl

[5] S. Eilenberg and M.-P. Schützenbeger, Rational sets in commutative monoids. J. Algebra 13 (1969) 173-191. | Zbl

[6] C.C. Elgot and J.E. Mezei, On Relations Defined by Finite Automata. IBM J. 10 (1965) 47-68. | Zbl

[7] P.C. Fischer and A.L. Rosenberg, Multitape one-way nonwriting automata. J. Comput. Syst. Sci. 2 (1968) 88-101. | Zbl

[8] S. Ginsburg and E.H. Spanier, Semigroups, Presburger formulas, and languages. Pacific J. Math. 16 (1966) 285-296. | Zbl

[9] E. Grädel, Subclasses of Presburger arithmetic and the polynomial hierarchy. Theoret. Comput. Sci. 56 (1989) 281-301. | Zbl

[10] O.H. Ibarra, The unsolvability of the equivalence problem for ϵ-free ngsm’s with unary input (output) alphabets and application. SIAM J. Comput. 7 (1978) 520-532. | Zbl

[11] L.P. Lisovik, The identity problem for regular events over the direct product of free and cyclic semigroups. Dok. Akad. Nauk Ukrainskoj RSR Ser. A. 6 (1979) 410-413. | Zbl

[12] M. Rabin and D. Scott, Finite automata and their decision problems. IBM J. Res. Develop (1959) 125-144. | Zbl

[13] J. Sakarovitch, Éléments de théorie des automates. Vuibert Informatique (2003).

[14] R.E. Stearns, A regularity test for pushdown machines. Inform. Control 11 (1967) 323-340. | Zbl

[15] L.G. Valiant, Regularity and related problems for deterministic pushdown automata. Inform. Control 22 (1975) 1-10. | Zbl

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