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Decision problems among the main subfamilies of rational relations

Published online by Cambridge University Press:  20 July 2006

Olivier Carton ,
Christian Choffrut  and
Serge Grigorieff
Olivier Carton
Affiliation:
LIAFA, Université Paris 7 & CNRS, 2 pl. Jussieu, 75251 Paris Cedex 05, France; Christian.Choffrut@liafa.jussieu.fr
Christian Choffrut
Affiliation:
LIAFA, Université Paris 7 & CNRS, 2 pl. Jussieu, 75251 Paris Cedex 05, France; Christian.Choffrut@liafa.jussieu.fr
Serge Grigorieff
Affiliation:
LIAFA, Université Paris 7 & CNRS, 2 pl. Jussieu, 75251 Paris Cedex 05, France; Christian.Choffrut@liafa.jussieu.fr
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Abstract

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.

Keywords

Multitape automataPresburger arithmeticsdecision problems.
Type
Research Article
Information
RAIRO - Theoretical Informatics and Applications , Volume 40 , Issue 2: Alberto Bertoni: Climbing summits , April 2006 , pp. 255 - 275
Copyright
© EDP Sciences, 2006

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References

J. Berstel, Transductions and context-free languages. B.G. Teubner (1979).
Bertoni, A. and Massazza, P., On the inclusion problem for finitely ambiguous rational trace languages. RAIRO: Inform. Théor. Appl. 32 (1998) 79–98.
S. Eilenberg, Automata, Languages and Machines, Vol. A. Academic Press (1974).
Eilenberg, S., Elgot, C.C. and Shepherdson, J.C., Sets recognized by n-tape automata. J. Algebra 3 (1969) 447–464. CrossRef
Eilenberg, S. and Schützenbeger, M.-P., Rational sets in commutative monoids. J. Algebra 13 (1969) 173–191. CrossRef
Elgot, C.C. and Mezei, J.E., Relations De, Onfined by Finite Automata. IBM J. 10 (1965) 47–68. CrossRef
Fischer, P.C. and Rosenberg, A.L., Multitape one-way nonwriting automata. J. Comput. Syst. Sci. 2 (1968) 88–101. CrossRef
Ginsburg, S. and Spanier, E.H., Semigroups, Presburger formulas, and languages. Pacific J. Math. 16 (1966) 285–296. CrossRef
Grädel, E., Subclasses of Presburger arithmetic and the polynomial hierarchy. Theoret. Comput. Sci. 56 (1989) 281–301.
Ibarra, O.H., 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. CrossRef
Lisovik, L.P., 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.
M. Rabin and D. Scott, Finite automata and their decision problems. IBM J. Res. Develop (1959) 125–144.
J. Sakarovitch, Éléments de théorie des automates. Vuibert Informatique (2003).
Stearns, R.E., A regularity test for pushdown machines. Inform. Control 11 (1967) 323–340. CrossRef
Valiant, L.G., Regularity and related problems for deterministic pushdown automata. Inform. Control 22 (1975) 1–10.