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On Shuffle Ideals

Published online by Cambridge University Press:  15 February 2003

Pierre-Cyrille Héam*
Laboratoire d'Informatique de Franche-Comté, Université de Franche-Comté, 16 route de Gray, 25030 Besancon Cedex, France;
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A shuffle ideal is a language which is a finite union of languages of the form A*a1A*...A*ak where A is a finite alphabet and the ai's are letters. We show how to represent shuffle ideals by special automata and how to compute these representations. We also give a temporal logic characterization of shuffle ideals and we study its expressive power over infinite words. We characterize the complexity of deciding whether a language is a shuffle ideal and we give a new quadratic algorithm for this problem. Finally we also present a characterization by subwords of the minimal automaton of a shuffle ideal and study the complexity of basic operations on shuffle ideals.


Research Article
© EDP Sciences, 2002

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A. Aho, J. Hopcroft and J. Ullman, The design and analysis of computer algorithms. Addison-Wesley (1974) 395-400.
Almeida, J., Implicit operations on finite $\mathcal{{J}}$ -trivial semigroups and a conjecture of I. Simon. J. Pure Appl. Algebra 69 (1990) 205-218. CrossRef
Arfi, M., Opération polynomiales et hiérarchies de concaténation. Theoret. Comput. Sci. 91 (1991) 71-84. CrossRef
J. Berstel and L. Boasson, Shuffle factorization is unique, Technical Report. LIAFA, Université Paris 7 (1999).
J. Berstel, Transductions and context-free languages. Teubner (1979) Verlag.
Cohen, J., Perrin, D. and Pin, J.-E., On the expressive power of temporal logic for finite words. J. Comput. System Sci. 46 (1993) 271-294. CrossRef
S. Eilenberg, Automata, Languages and Machines, Vol. A. Academic Press (1974).
D. Gabbay, A. Pnueli, S. Shelah and J. Stavi, On the temporal analysis of fairness, in 12th ACM Symp. on Principles of Programming Languages (1980) 163-180.
C. Glasser and H. Schmidt, Level 5/2 of the straubing-thérien hierarchy for two-letter alphabets, in Conference on Developments in Language Theory (DLT). Vienna (2001).
P.-C. Héam, Some complexity results for polynomial rational expressions. Theoret. Comput. Sci. (to appear).
G. Higman, Ordering by divisibility in abstract algebras, in Proc. of the London Mathematical Society, Vol. 2 (1952) 326-336.
Hagenah, C. and Muscholl, A., Computing ε-free nfa from regular expressions in O(nlog2(n)) time. RAIRO: Theoret. Informatics Appl. 34 (2000) 257-277.
J. Hopcroft and J. Ullman, Introduction to automata theory, languages, and computation. Addison-Wesley (1980).
J.A. Kamp, Tense logic and the theory of linear order, Ph.D. Thesis. University of California, Los Angeles (1968).
O. Katai, Completeness and the expressive power of next time temporal logical system by semantic tableau method, Technical Report RR-0109. Inria, Institut National de Recherche en Informatique et en Automatique (1981).
S.C. Kleene, Representation of events in nerve nets and finite automata. Princeton University Press, Automata Studies (1956) 3-42.
M. Lothaire, Combinatorics on words. Cambridge Mathematical Library (1983).
Nivat, M., Ramkumar, G.D.S., Pandu Rangan, C., Saoudi, A. and Sundaram, R., Efficient parallel shuffle recognition. Parallel Process. Lett. 4 (1994) 455-463. CrossRef
C.H. Papadimitriou, Computational complexity. Addison Wesley (1994).
M. Parigot, Automates, réseaux, formules, in Actes des journées ``Informatiques et Mathématiques". Luminy (1984).
Pradeep, B., Murthy, C. and Ram, S., A constant time string shuffle algorithm on reconfigurable meshes. Int. J. Comput. Math. 68 (1998) 251-259. CrossRef
A. Pnueli, The temporal logic of programs, in 18th FOCS (1977) 46-57.
Pin, J.-E. and Weil, P., Polynomial closure and unambiguous product. Theory Comput. Systems 30 (1997) 1-39. CrossRef
Radford, D.E., A natural ring basis for shuffle algebra and an application to group schemes. J. Algebra 58 (1979) 432-454. CrossRef
I. Simon, Piecewise testable events, in GI Conf. Springer-Verlag, Lecture Notes in Comput. Sci. 33 (1975) 214-222.
Spehner, J.-C., Le calcul rapide des mélanges de deux mots. Theoret. Comput. Sci. 47 (1986) 181-203. CrossRef
Straubing, H. and Thérien, D., Partially ordered finite monoids and a theorem of I. Simon. J. Algebra 119 (1985) 161-183.
Stern, J., Characterization of some classes of regular events. Theoret. Comput. Sci. 35 (1985) 17-42. CrossRef
Straubing, H., Finite semigroups varieties of the form V *s D. J. Pure Appl. Algebra 36 (1985) 53-94. CrossRef
Thérien, D., Classification of finite monoids: The language approach. Theoret. Comput. Sci. 14 (1981) 195-208. CrossRef
W. Thomas, Classifying regular events in symbolic logic. J. Comput. System Sci. 25 360-375.
D. Thérien and Th. Wilke, Temporal logic and semidirect products: An effective characterization of the until hierarchy, in Proc. of the 37th Annual Symposium on Foundations of Computer Science. IEEE (1996) 256-263.
Th. Wilke, Classifying discrete temporal properties, in STACS'99. Springer-Verlag, Lecture Notes in Comput. Sci. 1563 (1999) 32-46.
S. Yu, State complexity of regular languages, in Proc. of the International Workshop on Descriptional Complexity of Automata, Grammars and Related Structures (1999) 77-88.