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On the intersection of regex languages with regular languages. (English) Zbl 1167.68031

Summary: We revisit the semantics of extended regular expressions (regex), defined succinctly in the 90s by A. V. Aho [“Algorithms for finding patterns in strings”, in: J. van Leeuwen (ed.), Algorithms and complexity. Handbook of theoretical computer science. Vol. A. Amsterdam etc.: Elsevier Science Publishers. 255–300 (1990; Zbl 0900.68249)] and rigorously in 2003 by C. Câmpeanu, K. Salomaa and S. Yu [“A formal study of practical regular expressions”, Int. J. Found. Comput. Sci. 14, No. 6, 1007–1018 (2003; Zbl 1101.68443)], when the authors reported an open problem, namely whether regex languages are closed under the intersection with regular languages. We give a positive answer; and for doing so, we propose a new class of machines – regex automata systems (RAS) – which are equivalent to regex. Among others, these machines provide a consistent and convenient method of implementing regex in practice. We also prove, as a consequence of this closure property, that several languages, such as the mirror language, the language of palindromes, and the language of balanced words, are not regex languages.

MSC:

68Q45 Formal languages and automata
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References:

[1] Aho, A. V., Algorithms for finding patterns in strings, (van Leeuwen, Jan, Handbook of Theoretical Computer Science. Handbook of Theoretical Computer Science, Algorithms and Complexity, vol. A (1990), Elsevier and MIT Press), 255-300 · Zbl 0900.68249
[2] Câmpeanu, C.; Salomaa, K.; Yu, S., A formal study of practical regular expressions, IJFCS, 14, 6, 1007-1018 (2003) · Zbl 1101.68443
[3] C. Câmpeanu, N. Santean, On pattern expression languages, Technical Report CS-2006-20, David R. Cheriton School of Computer Science, University of Waterloo, Waterloo, ON, Canada, 2006; C. Câmpeanu, N. Santean, On pattern expression languages, Technical Report CS-2006-20, David R. Cheriton School of Computer Science, University of Waterloo, Waterloo, ON, Canada, 2006
[4] Câmpeanu, C.; Yu, S., Pattern expressions and pattern automata, IPL, 92, 267-274 (2004) · Zbl 1173.68546
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[6] Friedl, J. E.F., Mastering Regular Expressions (1997), O’Reilly & Associates, Inc.: O’Reilly & Associates, Inc. Cambridge · Zbl 0881.68017
[7] Hopcroft, J. E.; Motwani, R.; Ullman, J. D., Introduction to Automata Theory, Languages, and Computation (2006), Addison Wesley: Addison Wesley Reading Mass
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[10] Yu, S., Regular languages, (Salomaa, A.; Rozenberg, G., Handbook of Formal Languages (1997), Springer Verlag), 41-110
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