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On the Complexity of Decidable Cases of Commutation Problem for Languages

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Fundamentals of Computation Theory (FCT 2001)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2138))

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Abstract

We investigate the complexity of basic decidable cases of the commutation problem for languages: testing the equality XY = YX for two languages X, Y, given different types of representations of X, Y. We concentrate on (the most interesting) case when Y is an explicitly given finite language. This is motivated by a renewed interest and recent progress, see [12,1], in an old open problem posed by Conway [2]. We show that the complexity of the commutation problem varies from co-NEXPTIME-complete, through P-SPACE complete and co-NP complete, to deterministic polynomial time. Classical types of description are considered: nondeterministic automata with and without cycles, regular expressions and grammars. Interestingly, in most cases the complexity status does not change if instead of explicitly given finite Y we consider general Y of the same type as that of X. For the case of commutation of two finite sets we provide polynomial time algorithms whose time complexity beats that of a naive algorithm. For deterministic automata the situation is more complicated since the complexity of concatenation of deterministic automaton language X with a finite set Y is asymmetric: while the minimal dfa’s for XY would be polynomial in terms of dfa’s for X and Y, that for YX can be exponential.

Supported by Academy of Finland under grant 44087.

Supported in part by KBN grant 8T11C03915.

Supported in part by KBN grant 8T11C03915.

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Karhumäki, J., Plandowski, W., Rytter, W. (2001). On the Complexity of Decidable Cases of Commutation Problem for Languages. In: Freivalds, R. (eds) Fundamentals of Computation Theory. FCT 2001. Lecture Notes in Computer Science, vol 2138. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44669-9_20

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  • DOI: https://doi.org/10.1007/3-540-44669-9_20

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  • Print ISBN: 978-3-540-42487-1

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