Abstract
We attack the problem of deciding whether a finite collection of finite languages is a code, that is, possesses the unique decipherability property in the monoid of finite languages. We investigate a few subcases where the theory of rational relations can be employed to solve the problem. The case of unary languages is one of them and as a consequence, we show how to decide for two given finite subsets of nonnegative integers, whether they are the n-th root of a common set, for some n ≥ 1. We also show that it is decidable whether a finite collection of finite languages is a Parikh code, in the sense that whenever two products of these sets are commutatively equivalent, so are the sequences defining these products. Finally, we consider a nonunary special case where all finite sets consist of words containing exactly one occurrence of the specific letter.
This research was done during the second author’s visit at the LIAFA, CNRS UMR 7089, Université Paris Diderot, Novembre–December 2008. The second author was supported by the Academy of Finland under the grant 126211.
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Choffrut, C., Karhumäki, J. (2009). Unique Decipherability in the Monoid of Languages: An Application of Rational Relations. In: Frid, A., Morozov, A., Rybalchenko, A., Wagner, K.W. (eds) Computer Science - Theory and Applications. CSR 2009. Lecture Notes in Computer Science, vol 5675. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03351-3_9
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DOI: https://doi.org/10.1007/978-3-642-03351-3_9
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