On the Reversibility of Parallel Insertion, and Its Relation to Comma Codes

Authors:
Bo Cui;Lila Kari;Shinnosuke Seki
Affiliations:
Department of Computer Science, University of Western Ontario, London, Canada N6A 5B7;Department of Computer Science, University of Western Ontario, London, Canada N6A 5B7;Department of Computer Science, University of Western Ontario, London, Canada N6A 5B7
Venue:
CAI '09 Proceedings of the 3rd International Conference on Algebraic Informatics
Year:
2009

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Abstract

This paper studies conditions under which the operation of parallel insertion can be reversed by parallel deletion, i.e., when does the equality $(L_1 \Leftarrow L_2) \Rightarrow L_2 = L_1$ hold for languages L 1 and L 2 . We obtain a complete characterization of the solutions in the special case when both languages involved are singleton words. We also define comma codes , a family of codes with the property that, if L 2 is a comma code, then the above equation holds for any language $L_1 \subseteq {\it \Sigma}^*$. Lastly, we generalize the notion of comma codes to that of comma intercodes of index m . Besides several properties, we prove that the families of comma intercodes of index m form an infinite proper inclusion hierarchy, the first element which is a subset of the family of infix codes, and the last element of which is a subset of the family of bifix codes.