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

This paper studies conditions under which the operation of parallel insertion can be reversed by parallel deletion, i.e., when does the equality (L1 ⇐ L2) ⇒ L2 = L1 hold for languages L1 and L2. 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 L2 is a comma code, then the above equation holds for any language L1 ⊆ Σ∗. 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.