On Language Equations XXK = XXL and XM = N over a Unary Alphabet

It is shown that the recently discovered computational universality in systems of language equations over a unary alphabet occurs already in systems of the simplest form, with one unknown X and two equations XXK = XXL and XM = N, where K, L, M, N ⊆ a* are four regular constants. Every recursive (r.e., co-r.e.) set can be encoded in a unique (least, greatest) solution of a system of such a form. The proofs are carried out in terms of equations over sets of numbers.

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