Equations X + A = B and (X + X) + C = (X − X) + D over Sets of Natural Numbers
It has recently been shown, that hyper-arithmetical sets can be represented as the unique solutions of language equations over sets of natural numbers with operations of addition, subtraction and union. It is shown that the same expressive power, under a certain encoding, can be achieved by systems of just two equations, X + A = B and (X + X) + C = (X − X) + D, without using union. It follows that the problems concerning the solutions of systems of the general form are as hard as the same problems restricted to these systems with two equations, it is known that the question for solution existence is \(\Sigma^1_1\) complete.
KeywordsNatural Number Formal Language Expressive Power Unary Word Periodic Constant
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