ON EQUATIONS OVER SETS OF NUMBERS AND THEIR LIMITATIONS
Systems of equations of the form X = Y + Z and X C, in which the unknowns sets of natural numbers, "+" denotes elementwise sum of sets S + T = {m + n vertical bar m is an element of S, n is an element of T}, and C is an ultimately periodic constant, have recently been proved to be computationally universal (Jet, Okhotin, "Equations over sets of natural numbers with addition only", STACS 2009). This paper establishes some limitations of such systems. A class of sets of numbers that cannot be represented by unique, least or greatest solutions of systems of this form is defined, and a particular set in this class is constructed. The argument is then extended to equations over sets of integers.
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