On the Computational Completeness of Equations over Sets of Natural Numbers
ICALP '08 Proceedings of the 35th international colloquium on Automata, Languages and Programming, Part II
Functions Definable by Arithmetic Circuits
CiE '09 Proceedings of the 5th Conference on Computability in Europe: Mathematical Theory and Computational Practice
One-Nonterminal Conjunctive Grammars over a Unary Alphabet
CSR '09 Proceedings of the Fourth International Computer Science Symposium in Russia on Computer Science - Theory and Applications
Complex Algebras of Arithmetic
Fundamenta Informaticae
Least and greatest solutions of equations over sets of integers
MFCS'10 Proceedings of the 35th international conference on Mathematical foundations of computer science
Univariate Equations Over Sets of Natural Numbers
Fundamenta Informaticae
Parsing Boolean grammars over a one-letter alphabet using online convolution
Theoretical Computer Science
Complex Algebras of Arithmetic
Fundamenta Informaticae
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The problem of testing membership in the subset of the natural numbers produced at the output gate of a { $$\bigcup, \bigcap, ^-, +, \times$$ } combinational circuit is shown to capture a wide range of complexity classes. Although the general problem remains open, the case { $$\bigcup, \bigcap, +, \times$$ } is shown NEXPTIME-complete, the cases { $$\bigcup, \bigcap, ^-, \times$$ }, { $$\bigcup, \bigcap, \times$$ }, { $$\bigcup, \bigcap, +$$ } are shown PSPACE-complete, the case { $$\bigcup, +$$ } is shown NP-complete, the case {驴, +} is shown C=L-complete, and several other cases are resolved. Interesting auxiliary problems are used, such as testing nonemptyness for union-intersection-concatenation circuits, and expressing each integer, drawn from a set given as input, as powers of relatively prime integers of one's choosing. Our results extend in nontrivial ways past work by Stockmeyer and Meyer (1973), Wagner (1984) and Yang (2000).