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SIAM Journal on Computing
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Journal of Computer and System Sciences
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SIAM Journal on Computing
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SIAM Journal on Computing
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Theoretical Computer Science
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Theoretical Computer Science
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Fundamenta Informaticae
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Satisfiability of algebraic circuits over sets of natural numbers
Discrete Applied Mathematics
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FCT'05 Proceedings of the 15th international conference on Fundamentals of Computation Theory
Complex Algebras of Arithmetic
Fundamenta Informaticae
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FCT'07 Proceedings of the 16th international conference on Fundamentals of Computation Theory
Equivalence problems for circuits over sets of natural numbers
CSR'07 Proceedings of the Second international conference on Computer Science: theory and applications
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The problem of testing membership in the subset of the natural numbers produced at the output gate of a {驴, 驴,-,+, 脳} combinational circuit is shown to capture a wide range of complexity classes. Although the general problem remains open, the case {驴, 驴,+, 脳} is shown NEXPTIME-complete, the cases {驴, 驴,-, 脳}, {驴, 驴, 脳}, {驴, 驴,+} are shown PSPACE-complete, the case {驴,+} 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).