Counting complexity classes for numeric computations II: algebraic and semialgebraic sets
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
Optimization and approximation problems related to polynomial system solving
CiE'06 Proceedings of the Second conference on Computability in Europe: logical Approaches to Computational Barriers
The complexity of semilinear problems in succinct representation
FCT'05 Proceedings of the 15th international conference on Fundamentals of Computation Theory
Exotic quantifiers, complexity classes, and complete problems
ICALP'07 Proceedings of the 34th international conference on Automata, Languages and Programming
Complexity and algorithms for Euler characteristic of simplicial complexes
Journal of Symbolic Computation
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We define a counting class ${\rm \#P}_\add$ in the Blum--Shub--Smale setting of additive computations over the reals. Structural properties of this class are studied, including a characterization in terms of the classical counting class $\#{\sf P}$ introduced by Valiant. We also establish transfer theorems for both directions between the real additive and the discrete setting. Then we characterize in terms of completeness results the complexity of computing basic topological invariants of semilinear sets given by additive circuits. It turns out that the computation of the Euler characteristic is ${\rm FP}_{\rm add}^{{\rm \#P}_{\rm add}}$-complete, while for fixed $k$ the computation of the $k$th Betti number is ${\rm FPAR}_{\rm add}$-complete. Thus the latter is more difficult under standard complexity theoretic assumptions. We use all of the above to prove some analogous completeness results in the classical setting.