Abstract
We study the class \(\mathrm {\#AC^0}\) of functions computed by constant-depth polynomial-size arithmetic circuits of unbounded fan-in addition and multiplication gates. No model-theoretic characterization for arithmetic circuit classes is known so far. Inspired by Immerman’s characterization of the Boolean class \({\mathrm {AC^0}}\), we remedy this situation and develop such a characterization of \(\mathrm {\#AC^0}\). Our characterization can be interpreted as follows: Functions in \(\mathrm {\#AC^0}\) are exactly those functions counting winning strategies in first-order model checking games. A consequence of our results is a new model-theoretic characterization of \(\mathrm {TC}^0\), the class of languages accepted by constant-depth polynomial-size majority circuits.
Supported by DFG grant VO 630/8-1.
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Acknowledgements
We are grateful to Lauri Hella (Tampere) and Juha Kontinen (Helsinki) for helpful discussion, leading in particular to Definition 20. We also thank the anonymous referees for helpful comments.
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Haak, A., Vollmer, H. (2016). A Model-Theoretic Characterization of Constant-Depth Arithmetic Circuits. In: Väänänen, J., Hirvonen, Å., de Queiroz, R. (eds) Logic, Language, Information, and Computation. WoLLIC 2016. Lecture Notes in Computer Science(), vol 9803. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-52921-8_15
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DOI: https://doi.org/10.1007/978-3-662-52921-8_15
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