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Lower Bounds Against Weakly-Uniform Threshold Circuits

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Abstract

An ongoing line of research has shown super-polynomial lower bounds for uniform and slightly-non-uniform small-depth threshold and arithmetic circuits (Allender, in Chicago J. Theor. Comput. Sci. 1999(7), 1999; Koiran and Perifel, in Proceedings of the 24th Annual IEEE Conference on Computational Complexity (CCC 2009), pp. 35–40, 2009; Jansen and Santhanam, in Proceedings of the 38th International Colloquium on Automata, Languages and Programming (ICALP 2011), I, pp. 724–735, 2011).

We give a unified framework that captures and improves each of the previous results. Our main results are that Permanent does not have threshold circuits of the following kinds.

  1. (1)

    Depth O(1), n o(1) bits of non-uniformity, and size n O(1).

  2. (2)

    Depth O(1), polylog(n) bits of non-uniformity, and size s(n) such that for all constants c the c-fold composition of s, s (c)(n), is less than 2n.

  3. (3)

    Depth o(loglogn), polylog(n) bits of non-uniformity, and size n O(1).

(1) strengthens a result of Jansen and Santhanam (Jansen and Santhanam, in Proceedings of the 38th International Colloquium on Automata, Languages and Programming (ICALP 2011), I, pp. 724–735, 2011), who obtained similar parameters but for arithmetic circuits of constant depth rather than Boolean threshold circuits. (2) and (3) strengthen results of Allender (Allender, in Chicago J. Theor. Comput. Sci. 1999(7), 1999) and Koiran and Perifel (Koiran and Perifel, in Proceedings of the 24th Annual IEEE Conference on Computational Complexity (CCC 2009), pp. 35–40, 2009), respectively, who obtained results with similar parameters but for completely uniform circuits. Our main technical contribution is to simplify and unify earlier proofs in this area, and adapt the proofs to handle some amount of non-uniformity. We also develop a notion of circuits with a small amount of non-uniformity that naturally interpolates between fully uniform and fully non-uniform circuits. We use this notion, which we term weak uniformity, rather than the earlier and essentially equivalent notion of succinctness used by Jansen and Santhanam because the notion of weak uniformity more fully and easily interpolates between full uniformity and non-uniformity of circuits.

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Notes

  1. A plausible explanation of this “barrier” is given by the “natural proofs” framework of [30], who argue it is hard to prove lower bounds against the circuit classes that are powerful enough to implement cryptography.

  2. Unlike the nonuniform setting, where every n-variate Boolean function is computable by a circuit of size about 2n/n [23], uniform circuit lower bounds can be >2n.

  3. Fully uniform constant-depth poly-size threshold circuits can be simulated by threshold Turing machines running in polylog time, but the relaxed n o(1)-weak-uniformity translates to threshold Turing machines with longer running time.

  4. We note that f(n) is constructible in our sense if and only if 2f(n) is constructible according to Allender’s definition in [3].

  5. This is true for MAJ with an odd number of inputs, which is easily achieved by replacing MAJ(x 1,x 2,…,x k ) with MAJ(x 1,x 1,x 2,x 2,…,x k ,x k ,0). This presupposes our definition for MAJ—that MAJ is 1 if strictly more than half of the inputs is 1.

  6. Th d(n) TIME(T(n)) is closed under complement since the negation of MAJ is MAJ of negated inputs when MAJ has an odd number of inputs; the latter is easy to achieve by replacing MAJ(x 1,…,x k ) with MAJ(x 1,x 1,…,x k ,x k ,0). Allender [3] uses a lazy diagonalization argument [40] for nondeterministic TMs. However, that argument seems incapable of handling the amount of advice we need. Fortunately, the basic diagonalization argument we use here is sufficient for our purposes.

  7. We can assume all queries are the same size because there are paddable PP-complete languages, including language versions of Permanent. A language is paddable if queries of smaller length can efficiently, e.g. by a uniform AC 0 reduction, be made longer to match the longest query.

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Acknowledgements

While conducting this research, the first and second author were supported by NSERC Discovery Grant; the third author was partially supported by Indiana State University, University Research Council grants #11-07 and #12-18. The third author thanks Matt Anderson, Dieter van Melkebeek, and Dalibor Zelený for discussions that began this project, continued discussions since, and comments on early drafts of this work; and in particular thanks Matt Anderson for observations that refined the statement of Corollary 3. We also thank the reviewers for comments and suggestions that improved the exposition of the paper.

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  1. School of Computing Science, Simon Fraser University, Burnaby, BC, Canada

    Ruiwen Chen & Valentine Kabanets

  2. Indiana State University, Terre Haute, IN, USA

    Jeff Kinne

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  1. Ruiwen Chen
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Correspondence to Jeff Kinne.

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Chen, R., Kabanets, V. & Kinne, J. Lower Bounds Against Weakly-Uniform Threshold Circuits. Algorithmica 70, 47–75 (2014). https://doi.org/10.1007/s00453-013-9823-y

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