Computational Complexity· 2026Q2
A Technique for Hardness Amplification Against $$\textsf{AC}^0$$
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- 2026year
Short summary
A new technique shows that XORing multiple copies of a function amplifies its hardness against AC0 circuits, provided the hardness can be proven using a specific restriction/projection method.
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Key points
- Introduces a technique to amplify hardness of functions against AC0 circuits.
- The amplification works by XORing multiple copies of the function.
- This technique is applicable when hardness proofs involve random restrictions/projections and shallow decision trees.
- The method is applied to improve hardness results for the average-case depth hierarchy theorem and majority function inapproximability.
- A new XOR lemma for decision trees is developed as part of the analysis.
AI-generated from the title and abstract; the full text is not read.
Abstract
Abstract To show that a function h is hard to approximate for $$\textsf{AC}^0$$ AC 0 circuits, one method is to (a) design some distribution over random restrictions or random projections, (b) show that $$\textsf{AC}^0$$ AC 0 circuits simplify to shallow decision trees under these restrictions/projections, and finally (c) show that after applying the restriction/projection, h is hard to approximate for shallow decision trees with respect to an appropriate distribution. We show that (roughly speaking) if h can be proven to be hard to approximate by a proof with that structure, then XORing multiple copies of h amplifies its hardness. We apply our technique to two well-known average-case hardness results: the average-case depth hierarchy theorem (Håstad et al., 2017) and the inapproximability of the majority function. Our analysis involves a new kind of XOR lemma for decision trees, which might be of independent interest.
The authors' abstract, as published at the source. Computational Complexity, 2026 · DOI ↗
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Computational Theory and MathematicsComputer Science