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A Polynomial Separation for Adaptive Fourier-Overlap Parity Decision Trees

Primarily AI-generated textHuman understanding: no partscs.CC — Computational Complexity

Contributed by YURAN XIU ↗

Version 1 / Oct 17, 2026 / CC BY 4.0

Abstract

Consider the parity decision tree rule that, at every nonconstant restriction, queries a nonzero character maximizing the intersection of the current Fourier support with its translate. We construct an explicit infinite family for which every adaptive policy obeying this rule has polynomial worst-case depth, although an unrestricted parity decision tree has logarithmic depth. For every ℓ≥1\ell\geq1, let R=2ℓR=2^\ell. Our function has n=4ℓ−1+R2ℓ+Rn=4\ell-1+R^2\ell+R input bits and exactly k=R6−R5+Rk=R^6-R^5+R nonzero Fourier coefficients. Every history-dependent maximizing policy terminates and computes the function, but has a feasible path of length at least R>k1/6R>k^{1/6}; an explicit unrestricted tree has depth at most 5ℓ≤log⁡2k5\ell\leq\log_2 k. The construction places one pointer behind each cell of a finite-field affine partition, with all pointers accessing a shared data table. After any sequence of equal-answer table comparisons, exact signed Fourier coefficients force every maximizing query to compare two remaining table blocks. This invariant accommodates all tie choices and all resulting table partitions. The complete separation, including the specified polynomial-field representation and the existence of every operational policy's finite run, has been checked in Lean. The result concerns this particular support-overlap rule; it is not a lower bound against arbitrary parity decision trees. HAL deposit: hal-05800236v1.

References

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Version history

  1. v1Submitted by YURAN XIUInitial depositCurrentOct 17, 2026