On the Power of Parity Queries in Boolean Decision Trees

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9076)

Abstract

In an influential paper, Kushilevitz and Mansour (1993) introduced a natural extension of Boolean decision trees called parity decision tree (PDT) where one may query the sum modulo \(2,\) i.e., the parity, of an arbitrary subset of variables. Although originally introduced in the context of learning, parity decision trees have recently regained interest in the context of communication complexity (cf. Shi and Zhang 2010) and property testing (cf. Bhrushundi, Chakraborty, and Kulkarni 2013). In this paper, we investigate the power of parity queries. In particular, we show that the parity queries can be replaced by ordinary ones at the cost of the total influence aka average sensitivity per query. Our simulation is tight as demonstrated by the parity function.

At the heart of our result lies a qualitative extension of the result of O’Donnell, Saks, Schramme, and Servedio (2005) titled: Every decision tree has an influential variable. Recently Jain and Zhang (2011) obtained an alternate proof of the same. Our main contribution in this paper is a simple but surprising observation that the query elimination method of Jain and Zhang can indeed be adapted to eliminate, seemingly much more powerful, parity queries. Moreover, we extend our result to linear queries for Boolean valued functions over arbitrary finite fields.

Notes

Acknowledgements

We thank Rahul Jain, Supartha Poddar, Miklos Santha, and Avishay Tal for several helpful discussions. We also thank Ben vee Volk for pointing out that the super-linear separation in [27] works for PDTs as well.

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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Centre for Quantum TechnologiesThe National University of SingaporeSingaporeSingapore
  2. 2.Institute of Computing TechnologyChinese Academy of SciencesBeijingChina

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