# Query order in the polynomial hierarchy

Technical Contributions

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## Abstract

The study of query order was initiated by Hemaspaandra, Hempel, and Wechsung [HHW]. Their goal was to learn whether the order of access to information sources affects the class of problems that can be solved. They showed that in the boolean hierarchy over NP, order matters. In the present paper, we study the power of query order when accessing levels of the

*polynomial*hierarchy, and we show that here order does not matter. In particular, let P^{C:D}denote the class of languages computable by a polynomial-time machine that is allowed one query to*C*followed by one query to*D*[HHW]. We prove that the levels of the polynomial hierarchy are*order-oblivious*:$$P^{\sum _i^P :\sum _k^P } = P^{\sum _k^P :\sum _j^P } .$$

Yet, we also show that these ordered query classes form new levels in the polynomial hierarchy unless the polynomial hierarchy collapses. We prove that a wide range of other classes (UP, BPP, ⊕P, PP, etc.) inherit all order-obliviousness results that hold for deterministic polynomialtime transducers.

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