A composition theorem for parity kill number


In this work, we study the parity complexity measures C⊕min[f] and DT ⊕[f]. C⊕min[f] is the parity kill number of f, the fewest number of parities on the input variables one has to fix in order to “kill ” f, i.e. to make it constant. DT⊕[f] is the depth of the shortest parity decision tree which computes f. These complexity measures have in recent years become increasingly important in the fields of communication complexity [ZS09, MO09, ZS10, TWXZ13] and pseu-dorandomness [BSK12, Sha11, CT13]. Our main result is a composition theorem for C⊕min. The k-th power of f, denoted f ◦k, is the function which results from composing f with itself k times. We prove that if f is not a parity function, then C⊕min[f ◦k] ≥ Ω(Cmin[f]k). In other words, the parity kill number of f is essentially supermultiplicative in the normal kill number of f (also known as the minimum certificate complexity). As an application of our composition theorem, we show lower bounds on the parity com-plexity measures of Sort◦k and HI◦k. Here Sort is the sort function due to Ambainis [Amb06], and HI is Kushilevitz’s hemi-icosahedron function [NW95]. In doing so, we disprove a conjec-ture of Montanaro and Osborne [MO09] which had applications to communication complexity and computational learning theory. In addition, we give new lower bounds for conjectures of [MO09, ZS10] and [TWXZ13]. a

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