A composition theorem for parity kill number

In this work, we study the parity complexity measures ${\mathsf{C}^{\oplus}_{\min}}[f]$ and ${\mathsf{DT^{\oplus}}}[f]$. ${\mathsf{C}^{\oplus}_{\min}}[f]$ is the \emph{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. ${\mathsf{DT^{\oplus}}}[f]$ is the depth of the shortest \emph{parity decision tree} which computes $f$. These complexity measures have in recent years become increasingly important in the fields of communication complexity \cite{ZS09, MO09, ZS10, TWXZ13} and pseudorandomness \cite{BK12, Sha11, CT13}. Our main result is a composition theorem for ${\mathsf{C}^{\oplus}_{\min}}$. The $k$-th power of $f$, denoted $f^{\circ k}$, is the function which results from composing $f$ with itself $k$ times. We prove that if $f$ is not a parity function, then ${\mathsf{C}^{\oplus}_{\min}}[f^{\circ k}] \geq \Omega({\mathsf{C}_{\min}}[f]^{k}).$ In other words, the parity kill number of $f$ is essentially supermultiplicative in the \emph{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 complexity measures of $\mathsf{Sort}^{\circ k}$ and $\mathsf{HI}^{\circ k}$. Here $\mathsf{Sort}$ is the sort function due to Ambainis \cite{Amb06}, and $\mathsf{HI}$ is Kushilevitz's hemi-icosahedron function \cite{NW95}. In doing so, we disprove a conjecture of Montanaro and Osborne \cite{MO09} which had applications to communication complexity and computational learning theory. In addition, we give new lower bounds for conjectures of \cite{MO09,ZS10} and \cite{TWXZ13}

Simulation Theorems via Pseudorandom Properties

We generalize the deterministic simulation theorem of Raz and McKenzie [RM99], to any gadget which satisfies certain hitting property. We prove that inner-product and gap-Hamming satisfy this property, and as a corollary we obtain deterministic simulation theorem for these gadgets, where the gadget's input-size is logarithmic in the input-size of the outer function. This answers an open question posed by G\"{o}\"{o}s, Pitassi and Watson [GPW15]. Our result also implies the previous results for the Indexing gadget, with better parameters than was previously known. A preliminary version of the results obtained in this work appeared in [CKL+17]