Decision Trees Shalev Ben-David

Definition Given a function and oracle access to, determine f(x) with minimum number of queries E.g. f is OR on the bits of x – Grover search D(f) is the deterministic query complexity R(f) is the randomized query complexity Q(f) is the quantum query complexity What’s the relation between D(f), R(f), and Q(f)?

Variants Different input set: Promise: we only care about inputs from some subset Restrictions on f: o Invariant under some group action on the bits o Invariant under some group action on the values of the bits (i.e. exchanging the 0s and 1s) o Monotonicity o Read-only

Simplicial Complex for Monotone Functions A simplicial complex is a set system which is closed under subsets. Given monotone f, consider the set system consisting of sets of coordinates of the input such that, if those coordinates were the only 0s, then the value of the function would be 1. Example: f(xyz)=OR. Set system is {{1,2},{2,3},{3,1},{1},{2},{3},{}}

Simplicial Complex for Non-Monotone Functions Given f, the domain of the set system is the set of pairs (i,b) where i is a coordinate and b is either 0 or 1. A set of such pairs is in the set system iff o It does not contain both (i,0) and (i,1) o If we construct a partial input x whose bits are determined by the set, then we do not get a certificate.

Rejection Model Given boolean f, we allow the oracle to reject some queries, returning ‘*’ The goal is to determine if it’s possible to discern f(x) at all