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Understanding Query Complexity for Boolean Functions through Decision Trees

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This article explores the methods for determining the value of a function f(x) using oracle access with minimum queries. It defines deterministic (D(f)), randomized (R(f)), and quantum (Q(f)) query complexities, analyzing their interrelations. The paper discusses various input frameworks including restrictions and invariants pertaining to monotonicity and group actions, as well as the concept of simplicial complexes for both monotone and non-monotone functions. It also introduces the rejection model for boolean functions, examining query strategies under restrictive conditions.

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Understanding Query Complexity for Boolean Functions through Decision Trees

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  1. Decision Trees Shalev Ben-David

  2. 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)?

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

  4. 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},{}}

  5. 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 • It does not contain both (i,0) and (i,1) • If we construct a partial input x whose bits are determined by the set, then we do not get a certificate.

  6. 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

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