A survey of Evasiveness: Lower Bounds on the Decision-Tree Complexity of Boolean Functions

The decision tree complexity of a boolean function F of n arguments is the depth of a minimum-depth decision tree that computes F correctly on every input. A boolean function F of n arguments is cn-evasive if its decision tree complexity is cn, (c < 1). F is completely evasive, (n-evasive), if all of its arguments need to be probed in the worst case. When we restrict the properties and form of the boolean functions considered, interesting results on their decision tree complexity can be derived. One such restriction is boolean functions which represent the adjacency matrix of a graph. Rivest and Vuillemin proved the Aanderaa-Rosenberg conjecture which stated that every graph property on n node graphs is c n 2-evasive. Subsequently it was proved that all nontrivial monotone graph properties on a prime power number of nodes are completely evasive. However proving that all monotone graph properties are evasive seems a diicult task. In this paper we survey the proof techniques used to establish the evasiveness of boolean functions under various conditions of symmetry and monotonicity. Furthermore, we explore the impact of these conditions on the form and complexity of the boolean functions, aiming to understand the qualities that make a boolean function completely evasive.