Models of Query Complexity for Boolean Functions

In this thesis we study various models of query complexity. A query algorithm computes a function under the restriction that the input can be accessed only by making probes to the the bits of the input. The query complexity of a function $f$ is the minimum number of probes made by any query algorithm that computes $f$. In this thesis, we consider three different models of query complexity, (1) deterministic decision tree complexity (query complexity when the underlying algorithm is deterministic), (2) approximate decision tree complexity aka. property testing (query complexity when the underlying algorithm is probabilistic and only expected to "approximately" compute $f$) and quantum query complexity (query complexity when the underlying algorithm is allowed to make quantum queries).