Browse Theses

A language is in PP if there is a polynomial-time bounded probabilistic Turing machine that correctly determines whether a string is in the language with probability greater than 1/2. In his seminal paper on probabilistic Turing machines, John Gill asked whether PP was closed under intersection. Using ideas from rational approximation theory, we prove that it is. Extending the basic technique, we also prove closure under polynomial-time truth-table reductions and even constant-round truth-table reductions.

A modification of these techniques proves that the AND of several threshold gates can be computed by a single threshold gate whose inputs are ANDs of some of the original inputs. We also prove that any circuit containing boolean gates and few threshold gates can be simulated by a circuit with a single threshold gate at the root (although the depth increases by a constant, and the size increases doubly exponentially in the number of threshold gates). Using this and some known results on computing parity by a perceptron, we prove a lower bound on the number of threshold gates needed to compute parity in a circuit containing both boolean gates and threshold gates.

We also prove that the OR function can be simulated by a low-degree probabilistic polynomial. From this we show that any function computed by a family of bounded-depth, polynomial-size AND-OR circuits can be computed by a low-degree probabilistic polynomial. We also use this result to give a new proof that constant-depth boolean circuits computing parity require exponential size.