Acknowledgement I would like to thank my advisor Kenneth Regan for introducing me to the study of computational complexity. Thank you for your guidance and encouragement. The last six years have been great working together. Thank you also for teaching me how to teach. I would like to thank Alan Selman and Xin He for being on the Ph.D. committee and Martin Lotz for his work as the outside reader...

Recently, Razborov obtained superpolynomial lower bounds for monotone circuits that lect cliques in graphs. In particular, Razborov showed that detecting cliques of size s in a graph dh m vertices requires monotone circuits of size .Q(m-'/(log m) ~') for fixed s, and size rn ao°~') for ,. :[log ml4J. In this paper we modify the arguments of Razborov to obtain exponential lower bounds for ,moton...

A finite function f is a mapping of {1 , 2 ,. .. , m } into {1 , 2 ,. .. , m } ∪ { # } where # is a symbol to be thought of as ''undefined.'' This paper defines a measure M(f) of the difficulty of inverting a finite function f, which is given by M(f) = MIN    log 2 C(f) log 2 C(g) _ _________ : g an inverse of f    where C(f) is a circuit complexity measure of the difficulty of computing ...

The boolean circuit complexity classes AC ⊆ AC[m] ⊆ TC ⊆ NC have been studied intensely. Other than NC, they are defined by constant-depth circuits of polynomial size and unbounded fan-in over some set of allowed gates. One reason for interest in these classes is that they contain the boundary marking the limits of current lower bound technology: such technology exists for AC and some of the cl...

Circuit complexity, a subfield of computational complexity theory, can be used to analyze how the resource usage of neural networks scales with problem size. The computational complexity of discrete feedforward neural networks is surveyed, with a comparison of classical circuits to circuits constructed from gates that compute weighted majority functions.

Reversible single-target gates are a generalization of Toffoli gates which are a helpful formal representation for the description of synthesis algorithms but are too general for an actual implementation based on some technology. There is an exponential lower bound on the number of Toffoli gates required to implement any reversible function, however, there is also a linear upper bound on the nu...

We describe and motivate a proposed new approach to the problem of finding a “complexity formula” which lower-bounds the circuit complexity (over circuits in a given class) of an arbitrary boolean function, given its truth table, and which can be used to prove nontrivial lower bounds on the complexity of specific functions of interest. This requires addressing the “natural proofs barrier” [Razb...

The stabilizer class of circuits, introduced by Daniel Gottesman, consists of quantum circuits in which every gate is a controlled-NOT (CNOT), Hadamard or phase gate [5]. These circuits have several interesting properties. For example, they are robust enough to allow for entangled states, yet they are known to be simulable in polynomial time by a classical computer. These circuits also naturall...

We consider the size of circuits that perfectly hash an arbitrary subset S⊂{0, 1} of cardinality 2 into {0, 1}. We observe that, in general, the size of such circuits is exponential in 2k−m, and provide a matching upper bound.