4 research outputs found
Randomization in Non-Uniform Finite Automata
The non-uniform version of Turing machines with an extra advice input tape that depends on the length of the input but not the input itself is a well-studied model in complexity theory. We investigate the same notion of non-uniformity in weaker models, namely one-way finite automata. In particular, we are interested in the power of two-sided bounded-error randomization, and how it compares to determinism and non-determinism. We show that for unlimited advice, randomization is strictly stronger than determinism, and strictly weaker than non-determinism. However, when the advice is restricted to polynomial length, the landscape changes: the expressive power of determinism and randomization does not change, but the power of non-determinism is reduced to the extent that it becomes incomparable with randomization
Optimal Error Pseudodistributions for Read-Once Branching Programs
In a seminal work, Nisan (Combinatorica'92) constructed a pseudorandom
generator for length and width read-once branching programs with seed
length and error
. It remains a central question to reduce the seed length to
, which would prove that .
However, there has been no improvement on Nisan's construction for the case
, which is most relevant to space-bounded derandomization.
Recently, in a beautiful work, Braverman, Cohen and Garg (STOC'18) introduced
the notion of a pseudorandom pseudo-distribution (PRPD) and gave an explicit
construction of a PRPD with seed length . A PRPD is a relaxation of a pseudorandom
generator, which suffices for derandomizing and also implies a
hitting set. Unfortunately, their construction is quite involved and
complicated. Hoza and Zuckerman (FOCS'18) later constructed a much simpler
hitting set generator with seed length , but their techniques are restricted to hitting
sets.
In this work, we construct a PRPD with seed length This improves upon the
construction in [BCG18] by a factor, and is
optimal in the small error regime. In addition, we believe our construction and
analysis to be simpler than the work of Braverman, Cohen and Garg
The Complexity of Verifying Boolean Programs as Differentially Private
We study the complexity of the problem of verifying differential privacy for
while-like programs working over boolean values and making probabilistic
choices. Programs in this class can be interpreted into finite-state
discrete-time Markov Chains (DTMC). We show that the problem of deciding
whether a program is differentially private for specific values of the privacy
parameters is PSPACE-complete. To show that this problem is in PSPACE, we adapt
classical results about computing hitting probabilities for DTMC. To show
PSPACE-hardness we use a reduction from the problem of checking whether a
program almost surely terminates or not. We also show that the problem of
approximating the privacy parameters that a program provides is PSPACE-hard.
Moreover, we investigate the complexity of similar problems also for several
relaxations of differential privacy: R\'enyi differential privacy, concentrated
differential privacy, and truncated concentrated differential privacy. For
these notions, we consider gap-versions of the problem of deciding whether a
program is private or not and we show that all of them are PSPACE-complete.Comment: Appeared in CSF 202