2 research outputs found
A Variant of the VC-Dimension with Applications to Depth-3 Circuits
We introduce the following variant of the VC-dimension. Given and a positive integer , we define to be the
size of the largest subset such that the projection of on
every subset of of size is the -dimensional cube. We show that
determining the largest cardinality of a set with a given
dimension is equivalent to a Tur\'an-type problem related to the total number
of cliques in a -uniform hypergraph. This allows us to beat the
Sauer--Shelah lemma for this notion of dimension. We use this to obtain several
results on -circuits, i.e., depth- circuits with top gate OR and
bottom fan-in at most :
* Tight relationship between the number of satisfying assignments of a
-CNF and the dimension of the largest projection accepted by it, thus
improving Paturi, Saks, and Zane (Comput. Complex. '00).
* Improved -circuit lower bounds for affine dispersers for
sublinear dimension. Moreover, we pose a purely hypergraph-theoretic conjecture
under which we get further improvement.
* We make progress towards settling the complexity of the inner
product function and all degree- polynomials over in general.
The question of determining the complexity of IP was recently
posed by Golovnev, Kulikov, and Williams (ITCS'21)
Linear Branching Programs and Directional Affine Extractors
A natural model of read-once linear branching programs is a branching program
where queries are linear forms, and along each path, the queries
are linearly independent. We consider two restrictions of this model, which we
call weakly and strongly read-once, both generalizing standard read-once
branching programs and parity decision trees. Our main results are as follows.
- Average-case complexity. We define a pseudo-random class of functions which
we call directional affine extractors, and show that these functions are hard
on average for the strongly read-once model. We then present an explicit
construction of such function with good parameters. This strengthens the result
of Cohen and Shinkar (ITCS'16) who gave such average-case hardness for parity
decision trees. Directional affine extractors are stronger than the more
familiar class of affine extractors. Given the significance of these functions,
we expect that our new class of functions might be of independent interest.
- Proof complexity. We also consider the proof system
which is an extension of resolution with linear queries. A refutation of a CNF
in this proof system naturally defines a linear branching program solving the
corresponding search problem. Conversely, we show that a weakly read-once
linear BP solving the search problem can be converted to a
refutation with constant blow up