88 research outputs found
On optimal language compression for sets in PSPACE/poly
We show that if DTIME[2^O(n)] is not included in DSPACE[2^o(n)], then, for
every set B in PSPACE/poly, all strings x in B of length n can be represented
by a string compressed(x) of length at most log(|B^{=n}|)+O(log n), such that a
polynomial-time algorithm, given compressed(x), can distinguish x from all the
other strings in B^{=n}. Modulo the O(log n) additive term, this achieves the
information-theoretic optimum for string compression. We also observe that
optimal compression is not possible for sets more complex than PSPACE/poly
because for any time-constructible superpolynomial function t, there is a set A
computable in space t(n) such that at least one string x of length n requires
compressed(x) to be of length 2 log(|A^=n|).Comment: submitted to Theory of Computing System
On Algorithmic Statistics for space-bounded algorithms
Algorithmic statistics studies explanations of observed data that are good in
the algorithmic sense: an explanation should be simple i.e. should have small
Kolmogorov complexity and capture all the algorithmically discoverable
regularities in the data. However this idea can not be used in practice because
Kolmogorov complexity is not computable.
In this paper we develop algorithmic statistics using space-bounded
Kolmogorov complexity. We prove an analogue of one of the main result of
`classic' algorithmic statistics (about the connection between optimality and
randomness deficiences). The main tool of our proof is the Nisan-Wigderson
generator.Comment: accepted to CSR 2017 conferenc
Probabilistic Model Counting with Short XORs
The idea of counting the number of satisfying truth assignments (models) of a
formula by adding random parity constraints can be traced back to the seminal
work of Valiant and Vazirani, showing that NP is as easy as detecting unique
solutions. While theoretically sound, the random parity constraints in that
construction have the following drawback: each constraint, on average, involves
half of all variables. As a result, the branching factor associated with
searching for models that also satisfy the parity constraints quickly gets out
of hand. In this work we prove that one can work with much shorter parity
constraints and still get rigorous mathematical guarantees, especially when the
number of models is large so that many constraints need to be added. Our work
is based on the realization that the essential feature for random systems of
parity constraints to be useful in probabilistic model counting is that the
geometry of their set of solutions resembles an error-correcting code.Comment: To appear in SAT 1
Immunity and Simplicity for Exact Counting and Other Counting Classes
Ko [RAIRO 24, 1990] and Bruschi [TCS 102, 1992] showed that in some
relativized world, PSPACE (in fact, ParityP) contains a set that is immune to
the polynomial hierarchy (PH). In this paper, we study and settle the question
of (relativized) separations with immunity for PH and the counting classes PP,
C_{=}P, and ParityP in all possible pairwise combinations. Our main result is
that there is an oracle A relative to which C_{=}P contains a set that is
immune to BPP^{ParityP}. In particular, this C_{=}P^A set is immune to PH^{A}
and ParityP^{A}. Strengthening results of Tor\'{a}n [J.ACM 38, 1991] and Green
[IPL 37, 1991], we also show that, in suitable relativizations, NP contains a
C_{=}P-immune set, and ParityP contains a PP^{PH}-immune set. This implies the
existence of a C_{=}P^{B}-simple set for some oracle B, which extends results
of Balc\'{a}zar et al. [SIAM J.Comp. 14, 1985; RAIRO 22, 1988] and provides the
first example of a simple set in a class not known to be contained in PH. Our
proof technique requires a circuit lower bound for ``exact counting'' that is
derived from Razborov's [Mat. Zametki 41, 1987] lower bound for majority.Comment: 20 page
On an almost-universal hash function family with applications to authentication and secrecy codes
Universal hashing, discovered by Carter and Wegman in 1979, has many
important applications in computer science. MMH, which was shown to be
-universal by Halevi and Krawczyk in 1997, is a well-known universal
hash function family. We introduce a variant of MMH, that we call GRDH,
where we use an arbitrary integer instead of prime and let the keys
satisfy the
conditions (), where are
given positive divisors of . Then via connecting the universal hashing
problem to the number of solutions of restricted linear congruences, we prove
that the family GRDH is an -almost--universal family of
hash functions for some if and only if is odd and
. Furthermore, if these conditions are
satisfied then GRDH is -almost--universal, where is
the smallest prime divisor of . Finally, as an application of our results,
we propose an authentication code with secrecy scheme which strongly
generalizes the scheme studied by Alomair et al. [{\it J. Math. Cryptol.} {\bf
4} (2010), 121--148], and [{\it J.UCS} {\bf 15} (2009), 2937--2956].Comment: International Journal of Foundations of Computer Science, to appea
Empirical Bounds on Linear Regions of Deep Rectifier Networks
We can compare the expressiveness of neural networks that use rectified
linear units (ReLUs) by the number of linear regions, which reflect the number
of pieces of the piecewise linear functions modeled by such networks. However,
enumerating these regions is prohibitive and the known analytical bounds are
identical for networks with same dimensions. In this work, we approximate the
number of linear regions through empirical bounds based on features of the
trained network and probabilistic inference. Our first contribution is a method
to sample the activation patterns defined by ReLUs using universal hash
functions. This method is based on a Mixed-Integer Linear Programming (MILP)
formulation of the network and an algorithm for probabilistic lower bounds of
MILP solution sets that we call MIPBound, which is considerably faster than
exact counting and reaches values in similar orders of magnitude. Our second
contribution is a tighter activation-based bound for the maximum number of
linear regions, which is particularly stronger in networks with narrow layers.
Combined, these bounds yield a fast proxy for the number of linear regions of a
deep neural network.Comment: AAAI 202
Balancing Scalability and Uniformity in SAT Witness Generator
Constrained-random simulation is the predominant approach used in the
industry for functional verification of complex digital designs. The
effectiveness of this approach depends on two key factors: the quality of
constraints used to generate test vectors, and the randomness of solutions
generated from a given set of constraints. In this paper, we focus on the
second problem, and present an algorithm that significantly improves the
state-of-the-art of (almost-)uniform generation of solutions of large Boolean
constraints. Our algorithm provides strong theoretical guarantees on the
uniformity of generated solutions and scales to problems involving hundreds of
thousands of variables.Comment: This is a full version of DAC 2014 pape
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