1,903 research outputs found
Local Testing for Membership in Lattices
Motivated by the structural analogies between point lattices and linear error-correcting codes, and by the mature theory on locally testable codes, we initiate a systematic study of local testing for membership in lattices. Testing membership in lattices is also motivated in practice, by applications to integer programming, error detection in lattice-based communication, and cryptography. Apart from establishing the conceptual foundations of lattice testing, our results include the following: 1. We demonstrate upper and lower bounds on the query complexity of local testing for the well-known family of code formula lattices. Furthermore, we instantiate our results with code formula lattices constructed from Reed-Muller codes, and obtain nearly-tight bounds. 2. We show that in order to achieve low query complexity, it is sufficient to design one-sided non-adaptive canonical tests. This result is akin to, and based on an analogous result for error-correcting codes due to Ben-Sasson et al. (SIAM J. Computing 35(1) pp1-21)
The Covering Problem
An important endeavor in computer science is to understand the expressive
power of logical formalisms over discrete structures, such as words. Naturally,
"understanding" is not a mathematical notion. This investigation requires
therefore a concrete objective to capture this understanding. In the
literature, the standard choice for this objective is the membership problem,
whose aim is to find a procedure deciding whether an input regular language can
be defined in the logic under investigation. This approach was cemented as the
right one by the seminal work of Sch\"utzenberger, McNaughton and Papert on
first-order logic and has been in use since then. However, membership questions
are hard: for several important fragments, researchers have failed in this
endeavor despite decades of investigation. In view of recent results on one of
the most famous open questions, namely the quantifier alternation hierarchy of
first-order logic, an explanation may be that membership is too restrictive as
a setting. These new results were indeed obtained by considering more general
problems than membership, taking advantage of the increased flexibility of the
enriched mathematical setting. This opens a promising research avenue and
efforts have been devoted at identifying and solving such problems for natural
fragments. Until now however, these problems have been ad hoc, most fragments
relying on a specific one. A unique new problem replacing membership as the
right one is still missing. The main contribution of this paper is a suitable
candidate to play this role: the Covering Problem. We motivate this problem
with 3 arguments. First, it admits an elementary set theoretic formulation,
similar to membership. Second, we are able to reexplain or generalize all known
results with this problem. Third, we develop a mathematical framework and a
methodology tailored to the investigation of this problem
Space-Efficient Data Structures for Lattices
A lattice is a partially-ordered set in which every pair of elements has a
unique meet (greatest lower bound) and join (least upper bound). We present new
data structures for lattices that are simple, efficient, and nearly optimal in
terms of space complexity.
Our first data structure can answer partial order queries in constant time
and find the meet or join of two elements in time, where is
the number of elements in the lattice. It occupies bits of
space, which is only a factor from the -bit
lower bound for storing lattices. The preprocessing time is . This
structure admits a simple space-time tradeoff so that, for any , the data structure supports meet and join queries in
time, occupies bits of space, and can be
constructed in time.
Our second data structure uses bits of space and supports
meet and join in time, where is the maximum
degree of any element in the transitive reduction graph of the lattice. This
structure is much faster for lattices with low-degree elements.
This paper also identifies an error in a long-standing solution to the
problem of representing lattices. We discuss the issue with this previous work.Comment: Accepted in SWAT 202
- …