57,065 research outputs found
What's Decidable About Sequences?
We present a first-order theory of sequences with integer elements,
Presburger arithmetic, and regular constraints, which can model significant
properties of data structures such as arrays and lists. We give a decision
procedure for the quantifier-free fragment, based on an encoding into the
first-order theory of concatenation; the procedure has PSPACE complexity. The
quantifier-free fragment of the theory of sequences can express properties such
as sortedness and injectivity, as well as Boolean combinations of periodic and
arithmetic facts relating the elements of the sequence and their positions
(e.g., "for all even i's, the element at position i has value i+3 or 2i"). The
resulting expressive power is orthogonal to that of the most expressive
decidable logics for arrays. Some examples demonstrate that the fragment is
also suitable to reason about sequence-manipulating programs within the
standard framework of axiomatic semantics.Comment: Fixed a few lapses in the Mergesort exampl
Dynamic Relative Compression, Dynamic Partial Sums, and Substring Concatenation
Given a static reference string and a source string , a relative
compression of with respect to is an encoding of as a sequence of
references to substrings of . Relative compression schemes are a classic
model of compression and have recently proved very successful for compressing
highly-repetitive massive data sets such as genomes and web-data. We initiate
the study of relative compression in a dynamic setting where the compressed
source string is subject to edit operations. The goal is to maintain the
compressed representation compactly, while supporting edits and allowing
efficient random access to the (uncompressed) source string. We present new
data structures that achieve optimal time for updates and queries while using
space linear in the size of the optimal relative compression, for nearly all
combinations of parameters. We also present solutions for restricted and
extended sets of updates. To achieve these results, we revisit the dynamic
partial sums problem and the substring concatenation problem. We present new
optimal or near optimal bounds for these problems. Plugging in our new results
we also immediately obtain new bounds for the string indexing for patterns with
wildcards problem and the dynamic text and static pattern matching problem
Entanglement Entropy From Tensor Network States for Stabilizer Codes
In this paper, we present the construction of tensor network states (TNS) for
some of the degenerate ground states of 3D stabilizer codes. We then use the
TNS formalism to obtain the entanglement spectrum and entropy of these
ground-states for some special cuts. In particular, we work out the examples of
the 3D toric code, the X-cube model and the Haah code. The latter two models
belong to the category of "fracton" models proposed recently, while the first
one belongs to the conventional topological phases. We mention the cases for
which the entanglement entropy and spectrum can be calculated exactly: for
these, the constructed TNS is the singular value decomposition (SVD) of the
ground states with respect to particular entanglement cuts. Apart from the area
law, the entanglement entropies also have constant and linear corrections for
the fracton models, while the entanglement entropies for the toric code models
only have constant corrections. For the cuts we consider, the entanglement
spectra of these three models are completely flat. We also conjecture that the
negative linear correction to the area law is a signature of extensive ground
state degeneracy. Moreover, the transfer matrices of these TNS can be
constructed. We show that the transfer matrices are projectors whose
eigenvalues are either 1 or 0. The number of nonzero eigenvalues is tightly
related to the ground state degeneracy.Comment: 33+9 pages. 16+3 figure
Schwinger-Dyson operator of Yang-Mills matrix models with ghosts and derivations of the graded shuffle algebra
We consider large-N multi-matrix models whose action closely mimics that of
Yang-Mills theory, including gauge-fixing and ghost terms. We show that the
factorized Schwinger-Dyson loop equations, expressed in terms of the generating
series of gluon and ghost correlations G(xi), are quadratic equations S^i G = G
xi^i G in concatenation of correlations. The Schwinger-Dyson operator S^i is
built from the left annihilation operator, which does not satisfy the Leibnitz
rule with respect to concatenation. So the loop equations are not differential
equations. We show that left annihilation is a derivation of the graded shuffle
product of gluon and ghost correlations. The shuffle product is the point-wise
product of Wilson loops, expressed in terms of correlations. So in the limit
where concatenation is approximated by shuffle products, the loop equations
become differential equations. Remarkably, the Schwinger-Dyson operator as a
whole is also a derivation of the graded shuffle product. This allows us to
turn the loop equations into linear equations for the shuffle reciprocal, which
might serve as a starting point for an approximation scheme.Comment: 13 pages, added discussion & references, title changed, minor
corrections, published versio
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