1,245 research outputs found
Enumerating Segmented Patterns in Compositions and Encoding by Restricted Permutations
A composition of a nonnegative integer (n) is a sequence of positive integers
whose sum is (n). A composition is palindromic if it is unchanged when its
terms are read in reverse order. We provide a generating function for the
number of occurrences of arbitrary segmented partially ordered patterns among
compositions of (n) with a prescribed number of parts. These patterns
generalize the notions of rises, drops, and levels studied in the literature.
We also obtain results enumerating parts with given sizes and locations among
compositions and palindromic compositions with a given number of parts. Our
results are motivated by "encoding by restricted permutations," a relatively
undeveloped method that provides a language for describing many combinatorial
objects. We conclude with some examples demonstrating bijections between
restricted permutations and other objects.Comment: 12 pages, 1 figur
Unitary Representations of Wavelet Groups and Encoding of Iterated Function Systems in Solenoids
For points in real dimensions, we introduce a geometry for general digit
sets. We introduce a positional number system where the basis for our
representation is a fixed by matrix over \bz. Our starting point is a
given pair with the matrix assumed expansive, and
a chosen complete digit set, i.e., in bijective correspondence
with the points in \bz^d/A^T\bz^d. We give an explicit geometric
representation and encoding with infinite words in letters from .
We show that the attractor for an affine Iterated Function
System (IFS) based on is a set of fractions for our digital
representation of points in \br^d. Moreover our positional "number
representation" is spelled out in the form of an explicit IFS-encoding of a
compact solenoid \sa associated with the pair . The intricate
part (Theorem \ref{thenccycl}) is played by the cycles in \bz^d for the
initial -IFS. Using these cycles we are able to write down
formulas for the two maps which do the encoding as well as the decoding in our
positional -representation.
We show how some wavelet representations can be realized on the solenoid, and
on symbolic spaces
Quantum Information Encoding, Protection, and Correction from Trace-Norm Isometries
We introduce the notion of trace-norm isometric encoding and explore its
implications for passive and active methods to protect quantum information
against errors. Beside providing an operational foundations to the "subsystems
principle" [E. Knill, Phys. Rev. A 74, 042301 (2006)] for faithfully realizing
quantum information in physical systems, our approach allows additional
explicit connections between noiseless, protectable, and correctable quantum
codes to be identified. Robustness properties of isometric encodings against
imperfect initialization and/or deviations from the intended error models are
also analyzed.Comment: 10 pages, 1 figur
Relational Parametricity and Control
We study the equational theory of Parigot's second-order
λμ-calculus in connection with a call-by-name continuation-passing
style (CPS) translation into a fragment of the second-order λ-calculus.
It is observed that the relational parametricity on the target calculus induces
a natural notion of equivalence on the λμ-terms. On the other hand,
the unconstrained relational parametricity on the λμ-calculus turns
out to be inconsistent with this CPS semantics. Following these facts, we
propose to formulate the relational parametricity on the λμ-calculus
in a constrained way, which might be called ``focal parametricity''.Comment: 22 pages, for Logical Methods in Computer Scienc
A QUANTUM ALGORITHM FOR AUTOMATA ENCODING
Encoding of finite automata or state machines is critical to modern digital logic design methods for sequential circuits. Encoding is the process of assigning to every state, input value, and output value of a state machine a binary string, which is used to represent that state, input value, or output value in digital logic. Usually, one wishes to choose an encoding that, when the state machine is implemented as a digital logic circuit, will optimize some aspect of that circuit. For instance, one might wish to encode in such a way as to minimize power dissipation or silicon area. For most such optimization objectives, no method to find the exact solution, other than a straightforward exhaustive search, is known. Recent progress towards producing a quantum computer of large enough scale to surpass modern supercomputers has made it increasingly relevant to consider how quantum computers may be used to solve problems of practical interest. A quantum computer using Grover’s well-known search algorithm can perform exhaustive searches that would be impractical on a classical computer, due to the speedup provided by Grover’s algorithm. Therefore, we propose to use Grover’s algorithm to find optimal encodings for finite state machines via exhaustive search. We demonstrate the design of quantum circuits that allow Grover’s algorithm to be used for this purpose. The quantum circuit design methods that we introduce are potentially applicable to other problems as well
Z2SAL: a translation-based model checker for Z
Despite being widely known and accepted in industry, the Z formal specification language has not so far been well supported by automated verification tools, mostly because of the challenges in handling the abstraction of the language. In this paper we discuss a novel approach to building a model-checker for Z, which involves implementing a translation from Z into SAL, the input language for the Symbolic Analysis Laboratory, a toolset which includes a number of model-checkers and a simulator. The Z2SAL translation deals with a number of important issues, including: mapping unbounded, abstract specifications into bounded, finite models amenable to a BDD-based symbolic checker; converting a non-constructive and piecemeal style of functional specification into a deterministic, automaton-based style of specification; and supporting the rich set-based vocabulary of the Z mathematical toolkit. This paper discusses progress made towards implementing as complete and faithful a translation as possible, while highlighting certain assumptions, respecting certain limitations and making use of available optimisations. The translation is illustrated throughout with examples; and a complete working example is presented, together with performance data
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