2,092 research outputs found
A Crevice on the Crane Beach: Finite-Degree Predicates
First-order logic (FO) over words is shown to be equiexpressive with FO
equipped with a restricted set of numerical predicates, namely the order, a
binary predicate MSB, and the finite-degree predicates: FO[Arb] = FO[<,
MSB, Fin].
The Crane Beach Property (CBP), introduced more than a decade ago, is true of
a logic if all the expressible languages admitting a neutral letter are
regular.
Although it is known that FO[Arb] does not have the CBP, it is shown here
that the (strong form of the) CBP holds for both FO[<, Fin] and FO[<, MSB].
Thus FO[<, Fin] exhibits a form of locality and the CBP, and can still express
a wide variety of languages, while being one simple predicate away from the
expressive power of FO[Arb]. The counting ability of FO[<, Fin] is studied as
an application.Comment: Submitte
Logic Meets Algebra: the Case of Regular Languages
The study of finite automata and regular languages is a privileged meeting
point of algebra and logic. Since the work of Buchi, regular languages have
been classified according to their descriptive complexity, i.e. the type of
logical formalism required to define them. The algebraic point of view on
automata is an essential complement of this classification: by providing
alternative, algebraic characterizations for the classes, it often yields the
only opportunity for the design of algorithms that decide expressibility in
some logical fragment.
We survey the existing results relating the expressibility of regular
languages in logical fragments of MSO[S] with algebraic properties of their
minimal automata. In particular, we show that many of the best known results in
this area share the same underlying mechanics and rely on a very strong
relation between logical substitutions and block-products of pseudovarieties of
monoid. We also explain the impact of these connections on circuit complexity
theory.Comment: 37 page
Regular Representations of Uniform TC^0
The circuit complexity class DLOGTIME-uniform AC^0 is known to be a modest
subclass of DLOGTIME-uniform TC^0. The weakness of AC^0 is caused by the fact
that AC^0 is not closed under restricting AC^0-computable queries into simple
subsequences of the input. Analogously, in descriptive complexity, the logics
corresponding to DLOGTIME-uniform AC^0 do not have the relativization property
and hence they are not regular. This weakness of DLOGTIME-uniform AC^0 has been
elaborated in the line of research on the Crane Beach Conjecture. The
conjecture (which was refuted by Barrington, Immerman, Lautemann, Schweikardt
and Th{\'e}rien) was that if a language L has a neutral letter, then L can be
defined in first-order logic with the collection of all numerical built-in
relations, if and only if L can be already defined in FO with order.
In the first part of this article we consider logics in the range of AC^0 and
TC^0. First we formulate a combinatorial criterion for a cardinality quantifier
C_S implying that all languages in DLOGTIME-uniform TC^0 can be defined in
FO(C_S). For instance, this criterion is satisfied by C_S if S is the range of
some polynomial with positive integer coefficients of degree at least two. In
the second part of the paper we first adapt the key properties of abstract
logics to accommodate built-in relations. Then we define the regular interior
R-int(L) and regular closure R-cl(L), of a logic L, and show that the Crane
Beach Conjecture can be interpreted as a statement concerning the regular
interior of first-order logic with built-in relations B. We show that if B={+},
or B contains only unary relations besides the order, then R-int(FO_B)
collapses to FO with order. In contrast, our results imply that if B contains
the order and the range of a polynomial of degree at least two, then R-cl(FO_B)
includes all languages in DLOGTIME-uniform TC^0
When Can You Fold a Map?
We explore the following problem: given a collection of creases on a piece of
paper, each assigned a folding direction of mountain or valley, is there a flat
folding by a sequence of simple folds? There are several models of simple
folds; the simplest one-layer simple fold rotates a portion of paper about a
crease in the paper by +-180 degrees. We first consider the analogous questions
in one dimension lower -- bending a segment into a flat object -- which lead to
interesting problems on strings. We develop efficient algorithms for the
recognition of simply foldable 1D crease patterns, and reconstruction of a
sequence of simple folds. Indeed, we prove that a 1D crease pattern is
flat-foldable by any means precisely if it is by a sequence of one-layer simple
folds.
Next we explore simple foldability in two dimensions, and find a surprising
contrast: ``map'' folding and variants are polynomial, but slight
generalizations are NP-complete. Specifically, we develop a linear-time
algorithm for deciding foldability of an orthogonal crease pattern on a
rectangular piece of paper, and prove that it is (weakly) NP-complete to decide
foldability of (1) an orthogonal crease pattern on a orthogonal piece of paper,
(2) a crease pattern of axis-parallel and diagonal (45-degree) creases on a
square piece of paper, and (3) crease patterns without a mountain/valley
assignment.Comment: 24 pages, 19 figures. Version 3 includes several improvements thanks
to referees, including formal definitions of simple folds, more figures,
table summarizing results, new open problems, and additional reference
The Small Scale Structure of Space-Time: A Bibliographical Review
This essay is a tour around many of the lesser known pregeometric models of
physics, as well as the mainstream approaches to quantum gravity, in search of
common themes which may provide a glimpse of the final theory which must lie
behind them.Comment: 50 pages, TeX (two references added
Study of space battery accelerated testing techniques. Phase 2 report - Ideal approaches towards accelerated tests and analysis of data
Ideal approaches to accelerated life tests and data analysis applied to space batterie
Constraint treatment techniques and parallel algorithms for multibody dynamic analysis
Computational procedures for kinematic and dynamic analysis of three-dimensional multibody dynamic (MBD) systems are developed from the differential-algebraic equations (DAE's) viewpoint. Constraint violations during the time integration process are minimized and penalty constraint stabilization techniques and partitioning schemes are developed. The governing equations of motion, a two-stage staggered explicit-implicit numerical algorithm, are treated which takes advantage of a partitioned solution procedure. A robust and parallelizable integration algorithm is developed. This algorithm uses a two-stage staggered central difference algorithm to integrate the translational coordinates and the angular velocities. The angular orientations of bodies in MBD systems are then obtained by using an implicit algorithm via the kinematic relationship between Euler parameters and angular velocities. It is shown that the combination of the present solution procedures yields a computationally more accurate solution. To speed up the computational procedures, parallel implementation of the present constraint treatment techniques, the two-stage staggered explicit-implicit numerical algorithm was efficiently carried out. The DAE's and the constraint treatment techniques were transformed into arrowhead matrices to which Schur complement form was derived. By fully exploiting the sparse matrix structural analysis techniques, a parallel preconditioned conjugate gradient numerical algorithm is used to solve the systems equations written in Schur complement form. A software testbed was designed and implemented in both sequential and parallel computers. This testbed was used to demonstrate the robustness and efficiency of the constraint treatment techniques, the accuracy of the two-stage staggered explicit-implicit numerical algorithm, and the speed up of the Schur-complement-based parallel preconditioned conjugate gradient algorithm on a parallel computer
Computational methods and software systems for dynamics and control of large space structures
Two key areas of crucial importance to the computer-based simulation of large space structures are discussed. The first area involves multibody dynamics (MBD) of flexible space structures, with applications directed to deployment, construction, and maneuvering. The second area deals with advanced software systems, with emphasis on parallel processing. The latest research thrust in the second area involves massively parallel computers
- …