3,377 research outputs found
Combining Enumeration and Deductive Techniques in order to Increase the Class of Constructible Infinite Models
AbstractA new method for building infinite models for first-order formulae is presented. The method combines enumeration techniques with existing deductive (in a broad sense) ones. Its soundness and completeness w.r.t. the class of models that can be represented by equational constraints are proven. This shows that the use of enumeration techniques strictly increases the power of existing methods for building Herbrand models that are not complete in this sense. Some strategies are proposed to reduce the search space. We give examples and show how to use this approach for building interactively a model of a formula introduced by Goldfarb in his proof of the undecidability of the Gödel class with identity. This formula is satisfiable but has no finite model
SYK model with an extra diagonal perturbation: phase transition in the eigenvalue spectrum
We study the SYK model with an extra constant source, \.i.e. a constant
matrix or equivalently a diagonal matrix with only one non-zero entry
. By using methods from analytic combinatorics, we find exact
expressions for the moments of this model. We further prove that the spectrum
of this model can have a gap when , thus exhibiting a
phase transition in . In this case, a single isolated eigenvalue
splits off from SYK's eigenvalues distribution. We located this single
eigenvalue by analyzing the singular behavior of a supercritical functional
composition scheme. In certain limit our results recover the ones of random
matrices with non-zero mean entries.Comment: 22 pages (1 appendix and with references included), 7 figure
Linear Temporal Logic and Propositional Schemata, Back and Forth (extended version)
This paper relates the well-known Linear Temporal Logic with the logic of
propositional schemata introduced by the authors. We prove that LTL is
equivalent to a class of schemata in the sense that polynomial-time reductions
exist from one logic to the other. Some consequences about complexity are
given. We report about first experiments and the consequences about possible
improvements in existing implementations are analyzed.Comment: Extended version of a paper submitted at TIME 2011: contains proofs,
additional examples & figures, additional comparison between classical
LTL/schemata algorithms up to the provided translations, and an example of
how to do model checking with schemata; 36 pages, 8 figure
Incompleteness via paradox and completeness
This paper explores the relationship borne by the traditional paradoxes of set theory and semantics to formal incompleteness phenomena. A central tool is the application of the Arithmetized Completeness Theorem to systems of second-order arithmetic and set theory in which various “paradoxical notions” for first-order languages can be formalized. I will first discuss the setting in which this result was originally presented by Hilbert & Bernays (1939) and also how it was later adapted by Kreisel (1950) andWang (1955) in order to obtain formal undecidability results. A generalization of this method will then be presented whereby Russell’s paradox, a variant of Mirimano’s paradox, the Liar, and the Grelling-Nelson paradox may be uniformly transformed into incompleteness theorems. Some additional observations are then framed relating these results to the unification of the set theoretic and semantic paradoxes, the intensionality of arithmetization (in the sense of Feferman, 1960), and axiomatic theories of truth
Tameness and frames revisited
We study the problem of extending an abstract independence notion for types
of singletons (what Shelah calls a good frame) to longer types. Working in the
framework of tame abstract elementary classes, we show that good frames can
always be extended to types of independent sequences. As an application, we
show that tameness and a good frame imply Shelah's notion of dimension is
well-behaved, complementing previous work of Jarden and Sitton. We also improve
a result of the first author on extending a frame to larger models.Comment: 36 page
Finite Boolean Algebras for Solid Geometry using Julia's Sparse Arrays
The goal of this paper is to introduce a new method in computer-aided
geometry of solid modeling. We put forth a novel algebraic technique to
evaluate any variadic expression between polyhedral d-solids (d = 2, 3) with
regularized operators of union, intersection, and difference, i.e., any CSG
tree. The result is obtained in three steps: first, by computing an independent
set of generators for the d-space partition induced by the input; then, by
reducing the solid expression to an equivalent logical formula between Boolean
terms made by zeros and ones; and, finally, by evaluating this expression using
bitwise operators. This method is implemented in Julia using sparse arrays. The
computational evaluation of every possible solid expression, usually denoted as
CSG (Constructive Solid Geometry), is reduced to an equivalent logical
expression of a finite set algebra over the cells of a space partition, and
solved by native bitwise operators.Comment: revised version submitted to Computer-Aided Geometric Desig
Model Checking Spatial Logics for Closure Spaces
Spatial aspects of computation are becoming increasingly relevant in Computer
Science, especially in the field of collective adaptive systems and when
dealing with systems distributed in physical space. Traditional formal
verification techniques are well suited to analyse the temporal evolution of
programs; however, properties of space are typically not taken into account
explicitly. We present a topology-based approach to formal verification of
spatial properties depending upon physical space. We define an appropriate
logic, stemming from the tradition of topological interpretations of modal
logics, dating back to earlier logicians such as Tarski, where modalities
describe neighbourhood. We lift the topological definitions to the more general
setting of closure spaces, also encompassing discrete, graph-based structures.
We extend the framework with a spatial surrounded operator, a propagation
operator and with some collective operators. The latter are interpreted over
arbitrary sets of points instead of individual points in space. We define
efficient model checking procedures, both for the individual and the collective
spatial fragments of the logic and provide a proof-of-concept tool
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