4,079 research outputs found
A Complete Grammar for Decomposing a Family of Graphs into 3-connected Components
Tutte has described in the book "Connectivity in graphs" a canonical
decomposition of any graph into 3-connected components. In this article we
translate (using the language of symbolic combinatorics)
Tutte's decomposition into a general grammar expressing any family of graphs
(with some stability conditions) in terms of the 3-connected subfamily. A key
ingredient we use is an extension of the so-called dissymmetry theorem, which
yields negative signs in the grammar.
As a main application we recover in a purely combinatorial way the analytic
expression found by Gim\'enez and Noy for the series counting labelled planar
graphs (such an expression is crucial to do asymptotic enumeration and to
obtain limit laws of various parameters on random planar graphs). Besides the
grammar, an important ingredient of our method is a recent bijective
construction of planar maps by Bouttier, Di Francesco and Guitter.Comment: 39 page
Colored operads, series on colored operads, and combinatorial generating systems
We introduce bud generating systems, which are used for combinatorial
generation. They specify sets of various kinds of combinatorial objects, called
languages. They can emulate context-free grammars, regular tree grammars, and
synchronous grammars, allowing us to work with all these generating systems in
a unified way. The theory of bud generating systems uses colored operads.
Indeed, an object is generated by a bud generating system if it satisfies a
certain equation in a colored operad. To compute the generating series of the
languages of bud generating systems, we introduce formal power series on
colored operads and several operations on these. Series on colored operads are
crucial to express the languages specified by bud generating systems and allow
us to enumerate combinatorial objects with respect to some statistics. Some
examples of bud generating systems are constructed; in particular to specify
some sorts of balanced trees and to obtain recursive formulas enumerating
these.Comment: 48 page
Gibbs and Quantum Discrete Spaces
Gibbs measure is one of the central objects of the modern probability,
mathematical statistical physics and euclidean quantum field theory. Here we
define and study its natural generalization for the case when the space, where
the random field is defined is itself random. Moreover, this randomness is not
given apriori and independently of the configuration, but rather they depend on
each other, and both are given by Gibbs procedure; We call the resulting object
a Gibbs family because it parametrizes Gibbs fields on different graphs in the
support of the distribution. We study also quantum (KMS) analog of Gibbs
families. Various applications to discrete quantum gravity are given.Comment: 37 pages, 2 figure
Recovering Grammar Relationships for the Java Language Specification
Grammar convergence is a method that helps discovering relationships between
different grammars of the same language or different language versions. The key
element of the method is the operational, transformation-based representation
of those relationships. Given input grammars for convergence, they are
transformed until they are structurally equal. The transformations are composed
from primitive operators; properties of these operators and the composed chains
provide quantitative and qualitative insight into the relationships between the
grammars at hand. We describe a refined method for grammar convergence, and we
use it in a major study, where we recover the relationships between all the
grammars that occur in the different versions of the Java Language
Specification (JLS). The relationships are represented as grammar
transformation chains that capture all accidental or intended differences
between the JLS grammars. This method is mechanized and driven by nominal and
structural differences between pairs of grammars that are subject to
asymmetric, binary convergence steps. We present the underlying operator suite
for grammar transformation in detail, and we illustrate the suite with many
examples of transformations on the JLS grammars. We also describe the
extraction effort, which was needed to make the JLS grammars amenable to
automated processing. We include substantial metadata about the convergence
process for the JLS so that the effort becomes reproducible and transparent
ACE: A Cliché-based Program Structure Editor
ACE extends the syntax-directed paradigm of program editing by adding support for programming clichés. A programming cliché is a standard algorithmic fragment. ACE supports the rapid construction of programs through the combination of clichés selected from a cliché library.
ACE is also innovative in the way it support the basic structure editor operations. Instead of being based directly on the grammar for a programming language, ACE is based on a modified grammar which is designed to facilitate editing. Uniformity of the user interface is achieved by encoding the modified grammar as a set of clichés.MIT Artificial Intelligence Laborator
The Design & Implementation of an Abstract Semantic Graph for Statement-Level Dynamic Analysis of C++ Applications
In this thesis, we describe our system, Hylian, for statement-level analysis, both static and dynamic, of a C++ application. We begin by extending the GNU gcc parser to generate parse trees in XML format for each of the compilation units in a C++ application. We then provide verification that the generated parse trees are structurally equivalent to the code in the original C++ application. We use the generated parse trees, together with an augmented version of the gcc test suite, to recover a grammar for the C++ dialect that we parse. We use the recovered grammar to generate a schema for further verification of the parse trees and evaluate the coverage provided by our C++ test suite. We then extend the parse tree, for each compilation unit, with semantic information to form an abstract semantic graph, ASG, and then link the ASGs for all of the compilation units into a unified ASG for the entire application under study. In addition, to relieve the cognitive burden of information that may inundate a developer, we describe our development of extensions to Hylian to build abbreviated abstract semantic graphs, which incorporate information about user code, but not about compiler provided library code. Finally, we describe the various approaches that we adopted to provide assurance for the developer that the ASGs that Hylian builds, correctly represent the program under study
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