Automated Analytic Combinatorics

In this paper, we study the interleaving – or pure merge – operator that most often characterizes parallelism in concurrency theory. This operator is a principal cause of the so-called combinatorial explosion that makes the analysis of process behaviours e.g. by model-checking, very hard – at least from the point of view of computational complexity. The originality of our approach is to study this combinatorial explosion phenomenon on average, relying on advanced analytic combinatorics techniques. We study various measures that contribute to a better understanding of the process behaviours represented as plane rooted trees: the number of runs (corresponding to the width of the trees), the expected total size of the trees as well as their overall shape. Two practical outcomes of our quantitative study are also presented: (1) a linear-time algorithm to compute the probability of a concurrent run prefix, and (2) an efficient algorithm for uniform random sampling of concurrent runs. These provide interesting responses to the combinatorial explosion problem.

Download Full-text

Random Regular Expression Over Huge Alphabets

International Journal of Foundations of Computer Science ◽

10.1142/s012905412141001x ◽

2021 ◽

pp. 1-20

Author(s):

Cyril Nicaud ◽

Pablo Rotondo

Keyword(s):

Regular Expression ◽

Empty Word ◽

Regular Expressions ◽

Expected Number ◽

Analytic Combinatorics ◽

Transfer Theorem ◽

Leading Term

In this article, we study some properties of random regular expressions of size [Formula: see text], when the cardinality of the alphabet also depends on [Formula: see text]. For this, we revisit and improve the classical Transfer Theorem from the field of analytic combinatorics. This provides precise estimations for the number of regular expressions, the probability of recognizing the empty word and the expected number of Kleene stars in a random expression. For all these statistics, we show that there is a threshold when the size of the alphabet approaches [Formula: see text], at which point the leading term in the asymptotics starts oscillating.

Download Full-text

Algebraic combinatorics and QFT

Combinatorial Physics ◽

10.1093/oso/9780192895493.003.0008 ◽

2021 ◽

pp. 76-94

Author(s):

Adrian Tanasa

Keyword(s):

Hopf Algebra ◽

Hochschild Cohomology ◽

Algebraic Combinatorics ◽

Fourth Section ◽

Analytic Combinatorics ◽

Multi Scale ◽

Starting Point ◽

Feynman Graphs ◽

Perturbative Renormalization

We have seen in the previous chapter some of the non-trivial interplay between analytic combinatorics and QFT. In this chapter, we illustrate how yet another branch of combinatorics, algebraic combinatorics, interferes with QFT. In this chapter, after a brief algebraic reminder in the first section, we introduce in the second section the Connes–Kreimer Hopf algebra of Feynman graphs and we show its relation with the combinatorics of QFT perturbative renormalization. We then study the algebra's Hochschild cohomology in relation with the combinatorial Dyson–Schwinger equation in QFT. In the fourth section we present a Hopf algebraic description of the so-called multi-scale renormalization (the multi-scale approach to the perturbative renormalization being the starting point for the constructive renormalization programme).

Download Full-text

Analytic Combinatorics

10.1201/9781351036825 ◽

2019 ◽

Author(s):

Marni Mishna

Keyword(s):

Analytic Combinatorics

Download Full-text

Analytic combinatorics by Philippe Flajolet and Robert Sedgewick, published by Cambridge Press, 2009 824 pages, hardcover

ACM SIGACT News ◽

10.1145/1814370.1814373 ◽

2010 ◽

Vol 41 (2) ◽

pp. 11-14

Author(s):

Miklós Bóna

Keyword(s):

Analytic Combinatorics

Download Full-text

Combinatorial Analysis of Growth Models for Series-Parallel Networks

Combinatorics Probability Computing ◽

10.1017/s096354831800038x ◽

2018 ◽

Vol 28 (4) ◽

pp. 574-599

Author(s):

MARKUS KUBA ◽

ALOIS PANHOLZER

Keyword(s):

Stochastic Models ◽

Growth Process ◽

Growth Models ◽

Expected Number ◽

Asymptotic Results ◽

Tree Structures ◽

Analytic Combinatorics ◽

Source To Sink ◽

Parallel Networks ◽

Stochastic Growth Models

We give combinatorial descriptions of two stochastic growth models for series-parallel networks introduced by Hosam Mahmoud by encoding the growth process via recursive tree structures. Using decompositions of the tree structures and applying analytic combinatorics methods allows a study of quantities in the corresponding series-parallel networks. For both models we obtain limiting distribution results for the degree of the poles and the length of a random source-to-sink path, and furthermore we get asymptotic results for the expected number of source-to-sink paths. Moreover, we introduce generalizations of these stochastic models by encoding the growth process of the networks via further important increasing tree structures.

Download Full-text