Automated Analytic Combinatorics

2020 ◽  
pp. 263-304
Author(s):  
Stephen Melczer
Keyword(s):  
Analytic Combinatorics

scholarly journals A Quantitative Study of Pure Parallel Processes

10.37236/3977 ◽  
2016 ◽  
Vol 23 (1) ◽  
Author(s):  
O. Bodini ◽  
A. Genitrini ◽  
F. Peschanski
Keyword(s):  
Quantitative Study ◽  
Random Sampling ◽  
Linear Time ◽  
Time Algorithm ◽  
Point Of View ◽  
Total Size ◽  
Combinatorial Explosion ◽  
Analytic Combinatorics ◽  
Parallel Processes ◽  
Uniform Random Sampling

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.

Random Regular Expression Over Huge Alphabets

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.

Algebraic combinatorics and QFT

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).

Analytic Combinatorics

2019 ◽  
Author(s):  
Marni Mishna
Keyword(s):  
Analytic Combinatorics

scholarly journals Combinatorial Analysis of Growth Models for Series-Parallel Networks

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.

scholarly journals Philippe Flajolet, the Father of Analytic Combinatorics

2011 ◽  
Vol 412 (35) ◽  
pp. 4469-4470
Author(s):  
Bruno Salvy ◽  
Bob Sedgewick ◽  
Michèle Soria ◽  
Wojciech Szpankowski ◽  
Brigitte Vallée
Keyword(s):  
Analytic Combinatorics
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