scholarly journals Formulae and Asymptotics for Coefficients of Algebraic Functions

2014 ◽  
Vol 24 (1) ◽  
pp. 1-53 ◽  
Author(s):  
CYRIL BANDERIER ◽  
MICHAEL DRMOTA
Keyword(s):  
Generating Functions ◽  
Entire Functions ◽  
Limit Laws ◽  
Context Free Grammar ◽  
Algebraic Functions ◽  
Strongly Connected ◽  
Non Gaussian ◽  
Positive Coefficients ◽  
Dyadic Numbers ◽  
Context Free

We study the coefficients of algebraic functions ∑n≥0fnzn. First, we recall the too-little-known fact that these coefficientsfnalways admit a closed form. Then we study their asymptotics, known to be of the typefn~CAnnα. When the function is a power series associated to a context-free grammar, we solve a folklore conjecture: the critical exponents α cannot be 1/3 or −5/2; they in fact belong to a proper subset of the dyadic numbers. We initiate the study of the set of possible values forA. We extend what Philippe Flajolet called the Drmota–Lalley–Woods theorem (which states that α=−3/2 when the dependency graph associated to the algebraic system defining the function is strongly connected). We fully characterize the possible singular behaviours in the non-strongly connected case. As a corollary, the generating functions of certain lattice paths and planar maps are not determined by a context-free grammar (i.e., their generating functions are not ℕ-algebraic). We give examples of Gaussian limit laws (beyond the case of the Drmota–Lalley–Woods theorem), and examples of non-Gaussian limit laws. We then extend our work to systems involving non-polynomial entire functions (non-strongly connected systems, fixed points of entire functions with positive coefficients). We give several closure properties for ℕ-algebraic functions. We end by discussing a few extensions of our results (infinite systems of equations, algorithmic aspects).

scholarly journals Coefficients of algebraic functions: formulae and asymptotics

2013 ◽  
Vol DMTCS Proceedings vol. AS,... (Proceedings) ◽  
Author(s):  
Cyril Banderier ◽  
Michael Drmota
Keyword(s):  
Critical Exponents ◽  
Entire Functions ◽  
Limit Laws ◽  
Context Free Grammar ◽  
Algebraic Functions ◽  
Strongly Connected ◽  
International Audience ◽  
Planar Maps ◽  
Dyadic Numbers ◽  
Context Free

International audience This paper studies the coefficients of algebraic functions. First, we recall the too-little-known fact that these coefficients $f_n$ have a closed form. Then, we study their asymptotics, known to be of the type $f_n \sim C A^n n^{\alpha}$. When the function is a power series associated to a context-free grammar, we solve a folklore conjecture: the appearing critical exponents $\alpha$ can not be $^1/_3$ or $^{-5}/_2$, they in fact belong to a subset of dyadic numbers. We extend what Philippe Flajolet called the Drmota-Lalley-Woods theorem (which is assuring $\alpha=^{-3}/_2$ as soon as a "dependency graph" associated to the algebraic system defining the function is strongly connected): We fully characterize the possible critical exponents in the non-strongly connected case. As a corollary, it shows that certain lattice paths and planar maps can not be generated by a context-free grammar (i.e., their generating function is not $\mathbb{N}-algebraic). We end by discussing some extensions of this work (limit laws, systems involving non-polynomial entire functions, algorithmic aspects). Cet article a pour héros les coefficients des fonctions algébriques. Après avoir rappelé le fait trop peu connu que ces coefficients $f_n$ admettent toujours une forme close, nous étudions leur asymptotique $f_n \sim C A^n n^{\alpha}$. Lorsque la fonction algébrique est la série génératrice d'une grammaire non-contextuelle, nous résolvons une vieille conjecture du folklore : les exposants critiques $\alpha$ ne peuvent pas être $^1/_3$ ou $^{-5}/_2$ et sont en fait restreints à un sous-ensemble des nombres dyadiques. Nous étendons ce que Philippe Flajolet appelait le théorème de Drmota-Lalley-Woods (qui affirme que $\alpha=^{-3}/_2$ dès lors qu'un "graphe de dépendance" associé au système algébrique est fortement connexe) : nous caractérisons complètement les exposants critiques dans le cas non fortement connexe. Un corolaire immédiat est que certaines marches et cartes planaires ne peuvent pas être engendrées par une grammaire non-contextuelle non ambigüe (i. e., leur série génératrice n'est pas $\mathbb{N}-algébrique). Nous terminons par la discussion de diverses extensions de nos résultats (lois limites, systèmes d'équations de degré infini, aspects algorithmiques).

Framework for rare event detection using Artificial Neural Network based context free grammar

2020 ◽  
Vol 39 (6) ◽  
pp. 8463-8475
Author(s):  
Palanivel Srinivasan ◽  
Manivannan Doraipandian
Keyword(s):  
Neural Network ◽  
Artificial Neural Network ◽  
Event Detection ◽  
Performance Metrics ◽  
Rare Events ◽  
Rare Event ◽  
Video Stream ◽  
Context Free Grammar ◽  
Artificial Neural ◽  
Context Free

Rare event detections are performed using spatial domain and frequency domain-based procedures. Omnipresent surveillance camera footages are increasing exponentially due course the time. Monitoring all the events manually is an insignificant and more time-consuming process. Therefore, an automated rare event detection contrivance is required to make this process manageable. In this work, a Context-Free Grammar (CFG) is developed for detecting rare events from a video stream and Artificial Neural Network (ANN) is used to train CFG. A set of dedicated algorithms are used to perform frame split process, edge detection, background subtraction and convert the processed data into CFG. The developed CFG is converted into nodes and edges to form a graph. The graph is given to the input layer of an ANN to classify normal and rare event classes. Graph derived from CFG using input video stream is used to train ANN Further the performance of developed Artificial Neural Network Based Context-Free Grammar – Rare Event Detection (ACFG-RED) is compared with other existing techniques and performance metrics such as accuracy, precision, sensitivity, recall, average processing time and average processing power are used for performance estimation and analyzed. Better performance metrics values have been observed for the ANN-CFG model compared with other techniques. The developed model will provide a better solution in detecting rare events using video streams.

scholarly journals Searching for universal model of amyloid signaling motifs using probabilistic context-free grammars

2021 ◽  
Vol 22 (1) ◽  
Author(s):  
Witold Dyrka ◽  
Marlena Gąsior-Głogowska ◽  
Monika Szefczyk ◽  
Natalia Szulc
Keyword(s):  
Functional Relationship ◽  
High Sensitivity ◽  
Alternative Methods ◽  
Discriminative Power ◽  
Context Free Grammar ◽  
Protein Motifs ◽  
Functional Features ◽  
Universal Grammars ◽  
Context Free ◽  
Probabilistic Context

Abstract Background Amyloid signaling motifs are a class of protein motifs which share basic structural and functional features despite the lack of clear sequence homology. They are hard to detect in large sequence databases either with the alignment-based profile methods (due to short length and diversity) or with generic amyloid- and prion-finding tools (due to insufficient discriminative power). We propose to address the challenge with a machine learning grammatical model capable of generalizing over diverse collections of unaligned yet related motifs. Results First, we introduce and test improvements to our probabilistic context-free grammar framework for protein sequences that allow for inferring more sophisticated models achieving high sensitivity at low false positive rates. Then, we infer universal grammars for a collection of recently identified bacterial amyloid signaling motifs and demonstrate that the method is capable of generalizing by successfully searching for related motifs in fungi. The results are compared to available alternative methods. Finally, we conduct spectroscopy and staining analyses of selected peptides to verify their structural and functional relationship. Conclusions While the profile HMMs remain the method of choice for modeling homologous sets of sequences, PCFGs seem more suitable for building meta-family descriptors and extrapolating beyond the seed sample.

A generalization of the concept of a context-free grammar

Cybernetics ◽  
1974 ◽  
Vol 8 (3) ◽  
pp. 349-351
Author(s):  
A. A. Letichevskii
Keyword(s):  
Context Free Grammar ◽  
Context Free

scholarly journals Knowledge Sources for Constituent Parsing of German, a Morphologically Rich and Less-Configurational Language

2013 ◽  
Vol 39 (1) ◽  
pp. 57-85 ◽  
Author(s):  
Alexander Fraser ◽  
Helmut Schmid ◽  
Richárd Farkas ◽  
Renjing Wang ◽  
Hinrich Schütze
Keyword(s):  
State Of The Art ◽  
Lessons Learned ◽  
Knowledge Sources ◽  
Lexical Knowledge ◽  
Context Free Grammar ◽  
The Impact ◽  
Context Free ◽  
Probabilistic Context

We study constituent parsing of German, a morphologically rich and less-configurational language. We use a probabilistic context-free grammar treebank grammar that has been adapted to the morphologically rich properties of German by markovization and special features added to its productions. We evaluate the impact of adding lexical knowledge. Then we examine both monolingual and bilingual approaches to parse reranking. Our reranking parser is the new state of the art in constituency parsing of the TIGER Treebank. We perform an analysis, concluding with lessons learned, which apply to parsing other morphologically rich and less-configurational languages.

scholarly journals Context-free grammar forms with strict interpretations

1980 ◽  
Vol 21 (1) ◽  
pp. 110-135 ◽  
Author(s):  
H.A. Maurer ◽  
A. Salomaa ◽  
D. Wood
Keyword(s):  
Context Free Grammar ◽  
Context Free

scholarly journals Asymptotic Properties of Some Minor-Closed Classes of Graphs

2014 ◽  
Vol 23 (5) ◽  
pp. 749-795 ◽  
Author(s):  
MIREILLE BOUSQUET-MÉLOU ◽  
KERSTIN WELLER
Keyword(s):  
Asymptotic Properties ◽  
Systematic Investigation ◽  
Exact Enumeration ◽  
Limit Laws ◽  
Closed Class ◽  
Connected Case ◽  
Non Gaussian ◽  
A Minor ◽  
Excluded Minors

Let${\cal A}$be a minor-closed class of labelled graphs, and let${\cal G}_{n}$be a random graph sampled uniformly from the set ofn-vertex graphs of${\cal A}$. Whennis large, what is the probability that${\cal G}_{n}$is connected? How many components does it have? How large is its biggest component? Thanks to the work of McDiarmid and his collaborators, these questions are now solved when all excluded minors are 2-connected.Using exact enumeration, we study a collection of classes${\cal A}$excluding non-2-connected minors, and show that their asymptotic behaviour may be rather different from the 2-connected case. This behaviour largely depends on the nature of the dominant singularity of the generating functionC(z) that counts connected graphs of${\cal A}$. We classify our examples accordingly, thus taking a first step towards a classification of minor-closed classes of graphs. Furthermore, we investigate a parameter that has not received any attention in this context yet: the size of the root component. It follows non-Gaussian limit laws (Beta and Gamma), and clearly merits a systematic investigation.

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