10 research outputs found
Asymptotic study of subcritical graph classes
International audienceWe present a unified general method for the asymptotic study of graphs from the so-called subcritical graph classes, which include the classes of cacti graphs, outerplanar graphs, and series-parallel graphs. This general method works both in the labelled and unlabelled framework. The main results concern the asymptotic enumeration and the limit laws of properties of random graphs chosen from subcritical classes. We show that the number (resp. ) of labelled (resp. unlabelled) graphs on vertices from a subcritical graph class {\cG}=\cup_n {\cG_n} satisfies asymptotically the universal behaviour for computable constants , e.g. for unlabelled series-parallel graphs, and that the number of vertices of degree ( fixed) in a graph chosen uniformly at random from \cG_n, converges (after rescaling) to a normal law as
Statistical properties of lambda terms
We present a quantitative, statistical analysis of random lambda terms in the
de Bruijn notation. Following an analytic approach using multivariate
generating functions, we investigate the distribution of various combinatorial
parameters of random open and closed lambda terms, including the number of
redexes, head abstractions, free variables or the de Bruijn index value
profile. Moreover, we conduct an average-case complexity analysis of finding
the leftmost-outermost redex in random lambda terms showing that it is on
average constant. The main technical ingredient of our analysis is a novel
method of dealing with combinatorial parameters inside certain infinite,
algebraic systems of multivariate generating functions. Finally, we briefly
discuss the random generation of lambda terms following a given skewed
parameter distribution and provide empirical results regarding a series of more
involved combinatorial parameters such as the number of open subterms and
binding abstractions in closed lambda terms.Comment: Major revision of section 5. In particular, proofs of Lemma 5.7 and
Theorem 5.
Statistical properties of lambda terms
We present a quantitative, statistical analysis of random lambda terms in the De Bruijn notation. Following an analytic approach using multivariate generat-ing functions, we investigate the distribution of various combinatorial parameters of random open and closed lambda terms, including the number of redexes, head abstractions, free variables or the De Bruijn index value profile. Moreover, we con-duct an average-case complexity analysis of finding the leftmost-outermost redex in random lambda terms showing that it is on average constant. The main technical
ingredient of our analysis is a novel method of dealing with combinatorial paramet-ers inside certain infinite, algebraic systems of multivariate generating functions. Finally, we briefly discuss the random generation of lambda terms following a given skewed parameter distribution and provide empirical results regarding a series of more involved combinatorial parameters such as the number of open subterms and binding abstractions in closed lambda terms
Infinite Systems of Functional Equations and Gaussian Limiting Distributions
In this paper infinite systems of functional equations in finitely or infinitely many random variables arising in combinatorial enumeration problems are studied. We prove sufficient conditions under which the combinatorial random variables encoded in the generating function of the system tend to a finite or infinite dimensional limiting distribution