arXiv subjects

Bernhard Gittenberger

Publications and source records attributed to Bernhard Gittenberger.

24 records · Page 2Linked to original sources

arXiv2015

The relation between tree size complexity and probability for Boolean functions generated by uniform random trees

We consider a probability distribution on the set of Boolean functions in n variables which is induced by random Boolean expressions. Such an expression is a random rooted plane tree where the internal vertices are labelled with connectives And and OR and the leaves are labelled with variables or negated variables. We study limiting distribution when the tree size tends to infinity and derive a relation between the tree size complexity and the probability of a function. This is done by first expressing trees representing a particular function as expansions of minimal trees representing this function and then computing the probabilities by means of combinatorial counting arguments relying on generating functions and singularity analysis.

math.CO↗

arXiv2015

On the number of lambda terms with prescribed size of their De Bruijn representation

John Tromp introduced the so-called 'binary lambda calculus' as a way to encode lambda terms in terms of binary words. Later, Grygiel and Lescanne conjectured that the number of binary lambda terms with $m$ free indices and of size $n$ (encoded as binary words of length $n$) is $o(n^{-3/2} τ^{-n})$ for $τ\approx 1.963448\ldots$. We generalize the proposed notion of size and show that for several classes of lambda terms, including binary lambda terms with $m$ free indices, the number of terms of size $n$ is $Θ(n^{-3/2} ρ^{-n})$ with some class dependent constant $ρ$, which in particular disproves the above mentioned conjecture. A way to obtain lower and upper bounds for the constant near the leading term is presented and numerical results for a few previously introduced classes of lambda terms are given.

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arXiv2013

Enumeration of generalized $BCI$ lambda-terms

We investigate the asymptotic number of elements of size $n$ in a particular class of closed lambda-terms (so-called $BCI(p)$-terms) which are related to axiom systems of combinatory logic. By deriving a differential equation for the generating function of the counting sequence we obtain a recurrence relation which can be solved asymptotically. We derive differential equations for the generating functions of the counting sequences of other more general classes of terms as well: the class of $BCK(p)$-terms and that of closed lambda-terms. Using elementary arguments we obtain upper and lowerestimates for the number of closed lambda-terms of size $n$. Moreover, a recurrence relation is derived which allows an efficient computation of the counting sequence. $BCK(p)$-terms are discussed briefly.

math.CO↗

arXiv2013

Associative and commutative tree representations for Boolean functions

Since the 90's, several authors have studied a probability distribution on the set of Boolean functions on $n$ variables induced by some probability distributions on formulas built upon the connectors $And$ and $Or$ and the literals $\{x_{1}, \bar{x}_{1}, \dots, x_{n}, \bar{x}_{n}\}$. These formulas rely on plane binary labelled trees, known as Catalan trees. We extend all the results, in particular the relation between the probability and the complexity of a Boolean function, to other models of formulas: non-binary or non-plane labelled trees (i.e. Polya trees). This includes the natural tree class where associativity and commutativity of the connectors $And$ and $Or$ are realised.

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arXiv2011

The degree profile of Pólya trees

We investigate the profile of random Pólya trees of size $n$ when only nodes of degree $d$ are counted in each level. It is shown that, as in the case where all nodes contribute to the profile, the suitably normalized profile process converges weakly to a Brownian excursion local time. Moreover, we investigate the joint distribution of the number of nodes of degree $d_1$ and $d_2$ in the levels of the tree.

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arXiv2010

The Shape of Unlabeled Rooted Random Trees

We consider the number of nodes in the levels of unlabelled rooted random trees and show that the stochastic process given by the properly scaled level sizes weakly converges to the local time of a standard Brownian excursion. Furthermore we compute the average and the distribution of the height of such trees. These results extend existing results for conditioned Galton-Watson trees and forests to the case of unlabelled rooted trees and show that they behave in this respect essentially like a conditioned Galton-Watson process.

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