Papers by Antoine Genitrini
Applicable Analysis and Discrete Mathematics, 2016
We consider a probability distribution on the set of Boolean functions in n variables which is in... more 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.
Pointed versus singular Boltzmann samplers: a comparative analysis
Pure Mathematics and Applications, 2015
Probabilities of Boolean Functions given by Random Implicational Formulas
The Electronic Journal of Combinatorics, 2012
ABSTRACT We study the asymptotic relation between the probability and the complexity of Boolean f... more ABSTRACT We study the asymptotic relation between the probability and the complexity of Boolean functions in the implicational fragment which are generated by large random Boolean expressions involving variables and implication, as the number of variables tends to infinity. In contrast to models studied in the literature so far, we consider two expressions to be equal if they differ only in the order of the premises. A precise asymptotic formula is derived for functions of low complexity. Furthermore, we show that this model does not exhibit the Shannon effect.
Our study is mainly based on analytic combinatorics and extends the Kozik's pattern theory, first... more Our study is mainly based on analytic combinatorics and extends the Kozik's pattern theory, first developed for the fixed-k Catalan tree model.
Increasing Diamonds
Lecture Notes in Computer Science, 2016
In this paper, we study the interleaving -or pure merge -operator that most often characterizes p... more 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 very hard -at least from the point of view of computational complexity -the analysis of process behaviours e.g. by model-checking. 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. 1 When one is interested in a finite axiomatization of the pure merge operator, a left variant must be introduced, cf.
Algorithmica, 2016
This article is motivated by the following satisfiability question: pick uniformly at random an a... more This article is motivated by the following satisfiability question: pick uniformly at random an and{or Boolean expression of length n, built on a set of kn Boolean variables. What is the probability that this expression is satisfiable? asymptotically when n tends to infinity?
The Shannon effect states that "almost all" Boolean functions have a complexity close to the maxi... more The Shannon effect states that "almost all" Boolean functions have a complexity close to the maximal possible for the uniform probability distribution. In this paper we use some probability distributions on functions, induced by random expressions, and prove that this model does not exhibit the Shannon effect.
Within the language of propositional formulae built on implication and a finite number of variabl... more Within the language of propositional formulae built on implication and a finite number of variables k, we analyze the set of formulae which are classical tautologies but not intuitionistic (we call such formulae -Peirce's formulae). We construct the large family of so called simple Peirce's formulae, whose sequence of densities for different k is asymptotically equivalent to the sequence 1 2k 2 . We prove that the densities of the sets of remaining Peirce's formulae are asymptotically bounded from above by c k 3 for some constant c ∈ R. The result justifies the statement that in the considered language almost all Peirce's formulae are simple. The result gives a partial answer to the question stated in the recent paper by H. Fournier, D. Gardy, A. Genitrini and M. Zaionc -although we have not proved the existence of the densities for Peirce's formulae, our result gives lower and upper bound for it (if it exists) and both bounds are asymptotically equivalent to 1 2k 2 .
Associativity for Binary Parallel Processes: A Quantitative Study
Lecture Notes in Computer Science, 2015
Lecture Notes in Computer Science, 2009
We address the problem of quantitative comparison of classical and intuitionistic logics within t... more We address the problem of quantitative comparison of classical and intuitionistic logics within the language of the full propositional system. We apply two different approaches, to estimate the asymptotic fraction of intuitionistic tautologies among classical tautologies, obtaining the same results for both. Our results justify informal statements such as "about 5/8 of classical tautologies are intuitionistic".
We consider random balanced Boolean formulas, built on the two connectives and and or, and a fixe... more We consider random balanced Boolean formulas, built on the two connectives and and or, and a fixed number of variables. The probability distribution induced on Boolean func- tions is shown to have a limit when letting the depth of these formulas grow to infinity. By investigating how this limiting distribution depends on the two underlying probability dis- tributions, over the connectives and over the Boolean variables, we prove that its support is made of linear threshold functions, and give the speed of convergence towards this limiting distribution.
Quantitative logic has been the subject of an increasing interest since a seminal paper by which ... more Quantitative logic has been the subject of an increasing interest since a seminal paper by which presented the first Analytic Combinatorics approach of the subject. Since then, the understanding of random Boolean trees has been deeply widened, even if the question of Shannon effect remains open for the majority of the models. We focus in this paper on the original case of Catalan And/Or trees and propose a new specification of those objects that implies easier ways to describe large families of trees. Equipped with this specification, we prove that the model of Catalan And/Or binary trees do not exhibit Shannon effect, i.e. there exists a family of functions with small complexities, that have a positive probability.
Probabilities of Boolean Functions given by Random Implicational Formulas
The electronic journal of combinatorics
We study the asymptotic relation between the probability and the complexity of Boolean functions ... more We study the asymptotic relation between the probability and the complexity of Boolean functions in the implicational fragment which are generated by large random Boolean expressions involving variables and implication, as the number of variables tends to infinity. In contrast to models studied in the literature so far, we consider two expressions to be equal if they differ only in the order of the premises. A precise asymptotic formula is derived for functions of low complexity. Furthermore, we show that this model does not exhibit the Shannon effect.
The Combinatorics of Non-determinism
Lecture Notes in Computer Science, 2014
An and/or tree is usually a binary plane tree, with internal nodes labelled by logical connective... more An and/or tree is usually a binary plane tree, with internal nodes labelled by logical connectives, and with leaves labelled by literals chosen in a fixed set of k variables and their negations. In the present paper, we introduce the first model of such Catalan trees, whose number of variables kn is a function of n, the size of the expressions. We describe the whole range of the probability distributions depending on the functions kn, as soon as it tends jointly with n to infinity. As a by-product we obtain a study of the satisfiability problem in the context of Catalan trees.
Theoretical Computer Science, 2015
Since the 90's, several authors have studied a probability distribution on the set of Boolean fun... more 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 {x1,x1, . . . , xn,xn}. 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.
Lecture Notes in Computer Science, 2008
We consider the logical system of boolean expressions built on the single connector of implicatio... more We consider the logical system of boolean expressions built on the single connector of implication and on positive literals. Assuming all expressions of a given size to be equally likely, we prove that we can define a probability distribution on the set of boolean functions expressible in this system. We then show how to approximate the probability of a function f when the number of variables grows to infinity, and that this asymptotic probability has a simple expression in terms of the complexity of f . We also prove that most expressions computing any given function in this system are "simple", in a sense that we make precise.
In this paper, we study the interleaving -or pure merge -operator that most often characterizes p... more 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 very hard -at least from the point of view of computational complexity -the analysis of process behaviours e.g. by model-checking. 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. 1 When one is interested in a finite axiomatization of the pure merge operator, a left variant must be introduced, cf.
Enumeration and random generation of concurrent computations
ABSTRACT In this paper, we study the shuffle operator on concurrent processes (represented as tre... more ABSTRACT In this paper, we study the shuffle operator on concurrent processes (represented as trees) using analytic combinatorics tools. As a first result, we show that the mean width of shuffle trees is exponentially smaller than the worst case upper-bound. We also study the expected size (in total number of nodes) of shuffle trees. We notice, rather unexpectedly, that only a small ratio of all nodes do not belong to the last two levels. We also provide a precise characterization of what “exponential growth” means in the case of the shuffle on trees. Two practical outcomes of our quantitative study are presented: (1) a linear-time algorithm to compute the probability of a concurrent run prefix, and (2) an efficient algorithm for uniform random generation of concurrent runs.
Uploads
Papers by Antoine Genitrini