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Pierre Lescanne

Publications and source records attributed to Pierre Lescanne.

At least 19 recordsLinked to original sources
arXiv2024

Holonomic equations and efficient random generation of binary trees

Holonomic equations are recursive equations which allow computing efficiently numbers of combinatoric objects. R{é}my showed that the holonomic equation associated with binary trees yields an efficient linear random generator of binary trees. I extend this paradigm to Motzkin trees and Schr{ö}der trees and show that despite slight differences my algorithm that generates random Schr{ö}der trees has linear expected complexity and my algorithm that generates Motzkin trees is in O(n) expected complexity, only if we can implement a specific oracle with a O(1) complexity. For Motzkin trees, I propose a solution which works well for realistic values (up to size ten millions) and yields an efficient algorithm.

cs.DS↗
arXiv2022

Almost all classical theorems are intuitionistic

Canonical expressions are representative of implicative propositions upto renaming of variables. In this paper we explore, using a Monte-Carlo approach, the model of canonical expressions in order to confirm the paradox that says that asymptotically almost all classical theorems are intuitionistic.

cs.LO↗
arXiv2021

List types for resource aware languages: an implicit name approach

A novel formalisation of variable control in languages with implicit names based on de Bruijn indices is presented. We design and implement three languages: first, a restricted language with implicit names; then, a restricted calculus with implicit names and explicit substitution, and finally, an extended calculus with implicit names, implicit substitution and resource control. We propose a novel concept of list types, which are used to give a simple and manageable definition of linearity. We develop an implementation in Haskell.

cs.LO↗
arXiv2019

On the enumeration of closures and environments with an application to random generation

Environments and closures are two of the main ingredients of evaluation in lambda-calculus. A closure is a pair consisting of a lambda-term and an environment, whereas an environment is a list of lambda-terms assigned to free variables. In this paper we investigate some dynamic aspects of evaluation in lambda-calculus considering the quantitative, combinatorial properties of environments and closures. Focusing on two classes of environments and closures, namely the so-called plain and closed ones, we consider the problem of their asymptotic counting and effective random generation. We provide an asymptotic approximation of the number of both plain environments and closures of size $n$. Using the associated generating functions, we construct effective samplers for both classes of combinatorial structures. Finally, we discuss the related problem of asymptotic counting and random generation of closed environemnts and closures.

cs.LO↗
arXiv2018

Combinatorics of explicit substitutions

$λ\upsilon$ is an extension of the $λ$-calculus which internalises the calculus of substitutions. In the current paper, we investigate the combinatorial properties of $λ\upsilon$ focusing on the quantitative aspects of substitution resolution. We exhibit an unexpected correspondence between the counting sequence for $λ\upsilon$-terms and famous Catalan numbers. As a by-product, we establish effective sampling schemes for random $λ\upsilon$-terms. We show that typical $λ\upsilon$-terms represent, in a strong sense, non-strict computations in the classic $λ$-calculus. Moreover, typically almost all substitutions are in fact suspended, i.e. unevaluated, under closures. Consequently, we argue that $λ\upsilon$ is an intrinsically non-strict calculus of explicit substitutions. Finally, we investigate the distribution of various redexes governing the substitution resolution in $λ\upsilon$ and investigate the quantitative contribution of various substitution primitives.

cs.LO↗
arXiv2017

Dependent Types for Extensive Games

Extensive games are tools largely used in economics to describe decision processes ofa community of agents. In this paper we propose a formal presentation based on theproof assistant COQ which focuses mostly on infinite extensive games and theircharacteristics. COQ proposes a feature called "dependent types", which meansthat the type of an object may depend on the type of its components. For instance,the set of choices or the set of utilities of an agent may depend on the agentherself. Using dependent types, we describe formally a very general class of gamesand strategy profiles, which corresponds somewhat to what game theorists are used to.We also discuss the notions of infiniteness in game theory and how this can beprecisely described.

cs.GT↗
arXiv2017

Quantitative aspects of linear and affine closed lambda terms

Affine $λ$-terms are $λ$-terms in which each bound variable occurs at most once and linear $λ$-terms are $λ$-terms in which each bound variables occurs once. and only once. In this paper we count the number of closed affine $λ$-terms of size $n$, closed linear $λ$-terms of size $n$, affine $β$-normal forms of size $n$ and linear $β$-normal forms of ise $n$, for different ways of measuring the size of $λ$-terms. From these formulas, we show how we can derive programs for generating all the terms of size $n$ for each class. For this we use a specific data structure, which are contexts taking into account all the holes at levels of abstractions.

cs.DM↗
arXiv2016

Combinatorics of $λ$-terms: a natural approach

We consider combinatorial aspects of $λ$-terms in the model based on de Bruijn indices where each building constructor is of size one. Surprisingly, the counting sequence for $λ$-terms corresponds also to two families of binary trees, namely black-white trees and zigzag-free ones. We provide a constructive proof of this fact by exhibiting appropriate bijections. Moreover, we identify the sequence of Motzkin numbers with the counting sequence for neutral $λ$-terms, giving a bijection which, in consequence, results in an exact-size sampler for the latter based on the exact-size sampler for Motzkin trees of Bodini et alli. Using the powerful theory of analytic combinatorics, we state several results concerning the asymptotic growth rate of $λ$-terms in neutral, normal, and head normal forms. Finally, we investigate the asymptotic density of $λ$-terms containing arbitrary fixed subterms showing that, inter alia, strongly normalising or typeable terms are asymptotically negligible in the set of all $λ$-terms.

cs.LO↗
arXiv2016

A natural counting of lambda terms

We study the sequences of numbers corresponding to lambda terms of given sizes, where the size is this of lambda terms with de Bruijn indices in a very natural model where all the operators have size 1. For plain lambda terms, the sequence corresponds to two families of binary trees for which we exhibit bijections. We study also the distribution of normal forms, head normal forms and strongly normalizing terms. In particular we show that strongly normalizing terms are of density 0 among plain terms.

cs.LO↗
arXiv2015

The risk of divergence

We present infinite extensive strategy profiles with perfect information and we show that replacing finite by infinite changes the notions and the reasoning tools. The presentation uses a formalism recently developed by logicians and computer science theoreticians, called coinduction. This builds a bridge between economic game theory and the most recent advance in theoretical computer science and logic. The key result is that rational agents may have strategy leading to divergence .

cs.GT↗
arXiv2015

Counting and Generating Terms in the Binary Lambda Calculus (Extended version)

In a paper entitled Binary lambda calculus and combinatory logic, John Tromp presents a simple way of encoding lambda calculus terms as binary sequences. In what follows, we study the numbers of binary strings of a given size that represent lambda terms and derive results from their generating functions, especially that the number of terms of size n grows roughly like 1.963447954. .. n. In a second part we use this approach to generate random lambda terms using Boltzmann samplers.

cs.LO↗
arXiv2015

Intelligent escalation and the principle of relativity

Escalation is the fact that in a game (for instance in an auction), the agents play forever. The $0,1$-game is an extremely simple infinite game with intelligent agents in which escalation arises. It shows at the light of research on cognitive psychology the difference between intelligence (algorithmic mind) and rationality (algorithmic and reflective mind) in decision processes. It also shows that depending on the point of view (inside or outside) the rationality of the agent may change which is proposed to be called the principle of relativity.

cs.GT↗
arXiv2014

Boltzmann samplers for random generation of lambda terms

Randomly generating structured objects is important in testing and optimizing functional programs, whereas generating random $'l$-terms is more specifically needed for testing and optimizing compilers. For that a tool called QuickCheck has been proposed, but in this tool the control of the random generation is left to the programmer. Ten years ago, a method called Boltzmann samplers has been proposed to generate combinatorial structures. In this paper, we show how Boltzmann samplers can be developed to generate lambda-terms, but also other data structures like trees. These samplers rely on a critical value which parameters the main random selector and which is exhibited here with explanations on how it is computed. Haskell programs are proposed to show how samplers are actually implemented.

cs.DS↗
arXiv2014

Counting Terms in the Binary Lambda Calculus

In a paper entitled Binary lambda calculus and combinatory logic, John Tromp presents a simple way of encoding lambda calculus terms as binary sequences. In what follows, we study the numbers of binary strings of a given size that represent lambda terms and derive results from their generating functions, especially that the number of terms of size n grows roughly like 1.963447954^n.

cs.LO↗
arXiv2013

Counting and generating lambda terms

Lambda calculus is the basis of functional programming and higher order proof assistants. However, little is known about combinatorial properties of lambda terms, in particular, about their asymptotic distribution and random generation. This paper tries to answer questions like: How many terms of a given size are there? What is a "typical" structure of a simply typable term? Despite their ostensible simplicity, these questions still remain unanswered, whereas solutions to such problems are essential for testing compilers and optimizing programs whose expected efficiency depends on the size of terms. Our approach toward the afore-mentioned problems may be later extended to any language with bound variables, i.e., with scopes and declarations. This paper presents two complementary approaches: one, theoretical, uses complex analysis and generating functions, the other, experimental, is based on a generator of lambda-terms. Thanks to de Bruijn indices, we provide three families of formulas for the number of closed lambda terms of a given size and we give four relations between these numbers which have interesting combinatorial interpretations. As a by-product of the counting formulas, we design an algorithm for generating lambda terms. Performed tests provide us with experimental data, like the average depth of bound variables and the average number of head lambdas. We also create random generators for various sorts of terms. Thereafter, we conduct experiments that answer questions like: What is the ratio of simply typable terms among all terms? (Very small!) How are simply typable lambda terms distributed among all lambda terms? (A typable term almost always starts with an abstraction.) In this paper, abstractions and applications have size 1 and variables have size 0.

cs.LO↗
arXiv2013

A journey through resource control lambda calculi and explicit substitution using intersection types (an account)

In this paper we invite the reader to a journey through three lambda calculi with resource control: the lambda calculus, the sequent lambda calculus, and the lambda calculus with explicit substitution. All three calculi enable explicit control of resources due to the presence of weakening and contraction operators. Along this journey, we propose intersection type assignment systems for all three resource control calculi. We recognise the need for three kinds of variables all requiring different kinds of intersection types. Our main contribution is the characterisation of strong normalisation of reductions in all three calculi, using the techniques of reducibility, head subject expansion, a combination of well-orders and suitable embeddings of terms.

math.LO↗