Investigations on a Pedagogical Calculus of Constructions

In the last few years appeared pedagogical propositional natural deduction systems. In these systems, one must satisfy the pedagogical constraint: the user must give an example of any introduced notion. First we expose the reasons of such a constraint and properties of these "pedagogical" calculi: the absence of negation at logical side, and the "usefulness" feature of terms at computational side (through the Curry-Howard correspondence). Then we construct a simple pedagogical restriction of the calculus of constructions (CC) called CCr. We establish logical limitations of this system, and compare its computational expressiveness to Godel system T. Finally, guided by the logical limitations of CCr, we propose a formal and general definition of what a pedagogical calculus of constructions should be.Comment: 18 page

What are the best upland river characteristics for glass eel restocking practice?

peer reviewedThe fitness of restocked European eel (Anguilla anguilla), an endangered fish species, was studied in relation to the environmental variables of habitats in six upland rivers that are typologically different in terms of their hydromorphological and physicochemical characteristics, food resources and fish communities. These rivers received a total of 76,370 imported glass eels in 2017. During a three-year period, we monitored eels with respect to total length, annual growth rate, condition factor and density using capture-mark-recapture experiments to understand the effects of the characteristics of receiving rivers on restocking success levels. Our results showed the survival of the restocked eels in the six rivers and revealed significant differences between them in terms of total length, condition factor and density. Better performance in eel yield variableswas observed in a eutrophic alkaline river with greater roughness of riverbed substrates, dominant pool- and riffle-type flowfacies and lower brown trout density. The variables conductivity and total hardness had higher explanatory power and were strongly associated with increased eel density. This study suggests that a well-selected habitat/river in a restocking programme can be beneficial for the species and recommends restocking practice as a management tool to achieve eel conservation goals

About primitive recursive algorithms

AbstractIn the past few years, there has been a growing interest in the application of proof-theoretical methods to the design of functional programming languages [3, 11]. One approach relies on representation theorems [3, 8, 10], which show that a large class of general recursive functions can be encoded in a language where general recursion is replaced by primitive recursion with functions, functionals,… as parameters [13]. These results are however purely extensional in nature: they state that a large class of mathematical functions is representable in a given system, but they say nothing about the efficiency of such a representation. Although the intensional aspect is of primary concern for computer science, very little seems to be known about this question. This paper is a beginning in the study of this problem. We take as a case study the following computational model: a primitive recursive function is seen as defining a rewriting system which is evaluated in call-by-name. In this setting, we give a non-trivial necessary condition for an algorithm to be representable. As an application, we can show that the function inf (which computes the minimum of two integers in unary representation) cannot be programmed in complexity O(inf(n,p)). Our proof method uses some basic notions of denotational semantics

Vers un calcul des constructions pédagogique

Les systèmes pédagogiques sont apparus récemment à propos des calculs propositionnels (jusqu'à l'ordre supérieur), et consistent à donner systématiquement des exemples des notions (hypothèses) introduites. Formellement, cela signifie que pour mettre un ensemble Delta de formules en hypothèse, il est requis de donner une substitution sigma telle que les instances de formules sigma(Delta) soient démontrables. Cette nécessité d'exemplification ayant été pointée du doigt par Poincaré (1913) comme relevant du bon sens: une définition d'un objet par postulat n'ayant d'intérêt que si un tel objet peut être construit. Cette restriction appliquée à des systèmes formels intuitionnistes rejoint l'idée des mathématiques sans négation défendues par Griss (1946) au milieu du siècle dernier, et présentées comme une version radicale de l'intuitionnisme. À travers l'isomorphisme de Curry-Howard (1980), la contrepartie calculatoire est l'utilité des programmes définis dans les systèmes fonctionnels correspondant: toute fonction peut être appliquée à un argument clos. Les premiers résultats concernant les calculs propositionnels jusqu'au second ordre ont été publiés récemment par Colson et Michel (2007, 2008, 2009). Nous exposons dans ce rapport une tentative d'uniformisation et d'extension au Calcul des Constructions (CC) des précédents résultats. Tout d'abord une définition formelle et précise de sous-système pédagogique du Calcul des Constructions est introduite, puis différents tels sous-systèmes sont déclinés en exemplePedagogical formal systems have appeared recently for propositional calculus (up to the higher order), and it consists of systematically give examples of introduced notions (hypotheses). Formally, it means that to use a set Delta of formulas as hypotheses, one must first give a substitution sigma such that all the instances of formulas sigma(Delta) can be proved. This neccesity of giving examples has been pointed out by Poincaré (1913) as a common-sense practice: a definition of an object by means of assumptions has interest only if such an object can be constructed. This restriction applied to intuitionistic formal systems is consistent with the idea of negationless mathematics advocated by Griss (1946) in the middle of the past century, and shown as a more radical view of intuitionism. Through the Curry-Howard isomorphism (1980), the computational counterpart is the utility of programs defined in the associated functional systems: every function can be applied to a closed value. First results concerning propositional calculi up to the second-order has recently been published by Colson and Michel (2007, 2008, 2009). In this thesis we present an attempt to standardize and to extend to the Calculus of Constructions (CC) those previous results. First a formal and precise definition of pedagogical sub-systems of the Calculus of Constructions is introduced, and different such sub-systems are exhibited as examplesMETZ-SCD (574632105) / SudocNANCY1-Bib. numérique (543959902) / SudocNANCY2-Bibliotheque electronique (543959901) / SudocNANCY-INPL-Bib. électronique (545479901) / SudocSudocFranceF

Investigations on a Pedagogical Calculus of Constructions

In the last few years appeared pedagogical propositional natural deduction systems. In these systems one must satisfy the pedagogical constraint: the user must give an example of any introduced notion. In formal terms, for instance in the propositional case, the main modification is that we replace the usual rule (hyp) by the rule (p-hyp) where σ denotes a substitution which replaces variables of Γ with an example. This substitution σ is called the motivation of Γ. First we expose the reasons of such a constraint and properties of these "pedagogical" calculi: the absence of negation at logical side, and the "usefulness" feature of terms at computational side (through the Curry-Howard correspondence). Then we construct a simple pedagogical restriction of the calculus of constructions (CC) called CCr. We establish logical limitations of this system, and compare its computational expressiveness to Gödel system T. Finally, guided by the logical limitations of CCr, we give a formal and general definition of a pedagogical calculus of constructions

Pedagogical Natural Deduction Systems: the Propositional Case

Abstract: This paper introduces the notion of pedagogical natural deduction systems, which are natural deduction systems with the following additional constraint: all hypotheses made in a proof must be motivated by some example. It is established that such systems are negationless. The expressive power of the pedagogical version of some propositional calculi are studied. Key Words: mathematical logic, negationless mathematics, constructive mathematics, natural deduction, typed λ-calculu

Systèmes formels et systèmes fonctionnels pédagogiques

Cette thèse introduit la notion de systèmes pédagogiques, qui sont des systèmes de d éducation naturelle contraints de la manière suivante : toutes les hypothèses posées dans une démonstration doivent être motivées par un exemple. Ces systèmes sont par essence sans négation. Nous étudions les systèmes propositionnels pédagogiques du premier ordre, du second ordre et plus généralement tous les systèmes d'ordre supérieur. Nous présentons, quand cela est possible, le lamda-calcul associé à chaque système via l'isomorphisme de Curry-Howard ; la contrainte pédagogique y fait apparaître une nouvelle propriété que nous appelons l'utilité: un lamda-terme typé est utile quand son contenu algorithmique peut être utiliséThe present thesis introduces the notion of pedagogical systems, which are natural deduction systems with the following additional constraint: all hypotheses made in a proof must be motivated by an example. These systems are in essence negationless. We study _rst order, second order and higher order pedagogical propositional systems. We present when it is possible the _-calculi associated to these systems; the pedagogical constraint introduces a new notion we call usefulness: a _-term is usefull when it's algorithmic content can be used.METZ-SCD (574632105) / SudocSudocFranceF