The monitoring power of forcing program transformations

In this thesis, we are interested in semantical proofs of correctness results for complex programming languages. In particular, we advocate the need for a theoretical framework that allows one to:- design realizability semantics using basic blocks - use algebraic constructions to combine those blocks As a step towards this goal, we propose a new semantical framework, based on the composition of linear variants of Krivine realizability and Cohen forcing. The first ingredient of this framework is the Monitoring Abstract Machine: a computing environment that possesses special memory cells used to monitor the execution of programs, in the style of Miquel's KFAM. It is shown how this new machine emerges from a linear forcing program transformation. We then introduce the central notion of Monitoring Algebra and the associated realizability interpretation. Different monitoring algebras induce sound semantics of different programming languages. We then present an algebraic construction to combine different Monitoring Algebras (and the associated programming languages) based on the technique of forcing iteration. We present various results and first applications of our theory. We show that the forcing structure can be used to represent the consumption of resources, in particular time, but also step-indexing or the use of higher-order references. We finally apply our results to retrieve three complex soundness results:- we give the first semantical proof of the consistency of a contraction-free naive set theory, originally introduced by Grishin- we use our framework to obtain a polynomial time termination result for a light-logic based programming language featuring recursive types - we prove the soundness of a language with references that supports strong updates, based on a linear type system inspired by a work of Ahmed, Fluet and Morrisett.Dans cette thèse, nous nous intéressons aux preuves sémantiques de résultats de corrections pour des langages de programmation complexes. En particulier, nous mettons en évidencele besoin d'un nouveau cadre théorique permettant de:- concevoir des sémantiques de réalisabilité à partir de briques plus élémentaires.- combiner ces briques élémentaires grâce à des constructions algébriques.- prouver des théorèmes généraux réutilisables lors de preuves futures de correctionde langages de programmation. Nous proposons dans ce manuscrit un tel cadre sémantique, basé sur la composition de variantes linéaires de la réalisabilité de Krivine et du forcing de Cohen. Le premier ingrédient est la Monitoring Abstract Machine: un environnement de calcul qui utilise des cases mémoires réservées pour "surveiller" l'exécution des programmes, dans le style de la KFAM introduite par Miquel. Cette machine émerge naturellement d'une transformation de programme basée sur une variante linéaire du forcing de Cohen. Nous introduisons par la suite la notion centrale d'Algèbre de Monitoring et le modèle de réalisabilité associé. Chaque algèbre de monitoring induit une sémantique correcte pour un langage de programmation associé. Point crucial de cette thèse, nous définissons, en se basant sur la technique du forcing itéré, une construction algébrique permettant de combiner plusieurs algèbres de monitoring. Nous développons de nombreux résultats élémentaires à propos de notre théorie. En particulier, nous montrons que la structure de forcing peut être utilisée pour représenter la consommation de ressources (en particulier le temps), le step-indexing ou encore des références d'ordre supérieur. Finalement, nous appliquons notre théorie pour obtenir trois preuves de correction complexes:- nous donnons la première preuve sémantique connue de la cohérence d'une théorie des ensembles naïve sans contraction introduite originellement par Grishin dans les années 70- nous utilisons notre cadre pour obtenir un résultat de terminaison en temps polynomial pour un langage de programmation avec types récursifs basé sur une logique light- nous reprouvons la correction d'un langage avec références d'ordre supérieur et mise à jour forte, inspiré par un système de type introduit par Ahmed, Fluet et Morrisett

Foundations of Software Science and Computation Structures

This open access book constitutes the proceedings of the 24th International Conference on Foundations of Software Science and Computational Structures, FOSSACS 2021, which was held during March 27 until April 1, 2021, as part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2021. The conference was planned to take place in Luxembourg and changed to an online format due to the COVID-19 pandemic. The 28 regular papers presented in this volume were carefully reviewed and selected from 88 submissions. They deal with research on theories and methods to support the analysis, integration, synthesis, transformation, and verification of programs and software systems

On Induction, Coinduction and Equality in Martin-L\uf6f and Homotopy Type Theory

Martin L\uf6f Type Theory, having put computation at the center of logicalreasoning, has been shown to be an effective foundation for proof assistants,with applications both in computer science and constructive mathematics. Oneambition though is for MLTT to also double as a practical general purposeprogramming language. Datatypes in type theory come with an induction orcoinduction principle which gives a precise and concise specification of theirinterface. However, such principles can interfere with how we would like toexpress our programs. In this thesis, we investigate more flexible alternativesto direct uses of the (co)induction principles.As a first contribution, we consider the n-truncation of a type in Homo-topy Type Theory. We derive in HoTT an eliminator into (n+1)-truncatedtypes instead of n-truncated ones, assuming extra conditions on the underlyingfunction.As a second contribution, we improve on type-based criteria for terminationand productivity. By augmenting the types with well-foundedness information,such criteria allow function definitions in a style closer to general recursion.We consider two criteria: guarded types, and sized types.Guarded types introduce a modality ”later” to guard the availability ofrecursive calls provided by a general fixed-point combinator. In Guarded Cu-bical Type Theory we equip the fixed-point combinator with a propositionalequality to its one-step unfolding, instead of a definitional equality that wouldbreak normalization. The notion of path from Cubical Type Theory allows usto do so without losing canonicity or decidability of conversion.Sized types, on the other hand, explicitly index datatypes with size boundson the height or depth of their elements. The sizes however can get in theway of the reasoning principles we expect. Our approach is to introduce newquantifiers for ”irrelevant” size quantification. We present a type theory withparametric quantifiers where irrelevance arises as a “free theorem”. We alsodevelop a conversion checking algorithm for a more specific theory where thenew quantifiers are restricted to sizes.Finally, our third contribution is about the operational semantics of typetheory. For the extensions above we would like to devise a practical conversionchecking algorithm suitable for integration into a proof assistant. We formal-ized the correctness of such an algorithm for a small but challenging corecalculus, proving that conversion is decidable. We expect this development toform a good basis to verify more complex theories.The ideas discussed in this thesis are already influencing the developmentof Agda, a proof assistant based on type theory

Nominal Coalgebraic Data Types with Applications to Lambda Calculus

We investigate final coalgebras in nominal sets. This allows us to define types of infinite data with binding for which all constructions automatically respect alpha equivalence. We give applications to the infinitary lambda calculus

Aspects of emergent cyclicity in language and computation

This thesis has four parts, which correspond to the presentation and development of a theoretical framework for the study of cognitive capacities qua physical phenomena, and a case study of locality conditions over natural languages. Part I deals with computational considerations, setting the tone of the rest of the thesis, and introducing and defining critical concepts like ‘grammar’, ‘automaton’, and the relations between them . Fundamental questions concerning the place of formal language theory in linguistic inquiry, as well as the expressibility of linguistic and computational concepts in common terms, are raised in this part. Part II further explores the issues addressed in Part I with particular emphasis on how grammars are implemented by means of automata, and the properties of the formal languages that these automata generate. We will argue against the equation between effective computation and function-based computation, and introduce examples of computable procedures which are nevertheless impossible to capture using traditional function-based theories. The connection with cognition will be made in the light of dynamical frustrations: the irreconciliable tension between mutually incompatible tendencies that hold for a given dynamical system. We will provide arguments in favour of analyzing natural language as emerging from a tension between different systems (essentially, semantics and morpho-phonology) which impose orthogonal requirements over admissible outputs. The concept of level of organization or scale comes to the foreground here; and apparent contradictions and incommensurabilities between concepts and theories are revisited in a new light: that of dynamical nonlinear systems which are fundamentally frustrated. We will also characterize the computational system that emerges from such an architecture: the goal is to get a syntactic component which assigns the simplest possible structural description to sub-strings, in terms of its computational complexity. A system which can oscillate back and forth in the hierarchy of formal languages in assigning structural representations to local domains will be referred to as a computationally mixed system. Part III is where the really fun stuff starts. Field theory is introduced, and its applicability to neurocognitive phenomena is made explicit, with all due scale considerations. Physical and mathematical concepts are permanently interacting as we analyze phrase structure in terms of pseudo-fractals (in Mandelbrot’s sense) and define syntax as a (possibly unary) set of topological operations over completely Hausdorff (CH) ultrametric spaces. These operations, which makes field perturbations interfere, transform that initial completely Hausdorff ultrametric space into a metric, Hausdorff space with a weaker separation axiom. Syntax, in this proposal, is not ‘generative’ in any traditional sense –except the ‘fully explicit theory’ one-: rather, it partitions (technically, ‘parametrizes’) a topological space. Syntactic dependencies are defined as interferences between perturbations over a field, which reduce the total entropy of the system per cycles, at the cost of introducing further dimensions where attractors corresponding to interpretations for a phrase marker can be found. Part IV is a sample of what we can gain by further pursuing the physics of language approach, both in terms of empirical adequacy and theoretical elegance, not to mention the unlimited possibilities of interdisciplinary collaboration. In this section we set our focus on island phenomena as defined by Ross (1967), critically revisiting the most relevant literature on this topic, and establishing a typology of constructions that are strong islands, which cannot be violated. These constructions are particularly interesting because they limit the phase space of what is expressible via natural language, and thus reveal crucial aspects of its underlying dynamics. We will argue that a dynamically frustrated system which is characterized by displaying mixed computational dependencies can provide straightforward characterizations of cyclicity in terms of changes in dependencies in local domains

Systematics, diversification, and functional diversity of Russulaceae (Russulales)

The family Russulaceae is an iconic family of mushroom-forming Basidiomycetes both because of their importance as edible mushrooms in many parts of the world and their species richness in both temperate and tropical forested biomes. While much mycological research has been focused on this group, recent systematic and ecological research has failed to develop a comprehensive or cohesive organization by which to understand the evolutionary relationships, patterns of diversification, or functional importance of the group. Recently, interest in ectomycorrhizal fungi (EmF), of which Russulaceae is a key lineage, has greatly increased due to the recognition of the importance of EmF in carbon sequestration in the face of global climate change. By specifically taking a lineage-based approach to the study of Russulaceae, this work is an attempt to elucidate the biological importance of this group as a model for understanding important biological patterns in EmF. To this end, this dissertation work seeks to address five key questions: 1) What are the major systematic relationships in the Russula, Russulaceae, and their placement within Russulales? 2) What are the biogeographic and host patterns in Russula? 3) What factors have contributed to the high diversification of Russula? 4) What are the functional differences between major groups within Russulaceae? 5) To what extent have members of Russulaceae retained the ability to decompose soil organic matter? To address these main questions, my research has combined the collection and study of sporocarps with molecular phylogenetics and contemporary evolutionary analytics. These efforts have led to the first multi-gene phylogeny of the genus Russula with a clade-based classification system proposed. By applying ancestral area reconstruction methodologies and diversification analyses using state speciation-extinction (SSE) models, I have inferred a temperate origin associated with angiosperms for Russula. I have provided support for a higher net diversification rate in temperate species of Russula that is not a result of migration. Here I present a molecular systematic revision of the Roseinae clade and provide support for at least 5 new species. Finally, in a comparative genomic analysis I show that Russulaceae are widely diverse in gene content, indicating diverse functional roles