[HKMP22b]

Emmanuel Hainry, Bruce M. Kapron, Jean-Yves Marion, and Romain Péchoux. A tier-based typed programming language characterizing feasible functionals. Logical Methods in Computer Science, 18(1), 2022. [ bib | DOI | http | .pdf ]

[HKMP22a]

Emmanuel Hainry, Bruce M. Kapron, Jean-Yves Marion, and Romain Péchoux. Complete and tractable machine-independent characterizations of second-order polytime. In Foundations of Software Science and Computation Structures (FoSSaCS 2022), Lecture Notes in Computer Science, pages 368–388. Springer, 2022. [ bib | DOI | http | .pdf ]

[HJPZ21]

Emmanuel Hainry, Emmanuel Jeandel, Romain Péchoux, and Olivier Zeyen. Complexityparser: An automatic tool for certifying poly-time complexity of java programs. In Antonio Cerone and Peter Csaba Ölveczky, editors, ICTAC 2021 - 18th International Colloquium on Theoretical Aspects of Computing, volume 12819 of Lecture Notes in Computer Science, pages 357–365, Nur-Sultan, Kazakhstan, september 2021. Springer. [ bib | DOI | http | .pdf ]

[HP20]
Emmanuel Hainry and Romain Péchoux. Theory of Higher Order Interpretations and Application to Basic Feasible Functions. Logical Methods in Computer Science, 16(4):25, december 2020. [ bib | DOI | http | .pdf ]

Implicit computational complexity ; basic feasible functionals

[HMP20]

Emmanuel Hainry, Damiano Mazza, and Romain Péchoux. Polynomial time over the reals with parsimony. In FLOPS 2020 - International Symposium on Functional and Logic Programming, Akita, Japan, april 2020. [ bib | DOI | http | .pdf ]

[HKMP20]
Emmanuel Hainry, Bruce M. Kapron, Jean-Yves Marion, and Romain Péchoux. A tier-based typed programming language characterizing Feasible Functionals. In LICS ’20 - 35th Annual ACM/IEEE Symposium on Logic in Computer Science, pages 535–549, Saarbrücken, Germany, july 2020. ACM. [ bib | DOI | http | .pdf ]

Feasible functionals ; BFF ; implicit computational complexity ; tiering ; type-2 ; type system

[HP18]
Emmanuel Hainry and Romain Péchoux. A Type-Based Complexity Analysis of Object Oriented Programs. Information and Computation, 261(1):78–115, august 2018. [ bib | DOI | http | .pdf ]

Object Oriented Program ; Type system ; complexity ; polynomial time

[HP17]
Emmanuel Hainry and Romain Péchoux. Higher order interpretation for higher order complexity. In Thomas Eiter and David Sands, editors, LPAR-21. 21st International Conference on Logic for Programming, Artificial Intelligence and Reasoning, volume 46 of EPiC Series in Computing, pages 269–285, 2017. [ bib | DOI ]

We design an interpretation-based theory of higher-order functions that is well-suited for the complexity analysis of a standard higher- order functional language à la ml. We manage to express the interpretation of a given program in terms of a least fixpoint and we show that when restricted to functions bounded by higher-order polynomials, they characterize exactly classes of tractable functions known as Basic Feasible Functions at any order.

[HP15]

Emmanuel Hainry and Romain Péchoux. Objects in Polynomial Time. In Xinyu Feng and Sungwoo Park, editors, APLAS 2015, volume 9458 of Lecture Notes in Computer Science, pages 387–404, Pohang, South Korea, november 2015. Springer. [ bib | DOI | http | .pdf ]

[FHHP15]

Hugo Férée, Emmanuel Hainry, Mathieu Hoyrup, and Romain Péchoux. Characterizing polynomial time complexity of stream programs using interpretations. Theoretical Computer Science, 585:41–54, 2015. [ bib | DOI ]

[HMP13]
Emmanuel Hainry, Jean-Yves Marion, and Romain Péchoux. Type-based complexity analysis for fork processes. In Frank Pfenning, editor, Foundations of Software Science and Computation Structures (FoSSaCS 2013), volume 7794, pages 305–320, Rome, Italy, 2013. Springer. [ bib | DOI | http ]

Implicit Computational Complexity ; Tiering ; Secure Information Flow ; Concurrent Programming ; PSpace

[BGH13]
Olivier Bournez, Daniel Graça, and Emmanuel Hainry. Computation with perturbed dynamical systems. Journal of Computer and System Sciences, 79(5):714–724, 2013. [ bib | DOI | http ]

robustness ; Dynamical systems ; reachability ; computational power ; verification

[BGH11]
Olivier Bournez, Walid Gomaa, and Emmanuel Hainry. Algebraic Characterizations of Complexity-Theoretic Classes of Real Functions. International Journal of Unconventional Computing, 7(5):331–351, 2011. [ bib | http ]

Recursive Analysis ; Polynomial Time ; Algebraic Characterization ; Real Computation ; Oracle Turing Machines

[BGH10]
Olivier Bournez, Daniel S. Graça, and Emmanuel Hainry. Robust computations with dynamical systems. In Petr Hliněný and Antonín Kučera, editors, Mathematical Foundations of Computer Science, MFCS 2010, volume 6281 of Lecture Notes in Computer Science, pages 198–208. Springer, 2010. [ bib | DOI ]

In this paper we discuss the computational power of Lipschitz dynamical systems which are robust to infinitesimal perturbations. Whereas the study in [1] was done only for not-so-natural systems from a classical mathematical point of view (discontinuous differential equation systems, discontinuous piecewise affine maps, or perturbed Turing machines), we prove that the results presented there can be generalized to Lipschitz and computable dynamical systems. In other words, we prove that the perturbed reachability problem (i.e. the reachability problem for systems which are subjected to infinitesimal perturbations) is co-recursively enumerable for this kind of systems. Using this result we show that if robustness to infinitesimal perturbations is also required, the reachability problem becomes decidable. This result can be interpreted in the following manner: undecidability of verification doesn’t hold for Lipschitz, computable and robust systems. We also show that the perturbed reachability problem is co-r.e. complete even for C-systems.

[FHHP10]
Hugo Férée, Emmanuel Hainry, Mathieu Hoyrup, and Romain Péchoux. Interpretation of stream programs: characterizing type 2 polynomial time complexity. In Ottfried Cheong, Kyung-Wong Chwa, and Kunsoo Park, editors, International Symposium on Algorithms and Computation (ISAAC), volume 6506 of Lecture Notes in Computer Science, pages 291–303, Jeju Island, South Korea, 2010. Springer. [ bib | DOI ]

We study polynomial time complexity of type 2 functionals. For that purpose, we introduce a first order functional stream language. We give criteria, named well-founded, on such programs relying on second order interpretation that characterize two variants of type 2 polynomial complexity including the Basic Feasible Functions (BFF). These characterizations provide a new insight on the complexity of stream programs. Finally, we adapt these results to functions over the reals, a particular case of type 2 functions, and we provide a characterization of polynomial time complexity in Recursive Analysis.

[Hai09]
Emmanuel Hainry. Decidability and Undecidability in Dynamical Systems. Research report, CARTE - INRIA Lorraine - LORIA - CNRS : UMR7503 - INRIA - Université Henri Poincaré - Nancy I - Université Nancy II - Institut National Polytechnique de Lorraine, 2009. [ bib | http ]

A computing system can be modelized in various ways: one being in analogy with transfer functions, this is a function that associates to an input and optionally some internal states, an output ; another being focused on the behaviour of the system, that is describing the sequence of states the system will follow to get from this input to produce the output. This second kind of system can be defined by dynamical systems. They indeed describe the “local” behaviour of a system by associating a configuration of the system to the next configuration. It is obviously interesting to get an idea of the “global” behaviour of such a dynamical system. The questions that it raises can be for example related to the reachability of a certain configuration or set of configurations or to the computation of the points that will be visited infinitely often. Those questions are unfortunately very complex: they are in most cases undecidable. This article will describe the fundamental problems on dynamical systems and exhibit some results on decidability and undecidability in various kinds of dynamical systems.

[BGH09]
Olivier Bournez, Walid Gomaa, and Emmanuel Hainry. Implicit complexity in recursive analysis. In LCC’09 - Logic and Computational Complexity, Los Angeles États-Unis d’Amérique, august 2009. [ bib | http ]

Recursive analysis is a model of analog computation which is based on type 2 Turing machines. Various classes of functions computable in recursive analysis have recently been characterized in a machine independent and algebraical context. In particular nice connections between the class of computable functions (and some of its sub and sup-classes) over the reals and algebraically defined (sub- and sup-) classes of R-recursive functions à la Moore have been obtained. We provide in this paper a framework that allows to dive into complexity for functions over the reals. It indeed relates classical computability and complexity classes with the corresponding classes in recursive analysis. This framework opens the field of implicit complexity of functions over the reals. While our setting provides a new reading of some of the existing characterizations, it also provides new results: inspired by Bellantoni and Cook’s characterization of polynomial time computable functions, we provide the first algebraic characterization of polynomial time computable functions over the reals.

[Hai08a]
Emmanuel Hainry. Computing omega-limit sets in linear dynamical systems. In Cristian S. Calude, José Félix Costa, Rudolf Freund, Marion Oswald, and Grzegorz Rozenberg, editors, Unconventional Computing, UC 2008, volume 5204 of Lecture Notes in Computer Science, pages 83–95, 2008. [ bib | DOI ]

Dynamical systems allow to modelize various phenomena or processes by only describing their local behaviour. It is an important matter to study the global and the limit behaviour of such systems. A possible description of this limit behaviour is via the omega-limit set: the set of points that can be limit of subtrajectories. The omega-limit set is in general uncomputable. It can be a set highly difficult to apprehend. Some systems have for example a fractal omega-limit set. However, in some specific cases, this set can be computed. This problem is important to verify properties of dynamical systems, in particular to predict its collapse or its infinite expansion. We prove in this paper that for linear continuous time dynamical systems, it is in fact computable. More, we also prove that the ω-limit set is a semi-algebraic set. The algorithm to compute this set can easily be derived from this proof.

[Hai08b]
Emmanuel Hainry. Reachability in linear dynamical systems. In Arnold Beckmann, Costas Dimitracopoulos, and Benedikt Löwe, editors, CiE 2008: Logic and Theory of Algorithms, volume 5028 of Lecture Notes in Computer Science, pages 241–250, 2008. [ bib | DOI ]

Dynamical systems allow to modelize various phenomena or processes by only describing their local behaviour. The study of dynamical systems aims at knowing more on the global behaviour. Checking the reachability of a point is a fundamental problem. In this document, using results from the algebraic numbers theory such as Gelfond-Schneider’s theorem, we will show that this problem that is undecidable in the general case is in fact decidable for a natural class of continuous-time dynamical systems: linear systems.

[BH07]
Olivier Bournez and Emmanuel Hainry. On the computational capabilities of several models. In Jérôme Durand-Lose and Maurice Margenstern, editors, Machines, Computations, and Universality - MCU 2007, Orléans, France, volume 4664 of Lecture Notes in Computer Science, pages 12–23. Springer, 2007. [ bib | DOI ]

We review some results about the computational power of several computational models. Considered models have in common to be related to continuous dynamical systems.

[BCGH07]
Olivier Bournez, Manuel L. Campagnolo, Daniel S. Graça, and Emmanuel Hainry. Polynomial differential equations compute all real computable functions on computable compact intervals. Journal of Complexity, 23(3):317–335, 2007. [ bib | DOI ]

In the last decade, there have been several attempts to understand the relations between the many models of analog computation. Unfortunately, most models are not equivalent. Euler’s Gamma function, which is computable according to computable analysis, but that cannot be generated by Shannon’s General Purpose Analog Computer (GPAC), has often been used to argue that the GPAC is less powerful than digital computation. However, when computability with GPACs is not restricted to real-time generation of functions, it has been shown recently that Gamma becomes computable by a GPAC. Here we extend this result by showing that, in an appropriate framework, the GPAC and computable analysis are actually equivalent from the computability point of view, at least in compact intervals. Since GPACs are equivalent to systems of polynomial differential equations then we show that all real computable functions over compact intervals can be defined by such models.

Analog computation; Computable analysis; General Purpose Analog Computer; Church–Turing thesis; Differential equations

[Hai06]
Emmanuel Hainry. Modèles de calcul sur les réels, résultats de comparaison. PhD thesis, Institut National Polytechnique de Lorraine, december 2006. [ bib | .pdf ]

Il existe de nombreux modèles de calcul sur les réels. Ces différents modèles calculent diverses fonctions, certains sont plus puissants que d’autres, certains sont deux à deux incomparables. Le calcul sur les réels est donc de ce point de vue bien différent du calcul sur les entiers qui est unifié par la thèse de Church-Turing qui affirme que tous les modèles raisonnables calculent les m^emes fonctions.

Les résultats de cette thèse sont de deux sortes. Premièrement, nous montrons des équivalences entre les fonctions récursivement calculables et une certaine classe de fonctions R-récursives et entre les fonctions GPAC-calculables et les fonctions récursivement calculables. Ces deux résultats ne sont cependant valables que si les fonctions présentent quelques caractéristiques : elles doivent ^etre définies sur un compact et dans le premier cas ^etre de classe C2. Deuxièmement, nous montrons également une hiérarchie de classes de fonctions R-récursives qui caractérisent les fonctions élémentairement calculables, les fonctions En-calculables pour n>=3 (où les En sont les fonctions de la hiérarchie de Grzegorczyk), et des fonctions récursivement calculables. Ce résultat utilise un opérateur de limite dont nous avons prouvé la généralité en montrant qu’il transfère une inclusion sur la partie discrète des fonctions en une inclusion sur les fonctions sur les réels elles-m^emes.

Ces résultats constituent donc une avancée vers une éventuelle unification des modèles de calcul sur les réels.

Analyse récursive, calculabilité réelle, fonctions élémentaires, hiérarchie de Grzegorczyk, General Purpose Analog Computer

[BH06]
Olivier Bournez and Emmanuel Hainry. Recursive analysis characterized as a class of real recursive functions. Fundamenta Informaticae, 74(4):409–433, 2006. [ bib | http ]

Recently, using a limit schema, we presented an analog and machine independent algebraic characterization of elementarily computable functions over the real numbers in the sense of recursive analysis. In a dierent and orthogonal work, we proposed a minimization schema that allows to provide a class of real recursive functions that corresponds to extensions of computable functions over the integers. Mixing the two approaches we prove that computable functions over the real numbers in the sense of recursive analysis can be characterized as the smallest class of functions that contains some basic functions, and closed by composition, linear integration, minimization and limit schema.

[BCGH06]
Olivier Bournez, Manuel L. Campagnolo, Daniel S. Graça, and Emmanuel Hainry. The general purpose analog computer and computable analysis are two equivalent paradigms of analog computation. In Jin-Yi Cai, S. Barry Cooper, and Angsheng Li, editors, Theory and Applications of Models of Computation, TAMC 2006, volume 3959 of Lecture Notes in Computer Science, pages 631 – 643. Springer, 2006. [ bib | DOI ]

In this paper we revisit one of the first models of analog computation, Shannon’s General Purpose Analog Computer (GPAC). The GPAC has often been argued to be weaker than computable analysis. As main contribution, we show that if we change the notion of GPAC-computability in a natural way, we compute exactly all real computable functions (in the sense of computable analysis). Moreover, since GPACs are equivalent to systems of polynomial differential equations then we show that all real computable functions can be defined by such models.

[BH05a]
Olivier Bournez and Emmanuel Hainry. Elementary computable functions over the real numbers and R-sub-recursive functions. Theoretical Computer Science, 348(2-3):130–147, december 2005. [ bib | DOI ]

We present an analog and machine-independent algebraic characterization of elementarily computable functions over the real numbers in the sense of recursive analysis: we prove that they correspond to the smallest class of functions that contains some basic functions, and closed by composition, linear integration, and a simple limit schema. We generalize this result to all higher levels of the Grzegorczyk Hierarchy. This paper improves several previous partial characterizations and has a dual interest: * Concerning recursive analysis, our results provide machine-independent characterizations of natural classes of computable functions over the real numbers, allowing to define these classes without usual considerations on higher-order (type 2) Turing machines. * Concerning analog models, our results provide a characterization of the power of a natural class of analog models over the real numbers and provide new insights for understanding the relations between several analog computational models.

Analog computation; Recursive analysis; Real recursive functions; Computability; Analysis

[BH05b]
Olivier Bournez and Emmanuel Hainry. Real recursive functions and real extentions of recursive functions. In Maurice Margenstern, editor, Machines, Computations, and Universality, MCU 2004, volume 3354 of Lecture Notes in Computer Science, pages 116–127. Springer-Verlag, 2005. [ bib ]

Recently, functions over the reals that extend elementarily computable functions over the integers have been proved to correspond to the smallest class of real functions containing some basic functions and closed by composition and linear integration. We extend this result to all computable functions: functions over the reals that extend total recursive functions over the integers are proved to correspond to the smallest class of real functions containing some basic functions and closed by composition, linear integration and a very natural unique minimization schema.

[BH04b]
Olivier Bournez and Emmanuel Hainry. An analog characterization of elementary computable functions over the real numbers. In Josep Díaz, Juhani Karhumäki, Arto Lepistö, and Donald Sannella, editors, International Colloquium on Automata, Languages and Programming (ICALP 2004), volume 3142 of Lecture Notes in Computer Science, pages 269–280, 2004. [ bib ]

We present an analog and machine-independent algebraic characterization of elementarily computable functions over the real numbers in the sense of recursive analysis: we prove that they correspond to the smallest class of functions that contains some basic functions, and closed by composition, linear integration, and a simple limit schema. We generalize this result to all higher levels of the Grzegorczyk Hierarchy. Concerning recursive analysis, our results provide machine-independent characterizations of natural classes of computable functions over the real numbers, making it possible to define these classes without usual considerations on higher-order (type 2) Turing machines. Concerning analog models, our results provide a characterization of the power of a natural class of analog models over the real numbers.

[BH04a]
Olivier Bournez and Emmanuel Hainry. An analog characterization of elementarily computable functions over the real numbers. In 2nd APPSEM II Workshop - APPSEM’2004, Tallinn, Estonia, april 2004. [ bib | .ps ]

We present an analog and machine-independent algebraic characterizations of elementarily computable functions over the real numbers in the sense of recursive analysis: we prove that they correspond to the smallest class of functions that contains some basic functions, and closed by composition, linear integration, and a simple limit schema. We generalize this result to all higher levels of the Grzegorczyk Hierarchy. Concerning recursive analysis, our results provide machine-independent characterizations of natural classes of computable functions over the real numbers, allowing to define these classes without usual considerations on higher-order (type 2) Turing machines. Concerning analog models, our results provide a characterization of the power of a natural class of analog models over the real numbers.

analog models, complexity, computability

[Hai03]
Emmanuel Hainry. Fonctions réelles calculables et fonctions R-récursives. Stage de dea, ENS Lyon, july 2003. [ bib | .ps ]

On définit des opérateurs de limites sur les fonctions. A l’aide de ces opérateurs, on définit de nouvelles classes de fonctions par clôture. On compare ces classes avec les fonctions élémentairement calculables (définies à partir de machines de Turing). On obtient ainsi une caractérisation des fonctions élémentairement calculables sous forme de clôture.

computability, computation over reals, elementary functions, real rcomputable functions