[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, Corée du Sud, novembre 2015.

A type system based on non-interference and data ramification principles is introduced in order to capture the set of functions computable in polynomial time on OO programs. The studied language is general enough to capture most OO constructs and our characterization is quite expressive as it allows the analysis of a combination of imperative loops and of data ramification scheme based on Bellantoni and Cook’s safe recursion using function algebra

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[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.

This paper provides a criterion based on interpretation methods on term rewrite systems in order to characterize the polynomial time complexity of second order functionals. For that purpose it introduces a first order functional stream language that allows the programmer to implement second order functionals. This characterization is extended through the use of exp-poly interpretations as an attempt to capture the class of Basic Feasible Functionals (bff). Moreover, these results are adapted to provide a new characterization of polynomial time complexity in computable analysis. These characterizations give a new insight on the relations between the complexity of functional stream programs and the classes of functions computed by Oracle Turing Machine, where oracles are treated as inputs.

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

This paper analyzes the computational power of dynamical systems robust to infinitesimal perturbations. Previous work on the subject has delved on very specific types of systems. Here we obtain results for broader classes of dynamical systems (including those systems defined by Lipschitz/analytic functions). In particular we show that systems robust to infinitesimal perturbations only recognize recursive languages. We also show the converse direction: every recursive language can be robustly recognized by a computable system. By other words we show that robustness is equivalent to decidability.

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[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, volume 7794 of Lecture Notes in Computer Science, pages 305–320, 2013.

We introduce a type system for concurrent programs described as a parallel imperative language using while-loops and fork/wait instructions, in which processes do not share a global memory, in order to analyze computational complexity. The type system provides an analysis of the data-flow based both on a data ramification principle related to tiering discipline and on secure typed languages. The main result states that well-typed processes characterize exactly the set of functions computable in polynomial space under termination, confluence and lock-freedom assumptions. More precisely, each process computes in polynomial time so that the evaluation of a process may be performed in polynomial time on a parallel model of computation. Type inference of the presented analysis is decidable in linear time provided that basic operator semantics is known.

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

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[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.

Recursive analysis is the most classical approach to model and discuss computations over the real numbers. Recently, it has been shown that computability classes of functions in the sense of recursive analysis can be defined (or characterized) in an algebraic machine independent way, without resorting to Turing machines. 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 96 have been obtained. However, until now, this has been done only at the computability level, and not at the complexity level. In this paper we provide a framework that allows us to dive into the complexity level of real functions. In particular we provide the first algebraic characterization of polynomial-time computable functions over the reals. This framework opens the field of implicit complexity of analog functions, and also provides a new reading of some of the existing characterizations at the computability level.

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[BGH10]
Olivier Bournez, Daniel S. Graça, and Emmanuel Hainry. Robust computations with dynamical systems. In Petr Hlinand AntonKu, editors, Mathematical Foundations of Computer Science 2010, MFCS 2010, Brno, Czech Republic, volume 6281 of Lecture Notes in Computer Science, pages 198–208. Springer, 2010.

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.

[ bib | http ]
[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.

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.

[ bib | http ]
[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, août 2009.

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.

[ bib | http ]
[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.

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.

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[Hai08a]
Emmanuel Hainry. Reachability in linear dynamical systems. In Arnold Beckmann, Costas Dimitracopoulos, and Benedikt Löwe, editors, Computability in Europe 2008, Athènes, Grèce, volume 5028 of Lecture Notes in Computer Science, pages 241–250, 2008.

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.

[ bib | http ]
[Hai08b]
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 Computation, UC 2008, Vienne, Autriche, volume 5204 of Lecture Notes in Computer Science, pages 83–95, 2008.

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.

[ bib | http ]
[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.

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

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[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.

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

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

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.

[ bib | http ]
[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.

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.

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

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 être définies sur un compact et dans le premier cas être de classe C^2. Deuxièmement, nous montrons également une hiérarchie de classes de fonctions R-récursives qui caractérisent les fonctions élémentairement calculables, les fonctions E_n-calculables pour n≥3 (où les E_n 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

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[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, décembre 2005.

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

[ bib | http ]
[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.

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.

[ bib ]
[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, avril 2004.

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

[ bib | http ]
[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.

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.

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

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

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