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A survey on continuous time computations
 New Computational Paradigms
"... Abstract. We provide an overview of theories of continuous time computation. These theories allow us to understand both the hardness of questions related to continuous time dynamical systems and the computational power of continuous time analog models. We survey the existing models, summarizing resu ..."
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Abstract. We provide an overview of theories of continuous time computation. These theories allow us to understand both the hardness of questions related to continuous time dynamical systems and the computational power of continuous time analog models. We survey the existing models, summarizing results, and point to relevant references in the literature. 1
Robust computations with dynamical systems
"... Abstract. 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 notsonatural systems from a classical mathematical point of view (discontinuous differential equation systems, disco ..."
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Abstract. 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 notsonatural 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 corecursively 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 cor.e. complete even for C ∞systems.
DECISIVE MARKOV CHAINS
"... ABSTRACT. We consider qualitative and quantitative verification problems for infinitestate Markov chains. We call a Markov chain decisive w.r.t. a given set of target states F if it almost certainly eventually reaches either F or a state from which F can no longer be reached. While all finite Marko ..."
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ABSTRACT. We consider qualitative and quantitative verification problems for infinitestate Markov chains. We call a Markov chain decisive w.r.t. a given set of target states F if it almost certainly eventually reaches either F or a state from which F can no longer be reached. While all finite Markov chains are trivially decisive (for every set F), this also holds for many classes of infinite Markov chains. Infinite Markov chains which contain a finite attractor are decisive w.r.t. every set F. In particular, all Markov chains induced by probabilistic lossy channel systems (PLCS) contain a finite attractor and are thus decisive. Furthermore, all globally coarse Markov chains are decisive. The class of globally coarse Markov chains includes, e.g., those induced by probabilistic vector addition systems (PVASS) with upwardclosed sets F, and all Markov chains induced by probabilistic noisy Turing machines (PNTM) (a generalization of the noisy Turing machines (NTM) of Asarin and Collins). We consider both safety and liveness problems for decisive Markov chains. Safety: What is the probability that a given set of states F is eventually reached. Liveness: What is the probability that a given set of states is reached infinitely often. There are three variants of these questions. (1) The qualitative problem, i.e., deciding if the probability is one (or zero); (2) the approximate quantitative
Eager Markov chains
 In Proc. ATVA ’06, 4Ø�Int. Symp. on Automated Technology for Verification and Analysis
, 2006
"... Abstract. We consider infinitestate discrete Markov chains which are eager: the probability of avoiding a defined set of final states for more thanÒsteps is bounded by some exponentially decreasing function�(Ò). We prove that eager Markov chains include those induced by Probabilistic Lossy Channel ..."
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Abstract. We consider infinitestate discrete Markov chains which are eager: the probability of avoiding a defined set of final states for more thanÒsteps is bounded by some exponentially decreasing function�(Ò). We prove that eager Markov chains include those induced by Probabilistic Lossy Channel Systems, Probabilistic Vector Addition Systems with States, and Noisy Turing Machines, and that the bounding function�(Ò) can be effectively constructed for them. Furthermore, we study the problem of computing the expected reward (or cost) of runs until reaching the final states, where rewards are assigned to individual runs by computable reward functions. For eager Markov chains, an effective path exploration scheme, based on forward reachability analysis, can be used to approximate the expected reward upto an arbitrarily small error. 1
Computation with perturbed dynamical systems
, 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 fu ..."
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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. 1
A Turing Machine Resisting Isolated Bursts Of Faults
, 2012
"... We consider computations of a Turing machine under noise that causes violations of the transition function. Given an upper bound β on the size of bursts of faults, we construct a Turing machine M(β) subject to faults that can simulate any faultfree machine under the condition that that the time bet ..."
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We consider computations of a Turing machine under noise that causes violations of the transition function. Given an upper bound β on the size of bursts of faults, we construct a Turing machine M(β) subject to faults that can simulate any faultfree machine under the condition that that the time between bursts is lowerbounded by V for an appropriate V = O(β 2). 1
Thèse de Church. Autres Modèles de Calculs
, 2009
"... Un des résultats fondamentaux les plus inattendus du vingtième siècle est le théorème d’incomplétude de Gödel, qui affirme qu’aucun système de preuve ne peut capturer pleinement le raisonnement mathématique: toute théorie suffisante pour capturer les raisonnements arithmétiques est nécessairement in ..."
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Un des résultats fondamentaux les plus inattendus du vingtième siècle est le théorème d’incomplétude de Gödel, qui affirme qu’aucun système de preuve ne peut capturer pleinement le raisonnement mathématique: toute théorie suffisante pour capturer les raisonnements arithmétiques est nécessairement incomplète, c’estàdire telle qu’il existe des énoncés qui ne sont pas démontrables et dont la négation n’est pas non plus démontrable. En particulier, on peut exprimer la cohérence d’une théorie mathématique par un énoncé, qui ne peut être démontré, ou infirmé. Les arguments de Kurt Gödel dans l’article original [1] sont en fait très intimement basés sur une notion (informelle) de déduction algorithmique. Alan Turing, travaillant sur le problème de la décision de Hilbert (Entscheidungsproblem, formulé ainsi par Turing: “peuton décider mécaniquement si un énoncé est démontrable ou non”) proposa dans l’article [52] son célèbre modèle de machine, capable de capturer la déduction dans les systèmes formels, et en particulier la notion de déduction utilisée par Gödel dans sa preuve.
ACTA UNIVERSITATIS UPSALIENSIS Uppsala Dissertations from the Faculty of Science and Technology
"... A major current challenge consists in extending formal methods in order to handle infinitestate systems. Infiniteness stems from the fact that the system operates on unbounded data structure such as stacks, queues, clocks, integers; as well as parameterization. Systems with unbounded data structure ..."
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A major current challenge consists in extending formal methods in order to handle infinitestate systems. Infiniteness stems from the fact that the system operates on unbounded data structure such as stacks, queues, clocks, integers; as well as parameterization. Systems with unbounded data structure are natural models for reasoning about communication protocols, concurrent programs, realtime systems, etc. While parameterized systems are more suitable if the system consists of an arbitrary number of identical processes which is the case for cache coherence protocols, distributed algorithms and so forth. In this thesis, we consider model checking problems for certain fundamental classes of probabilistic infinitestate systems, as well as the verification of safety properties in parameterized systems. First, we consider probabilistic systems with unbounded data structures. In particular, we study probabilistic extensions of Lossy Channel Systems (PLCS), Vector addition Systems with States (PVASS) and Noisy Turing Machine (PNTM). We show how we can describe the semantics of such models by infinitestate Markov chains; and then define certain abstract properties, which allow model checking several qualitative and quantitative problems.
Computation with perturbed dynamical systemsI
"... 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 fu ..."
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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.