Descriptive Set Theory

Reactive distributed systems have pervaded everyday life and objects, but often lack measures to ensure adequate behaviour in the presence of unforeseen events or even errors at runtime. As interactions and dependencies within distributed systems increase, the problem of detecting failures which depend on the exact situation and environment conditions they occur in grows. As a result, not only the detection of failures is increasingly difficult, but also the differentiation between the symptoms of a fault, and the actual fault itself, i.e., the cause of a problem. In this paper, we present a novel and efficient approach for analysing reactive distributed systems at runtime, in that we provide a framework for detecting failures as well as identifying their causes. Our approach is based upon monitoring safety-properties, specified in the linear time temporal logic LTL (respectively, TLTL) to automatically generate monitor components which detect violations of these properties. Based on the results of the monitors, a dedicated diagnosis is then performed in order to identify explanations for the misbehaviour of a system. These may be used to store detailed log files, or to trigger recovery measures. Our framework is built modular, layered, and uses merely a minimal communication overhead - especially when compared to other, similar approaches. Further, we sketch first experimental results from our implementations, and describe how it can be used to build a variety of distributed systems using our techniques.

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Model-based runtime analysis of distributed reactive systems

We show that the Borel hierarchy of the class of context free $\omega$-languages, or even of the class of $\omega$-languages accepted by Buchi 1-counter automata, is the same as the Borel hierarchy of the class of $\omega$-languages accepted by Turing machines with a Buchi acceptance condition. In particular, for each recursive non-null ordinal $\alpha$, there exist some ${\bf \Sigma}^0_\alpha$-complete and some ${\bf \Pi}^0_\alpha$-complete $\omega$-languages accepted by Buchi 1-counter automata. And the supremum of the set of Borel ranks of context free $\omega$-languages is an ordinal $\gamma_2^1$ that is strictly greater than the first non-recursive ordinal $\omega_1^{\mathrm{CK}}$. We then extend this result, proving that the Wadge hierarchy of context free $\omega$-languages, or even of $\omega$-languages accepted by Buchi 1-counter automata, is the same as the Wadge hierarchy of $\omega$-languages accepted by Turing machines with a Buchi or a Muller acceptance condition.

/pdf/borel-ranks-and-wadge-degrees-of-context-free-omega-4c94iq8a0s.pdf

Borel ranks and Wadge degrees of context free $\omega$-languages

Fine hierarchies and m-reducibilities in theoretical computer science

In this article, we give a characterization of a class of bounded Fredholm operators on a Banach space which is developed to present some general existence results of the operators equation of the second kind. The obtained results are used to describe the Riesz–Schauder theory of compact operators in the more general setting of polynomially compact operators.

Fredholm Operators and Riesz Theory for Polynomially Compact Operators

We study the links between the topological complexity of an omega context free language and its degree of ambiguity. In particular, using known facts from classical descriptive set theory, we prove that non Borel omega context free languages which are recognized by Buchi pushdown automata have a maximum degree of ambiguity. This result implies that degrees of ambiguity are really not preserved by the operation of taking the omega power of a finitary context free language. We prove also that taking the adherence or the delta-limit of a finitary language preserves neither unambiguity nor inherent ambiguity. On the other side we show that methods used in the study of omega context free languages can also be applied to study the notion of ambiguity in infinitary rational relations accepted by Buchi 2-tape automata and we get first results in that direction.

/pdf/topology-and-ambiguity-in-omega-context-free-languages-3b2vdy9c3r.pdf

Topology and Ambiguity in Omega Context Free Languages

Let X be a separable Banach space and denote by L(X) (resp. K(C)) the set of all bounded linear operators on X (resp. the set of all compact subsets of C). We show that the maps from L(X) into K(C) which assign to each element of L(X) its spectrum, approximate point spectrum, essential spectrum, Weyl essential spectrum, Browder essential spectrum, respectively, are Borel maps, where L(X) (resp. K(C)) is endowed with the strong operator topology (resp. Hausdorff topology). This enables us to derive the topological complexity of some subsets of L(X) and to discuss the properties of a class of strongly continuous semigroups. We close the paper by giving a characterization of strongly continuous semigroups on hereditarily indecomposable Banach spaces.

Some facts from descriptive set theory concerning essential spectra and applications

We investigate the topological complexity of non Borel recognizable tree languages with regard to the difference hierarchy of analytic sets. We show that, for each integer n g 1, there is a D$_{ω^n}$(Σ$^1_1$)-complete tree language L$_n$ accepted by a (non deterministic) Muller tree automaton. On the other hand, we prove that a tree language accepted by an unambiguous Buchi tree automaton must be Borel. Then we consider the game tree languages W$_{(i,κ)}$, for Mostowski-Rabin indices (iκ). We prove that the D$_{ω^n}$(Σ$^1_1$)-complete tree languages L$_n$ are Wadge reducible to the game tree language W$_{(i,κ)}$ for κ−ig 2. In particular these languages W$_{(i,κ)}$ are not in any class D$_{α}$(Σ$^1_1$) for α<ω$^{ω}$.

On Recognizable Tree Languages Beyond the Borel Hierarchy

Let X be a Banach space and $$(T(t))_{t \in {\mathbb{R}}}$$
 a strongly continuous group of linear operators on X. Set $$\sigma^1(T(t))
 := \{ \frac{\lambda}{\mid \lambda \mid}\, : \,\lambda\, \in\, \sigma(T(t)) \}$$
 and $$\chi(T) := \{t \in {\mathbb{R}} : \sigma^1(T(t)) \neq {\mathbb{T}}\}$$
 where $${\mathbb{T}}$$
 is the unit circle and $$\sigma(T(t))$$
 denotes the spectrum of T(t). The main result of this paper is: $$(T(t))_{t \in {\mathbb{R}}}$$
 is uniformly continuous if and only if $$\chi(T)$$
 is non-meager. Similar characterizations in terms of the approximate point spectrum and essential spectra are also derived.

A spectral characterization of the uniform continuity of strongly continuous groups

In his thesis Baire defined functions of Baire class 1. A function f is of Baire class 1 if it is the pointwise limit of a sequence of continuous functions. Baire proves the following theorem. A function f is not of class 1 if and only if there exists a closed nonempty set F such that the restriction of f to F has no point of continuity. We prove the automaton version of this theorem. An ω-rational function is not of class 1 if and only if there exists a closed nonempty set F recognized by a Buchi automaton such that the restriction of f to F has no point of continuity. This gives us the opportunity for a discussion on Hausdorff's analysis of Δ°2, ordinals, transfinite induction and some applications of computer science.

https://hal.inria.fr/hal-00966518/document

Pierre Simonnet

Papers

Topology and Ambiguity in Omega Context Free Languages

Some facts from descriptive set theory concerning essential spectra and applications

On Recognizable Tree Languages Beyond the Borel Hierarchy

A spectral characterization of the uniform continuity of strongly continuous groups

Baire and automata