communicating **finite** **state** **machines** [DY12], and were shown to enforce various desirable properties.
Several notions similar to the one of synchronisability have also been studied in different context. Slack elasticity seems to be the most general name given to a the property that a given distributed system with asynchronous communications “behaves the same” whatever the slack of communications is. This property has been studied in hardware design [MM98], with the goal of ensuring that some code transformations are semantic-preserving, in high performance computing, for ensuring the absence of deadlocks and other bugs in MPI programs [Sie05, VVGK10], but also for communicating **finite** **state** **machines**, like in this work, with a slightly different way of comparing the behaviours of the system at different buffer bounds. Genest et al introduced the notion of existentially bounded systems of communicating **finite** **state** **machines**, that is defined on top of Mazurkiewicz traces, aka message sequence charts in the context of communicating **finite** **state** **machines** [GKM06]. Finally, a notion similar to the one of existentially bounded systems has been recently introduced and christened “k-synchronous systems” [BEJQ18]. Existential boundedness, k-synchronous systems, and synchronizability are further compared in Section 5.3.

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II. B ACKGROUND
A. PLCs behavior specification with FSMs
Among the numerous formalisms developed to describe the behavior of discrete event systems (DES), **finite** **state** **machines** (FSMs) with inputs/outputs, e.g. Mealy **machines**, are well suited to formal specification of PLCs that receive and send logic data from/to the plant. This formalism cannot be used by control engineers obviously, but it is possible to translate models in industrial standardized languages in this kind of formal models, as presented in [1].

2 LSV, ENS Cachan, CNRS, Cachan, France lozes@lsv.fr
Abstract
A system of communicating **finite** **state** **machines** is synchronizable [1, 4] if its send trace semantics, i.e. the set of sequences of sendings it can perform, is the same when its communications are FIFO asynchronous and when they are just rendez-vous synchronizations. This property was claimed to be decidable in several conference and journal papers [1, 4, 3, 2] for either mailboxes (∗-1) or peer-to-peer (1-1) communications, thanks to a form of small model property. In this paper, we show that this small model property does not hold neither for mailbox communications, nor for peer-to-peer communications, therefore the decidability of synchronizability becomes an open question. We close this question for peer-to-peer communications, and we show that syn- chronizability is actually undecidable. We show that synchronizability is decidable if the topology of communications is an oriented ring. We also show that, in this case, synchronizability implies the absence of unspecified receptions and orphan messages, and the channel-recognizability of the reachability set.

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ism is observable if each input/output pair i/o uniquely identifies the successor of each FSM **state** (if it exists). Given a **state** s of S and an input/output sequence α/β, the α/β-
successor of **state** s is the set of all states that are reached from s via an application of
α when an output reaction β is produced. Note that for an observable FSM S, the car- dinality of the α/β-successor of **state** s is at most one for any input/output sequence α/β.

3 Superadditive complexity measures
The **finite**-**state** dimension is a scaled-down version of effective Hausdorff dimen- sion [8]. The effective Hausdorff dimension of a sequence α = α0α1 . . . can be equivalently defined as the lim inf C(α0 . . . αN −1)/N , where C stands for the Kol- mogorov complexity function [9,10]. We use here plain complexity, but prefix, a priori or monotone complexity (see, e.g., [14, Chapter 6]) will work as well, since they all differ only by O(log n) for n-bit strings (see, e.g., [14] for more details about Kolmogorov complexity and effective dimension). It is natural to look for a similar characterization of **finite**-**state** dimension in terms of compress- ibility. Such a characterization was given in [7, Section 7]. However, it did not use a complexity notion that can replace C in the definition of effective Haus- dorff dimension, using **finite**-**state** compressors instead. A suitable complexity notion was introduced in [13], and it indeed gives the desired characterization. We may also use superadditive upper bounds for Kolmogorov complexity. In this extended abstract we present only a version that does not mention Kolmogorov complexity or **finite**-**state** **machines** at all.

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and a semantic step, and on the other hand such a multi-step approach unnecessarily complicates the formalization and use of **state** **machines**.
Latella et al. [15, 17, 18] follow (as we do) a ”semantics-first” approach in which a sound basic kernel of the notation is considered and extended, only if the main features are investigated. In contrast to our work they do not consider complex structured data (i.e., interpret a **state** **machines** data space or event parameters). They also refer to an older UML standard and do not consider different transition orders, which become rel- evant if data are regarded. Offutt et al. [19, 16] present techniques to generate test cases from UML **state** diagrams on class-level testing. In contrast to our work, they have a data-centric view and focus on change events and boolean variables. The generated test suites are related to full-predicate coverage. Moreover, it is not clear how far the ap- plied semantics follows the UML standard. The work around the AutoFocus tool [20] is interesting but they use proprietary notation which does not include all aspects of UML **state** **machines** and a formal, but synchronous, semantics. Another mentionable industrial approach are the AGEDIS tools [21], but the used semantics is not completely clear. Spec Explorer developed at Microsoft Research ([22] and related publications) is an industrial approach that uses **finite** **state** **machines** as the underlying model for au- tomated testing. They also test against nondeterministic systems and address problems of data instantiation. The general principle to explore the specifications **state** space is comparable to our approach. In contrast, the focus is on (synchronous) method calls.

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An early work on testing on the basis of transition systems is the work of Chow [6] (W-Method). Bourhfir et al. [3] presented sev- eral approaches for conformance testing based on extended **finite** **state** **machines**. Ural [25] reviewed several methods for generating test sequences from **finite** **state** machine based specification. De Nicola and Hennessy [9] introduce a formal theory of testing on which (later on) Brinksma [4] and Tretmans [22] build approaches to derive test cases from a formal specification. In contrast to our work, the approaches of Brinksma and Tretmans assume the test- ing process to communicate synchronously with the system under test. Tretmans developed a tool called TorX which allows confor- mance testing of reactive systems [23]. The used internal represen- tation is based on (input/output) labeled transition systems. TorX is mainly used to perform on-the-fly testing. Newer work also ad- dresses problems of selecting inputs and data, batch test case gener- ation or asynchronous behavior [10, 12]. A detailed overview of the fundamental literature for classical formal testing can be found in Brinksma’s and Tretmans’ annotated bibliography [5]. Grieskamp et al. at Microsoft Research use **finite** **state** **machines** as the under- lying model for automated testing [14]. The approach is based on a AsmL [13] specification for which a FSM model is generated. In contrast to our work, the test approach uses synchronous method calls to test the system under test. The needed data to instantiate the methods parameter are automatically chosen. The **state** space ex- ploration is controlled by so-called test properties and filters. The general principle to explore the **state** space is comparable to our approach. Belli et al. (see [2] and the work cited there) base their testing methodology on a variant of **state** **machines**. In contrast to our approach, they do test against a fault model that has to be set up explicitly. They do not execute the **state** **machines** directly, but represent them as event sequence graphs. Auguston et al. [1]

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Despite formal verification can guarantee that the properties hold for the model, some implementation faults might still manifest. Therefore, active test- ing has been applied as well to the data plane (and the switch) where the latter is stimulated by test in- puts. These works can be divided into two main groups. The approaches of the first group propose testing based on mechanisms of formal verification (Pereˇs´ıni et al., 2015), (Bu et al., 2016), (Zeng et al., 2014), (Fayaz et al., 2016), (Stoenescu et al., 2016); in particular, Kuzniar et al. (Kuzniar et al., 2012) have proposed interoperability testing for the switch- to-controller interaction. The approaches of the sec- ond group rely on model based testing techniques when formal models are used to describe the desired behaviour of the data plane/switch under test (Zhao et al., 2017), (Fayaz and Sekar, 2014), (Alsmadi et al., 2015), (L´opez et al., 2018), (Yao et al., 2014a), (Yao et al., 2017). In particular, the closest research to our work has been conducted in (Yao et al., 2014b) and (Zhang et al., 2016). Indeed, the authors in these works have also investigated the possibility of mod- elling the switch behaviour via a **state** machine, how- ever, in contrast to our approach, they rather model the pipeline processing in a switch. Moreover, their approaches rely on a composition of **finite** **state** mod- els and verify the packet processing while we are in- terested in testing the switch-to-controller communi- cation.

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The previous point is related with the following question: does the minimal time TC (y 0 , y 1 ) defined with L ∞ controls coincide with the minimal time defined with controls chosen in a wider class? To address this issue, a first step could be to consider Radon measure controls instead of L ∞ controls, this ensures that the corresponding **state** trajectory belongs to L ∞ and hence the **state** constraint y (t) ∈ C still makes sense. But as we can see in the example of Section 2.6, the target **state** (0, 1, 1) ⊺ can be reached instantaneously from the initial **state** (0, 1, 0) ⊺ with a Dirac impulse whereas this requires a waiting time greater than 1 when using classical L ∞ controls. Consequently, the minimal control time with Radon measure controls may differ from the minimal time with L ∞ controls. The study of conditions ensuring that these two minimal controllability times coincide is an interesting problem which would deserve more attention. We refer to [11, 30] for studies of a possible gap of the value functions respectively associated to classical L ∞ controls or to relaxed measure controls.

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Unit´e de recherche INRIA Lorraine, Technopˆole de Nancy-Brabois, Campus scientifique, ` NANCY 615 rue du Jardin Botanique, BP 101, 54600 VILLERS LES Unit´e de recherche INRIA Rennes, Ir[r]

Borja Balle 1 Odalric-Ambrym Maillard 2
Abstract
We present spectral methods of moments for learning sequential models from a single trajec- tory, in stark contrast with the classical litera- ture that assumes the availability of multiple i.i.d. trajectories. Our approach leverages an efficient SVD-based learning algorithm for weighted au- tomata and provides the first rigorous analysis for learning many important models using depen- dent data. We **state** and analyze the algorithm un- der three increasingly difficult scenarios: proba- bilistic automata, stochastic weighted automata, and reactive predictive **state** representations con- trolled by a **finite**-**state** policy. Our proofs in- clude novel tools for studying mixing properties of stochastic weighted automata.

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with the x and y axes ( Fig. 10 (b)). For each investigated spin speed O , the linear response signal is
captured and recorded by the sensors. Experimental results consist of the average of 10 recordings. The free whirling frequency spectrum is ﬁnally obtained by using a simple signal processing system and modal analysis software.
Two experimental setup are investigated in the following subsections. In the ﬁrst particular case, the displacements of the supporting plate are forbidden. In the second case, the supporting plate is free to move. For both, comparison of the experimental results with the numerical ones is provided. Numerical results are obtained with the **Finite** Element freeware

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This paper presents a methodology that allows taking advantage of the geometrical periodicity of electrical **machines** together with the modeling of rotor motion. It enables by means of the discrete Fourier transform (DFT) to reduce the large-scale system obtained from the **finite**-element model to several smaller independent subsystems, allowing a shortening of the computational time. Due to DFT properties, the computational time can be more reduced especially when we consider the inter-dependence of the spectral components under either balanced or unbalanced supply condition. In addition, a further reduction is possible in the case of balanced regimes where the distribution of the eventual numerical solution is governed by a limited number of prevailing harmonics.

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This is the representation we have used for the
onstru
tion of the MovistarBot grammars.
5
Graphs basi
ally
onsist in a set of linked boxes, where boxes
orrespond to transitions and links
or- respond to states. Ea
h box represents a set of transitions, one transition per box line, whi
h is shared among every pair of states represented by an in
oming and an outgoing link (e.g.: box `<TOKEN> ' of gure 7.1(a)
or- responds to the 4 `%<TOKEN>' transitions of gure 7.1(b) ). Start symbols `' and `%' of lexi
al masks within boxes are not spe
ied:
ase-sensitive masks are to be quoted, and
ase-insensitive masks are not. Moreover, a box entry may
ontain a sequen
e of lexi
al masks rather than a single one, in whi
h
ase represents an alternating sequen
e of transitions and states rather than a simple transition (e.g.: a box entry `"Feliz Navidad"' represents two
ase-sensitive masks whi
h are to be applied in that order). The dire
tion of the transitions represented by a box is given by a triangular arrowhead at- ta
hed to one side of the box. Graphs are meant to be read in the text sense, whi
h depends on the language (e.g.: left-to-right in English, right-to-left in Arabi
, et
.), thus the arrowhead is always atta
hed to the same side of the boxes. Links between boxes do not
arry **state** labels: **state** labels are not to be expli
itly dened sin
e they have no impa
t in the represented gram- mar. Graphs dene a unique initial **state** represented by a link
onne
ted to a single box. T o make the initial **state** more expli
it, an empty box (a box having only the arrowhead) is inserted right after the link, though this is not ne
essary; indeed, boxes having a single entry `<E>' (the blank-insensitive

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Knowing that p s ±pm=±p mod , the power balance can be observed.
4.9 Torque production in a machine with rotating PMs
Two analytical frameworks are proposed here to model magnetically-geared **machines** in which the Pm ring are the rotor. Fig. 4.41 shows the block diagram of the first method in which the developed torque is obtained Kelvin force. The stator current produces a MMF. Also, the MMF produced by permanent magnets and an equivalent surface charge density is derived. The radial component of the magnetic flux density is also obtained using the total MMF and the air-gap reluctance derived from flux tube modeling. Carter coefficient is used in the air-gap length corrections as well. Having the equivalent surface charge and the tangential component of the magnetic field on the surface of PMs, the shear stress and subsequently the developed torque on the PM side is determined using Kelvin force.

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where P (z) is the Bernoulli distribution on B-strings based on the frequencies of letters in z. In the language of Kolmogorov complexity, we may use KA(z) instead of K(z).
Let us comment on the history of the notion of strong **finite**-**state** dimension of a bit sequence. Originally it appeared in the paper of Lempel and Ziv [65] where it was defined in terms of variable-rate block encoders with **finite** memory. This is close to the auto- matic complexity definition; however, definition in [65] did not introduce the notion of automatic complexity and used instead **finite**-**state** compressors such that decompression is unique if the initial and **finite** states are given in addition to the compressed output. Theorem 3 from their paper [65, p. 534] says that this notion, denoted there by ρ(·), can be equivalently defined in terms of lim sup of entropies of unaligned blocks. Much later, in 2002 (the year when the arxiv version was published, the journal version appeared in 2007) Athreya, Hitchcock, Lutz and Mayordomo noted [4, Theorem 6.18] that the notion introduced by Ziv and Lempel (Athreya et al. refer to Ziv’s paper [63], not to [65], but this is most probably a typo) coincides with the **finite**-**state** strong dimension defined in terms of gales. They also note that one can use both 1-account and k-account gales in this equivalence proof (though they do not consider more general notion of output distri- bution of a **finite**-**state** probabilistic process). In a later paper (2005) Bourke, Hitchcock and Vinodchandran [14, Theorem 5.3] cite the result of Ziv and Lempel but strangely use aligned blocks (without explaining why aligned and non-aligned blocks give the same notion of strong **finite**-**state** dimension). Theorem 19 includes all these results and shows that they may be obtained almost for free by using the notions of automatic complexity, **finite**-**state** a priori probability and their superadditivity, while the original proofs were rather technical and were scattered among several papers.

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4. **Finite**-Time and Fixed-Time ISS Lyapunov Functions: Implicit Approach
This section introduces the tools to determine the FTISS and FXISS properties using the Implicit Lya- punov approach. Let us insist on the fact that this approach allows to circumvent explicit knowledge of a Lyapunov function V and its derivative; instead, knowledge of an implicit function Q(V, x) and its derivative is required. This new conditions on the function Q may be more difficult to verify in the general case; how- ever, as shown in Example 5, a combination with a suitable proposition of Q can ease the calculations and succeed in determining non-asymptotic ISS properties where the explicit approach fails to do so.

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L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignemen[r]

Figure 11. Courant dans la machine homopolaire, MLI intersective
Il apparaît alors que le taux d’ondulation des courants dans la machine principale est faible, conformément au rapport entre période de MLI et constante de temps L c1 /R. Par contre, il est manifeste que les constantes de temps des **machines** secondaire et homopolaire L c2 /R et L 0 /R sont plus faibles que celle de la machine principale. Même si ces dernières ne sont pas alimentées aux valeurs moyennes, elles le sont aux valeurs instantanées, des courants parasites s’y développant alors.

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marches dégradées lors d'une mise en défaut d'un bobinage ou d'un composant de puissance [ J AH -80 ]. Cependant, ces divers avantages ne doivent pas occulter la
complexité de leur commande, tant en mode normal qu'en mode dégradé [ T OL -00 ].
Le projet SMM (Systèmes Multimachines Multiconvertisseurs) du GdR SDSE puis du GdR ME2MS 1 , a travaillé sur l'étude, la représentation synthétique et la commande de systèmes de conversion électromécanique composés de plusieurs **machines** et / ou convertisseurs statiques [ SMM-00 A ] . Dans la classification des

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