Daniel Monceau
email: daniel.monceau@toulouseinp.fr

Michel Vilasi

High Temperature Oxidation of Additively Manufactured Structural Alloys

Metal alloys produced by additive manufacturing have specifi c microstruc tures and compositions. Their microstructure is generally out of equilibrium, textured, and inhomogeneities can be observed. They can have pores and nano-inclusions. The contents of volatile elements may deviate from the nominal composition. These alloys often experience signifi cant interna! stresses and present fairly rough surfaces depending on the process used. Hipping these alloys modifies their microstructure and defects. This literature review shows that open porosity can lead to much higher oxidation kinetics than dense materials, with very deep interna! oxide penetrations. Neverthe less, when fabricated with optimized parameters, LBM-or EBM-alloys, such as TA6V, 718, or 316L, can behave as well as, or even better than, wrought alloys. The roughness of the surface is not necessarily problematic, but it can lead to local breakaway phenomena and premature spalling on some alloys. It has been verifi ed that the nature of the grain boundaries strongly affects the intergranular oxidation. The effect of chemical segregations on the protective nature of the outer oxide layer has also been reported. Today, too few studies have been devoted to the effect of raw and pre-oxidized surface states after HIP on cyclic oxidation and on hot corrosion of these alloys.

INTRODUCTION

The alloys produced by additive manufacturing (AM) 1 • 2 may have been designed for specific pro cesses, but in general, they use standard chemical compositions of alloys already used in industry, and produced by casting or forging. However, these alloys have special characteristics that can affect their resistance to oxidation at high temperatures. Therefore, most of the work published to date reports standard characterizations of resistance to oxidation at high temperature, intended to demon strate that the material does not have strongly degraded properties compared to those produced by conventional processes. Most of these studies deal with alloys fabricated by laser beam melting (LBM) and electron beam melting (EBM) of powder bed processes. In this article, we present the chemical and microstructural characteristics of alloys result ing from additive manufacturing that we believe to be important in relation to resistance to oxidation. We then review the experimental results which do or do not demonstrate an influence of these charac teristics. W e also highlight topics that are not yet well documented.

SPECIAL CHA.RACTERISTICS OF AM ALLOYS THAT CAN INFLUENCE OXIDATION AT HIGH TEMPERATURE

The particular characteristics of metal alloy parts resulting from additive manufacturing relate to the chemical composition and its homogeneity, the microstructure, the surface condition and the sur face particular morphology, the complex geometry of the parts, and, in particular, the possible thinness of their walls, the residual stresses, and fi nally certain characteristic defects. 1 • 2 Figure 1 illustrates some characteristics of AM alloys with their consequences on high-temperature oxidation. Nominal compositions of all the alloys cited in this paper are given in Table I.

EFFECT OF CHEMICAL COMPOSITION

The various atomization processes (by gas jet, water jet, etc.) can lead to significantly different compositions from the alloy powder intended for additive manufacturing. If the contents of the major elements comply with the desired characteristic values, those of the minor elements may, on the other hand, diverge depending on the process for making the powder. This is, for example, the case with manganese, the content of which varied from 0.1 wt.% to 0.5 wt.% within two types of powder used by Tobar et al. 7 to produce 316L steels by laser fusion (Table I). In addition, the chemical composition of LBM or EBM alloys may differ slightly from the composition of the precursor powder. Some elements, which can volatilize, are important for oxidation and corrosion. Note in particular the case of aluminum, which allows the formation of a protective oxide layer of alumina, provided it is in sufficient concentration in the alloy. Juechter et al. studied the compositional variations during EBM of a TA6V titanium alloy, and showed loss of Al during processing. 8 Finally, during manufacturing, it is difficult to guarantee that the gas of the chamber is completely inert.

Additive manufacturing processes use a flow of Ar (or even N 2 ) directed to the melting zone, or are carried out under secondary vacuum. Nevertheless, gases such as O 2 , H 2 , H 2 O, N 2 , CO, and CO 2 can cause contamination of the material and powder particles which will be recycled. 9 These gases are impurities in the shielding gases used, but they can also desorb from the metal during its melting. Given the nature of the gases and the vacuum levels used, one can expect partial oxygen pressures of 10 -5 atm to 10 -8 atm, which are sufficient to oxidize most metals and, in particular, Al, Ti, Zr, Si, and Cr; however, the oxidation times are short. Yu et al. 10 demonstrated that fine particles of Al 2 O 3 may have formed during the fabrication of alloy 718 by the EBM and LBM processes. These particles have been identified as being sites for the nucleation of Ti-Nb carbides.

Titanium alloys can be expected to be sensitive to the printing atmosphere, due to the reactivity of titanium with oxygen, nitrogen, and carbon. This has been shown for the TA6V alloy produced by the EBM process. An increase of 0.11 wt.% oxygen was measured on TA6V powder recycled 21 times. 11 This is important for titanium alloys, whose microstructure and mechanical properties are strongly affected by a few hundred ppm of oxygen. 12 The recycled powder particles accumulate pollution by oxidation cycle after cycle. The recyclability of powders has been the subject of numerous studies, for example, 4 (c) unmelted powder particles fully oxidized (A), TA6V EBM (SEM-SE), 5 (d) special grain boundaries (B in blue, green, red, yellow) without any intergranular oxidation (C in black) (electron backscatter diffraction). 6 for titanium alloy TA6V and superalloy 718. 11,[13][14][15] The results show overall that it is possible to maintain a very low pollution with the LBM and EBM processes for these two alloys, even after dozens of cycles.

Impurities are very important for certain properties of alloys. In the field of high-temperature oxidation, with conventionally processed materials, it is well known that the sulfur content, at the ppm or even at the sub-ppm level, affects the adhesion of alumina to Ni-based superalloys (e.g., 16 ) As a consequence, the kinetics of cyclic oxidation increases with the level of S (in terms of metal consumption). Carbon can also play a role. 17 The reactive elements (Hf, Y, Zr, Ce, and La) interact with these species, 18 and then also affect the oxidation behavior in thermal cycling. It has also been shown that phosphorus can affect intergranular oxidation in alloy 718. 19 In 2014, Unocic et al. 20 published a first study on the cyclic oxidation characterization of AM-alloy 718. Following these pioneering works carried out at ORNL, Romedenne et al. 21 studied the cyclic oxidation in air at 950°C of the Hastelloy X alloy, produced by LBM and EBM, by comparing their behavior with that of the wrought alloy. They found that the oxide scale of the Hastelloy X alloy produced by EBM and LBM spalled faster than the wrought alloy. This result was attributed to the formation of a thin layer of SiO 2 at the metal/Cr 2 O 3 interface.

In 2021, Sanviemvongsak 22 showed that the LBM 718 alloy exhibited a highly degraded cyclic oxidation resistance compared to the EBM 718 and the wrought 718 alloys, while these alloys exhibit similar isothermal oxidation behaviors. No explanation could be found in the microstructural differences of the three alloys. It was instead proposed that the level of free sulfur (S dissolved in the gamma phase) was the explanation, and it was shown by a thermodynamic calculation that the small concentration of Mn could explain the fact that the alloy LBM had a higher level of free S (Fig. 2). These examples show that, first, cyclic oxidation experiments are required in addition to isothermal experiments, and, second, the level of impurities should be controlled at the level of 0.1-100 ppm, depending on the elements. This includes the level of S and C in Ni-base superalloys and steels, and also of reactive elements known to traps S and C, and of some elements with the same properties, such as Mn. This includes also Al, Ti, and Si elements in Ni-or Fe-based chromia formers because they can form internal and intergranular oxides (Fig. 3). 23

EFFECT OF MICROSTRUCTURE AND CHEMICAL HOMOGENEITY

Even when their level of porosity is very low, alloys from additive manufacturing can have specific microstructures that can affect high-temperature Morphological anisotropies and crystallographic texturing are also often observed. Parts can be used as raw materials, as processed or annealed, or exposed to high isostatic pressing (HIP). HIP can erase all or part of these microstructural peculiarities. These differences in microstructures can influence the homogeneity of the outer oxide layer, intergranular oxidation, or even the diffusion of oxygen into the metal matrix in the case of titanium alloys. 26 For example, it has been shown for standard manufacturing processes that the texturing of alloys influences the number of special grain boundaries, and that these are much more resistant to intergranular oxidation. 27,28 In particular, R3 grain boundaries have better resistance to intergranular oxidation. Sanviemvongsak et al. 4 studied the link between the nature of grain boundaries resulting from additive manufacturing and the kinetics of intergranular oxidation. Samples of alloy 718 from LBM-or EBM-optimized fabrication showed external oxidation kinetics and mass gains similar to wrought samples. On the other hand, intergranular oxidation differences were noted. More recently, it was shown by electron backscatter diffraction analyzes on samples oxidized over long periods of time Fig. 2. Effect of alloy 718 impurities on cyclic oxidation kinetics. The worst behavior is obtained for the LBM-718 alloy which may not have enough Mn to compensate for its S level. The best behavior is obtained for the forged alloy 718 which has a low S concentration and a lot of Mn (see text). 22 HT heat-treated, P600 ground to P600 paper finish, raw not ground. Fig. 3. Effect of powder composition and AM processing on the composition on internal and intergranular oxidation of alloy 625 (after). 23 Less Al in LBM-625 than in wrought 625 leads to less internal and intergranular oxidation at 900°C and 1050°C. at 850°C, that these differences were in part due to different special grain boundary densities in these different samples (Fig. 4). 6 Ramenatte et al. 23 compared the air oxidation of alloy 625 produced by LBM or casting and rolling. The LBM alloy exhibits a cellular microstructure with significant segregations of Nb and Mo. At 900°C, this does not impact the resistance to oxidation, but at 1050°C, a marked deterioration in the oxidation behavior of the LBM alloy is observed, with the formation of numerous pores under the oxide layer and a significant amount of Nb 1.5 Cr 0.5 O 4 rutile in the internal part of the oxide layer (Fig. 5). Juillet et al. 29 had previously found an opposite effect when comparing cast and LBM alloys. The higher d-Ni 3 Nb phase level in the cast alloy resulted in the presence of more Nb 2 O 5 in the oxide layer which was then less protective. Siri et al. 30 have compared the isothermal oxidation behavior of 316L steel made by LBM to the isothermal oxidation behavior of a wrought alloy. They found slower oxidation kinetics for the LBM alloy between 700°C and 1000°C, and attributed this improvement to an enhanced diffusion of Cr to the surface.

Regarding the TA6V alloy, the microstructure is closely dependent on the AM technique used, as illustrated in Fig. 2, showing the microstructures of two materials studied by Casadebaigt et al., 5,26 produced, respectively, by EBM and LBM. The main problem with titanium alloys is their embrittlement due to oxygen dissolution at high temperature. 12 The temperature of use of TA6V (<500°C) is low compared to its melting temperature, so we can therefore expect a significant contribution of shortcircuit diffusion at internal interfaces on the diffusion of oxygen in the metal matrix. Despite this, Casadebaigt et al. 26 showed that the fineness of the microstructure did not influence the kinetics of oxygen penetration. This was shown by comparing the behavior of TA6V produced by conventional methods, by LBM and EBM, with thin microstructures and after HIP with a much coarser microstructure.

EFFECT OF PROCESSING DEFECTS AND HIP

The manufacture of an alloy with non-optimized parameters can lead to the presence of defects which can greatly influence the oxidation kinetics. Jia et al. 3 studied the high-temperature oxidation of samples of the LBM-718 alloy with high energy densities, which led to the presence of open porosity in the samples and to mass gains very important during high-temperature oxidation. Sanviemvongsak et al. 4 also studied the oxidation of alloy 718 manufactured by LBM and EBM, but with optimized parameters, and few defects were observed. They showed that the LBM, EBM, and forged alloys then had very similar oxidation kinetics when the surface conditions were equivalent (P600 polishing). For a material with pores, a HIP treatment can lead to an improvement in the resistance to oxidation at high temperature by densifying the material, but the HIP can also lead to microstructural changes, such as precipitation at the grain boundaries which can affect intergranular oxidation, as has been shown on alloy 718 fabricated by LBM. 31

EFFECT OF SURFACE ROUGHNESS

The powder particles used for additive manufacturing have diameters generally between 15 lm and 120 lm, with a typical particle size of a TA6V powder for the EBM process being 70 lm ± À25/+50 lm and a typical particle size of a Ni-base alloy powder for the LBM process being 30 lm ± 20 lm. 32 This leads to surface roughness which can be high (R a of several tens of micrometers). There are ''staircase'' effects due to the fabrication by layer, with an effect of the fabrication angle. Sometimes, the presence of unmelted powder particles sintered on the surface (''satellites''), or metal droplets of very variable sizes, ejected from the molten bath to a previously molten and solidified layer of metal (''spatters''), are observed. It has been shown that, for alloy 718, roughness of EBM samples could lead to reactive surface areas 2-4 times higher than a flat surface. 4 When measuring the reactive surface area, one must be careful of hidden surfaces if a 3Dimage analysis is used. 5,33 When the oxidation or corrosion kinetics are studied, it is therefore necessary to be particularly attentive to this parameter in order not to reach erroneous conclusions. For example, in, 4 as processed additive manufacturing alloys exhibit higher oxidation kinetics than alloys forged or polished in terms of mass gains. However, this is only due to their higher reactive surface area. If the oxidation kinetics are characterized by the thickness of the oxide layer formed, or the thickness of the metal consumed, and this is what ultimately matters, there is no longer any difference between the 718 alloys forged, LBM or EBM, under isothermal oxidation conditions. These authors even noted that the EBM alloy exhibited slightly better behavior than forged, from measurements on polished samples. Ramenatte et al. 23 confirmed these results on alloy 625.

Casadebaigt et al. 5 studied the effect of surface roughness on a TA6V alloy obtained from LBM, EBM, or rolled. However, they also showed a specific behavior for this titanium alloy. Indeed, on a raw surface after additive manufacturing, the partially melted powder particles attached to the surface oxidize catastrophically. After an incubation time, the anionic growth of the rutile TiO 2 layer and the strong increase in volume induced by the oxidation of titanium generated very high mechanical stresses in the powder particles, which caused their cracking and the acceleration of their oxidation. In this study, it was shown that the very rapid oxidation of the powder particles contributed about half of the mass gains recorded (Fig. 6).

The catastrophic oxidation of the Ti powder at the surface of AM part is a phenomenon which does not affect the health of the material. However, sometimes, the surface geometry can affect the nature, the kinetics, and the growth constraints of the layer and its adherence. In an atmosphere of catastrophic powdery carburization, called metal dusting, Vernouillet et al. 34 showed that the samples of alloy 625 LBM and LBM + HIP with rough surfaces, suffered greater mass losses than polished samples. In addition, the raw HIP samples showed large variations in behavior. Indeed, the significant roughness due to additive manufacturing, and the high grain sizes due to HIP, can cause greater localized depletion of Cr and thus lead to the formation of a spinel on the surface, which catalyzes the decomposition of CO and is less protective with respect to oxidation and carburization (Fig. 4). The spinel formation in convex surface areas has also been observed after 500 h of oxidation at 800°C of EBM-Hastelloy X. 33 The roughness also affects the state of stress in the oxide layer and can lead to premature spalling of the oxide layers. 35 During their study on the cyclic oxidation kinetics of alloy 718, Sanviemvongsak et al. 22 observed that grinding with P600 paper of the surface of the LBM alloy delayed the catastrophic spallation of oxides during thermal cycles of 1 h at 900°C, but did not prevent it.

EFFECT OF RESIDUAL STRESSES

Due to the large temperature gradients during additive manufacturing, parts are subjected to high levels of residual elastic stresses. These can influence the reactivity of surfaces, and the diffusion to the core of the material. In particular, internal and intergranular oxidation, which itself generates significant strain due to the large increase in volume associated with oxidation, may depend on the initial presence of stresses from the manufacturing process. This has been shown by Bertali et al. 36 for a conventional 600 alloy tested at 480°C under an H 2 + H 2 O mixture, but this does not seem to have been reported to date for additive manufacturing materials.

EFFECT OF COMPLEX GEOMETRIES AND THIN WALLS

One of the major interests of additive manufacturing is the possibility of manufacturing parts of complex geometry, with hidden surfaces inaccessible to machining. These complex geometries can have thin elements and thin walls, which can have several consequences on their chemical reactivity and mechanical behavior. First, hidden surfaces cannot receive all types of surface treatments. For example, grit blasting, shot blasting, and coating application can be difficult or impossible inside parts. This of course has an effect on the reactivity of the internal surfaces. Then, the morphology and the surface defects depend on their angle with respect to the direction of construction of the part. This influences the oxidation. For example, Li et al. 28 seem to show during preliminary results on an alloy 718 produced by EBM, that the fabrication angle affects the orientation of the grains and the density of grain boundaries, which would have a slight repercussion on the resistance to oxidation at high temperature. Finally, the fineness of some AM structures can make them susceptible to premature exhaustion of alloying elements that protect against oxidation or corrosion at high temperature. In isothermal oxidation, if a protective layer of Al 2 O 3 or Cr 2 O 3 forms on a surface, this creates a Cr or Al depletion under the oxide layer. If the wall of the part is thin and at high temperature, the depletion profiles meet from the two surfaces of the wall, the core concentration decreases, and consequently that under the oxide layer also decreases. This can cause a ''breakaway'' phenomenon, that is, the formation of a less protective oxide layer formed from the other elements of the alloy, and faster growing by several orders of magnitude. [START_REF] Chyrkin | Predicting subsurface enrichment de pletion processes during high temperature oxidation of alloy 625 thin foils[END_REF][START_REF] Romain | [END_REF] In the case of a titanium alloy, another thin wall effect exists, as oxygen diffuses in large quantities into the metal under the oxide layer and weakens it. 12 This effect is, of course, more dangerous on a thin wall. [START_REF] Texier | 14th World Conference on Titanium[END_REF]

EFFECT OF PREOXIDATION DURING HIGH ISOSTATIC PRESSURE TREATMENT (HIP)

The purpose of the HIP treatment carried out at high temperature and high hydrostatic pressure is to close the pores and defects formed during additive manufacturing. This heat treatment will also reduce or eliminate the internal stresses 31 and modify the microstructure (nature of the phases and grain size). It is carried out at a high temperature and under a very high pressure, typically 1000 bars. If the gas used is nitrogen or argon, these contain impurities of O 2 and H 2 O, which, due to the high pressure,can lead to significant surface oxidation. Nitrogen can also cause internal nitridation. Since the HIP treatment is carried out in a closed chamber, which may contain metal walls, graphite tools, and ceramic insulators, the reactivity of the metal alloys will also depend on the furnace used and on the total surface area of the samples treated at the same time in the chamber. No systematic study on this subject is available in the literature to our knowledge.

ALLOYS DEVELOPED SPECIFICALLY FOR AM

New alloys with improved mechanical and chemical properties have been designed by exploiting the specificities of AM powder bed fusion processes to develop novel microstructures. This is, for example, the high mechanical performance Scalmalloy Ò aluminum alloy, which has been patented by Airbus-APWorks GmbH, and which has been the subject of a comprehensive metallurgical study by Spierings et al. 40 Another example is given by Hong et al. 41 who reported a comparative study of the hightemperature oxidation of alloy 625 and of a 625/TiC composite, both developed by LMD. These authors report a surprising beneficial effect of the presence of TiC, but the oxidation kinetics are not compared to a forged 625 alloy, and the oxide layers are not observed in section to verify their thicknesses. These preliminary works need more extensive studies on reactivity.

SUMMARY OF AM ALLOY PERFORMANCE AND CONCLUSIONS

1. Porosity and defects resulting from a lack of powder fusion, and also from gas trapping, lead to significant degradation of oxidation behavior 3 if these defects are connected to the external surface. High-temperature oxidation reveals the open porosity. The oxidation kinetics increase, and also the consumption of protective alloying elements. This can lead to a premature breakaway phenomenon. In addition, the oxidation of cracks with the very significant increase in volume when the metal is transformed into oxide are sites of stress concentration and can lead to rupture. 2. The surface roughness, generally linked to the particle size distribution of the powders, increases the area of the reactive surface and therefore the quantity of oxide formed for fairly short times. The reactive surface area was observed to be multiplied by 4 for EBM-718 in comparison to a flat surface. 4 . This increase in reactivity is transient and not necessarily detrimental to the part in terms of affected thickness, as has been seen in high-temperature oxidation of alloy 718, 4 but may lead to local breakaway, as observed on alloy 625 23,34 or Hastelloy X. 33 3. The complexity of microstructures is inherent to the rate of the melting and cooling processes, which is conducive to elemental segregation. During high-temperature oxidation, these segregations could lead to the formation of nonprotective oxides, 23 but there is a lack of experimental data on this point. 4. The nature of grain boundaries may be different from those observed in materials from conventional elaboration processes, due to increased segregation, pore closures during HIP, texturing, and internal stresses, and this may affect high-temperature intergranular oxidation kinetics. 6 5. The resistance to cyclic oxidation and hot corrosion has been scarcely reported in the literature. [20][21][22] On this subject, special attention must be paid to impurities and minor elements that can affect the adhesion of the oxide layers (S, C, reactive elements, sulfide-forming elements).

The geometry of the surfaces may also greatly affect the stress state during thermal cycling. Furthermore, surface roughness could affect hot corrosion with melted salts. Here again, experimental data are missing.

To improve the performance of AM parts, it is first mandatory to produce materials without open porosity. Roughness is not necessarily a problem, but it must be evaluated with long-term oxidation tests until the appearance of the ''breakaway'' phenomenon, and under thermal cycling conditions to increase the level of stress in the oxide scale. Then, it is possible to consider heat treatments intended to reduce the residual stresses. HIP treatment helps to homogenize the chemical composition and at the same time reduces manufacturing defects. Chemical or electrochemical surface treatments can also be considered to reduce surface roughness when necessary, with the difficulty of accessing hidden surfaces.

Fig. 1 .

 1 Fig. 1. Effect of the microstructural characteristics of materials from AM on their oxidation behavior: (a) open porosity, 718 LBM 3 (SEM-SE), (b) roughness, 718 EBM (SEM-SE),4 (c) unmelted powder particles fully oxidized (A), TA6V EBM (SEM-SE),5 (d) special grain boundaries (B in blue, green, red, yellow) without any intergranular oxidation (C in black) (electron backscatter diffraction).6

*

  TableI. Nominal composition of alloys referenced in this paper (wt.Max, **Min oxidation. Non-equilibrium microstructures can be generated: for example, a' martensite in Ti alloys, or the absence of hardening precipitates in Ni-based alloys, or chemical segregations, for example, Nb, Ti, and Mo in interdendritic zones of alloy 718, which can cause precipitation of Laves phases.[START_REF] Sames | Effect of process control and powder quality on inconel 718 produced using electron beam melting[END_REF][START_REF] Tao | [END_REF]

Fig. 4 . 6 Fig. 5 .

 465 Fig.4. Intergranular oxidation of alloy 718. Statistical evaluations showed that AM alloys present thicker intergranular oxidation, the same average depth of intergranular oxidation than forged alloy, but with deeper occurrences.6

Fig. 6 .

 6 Fig. 6. Effect of TA6V powder oxidation at the surface of LBM-TA6V. Catastrophic oxidation of titanium powder leads to an initial large increase of mass gain. (after 5 ).

ACKNOWLEDGEMENTS

Tom Sanviemvongsak, Nicolas Ramenatte, Annabelle Vernouillet, Antoine Casadebaigt, Aure ´lie Vande Put, Ste ´phane Mathieu, Jonathan Hugues, Eric Andrieu, Se ´bastien Doublet are thanked for fruitful discussions.

CONFLICT OF INTEREST

On behalf of all authors, the corresponding author states that there is no conflict of interest.

Publisher's Note Springer Nature remains neutral with re gard to jurisdictional claims in published maps and institutional affiliations.


Akbaraly Tasnime
email: tasnimeakbaraly<t.akbaraly@ucl.ac.uk>

Kivim Ki Ä Mika

Eric J Brunner

Chandola Tarani

Singh-Manoux Archana

Jane E Ferrie

Association between metabolic syndrome and depressive symptoms in middle-aged adults: results from the Whitehall II study

Objective

Although it is possible that the association between depression and the metabolic syndrome is a two-way street , the metabolic syndrome

as a predictor of depression has been little investigated. We examined whether the metabolic syndrome is associated with the onset of depressive symptoms in a cohort of middle-aged British civil servants.

Research Design and Methods

Analyses

included 5232 participants (41 61 years) from the Whitehall II prospective cohort study. Depressive symptoms were assessed in -1991 93 and again 6-years later using the depression subscale from the 30-item General Health Questionnaire. Metabolic syndrome was assessed in 1991 93, according to National Cholesterol Education Program criteria. -Results Presence of the metabolic syndrome was associated with an increased risk of future depressive symptoms, odds ratio 1.38 95 = [ % Confidence Interval, 1.02 1.96 after adjustment for potential confounders. Of the 5 components, only central obesity, high triglyceride -]

levels and low HDL cholesterol levels predicted depressive symptoms. These components explained most of the association between the metabolic syndrome and the onset of depressive symptoms.

Conclusion

Our results suggest that the metabolic syndrome, in particular the obesity and dyslipidemia components, is predictive of depressive symptoms.

Author Keywords Metabolic syndrome ; Depressive symptoms ; Prospective study ; Middle-aged population Evidence of an association between depression and cardiovascular diseases (CVD) ( ; ) has led some studies to investigate whether the 1 2 metabolic syndrome may constitute an explanatory mechanism. The metabolic syndrome is defined as a clustering of risk factors which predispose an individual to cardiovascular morbidity and mortality ( ; ). It is characterised by elevated abdominal obesity, high triglycerides, 3 4 high blood pressure, high fasting glucose and low high-density lipoproteins (HDL) cholesterol, and represents an important risk factor for CVD ( ). Cross-sectional studies have shown an association between depression and the metabolic syndrome in young adults ( ) and in middle aged 5 6 populations ( ). Cross-sectional associations between depressive symptoms and individual components of the metabolic syndrome have been 7-9

observed in a North American study of elderly male twins, ( ) but were not evident in a middle-aged cohort from northern Finland ( ). The 10 11 focus of longitudinal studies essentially has been to investigate whether depression predicts the metabolic syndrome and some have shown that, in middle aged populations, depressive symptoms appear to be associated with an increased risk of the metabolic syndrome in women but not in men ( ; ). 12 13 Although in most existing research the assumption has been that depression predicts the metabolic syndrome, depression could also be a consequence of the metabolic syndrome. To date, this reverse association has been little investigated. However, a two-way street between

" "
depression and the metabolic syndrome was suggested by findings from a population-based prospective cohort study in which women with the metabolic syndrome in childhood and adulthood had the highest level of depressive symptoms in adulthood ( ). This finding was not observed 14 for men. Further evidence was provided from a cohort of middle aged women, in which Raikkonen et al. showed that anger and anxiety increased in response to the metabolic syndrome ( ). Only one report published recently has investigated the metabolic syndrome as a risk factor for depressive symptoms in middle aged adults. It showed that the metabolic syndrome may be an important factor for the development of depression in women but not in men ( ). 15

Our objective is to investigate prospectively whether the metabolic syndrome is associated with the onset of depressive symptoms in a cohort of middle-aged British adults free of such symptoms at baseline, after taking into account potential confounders. circumference below the costal margin. Resting blood pressure was measured using the Hawksley Random Zero Sphygmomanometer. Serum triglycerides, HDL cholesterol and fasting blood glucose were analyzed as previously described ( ). 18

RESEARCH DESIGN AND METHODS

Assessment of depressive symptoms at phases 3 and 5: We assessed depressive symptoms using the four-item depression subscale of the 30-item General Health Questionnaire (GHQ) ( ). The items, obtained on the basis of factor analysis and comparison with the items of the 19 depression subscale of the 28 item GHQ ( ), are: thinking of yourself as a worthless person , felt life is entirely hopeless , felt life isn t 20

" " " " " '

worth living , found at times you couldn t do anything because your nerves were too bad . All items were scored on a Likert scale from 0 3

" " ' " and then summed (range 0 12). Respondents were categorised as free of depressive symptoms if they scored 0 3 or having depressive -symptoms if they scored 4 or more. New onset depressive symptoms refers to participants who developed symptoms of depression between phases 3 and 5, after excluding participants with depressive symptoms at phase 3. Depressive symptoms, assessed with GHQ, cannot be equated with clinically diagnosed depression ( ). 21

Assessment of covariates at phase 3: consisted of age, sex, ethnicity (white/South Asian/black), marital Socio-demographic variables status, employment grade and education. Employment grade, defined on the basis of salary, ranges from 1 to 3, with grade 1 representing the highest level and grade 3 the lowest. Educational attainment was grouped into five levels (no academic qualification, lower secondary, higher secondary, university degree, higher university degree). measured were current smoker (yes/no), alcohol consumption Health behaviors (grams/day), and intensity of physical activity (based on questions on frequency and duration of physical activity and categorised as high ,

"

" " medium and low ). Prevalent coronary heart disease (CHD), identified using clinically verified events, included non fatal myocardial

" " "
infarction and definite angina as described previously ( ). 22

Statistical Methods

Logistic regression was used to model the association between the metabolic syndrome and each of its components at phase 3 and the onset of depressive symptoms between phase 3 and phase 5. In the first model (Model 1), the analyses were adjusted for age, sex and ethnicity; in the second model (Model 2) they were also adjusted for employment grade, educational level, marital status, smoking habit, alcohol consumption, physical activity and CHD. Interactions between the metabolic syndrome and the covariates (including sex) were tested and were not found to be statistically significant at p<0.05.

To assess the extent to which components of the metabolic syndrome drive the association with depressive symptoms we calculated the percentage attenuation in the metabolic syndrome-depressive symptoms association after the components separately associated with depressive symptoms were added to the model. The percentage attenuation in the association between metabolic syndrome and depressive symptoms was determined using the formula: (( )/ ) 100), where is the beta coefficient estimated from the logistic regression model.

% =[ β x -β x adjusted for w β x ]* β
In order further to understand the role of the metabolic syndrome as a risk factor for the onset of depressive symptoms, we performed analyses to explore whether its components act synergistically by testing interactions between them. We used a backward elimination approach that removed all of 5-, 4-, 3-and 2-way interaction terms with p>0.05 from a saturated model that included all of the metabolic syndrome components and their interaction terms. A new variable combining obesity, high triglyceride and low HDL cholesterol components (7 modalities: 0 none of the component, 1 only obesity, 2 only high triglyceride; 3 only low HDL cholesterol, 4 obesity and high

= = = = =
triglyceride, 5 obesity and low HDL cholesterol, 6 high triglyceride and low HDL cholesterol, 7 all three components) was constructed to

= = =
test specifically the combined effects of these three components on onset of depressive symptoms.

In sensitivity analyses, we excluded participants with CHD at phase 3 to explore the contribution of CHD to the association between the metabolic syndrome and onset of depressive symptoms. To assess potential bidirectional effects, we excluded participants free of the metabolic syndrome at phase 3 and we performed logistic regression analyses examining associations between depressive symptoms at phase 3 and onset of the metabolic syndrome between phase 3 and phase 5. All analyses were conducted using the SAS software, version 9 (SAS Institute).

RESULTS

Overall, 7263 individuals participated in Phase 5. However, complete data on depressive symptoms at phases 3 and 5, the metabolic syndrome and all the covariates were available for 5562 participants (1604 women and 3958 men) aged 49.5 6.1 years at phase 3. Three ± hundred and thirty of these participants had depressive symptoms at phase 3 and were excluded in order to identify new onset depressive symptoms between phase 3 and phase 5. Compared with those excluded, participants included in the analysis (n 5232) were more likely to be = men (71.5 vs. 66.2 ) and younger (49.6 6.08 vs. 49.9 6.06 years). However they did not differ on educational level ( of participants

% % ± ± %
with no academic qualification: 13.1 for participants included vs. 13.7 for those excluded) or prevalence of the metabolic syndrome (10.4

% %
for participants included vs. 11.4 for those excluded). Prevalence of the metabolic syndrome in participants who already had depressive % % symptoms at phase 3 was 9.7 and was not significantly different from that in participants free from depressive symptoms both at phase 3 and % at phase 5 (10.2 ).

%

Characteristics of the participants as a function of the presence of metabolic syndrome are shown in . Prevalence of the metabolic Table 1 syndrome was 10.4 (n 547). Compared to participants without the metabolic syndrome, the proportion of participants with the metabolic % = syndrome was lower in women and in white participants , but was higher in those with a low education level and among those who had low " " physical activity. As expected, participants with the metabolic syndrome had higher waist circumference, triglycerides, blood pressure and fasting blood glucose levels and a lower HDL cholesterol concentration compared to participants without the metabolic syndrome.

Four hundred and twenty eight participants (8.2 ) developed depressive symptoms after phase 3, 7.8 of the men and 9.2 of the women.

% % %

New onset depressive symptoms were more common among those who had the metabolic syndrome at phase 3 (13.5 vs. 10.2 ).

% %

Furthermore, a higher rate of new onset depressive symptoms was observed in participants with the central obesity (14.3 vs. 10.4 ), high % % triglycerides (29.4 vs. 24.9 ) and low HDL-cholesterol (22.7 vs. 17.8 ) components of the metabolic syndrome.

% % % %
shows multivariate models of the association between the metabolic syndrome at phase 3 and onset of depressive symptoms Table 2 between phase 3 and phase 5. In analyses adjusted for sex, ethnicity and age, the odds of new-onset depressive symptoms was 1.47 95 CI:

[ % 1.09 1.99 times higher for participants with the metabolic syndrome than for those without. This association remained statistically significant

- ]
after further adjustment for other socio-demographic factors, health behaviour and CHD; odds ratio (OR) 1.38 1.02 1.87 . Multivariate

= [ - ]
analyses of the association between each component of the metabolic syndrome and onset of depressive symptoms showed that of the 5 components, only central obesity, high triglyceride levels and low HDL cholesterol levels were related to a greater odds of onset of depressive symptoms between phase 3 and phase 5, while there was no evidence of an association between hypertension and new-onset depressive symptoms. Unexpectedly, participants with elevated fasting blood glucose levels were less likely to develop depressive symptoms during the 6-years follow-up. In order to examine if the components of the metabolic syndrome had an independent effect we included them all simultaneously in a model that included only participants with no missing data on any of the metabolic syndrome components (n 4983 The fact that nearly half the participants with metabolic syndrome were obese (49.8 ) raises the possibility that the association we % observed between the metabolic syndrome and depressive symptoms is driven mainly by obesity. The percentage attenuation of the association between metabolic syndrome and depressive symptoms by its individual components was as follows: 37. depressive symptoms ( ). These findings provide support for the hypothesis that the association between the metabolic syndrome and Table 3 depressive symptoms is partly driven by central obesity, but that dyslipidemia is also a major driver.

To test for synergistic effects between the components of metabolic syndrome on depressive symptoms, we examined multiple interaction terms but none of these were statistically significant (results not shown but available on request). However, using a unique variable with all possible combinations of the obesity, high triglycerides and low HDL components, we found that participants with all three components were at the greatest risk of new-onset depressive symptoms, odds ratio 2.71 (95 CI: 1.73 4.24).

% -

Sensitivity analyses

Exclusion of participants with CHD at phase 3 (n 136) had little effect on the results: After adjusting for sex, ethnicity and age, the odds of = new-onset depressive symptoms were 1.37 1.00 1.87 times higher for participants with the metabolic syndrome than for those without. No support was found for the hypothesis that depressive symptoms were a consequence rather than a cause of the metabolic syndrome as there was no association between presence of depressive symptoms at phase 3 and new onset cases of the metabolic syndrome in participants free of metabolic syndrome at phase 3, odds ratio 0.89 (95 CI 0.60 1.33), p 0.57 after controlling for sex, age and ethnicity and 0.81 (95 CI 054

% - = % -
1.22) after adjusting for socio-demographic factors, health behaviours and CHD (results not shown but available on request).

CONCLUSIONS

We examined associations between presence of the metabolic syndrome and the onset of depressive symptoms in a middle aged population free of such symptoms at baseline. After adjusting for demographic measures, socio-economic factors, health behaviours and prevalent coronary heart disease, participants with the metabolic syndrome had a higher risk of developing depressive symptoms six years later. Of the components of the metabolic syndrome, central obesity, high triglycerides and low HDL cholesterol were the main contributing factors to this association. There was no evidence to support bidirectional associations between the metabolic syndrome and depressive symptoms.

Among the 5562 participants in this study in 1993, prevalence of the metabolic syndrome according to NCEP criteria was 10.4 . Similar % levels of prevalence were found in a cohort of healthy middle aged women in the Healthy Women study (9.3 in 1988) ( ) and in a cohort of % 13 middle aged men participating in the Kuopio Ischemic Heart Disease Risk Factor Study (11 in 1989) ( ). Prevalent CHD is a potential % 23 source of depression, but in the present study the association between the metabolic syndrome and new-onset depressive symptoms remained significant after controlling for prevalent CHD, and in a sub-cohort that excluded participants with this disease. This suggests that the association between the metabolic syndrome and depressive symptoms is not driven by depressive symptoms generated by manifest CHD.

Our findings are consistent with the hypothesis that depressive symptoms may be a consequence rather than a cause of the metabolic syndrome. Prevalence of the metabolic syndrome was no higher in participants with depressive symptoms at baseline than among participants who did not have such symptoms at baseline or at follow-up. This lack of evidence was confirmed in further analyses showing no association between presence of depressive symptoms at baseline and development of the metabolic syndrome during the follow-up. A corresponding finding was observed in the Northern Finland 1966 Birth cohort Study in which the metabolic syndrome defined according NCEP criteria was not associated with depressiveness and anxiousness in 31-year-old adults ( ). 11

In a middle-aged population-based sample, Koponen et al found the metabolic syndrome, assessed using NCEP criteria, was associated with a higher probability of depressive symptoms (measured using the Beck Depression Inventory) after a 7-year follow-up in women but not in men ( ). In another Finnish population-based study women with the metabolic syndrome in childhood had a higher mean depressive 15 symptom score (Beck Depression Inventory) in adulthood but again no association was found in men ( ). The reasons why these associations 14 should be sex-specific remain unclear and are in contrast to our findings of an association between the metabolic syndrome and onset of depressive symptoms 6 years later in both sexes.

An important aspect of our study is the exploration of the relationship between each component of the metabolic syndrome and subsequent onset of depressive symptoms. We found good evidence that central obesity and abnormal lipids were associated with the onset of depressive symptoms. However, the hypertension component was not associated with subsequent depressive symptoms and, unexpectedly, high levels of fasting blood glucose were associated with a lower rather than a higher probability of depressive symptoms. The results for central obesity are in concordance with the Alameda County Study which showed that obesity was associated with an increased risk of depression 5-years later (

). According to Ross et al. ( ) one explanation of the obesity-depression relationship could be devaluation and stigma which may cause 24 25 overweight and obese individuals to suffer from lower self-esteem and a higher level of depression. Another potential explanation involves unsuccessful weight control by dieting among obese and overweight people which could be more stressful than the obesity per se ( ).

It has been argued that the metabolic syndrome is driven almost entirely by central obesity. Our results do not support this position in relation to either the basic association between the metabolic syndrome and depressive symptoms, or the associations between the high triglycerides and low HDL cholesterol components of the metabolic syndrome and the onset of depressive symptoms. Thus, the metabolic syndrome-depressive symptoms association does not appear to be simply an artefact of the preponderance of obesity in individuals with the syndrome. While we found no evidence of synergy among the metabolic syndrome components in predicting depressive symptoms, our results

showed that central obesity, high triglycerides and low HDL cholesterol components had a cumulative effect on their onset. These findings suggest that a combination of central obesity and abnormal lipids constitutes a risk factor for the onset of depressive symptoms. Further research on prospective associations between components of the metabolic syndrome and the development of depression is needed to confirm our findings and to explore the mechanisms that underlie the association.

Our study has several potential limitations. First, depressive symptoms were measured using a short scale that is not a measure of clinically recognized psychiatric disorder. Although the symptom scale is reliable it does not indicate the severity or the chronicity of the depression. A second drawback concerns the limited generalizability of our findings. Whitehall II study participants are mainly white, office-based civil servants who are not fully representative of the British population ( ). Third, despite the extensive level of adjustment in our analyses, with 16

observational data the possibility remains that unmeasured confounders may explain part of the association between the metabolic syndrome and depressive symptoms.

In conclusion, this is apparently the first study to show that the probability of new onset depressive symptoms 6-years later is higher amongst men and women with the metabolic syndrome; an association that remains after taking into account a large range of potential confounders. Further research is needed to examine whether prevention of the metabolic syndrome, in particular its obesity and dyslipidemia components, might reduce the onset of depressive symptoms.

  The target population for the Whitehall II study was all London-based office staff, aged 35 55 years, working in 20 civil service

	-
	departments ( ). Baseline screening (Phase 1) took place during 1985 1988 (N 10,308), and involved a clinical examination and a 16 -=
	self-administered questionnaire containing sections on demographic characteristics, health, lifestyle factors, work characteristics, social support
	and life events. The clinical examination included measures of blood pressure, anthropometry, biochemical measurements, neuroendocrine
	function, and subclinical markers of cardiovascular disease. Subsequent phases of data collection have alternated between postal questionnaire
	alone Phases 2 (1988 1990), 4 (1995 1996), 6 (2001) and 8 (2006) and postal questionnaire accompanied by a clinical examination Phases 3 [ --] [
	(1991 1993), 5 (1997 1999) and 7 (2002 2004) . ---]
	Assessment of metabolic syndrome at phases 3 and 5: The metabolic syndrome was defined according to the criteria of the National
	Cholesterol Education Program (NCEP) ( ), based on the presence of 3 or more of the following: waist circumference >102 cm in men and 17
	>88 cm in women; serum triglycerides 1.7 mmol/L; HDL cholesterol <1.04 mmol/L in men and < 1.29 mmol/L in women; blood pressure ≥	≥
	130 mmHg/ 85 mmHg systolic over diastolic pressure; fasting glucose 6.1 mmol/L. Waist circumference was taken as the smallest ≥ ≥

  20.4 of the relationship between high triglycerides and depressive symptoms and 17.8 of the association between low HDL cholesterol and

5 for obesity, 38.0 for % % triglycerides, and 23.4 for low HDL cholesterol. All three components together explained 93 of the association. Obesity explained only % % % %

Table 1

 1 Characteristics of Whitehall participants according to presence of the metabolic syndrome; phase3 (1991 1993)

	-
	Metabolic syndrome*
	No		Yes
	N 4685 =		N 547 =
	% or M SD ±	**	% or M SD ±	**	P

± 131.2 11.6 * Diagnosis of the metabolic syndrome was based on the NCEP definition ( ) 17 ** For continuous variables, means standard deviations are given.

+

Ackowledgements:

We thank all of the participating civil service departments and their welfare, personnel, and establishment officers; the British Occupational Health and Safety Agency; the British Council of Civil Service Unions; all participating civil servants in the Whitehall II study; and all members of the Whitehall II study team.

MK is supported by the Academy of Finland (projects 117604, 124322 and 124271). AS-M is supported by a European Young Investigator

"

Award from the European Science Foundation. MM is supported by an MRC research professorship. J.E.F. is supported by the Medical Research

Footnotes:

Authors have no conflict of interest to disclose.

Table 2

Association between the metabolic syndrome at phase 3 and the onset of depressive symptoms between phase 3 (1991 93) and phase 5 (1997 99)


Mika Hirvensalo

Etienne Moutot
email: etienne.moutot@math.cnrs.fr

Abuzer Yakaryılmaz
email: abuzer@lu.lv

Computational limitations of affine automata and generalized affine automata

Keywords: non-classical models of automata, affine automata, logarithmic space, generalized automata, cutpoint languages, bounded error

We present new results on the computational limitations of affine automata (AfAs). First, we show that using the endmarker does not increase the computational power of AfAs. Second, we show that the computation of boundederror rational-valued AfAs can be simulated in logarithmic space. Third, we identify some logspace unary languages that are not recognized by algebraic-valued AfAs. Fourth, we show that using arbitrary real-valued transition matrices and state vectors does not increase the computational power of AfAs in the unboundederror model. When focusing only the rational values, we obtain the the same result also for bounded error. As a consequence, we show that the class of bounded-error affine languages remains the same when the AfAs are restricted to use rational numbers only.

Introduction

Finite automata are interesting models to study since they express a very natural limitation of finite memory. They are also an interesting starting point for many computational models, since they are simpler than many others like pushdown automata or Turing machines. Due to this simplicity, there exist many different models of finite automata, all trying to express different computational settings. Deterministic [START_REF] Sipser | Introduction to the Theory of Computation, 3rd edition[END_REF], probabilistic [START_REF] Paz | Introduction to Probabilistic Automata[END_REF] and quantum [START_REF] Ambainis | Automata: From Mathematics to Applications, chap. Automata and Quantum Computing[END_REF] finite automata (DFAs, PFAs, and QFAs, respectively) have been studied to try to understand better the computational limitations inherent to all these cases.

Recently, Díaz-Caro and Yakaryılmaz introduced a new computational concept, called affine computation [START_REF] Díaz-Caro | Affine computation and affine automaton[END_REF]. As a non-physical model, the goal of affine computation is to investigate the power of interference caused by negative amplitudes in the computation, like in the quantum case. But unlike QFAs, affine finite automata (AfAs) have unbounded state set and the final operation corresponding to quantum measurement cannot be interpreted as linear. The final operation in AfAs is analogous to renormalization in Kondacs-Watrous [START_REF] Kondacs | On the power of quantum finite state automata[END_REF] or Latvian [START_REF] Ambainis | Algebraic results on quantum automata[END_REF] quantum automata models.

AfAs and their certain generalizations have been investigated in a series of works [START_REF] Belovs | On a conjecture by Christian Choffrut[END_REF][START_REF] Díaz-Caro | Affine computation and affine automaton[END_REF][START_REF] Hirvensalo | On the computational power of affine automata[END_REF][START_REF] Ibrahimov | Error-free affine, unitary, and probabilistic OBDDs[END_REF][START_REF] Nakanish | Exact affine counter automata[END_REF][START_REF] Villagra | Language recognition power and succinctness of affine automata[END_REF]. In most of the cases, affine models (e.g., bounded-error and unbouded-error AfAs, zero-error affine OBDDs, zero-error affine counter automata, etc.) have been shown more powerful than their classical or quantum counterparts. On the other hand, we still do not know too much regarding the computational limitations of AfAs. Towards this direction, we present new results. First, we show that using end-marker does not increase the computational power of affine automata with unbounded error or bounded error. Second, we show that the computation of bounded-error rational-valued affine automata is simulated in logarithmic space, and so we answer positively one of the open problems in [START_REF] Díaz-Caro | Affine computation and affine automaton[END_REF]. Third, we give an impossibility result for algebraic-valued AfAs, and, as a result, we identify some unary languages (in logarithmic space) that are not recognized by algebraic-valued AfAs with cutpoints, improving a previous result showing that the same languages cannot be recognized with bounded error [START_REF] Hirvensalo | On the computational power of affine automata[END_REF].

Moreover, we give the formal definition of generalized AfAs by allowing to use arbitrary real-valued transition matrices and state vectors. Fourth, we show that such generalization does not increase the computational power of AfAs with cutpoint language recognition. If we restricted these generalized AfAs to use only rational numbers, we obtain the same result also for bounded error language recognition. As a consequence, we show that the class of bounded-error affine languages remains the same when the AfAs are restricted to use rational numbers or only integers.

We provide all definitions in the next section and our results regarding using end-marker (Section 2.4). Our logarithmic space simulation is given in Section 3. Our impossibility result is given in Section 4. Our results related to generalized AfAs are given in Section 5.

A preliminary version of this paper was presented in UCNC2019 [START_REF] Hirvensalo | On the computational power of affine automata[END_REF]. In this version, Sections 2.4 and 5 are completely new.

Preliminaries

Throughout the paper, Σ denotes the input alphabet -not containing letter $ (we fix it as the right end-marker wherever it is used) , and Σ = Σ ∪ {$}. The empty word is represented as ε. The set of words written using the alphabet Σ is denoted Σ * . For any given word w ∈ Σ * , |w| is the length of w, we define w = w$, and, if w = ε, w i represents its i-th letter, where 1 ≤ i ≤ |w|.

For any given class C, C Q and C A denote the classes defined by the machines restricted to have rational-valued and algebraic-valued components, respectively. The logarithmic and polynomial space classes are denoted as L and PSPACE, respectively. We assume that the reader is familiar with the basic notions of automata theory.

Models

As a probability distribution (also known as a stochastic vector) we understand a (column) vector with nonnegative entries summing up to one, and a stochastic matrix (also known as a Markov matrix) here stands for a square matrix whose all columns are probability distributions.

A k-state probabilistic finite automaton (PFA) P over alphabet Σ is a triplet P = (x, {M i | i ∈ Σ}, y) where x ∈ R k is a stochastic vector called initial distribution, each M i ∈ R k×k is a stochastic matrix, and y ∈ {0, 1} k is the final vector (each 1 in y represents an accepting state).

For any input word w ∈ Σ * with length n, P has a probability distribution of states as follows:

v f = M w x = M w n • • • M w 1 x.
The accepting probability corresponds to the probability of P being in an accepting state after reading w, which is given by

f P (w) = y T M w x. (1)
Affine finite automaton (AfA) is a generalization of PFA allowing negative transition values. Only allowing negative values in the transition matrices does not add any power (generalized PFAs are equivalent to usual ones, see [START_REF] Turakainen | Generalized automata and stochastic languages[END_REF]), but affine automata introduce also a non-linear behaviour. The automaton acts like a usual generalized probabilistic automaton until the last operation, which is a non-linear operation called a weighting operation.

A vector v ∈ R k is an affine vector if and only if its coordinates sum up to 1. A matrix M is an affine matrix if and only if all its columns are affine vectors. It is easy to verify that the multiplication of an affine matrix with an affine vector is also an affine vector, which ensures that affine automata are well defined.

A k-state AfA A over alphabet Σ is a triplet A = (x, {M i | i ∈ Σ}, F ), where x is an initial affine vector, each M i is an affine transition matrix, and F = diag(δ 1 , . . . , δ n ) is the final projection matrix, where each δ i ∈ {0, 1}.

The value computed by an affine automaton can be defined most conveniently via the following notion: Notation |v| = i |v i | stands for the usual L 1 norm. The final value of the affine automaton A of is

f A (w) = |F M w x| |M w x| . (2)
Clearly f A (w) ∈ [0, 1] for any input word w ∈ Σ * . Remark that the final value for PFAs (1) is defined as matrix product v f → y T v f , which is a linear operation on v f . On the other hand, computing final value

from v f as in (2) involves nonlinear operations v f → |F v f | |v f | such as L 1 -norm and normalization (division).

Language recognition

Given a function f : Σ * → [0, 1] computed by an automaton (stochastic or affine), there are different ways of defining the language recognized by this automaton.

A language L ⊆ Σ * is recognized by an automaton A with cutpoint λ ∈ [0, 1) if and only if L = {w ∈ Σ * | f A (w) > λ}. These languages are called cutpoint languages.

A language L ⊆ Σ * is recognized by an automaton A with exclusive cutpoint λ ∈ [0, 1] if and only if L = {w ∈ Σ * | f A (w) = λ}. These languages are called exclusive cutpoint languages.

A stronger condition is to impose that accepted and rejected words are separated by a gap: the cutpoint is said to be isolated. A language L is recognized by an automaton A with isolated cutpoint λ if and only if there exist δ > 0 such that ∀w ∈ L, f A (w) ≥ λ + δ and ∀w / ∈ L, f A (w) ≤ λ -δ. By fixing λ = 1 2 , we define language recognition with bounded error: A language L is recognized by an automaton A with bound error if and only if there exists an error bound ∈ [0, 1/2) such that ∀w ∈ L, f A (w) ≥ 1 -and ∀w / ∈ L, f A (w) ≤ . It is known that [START_REF] Hirvensalo | On the computational power of affine automata[END_REF] if a language recognized by a AfA (or PFA) with bounded error, then the error bound can be arbitrarily close to 0.

Language classes

In the case of probabilistic (resp., affine automata), the set of cut-point languages are called stochastic languages (resp., affine languages) and denoted by SL (resp., AfL). We remark that fixing the cutpoint in the interval (0, 1) does not change the classes SL and AfL [START_REF] Díaz-Caro | Affine computation and affine automaton[END_REF][START_REF] Paz | Introduction to Probabilistic Automata[END_REF].

In the case of probabilistic (resp., affine automata), the set of exclusive cutpoint languages are called exclusive stochastic languages (resp., exclusive affine languages) and denoted by SL = (resp., AfL = ). The complements of the languages in SL = (resp., AfL = ) form SL = (resp., AfL = ). (Fixing the cutpoint in the interval (0, 1) does not change the classes SL = , SL = , AfL = , and AfL = [START_REF] Díaz-Caro | Affine computation and affine automaton[END_REF][START_REF] Paz | Introduction to Probabilistic Automata[END_REF][START_REF] Yakaryılmaz | Languages recognized by nondeterministic quantum finite automata[END_REF].)

The set of languages recognized with bounded error (or isolated cutpoint, which is the same) by affine automata is denoted by BAfL.

A classical result by Rabin [START_REF] Rabin | Probabilistic automata[END_REF] shows that isolated cutpoint stochastic languages are regular. Rabin's proof essentially relies on two facts: 1) the function mapping the final vector into [0, 1] is a contraction, and 2) the state vector set is bounded. By modifying Rabin's proof, it is possible to show that also many quantum variants of stochastic automata obey the same principle [START_REF] Ambainis | Automata: From Mathematics to Applications, chap. Automata and Quantum Computing[END_REF]: bounded-error property implies the regularity of the accepted languages. In fact, E. Jeandel generalized Rabin's proof by demonstrating that the compactness of the state vector set together with the continuity of the final function are sufficient to guarantee the regularity of the accepted language if the cutpoint is isolated [START_REF] Jeandel | Topological automata[END_REF]. Affine automata do not have these properties, and in fact, they can recognize more than regular languages with bounded error [START_REF] Díaz-Caro | Affine computation and affine automaton[END_REF].

2.4 Models using the right end-marker A PFA or AfA can be defined by reading an extra letter (M $ ) for post-processing after reading the whole input. That is, the automaton reads w = w$ for a given input word w ∈ Σ * . Any such AfA (the definition of any such PFA is similar) can be formally defined as A = (x, {M i | i ∈ Σ}, F ), and the accepting probability of the input w is calculated as

f A (w) = |F M w x| |M w x| . Moreover, v f = M w x = M $ M w x.
It is known that, for any k-state PFA using the right end-marker, there is an equivalent k 2 -state PFA without using the right end-marker such that, for any input word, both automata have the same accepting probabilities [START_REF] Turakainen | Generalized automata and stochastic languages[END_REF]. Even though we do not know whether this result is valid for AfAs or not, we can still show that post-processing does not increase the computational power of AfAs in the case of recognition with cutpoint or bounded error.

Theorem 1 For a given k-state AfA A = (x, {M i | i ∈ Σ}, F ) using the endmarker and for a given cutpoint λ ∈ [0, 1], there is a 4k-state AfA A = (x , {M i | i ∈ Σ}, F ) not using the end-marker such that, for any w ∈ Σ * , both of f A (w) and f A (w) are greater than λ or equal to λ or less than λ.

Proof Let w ∈ Σ * be the given input of length n ≥ 0. Let v 0 = x and u 0 = M $ v 0 , and similarly, whenever n > 0, let

v l = M w l M w l-1 • • • M w 1 x and u l = M $ v l , where 1 ≤ l ≤ n. Remark that v f = u n . We define v 0 = x 0 =     λv 0 (1 -λ)v 0 u 0 -u 0     .
It is clear that the summation of entries are 1 and so v 0 is an affine state. For any i ∈ Σ, M i is defined as

    M i 0 I I 0 M i 0 0 M $ M i M $ M i 0 0 -M $ M i -M $ M i 0 0     .
It is easy to see that the entry summation of each column of M i is equal to 1, and so M i is an affine transition matrix. The multiplication of transition matrices with state vectors is trivial, and so we can easily obtain that

v f = v n =     λv n (1 -λ)v n u n -u n     =     λv n (1 -λ)v n v f -v f     . Let f A (w) = |F v f | |v f | = λ + d for some real number d. We can derive that |F v f | = |v f |(λ + d). We define F as     I 0 0 0 0 0 0 0 0 0 F 0 0 0 0 F     . Then, we can calculate f A (w)
as follows:

f A (w) = |λv n | + |F v f | + | -F v f | |λv n | + |(1 -λ)v n | + 2|v f | = λ|v n | + 2λ|v f | + 2d|v f | |v n | + 2|v f | = λ + d 2|v f | |v n | + 2|v f | = λ + d ,
where either d = d = 0 or both d and d have the same sign.

Corollary 1 Any language recognized by an AfA using the right end-marker with a cutpoint (or an exclusive cutpoint) can be recognized by another AfA not using the right end-marker with the same cutpoint.

Theorem 2 Any language L recognized by a k-state AfA A = (x, {M i | i ∈ Σ}, F ) using the right end-marker with error bound 1 10 can be recognized by a 3k-state AfA A = (x , {M i | i ∈ Σ}, F ) not using the right end-marker with error bound 2 10 .

Proof We use the same terminology in the previous proof. Let m = |x| and let m > 1 be a real number satisfying |M i v| < m|v| for any i ∈ Σ and for any affine vector v.

Let w ∈ Σ * be an input of length n ≥ 0. We define x =   x 5mm M $ x -5mm M $ x   and M i =   mM i I I 5m mM $ M i 0 0 -5m mM $ M i 0 0   for any i ∈ Σ. Then, we obtain v f =   m n v n 5m m n v f -5m m n v f   . We define F =   0 0 0 0 F 0 0 0 F   . We know that |v n | < m m n . The accepting probabil- ity of A on w is f A (w) = |F v f | |v f | = 10m m n |F v f | |v n | + 10m m n |v f | . If w ∈ L, then 10|F v f | ≥ 9|v f | and f A (w) ≥ 9m m n |v f | |v n | + 10m m n |v f | > 9m m n |v f | 11m m n |v f | > 0.8181. If w / ∈ L, then 10|F v f | ≤ |v f | and f A (w) ≤ m m n |v f | |v n | + 10m m n |v f | < m m n |v f | 10m m n |v f | = 0.1
Therefore, L is recognized by A with error bound 2 10 .

Logarithmic simulation

Macarie [START_REF] Macarie | Space-efficient deterministic simulation of probabilistic automata[END_REF] proved that SL = Q ⊆ L and SL Q ⊆ L. That is, the computation of any rational-valued probabilistic automaton can be simulated by an algorithm using only logarithmic space. However, this logarithmic simulation cannot be directly generalized for rational-valued affine automata due to the non-linearity of their last operation. In order to understand why, we will first reproduce the proof.

Before that, let us introduce the most important space-saving technique:

Definition 1 Notation (b mod c) stands for the least nonnegative integer a satisfying a ≡ b (mod c). If x = (x 1 , . . . , x r ) and n = (n 1 , . . . , n r ) ∈ Z r , we define x (mod n) = ((x 1 mod n 1 ), . . . , (x r mod n r )). Analogously, for any matrix

A ∈ Z k×k , we define (A (mod n)) ij = (A ij mod n).
The problem of recovering x from the residue representation ((x mod n 1 ), . . . , (x mod n r )) is practically resolved by the following well-known theorem.

Theorem 3 (The Chinese Remainder Theorem) Let n 1 , . . . , n r be pairwise coprime integers, a 1 , . . . , a r arbitrary integers, and N = n 1 • • • n r . Then there exists an integer x such that

x ≡ a 1 (mod n 1 ), . . . , x ≡ a r (mod n r ), (3)
and any two integers x 1 and x 2 satisfying (3) satisfy also

x 1 ≡ x 2 (mod N ).
Remark 1 The Chinese Remainder Theorem implies that the integer ring operations (+, •) can be implemented using the residue representation, and that the integers can be uncovered from the residue representations provided that 1) n = (n 1 , . . . , n r ) consists of pairwise coprime integers and 2) The integers stay in interval of length N -1, where

N = n 1 • • • • • n r .
Remark 2 In order to ensure that n = (n 1 , . . . , n r ) consists of pairwise coprime integers, we select numbers n i from the set of prime numbers. For the reasons that will become obvious later, we will however omit the first prime 2.

Definition 2 p r is an r-tuple p r = (3, 5, 7, . . . , p r ) consisting of r first primes by excluding 2. For this selection, a consequence of the prime number theorem is that, asymptotically, 1))r ln r .

P r = 3 • 5 • 7 • • • • • p r = 1 2 e (1+o(
Definition 3 Let p r be as before. Then for any integer x, the residual representation Res p r (x) stands for an integer vector of the residues: (x (mod 3), x (mod 5), x (mod 7), . . . , x (mod p r )).

Theorem 4 (Macarie [START_REF] Macarie | Space-efficient deterministic simulation of probabilistic automata[END_REF])

SL = Q ⊆ L Proof For a given alphabet Σ, let L ⊆ Σ * be a language in SL = Q and P = (x, {M i | i ∈ Σ}, y) be a k-state rational-valued PFA over Σ such that L = w ∈ Σ * | f P (w) = 1 2 .
We remind that, for any input word

w = w 1 • • • w n ∈ Σ * , we have f P (w) = y T M w n • • • M w 1 x. (4)
Since each M i ∈ Q k×k , there exists an integer D such that all entries of each matrix M i = DM i are integers, and (4) can be rewritten as

f P (w) = 1 D n y T M w n • • • M w 1 x f P (w)
, and the language L can be characterized as

L = {w ∈ Σ * | 2f P (w) = D n }. (5)
Since the original matrices M i are stochastic, meaning that their entries are in [0, 1], it follows that each matrix M i = DM i has integers entries in [0, D]. Moreover, f P (w) ∈ [0, 1] implies that f P (w) ∈ [0, D n ] for every input word w ∈ Σ n . As now f P (w) can be computed by multiplying k × k integer matrices, the residue representation will serve as a space-saving technique.

We will fix r later, but the description of the algorithm is as follows: For each entry p of p r = (3, 5, 7, . . . , p r ), we let M (p) i = M i mod p, and compute

(2f P (w) mod p) = y T M (p) w n • • • M (p) w 1 x. (6)
As all the products are computed modulo p, k 2 log p bits are needed to compute [START_REF] Hansel | Une démonstration simple du théorème de Skolem-Mahler-Lech[END_REF]. Likewise, (D n mod p) can be computed in space O(log p) for each coordinate p of p r . The comparison 2f P (w) ≡ D n (mod p) can be hence done in O(log p) space.

Reusing the space, the comparison can be made sequentially for each coordinate of p r , and if any comparison gives a negative outcome, we can conclude that 2P (w) = D n .

To conclude the proof, it remains to fix r so that both 2f P (w) and D n are smaller than

P r = 3 • 5 • 7 • • • • • p r .
If no congruence test is negative, then the Chinese Remainder Theorem ensures that 2f P (w) = D n . Since f P (w) ≤ D n , we need to select r so that P r > 2D n , which is equivalent to As a final remark let us note that p cn , the cn -th prime, can be generated in logarithmic space and the prime number theorem implies that O(log n) bits are enough to present p cn , since c is a constant.

To extend the above theorem to cover SL Q as well, auxiliary results are used.

Lemma 1 (Macarie [START_REF] Macarie | Space-efficient deterministic simulation of probabilistic automata[END_REF]) If N is an odd integer and x, y ∈ [0, N -1] are also integers, then x ≥ y iff x -y has the same parity as ((x -y) mod N ).

Proof As x, y ∈ [0, N -1], it follows that (x -y mod N ) = x -y if x ≥ y N + x -y if x < y,
which shows that the parity changes in the latter case since N is odd. The problem of using the above lemma is that, in modular computing, numbers x and y are usually known only by their residue representations Res p r (x) and Res p r (y), and it is not straightforward how to compute the parity from the modular representation in logarithmic space. Macarie solved this problem not only for parity but also for a more general modulus (not necessarily equal to 2).

Lemma 2 (Claim modified from [START_REF] Macarie | Space-efficient deterministic simulation of probabilistic automata[END_REF]) For any integer x and modulus p r = (3, 5, 7, . . . , p r ), there is a deterministic algorithm that given Res p r (x) and M ∈ Z as input, produces the output x (mod M ) in space O(log p r + log M ).

As a corollary of the previous lemmata, Macarie presented a conclusion which implies the logarithmic space simulation of rational stochastic automata.

Lemma 3 (Claim modified from [START_REF] Macarie | Space-efficient deterministic simulation of probabilistic automata[END_REF]) Let p r = (3, 5, 7, . . . , p r ) and

P r = 3 • 5 • 7 • • • • • p r .
Given the residue representations of integers x, y ∈ [0, P r -1], the decisions x > y, x = y or x < y can be made in O(log p r ) space.

Proof The equality test can be done as in the proof of Theorem 4, testing the congruence sequentially for each prime. Testing x ≥ y is possible by Lemmata 1 and 2: First compute Res p r (z) = Res p r (x) -Res p r (y) (mod p r ), then compute the parities of x, y, z using Lemma 2 with M = 2.

The following theorem is a straightforward corollary from the above:

Theorem 5 SL Q ⊆ L.
When attempting to prove an analogous result to affine automata, there is at least one obstacle: computing the final value includes the absolute values, but the absolute value is not even a well-defined operation in the modular arithmetic. For example, 2 ≡ -3 (mod 5), but |2| ≡ |-3| (mod 5). This is actually another way to point out that, in the finite fields, there is no order relation compatible with the algebraic structure.

Hence for affine automata with matrix entries of both signs, another approach must be adopted. One obvious approach is to present an integer n as a pair (|n| , sgn(n)), and apply modular arithmetic to |n|. The signum function and the absolute value indeed behave smoothly with respect to the product, but not with the sum, which is a major problem with this approach, since to decide the sign of the sum requires a comparison of the absolute values, which seems impossible without having the whole residue representation. The latter, in its turn seems to cost too much space resources to fit the simulation in logarithmic space.

Hence the logspace simulation for automata with matrices having both positive and negative entries seems to need another approach. It turns out that we can use that introduced by Turakainen already in 1969 [START_REF] Turakainen | On probabilistic automata and their generalizations[END_REF][START_REF] Turakainen | Generalized automata and stochastic languages[END_REF].

Theorem 6 AfL Q ⊆ L.
Proof For a given alphabet Σ, let L ∈ Σ * be a language in AfL Q and A = (x, {M i | i ∈ Σ}, F ) be a k-state rational-valued AfA over Σ such that

L = w ∈ Σ * | f A (w) > 1 2 .
For each M i ∈ Q k×k , we define a new matrix as For the next step, we introduce an (k + 2) × (k + 2) matrix E, whose each element is 1. It is then clear that E n = (k + 2) n-1 E and B i E = EB i = 0. Now we define

B i =   0 0 T 0 c i M i 0 e i d T i 0   ,
C i = B i + mE,
where m ∈ Z is selected large enough to ensure the nonnegativity of the matrix entries of each C i . It follows that

C w = B w + m |w| (k + 2) |w|-1 E,
and

C w x = B w x + m |w| (k + 2) |w|-1 Ex .
Similarly,

F C w x = F B w x + m |w| (k + 2) |w|-1 F Ex . Now |F M w x| |M w x| = F B w x |B w x| = F C w x -m |w| (k + 2) |w|-1 F Ex C w x -m |w| (k + 2) |w|-1 Ex
which can further be modified by expanding the denominators away: For an integer g large enough all matrices D i = gC i will be integer matrices and the former equation becomes

|F M w x| |M w x| = F B w x |B w x| = F D w x -m |w| (k + 2) |w|-1 g |w| F Ex D w x -m |w| (k + 2) |w|-1 g |w| Ex . ( 7
)
Hence the inequality

|F M w x| |M w x| ≥ 1 2 is equivalent to 2 F D w x -m |w| (k + 2) |w|-1 g |w| F Ex ≥ D w x -m |w| (k + 2) |w|-1 g |w| Ex . (8)
In order to verify inequality [START_REF] Hirvensalo | On the computational power of affine automata[END_REF] in logarithmic space, it is sufficient to demonstrate that the residue representations of both sides can be obtained in logarithmic space.

For that end, the residue representation of vector a = F D w x ∈ R k+2 can be obtained in logarithmic space as in the proof of Theorem 4.

Trivially, the residue representation of b = m |w| (k + 2) |w|-1 g |w|+1 F Ex ∈ R k+2 can be found in logarithmic space, as well. In order to compute the residue representation of

|a -b| = |a 1 -b 1 | + • • • + |a k -b k | ,
it is sufficient to decide whether a i ≥ b i holds. As the residue representations for each a i and b i is known, all the decisions can be made in logspace, according to Lemma 3. The same conclusion can be made for the right hand side of (8).

A Non-affine Language

As we saw in the previous section, AfL Q ⊆ L, and hence languages beyond L, are good candidates for non-affine languages. 1 In this section, we will however demonstrate that the border of non-affinity may lie considerably lower: There are languages in L which are not affine.

In an earlier work [START_REF] Hirvensalo | On the computational power of affine automata[END_REF], we applied the method of Turakainen [START_REF] Turakainen | On nonstochastic languages and homomorphic images of stochastic languages[END_REF] to show that there are languages in L which however are not contained in BAfL. Here we will extend the previous result to show that those languages are not contained even in AfL A .

Definition 4 (Lower density)

Let L ⊆ a * be a unary language. We call lower density of L the limit

dens(L) = lim inf n→∞ {a k ∈ L | k ≤ n} n + 1 . Definition 5 (Uniformly distributed sequence) Let (x n ) be a sequence of vectors in R k and I = [a 1 , b 1 )ו • •×[a k , b k ) be an interval in R k . We define C(I, n) as C(I, n) = |{x i mod 1 ∈ I | 1 ≤ i ≤ n}|.
We say that (x n ) is uniformly distributed mod 1 if and only if for any I of such type,

lim n→∞ C(I, n) n = (b 1 -a 1 ) • • • (b k -a k ).
Theorem 7 If L ⊆ a * satisfies the following conditions:

1. dens(L) = 0. 2. For all N ∈ N, there exists r ∈ N and an ascending sequence (m i ) ∈ N such that a r+m i N ⊆ L and for any irrational number α, the sequence ((r + m i N )α) is uniformly distributed mod 1.

Then L is not in AfL A .

Proof Let's assume for contradiction that L ∈ AfL A . Then there exists an AfA A with s states, matrix M and initial vector v such that the acceptance value of A is

f A (a n ) = |F M n v| |M n v| . ( 9
)
Without loss of generality, we can assume that the cutpoint equals to 1 2 , and hence w ∈ L ⇔ f A (w) > 1 2 . Using the Jordan decomposition M = P JP -1 , one has M n = P J n P -1 . So the coordinates of M n v have the form

(M n v) j = s k=1 p jk (n)λ n k , (10)
where λ k are the eigenvalues of M and p jk are polynomials of degree less than the degree of the corresponding eigenvalue. For short, we denote

F (n) = f A (a n ), and let λ k = |λ k | e 2iπθ k .
When studying expression ( 9), we can assume without loss of generality, that all numbers θ k are irrational. In fact, replacing matrix M with αM , where α = 0 does not change [START_REF] Hirvensalo | On the computational power of affine automata[END_REF], since

|F (αM ) n v| |(αM ) n v| = |α n F M n v| |α n M n v| = |F M n v| |M n v| .
Selecting now α = e 2πiθ (where θ ∈ R) implies that the eigenvalues of M are λ k e 2iπ(θ k +θ) . The field extension Q(θ 1 , . . . , θ s ) is finite, and hence there is always an irrational number θ / ∈ Q(θ 1 , . . . , θ s ). It follows directly that all numbers θ k + θ are irrational. Hence we can assume that all the numbers θ k are irrational in the first place. 2By restricting to an arithmetic progression n = r + mN (m ∈ N) we can also assume that no λ i /λ j is a root of unity for i = j. In fact, selecting

N = lcm{ord(λ i /λ j ) | i = j and λ i /λ j is a root of unity}, (11) equation (10) becomes (
M r+mN v) j = s k=1 p jk (r + mN )λ r k (λ k ) N m = s k=1 q jk (m)µ m k , (12)
where {µ 1 , . . . , µ s } are the distinct elements of set {λ N 1 , . . . , λ N s } Now for i = j µ i /µ j cannot be a root of unity, since (µ i /µ j ) t = 1 would imply (λ i /λ j ) N t = 1, which in turn implies (λ i /λ j ) N = 1 and hence µ i = λ N i = λ N j = µ j , which contradicts the assumption µ i = µ j .

We can now write the acceptance condition f A (a n ) > 1 2 equivalently as

f A (a n ) > 1 2 ⇔ 2 |P M n v| > |M n v| ⇔ 2 j∈E a |(M n v) j | > j∈E |(M n v) j | ⇔ j∈E a |(M n v) j | - j∈E a |(M n v) j | g(n) > 0,
Where E is the set of states of A, E a ⊆ E its set of accepting states, and E a the complement of E a . According to [START_REF] Ibrahimov | Error-free affine, unitary, and probabilistic OBDDs[END_REF]

, g(n) := j∈E a |(M n v) j |-j∈E a |(M n v) j |
consists of combinations of absolute values of linear combination of functions of type n d λ n .

We say that n

d 1 λ n 1 is of larger order than n d 2 λ n 2 , if |λ 1 | > |λ 2 |; and in the case |λ 1 | = |λ 2 |, if d 1 > d 2 . If |λ 1 | = |λ 2 |, we say that n d λ n
1 and n d λ n 2 and of the same order. It is clear that if term t 1 (n) is of larger order than t 2 (n), then

lim n→∞ t 2 (n) t 1 (n) = 0.
We can organize the terms in expression [START_REF] Ibrahimov | Error-free affine, unitary, and probabilistic OBDDs[END_REF] as

(M n v) j = s k=1 p jk (n)λ n k = Λ (N ) j (n) + Λ (N -1) j (n) + • • • + Λ (0) j (n), (13)
where each Λ (m) j

(n) consists of terms with equal order multiplier:

Λ (m) j (n) = m j k=1 c mk n d m λ mk n = n d m λ n m m j k=1 c mk e 2πinθ mk (14)
(for notational simplicity, we mostly omit the dependency on j in the right hand side of ( 14)). Here λ m ∈ R + is the common absolute value of all eigenvalues λ mk = λ m e 2πiθ mk , and expression ( 13) is organized in descending order:

Λ (N ) j
is the sum of terms of the highest order multiplier, Λ

(N -1) j contains the terms of the second highest order multiplier, etc. We say that Λ

(k 2 ) j is lower than Λ (k 1 ) j if k 2 < k 1
We will then fix a representation

g(n) = j∈E a s k=1 p jk (n)λ n k - j∈E a s k=1 p jk (n)λ n k = j∈E a |A j (n) + B j (n) + C j (n)| - j∈E a |A j (n) + B j (n) + C j (n)| , (15)
where A j (n)+B j (n)+C j (n) is a grouping of all Λ-terms in (13) defined as follows:

1. A j (n) = m k=0 Λ (N -k) j (n), where m ∈ [-1, N ] ∩ Z is chosen as the maximal number so that A = j∈E a |A j (n)| - j∈E a |A j (n)| (16)
is a constant function N → R. Such an m exists, since for m = -1, the sum is regarded empty and A j (n) = 0, but for m = N , all Λ-terms are included, and then ( 16) becomes f A (a n ), which is not constant (otherwise condition 1 or 2 of the theorem would be false). 2. B j (n) consists a single Λ-term immediately lower than those in A j (n)4, and 3. C j (n) contains the rest of the Λ-terms, lower than B j (n)

Lemma 4 If A = 0, then ∀z ∈ C, |A + z| = |A| + Re |A| A z + O( z 2 A ).
Proof Denote z = x + iy. Because |Re z| ≤ |z|, we have

|1 + z| = |1 + x + iy| = (1 + x) 2 + y 2 = 1 + 2 Re z + |z| 2 = 1 + Re z + O(z 2 ). Now |A + z| = |A| 1 + z A = |A| 1 + Re z A + O( z A 2 ) = |A| + Re |A| A z + O( z 2 A ).
We choose λ ∈ R + and d so that the highest Λ-term in B(n) is of order n d λ n and define

A j (n) = n -d λ -n A j (n), B j (n) = n -d λ -n B j (n), g (n) = g(n)n -d λ -n .
Then clearly g (n) > 0 if and only if g(n) > 0 and each B j (n) remains bounded as n → ∞. To simplify the notations, we omit the primes and recycle the notations to have a new version of g(n) of ( 15) where A j -terms may tend to infinity but B j -terms remain bounded.

Recall that we may assume (by restricting to a arithmetic progression) that no λ i /λ j is a root of unity. By Skolem-Mahler-Lech theorem [START_REF] Hansel | Une démonstration simple du théorème de Skolem-Mahler-Lech[END_REF], this implies that functions A j can have only a finite number of zeros, and in the continuation we assume that n is chosen so large that no function A j becomes zero. Furthermore, by the main theorem of [START_REF] Evertse | On sums of S-units and linear recurrences[END_REF], then |A j (n)| = Ω(n d λ n-) for each > 0. 3 As each B j remains bounded, we find that B 2 j /A j tend to zero as n → ∞, and hence by Lemma 4, defining

g 1 (n) = j∈E a |A j (n)| + Re( |A j (n)| A j (n) B j (n)) - j∈E a |A j (n)| + Re( |A j (n)| A j (n) B j (n)) = j∈E a |A j (n)| - j∈E a |A j (n)| h(n) + j∈E a Re( |A j (n)| A j (n) B j (n)) + j∈E a Re( |A j (n)| A j (n) B j (n))
we have a function g 1 (n) with the property g 1 (n) -g(n) → 0 (C-terms are lower than B-terms, so they can be dropped without violating this property), when n → ∞. Also by the construction it is clear that h(n) = C • n d λ n , where C is a constant, and by the conditions of the theorem, this is possible only if C = 0. Notice tat g 1 (n) is not a constant function by construction. Also, each B j is a linear combination of functions of form e 2πiθ k n , each θ k can be assumed irrational, and ||A j (n)| A j (n) = 1|, so we can conclude that g 1 (n) is a continuous function formed of terms of form ce iθ k n and of ratios |A j | /A j . In these terms, however the behaviour is asymptotically determined by the highest Λ-terms, so the conclusion remains even if we drop the lower terms.

By assumption, for all k, the sequence (r + mN )θ k is uniformly distributed modulo 1. It follows that the values e 2iπ(r+mN )θ k are dense in the unit circle. If for some m, g 1 (r + mN ) < 0, then g 1 (r + N m) ≤ -ε for some > 0. Then, because of the density argument, there are arbitrarily large values of i for which g 1 (r+m i N ) ≤ 0 contradicting condition 2 of the statement. Hence g 1 (r+mN ) ≥ 0 for each m large enough. As g 1 is not a constant, there must be some m 0 so that g 1 (m 0 ) ≥ > 0.

Next, let R(x 1 , . . . , x s ) be a function obtained from g 1 by replacing each occurrence of e iθ k n by a variable x k , hence each x k will assume its value in the unit circle. Moreover, by the assumptions of the theorem, the values of x k will be uniformly distributed in the unit circle.

Note that g 1 (n) = R((e 2iπ(r+m i N )θ k ) k∈A ). Then, because the sequences ((r + m i N )θ k ) i are uniformly distributed modulo 1, it follows that any value obtained by the function R((e 2iπy k ) k∈A ) can be approximated by some g 1 (r + m i M ) with arbitrary precision. The function R is continuous, therefore there exists an interval

I = (x 1 , y 1 , ...) = ((x k , y k )) k∈A on which R((x k )) > ε 2 . So, if m i is large enough and satisfies ((r + m i N )θ 1 mod 1, . . . ) = ((r + m i M )θ k mod 1) k∈A ∈ I, then g 1 (r + m i N ) > ε
2 , which implies f A (r + m i N ) > 0 and hence a r+m i N ∈ L. Now we just have to prove that the sequence (r + m i N ) is "dense enough" to have dens(L) > 0, contradicting again condition 1. Then, because of uniform distribution imposed by condition 2, one has

d = lim i→∞ C(I, r + mN ) r + mN = k∈A (y k -x k )
And so for i large enough, C(I,r+m i N )

r+m i N ≥ d 2 , with a h+n i Q ∈ L, implying dens(L) > 0, a contradiction.
Corollary 2 Let P be any polynomial with nonnegative coefficients and deg(P ) > 2.

The language {a P (n) | n ∈ N} is not in AfL A .

Corollary 3 The language {a p | p prime} is not in AfL A .

Proof (Proof of Corollary 2 and Corollary 3.) Turakainen proved that these two languages satisfies the two conditions of Theorem 7 [START_REF] Turakainen | On nonstochastic languages and homomorphic images of stochastic languages[END_REF]. Therefore, these two languages not in AfL A .

Generalized Affine Automata

In this section, we show that using arbitrary real state vector and transition matrices does not increase the computational power of AfAs. A generalized affine finite automaton (GAfA) is a 3-tuple G = (x, {M i |i ∈ Σ}, F ), where, different from an AfA, {M i |i ∈ Σ} is the set of real-valued transition matrices without any restriction on the column summations and x is the real-valued initial state vector. The final affine state of G on the given input w ∈ Σ * for some n ≥ 0 is

v f = M $ M w x = M $ M w n • • • M w 1 x,
where M ε = I. It must be guaranteed that at least one entry of v f is non-zero for any possible input. The accepting probability of G on w is calculated in the same way of an AfA:

f G (w) = |F v f | |v f | .
We start with proving that GAfAs with cutpoint define the same class of languages as AfAs with cutpoint.

Theorem 8 Any language L recognized by a k-state GAfA G = (x, {M i |i ∈ Σ}, F ) with cutpoint λ ∈ [0, 1) is recognized by a (k+2)-state AfA A = (x , {M i |i ∈ Σ}, F ) with cutpoint λ.

Proof Let t 0 = 1 -k i=1 x i . We define x =   x λt 0 (1 -λ)t 0   .
For letter i ∈ Σ, let c j be the j-th column summation of M i and d j = 1 -c j . We define M i based on M i :

      M i 0 λd 1 λd 2 • • • λd k (1 -λ)d 1 (1 -λ)d 2 • • • (1 -λ)d k 1 0 0 1       ,
where each column summation is 1. Then, we can calculate v f , for a given input

w ∈ Σ * , as   v f λt f (1 -λ)t f   , where t f = 1 -k i=1 (v f ) i . We define F =   F 0 0 0 1 0 0 0 0   . Let t = |t f |. We have f G (w) = |F v f | |v f | = λ + d for some real number d. Then the accepting probability of w by A is f A (w) = |F v f | |v f | = |F v f | + λt |v f | + t = (λ + d)|v f | + λt |v f | + t = λ + d |v f | + t .
Thus, both of f G (w) and f A (w) are greater than λ or equal to λ or less than λ.

Remark that when the cutpoint is 0, then the constructed AfA can indeed use one state less in the above proof.

We can obtain the same result for bounded error case when focusing on the rational numbers. First we show that there is no difference between using rational numbers and integers.

Lemma 5 For any given GAfA G 1 = (x, {M i |i ∈ Σ}, F ) with rational number components, there is a GAfA G 2 with integer number components such that they have the same accepting probability on any input string.

Proof Let z be sufficiently big integer such that zM i for each i ∈ Σ and zx contains only integers. Then, G 2 is defined as (zx, {zM i |i ∈ Σ}, F ). Due to linearity, if the final vector of G 1 on a given input w ∈ Σ * is v f , then, the final vector of G 2 on a any given input is z | w|+1 v f . Thus, f G 1 (w) = f G 2 (w). since we have two copies of v f where one is multiplied by m 2 and the other is multiplied by -m 2 . For any w / ∈ L, it is straightforward that

f A (w) = 2m 2 a 2m 2 a + 2m 2 r + 1 ≤ 2m 2 a 2m 2 a + 2m 2 r = a a + r = f G (w).
In the remaining part, we focus on only the members: w ∈ L and a a+r = 1 2 + c for some 1 2 ≥ c ≥ 1 m . From the equation of f G (w), we can obtain a + r = 2a 1+2c and we can substitute a + r with 2a 1+2c in equation ( 17):

f A (w) = 2m 2 a 4m 2 a 1+2c + 1 = (1 + 2c)2m 2 a 4m 2 a + 2c + 1 = (1 + 2c)(2m 2 a + c + 1 2 -c - 1
2 ) 4m 2 a + 2c + 1 .

After simplification, we have We know that a ≥ 1 (a = 0 for w ∈ L) and c ≤ 1 2 . Thus, we can easily follow that (2c + 1) 2 8m 2 a + 4c + 2

< 4 8m 2 = 1 2m 2 .
Hence, we can bound the accepting probability of any member from below as

f A (w) > 1 2 + 1 m - 1 2m 2 = 1 2 + 2m -1 2m 2 .
Since there is a constant gap for every member, we conclude that A recognizes L with bounded error.

In [START_REF] Villagra | Language recognition power and succintness of affine automata[END_REF], it was shown that one-sided error (either all members are accepted with probability 1 or all non-members are accepted with probability 0) versions of BAfL are the identical if they are defined by AfAs with rational number components or by AfAs with integer components. By using the above results, we can follow that the same result is valid also for (two-sided error class) BAfL.

log 1 2 + ( 1 +

 21 o(1))r ln r > log 2 + n log D. This inequality is clearly satisfied with r = n for large enough n, and for each n ≥ 1 by choosing r = c • n, where c is a positive constant (depending on D).



  where c i , d i , and e i are chosen so that the column and row sums of B i are zero. We define as the new initial state. For the projection matrix F , we define an extension F = It is straightforward to see that B w x = |M w x| as well as F B w x = |F M w x|.

Theorem 9 1  1  m 2 v f -m 2 v f 1 

 9111 Any language L recognized by a k-state GAfA G = (x, {M i |i ∈ Σ}, F ) with bounded error can be recognized by a (2k+1)-state AfA A = (x , {M i |i ∈ Σ}, F ) with bounded error, where both automata have only integer components. Proof Let 1 2 -1 m for m ≥ 2 be the error bound and w ∈ Σ * be the given input. . For each i ∈ Σ, we define M i = , where r i and r $ are row vectors guaranteeing that the entry summation of each corresponding column is 1. The final state vector of A on w can be easily obtained as v f =  , where v f is the final state vector of G on w. a = |F v f | and r = |v f | -a. Remark that a and r can be only non-negative integers. The accepting probability of A on w is f A (w) = 2m 2 a 2m 2 a + 2m 2 r + 1 (17)

It is known that L PSPACE, so it is clear that PSPACE-complete languages are not in AfL Q .

Note that the new matrix obtained may not be affine, so it would be wrong to assume that all AfAs have admit an equivalent one with only irrational eigenvalues. However, this does not affect this proof, since we do not require the new matrix to be affine, we only study the values that the fraction|P (αM ) n v| |(αM ) n v| = |P M n v||M n v| take.

Acknowledgments

We thank to the anonymous reviewers for their helpful corrections and comments.

M. Hirvensalo was partially supported by the Väisälä Foundation. E. Moutot was partially supported by ANR project CoCoGro (ANR-16-CE40-0005). A. Yakaryılmaz was partially supported by the ERDF project 1.1.1.5/18/A/020, Akadēmiskā personāla atjaunotne un kompetenču pilnveide Latvijas Universitātē līg Nr. 8.2.2.0/18/A/010 LU re ' gistrācijas Nr. ESS2018/289, and ERDF project Nr. 1.1.1.5/19/A/005 "Quantum computers with constant memory".

Corollary 4 BAfL


Keywords: control system, eco-feedback, prototyping, sketching

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  Designing the engaging Energy-Box Bridging the gap between energy control systems and users' energy awareness Emmanuel Dubois, Fabio Pittarello To cite this version: Emmanuel Dubois, Fabio Pittarello. Designing the engaging Energy-Box Bridging the gap between energy control systems and users' energy awareness. Workshop on Smart and Sustainable Cities, part of the SmartWorld 2017 Congress (WSSC/SmartWorld 2017), Aug 2017, San Francisco, United States. pp.43-50. hal-03658182