domain: sde chim

text:
Minghui Hong

Xuerui Yang

Xuewei Zhang

Yuefei Ji

Lei Zhou
email: zhoulei@ecust.edu.cn

Guangli Xiu

Zhigang Ni

Claire Richard

Aqueous photodegradation of the benzophenone fungicide metrafenone: Carbon-bromine bond cleavage mechanism

Keywords: Metrafenone, Photolysis, Debromination, Homolytic cleavage, Bond dissociation energy

Metrafenone (MF), as a new type of benzophenone fungicide, has been widely used in agriculture and is persistent in the environment. Understanding its photochemical fate is essential for the comprehensive evaluation of its ecological risk.

In the present work, we reported a detailed study on the photochemical transformation of MF in aqueous solution under irradiation (at λ > 290 nm using a high pressure mercury lamp). MF was relatively photo-reactive showing a low polychromatic quantum yield of photolysis (1.06×10 -4 ) counterbalanced by a significant light absorption above 290 nm. A series of photoproducts were identified by high resolution mass spectrometry (HR-MS) analysis, and three different pathways, including oxidation of the methyl group, debromination and replacement of bromine by hydroxyl group were proposed. Among them, debromination was identified as the dominating process that could be achieved via homolytic C-Br bond cleavage from singlet and triplet MF, as confirmed by laser flash photolysis (LFP) experiments and density functional theory (DFT) calculations. Toxicity assessment revealed that photochemical degradation reduced the ecotoxicity of MF efficiently. Nitrate ions and humic acid promoted the MF photolysis, while bicarbonate exhibited no effect.

Results obtained in this work would increase our understanding on the environmental fate of MF in sunlit surface waters.

Introduction

Fungicides are widely applied to control fungal pathogens on growing crops to improve yields [START_REF] Cintia | Evaluation of Fusarium mycotoxins and fungicide residues in barley grain produced in Uruguay[END_REF][START_REF] Lin | The distributions of three fungicides in vegetables and their potential health risks in Zhejiang, China: A 3-year study (2015-2017[END_REF][START_REF] Miller | Exploring fungicides and sealants for management of Cytospora plurivora infections in western Colorado peach production systems[END_REF][START_REF] Nermine | Determination of fungicides' residues and their degradation kinetics in orange tree fruits using liquid chromatography -Tandem mass spectrometry coupled with QuEChERS method[END_REF][START_REF] Patrick | Environmentally relevant fungicide levels modify fungal community composition and interactions but not functioning[END_REF][START_REF] Regueiro | Toxicity evaluation of new agricultural fungicides in primary cultured cortical neurons[END_REF]. Metrafenone (MF) is a new type of benzophenone derivative fungicide, which is a brand-new active ingredient in the field of phytochemical protection (structure in Fig. 1). Studies have demonstrated that MF is effective in reducing spore germination, preventing appressorium formation, and inhibiting powdery mildew infestation [START_REF] Franck | Investigating the mode of action of natural phytotoxins[END_REF][START_REF] Krystina | Metrafenone: studies on the mode of action of a novel cereal powdery mildew fungicide[END_REF].

Hence, it has been widely used to prevent and treat powdery mildew in crops such as wheat and grapes. However, repeated application of fungicides over a longtime period would inevitably lead to their accumulation in the environment. These toxic compounds can leach out of the soil, and contaminate surface water and groundwater, inducing potential health risks to residents [START_REF] Freya | Association of pesticide exposure with neurologic dysfunction and disease[END_REF][START_REF] Pimmata | Comparative bioremediation of carbofuran contaminated soil by natural attenuation, bioaugmentation and biostimulation[END_REF]. In agricultural production, MF is normally pre-applied and there is a tendency to increase the amount applied as microbial resistance increases, resulting in its persistence in the environment [START_REF] Kunova | Metrafenone resistance in a population of Erysiphe necator in northern Italy[END_REF][START_REF] Leroux | Fungicide resistance status in French populations of the wheat eyespot fungi Oculimacula acuformis and Oculimacula yallundae[END_REF]. Residual MF has been widely detected in grapes, juices, and wines after its application to vines, which eventually affect the human health [START_REF] Briz-Cid | Effect of Two Anti-Fungal Treatments (Metrafenone and Boscalid Plus Kresoxim-methyl) Applied to Vines on the Color and Phenol Profile of Different Red Wines[END_REF][START_REF] Cabras | Pesticide residues in grapes, wine, and their processing products[END_REF][START_REF] Noelia | Fungicide residues affect the sensory properties and flavonoid composition of red wine[END_REF][START_REF] Noguerol | Influence of tebuconazole residues on the aroma composition of Mencia red wines[END_REF]. [START_REF] Regueiro | Toxicity evaluation of new agricultural fungicides in primary cultured cortical neurons[END_REF] once evaluated the toxicity of MF on cortical neurons in mice, they reported that MF induces depolarization of mitochondrial membranes, reduces cellular activity, and exhibits toxic effects after 7 days of exposure [START_REF] Regueiro | Toxicity evaluation of new agricultural fungicides in primary cultured cortical neurons[END_REF].

Fig. 1.

In order to accurately assess the ecological risks of fungicides, it is essential to investigate their transformation and fate in the environment [START_REF] Baćmaga | The biochemical activity of soil contaminated with fungicides[END_REF].

Photochemical transformation is among the most important processes that affect the fate of organic compounds in surface water [START_REF] Ge | Light-Source-Dependent Effects of Main Water Constituent on Photodegradation of Phenicol Antibiotics: Mechanism and Kinetics[END_REF][START_REF] Yong | Indirect Photodegradation of Amine Drugs in Aqueous Solution under Simulated Sunlight[END_REF]. Direct photolysis of fungicides is caused by direct absorption of photons, while indirect photolysis is triggered by the absorption of light by photosensitizers. Active components in water (such as dissolved organic matter and ions) can interact with organic pollutants through energy transfer of excited substances or through photoproduction of reactive transient species, which affects the persistence and fate of fungicides in the environment [START_REF] Avetta | Phototransformation pathways of the fungicide dimethomorph ((E,Z) 4-[3-(4-chlorophenyl)-3-(3,4-dimethoxyphenyl)-1-oxo-2-propenyl]morpholine ), relevant to sunlit surface waters[END_REF]Fréneau et al., 2019;[START_REF] Jennifer | Photodegradation of Fludioxonil and Other Pyrroles: The Importance of Indirect Photodegradation for Understanding Environmental Fate and Photoproduct Formation[END_REF][START_REF] Liu | Photodegradation of the azole fungicide climbazole by ultraviolet irradiation under different conditions: Kinetics, mechanism and toxicity evaluation[END_REF]. As shown in Fig. 1, the molecular structure of MF contains chromophores such as benzene rings and carbonyl groups which can absorb ultraviolet or visible light, indicating that direct photolysis of MF is likely upon irradiation [START_REF] Turro | Modern Molecular Photochemistry[END_REF]. [START_REF] López-Fernández | Modelling the isothermal degradation kinetics of metrafenone and mepanipyrim in a grape juice analog[END_REF] confirmed that photolysis is an effective route for the removal of MF from grape juice [START_REF] López-Fernández | Modelling the isothermal degradation kinetics of metrafenone and mepanipyrim in a grape juice analog[END_REF]. However, to our knowledge, specific studies on the photolytic fate of MF in natural surface waters is quite limited.

In the current study, the photolytic behavior of MF in aqueous solution is investigated systematically, the main degradation pathways and mechanisms are proposed. For this, steady-state irradiations are performed to measure the quantum yield of MF photolysis, the second order rate constant

Materials and methods

Materials

Metrafenone (MF, 99.7%) and humic acid (HA, 99%) were purchased from Sigma-Aldrich (St. Louis, MO, USA). Phosphoric acid with chromatographic purity and sorbic acid (99.8%) were obtained from Aladdin (Shanghai, China). Deuterium oxide (D 2 O, 99.9%) was obtained from Energy Chemical (Shanghai, China).

Isopropanol (IPA, 99.9%) was obtained from Thermo Fisher Scientific (Waltham, USA). Sodium nitrate (≥98.5%) and sodium bicarbonate (≥99.5%) were purchased from Sinopharm Chemical Reagent Co., Ltd and Richjoint, respectively. HPLC grade acetonitrile (ACN, 99%) was obtained from J&K Scientific Ltd (Shanghai, China).

Other reagents were at least of analytical grade and used as received.

Photolysis experiments

Steady-state photolysis experiments

Steady-state photolysis experiments were conducted in a device equipped with a 500 W high pressure mercury lamp (Beijing Zhongjiao Jinyuan Technology Co., Ltd, China) surrounded by a 290 nm cut-off filter as the light source. The mercury lamp was turned on preliminarily for 30 min. After stabilization, 25 mL solution to be irradiated containing target compounds (20 μM) were pipetted into a Pyrex glass reactor. During the irradiation process, Pyrex glass reactors were transferred to the water-cooled immersion well in the device to maintain the temperature at 20 ○ C. Dark controls were also performed under the same conditions. During the irradiated process, 0.5 mL aliquot of the solution to be irradiated was withdrawn at predetermined time intervals for analysis. To ensure the reliability of the data, all experiments were carried out at least in duplicates, and the error bars in the figures represent the standard deviation (SD).

Laser flash photolysis experiments

Laser flash photolysis (LFP) experiments were carried out using a Quanta Ray LAB-150-10 Nd:YAG laser at an excitation wavelength of 266 nm to investigate the absorbance and disappearance of the transient species in ACN solutions of MF.

Solutions were purged with purified N 2 prior to the experiments to evaluate the effect of O 2 (triplet quencher) on the lifetime of transient species. Different time windows were used to fit the decay of the transient species of MF.

Analytical methods

The concentration of MF was measured using a high performance liquid chromatograph (HPLC, Waters) equipped with a separation module (Waters 2695) and a photodiode array detector (Waters 2998). Detailed operation conditions are provided in Text. S1.

The transformation products were identified by a high resolution mass spectrometry (HR-MS), which was coupled to a HPLC Waters Alliance system for chemical separation. The details of HR-MS analysis are given in Text. S2.

The bromine ion concentration was measured by ion chromatograph and the details are provided in Text. S3.

The details for electron paramagnetic resonance spectroscopy analysis (EPR) are illustrated in Text. S4.

Quantum yield

The polychromatic quantum yields of MF (Φ MF ) direct photolysis in aqueous solutions were determined using p-nitroanisole (PNA)/pyridine (pyr) as an actinometer according to Eq. ( 1).

λ λ λ,MF MF MF PNA PNA λ λ λ,PNA (L ε ) (L ε ) k k        (1)
where k MF and k PNA were the observed photolysis rate constant of MF and PNA, respectively. ε λ (mol -1 cm -1 ) represented the molar absorptivity and L λ (cm -2 s -1 ) was the incident light intensity.

( λ ε λ,MF ) λ and ( λ ε λ,P A ) λ were the light absorption for MF and actinometer, respectively. Φ PNA was the quantum yield of PNA and was calculated according to Φ PNA = 0.29[pyr] + 0.00029 ( aszakovits et al., 2016). The ε λ values of aqueous MF and PNA solutions were measured using a UV-Vis spectrophotometer and specific pathlength quartz cells.

DFT calculations

In this study all the density functional theory (DFT) calculations were conducted by using Gaussian 16 package with hybrid density functional B3LYP method. The details for DFT calculation were illustrated in Text. S6.

Ecotoxicity assessment

The aquatic toxicity data was an essential judgment basis for environmental hazard classification and ecological risk assessment of chemicals [START_REF] Cash | Prediction of chemical toxicity to aquatic organisms: ECOSAR vs. Microtox® Assay[END_REF]. In the present study, the ecotoxicity of MF and photoproducts was evaluated by the Ecological Structure Activity Relationship (ECOSAR 2.0). Three classical aquatic organisms (green algae, daphnia and fish) were assessed for acute toxicity (half-lethal and half-effect concentration (LC50, EC50)) and chronic toxicity values (Chv) [START_REF] Wang | Distribution, transformation and toxicity evaluation of 2,6-Di-tert-butyl-hydroxytotulene in aquatic environment[END_REF].

Results and discussion

Direct photolysis of MF

The UV-Vis absorption spectrum of aqueous MF exhibits a broad absorption band ranging from 260 to 350 nm (λ max = 292 nm, ε 292 = 7750 mol -1 cm -1 ) thus well overlapping with the emission spectrum of the mercury lamp (Fig. 2) and of the solar spectrum, implying that direct photodegradation can potentially take place for MF in natural conditions. The absorption band of MF is mainly due to the presence of chromophores such as benzene and ketone groups that are capable of absorbing UV-Vis light. In the present study, the absorption band corresponds to the low energy π→π* excitation due to the presence of the benzene chromophore rather than the ketone chromophore (n→π* excitation) [START_REF] Turro | Modern Molecular Photochemistry[END_REF]. This could be further confirmed by TD-DFT calculations, as shown in Fig. S1, the broad band ranging from 260 nm to 350 nm is attributed to the S1 to S5 absorptions, among which the S4 state has the largest oscillator strength (Table. S1). We thus analyzed the molecular orbital (MO) transition contributions of the S4 state. The two main transitions are the highest occupied molecular orbital (HOMO)-2 to the lowest unoccupied molecular orbital ( UMO) and HOMO-1 to UMO. Based on the isosurfaces of HOMO-2 (π), HOMO-1 (π), and UMO (π*) (Fig. S2), the excitation can be as assigned to a π→π* transition. In addition, solvatochromic experiments by using ACN as a solvent could also demonstrate this phenomenon of absorption band distribution (Fig. 2). Molecules that undergo a π→π* transition are believed to have excited states more strongly polarized than the ground state. Therefore, a strongly polar solvent (such as water) has a greater stabilizing effect on the excited state and can reduce the energy difference between the excited state and the ground state. Therefore, as polarity of the solvent decreased with the solvent substitution of H 2 O by ACN, the absorption was hypochromatically shifted, which implied that the energy required from the ground state to the excited state became larger, demonstrating that the MF molecule underwent a π→π* transition [START_REF] Da | Photophysics and photochemistry of azole fungicides: triadimefon and triadimenol[END_REF]. The n→π* excitation was not observed (or was weak) because of the poor spatial overlap of the n and π* orbitals, resulting in a weak vibronic intensity, which should theoretically be at 350 nm (Lim and Jack, 1966).

Fig. 2.

Dark control experiments showed no loss of MF, therefore, hydrolytic or microbial degradations were not further considered in this study. Upon irradiation, a significant MF decay was observed, implying that photolysis limited the persistence of MF in aqueous solutions. The decay of MF could be fitted by a pseudo-first-order kinetics, and the observed rate constant (k obs ) was calculated by linear regression of logarithmic concentration values (ln([MF] t / [MF] 0 )) determined as a function of time (Eq. ( 2)). Half-life (t 1/2 ) was also calculated according to Eq. ( 3). In the present work, k obs was determined to be 0.056 ± 0.003 h -1 under simulated sunlight, corresponding to a half-life (t 1/2 ) of 12.4 ± 0.6 h. These values were comparable with that reported values (0.072 ± 0.005 h -1 (k obs ) and 9.6 ± 0.7 h (t 1/2 )) [START_REF] López-Fernández | Modelling the isothermal degradation kinetics of metrafenone and mepanipyrim in a grape juice analog[END_REF].

European Food Safety Authority (EFSA) concluded that the half-life of MF in the dark aerobic water sediment system (biological degradation) is 8.5-9.2 days [START_REF] Food | Conclusion regarding the peer review of the pesticide risk assessment of the active substance metrafenone[END_REF]. It clearly shows that MF is difficult to biodegrade, therefore, we can conclude that photolysis plays an important role in the elimination of MF in the aquatic environment.

ln ( [MF] t [MF] 0 ) =obs

(2)

1 2 = ln2 obs (3)
Quantum yields are important parameters in photochemistry. The quantum of a given process or of a global reaction corresponds to the ratio of photons triggering a photochemical process or the reaction to the total number of absorbed photons [START_REF] Rene | Environmental Organic Chemistry[END_REF] and therefore gives the efficiency of absorbed photons to achieved the process or the reaction. Here, the polychromatic quantum yield of MF photolysis (Ф MF ) at 20 µM was determined to be 1.06×10 -4 , which is higher than the quantum yields of photolysis of other benzophenone derivatives that do not contain halogene atoms such as benzophenone-3 (1.51×10 -5 ) and 4-hydroxy-benzophenone (7.91×10 -6 ) [START_REF] Li | Photochemical transformation of sunscreen agent benzophenone-3 and its 30 metabolite in surface freshwater and seawater[END_REF]. This result suggests that the presence of a Br atom in MF may favor the photodegradation.

The potential impact of pH solution on MF photolysis was also tested. Our results indicated that the rate of photodegradation was barely affected by the solution pH between 4 and 10 (Fig. S4). This phenomenon is reasonable based on the pH-independent absorption spectra of MF (Fig. S5) and on the absence of dissociable functional groups in the molecular structure of MF.

Identification of potential reactive intermediates

The triplet excited state of carbonyls has attracted a wide interest as a special oxidant in photochemical reactions. Upon irradiation, MF molecules were likely to absorb photons to make an energy level transition from the ground state to the singlet excited state ( 1 MF*), followed by an intersystem crossing (ISC) process to generate the triplet excited state ( 3 MF* containing two electrons with the same spin direction and unpaired) (Eq. ( 4)) [START_REF] Nicholas | Modern Molecular Photochemistry of Organic Molecules[END_REF].

The presence of a bromine atom in MF molecule is expected to enhance the spin-orbit coupling due to the heavy halogen effect, thus favoring the ISC process.

Therefore, the generation of 3 MF* and its involvement in the photolysis process is likely. Sorbic acid (a triplet excited state quencher) was added to the MF solution to test the involvement of 3 MF* [START_REF] Zhou | Aquatic photodegradation of sunscreen agent p-aminobenzoic acid in the presence of dissolved organic matter[END_REF]. As shown in Fig. 3, the presence of sorbic acid (0.4 mM) partially inhibited the reaction, confirming the involvement of 3 MF* in the photo-transformation.

1 3 MF MF MF hv ISC      (4)
It has been reported that the energy required to excite many aromatic compounds to their triplet excited states was greater than 250 kJ mol -1 , whereas the energy for 1 O 2 formation was only 94 kJ mol -1 [START_REF] Zepp | Nitrate-induced photooxidation of trace organic chemicals in water[END_REF]. Therefore, under irradiation, energy transfer between the triplet excited state of organic compounds and dissolved oxygen may occur, resulting in the production of 1 O 2 . For example, the triplet excited state energy of benzophenone was as high as 288 kJ mol -1 , which was sufficient to transfer energy to dissolved oxygen and generate 1 O 2 with a quantum yield of 0.34 [START_REF] Turro | Modern Molecular Photochemistry of Organic Molecules[END_REF]. 4) and ( 5)). Once formed, 1 O 2 could react with MF (Eq (6)). We therefore further measured the bimolecular rate constant of this reaction, 1 2 O ,MF k . The value found: 5.6 × 10 7 L mol -1 s -1 demonstrated the easy oxidation of MF by 1 O 2 .

Furthermore, the irradiation of MF in the presence of isopropanol (0, 8 and 20 mM) used as an •OH radical scavenger ruled out the involvement of this radical in the photolysis of MF (Fig. S6).

3 1 2 2 MF O MF O     (5) 1 O 2 + MF  oxidation products (6) Fig. 3.
In the case MF disappears through Eq (6), the rate of MF loss can be written:

-d[MF]/dt = Ia ×Ф SO × ( 1 2 O ,MF k .[MF]/ ( 1 2 O ,MF k .[MF]+ k d ) ( 7
)
Where Ia is the rate of light absorption by MF, Ф SO , the quantum yield of 1 O 2 formation and k d , the rate of 1 O 2 desactivation in water (2.5 × 10 5 s -1 , Rodgers and Snowden, 1982). Moreover, Ф MF is equal to : ) with retention time of 1.63/1.90 and 1.22/1.41 min, respectively. Notably, TP3, TP4 and TP5 were also observed in a previous study [START_REF] López-Fernández | Modelling the isothermal degradation kinetics of metrafenone and mepanipyrim in a grape juice analog[END_REF]. Moreover, TP6 with m/z = 347.1480 ([M+H] + ) was identified as the OH substitution product, which was another major debromination product (Fig. S7). To the best of our knowledge, the formation of debrominated-hydroxylated photoproduct in the photolysis of MF was reported for the first time. TP7 and TP8 (m/z = 361.1273 and 377.1221, [M+H] + ) were assigned to the methyl oxidized from of TP6.

Ф MF = -d[MF]/dt/Ia = Ф SO ×( 1 2 O ,MF k .[MF]/( 1 2 O ,MF k .[MF]+ k d ) -(8)
To gain deeper insight into the debromination process, the evolution of releasing bromide was recorded during the photolysis of MF, and the yield of bromide was calculated according to Eq. ( 9) [START_REF] Xu | Aqueous photodecomposition of the emerging brominated flame retardant tetrabromobisphenol S (TBBPS)*[END_REF]:

Yield of bromide = [Br -] [MF] 0 [MF] ×100% (9)
where [Br -] t is the concentration of bromide ion detected at irradiation time t; [MF] 0 and [MF] t are the concentrations of MF at irradiation time zero and t, respectively.

Based on the results of ion chromatography, we obtained an average bromine yield of 731% for MF direct photolysis (Fig. S8), which confirmed our conclusion that debromination is a major pathway in the direct photolysis of MF.

Photolysis pathways

Based on the detected intermediates, several photodegradation pathways of MF are proposed (Fig. 4). Pathway I corresponded to the oxidation of the β-methyl group, giving rise to a phenylacetaldehyde product (TP1), followed by a further oxidation process to convert the aldehyde group to a carboxyl group. Pathway II corresponded to the debromination process, leading to the generation of TP3. It was noteworthy that TP3 exhibited a relatively strong signal in the mass spectrum (Fig. S7), indicating that the yield of this product was relatively high during MF photolysis. Subsequently, TP3 could further undergo methyl oxidation and aldehyde oxidation to produce TP4 and TP5 sequentially. Such process has been previously reported by [START_REF] López-Fernández | Modelling the isothermal degradation kinetics of metrafenone and mepanipyrim in a grape juice analog[END_REF] in the photolysis process of MF. Pathway III corresponded to the substitution of bromine by a hydroxyl group to give TP6, then, the methyl group on TP6 could sequentially be oxidized to form TP7 and TP8.

Furthermore, it has been well documented that for the photochemical transformation of carbonyl chemicals, Norrish type I and II reactions were proposed to take place through which fission of the carbon-carbon bond would occur at the αand γ-positions, leading to the formation of bond cleavage or intramolecular cyclization products, respectively [START_REF] Turro | Modern Molecular Photochemistry[END_REF]. However, in the present study, none of the above-mentioned products were identified by mass searching from the total ion chromatography (TIC) spectra; instead, the dominated photolysis pathway of MF was proposed as debromination process, which would be discussed in detail in the following section.

Photo-induced debromination

As mentioned, debrominated product (TP3) were found to be the major photoproduct of MF, indicating that C-Br bond dissociation was an important path of mechanistic explanation. This bond cleavage might take place either via a homolytic process to give a carbon-based benzoylbenzyl radical and bromine atom, or through a heterolytic process to generate a corresponding benzyl cation and bromide anion.

In order to gain a detailed insight into the photolysis mechanism of MF, LFP experiments were conducted to identify the potential transient intermediates involved in the photochemical process. Fig. 5 exhibited the time-dependent transient absorption spectrum of 0.25 mM MF in ACN after 266 nm laser excitation. As can be seen, a broad transient absorption band with maximum absorption around 345 nm was obtained at the pulse end, and the presence of two different triplet scavengers, namely, oxygen and acrylamide were found to hardly affect its formation, indicating that this species could not be assigned to the triplet excited state of MF (Fig. S10). In accordance, the triplet excited states of benzophenone (BP) and some of its derivatives were reported to show maximum absorption around 525 nm (Ryuzi et al., 1997) (Fig. 5(d)). We looked after a signal in this wavelength range, but absorbances were very weak (Fig. 5(c)). On the other hand, the spectrum analogy of our transient (λ max = 345 nm) with those of m-benzoylbenzyl radical (m-BBR, λ max = 335 nm) and p-benzoylbenzyl radical (p-BBR, λ max = 320 nm) [START_REF] Yamaji | Photoinduced w-bond dissociation of m-halomethylbenzophenones studied by laser photolysis techniques and DFT calculations.Substituted position effects[END_REF] prompted us to assign it to the benzoylbenzyl radical of MF (MBR). [START_REF] Yamaji | Photoinduced w-bond dissociation of m-halomethylbenzophenones studied by laser photolysis techniques and DFT calculations.Substituted position effects[END_REF] reported that upon 266 nm excitation in LFP experiments, m-bromomethylbenzophenone (m-BMBP) would be excited to generate m-BBR, and this process could be achieved via an extremely fast ω-cleavage of both lowest singlet (S1) and triplet (T1) m-BMBP.

In their case also, triplet m-BMBP could be hardly be detected by LFP [START_REF] Yamaji | Photoinduced w-bond dissociation of m-halomethylbenzophenones studied by laser photolysis techniques and DFT calculations.Substituted position effects[END_REF]. However, differently from m-BBR, MBR was found to be short-lived, undergoing a fast decay within 0.05 μs to give a relatively long-lived secondary intermediate (Fig. 5(a)). The shorter life-time of MBR might be attributed to the presence of methyl and methoxy substituents allowing fast intramolecular reactions to take place.

To further support the formation of MBR, DFT calculation were conducted to obtain the theoretical UV-Vis absorption of two potential intermediates, namely, MBR and the benzyl cation. The validation of this method was firstly checked by calculating the absorption of parent MF. As seen, theoretical calculation exhibited a maximum absorption peak at 290 nm (Fig. S1), which was in a good agreement with the experimentally measured absorption peak at 292 nm (Fig. 2). In the case of MBR, calculation showed a maximum absorption peak around 350 nm (Fig. S11), matching well with the spectrum observed by LFP experiments (Fig. 5(a)). For the benzyl cation, the maximum absorption was calculated to be around 460 nm. Therefore, we have confirmed the homolytic cleavage of the C-Br bond of MF.

In addition, for the homolytic cleavage process, we further calculated the bond dissociation energy (BDE) of C-Br bond on the basis of △ H derived from thermochemical parameters from the optimized MF, MBR and bromine atom at an ideal condition (298.15 K and 1 atm). A BDE value for C-Br bond cleavage of MF was estimated to be 67.9 kcal mol -1 , which was lower than that of the lowest singlet state (S1, 89.1 kcal mol -1 ) and triplet state (T1, 71.2 kcal mol -1 ). Based on the

Toxicity prediction during MF transformation

In the present study, the acute and chronic toxicity of MF and photoproducts to fish, daphnia and green algae was evaluated through the ECOSAR procedure [START_REF] Wei | Mechanism and kinetic of nitrate radical-initiated atmospheric reactions of guaiacol (2-methoxyphenol)[END_REF]. The toxicity was classified according to the Globally Harmonized System of Classification and Labelling of Chemicals [START_REF] Liu | Oxidative degradation of chlorpyrifos using ferrate(VI): Kinetics and reaction mechanism[END_REF].

The acute toxicity values of MF were determined to be 1.20 mg L -1 , 0.88 mg L -1 and 1.85 mg L -1 for fish, daphnia and green algae, respectively. These results showed that MF was toxic for aquatic species, especially with lower trophic organisms such as daphnia (< 1.0 mg L -1 ). The chronic toxicity values (Chv) of MF were predicted to be 0.16 mg L -1 , 0.17 mg L -1 and 0.85 mg L -1 for fish, daphnia and green algae, respectively, suggesting that MF was more likely to have a chronic toxic effect on organisms. Simultaneously, the trend in chronic toxicity for the three different trophic levels was consistent with acute toxicity. In summary, MF was obviously toxic to aquatic species and can disrupt the ecological balance of the aquatic environment, thus the detection and control of MF in the environment should be given more attention.

Predicting the toxicities of intermediate products during MF transformation was also particularly important [START_REF] Akin | Degradation of chlorophenols and alkylphenol ethoxylates, two representative textile chemicals, in water by advanced oxidation processes: The state of the art on transformation products and toxicity[END_REF]. Therefore, we calculated the toxicity changes of the intermediate products in the three MF photolysis pathways for fish, Daphnia and green algae, and the results were shown in Fig. 6. In ECOSAR predictions, toxicity actually decreases if the toxicity parameter (e.g., Lethal

Concentration 50 (LC50)) decreases by at least one order of magnitude. From this perspective, the acute toxicity of intermediate products showed a decreasing trend only for Daphnia. For fish, all the debromination products (TP3 and TP6) and the methyl aldolization products (TP1, TP4 and TP7) were in the same order of magnitude of acute toxicity as MF. However, the acute toxicity of the methyl carboxylation products (TP2, TP5 and TP8) was directly decreased by two orders of magnitude. The acute toxicity variation for green algae was similar with fish, but the toxicity of both TP6 and TP7 in the OH-debromination pathway was one order of magnitude higher than MF, indicating that toxicity was enhanced. A little different from the acute toxicity, the chronic toxicity of both debromination pathways showed a continuous decreasing trend only for green algae (except TP1). For fish and daphnia, the chronic toxicity of TP1, TP3, TP4, TP6 and TP7 remained in the same order of magnitude as MF. Aldehyde groups are known to be toxic to organisms through chemical bonding with amino bearing biomolecules such as enzymes, proteins, condensates, nucleic acid derivatives etc. via Schiff bases [START_REF] Slomkowski | Polyacrolein containing microspheres: synthesis, properties and possible medical applications[END_REF][START_REF] Sokolsky | Impact of aldehyde content on amphotericin B-dextran imine conjugate toxicity[END_REF]. This might be responsible to higher toxicity of intermediate products containing aldehyde groups. Notably, further oxidation of the aldehyde products to the carboxylate products decreased the toxicity rapidly. All the above results indicated that the toxicity of some intermediate products of three photolysis pathways may maintain or even increase, but the toxicity of the final carboxyl-containing product is greatly reduced, photodegradation is still a method that can safely reduce the ecotoxicity of MF in aquatic environment.

Fig. 6.

Effects of natural water constituents

Effect of nitrate

Nitrate is one of the most abundant anions in the aquatic environment whose photochemical reactivity can affect the photochemical fate of the organic contaminants [START_REF] Mack | Photochemistry of nitrite and nitrate in aqueous solution: a review[END_REF][START_REF] Zuo | Occurrence and photochemical degradation of 17alpha-ethinylestradiol in Acushnet River Estuary[END_REF]. In the present study, the photodegradation experiments of MF were conducted in 25 mL aqueous solutions containing 20 μM substrates with different nitrate concentrations (0, 5, 20 and 50 mM) under irradiation for 8 h and the results are presented in Fig. 7(a).

The photolysis rate of MF was observed to be significantly accelerated as the nitrate concentration increased. The rate constants were 0.055, 0.080, 0.143 and 0.167 h -1 for nitrate concentrations of 0, 5, 20 and 50 mM, respectively. •OH generated by nitrate photolysis is one of the major ROS accounting for the attenuation of pollutants in natural waters (George et al., 1988;[START_REF] Monica | The impact of the hydroxyl radical photochemical sources on the rivastigmine drug transformation in mimic and natural waters[END_REF]. Eqs ( 9)-( 13) show how solar irradiation of nitrate leads to the formation of • O 2 and •OH directly and indirectly via its photoisomerization to peroxynitrite/peroxynitrous acid [START_REF] Kotnik | Photolytic fate and genotoxicity of benzophenone-derived compounds and their photodegradation mixtures in the aqueous environment[END_REF][START_REF] Vione | Inhibition vs. enhancement of the nitrate-induced phototransformation of organic substrates by the bullOH scavengers bicarbonate and carbonate[END_REF]. In this work, the •OH generated by nitrate photolysis was identified by EPR analysis (Fig. S12(a)), thus was proposed to be responsible for the decay of MF.

3 2 NO NO + O • • hv     (9) • O + H 2 O → • OH + OH (10) O 3 - O OO - (11)
+ ONOO H HOONO   € (12) 2 HOONO N • O + H • O  (13)

Effect of bicarbonate

Bicarbonate is the major anion responsible for water alkalinity. Bicarbonate has been reported to play an important influence on the indirect photolysis of organic pollutants in natural waters [START_REF] Huang | Steady-state concentrations of carbonate radicals in field waters[END_REF]. Experimental results show that the presence of bicarbonate barely affected the photolysis process of 20 μM MF (Fig.

7(b))

. The corresponding rate constants were found to be equal to 0.059, 0.064, 0.059 and 0.062 h -1 as the bicarbonate concentration increased from 0 to 20 mM. Bicarbonate can interact with •OH to form carbonate radicals (CO 3 • -) (Eq. ( 14)) [START_REF] Huang | Steady-state concentrations of carbonate radicals in field waters[END_REF], which is a strong electrophile that can rapidly oxidize phenolic compounds [START_REF] Neta | Rate constants for reactions of inorganic radicals in aqueous solution[END_REF]. However, our quenching experiments ruled out the involvement of •OH in the direct photolysis of MF.

HCO 3 -+ •OH→CO 3 • -+ H 2 O (14)
In addition, the presence of bicarbonate can also affect the photochemical transformation of organic compounds via altering the solution pH [START_REF] Ji | Nitrate-induced photodegradation of atenolol in aqueous solution: Kinetics, toxicity and degradation pathways[END_REF].

However, this effect is expected to be not important due to the pH-independent photolysis of MF (vide supra).

Effect of humic acid

Humic substances are ubiquitous in soils and surface waters which are formed by decomposition of living organisms. In the aquatic environment, they are able to compete with pollutants to absorb photons for photolysis as an inner filter (light screener) [START_REF] Luciano | Relationship between Photosensitizing and Emission Properties of Peat Humic Acid Fractions Obtained by Tangential Ultrafiltration[END_REF][START_REF] Monica | PhetoFate: a new approach in accounting for the contribution of indirect photolysis of pesticides and pharmaceuticals in surface waters[END_REF]. Meanwhile, as a photosensitizer, they are capable of affecting the persistence of organic pollutants in the environment through energy transfer or reaction with photochemically produced reactive intermediates (PPRI) [START_REF] Manjun | Photosensitized degradation of bisphenol A involving reactive oxygen species in the presence of humic substances[END_REF][START_REF] Richard | Transformation of monuron photosensitized by soil extracted humic substances: energy or hydrogen transfer mechanism[END_REF][START_REF] Yong | Indirect Photodegradation of Amine Drugs in Aqueous Solution under Simulated Sunlight[END_REF][START_REF] Zepp | Photosensitized transformations involving electronic energy transfer in natural waters: role of humic substances[END_REF]. In view of this, the photolysis of 20 μM MF was conducted in the presence of humic acids (HA) at a concentration ranging from 0 and 20 mg L -1 , and the results are shown in Fig. 7(c).

The kinetic rate constants were determined to be 0.056, 0.064, 0.094, and 0.139 h -1 at HA concentrations 0, 2, 5 and 20 mg L -1 , respectively. Compared with ultrapure water controls, the photodegradation rate of MF in the presence of HA was significantly higher. HA was a macromolecule with a complex structure, containing a large number of chromophores comprised of benzene ring, carboxyl, carbonyl, and methoxyl moieties [START_REF] Laurentiis | Optical and Photochemical Characterization of Chromophoric Dissolved Organic Matter from Lakes in Terra Nova Bay, Antarctica. Evidence of Considerable Photoreactivity in an Extreme Environment[END_REF]. The HA-enhanced photolysis of MF may occur either through energy transfer from the excited HA to the ground state MF, or via oxidation by PPRI, such as •OH and 1 O 2 , which promote the photolysis of MF [START_REF] Rajib | Photodegradation of 17β-estradiol in aquatic solution under solar irradiation: Kinetics and influencing water parameters[END_REF]. The involvement of •OH and 1 O 2 in the photolysis of MF in the presence of HA was confirmed by our EPR analyses, as illustrated in Fig. S12(b).

From this section, it is clear that natural water constituents such as nitrate and humic acid have significant impacts on the photodegradation rate of MF. Therefore, direct photolysis is not expected to be the main photodegradation pathway of MF in the environment, and further studies are required to better understand its photoreactivity.

Fig. 7.

Conclusions

In this study, the photodegradation of MF in aqueous solution under simulated solar irradiation was investigated, including photolysis kinetics, reactive intermediates, products identification, photolysis pathways and mechanisms. Under simulated sunlit, direct photolysis rate constant of MF in aqueous solution was 0.056 ± 0.003 h -1 , corresponding to a half-life of 12.4 ± 0.6 h, indicating that photochemical degradation was an important pathway for the removal of the fungicide MF in nature. Three different pathways were proposed based on HR-MS analysis, including methyl oxidation, debromination and replacement of Br by OH group. Among them debromination process was identified as the main path. Homolytic cleavage of the C-Br bond from both 1 MF* and 3 MF* is likely to take place. MF is also likely oxidized by 1 O 2 produced by energy transfer from 3 MF* to O 2 . NO 3 -and humic acid were found to enhance the decay of MF, while HCO 3 -exhibited no significant effect on MF photolysis. Toxicity assessment revealed that photochemical transformation was an efficient process to reduce the ecotoxicity of MF. This study provided important information for the understanding of photolytic behavior of MF in the aqueous environment.

Declaration of competing interest

There is no declaration of interest in this investigation. The authors declare no competing financial interest.

  ( 1 O 2 ) and MF and test the effect of pH and quenchers. In addition, laser flash photolysis (LFP) experiments are conducted to identify the transient species involved in the reaction and density functional theory (DFT) calculations are adopted to confirm the hypotheses. The and MF and are also measured. The photodegradation products of MF are determined by high resolution mass spectrometry (HR-MS) and their toxicity is evaluated by Ecological Structure Activity Relationship (ECOSAR 2.0) software. Subsequently, the effects of water matrices including nitrate ions, bicarbonate, and humic acid on MF photodegradation are evaluated. Results obtained here will provide a scientific basis for a comprehensive understanding of the photochemical fate of MF in the aquatic environment.

  Since replacing H 2 O with D 2 O extends the lifetime of 1 O 2 13-fold (Rodgers and Snowden, 1982) a D 2 O-H 2 O mixture (50:50, v:v) was employed as a solvent to investigate whether 1 O 2 was produced in the reaction and was responsible for MF oxidation. The reaction rate constant increased by a factor of 1.64 when the solvent was changed to the mixture of D 2 O-H 2 O (Fig. 3), indicating that 1 O 2 was generated in the photolysis of MF by photosensitization. Therefore, these results are consistent with the formation of 3 MF* and the generation of 1 O 2 by energy transfer to O 2 (Eq. (

Fig

  Fig. 4.

Fig

  Scheme. 1(b). Specifically, upon irradiation, MF would be excited to singlet state,

  In this work, we systematically investigated intermediate photoproducts of MF via HR-MS analysis after 22 h irradiation. The major intermediate photoproducts are listed in Table. S2. Bromine atom has two isotopes (relative atomic mass 78.9183 and 80.9163), based on this property, isotopic features were used as a diagnostic tool for the identification of photoproducts containing bromine atoms. For instance, the parent compound MF had a molecular ion cluster corresponding to m/z = 409.0634/411.0612 ([M+H] + ) with retention time of 4.91 min, suggesting that the molecular structure of MF contained one bromine atom. Reviewing Table. S2, it can be seen that eight

	photoproduct intermediates were detected. Specifically, TP1 with m/z =
	423.0426/425.0404 ([M+H] + ) at retention time of 1.97/2.65 min was assigned as the
	oxidation product in which a methyl group was oxidized into aldehyde. The m/z =
	439.0375/441.0353 ([M+H] + ) molecular ion was labeled as the secondary product
	(TP2), which was generated by oxidation of the methyl group to carboxyl with
	concentrations ranging from 20 to 5 µM. Ф MF decreased from 1.06×10 -4 for 20 µM to retention time of 1.28/1.6 min. Based on the total ion currents (TICs), debrominated
	0.52×10 -4 for 5 µM, in accordance with a contribution of 1 O 2 to the product (TP3) was a major intermediate product generated during MF photolysis (Fig.
	phototransformation of MF. S7), which was eluted at 2.56 min with m/z = 331.1530. Furthermore,
	debrominated-methyl oxidation products (TP4 and TP5) were identified based on m/z
	M. A. J. Rodgers, P. T. Snowden. Lifetime of ( 1 O 2 ) in Liquid Water As Determined by = 345.1325 and 361.1273 ([M+H] +
	Time-Resolved Infrared Luminescence Measurements. J. Am. Chem. Soc. 1982,
	104, 5541-5543.doi-org.ezproxy.uca.fr/10.1021/ja00384a070
	3.3 Photoproducts identification

It means that Ф MF depends on the [MF], in particular when

1 2 O ,MF k .[MF] << k d , that is

the case at the chosen concentrations. To check this, we measured Ф MF for several

Acknowledgement

This work was financially supported by the National Natural Science Foundation of China (No. 22176059). The authors would also like to thank Dr. Ting Wu from Research Center of Analysis and Test of East China University of Science and Technology for the technical support on photoproducts identification.


Olivier Finkel
email: olivier.finkel@ens-lyon.fr

Highly Undecidable Problems about Recognizability by Tiling Systems

Keywords: Languages of infinite pictures, recognizability by tiling systems, decision problems, highly undecidable problems, analytical hierarchy

 that it is undecidable whether a Büchirecognizable language of infinite pictures is E-recognizable (respectively, A-recognizable). We show here that these two decision problems are actually Π 1 2 -complete, hence located at the second level of the analytical hierarchy, and "highly undecidable". We give the exact degree of numerous other undecidable problems for Büchi-recognizable languages of infinite pictures. In particular, the nonemptiness and the infiniteness problems are Σ 1 1 -complete, and the universality problem, the inclusion problem, the equivalence problem, the determinizability problem, the complementability problem, are all Π 1 2 -complete. It is also Π 1 2 -complete to determine whether a given Büchi recognizable language of infinite pictures can be accepted row by row using an automaton model over ordinal words of length ω 2 .

Introduction

Languages of infinite words accepted by finite automata were first studied by Büchi to prove the decidability of the monadic second order theory of one successor over the integers. Since then regular ω-languages have been much studied and many applications have been found for specification and verification of non-terminating systems, see [START_REF] Thomas | Automata on infinite objects[END_REF][START_REF] Staiger | ω-languages[END_REF][START_REF] Perrin | Infinite words, automata, semigroups, logic and games[END_REF] for many results and references.

Address for correspondence: E Mail: Olivier.Finkel@ens-lyon.fr In a recent paper, Altenbernd, Thomas and Wöhrle have considered acceptance of languages of infinite two-dimensional words (infinite pictures) by finite tiling systems, with the usual acceptance conditions, such as the Büchi and Muller ones, firstly used for infinite words. This way they extended both the classical theory of ω-regular languages and the classical theory of recognizable languages of finite pictures, [START_REF] Giammarresi | Two-Dimensional Languages[END_REF], to the case of infinite pictures.

Many classical decision problems are studied in formal language theory and in automata theory and arise now naturally about recognizable languages of infinite pictures. We proved in [START_REF] Finkel | On recognizable languages of infinite pictures[END_REF] that many decision problems for Büchi-recognizable languages of infinite pictures are undecidable. In particular, we showed, using topological arguments, that it is undecidable whether a Büchi-recognizable language of infinite pictures is E-recognizable (respectively, A-recognizable), giving the answer to two questions raised in [START_REF] Altenbernd | Tiling Systems over Infinite Pictures and Their Acceptance Conditions[END_REF]. We proved also several other undecidability results as the following ones: one cannot decide whether a Büchi-recognizable language of infinite pictures can be recognized by a deterministic Büchi or Muller tiling system, or whether it can be accepted row by row using an automaton model over ordinal words of length ω 2 .

Using the Π 1 2 -completeness of the universality problem for ω-languages of non deterministic Turing machines which was proved by Castro and Cucker in [START_REF] Castro | Nondeterministic ω-Computations and the Analytical Hierarchy[END_REF], and some topological arguments, we show in this paper that the above decision problems are actually Π 1 2 -complete, hence located at the second level of the analytical hierarchy, and "highly undecidable". Using other results of [START_REF] Castro | Nondeterministic ω-Computations and the Analytical Hierarchy[END_REF], we give also the exact degree of numerous other undecidable problems for Büchi-recognizable languages of infinite pictures. In particular, the non-emptiness and the infiniteness problems are Σ 1 1 -complete, and the universality problem, the inclusion problem, the equivalence problem, the complementability problem, are all Π 1 2complete. This gives new natural examples of decision problems located at the first or at the second level of the analytical hierarchy. We show also that topological properties of Büchi-recognizable languages of infinite pictures are highly undecidable.

The paper is organized as follows. In Section 2 we recall definitions for pictures and tiling systems. The definition and properties of the analytical hierarchy are introduced in Section 3. We recall in Section 4 some notions of topology, including the definitions of Borel and analytic sets. We prove high undecidability results in Section 5. Concluding remarks are given in Section 6.

Tiling Systems

We assume the reader to be familiar with the theory of formal (ω)-languages [START_REF] Thomas | Automata on infinite objects[END_REF][START_REF] Staiger | ω-languages[END_REF]. We recall usual notations of formal language theory. When Σ is a finite alphabet, a non-empty finite word over Σ is any sequence x = a 1 . . . a k , where a i ∈ Σ for i = 1, . . . , k , and k is an integer ≥ 1. The length of x is k, denoted by |x|. The empty word has no letter and is denoted by λ; its length is 0. Σ ⋆ is the set of finite words (including the empty word) over Σ. The first infinite ordinal is ω. An ω-word over Σ is an ω -sequence a 1 . . . a n . . ., where for all integers i ≥ 1, a i ∈ Σ. When σ is an ω-word over Σ, we write σ = σ(1)σ(2) . . . σ(n) . . ., where for all i, σ(i) ∈ Σ, and σ[n] = σ(1)σ(2) . . . σ(n) for all n ≥ 1 and σ[0] = λ. The usual concatenation product of two finite words u and v is denoted u.v (and sometimes just uv).

This product is extended to the product of a finite word u and an ω-word v: the infinite word u.v is then the ω-word such that:

(u.v)(k) = u(k) if k ≤ |u| , and (u.v)(k) = v(k -|u|) if k > |u|.
The set of ω-words over the alphabet Σ is denoted by Σ ω . An ω-language over an alphabet Σ is a subset of Σ ω .

We now define two-dimensional words, i.e. pictures. Let Σ be a finite alphabet and # be a letter not in Σ and let Σ = Σ∪{#}. If m and n are two integers > 0 or if m = n = 0, a picture of size (m, n) over Σ is a function p from {0, 1, . . . , m+1}×{0, 1, . . . , n+1} into Σ such that p(0, i) = p(m + 1, i) = # for all integers i ∈ {0, 1, . . . , n + 1} and p(i, 0) = p(i, n+1) = # for all integers i ∈ {0, 1, . . . , m+1} and p(i, j) ∈ Σ if i / ∈ {0, m+1} and j / ∈ {0, n+1}. The empty picture is the only picture of size (0, 0) and is denoted by λ. Pictures of size (n, 0) or (0, n), for n > 0, are not defined. Σ ⋆,⋆ is the set of pictures over Σ. A picture language L is a subset of Σ ⋆,⋆ .

An ω-picture over Σ is a function p from ω × ω into Σ such that p(i, 0) = p(0, i) = # for all i ≥ 0 and p(i, j) ∈ Σ for i, j > 0. Σ ω,ω is the set of ω-pictures over Σ. An ω-picture language L is a subset of Σ ω,ω . For Σ a finite alphabet we call Σ ω 2 the set of functions from ω × ω into Σ. So the set Σ ω,ω of ω-pictures over Σ is a strict subset of Σω 2 .

We shall say that, for each integer j ≥ 1, the j th row of an ω-picture p ∈ Σ ω,ω is the infinite word p(1, j).p(2, j).p(3, j) . . . over Σ and the j th column of p is the infinite word p(j, 1).p(j, 2).p(j, 3) . . . over Σ. As usual, one can imagine that, for integers j > k ≥ 1, the j th column of p is on the right of the k th column of p and that the j th row of p is "above" the k th row of p.

We introduce now tiling systems as in the paper [START_REF] Altenbernd | Tiling Systems over Infinite Pictures and Their Acceptance Conditions[END_REF]. A tiling system is a tuple A=(Q, Σ, ∆), where Q is a finite set of states, Σ is a finite alphabet, ∆ ⊆ ( Σ × Q) 4 is a finite set of tiles. A Büchi tiling system is a pair (A,F ) where A=(Q, Σ, ∆) is a tiling system and F ⊆ Q is the set of accepting states. A Muller tiling system is a pair (A, F) where A=(Q, Σ, ∆) is a tiling system and F⊆ 2 Q is the set of accepting sets of states.

Tiles are denoted by

(a 3 , q 3 ) (a 4 , q 4 ) (a 1 , q 1 ) (a 2 , q 2 ) with a i ∈ Σ and q i ∈ Q, and in general, over an alphabet Γ, by

b 3 b 4 b 1 b 2 with b i ∈ Γ.
A combination of tiles is defined by:

b 3 b 4 b 1 b 2 • b ′ 3 b ′ 4 b ′ 1 b ′ 2 = (b 3 , b ′ 3 ) (b 4 , b ′ 4 ) (b 1 , b ′ 1 ) (b 2 , b ′ 2 )
A run of a tiling system A=(Q, Σ, ∆) over a (finite) picture p of size (m, n) over Σ is a mapping ρ from {0, 1, . . . , m + 1} × {0, 1, . . . , n + 1} into Q such that for all (i, j) ∈ {0, 1, . . . , m} × {0, 1, . . . , n} with p(i, j) = a i,j and ρ(i, j) = q i,j we have a i,j+1 a i+1,j+1 a i,j a i+1,j • q i,j+1 q i+1,j+1 q i,j q i+1,j ∈ ∆.

A run of a tiling system A=(Q, Σ, ∆) over an ω-picture p ∈ Σ ω,ω is a mapping ρ from ω × ω into Q such that for all (i, j) ∈ ω × ω with p(i, j) = a i,j and ρ(i, j) = q i,j we have a i,j+1 a i+1,j+1 a i,j a i+1,j • q i,j+1 q i+1,j+1 q i,j q i+1,j ∈ ∆.

We now recall acceptance of finite or infinite pictures by tiling systems: Definition 2.1. Let A=(Q, Σ, ∆) be a tiling system, F ⊆ Q and F⊆ 2 Q .

• The picture language recognized by A is the set of pictures p ∈ Σ ⋆,⋆ such that there is some run ρ of A on p.

• The ω-picture language A-recognized (respectively, E-recognized, Büchi-recognized) by (A,F ) is the set of ω-pictures p ∈ Σ ω,ω such that there is some run ρ of A on p and ρ(v) ∈ F for all (respectively, for at least one, for infinitely many) v ∈ ω 2 . It is denoted by L A ((A,F )) (respectively, L E ((A,F )), L B ((A,F ))).

• The ω-picture language Muller-recognized by (A, F) is the set of ω-pictures p ∈ Σ ω,ω such that there is some run ρ of A on p and Inf (ρ) ∈ F where Inf (ρ) is the set of states occurring infinitely often in ρ. It is denoted by L M ((A,F)).

Notice that an ω-picture language L ⊆ Σ ω,ω is recognized by a Büchi tiling system if and only if it is recognized by a Muller tiling system, [START_REF] Altenbernd | Tiling Systems over Infinite Pictures and Their Acceptance Conditions[END_REF].

We shall denote T S(Σ ω,ω ) the class of languages L ⊆ Σ ω,ω which are recognized by some Büchi (or Muller) tiling system.

We recall now an interesting variation of the above defined acceptance conditions for infinite pictures, introduced in [START_REF] Altenbernd | Tiling Systems over Infinite Pictures and Their Acceptance Conditions[END_REF]. This variation uses the diagonal of an ω-picture.

The diagonal of an ω-picture p is the set of vertices

Di(p) = {(i, i) | i ∈ ω}.
The ω-picture language A-recognized (respectively, E-recognized, Büchi-recognized) by (A,F ) on the diagonal is the set of ω-pictures p ∈ Σ ω,ω such that there is some run ρ of A on p and ρ(v) ∈ F for all (respectively, for at least one, for infinitely many) v ∈ Di(p).

We define similarly the notion of ω-picture language Muller-recognized on the diagonal by (A, F), replacing Inf (ρ) by the set of states Inf (Di(ρ)) occurring infinitely often on the diagonal of ρ.

The following result was stated in [START_REF] Altenbernd | Tiling Systems over Infinite Pictures and Their Acceptance Conditions[END_REF].

Theorem 2.2. An ω-picture language L ⊆ Σ ω,ω is A-recognized (respectively, E-recognized, Büchirecognized, Muller-recognized) by a tiling system if and only if it is A-recognized (respectively, Erecognized, Büchi-recognized, Muller-recognized) on the diagonal by a tiling system.

We wish now to see links with classical notions of tiling of the (quarter of the) plane, see for instance [START_REF] Ballier | Tilings and Model Theory[END_REF].

We denote Γ = Σ × Q where Σ is the alphabet of pictures and Q is the set of states of a tiling system A=(Q, Σ, ∆). We consider configurations which are elements of Γ ω×ω . One can imagine that each cell of the quarter of the plane contains a letter of the alphabet Γ.

Let ∆ ⊆ ( Σ × Q) 4 = Γ 4 be a finite set of tiles. We denote its complement by ∆ -= Γ 4 -∆. A tiling of the (quarter of the) plane with ∆ -as set of forbidden patterns is simply a configuration c ∈ Γ ω×ω such that for all integers i, j ∈ ω:

c(i, j + 1) c(i + 1, j + 1) c(i, j) c(i + 1, j) ∈ ∆.
Then the ω-picture language L ⊆ Σ ω,ω which is A-recognized (respectively, E-recognized, Büchirecognized) on the diagonal by the tiling system (A,F ) is simply the set of ω-pictures p ∈ Σ ω,ω which are projections of configurations c ∈ Γ ω×ω which are tilings of the (quarter of the) plane with ∆ -as set of forbidden patterns such that for all (respectively, for at least one, for infinitely many) i ∈ ω the second component of c(i, i) is in F . A similar characterization can be given for the Muller acceptance condition.

We can also easily state similar characterizations for global recognizability, i.e. not on the diagonal, by tiling systems.

The Analytical Hierarchy

The set of natural numbers is denoted by N and the set of all mappings from N into N will be denoted by F.

We assume the reader to be familiar with the arithmetical hierarchy on subsets of N. We now recall the notions of analytical hierarchy and of complete sets for classes of this hierarchy which may be found in [START_REF] Rogers | Theory of Recursive Functions and Effective Computability[END_REF].

Definition 3.1. Let k, l > 0 be some integers. Φ is a partial recursive function of k function variables and l number variables if there exists z ∈ N such that for any

(f 1 , . . . , f k , x 1 , . . . , x l ) ∈ F k × N l , we have Φ(f 1 , . . . , f k , x 1 , . . . , x l ) = τ f 1 ,...,f k z (x 1 , . . . , x l ),
where the right hand side is the output of the Turing machine with index z and oracles f 1 , . . . , f k over the input (x 1 , . . . , x l ). For k > 0 and l = 0, Φ is a partial recursive function if, for some z,

Φ(f 1 , . . . , f k ) = τ f 1 ,...,f k z (0).
The value z is called the Gödel number or index for Φ.

Definition 3.2. Let k, l > 0 be some integers and R ⊆ F k × N l . The relation R is said to be a recursive relation of k function variables and l number variables if its characteristic function is recursive.

We now define analytical subsets of N l .

Definition 3.3. A subset R of N l is analytical if it is recursive or if there exists a recursive set S ⊆ F m × N n , with m ≥ 0 and n ≥ l, such that R = {(x 1 , . . . , x l ) | (Q 1 s 1 )(Q 2 s 2 ) . . . (Q m+n-l s m+n-l )S(f 1 , . . . , f m , x 1 , . . . , x n )},
where

Q i is either ∀ or ∃ for 1 ≤ i ≤ m + n -l,
and where s 1 , . . . , s m+n-l are f 1 , . . . , f m , x l+1 , . . . , x n in some order. The expression

(Q 1 s 1 )(Q 2 s 2 ) . . . (Q m+n-l s m+n-l )S(f 1 , . . . , f m , x 1 , . . . , x n ) is called a predicate form for R.
A quantifier applying over a function variable is of type 1, otherwise it is of type 0. In a predicate form the (possibly empty) sequence of quantifiers, indexed by their type, is called the prefix of the form. The reduced prefix is the sequence of quantifiers obtained by suppressing the quantifiers of type 0 from the prefix.

The levels of the analytical hierarchy are distinguished by considering the number of alternations in the reduced prefix.

Definition 3.4. For n > 0, a Σ 1 n -prefix is one whose reduced prefix begins with ∃ 1 and has n -1 alternations of quantifiers. A Σ 1 0 -prefix is one whose reduced prefix is empty. For n > 0, a Π 1 n -prefix is one whose reduced prefix begins with ∀ 1 and has n -1 alternations of quantifiers. A Π 1 0 -prefix is one whose reduced prefix is empty.

A predicate form is a Σ 1 n (Π 1 n )-form if it has a Σ 1 n (Π 1 n )-prefix.
The class of sets in some N l which can be expressed in Σ 1 n -form (respectively, Π 1 n -form) is denoted by Σ 1 n (respectively, Π 1 n ). The class Σ 1 0 = Π 1 0 is the class of arithmetical sets.

We now recall some well known results about the analytical hierarchy.

Proposition 3.5. Let R ⊆ N l for some integer l. Then R is an analytical set iff there is some integer

n ≥ 0 such that R ∈ Σ 1 n or R ∈ Π 1 n .
Theorem 3.6. For each integer n ≥ 1,

(a) Σ 1 n ∪ Π 1 n Σ 1 n+1 ∩ Π 1 n+1 . (b) A set R ⊆ N l is in the class Σ 1 n iff its complement is in the class Π 1 n . (c) Σ 1 n -Π 1 n = ∅ and Π 1 n -Σ 1 n = ∅.
Transformations of prefixes are often used, following the rules given by the next theorem.

Theorem 3.7. For any predicate form with the given prefix, an equivalent predicate form with the new one can be obtained, following the allowed prefix transformations given below :

(a) . . . ∃ 0 ∃ 0 . . . → . . . ∃ 0 . . . , . . . ∀ 0 ∀ 0 . . . → . . . ∀ 0 . . . ; (b) . . . ∃ 1 ∃ 1 . . . → . . . ∃ 1 . . . , . . . ∀ 1 ∀ 1 . . . → . . . ∀ 1 . . . ; (c) . . . ∃ 0 . . . → . . . ∃ 1 . . . , . . . ∀ 0 . . . → . . . ∀ 1 . . . ; (d) . . . ∃ 0 ∀ 1 . . . → . . . ∀ 1 ∃ 0 . . . , . . . ∀ 0 ∃ 1 . . . → . . . ∃ 1 ∀ 0 . . . ;
We can now define the notion of 1-reduction and of Σ 1 n -complete (respectively, Π 1 n -complete) sets. Notice that we give the definition for subsets of N but one can easily extend this definition to the case of subsets of N l for some integer l. Definition 3.8. Given two sets A, B ⊆ N we say A is 1-reducible to B and write A ≤ 1 B if there exists a total computable injective function f from

N to N such that A = f -1 [B]. Definition 3.9. A set A ⊆ N is said to be Σ 1 n -complete (respectively, Π 1 n -complete) iff A is a Σ 1 n -set (respectively, Π 1 n -set) and for each Σ 1 n -set (respectively, Π 1 n -set) B ⊆ N it holds that B ≤ 1 A. For each integer n ≥ 1 there exist some Σ 1 n -complete set E n ⊆ N. The complement E - n = N -E n is a Π 1
n -complete set. These sets are precisely defined in [START_REF] Rogers | Theory of Recursive Functions and Effective Computability[END_REF] or [START_REF] Castro | Nondeterministic ω-Computations and the Analytical Hierarchy[END_REF].

Borel Hierarchy and Analytic Sets

We assume now the reader to be familiar with basic notions of topology which may be found in [START_REF] Moschovakis | Descriptive set theory[END_REF][START_REF] Lescow | Logical specifications of infinite computations, A Decade of Concurrency[END_REF][START_REF] Kechris | Classical descriptive set theory[END_REF][START_REF] Staiger | ω-languages[END_REF][START_REF] Perrin | Infinite words, automata, semigroups, logic and games[END_REF].

There is a natural metric on the set Σ ω of infinite words over a finite alphabet Σ containing at least two letters which is called the prefix metric and defined as follows. For u, v ∈ Σ ω and u = v let δ(u, v) = 2 -l pref(u,v) where l pref(u,v) is the first integer n such that the (n + 1) st letter of u is different from the (n + 1) st letter of v. This metric induces on Σ ω the usual Cantor topology for which open subsets of Σ ω are in the form W.Σ ω , where W ⊆ Σ ⋆ . A set L ⊆ Σ ω is a closed set iff its complement Σ ω -L is an open set. Define now the Borel Hierarchy of subsets of Σ ω : Definition 4.1. For a non-null countable ordinal α, the classes Σ 0 α and Π 0 α of the Borel Hierarchy on the topological space Σ ω are defined as follows: Σ 0 1 is the class of open subsets of Σ ω , Π 0 1 is the class of closed subsets of Σ ω , and for any countable ordinal α ≥ 2:

Σ 0 α is the class of countable unions of subsets of Σ ω in γ<α Π 0 γ . Π 0 α is the class of countable intersections of subsets of Σ ω in γ<α Σ 0 γ . For a countable ordinal α, a subset of Σ ω is a Borel set of rank α iff it is in Σ 0 α ∪ Π 0 α but not in γ<α (Σ 0 γ ∪ Π 0 γ ).
There are also some subsets of Σ ω which are not Borel. Indeed there exists another hierarchy beyond the Borel hierarchy, which is called the projective hierarchy and which is obtained from the Borel hierarchy by successive applications of operations of projection and complementation. The first level of the projective hierarchy is formed by the class of analytic sets and the class of co-analytic sets which are complements of analytic sets. In particular the class of Borel subsets of Σ ω is strictly included into the class Σ 1 1 of analytic sets which are obtained by projection of Borel sets.

Definition 4.2.

A subset A of Σ ω is in the class Σ 1 1 of analytic sets iff there exists another finite set Y and a Borel subset B of (Σ × Y ) ω such that x ∈ A ↔ ∃y ∈ Y ω such that (x, y) ∈ B, where (x, y) is the infinite word over the alphabet Σ × Y such that (x, y)(i) = (x(i), y(i)) for each integer i ≥ 1.

We now define completeness with regard to reduction by continuous functions. For a countable ordinal α ≥ 1, a set F ⊆ Σ ω is said to be a Σ 0 α (respectively, Π 0 α , Σ 1 1 )-complete set iff for any set E ⊆ Y ω (with Y a finite alphabet):

E ∈ Σ 0 α (respectively, E ∈ Π 0 α , E ∈ Σ 1 1 ) iff there exists a continuous function f : Y ω → Σ ω such that E = f -1 (F ). Σ 0 n (respectively Π 0 n )-complete sets,
with n an integer ≥ 1, are thoroughly characterized in [START_REF] Staiger | Hierarchies of recursive ω-languages[END_REF].

In particular R = (0 ⋆ .1) ω is a well known example of Π 0 2 -complete subset of {0, 1} ω . It is the set of ω-words over {0, 1} having infinitely many occurrences of the letter 1.

Its complement {0, 1} ω -(0 ⋆ .1) ω is a Σ 0 2 -complete subset of {0, 1} ω .
For Γ a finite alphabet having at least two letters, the set Γ ω×ω of functions from ω × ω into Γ is usually equipped with the product topology of the discrete topology on Γ. This topology may be defined by the following distance d. Let x and y in Γ ω×ω such that x = y, then

d(x, y) = 1 2 n where n = min{p ≥ 0 | ∃(i, j) x(i, j) = y(i, j) and i + j = p}.
Then the topological space Γ ω×ω is homeomorphic to the topological space Γ ω , equipped with the Cantor topology. Borel subsets of Γ ω×ω are defined from open subsets as in the case of the topological space Γ ω . Analytic subsets of Γ ω×ω are obtained as projections on Γ ω×ω of Borel subsets of the product space Γ ω×ω × Γ ω . The set Σ ω,ω of ω-pictures over Σ, viewed as a topological subspace of Σω×ω , is easily seen to be homeomorphic to the topological space Σ ω×ω , via the mapping ϕ : Σ ω,ω → Σ ω×ω defined by ϕ(p)(i, j) = p(i + 1, j + 1) for all p ∈ Σ ω,ω and i, j ∈ ω.

Highly Undecidable Problems

We are now going to study decision problems about recognizable languages of infinite pictures. We shall use some results of Castro and Cucker who studied degrees of decision problems for ω-languages accepted by Turing machines and proved that many of them are highly undecidable, [START_REF] Castro | Nondeterministic ω-Computations and the Analytical Hierarchy[END_REF].

So we now recall the notion of acceptance of infinite words by Turing machines considered by Castro and Cucker in [START_REF] Castro | Nondeterministic ω-Computations and the Analytical Hierarchy[END_REF].

Definition 5.1. A non deterministic Turing machine M is a 5-tuple M = (Q, Σ, Γ, δ, q 0 ), where Q is a finite set of states, Σ is a finite input alphabet, Γ is a finite tape alphabet satisfying Σ ⊆ Γ, q 0 is the initial state, and δ is a mapping from Q × Γ to subsets of Q × Γ × {L, R, S}. A configuration of M is a 3-tuple (q, σ, i), where q ∈ Q, σ ∈ Γ ω and i ∈ N. An infinite sequence of configurations r = (q i , α i , j i ) i≥1 is called a run of M on w ∈ Σ ω iff:

(a) (q 1 , α 1 , j 1 ) = (q 0 , w, 1), and (b) for each i ≥ 1, (q i , α i , j i ) ⊢ (q i+1 , α i+1 , j i+1 ),

where ⊢ is the transition relation of M defined as usual. The run r is said to be complete if the limsup of the head positions is infinity, i.e. if (∀n ≥ 1)(∃k ≥ 1)(j k ≥ n). The run r is said to be oscillating if the liminf of the head positions is bounded, i.e. if (∃k ≥ 1)(∀n ≥ 1)(∃m ≥ n)(j m = k).

Definition 5.2. Let M = (Q, Σ, Γ, δ, q 0 ) be a non deterministic Turing machine and F ⊆ Q. The ω-language accepted by (M, F ) is the set of ω-words σ ∈ Σ ω such that there exists a complete non oscillating run r = (q i , α i , j i ) i≥1 of M on σ such that, for all i, q i ∈ F.

The above acceptance condition is denoted 1 ′ -acceptance in [START_REF] Cohen | ω-computations on Turing machines[END_REF]. Another usual acceptance condition is the now called Büchi acceptance condition which is also denoted 2-acceptance in [START_REF] Cohen | ω-computations on Turing machines[END_REF]. We now recall its definition.

Definition 5.3. Let M = (Q, Σ, Γ, δ, q 0 ) be a non deterministic Turing machine and F ⊆ Q. The ωlanguage Büchi accepted by (M, F ) is the set of ω-words σ ∈ Σ ω such that there exists a complete non oscillating run r = (q i , α i , j i ) i≥1 of M on σ and infinitely many integers i such that q i ∈ F.

Recall that Cohen and Gold proved in [START_REF] Cohen | ω-computations on Turing machines[END_REF]Theorem 8.6] that one can effectively construct, from a given non deterministic Turing machine, another equivalent non deterministic Turing machine, equipped with the same kind of acceptance condition, and in which every run is complete non oscillating. Cohen and Gold proved also in [4, Theorem 8.2] that an ω-language is accepted by a non deterministic Turing machine with 1 ′ -acceptance condition iff it is accepted by a non deterministic Turing machine with Büchi acceptance condition.

From now on, we shall denote M z the non deterministic Turing machine of index z, (accepting words over Σ = {a, b}), equipped with a 1 ′ -acceptance condition. In a similar way we shall denote T z the non deterministic tiling system of index z, (accepting pictures over Σ = {a, b}), equipped with a Büchi acceptance condition.

For σ ∈ Σ ω = {a, b} ω we denote σ a the ω-picture whose first row is the ω-word σ and whose other rows are labelled with the letter a. For an ω-language L ⊆ Σ ω = {a, b} ω we denote L a the language of infinite pictures {σ a | σ ∈ L}.

We can now recall a result proved in [START_REF] Finkel | On recognizable languages of infinite pictures[END_REF] which will be useful later.

Lemma 5.4. ([9])

If L ⊆ Σ ω is accepted by some Turing machine (in which every run is complete non oscillating) with a Büchi acceptance condition, then L a is Büchi recognizable by a finite tiling system.

Proof. Let L ⊆ Σ ω be an ω-language accepted by some Turing machine M = (Q, Σ, Γ, δ, q 0 ) with a Büchi acceptance condition, where F ⊆ Q is the set of accepting states. We assume that the Turing machine has a single semi-infinite tape, with one reading head which may also write on the tape.

Cohen and Gold proved that one can consider only such a restricted model of Turing machines [START_REF] Cohen | ω-computations on Turing machines[END_REF]. An instantaneous configuration of M is given by an infinite word u.q.v where u ∈ Γ ⋆ , q ∈ Q, v ∈ Γ ω , and the first letter of v is the one scanned by the head of M.

The initial configuration of M reading the infinite word σ ∈ Σ ω is q 0 .σ.

A computation of M reading σ ∈ Σ ω is an infinite sequence of configurations α 0 , α 1 , α 2 , . . . , α i , . . . , where α 0 = q 0 .σ is the initial configuration and for all integers i ≥ 0, α i = u i .q i .v i is the (i + 1) th configuration.

The computation is successful if and only if there exists a final state q f ∈ F and infinitely many integers i such that q i = q f . Using a similar reasoning as in the classical proof of the undecidability of the emptiness problem for recognizable languages of finite pictures, [11, p. 34], we can define a set of tiles ∆ in such a way that for σ ∈ Σ ω , a run ρ of the tiling system T =(Σ, Γ ∪ Q, ∆, F ) over the infinite picture σ a satifies:

for each integer i ≥ 0 ρ(0, i).ρ(1, i).ρ(2, i) . . . = α i = u i .q i .v i i.e. ρ(0, i).ρ(1, i).ρ(2, i) . . . is the (i + 1) th configuration of M reading the ω-word σ ∈ Σ ω . Thus the Büchi tiling system (T ,F ) recognizes the language L a .

Notice that the above cited constructions of [START_REF] Cohen | ω-computations on Turing machines[END_REF] and of the proof of Lemma 5.4 are effective and that they can be achieved in an injective way. This is expressed by the following lemma.

Lemma 5.5.

There is an injective computable function K from N into N satisfying the following property.

If M z is the non deterministic Turing machine (equipped with a 1 ′ -acceptance condition) of index z, and if T K(z) is the tiling system (equipped with a Büchi acceptance condition) of index K(z), then

L(M z ) a = L B (T K(z) )
Recall that Castro and Cucker proved in [START_REF] Castro | Nondeterministic ω-Computations and the Analytical Hierarchy[END_REF] that the non-emptiness problem and the infiniteness problem for ω-languages of Turing machines are both Σ 1 1 -complete. We can now easily infer from Lemma 5.5 a similar result for recognizable languages of infinite pictures.

Theorem 5.6. The non-emptiness problem and the infiniteness problem for Büchi-recognizable languages of infinite pictures are Σ 1 1 -complete, i.e. :

1. {z ∈ N | L B (T z ) = ∅} is Σ 1 1 -complete. 2. {z ∈ N | L B (T z ) is infinite } is Σ 1 1 -complete.
Proof. We first show that these two decision problems are in the class Σ 1 1 . Notice first that, using a recursive bijection b : (N -{0}) 2 → N -{0}, one can associate to each ω-word σ ∈ Σ ω a unique ω-picture p σ ∈ Σ ω,ω which is simply defined by p σ (i, j) = σ(b(i, j)) for all integers i, j ≥ 1.

On the other hand a run of a tiling system A=(Q, Σ, ∆) over an ω-picture p ∈ Σ ω,ω is a mapping ρ from ω × ω into Q, i.e. an element of Q ω×ω . Using again a recursive bijection between (N) 2 and N, we can identify a run ρ with an element of Q ω and finally with a coding of this element over the alphabet {0, 1}. So the run ρ can be identified with its code ρ ∈ {0, 1} ω . Assume now that the tiling system A=(Q, Σ, ∆) is equipped with a set of accepting states F ⊆ Q. It is then easy to see that for σ ∈ Σ ω and ρ ∈ {0, 1} ω , "ρ is a Büchi accepting run of (A, F ) over the ω-picture p σ " can be expressed by an arithmetical formula, see also [START_REF] Altenbernd | Tiling Systems over Infinite Pictures and Their Acceptance Conditions[END_REF]Section 2.4].

We can now express "L B (T z ) = ∅" by "∃σ ∈ Σ ω ∃ρ ∈ {0, 1} ω [ ρ is a Büchi accepting run of T z over the ω-picture p σ ]" which is a Σ 1 1 -formula.

In order to show that "L B (T z ) is infinite " can be also expressed by a Σ 1 1 -formula, we shall use again the bijection b : (N -{0}) 2 → N -{0}. We can consider an infinite word σ ∈ Σ ω as a countably infinite family of infinite words over Σ : the family of ω-words (σ i ) such that for each i ≥ 1, σ i is defined by σ i (j) = σ(b(i, j)) for each j ≥ 1. In a similar manner an ω-word ρ ∈ {0, 1} ω can be considered as a countably infinite family of infinite words (ρ i ) defined, for each i ≥ 1, by ρ i (j) = ρ(b(i, j)) for each j ≥ 1. We can now express "L B (T z ) is infinite " by the formula "∃σ ∈ Σ ω ∃ρ ∈ {0, 1} ω [ ( all ω-words σ i are distinct ) and (for each integer i ≥ 1, ρ i is a Büchi accepting run of T z over the ω-picture p σ i ) ]" . This is a Σ 1 1 -formula because "all ω-words σ i are distinct" can be expressed by the arithmetical formula:

"(∀j > k ≥ 1)(∃i ≥ 1) σ j (i) = σ k (i)".
Using the reduction K given by Lemma 5.5 we can easily see that L(M z ) is empty (respectively, infinite) if and only if L B (T K(z) ) = L(M z ) a is empty (respectively, infinite). This proves that

{z ∈ N | L(M z ) = ∅} ≤ 1 {z ∈ N | L B (T z ) = ∅} {z ∈ N | L(M z ) is infinite } ≤ 1 {z ∈ N | L B (T z ) is infinite }
and then the completeness result follows from the Σ 1 1 -completeness of the non-emptiness problem and of the infiniteness problem for ω-languages of Turing machines.

On the other hand it is easy to see that the language Σ ω,ω -(Σ ω ) a of ω-pictures is Büchi recognizable. But the class T S(Σ ω,ω ) is closed under finite union, so we get the following result. Lemma 5.7. If L ⊆ Σ ω is accepted by some Turing machine with a Büchi acceptance condition, then

L a ∪ [Σ ω,ω -(Σ ω ) a ]
is Büchi recognizable by a finite tiling system.

Notice that the constructions are effective and that they can be achieved in an injective way, so we can now state the following lemma, asserting the existence of a computable function H which will be often used in the sequel.

Lemma 5.8.

There is an injective computable function H from N into N satisfying the following property. If M z is the non deterministic Turing machine (equipped with a 1 ′ -acceptance condition) of index z, and if T H(z) is the tiling system (equipped with a Büchi acceptance condition) of index H(z), then

L(M z ) a ∪ [Σ ω,ω -(Σ ω ) a ] = L B (T H(z) )
recognizable language of infinite pictures is E-recognizable (respectively, A-recognizable). We are going to show that these problems are actually Π 1 2 -complete, using again some topological arguments.

Theorem 5.11. The problem to determine whether a given Büchi-recognizable language of infinite pictures is E-recognizable (respectively, A-recognizable) is Π 1 2 -complete, i.e. :

1. {z ∈ N | L B (T z ) is E-recognizable } is Π 1 2 -complete. 2. {z ∈ N | L B (T z ) is A-recognizable } is Π 1 2 -complete.
Proof. We first prove that the set

{z ∈ N | L B (T z ) is E-recognizable } is a Π 1 2 -set.
The sentence "L B (T z ) is E-recognizable" can be expressed by "∃y L B (T z ) = L E (T y )". The assertion "L B (T z ) = L E (T y )" can be expressed by a Π 1 2 -formula in a very similar manner as "L B (T z ) = L B (T y )" was (see the proof of Theorem 5.10), because for σ ∈ Σ ω and ρ ∈ {0, 1} ω the sentence "(ρ is a E-accepting run of T z over the ω-picture p σ )" can be expressed by an arithmetical formula. Moreover the quantifier ∃y is of type 0 thus "L B (T z ) is E-recognizable" can be expressed by a Π 1 2 -formula. We prove in a very similar manner that "L B (T z ) is A-recognizable" can be expressed by a Π 1 2 -formula. Details are here left to the reader.

We now prove the completeness part of the result. We first define a simple operation over ω-languages. For two ω-words x, x ′ ∈ Σ ω the ω-word x ⊗ x ′ is just the shuffle of the two ω-words x and x ′ which is simply defined by : for every integer n

≥ 1 (x ⊗ x ′ )(2n -1) = x(n) and (x ⊗ x ′ )(2n) = x ′ (n). For two ω-languages L, L ′ ⊆ Σ ω , the ω-language L ⊗ L ′ is defined by L ⊗ L ′ = {x ⊗ x ′ | x ∈ L and x ′ ∈ L ′ }.
We shall use the following construction. We know that there is a simple example of Σ 1 1 -complete set L ⊆ Σ ω accepted by a 1-counter automaton, hence by a Turing machine with 1 ′ acceptance condition, see [START_REF] Finkel | Borel hierarchy and omega context free languages[END_REF]. Then it is easy to define an injective computable function θ from N into N such that, for every integer z ∈ N, it holds that L(M θ(z) ) = (L ⊗ Σ ω ) ∪ (Σ ω ⊗ L(M z )).

We are going to use now the reduction H already considered above. We have seen that

L(M z ) = Σ ω if and only if L(T H(z) ) = Σ ω,ω
and we can easily see that

L(M θ(z) ) = Σ ω if and only if L(M z ) = Σ ω because L = Σ ω .
The reduction H • θ is an injective computable function from N into N. We consider now two cases.

First case. L(M

z ) = Σ ω . Then L(M θ(z) ) = Σ ω and L B (T H•θ(z) ) = Σ ω,ω . In particular L B (T H•θ(z)
) is E-recognized (respectively, A-recognized) by a tiling system. Second case. L(M z ) = Σ ω . Then there is an ω-word

x ∈ Σ ω such that x / ∈ L(M z ). But L(M θ(z) ) = (L ⊗ Σ ω ) ∪ (Σ ω ⊗ L(M z )) thus {σ ∈ Σ ω | σ ⊗ x ∈ L(M θ(z) )} = L is a Σ 1 1 -complete set. The function ψ x : σ → σ ⊗ x is continuous. This implies that L(M θ(z) ) is not a Borel set because otherwise L = {σ ∈ Σ ω | σ ⊗ x ∈ L(M θ(z) )} = ψ -1
x (L(M θ(z) )) would be also Borel as the inverse image of a Borel set by a continuous function [START_REF] Kechris | Classical descriptive set theory[END_REF]. Then it is easy to see that

L B (T H•θ(z) ) = L(M θ(z) ) a ∪ [Σ ω,ω -(Σ ω ) a ]
is not a Borel set. But it was proved in [START_REF] Finkel | On recognizable languages of infinite pictures[END_REF]Lemma 5.2] that every E-recognized language of infinite pictures is a Σ 0 2 -set and in [9, Lemma 5.3] that every A-recognized language of infinite pictures is a closed set. Thus in that case the ω-picture language L B (T H•θ(z) ) is neither E-recognizable nor A-recognizable.

Finally, using the reduction H • θ, we have proved that :

{z ∈ N | L(M z ) = Σ ω } ≤ 1 {z ∈ N | L B (T z ) is E-recognizable } {z ∈ N | L(M z ) = Σ ω } ≤ 1 {z ∈ N | L B (T z ) is A-recognizable }
and this ends the proof.

As in [START_REF] Finkel | On recognizable languages of infinite pictures[END_REF] we are going to infer from the proof of (high) undecidability of E-recognizability (respectively, A-recognizability) some other (high) undecidability results.

It was proved in [START_REF] Finkel | On recognizable languages of infinite pictures[END_REF] that for any Borel class Σ 0 α or Π 0 α , it is undecidable whether a given Büchirecognizable language of ω-pictures is in Σ 0 α (respectively, is in Π 0 α , is a Borel set). We can deduce from the above proof that the topological complexity of recognizable languages of infinite pictures is in fact highly undecidable. Theorem 5.12. Let α be a non-null countable ordinal. Then

1. {z ∈ N | L B (T z ) is in the Borel class Σ 0 α } is Π 1 2 -hard. 2. {z ∈ N | L B (T z ) is in the Borel class Π 0 α } is Π 1 2 -hard. 3. {z ∈ N | L B (T z ) is a Borel set } is Π 1 2 -hard.
Proof. We can use the same reduction H • θ as in the proof of Theorem 5.11. We have seen that there are two cases.

First case. L(M

z ) = Σ ω . Then L(M θ(z) ) = Σ ω and L B (T H•θ(z) ) = Σ ω,ω . In particular L B (T H•θ(z) ) = Σ ω,ω
is an open and closed subset of Σ ω,ω and it belongs to all Borel classes Σ 0 α and Π 0 α . Second case. L(M z ) = Σ ω . Then we have seen that L B (T H•θ(z) ) is not a Borel set. Finally, using the reduction H • θ, the result follows from the Π 1 2 -completeness of the universality problem for ω-languages of Turing machines.

We now come to the complementability problem. The class of Büchi-recognizable languages of infinite pictures is not closed under complement [START_REF] Altenbernd | Tiling Systems over Infinite Pictures and Their Acceptance Conditions[END_REF]. Thus the question naturally arises: "can we decide whether the complement of a Büchi-recognizable language of infinite pictures is Büchi-recognizable?". It has been proved in [START_REF] Finkel | On recognizable languages of infinite pictures[END_REF] that this problem is undecidable. We are going to prove that it is in fact Π 1 2 -complete.

Another classical problem is the determinizability problem: "can we decide whether a given recognizable language of infinite pictures is recognized by a deterministic tiling system?". Again this problem

We gave in [START_REF] Finkel | On recognizable languages of infinite pictures[END_REF] a solution to a question of [START_REF] Altenbernd | Tiling Systems over Infinite Pictures and Their Acceptance Conditions[END_REF], showing that all languages of infinite pictures which are accepted row by row by Büchi or Choueka automata reading words of length ω 2 are Büchi recognized by a finite tiling system, but the converse is not true. Then we showed that one cannot decide whether a given Büchi-recognizable language of infinite pictures is accepted row by row by a Büchi or Choueka automaton reading words of length ω 2 . We are going to show now that this decision problem is actually also Π 1 2 -complete.

Recall that an ω 2 -word x over the alphabet Σ is a sequence of length ω 2 of letters in Σ. It is denoted by (x(i)) 0≤i<ω 2 = x(0).x(1).x(2) . . . x(i) . . . , where for all i, 0 ≤ i < ω 2 , x(i) is a letter in Σ.

The set of ω 2 -words over Σ is denoted by Σ ω 2 . An ω 2 -language over Σ is a subset of Σ ω 2 .

To define a notion of acceptance row by row of an ω-picture we first associate, to an infinite picture p ∈ Σ ω,ω , an ω 2 -word p ∈ Σ ω 2 which is defined by p(ω.n + m) = p(m + 1, n + 1) for all integers n, m ≥ 0. This can be extended to languages of infinite pictures: for L ⊆ Σ ω,ω we denote L = {p | p ∈ L} so L is an ω 2 -language over Σ.

We refer the reader to [START_REF] Finkel | On recognizable languages of infinite pictures[END_REF] for a precise definition of generalized Büchi automaton acceptings words of ordinal length. We recall now the following definition.

Definition 5.14. A language of infinite pictures L ⊆ Σ ω,ω is accepted row by row by an ordinal Büchi automaton if and only if the ω 2 -language L is regular, i.e. is accepted by an ordinal Büchi automaton. We denote BA(Σ ω,ω ) the set of languages L ∈ Σ ω,ω such that L is regular.

We can now state the following result.

Theorem 5.15. The problem to determine whether a given Büchi-recognizable language of infinite pictures is accepted row by row by an ordinal Büchi automaton, is Π 1 2 -complete, i.e. :

{z ∈ N | LB (T z ) is regular } is Π 1 2 -complete.
Proof. Recall that, for each language of infinite pictures which is accepted row by row by a Büchi automaton reading words of length ω 2 , it was constructed in [START_REF] Finkel | On recognizable languages of infinite pictures[END_REF] a Muller tiling system accepting it. Then, using [1, Theorem 1], one can effectively construct a Büchi tiling system accepting the same language. The set T R of indices of Büchi tiling systems constructed from the proof of [9, Theorem 4.1] and [1, Theorem 1] is easily seen to be recursive. Notice that T R does not contain all indices of Büchi tiling systems accepting languages in BA(Σ ω,ω ). But for each language L in BA(Σ ω,ω ) there is an index z ∈ T R such that L = L B (T z ).

We can then express " LB (T z ) is regular " by the formula "∃y[( y ∈ T R ) and L B (T z ) = L B (T y )". This is a Π 1 2 -formula because "L B (T z ) = L B (T y )" can be expressed by a Π 1 2 -formula and the quantifier ∃y is of type 0.

To prove completeness we can use the same reduction H • θ as in the proof of Theorem 5.11. We have seen that there are two cases. First case. L(M z ) = Σ ω and then L B (T H•θ(z) ) = Σ ω,ω . In particular, L B (T H•θ(z) ) = Σ ω,ω is accepted row by row by an ordinal Büchi automaton. Second case. L(M z ) = Σ ω . Then we have seen that L B (T H•θ(z) ) is not a Borel set. Thus in that case LB (T H•θ(z) ) is not a regular ω 2 -language because otherwise L B (T H•θ(z) ) would be a Borel set (of rank smaller than or equal to 5), see [START_REF] Finkel | On recognizable languages of infinite pictures[END_REF]Proposition 4.2].

Finally, using the reduction H • θ, the completeness result follows from the Π 1 2 -completeness of the universality problem for ω-languages of Turing machines.

Concluding Remarks

We have given in this paper the exact degree of numerous natural decision problems for recognizable languages of infinite pictures. This way we have given examples of natural highly undecidable problems which are complete at the first or at the second level of the analytical hierarchy. Notice that many examples of Σ 1 1 -complete problems are already known, such as the recurring tiling problem, see for instance [START_REF] Harel | Recurring Dominoes: Making the Highly Undecidable Highly Understandable, Fundamentals of Computation Theory[END_REF][START_REF] Harel | Effective transformations on infinite trees, with applications to high undecidability, dominoes, and fairness[END_REF][START_REF] Hirst | Taking It to the Limit: On Infinite Variants of NP-Complete Problems[END_REF]. But it seems that very few natural problems, except some problems about ωlanguages of Turing machines, are known to be Π 1 2 -complete. One of the motivation of Castro and Cucker in [START_REF] Castro | Nondeterministic ω-Computations and the Analytical Hierarchy[END_REF] was actually to "give natural complete problems for the lowest levels of the analytical hierarchy which constitute an analog of the classical complete problems given in recursion theory for the arithmetical hierarchy". So we have added in this paper many new examples which complete the work of [START_REF] Castro | Nondeterministic ω-Computations and the Analytical Hierarchy[END_REF].

Notice that in another paper we have also given many natural Π 1 2 -complete problems about the infinite behaviour of very simple finite machines like 1-counter automata or 2-tape automata, [START_REF] Finkel | Highly Undecidable Problems for Infinite Computations[END_REF].

We hope also that our results could be useful in other connected areas, for instance in the study of the infinite behaviour of cellular automata, [START_REF] Wolfram | Theory and Applications of Cellular Automata[END_REF][START_REF] Delorme | Cellular Automata, a parallel Model[END_REF].

Acknowledgements.

Thanks to the anonymous referees for useful comments on a preliminary version of this paper.

We can now prove that the universality problem for Büchi-recognizable languages of infinite pictures is highly undecidable and give its exact degree.

Theorem 5.9. The universality problem for Büchi-recognizable languages of infinite pictures is Π 1 2complete, i.e. : {z ∈ N | L B (T z ) = Σ ω,ω } is Π 1 2 -complete.

Proof. We first check that the set {z ∈ N | L B (T z ) = Σ ω,ω } is in the class Π 1 2 . We can write that L B (T z ) = Σ ω,ω if and only if "∀ σ ∈ Σ ω ∃ρ ∈ {0, 1} ω ( ρ is a Büchi-accepting run of T z over p σ )". The two quantifiers of type 1 are followed by an arithmetical formula. Thus the set {z

In order to prove completeness we use the corresponding result for Turing machines proved in [START_REF] Castro | Nondeterministic ω-Computations and the Analytical Hierarchy[END_REF]: the

Consider now the injective computable function H from N into N given in Lemma 5.8. It is easy to see that for any Turing machine M z it holds that

We now consider the inclusion and the equivalence problems for Büchi-recognizable languages of infinite pictures.

Theorem 5.10. The inclusion and the equivalence problems for Büchi-recognizable languages of infinite pictures are Π 1 2 -complete, i.e. :

Proof. We first prove that the set {(y, z)

On the other hand it is easy to check that for all integers y, z, it holds that

). Thus using the reduction H we see that

The Π 1 2 -completeness follows then from the Π 1 2 -completeness of the inclusion and the equivalence problems for ω-languages of Turing machines proved in [START_REF] Castro | Nondeterministic ω-Computations and the Analytical Hierarchy[END_REF].

We are going to consider now the decision problems studied in [START_REF] Finkel | On recognizable languages of infinite pictures[END_REF]. Using topological arguments, we gave in [START_REF] Finkel | On recognizable languages of infinite pictures[END_REF] the answer to two questions raised in [START_REF] Altenbernd | Tiling Systems over Infinite Pictures and Their Acceptance Conditions[END_REF], showing that it is undecidable whether a Büchi has been proved to be undecidable in [START_REF] Finkel | On recognizable languages of infinite pictures[END_REF] and we shall prove it is in fact Π 1 2 -complete. Recall that a tiling system is called deterministic if on any picture it allows at most one tile covering the origin, the state assigned to position (i + 1, j + 1) is uniquely determined by the states at positions (i, j), (i + 1, j), (i, j + 1) and the states at the border positions (0, j + 1) and (i + 1, 0) are determined by the state (0, j), respectively (i, 0), [START_REF] Altenbernd | Tiling Systems over Infinite Pictures and Their Acceptance Conditions[END_REF]. As remarked in [START_REF] Altenbernd | Tiling Systems over Infinite Pictures and Their Acceptance Conditions[END_REF], the hierarchy proofs of the classical Landweber hierarchy defined using deterministic ω-automata "carry over without essential changes to pictures". In particular it is easy to see that a language of ω-pictures which is Büchi-recognized by a deterministic tiling system is a Π 0 2 -set and that a language of ω-pictures which is Muller-recognized by a deterministic tiling system is a boolean combination of Π 0 2 -sets, hence a ∆ 0 3 -set.

We can now state the following results.

Theorem 5.13. The determinizability problem and the complementability problem for Büchi-recognizable languages of infinite pictures are Π 1 2 -complete, i.e. :

It is easy to see that the set D of indices of deterministic tiling systems equipped with a Büchi acceptance condition is recursive. The formula ∃y ∈ D L B (T z ) = L B (T y ) can be written : "∃y[y ∈ D and L B (T z ) = L B (T y )]" and it can be expressed by a Π 1 2 -formula because the quantifier ∃y is of type 0 and "L B (T z ) = L B (T y )" can be expressed by a Π 1 2 -formula. Thus the set {z ∈ N | ∃y ∈ D L B (T z ) = L B (T y )} is in the class Π 1 2 . The case of deterministic tiling systems with Muller acceptance condition is very similar. Details are here left to the reader. On the other hand "Σ ω,ω -L B (T z ) = L B (T y )" can be expressed by a Π 1 2 -formula so " ∃y Σ ω,ω -L B (T z ) = L B (T y )" can be expressed by a Π 1 2 -formula because the quantifier ∃y is of type 0. Thus the

To prove completeness, we use the same reduction H • θ as in the proof of Theorem 5.11. We have seen that there are two cases.

First case. L(M

is accepted by a Büchi deterministic tiling system and also by a Muller deterministic tiling system. Morever its complement is empty so it is Büchi (or Muller) recognized by a tiling system. Second case. L(M z ) = Σ ω . Then we have seen that L B (T H•θ(z) ) is not a Borel set. Thus in that case L B (T H•θ(z) ) cannot be accepted by any deterministic tiling system with Büchi or Muller acceptance condition. Moreover its complement Σ ω,ω -L B (T H•θ(z) ) is not a Σ 1 1 -subset of Σ ω,ω because otherwise L B (T H•θ(z) ) would be in ∆ 1 1 = Σ 1 1 ∩ Π 1 1 which is the class of Borel sets by Suslin's Theorem. Thus Σ ω,ω -L B (T H•θ(z) ) cannot be Büchi-recognizable because it is not a Σ 1 1 -subset of Σ ω,ω and T S(Σ ω,ω ) ⊆ Σ 1 1 ⊆ Σ 1 1 , see [START_REF] Altenbernd | Tiling Systems over Infinite Pictures and Their Acceptance Conditions[END_REF]. Finally, using the reduction H • θ, the result follows from the Π 1 2 -completeness of the universality problem for ω-languages of Turing machines.