Ruijie J Yang
email: ruijyang@cityu.edu.hk

Liang Mei
email: liangmei2-c@my.cityu.edu.hk

Qingyong Zhang

Yingying Fan

Suk Shin
email: shin@unist.ac.kr

Damien Voiry
email: damien.voiry@umontpellier.fr

Dr Zhiyuan Y Zeng
email: zhiyzeng@cityu.edu.hk

Prof. H S Shin

High-yield production of mono-or few-layer transition metal dichalcogenides nanosheets by electrochemical lithium ion-intercalation and exfoliation method

Transition metal dichalcogenides (TMDs) nanomaterials, especially the mono-or fewlayer ones, have received extensive research interest owing to their suitable band-gap, unique electronic structure, outstanding biocompatibility, excellent stability, and promising catalytic properties. Therefore, the reliable production of these nanomaterials with atomically thin thickness and laterally uniform dimension are essential for their promising applications in transistors, photodetectors, electroluminescent devices, catalysis, energy conversion, environment remediation, biosensing, bioimaging, and so on. Recently, the electrochemical lithium ion-intercalation and exfoliation method has emerged as a mature, efficient and promising strategy for the high-yield production of mono-or few-layer TMDs nanosheets; mono-layer MoS2 (yield of 92%), mono-layer TaS2 (yield of 93%) and bi-layer TiS2 (yield of 93%) with lateral dimensions of around 1 µm. This Protocol describes the details of experimental procedures for the high-yield synthesis of mono-or few-layer TMDs and other inorganic nanosheets such as MoS2, WS2, TiS2, TaS2, ZrS2, graphene, h-BN, NbSe2, WSe2, Sb2Se3, and Bi2Te3 by using the electrochemical lithium ion-intercalation and exfoliation method which involves the electrochemical intercalation of lithium ions into the layered inorganic crystals and a mild sonication. In addition, this method offers some more advantages such as the control over the nature of the crystal polytypes. The whole protocol takes 26-38 h for the successful production of the final ultrathin inorganic nanosheets.

Introduction

Transition metal dichalcogenides (TMDs) nanomaterials, especially the mono-or fewlayered crystals, have received extensive research interest owing to their unique properties associated with various kinds of promising applications in transistors, photodetectors, electroluminescent devices, catalysis, energy conversion, environment remediation, biosensing, bioimaging, and so on [START_REF] Wang | Electronics and optoelectronics of two-dimensional transition metal dichalcogenides[END_REF][START_REF] Xu | Spin and pseudospins in layered transition metal dichalcogenides[END_REF][START_REF] Deng | 3D Ordered Macroporous MoS2@C Nanostructure for Flexible Li-Ion Batteries[END_REF][START_REF] Desai | /sub> transistors with 1-nanometer gate lengths[END_REF][START_REF] Mak | Photonics and optoelectronics of 2D semiconductor transition metal dichalcogenides[END_REF][START_REF] Joensen | Single-layer MoS2. Materials structures of tungsten disulphide[END_REF][7][START_REF] Wilson | The transition metal dichalcogenides discussion and interpretation of the observed optical, electrical and structural properties[END_REF][START_REF] Koppens | Photodetectors based on graphene, other twodimensional materials and hybrid systems[END_REF][START_REF] Manzeli | 2D transition metal dichalcogenides[END_REF][START_REF] Zhao | Engineering covalently bonded 2D layered materials by selfintercalation[END_REF][START_REF] Liu | Self-optimizing, highly surface-active layered metal dichalcogenide catalysts for hydrogen evolution[END_REF][START_REF] Voiry | Enhanced catalytic activity in strained chemically exfoliated WS2 nanosheets for hydrogen evolution[END_REF][START_REF] Liu | Spontaneous self-intercalation of copper atoms into transition metal dichalcogenides[END_REF][START_REF] Yu | Gate-tunable phase transitions in thin flakes of 1T-TaS2[END_REF][START_REF] Ye | Superconducting Dome in a Gate-Tuned Band Insulator[END_REF][START_REF] Zhao | High-order superlattices by rolling up van der Waals heterostructures[END_REF][START_REF] Li | Van der Waals epitaxial growth of air-stable CrSe2 nanosheets with thickness-tunable magnetic order[END_REF][START_REF] Li | General synthesis of two-dimensional van der Waals heterostructure arrays[END_REF][START_REF] Kong | Doping-free complementary WSe2 circuit via van der Waals metal integration[END_REF] . Their unique properties provide opportunities to go beyond those of graphene [START_REF] Chhowalla | The chemistry of two-dimensional layered transition metal dichalcogenide nanosheets[END_REF] . Therefore, the reliable production of these nanomaterials with atomically thin and uniform properties is of high importance both fundamentally and practically. Recent advances in synthesis technologies of ultrathin 2D nanomaterials have enabled the fabrication of mono-or few-layered TMDs nanomaterials via various synthesis approaches, including mechanical cleavage [START_REF] Xi | Ising pairing in superconducting NbSe2 atomic layers[END_REF][START_REF] Radisavljevic | Singlelayer MoS2 transistors[END_REF][START_REF] Mak | Atomically Thin ${\mathrm{MoS}}_{2}$: A New Direct-Gap Semiconductor[END_REF][START_REF] Splendiani | Emerging Photoluminescence in Monolayer MoS2[END_REF] , chemical vapour deposition (CVD) growth [START_REF] Zhou | A library of atomically thin metal chalcogenides[END_REF][START_REF] Wang | High-quality monolayer superconductor NbSe2 grown by chemical vapour deposition[END_REF][START_REF] Lin | Growth of environmentally stable transition metal selenide films[END_REF][START_REF] Najmaei | Vapour phase growth and grain boundary structure of molybdenum disulphide atomic layers[END_REF][START_REF] Lee | Synthesis of Large-Area MoS2 Atomic Layers with Chemical Vapor Deposition[END_REF][START_REF] Feldman | High-Rate, Gas-Phase Growth of MoS2 Nested Inorganic Fullerenes and Nanotubes[END_REF][START_REF] Van Der Zande | Grains and grain boundaries in highly crystalline monolayer molybdenum disulphide[END_REF][START_REF] Shi | Two-dimensional metallic tantalum disulfide as a hydrogen evolution catalyst[END_REF][START_REF] Gong | Spatially controlled doping of two-dimensional SnS2 through intercalation for electronics[END_REF] , molecular beam epitaxy (MBE) [START_REF] Ugeda | Characterization of collective ground states in single-layer NbSe2[END_REF][START_REF] Ge | Superconductivity above 100 K in single-layer FeSe films on doped SrTiO3[END_REF][START_REF] Barja | Charge density wave order in 1D mirror twin boundaries of single-layer MoSe2[END_REF][START_REF] Ugeda | Giant bandgap renormalization and excitonic effects in a monolayer transition metal dichalcogenide semiconductor[END_REF] , wet-chemical synthesis [START_REF] Mahler | Colloidal Synthesis of 1T-WS2 and 2H-WS2 Nanosheets: Applications for Photocatalytic Hydrogen Evolution[END_REF][START_REF] Yoo | Chemical Synthetic Strategy for Single-Layer Transition-Metal Chalcogenides[END_REF] , liquid intercalation [START_REF] Voiry | Enhanced catalytic activity in strained chemically exfoliated WS2 nanosheets for hydrogen evolution[END_REF][START_REF] Lin | Solution-processable 2D semiconductors for high-performance large-area electronics[END_REF][START_REF] Cullen | Ionic solutions of two-dimensional materials[END_REF][START_REF] Eda | Photoluminescence from Chemically Exfoliated MoS2[END_REF][START_REF] Kappera | Phase-engineered low-resistance contacts for ultrathin MoS2 transistors[END_REF][START_REF] Li | Printable two-dimensional superconducting monolayers[END_REF][START_REF] Zheng | High yield exfoliation of two-dimensional chalcogenides using sodium naphthalenide[END_REF][START_REF] Chou | Controlling the Metal to Semiconductor Transition of MoS2 and WS2 in Solution[END_REF][START_REF] Chou | Chemically Exfoliated MoS2 as Near-Infrared Photothermal Agents[END_REF][START_REF] Chou | Ligand Conjugation of Chemically Exfoliated MoS2[END_REF][START_REF] Coleman | Two-Dimensional Nanosheets Produced by Liquid Exfoliation of Layered Materials[END_REF][START_REF] Feng | Metallic Few-Layered VS2 Ultrathin Nanosheets: High Two-Dimensional Conductivity for In-Plane Supercapacitors[END_REF][START_REF] Wan | Flexible n-type thermoelectric materials by organic intercalation of layered transition metal dichalcogenide TiS2[END_REF] , and so on. However, preparing high-quality and high-yield mono-or few-layered TMDs nanosheets under mild conditions remains a challenge. For example, ultrathin 2D TMDs nanosheets fabricated using mechanical cleavage are plagued by the low-yield of the products and the difficulty for the controlling of the thickness, size, and shape of the 2D TMDs nanosheets. In terms of CVD growth, its working environment is relatively harsh, always needs high temperature and high vacuum, which are not always compatible with the targeted applications. Regarding wet-chemical synthesis, the limitation is that the synthesis process is easily affected by the reaction conditions (e.g., temperature, time, types and concentrations of precursors, solvents, and surfactant), which leads to the uncontrollability for fabrication of the 2D TMDs nanosheets. Besides, TMDs fabricated by liquid exfoliation show an intrinsically broad thickness distribution, leading to poor film quality and unsatisfactory thin-film electrical performance.

Motivated by these, our group has developed an electrochemical lithium ionintercalation and exfoliation strategy for the fabrication of mono-layer TMDs nanomaterials with better quality and high-yield (> 90%) under mild conditions (at room temperature), also with relative short time [START_REF] Zeng | Single-Layer Semiconducting Nanosheets: High-Yield Preparation and Device Fabrication[END_REF][START_REF] Zeng | An Effective Method for the Fabrication of Few-Layer-Thick Inorganic Nanosheets[END_REF][START_REF] Zeng | Growth of noble metal nanoparticles on single-layer TiS2 and TaS2 nanosheets for hydrogen evolution reaction[END_REF][START_REF] Tan | Liquid-Phase Epitaxial Growth of Two-Dimensional Semiconductor Hetero-nanostructures[END_REF] . In 2011, our group first reported the electrochemical lithium ion-intercalation and exfoliation synthesis of mono-layer MoS2, WS2, TiS2, TaS2, ZrS2 and graphene nanosheets, and achieved the 92% yield for mono-layer MoS2 [START_REF] Zeng | Single-Layer Semiconducting Nanosheets: High-Yield Preparation and Device Fabrication[END_REF] . Meanwhile, by regulating the cut-off voltage and discharge current in the synthesis process, few-layer-thick BN, NbSe2, WSe2, Sb2Se3, and Bi2Te3 were also obtained [START_REF] Zeng | An Effective Method for the Fabrication of Few-Layer-Thick Inorganic Nanosheets[END_REF] . In addition, the high-yield production of mono-layer TaS2 (93%) and bi-layer TiS2 (93%) was achieved via the optimization of the electrochemical intercalation condition [START_REF] Zeng | Growth of noble metal nanoparticles on single-layer TiS2 and TaS2 nanosheets for hydrogen evolution reaction[END_REF] . Note that the statistics atomic force microscope (AFM) measurements were conducted by testing large quantity (> 100 pieces for each TMD) of TMDs nanosheets, and the results shows that 92% of MoS2 and 93% of TaS2 nanosheets are mono-layers (< 1 nm), and 93% of TiS2 nanosheets are bi-layers (< 2 nm). To the best of our knowledge, until now, besides our work, no other route of exfoliation allows obtaining such high yield production of mono-layer MoS2, TaS2 and bi-layer TiS2 through a general exfoliation method. Here, the key parameter to determine whether producing mon-, bi-or few-layer TMDs nanosheets is the lithium intercalation extent in layers, if every layer is intercalated with lithium ions, then monlayer will be produced after exfoliation, if only parts of the layers are intercalated with lithium ions, then bi-or few-layer will be synthesized. Furthermore, TiS2-based heteronanostructures, such as TiS2-CuS, TiS2-ZnS and TiS2-Ni3S2 have also been obtained via this method, by using the layered TiS2 coated copper (for preparation of TiS2-CuS), zinc (for preparation of TiS2-ZnS) and nickel (for preparation of TiS2-Ni3S2) foil discs as cathodes, respectively [START_REF] Tan | Liquid-Phase Epitaxial Growth of Two-Dimensional Semiconductor Hetero-nanostructures[END_REF] .

Importantly, compared with the traditional chemical ion-intercalation and exfoliation method for the fabrication of mono-or few-layer TMDs nanosheets, our developed electrochemical ion-intercalation and exfoliation method is relatively simple, straightforward and offer a higher degree of control under a mild condition. The traditional chemical ion-intercalation and exfoliation method, which uses n-butyl lithium in hexane as the intercalation agent, needs to be carried out at relatively high temperature (e.g., 100 o C) for long time (e.g., 3 days) and lacks the controllability over the amount of Li insertion. Incomplete lithium intercalation will result in low yield of mono-layer production, while excessive lithium insertion will lead to the chemical decomposition of the crystals. Our developed electrochemical ion-intercalation and exfoliation method for the high-yield preparation of mono-layer TMDs nanosheets can be easily conducted at room temperature within 26 h. In addition, the Li insertion can be monitored and finely controlled using the battery testing equipment and exhibited in the form of galvanostatic discharge curves, and the galvanostatic discharge can be stopped at a proper Li content to avoid decomposition of the intercalated compounds.

Besides the improved feasibility and effectiveness in the methodology, our procedure is also scalable, and we believe that further scale-up production of mono-layer TMDs nanosheets for industry applications can be realized by increasing the bulk TMDs amount from milligrams (mg) to gram (g), or even to ton. Also, the lateral size of the exfoliated mon-layer nanosheets can be further increased from around 1 µm, to higher than 1 µm by increasing the lateral size of TMDs bulk materials. Interestingly, the phase transition of TMDs from the semiconducting 2H phase to the metallic 1T phase during the exfoliation process was achieved, which offers great opportunities towards phase engineering of TMDs, in order to realize superior performance for some emerging applications in electronics or electro-catalysis.

In this Protocol, we systematically describe the detailed experimental procedures of our established electrochemical lithium ion-intercalation and exfoliation strategy for the synthesis of mono-or few-layer inorganic nanosheets, including MoS2, WS2, TiS2, TaS2, graphene, h-BN, NbSe2, WSe2, Sb2Se3, and Bi2Te3. Our aim is to increase the general understanding of this exfoliation route for the efficient synthesis of ultrathin TMDs nanosheets and thus benefit the community from their various potential applications. In addition, it will also promote the fabrication of some other ultrathin 2D nanomaterials with atomic level thickness besides TMDs, such as black phosphorus, g-C3N4, metal oxides [START_REF] Kumbhakar | Emerging 2D metal oxides and their applications[END_REF][START_REF] Zhu | Layer-by-Layer Assembled 2D Montmorillonite Dielectrics for Solution-Processed Electronics[END_REF][START_REF] Yang | MnO2-Based Materials for Environmental Applications[END_REF] etc. We also believe that this protocol will attract a broad research interest from chemists, material scientists, crystallographers, photoelectric device manufacturer, catalysis researcher, bioscientist etc.

Applications:

Our developed electrochemical lithium ion-intercalation and exfoliation method for the high-yield production of mono-or few-layer TMDs nanosheets has opened a new direction for basic and applied research, attracting the attention of both academia and industry [START_REF] Zhang | Intercalation and exfoliation chemistries of transition metal dichalcogenides[END_REF] . The prepared mono-or few-layer TMDs nanosheets by this method has been wildly applied in various fields such as gas-sensing [START_REF] He | Fabrication of Flexible MoS2 Thin-Film Transistor Arrays for Practical Gas-Sensing Applications[END_REF] , memory devices [START_REF] Yin | Memory Devices Using a Mixture of MoS2 and Graphene Oxide as the Active Layer[END_REF] , detection of biomolecules [START_REF] Zhu | Single-Layer MoS2-Based Nanoprobes for Homogeneous Detection of Biomolecules[END_REF] , electrocatalytic hydrogen evolution [START_REF] Zeng | Growth of noble metal nanoparticles on single-layer TiS2 and TaS2 nanosheets for hydrogen evolution reaction[END_REF][START_REF] Huang | Solution-phase epitaxial growth noble metal nanostructures on dispersible single-layer molybdenum disulfide nanosheets[END_REF][START_REF] Mei | Size-selective synthesis of platinum nanoparticles on transitionmetal dichalcogenides for the hydrogen evolution reaction[END_REF] , light-emitting diodes [START_REF] Yin | Preparation of MoS2-MoO3 Hybrid Nanomaterials for Light-Emitting Diodes[END_REF] , and lithium-ion battery [START_REF] Tan | Liquid-Phase Epitaxial Growth of Two-Dimensional Semiconductor Hetero-nanostructures[END_REF][START_REF] Zeng | In Situ Study of Lithiation and Delithiation of MoS2 Nanosheets Using Electrochemical Liquid Cell Transmission Electron Microscopy[END_REF] . Based on a literature survey using Web of Science, the past few years have witnessed a continuation in the upward trend to adopt our developed method for the fabrication of mono-or few-layer TMDs nanosheets, due to the specificity, timeliness and innovation of this method.

Limitations:

To date, mono layer (< 1 nm) MoS2, WS2, TiS2, TaS2, ZrS2, and graphene nanosheets as well as several multi-layer (< 4 nm) BN, NbSe2, WSe2, Sb2Se3, and Bi2Te3 nanosheets have been successfully fabricated by our electrochemical lithium ion-intercalation and exfoliation method. While, as for the synthesis of other mono-or few-layered 2D nanomaterials, such as MoSe2, VSe2, black phosphorus, g-C3N4, substantial modifications of the protocol might be necessary.

Furthermore, in spite of the interesting phase transformation of TMDs from the semiconducting 2H phase to the metallic 1T or distorted 1T' phase during the exfoliation process, the switch of phase transition has not been fully controlled.

Previous crystal field theory explained that the intercalation of ions involves the transfer of electron from the s orbitals of guest ions to the d orbitals of the host transition-metal atoms [START_REF] Yang | Structural and quantum-state phase transitions in van der Waals layered materials[END_REF][START_REF] Voiry | Phase engineering of transition metal dichalcogenides[END_REF][START_REF] Li | Phase transitions in 2D materials[END_REF] . The injection of electron beyond a certain threshold causes the stability of 2H phase to be lower than that of 1T or 1T′ phase, inducing the corresponding phase transitions. Although the mechanism of phase transformation is clear, the control is still challenging. We will continue devoting our efforts toward the precise and purposeful control of phase modulation during the preparation of mono-or few-layer TMDs nanosheets by our developed electrochemical lithium ion-intercalation and exfoliation method, to realize the controllable synthesis of high purity phase.

Experimental design:

The mono-or few-layer inorganic nanosheets in this protocol, are synthesized by the electrochemical lithium ion-intercalation and exfoliation method, which involves the electrochemical intercalation of lithium ion into layered bulk materials, followed by a mild sonication and exfoliation process in solvent (Fig. 1).

The electrochemical lithium ion-intercalation process of the layered bulk materials is carried out in a test coin cell using Li foil as anode, copper (Cu) foil with uniformly coated layered bulk material as cathode, polypropylene (pp) film as the separator, and 1.0 M LiPF6 in a mixture of ethyl carbonate (EC) and dimethyl carbonate (DMC) (with 1:1 volume ratio)) solution as electrolyte. The prepared coin cell is discharged by connecting it to a Neware battery test system with a galvanostatic discharge current of 0.05 mA or 0.025 mA and a controlled cut off voltage at room temperature. During such a discharge process, lithium ions are inserted into the interlayer of the layered bulk materials, resulting in the enlarging of interlayer spacing and the weakening of interlayer van der Waals interaction. Such changes of interlaminar properties will make the subsequent exfoliation process easier to realize. Importantly, the amount of lithium intercalation can be effectively controlled by tuning the cut-off voltage (or the discharge capacity). This superior feature can make the lithium intercalation process stop at an appropriate lithium amount, avoiding the decomposition of lithium intercalated compounds.

After lithium intercalation, dismantle the coin cell and take out the Cu foil, whose surface coated with lithium intercalation compound materials, then wash it with acetone to remove the residual electrolyte (i.e., LiPF6), and dry it. The lithiumintercalated compound materials on the surface of Cu foil are then immersed in deionized (DI) water (or ethanol) and sonicated for 5 -10 min. After that, the opaque suspension of the exfoliated materials will be obtained.

During the lithium ion battery discharging process, the lithium ions, released from lithium metal anode, inserted into the interlayer spaces of layered bulk cathode materials, play dual roles. First, after intercalation, the lithium ions get electrons from lithium metal anode through external circuit (battery testing equipment) and reduced to lithium atom between the Van der Waals gaps of the layered materials.

Second, the intercalated lithium metal atoms in between layered materials react with DI water (H2O) to form lithium hydroxide (LiOH), along with the releasing of hydrogen (H2) gas (evidently bubbles can be observed during the ultrasound process in water), which will expand the interlayer distance and weaken the van der Waals interactions between the layers and thus help the layers separate. Note that, there are remaining charges (lithium ions) on the nanosheets after the exfoliation process, which has been confirmed using Zeta potential measurements and estimated to be 15 to 20 % in the case of group-6 TMDs [START_REF] Heising | Structure of Restacked MoS2 and WS2 Elucidated by Electron Crystallography[END_REF] . Such remaining charges of the exfoliated materials induces a stabilization of the suspension by increasing the entropy of the system 72 .

After exfoliation of the intercalated layered materials in H2O, the obtained opaque suspensions are centrifuged and washed 1 ~2 times to get the final 2D nanosheets for transmission electron microscope (TEM), atomic force microscope (AFM), Xray photoelectron spectroscopy (XPS), Raman, and ultraviolet-visible spectroscopy (UV-Vis) characterizations. TEM is used to investigate the crystal structure, morphology and size of the exfoliated materials. AFM can directly measure the thickness and size of the 2D nanosheets and help to determine the mono-layer yield. XPS is used to determine the elemental composition and valence of the exfoliated nanosheets. Raman is used to deduce electronic structure. UV-Vis can provide information about the photon absorption property. In this protocol, we describe how to prepare this mono-or few-layer nanosheets for these characterizations as well as how to use these analytical methods to analyze the properties of the exfoliated counterparts.

Materials

Reagents:

! CRITICAL: Store the bulk inorganic layered materials (e.g., MoS2, WS2, TiS2, TaS2, Natural graphite, BN, NbSe2, WSe2, Sb2Se3, and Bi2Te3) in Ar-filled glove box, because they are easily deliquesced in air. Meanwhile, keep them away from light by wrapping the bottle of bulk inorganic layered materials in aluminum foil.

• Molybdenum disulfide (MoS2; 10-30 μm, Rose Mill, West Hartford, USA, cat. no. C10404842).

• Tungsten (IV) disulfide (WS2, Sigma, Steinheim, Germany, cat. no. C10534199)

• Titanium (IV) sulfide (TiS2, Sigma, Steinheim, Germany, cat. no. MKCH8948)

• Tantalum (IV) sulfide (TaS2, Alfa Aesar, Massachusetts, USA, cat. no. 808717) (with 1:1 volume ratio)) solution as electrolyte. Note that the coin cell should be assembled in a glove box filled with Ar. The prepared coin cell is connected with the Neware battery test system and discharged with a galvanostatic discharge current of 0.05 or 0.025 mA and a cut-off voltage of 0.9 V (ZrS2: 0.7 V, BN: 0.4 V, Graphite 0.1V) at room temperature. A typical setup is shown in Fig. 2. Note that, the amount of intercalated lithium ions in layered materials (or the discharge capacity) can be effectively controlled by tuning the cut-off voltage according to the discharge profile. Because the discharge capacity can transform into lithium intercalation amount by calculating and substituting the discharge capacity that corresponding to 1 Moore lithium intercalation for different kinds of inorganic layered materials, as shown in Supplementary Table 1.

TEM setup • In this protocol, TEM characterization is carried out on a JEOL JEM-2100F

TEM instrument under the accelerating voltage of 200 kV. The TEM samples are prepared by dropping the as-prepared mono-or few-layer inorganic nanosheets solution onto lacey carbon 300 mesh copper TEM grids and then dried in air.

AFM setup

• In this protocol, AFM characterization is carried out on a Veeco Dimension 3100 AFM instrument under the tapping mode and with a Si tip (spring constant: 42 N/m; resonance frequency: 320 kHz). The AFM samples are prepared by dropping the as-prepared mono-or few-layer inorganic nanosheets solution onto an APTES-modified Si/SiO2 substrate and then dried in air.

Procedure

Synthesis of mono-or few-layer TMDs nanosheets (Timing 26 h):

1. Prepare the mixed slurry of layered bulk material, acetylene black, PVDF and NMP. Weigh out 200 mg of layered bulk material (i.e., MoS2, WS2, TiS2, TaS2, ZrS2, graphite, BN, NbSe2, WSe2, Sb2Se3, or Bi2Te3), 25 mg of acetylene black, and 25 mg PVDF into a 10-mL glass vial, and then add 1.6 mL NMP and a rotor.

After heating the vial in 60 °C oven for 30 min, put the mixed slurry contained 10-mL glass vial on magnetic stir plate and stir the mixed slurry for 7.5 h. (Timing 8 h) ! CRITICAL STEP: The mixed slurry needs to be stirred in a closed vial with good sealability because the slurry is easy to adsorb water.

2. Uniformly coat the obtained mixed slurry on one side of a copper foil using an infrared heating tape casting coater. Then put it into a vacuum drying oven and dry it at 100 °C under vacuum for 8 h. After drying, using precision disc cutting machine cut it into disc with diameter of 1.2 cm for next coin cell assembling.

(Timing 8 h) 3. Assemble the coin cell in an Ar-filled glove box (Fig. 3a). (Timing 1 h) (Ⅰ) Put the stainless steel cap on a piece of white paper.

(II) Put the TMDs bulk materials coated copper discs (with diameter of 1.2 cm), in the cap.

(III) Put the polypropylene (pp) film (with diameter of 1.6 cm) on the top of Cu discs.

(IV) Use 20 µL pipette gun to apply 4µL electrolyte on the polypropylene (pp) film.

(V) Put the lithium foil (with diameter of 1.4 cm), gasket, stainless steel shrapnel and aluminum coated cup on the top in turn.

(Ⅷ) Press it using punching machine with 500 psi, and then the cell is ready for testing.

Take out the pressed coin cell and let the electrolyte soak the electrode thoroughly for 4 hours.

! CRITICAL STEP: The water content in glove box should be extremely low,

because the lithium foil is very sensitive to water.

4.

Started the galvanostatic discharge process (Fig. 3b). Connect the obtained coin cell to the battery test system. Then, set up current density at 0.05 mA or 0.025 mA, and cut-off voltage at 0.9 V (ZrS2: 0.7 V, BN: 0.4 V, Graphite 0.1V). Press the start button to start the discharge process. After reaching the cut-off voltage, remove the coin cell from the battery test system and let the lithium intercalated electrode relax for 1 h. (Timing 7 h)

! CRITICAL STEP: The discharge current and cutoff voltage are very important for electrochemical Li intercalation process. High discharge current may induce the structure degradation but can shorten reaction time. Low discharge current will require much time to finish the intercalation but can avoid structure degradation.

High cutoff voltage may cause the insufficient lithium intercalation and thus not good exfoliation efficiency. While low cutoff voltage will result in the structure degradation. In this protocol, for MoS2, discharge current is 0.05 mA, cutoff voltage is 0.9 V. For TaS2, TiS2, WS2, discharge current is 0.025 mA, cutoff voltage is 0.9 V. For ZrS2, discharge current is 0.025 mA, cutoff voltage is 0.7 V. For graphene, discharge current is 0.025 mA, cutoff voltage is 0.1 V. For BN, discharge current is 0.025 mA, cutoff voltage is 0.4 V. For NbSe2, discharge current is 0.025 mA, cutoff voltage is 0.7 V. The corresponding galvanostatic discharge curves are shown in Supplementary Fig. 1. About how to determine these cut-off voltages for different materials, our explanation is as follows: for the materials that show obvious discharge plateau in galvanostatic discharge curve (such as MoS2. etc), the cut-off voltage should be a little lower than the platform voltage. While, for those without discharge plateau in discharge curve, SEM, TEM and AFM should be systematic combined for characterization of the product obtained under different cut-off voltage, to see its morphology (nanosheets or decomposed nanoparticles) and thickness, thus to determine whether the selected cut-off voltage is suitable for this product.

5. Disassemble the coin cell and take out the TMDs coated copper foil after Liintercalation (Fig. 3c). Then wash it immediately with acetone to remove any residual electrolyte (LiPF6/EC/DMC). Then immerse the TMDs coated copper foil in a 10-mL glass vial containing 10 mL distilled water or ethanol (Fig. 3d). Then, put the 10-mL glass vial into the ultrasonic pool and ultrasonic for 5 min with 28 KHz. After that, an opaque suspension of the exfoliated mono or few layer nanosheets is formed. (Timing 10 min) ! CRITICAL STEP: After taking out the copper foil loaded with Li-intercalated sample from the coin cell, this washed process should be quick, and immediately perform the next exfoliation step, because the Li-intercalated sample is sensitive to the environment. For NbSe2, WSe2, Sb2Se3, and Bi2Te3, it is better do N2 bubbles treatment for DI water in the glass vial before the ultrasonication process, this process can remove the dissolved oxygen in DI water and alleviate the surface oxidation for the exfoliated TMDs nanosheets in the dispersion. ! TROUBLESHOOTING 6. Started the centrifugal washing process (Fig. 3e). Put the obtained opaque suspension into 20-mL centrifuge tube. Centrifuge it for 15 min with 2000 rpm.

Then collect the supernatant into a new 20-mL centrifuge tube. Centrifuge it for 15 min with 10000 rpm. Then pour the supernatant away, leaving the bottom sedimentation, followed by re-disperse and the sedimentation in newly added DI water and doing 10000 rpm centrifuges for 1 ~ 2 times. Then collect the bottom sedimentation of the last centrifugation (the final products) into a 10-mL glass vial.

Finally, redisperse the collected sediment into opaque suspension by adding water or ethanol, and store this opaque suspension for use.

! CRITICAL STEP: The centrifugal speed affects the quality of final products. Too low centrifugation speed in first step is not enough to remove the un-exfoliated

TMDs materials, acetylene black and PVDF. A too high centrifugation speed in first step causes the mono or few layer nanosheets loss as well. A too low centrifugation speed in second step induces the mono or few layer nanosheets loss, since they all not completely centrifuged to bottom and exited as sedimentation.

While a too high centrifugation speed in second step results in obtained nanosheets too small in lateral size.

! PAUSE POINT: The obtained mono or few layer TMDs nanosheets can be stored at room temperature under dry and dark condition for several months without noticeable change.

TEM characterization of the obtained mono-or few-layer TMDs nanosheets (Timing 3 h):

7. Take 0.5 mL mono or few layer TMDs nanosheets collected from step 6 into 2-mL glass vial and add 1.5 mL water to dilute the TMDs nanosheets dispersion.

Then, shake the glass vial to obtain a homogeneous TMDs nanosheets solution.

8. Deposit 5 μL of this solution onto a lacey carbon-coated copper grids and let it dry in air for 15 min before the characterization. 9. Image the exfoliated mono or few layer TMDs nanosheets by using a JEOL JEM-2100F TEM instrument.

AFM characterization of the obtained mono-or few-layer TMDs nanosheets (Timing 5 h):

10. Prepare APTES-modified Si/SiO2 substrate.

(Ⅰ) Put Si/SiO2 substrate (10 mm × 8 mm) into acetone and sonicate for 5 min.

Take the substrate out from acetone and rinse it with DI water.

(Ⅱ) Then put the Si/SiO2 substrate into piranha solution (volume ratio H2SO4: H2O2 = 7:3) and heat the mix solution to boiling point for 30 min (hot plate temperate 120 °C).

(ⅡI) Take out the Si/SiO2 substrate, rinse it with DI water using wash bottle and blow it using N2 gas to dry the substrate.

(IV) Immerse the obtained Si/SiO2 substrate in APTES solution (volume ration APTES: H2O = 1:100), and put it into the vacuum condition for 10 min. Then take out the Si/SiO2 substrate, rinse it with DI water and blow it using the N2 gas.

(V) Then put this Si/SiO2 substrate in 120 ℃ oven for 2 h to get the final APTESmodified Si/SiO2 substrate. ! CRITICAL STEP: APTES functionalization is critical for the successful AFM measurement. APTES is positively charged on the surface, which is beneficial for electrostatic adsorption of the negatively charged TMDs nanosheets. Otherwise, the TMDs nanosheets are easily aggregated through drop casting on Si/SiO2 substrate. Without APTES functionalization, the AFM measurement of TMDs nanosheets is not easily succeed, especially for the single-layer or few layers nanosheets.

11. Take 0.5 mL mono or few layer TMDs nanosheets collected in step 6 into 2-mL glass vial and add 1.5 mL DI water. Then, shake the glass vial to obtain a homogeneous TMDs nanosheets solution.

12. Put the APTES-modified Si/SiO2 substrate (Prepared in step 10) into the 2-mL glass vial containing homogeneous TMDs nanosheets solution (prepared in step 11) and immerse it in solution for 15 min. Then, take Si/SiO2 substrate out and rinse it with DI water. After that, blow substrate use the N2 gas for 5 min.

Troubleshooting

Troubleshooting advice can be found in Table 1.

Anticipated results

After sonication treatment (Step 5), an opaque suspension of the exfoliated materials should be obtained (Fig. 4). After the whole electrochemical lithium ion-intercalation and exfoliation processes, the mono layer (thickness < 1 nm) MoS2, WS2, TiS2, TaS2, and multi-layers (thickness < 4 nm) BN, NbSe2 nanosheets can be obtained (Fig. 5 and Fig. 6) with high yield (MoS2 and TaS2 mono layer & TiS2 double layer yield > 90%) (Fig. 7) (Supplementary Table 2). Besides, mono layer (thickness < 1 nm) ZrS2 and graphene nanosheets, as well as multi-layers (thickness < 4 nm) WSe2, Sb2Se3, Bi2Te3 nanosheets can also been obtained (Supplementary Fig. 2). The elemental composition and valence (Fig. 8), electronic structure (Fig. 9a-d), and photon absorption property (Fig. 9e-h) of the exfoliated nanosheets are also provided. with permission from ref. [START_REF] Zeng | Single-Layer Semiconducting Nanosheets: High-Yield Preparation and Device Fabrication[END_REF][START_REF] Zeng | An Effective Method for the Fabrication of Few-Layer-Thick Inorganic Nanosheets[END_REF] . Adapted with permission from ref. [START_REF] Zeng | Single-Layer Semiconducting Nanosheets: High-Yield Preparation and Device Fabrication[END_REF][START_REF] Zeng | An Effective Method for the Fabrication of Few-Layer-Thick Inorganic Nanosheets[END_REF][START_REF] Zeng | Growth of noble metal nanoparticles on single-layer TiS2 and TaS2 nanosheets for hydrogen evolution reaction[END_REF] . Importantly, in the Raman spectra of MoS2 nanosheets, three weak but distinct new peaks J1 (153 cm -1 ), J2 (225 cm -1 ) and J3 (327 cm -1 ) appeared, illustrating that in addition to pristine 2H phase, the 1T phase also existed in the exfoliated MoS2 nanosheets. (b)

Raman spectra of WS2. 351 and 415 cm -1 : E2g and A1g modes of WS2 nanosheets with FWHM of 24.6 and 22.3 cm -1 , respectively. 352 and 417 cm -1 : E2g and A1g modes of bulk WS2 with FWHM of 10 and 9.2 cm -1 , respectively. (c) Raman spectra of TiS2.

225.2 and 331.

  TAE11Y1708TH)• Ethanol (>99.9 %, Merck, Darmstadt, Germany, cat. no. C10654595)• Deionized water (H2O; Milli-Q System; Millipore)Equipments:• Digital hotplate stirrers (Stuart Equipment, cat. no. UC152D)• Vacuum drying oven (Shanghai Yiheng Scientific Instrument Company, cat. no.190728051)• Precision disc cutting machine (Shenzhen Kejngstar Technology Company, cat. no. PE01219G2163) • Infrared heating tape casting coater (Shenzhen Kejngstar Technology Company, model no. MSK-AFA-ES200) • Compact hydraulic crimping machine (Shenzhen Kejngstar Technology Company, cat. no. PE00320L3807) • Coin cell shell (Taizhou Yajun Battery Material Co., Ltd, China, cat. no. CR2032) • Glove box (Vigor Gas Purification Technologies (Suzhou) Co., Ltd, model no. SG2400/750TS) • NEWARE battery test system (CT-ZWJ-4'S-T-1U, cat. no. T1906-200654) • Glass vials (5 and 25 ml) with appropriate polyethylene snap caps (VWR Collection, cat. nos. 548-0141 and 548-0144) • Parafilm (Parafilm, cat. no. PM996) • Disposable plastic dropper (Kangjian, cat. no. KJ 619) • Scissors • Three micropipettes (ranges: 0.5-10, 10-100 and 100-1,000 μl) with appropriate disposable tips • Polypropylene microcentrifuge tubes (1.5 and 2.0 ml; Axygen, cat. nos. MCT-150-C and MCT-200-C) • Microcentrifuge (Thermo Scientific, cat. no. 75002466) • Ultrasonic cleaning bath (Branson Ultrasonic, cat. no. CPX-952-238R) • Transmission electron microscope (TEM; JEOL, model no. JEOL-2100F) • Holey carbon-coated copper grids (200 mesh) for TEM sample preparation • Atomic force microscope (AFM; Veeco, model no. Dimension 3100) • APTES-modified Si/SiO2 substrate for AFM characterization • X-ray photoelectron spectrometer (Ulvac-Phi, model no. PHI5000 Versaprobe III) • Raman spectrometer (HORIBA Jobin Yvon S.A.S., model no. LabRAM HR Evolution) • Ultraviolet-visible spectrophotometer (Shimadzu, model no. Ultraviolet 2450) Equipment Setups: Electrochemical lithium ion-intercalation setup • This setup consists of a coin cell, and a Neware battery test system. Such a coin cell uses Li foil as anode, copper (Cu) foil with uniformly coated layered bulk material as cathode, polypropylene (pp) film as the separator, and 1.0 M LiPF6 in a mixture of ethyl carbonate (EC) and dimethyl carbonate (DMC)

13 .

 13 Image the obtained mono or few layer TMDs nanosheets by using a Veeco Dimension 3100 AFM instrument.! TROUBLESHOOTINGXPS and Raman characterizations of the obtained mono-or few-layer TMDs nanosheets (Timing 3 h):14. Take 0.5 mL opaque suspension of mono or few layer TMDs nanosheets collected from step 6 onto Si/SiO2 substrate (5 mm * 5 mm) and let it dry in air for 2 h before the XPS and Raman characterizations.15. Test the XPS curves of the exfoliated mono or few layer TMDs nanosheets using a PHI5000 Versaprobe III X-ray photoelectron spectrometer.16. Test the Raman curves of the exfoliated mono or few layer TMDs nanosheets using a LabRAM HR Evolution Raman spectrometer.UV-Vis characterization of the obtained mono-or few-layer TMDs nanosheets (Timing 0.5 h):17. Take 3.0 mL opaque suspension of mono or few layer TMDs nanosheets collected from step 6 into 3.5 mL quartz cuvette.18. Test the UV-Vis curves of the exfoliated mono or few layer TMDs nanosheets using an Ultraviolet-2450 ultraviolet-visible spectrophotometer.

6 , 20 Steps 17 -

 62017 Synthesis of mono-or few-layer inorganic nanosheets: 26 hSteps 7-9, TEM characterization of the obtained mono-or few-layer inorganic nanosheets: 3 h Steps 10-13, AFM characterization of the obtained mono-or few-layer inorganic nanosheets: 5 h Steps 14-16, XPS and Raman characterizations of the obtained mono-or few-layer inorganic nanosheets: 3.5 h 18, UV-Vis characterization of the obtained mono-or few-layer inorganic nanosheets: 0.5 h

Figure 1

 1 Figure 1 Schematic illustrations of the electrochemical lithium ion-intercalation and exfoliation process.

Figure 2 A

 2 Figure 2 A photograph of the electrochemical lithium ion-intercalation experimental setup (battery test system).

Figure 3

 3 Figure 3 Schematic illustrations of the whole procedures for the preparation of monoor few-layer TMDs nanosheets.

Figure 4

 4 Figure 4Photographs of the opaque suspension of the exfoliated MoS2, WS2, TiS2, TaS2, BN, and NbSe2 nanosheets. Adapted with permission from ref.[START_REF] Zeng | Single-Layer Semiconducting Nanosheets: High-Yield Preparation and Device Fabrication[END_REF][START_REF] Zeng | An Effective Method for the Fabrication of Few-Layer-Thick Inorganic Nanosheets[END_REF] .

Figure 5

 5 Figure 5 TEM images, selected area electron diffraction (SAED) patterns, and highresolution transmission electron microscopy (HRTEM) images of the exfoliated MoS2

Figure 6

 6 Figure 6 AFM images of the exfoliated MoS2 (a), WS2 (b), TiS2 (c), TaS2 (d), BN (e), NbSe2 (f) nanosheets, giving an average thickness for MoS2, WS2, TiS2, TaS2, BN, NbSe2 nanosheets of ~1.0, 1.0, 0.7, 1.0, 4.0, 2.4 nm, respectively, confirming that mono or few layer inorganic nanosheets were successfully produced by our method. Adapted

Figure 7

 7 Figure 7 AFM images of large-area MoS2 (a), TaS2 (b), TiS2 (c) nanosheets. Bymeasuring 100 pieces of MoS2, TaS2, and TiS2 nanosheets, respectively. It is found that 92% of MoS2 are mono layer with the average thickness of 0.99 ± 0.17 nm; 93% of TaS2 are mono layer with the average thickness of 0.88 ± 0.08 nm; 93% of TiS2 are double layers with the average thickness of 1.63 ± 0.23 nm. Adapted with permission from ref.53,55 .

Figure 8

 8 Figure 8 XPS spectra of the exfoliated (a, b) MoS2. 229.0-232.1 eV: Mo3d oxidation state of LixMoS2. 229.7-232.9 eV: Mo 4+ of MoS2. 233.0-236.2 eV: Mo 6+ of MoO3. 162.7-163.9 eV: S 2-of MoS2. 163.8-165.0 eV: S2 2-of MoS3. 162.0-163.2 eV: S2p oxidation state as in LixMoS2. (c, d) WS2. 33.5 -35.7 eV: W 4+ of WS2. 36.6-38.7 eV: W 6+ of WO3. 163.0-164.2 eV: S 2-of WS2. (e, f) TiS2. 456.2-462.2 eV: Ti2p oxidation state as in LixTiS2. 457.7-463.7 eV: Ti 4+ of TiS2. 458.9-464.9 eV: Ti2p oxidation state as in mixed environment; 459.3-465.3 eV: Ti 4+ of TiO2. 161.2-162.3 eV: S2p oxidation state as in LixTiS2. 161.8-162.9 eV: S 2-of TiS2. 162.6-163.7 eV: S 4/y of TiSy (1 < y < 2). 163.4-164.5 eV: S2 2-of TiS3. (g, h) TaS2, 24.1-26.0 eV: Ta4f oxidation state as in LixTaS2. 24.7-26.6 eV: Ta 4+ of TaS2. 26.6-28.5 eV: Ta 5+ of Ta2O5. 162.1-163.3 eV: S2p oxidation

Figure 9

 9 Figure 9 (a) Raman spectra of MoS2. 379.8 and 406.5 cm -1 : E2g and A1g modes of MoS2 nanosheets with full width at half maximum (FWHM) of 10.4 and 16.0 cm -1 , respectively. 379.8 and 406.5 cm -1 : E2g and A1g modes of bulk MoS2 with FWHM of 8.6 and 14.3 cm -1 , respectively. MoS2 nanosheet shows much broader bands compared to the bulk MoS2, which is owing to the phonon confinement in the ultra-thin structure.

  Hydrogen peroxide solution (H2O2, Sigma-Aldrich, Germany, cat. no. 107209)• Lithium-ion battery electrolyte (Suzhou Duo Duo Chemical Technology Co., Ltd,

	China, cat. no. LB064)
	• Lithium foil (Fisher Scientific Company, Canada, cat. no. AA1076914)
	• The copper foil (Aritech Chemazone Private Limited, India, cat. no. NCZ-MN-
	177/20)
	• Zirconium (IV) sulfide (ZrS2, Strem, Massachusetts, USA, cat. no. 808679)
	• Acetone (Anaqua Global International Inc. Limited, USA, cat. no.
	• Natural graphite (Bay carbon Inc, Michigan, USA, cat. no. 43319)
	• Boron nitride (BN, Sigma, Steinheim, Germany, cat. no. 112073)
	• Niobium selenide (NbSe2, Alfa Aesar, Massachusetts, USA, cat. no. S08E024)
	• Tungsten (IV) selenide (WSe2, Alfa Aesar, Massachusetts, USA, cat. no. 808822)
	• Antimony triselenide (Sb2Se3, Sigma, Steinheim, Germany, cat. no. 401196)
	• Bismuth telluride (Bi2Te3, Sigma, Steinheim, Germany, cat. no. 733482)
	• Poly(vinylidene fluoride) (PVDF, Sigma, Steinheim, Germany, cat. no.
	FCB036778)
	• Polypropylene (pp) film (Celgard 2300, North Carolina, USA)
	• N-methylpyrrolidone (NMP, Sigma, Steinheim, Germany, cat. no. LI40T113)
	• 3-aminopropyltriethoxysilane (APTES, Sigma, Steinheim, Germany, cat. no.
	C2002164)
	• Sulphuric acid (H2SO4, Sigma-Aldrich, Germany, cat. no. 109072)

! CAUTION: Personal protective equipment must be worn during experiments involving H2SO4, as it is strong acids and can cause skin burns.

•

Table 1

 1 Troubleshooting table

	Step		Problem	Possible causes	Solution
	5	Sometimes,	the
		layered bulk materials
		are	not	fully
		exfoliated, some of
		the bulk particles can
		be clearly seen in
		solution.

  3 cm -1 : Eg and A1g modes of TiS2 nanosheets with FWHM of 36.3 and 34.6 cm -1 , respectively. 235.3 and 334.4 cm -1 : Eg and A1g modes of bulk TiS2 with FWHM of 16.2 and 31.3 cm -1 , respectively. (d) Raman spectra of TaS2. 300.1 and 395.2 cm -1 : E2g and A1g modes of TaS2 nanosheets with FWHM of 33.2 and 35.0 cm -1 , respectively. 289.3 and 401.2 cm -1 : E2g and A1g modes of bulk TaS2 with FWHM of 17.2 and 37.2 cm -1 , respectively. Absorbance spectrum for exfoliated dispersions of (e) MoS2, (f) WS2, (g) TiS2, (h) TaS2 nanosheets. No obvious absorption peak is observed for mono-layer MoS2 dispersion, but the absorption increases gradually as the wavelength decreases down to 400 nm. Four peaks are observed for mono-layer WS2 dispersion, which are located at 437, 518, 614, 634 nm, respectively. For TiS2, three peaks: 561, 930, 1043 nm are presented. The same, there are three peaks located at 689, 936, 1095 nm, which are attributed to exfoliated TaS2 nanosheets. Adapted with permission from ref.[START_REF] Zeng | Single-Layer Semiconducting Nanosheets: High-Yield Preparation and Device Fabrication[END_REF] .

Tenne, R., Margulis, L., Genut, M. & Hodes, G. Polyhedral and cylindrical

Acknowledgements

This work was supported by a Start-Up Grant from CityU (CityU9610435) and ECS scheme (CityU9048163) from RGC in Hong Kong. 72 Voiry, D., Drummond, C. & Pénicaud, A. Portrait of carbon nanotube salts as

Author contributions

Z.Y. Zeng designed and performed the research; and all authors contributed to writing the manuscript.

Competing interests

The authors declare no competing interests. https://onlinelibrary.wiley.com/doi/10.1002/anie.201106004.


Emiliano Lorini
email: emiliano.lorini@irit.fr

Elise Perrotin
email: elise.perrotin@cril.fr

Epistemic Actions: Comparing Multi-agent Belief Bases with Action Models

We compare the syntactic multi-agent belief base approach, and the dynamic epistemic logic possible world semantic approach. In the belief base approach, the language provides an implicit and an explicit belief operators, plus a dynamic modality for actions consisting in adding formulae to bases. For the semantic approach, we rely on action models of Dynamic Epistemic Logic (DEL). We first show how to translate a formula of the belief base approach into DEL: in particular, we provide a specific action model scheme corresponding to the addition of a formula in a belief base. Conversely, we identify a fragment of DEL that can be translated in the multiagent belief base language.

Introduction

The formalization of epistemic states and their dynamics is one of the key topics in the area of knowledge representation and reasoning. There are two traditions in this area. On the one hand, there is epistemic logic which started with the seminal work of Hintikka [START_REF] Hintikka | Knowledge and Belief[END_REF]) on the logics of knowledge and belief. It was extended to the multiagent setting at a later stage [START_REF] Fagin | Reasoning about Knowledge[END_REF][START_REF] Meyer | Epistemic Logic for AI and Theoretical Computer Science[END_REF] and, more recently, to modeling knowledge and belief change with growing research on dynamic epistemic logic (DEL) [START_REF] Baltag | The logic of public announcements, common knowledge and private suspicions[END_REF][START_REF] Baltag | Logics for epistemic programs[END_REF]van Ditmarsch, van der Hoek, and Kooi 2007a). The standard approach to epistemic logic is extensional. Its formal semantics exploits the so-called multi-agent Kripke models, namely, multi-relational structures equipped with valuation functions for the interpretation of atomic formulas. Binary relations in a multi-agent Kripke model are called epistemic (or doxastic) accessibility relations and are used to describe the agents' epistemic states and uncertainty.

On the other hand, we have the so-called syntactic approach. It includes, for instance, work on belief base and knowledge base revision [START_REF] Hansson | Theory contraction and base contraction unified[END_REF][START_REF] Hansson | A Textbook of Belief Dynamics: Theory Change and Database Updating[END_REF][START_REF] Benferhat | A practical approach to revising prioritized knowledge bases[END_REF], belief base merging [START_REF] Konieczny | Merging information under constraints: a logical framework[END_REF] and input-output logic [START_REF] Makinson | Input/output logics[END_REF]. The syntactic approach typically leverages belief bases, or more generally knowledge bases, for representing what an agent knows or believes. A natural distinction in this approach is between explicit and implicit belief. An agent's explicit belief is seen as a piece of information in the agent's belief base, while an implicit belief corresponds to a piece of information that is derivable from the agent's belief base (i.e., included in the deductive closure of the agent's belief base). The syntactic approach was put forward among other things as a solution to the logical omniscience problem in virtue of the fact that an agent's beliefs are described either by a set of formulas which is not necessarily closed under deduction [START_REF] Eberle | A logic of believing, knowing and inferring[END_REF][START_REF] Moore | Computational models of belief and the semantics of belief sentences[END_REF] or by a set of formulas obtained by the application of an incomplete set of deduction rules [START_REF] Konolige | A deduction model of belief[END_REF][START_REF] Jago | Epistemic logic for rule-based agents[END_REF].

The two approaches have been recently reconciled in [START_REF] Lorini | In praise of belief bases: Doing epistemic logic without possible worlds[END_REF][START_REF] Lorini | Rethinking epistemic logic with belief bases[END_REF] in which a semantics for multi-agent epistemic logic using belief bases was proposed. The central idea of this semantics is that an agent's epistemic indistinguishability relation should be computed from the agent's belief base by stipulating that a state is considered possible by the agent if and only if it satisfies all information in the agent's belief base. Moreover, at a dynamic level, the dynamics of the agents' explicit and implicit beliefs are supposed to depend on how their belief bases change over time. For instance, by privately expanding its belief base with a new information α, an agent will start to explicitly believe α and, consequently, it will be able to deduce new facts from its expanded belief base.

In this paper we push forward the comparison between the two approaches by investigating the connection between the update semantics for DEL using so-called action models [START_REF] Baltag | The logic of public announcements, common knowledge and private suspicions[END_REF] and the update semantics for epistemic logic using belief bases introduced in [START_REF] Lorini | Rethinking epistemic logic with belief bases[END_REF]. We believe comparing the two approaches is useful since each of them has its own advantages and disadvantages and it is important to have a clear understanding of their relationship. For instance, the extensional approach using multi-relational Kripke models and action models is general, as it allows us to model a large variety of multiagent information dynamics, but it is not compact: even simple situations require big models. On the contrary, the syntactic approach is less general, but it represents information in a more compact way with the help of belief bases. We will answer the following questions: what is the extensional counterpart in terms of action model of the notion of belief base expansion? Which class of multi-agent information dynamics can be represented through update opera-tions on belief bases? Both directions will be explored: from multi-agent belief bases to action models and back from action models to multi-agent belief bases. On the one hand, we will show how to translate the notion of private belief base expansion into a specific class of action models representing private information change. On the other hand, we will identify a specific class of "ruby" actions models that can be translated into the belief base semantics. In a "ruby" action model each agent privately learns a new fact and updates its beliefs accordingly assuming that the others do not learn anything. The non-trivial aspect of the construction from the DEL semantics to the belief base semantics lies in syntactically representing action models with belief bases.

The paper is organized as follows. 1 In Section 2 we present the background material on the language and the semantics for epistemic logic exploiting belief bases introduced in [START_REF] Lorini | Rethinking epistemic logic with belief bases[END_REF]. Section 3 presents the background material on DEL: the notions of action model and product update between an epistemic model and an action model. In Section 4, we study the connection between the update semantics for epistemic logic using multi-agent belief bases and the DEL semantics using action models. We provide a polynomial embedding of the former into the latter and then we show how to represent "ruby" action models through the notion of private belief base expansion. In Section 5, we discuss the two semantics under the assumption of epistemic introspection. In Section 6, we discuss some related work.

Background on Multi-agent Belief Bases

This section presents the dynamic epistemic language for beliefs of both explicit and implicit types (Lorini 2020).

Language

Assume a countably infinite set of atomic propositions Ap = {p, q, . . .} and a finite set of agents Ag = {1, . . . , n}. Let 2 Ag * = 2 Ag \ {∅} be the set of non-empty coalitions.

We define the language in two steps. First define the language L 0 by:

L 0 def = α ::= p | ¬α | α ∧ α | i α
, where p ranges over Ap and i ranges over Ag. L 0 is the language used to represent explicit beliefs. The formula i α reads "agent i has the explicit belief that α". The language L B extends L 0 and is defined by:

L B def = ϕ ::= α | ¬ϕ | ϕ ∧ ϕ | i ϕ | [+ J α]ϕ,
where α ranges over L 0 , i over Ag and J over 2 Ag * . The other Boolean constructions , ⊥, ∨, → and ↔ are defined from p, ¬ and ∧ in the standard way. The formula i ϕ reads "agent i implicitly believes that ϕ". The abbreviation ♦ i ϕ def = ¬ i ¬ϕ defines the concept of belief compatibility. The formula ♦ i ϕ reads "ϕ is compatible with agent i's explicit beliefs". The operator [+ J α] is used to model private belief expansion. Specifically, the formula [+ J α]ϕ reads "ϕ holds after every agent in the coalition J has expanded its belief base with α".

1 The supplementary material contains the long version of this submission.

Belief Base Semantics

The formal semantics for the language L B exploits belief bases. Unlike the standard Kripke semantics in which possible worlds and epistemic alternatives are primitive, they are here defined from the primitive concept of belief base. Definition 1 (State) A state is a tuple B = (B i ) i∈Ag , V where B i ⊆ L 0 is agent i's belief base, and V ⊆ Ap is the actual environment. The set of all states is noted S.

The following definition specifies truth conditions for formulas in the sublanguage L 0 . Definition 2 (Satisfaction relation) For any state B = (B i ) i∈Ag , V ∈ S:

B |= p ⇐⇒ p ∈ V , B |= ¬α ⇐⇒ B |= α, B |= α 1 ∧ α 2 ⇐⇒ B |= α 1 and B |= α 2 , B |= i α ⇐⇒ α ∈ B i .
Observe the set-theoretic interpretation of the explicit belief operators in the previous definition: agent i has the explicit belief that α if and only if α is included in its belief base. The following definition introduces the notion of epistemic alternative. Definition 3 (Epistemic alternatives) Let i ∈ Ag. Then, R i is the binary relation on the set S such that, for all

B = (B i ) i∈Ag , V , B = (B i ) i∈Ag , V ∈ S: BR i B if and only if ∀α ∈ B i : B |= α. BR i B
means that B is an epistemic alternative for agent i at B, that is to say, B is a state that agent i considers possible at B. The idea of the previous definition is that B is an epistemic alternative for agent i at B if and only if, B satisfies all facts that agent i explicitly believes at B.

A multi-agent belief model (MAB), or simply model, is defined to be a state supplemented with a set of states, called context. The context Cxt is not necessarily equal to the set of all states S, since there could be states in S incompatible with the general "laws of the domain" and, consequently, with the agents' epistemic states. For example, we might want to exclude from the context Cxt all states in which the propositions "1+1=2" and "1+1=3" are true concomitantly. Definition 4 (Multi-agent belief model) A multi-agent belief model (MAB) is a pair (B, Cxt), where B ∈ S and Cxt ⊆ S. The class of MABs is noted M. Note that in Definition 4 we do not require B ∈ Cxt. The following definition extends Definition 2 to the full language L B . Its formulas are interpreted with respect to MABs. (We omit Boolean cases, as they are defined in the usual way.) Definition 5 (Satisfaction relation (cont.)) Let (B, Cxt) ∈ M. Then:

(B, Cxt) |= α ⇐⇒ B |= α, (B, Cxt) |= i ϕ ⇐⇒ ∀B ∈ Cxt, if BR i B then (B , Cxt) |= ϕ, (B, Cxt) |= [+ J α]ϕ ⇐⇒ (B + J α , Cxt) |= ϕ, with V + J α = V , B + J α i = B i ∪
{α} for all i ∈ J, and B + J α j = B j for all j ∈ J.

According to the previous definition, agent i implicitly believes that ϕ if and only if ϕ is true at all states in the context that i considers possible. Moreover, every agent in coalition J privately expands its belief base with α if every agent in J adds the information α to its belief base, while all agents outside of J keep their beliefs unchanged.

Let ϕ ∈ L B . Formula ϕ is said to be valid relative to the class M, noted |= M ϕ, if and only if, for every (B, Cxt) ∈ M, we have (B, Cxt) |= ϕ. Formula ϕ is said to be satisfiable for the class M if and only if ¬ϕ is not valid for the class M.

Background on DEL

An epistemic model is a Kripke structure (i.e. a graph) in which nodes are possible worlds and edges are labelled by agents.

Definition 6 An epistemic model M = (W, (R i ) i∈Ag , V ) is a tuple where:

• W is a non-empty finite set of possible worlds, • R i ⊆ W × W is an accessibility relation for agent i, • V : W -→ 2 Ap is a valuation function.
A static epistemic situation is then classically represented by a so-called pointed epistemic model, which is a pair M, w where w is the current world.

Example 1 In Figure 1 (on the left), the epistemic model M contains two worlds w and u that are indistinguishable for both agents i and j. Moreover, the current world is w.

To model epistemic actions (public/private announcements, etc.), DEL provides the notion of action model (Baltag, [START_REF] Baltag | The logic of public announcements, common knowledge and private suspicions[END_REF]). An action model is also a Kripke structure. Nodes are atomic actions, also called events, labelled by a pair of pre-and post-conditions. The precondition pre(a) of an atomic action a is the epistemic formula that should be true before the execution of a, and the postcondition function post(a) assigns to each proposition p a new truth value obtained by the evaluation of the formula post(a)(p).

Definition 7 An action model is a tuple

A = (A, (R A i ) i∈Ag , pre, post) where: • A is a non-empty finite set of possible atomic actions, • R A i ⊆ A × A is the accessibility relation on A for i, • pre : A -→ L EL is a precondition function, • post : A × Ap -→ L EL is a postcondition function.
Example 2 In Figure 1 (top), the action model A contains two atomic actions a and a . For a to be executed, its precondition p should be true. Then, action a assigns p to false. Action a is always executable since its precondition is , and it does not change the value of any proposition. Action a is the current event and the sole possible action for agent i, while agent j considers that the trivial action a happens. The pointed action model A, a corresponds to the private announcement of p to agent i and the private assignment of p to false for agent i (that is, only i sees that p has been assigned to false). a :

pre : p post : p ← ⊥ a : pre : post : / j i i, j A w : {p} u : ∅ i, j i, j i, j (w, a) : ∅ (w, a ) : {p} (u, a ) : ∅ j j i i, j i, j i, j M M × A Figure 1: Example of product M × A.
Now we recall the definition of the product M × A which corresponds to the epistemic model obtained by executing the action model A in the initial epistemic model M.

Definition 8 Let M = (W, (R i ) i∈Ag , V ) be a Kripke model. Let A = (A, (R A i ) i∈Ag , pre, post) be an ac- tion model. The product of M and A is M × A = (W , (R i ) i∈Ag , V ) where: 1. W = {(w, a) ∈ W × A | M, w |= pre(a)}; 2. (w, a)R i (w , a ) iff wR i w and aR A i a ; 3. V ((w, a)) = {p ∈ Ap | M, w |= post(a, p)}.
Worlds in M × A are pairs (w, a) in which the precondition of a holds in M, w.

Example 3 Figure 1 shows the resulting product M × A by executing A from M. Note that the pair (u, a ) is not present because the precondition p is not true in u. Note also that p is false in (w, a), due to the postcondition of a.

The product of a pointed epistemic model (M, w) with a pointed event model (A, a) is defined as (M, w)×(A, a) := (M × A, (w, a)). It is defined only if M, w |= pre(a).We recall the language of DEL, noted L DEL , by extending standard epistemic logic with dynamic modalities A, a :

ϕ ::= | p | ¬ϕ | ϕ ∧ ϕ | K i ϕ | A, a ϕ
Formula K i ϕ is read "agent i knows that ϕ". Formula A, a ϕ is read "action A, a is executable, and after having executed it, ϕ holds".

Definition 9

The truth conditions M, w |= ϕ are defined as follows (Boolean cases are omitted):

• M, w |= p if p ∈ V (w); • M, w |= K i ϕ if for all u such that wR i u, M, u |= ϕ. • M, w |= A, a ϕ if M, w |= pre(a) and M × A, (w, a) |= ϕ.
We finish this section by recalling the notion of modal depth md(ϕ) of a formula ϕ which is the maximum of the number of nested knowledge operators. It is defined by induction: md(p) := 0; md(¬ϕ) := md(ϕ); md(ϕ ∨ ψ) := max(md(ϕ), md(ψ)); md(K i ϕ) = 1 + md(ϕ) and finally md( A, a )ϕ is the maximum of the modal depths of ϕ and the formulas written in A.

a 1 : pre : post : p i α ← for all i ∈ J a 2 : pre : tr BA (α) post : / a 3 : pre : post : / J Ag \ J Ag Ag
The notion of bisimulation in modal logic has been refined to take the modal depth of formulas into account: the notion of n-bimilation noted n . We have that M, w n M , w iff M, w and M , w agree on formulas of modal depth at most n (see Prop. 2.31 in [START_REF] Blackburn | Modal Logic[END_REF] and Section 6.5 in (Van Ditmarsch, van Der Hoek, and Kooi 2007b)).

The Two Translations

In this section, we explore the connection between the DEL semantics using action models and the semantics using belief bases. We first investigate the direction from multi-agent belief bases to action models. Then, we investigate the other direction, from action models to multi-agent belief bases.

From Belief Base Operations to Action Models

We show how to embed the belief base semantics into the DEL action model semantics. In particular, we provide a polynomial satisfiability preserving translation tr BA of the language L B into L DEL . It is defined as follows:

tr BA (p) = p, tr BA (¬ϕ) = ¬tr BA (ϕ), tr BA (ϕ 1 ∧ ϕ 2 ) = tr BA (ϕ 1 ) ∧ tr BA (ϕ 2 ), tr BA ( i α) = p i α ∧ K i tr BA (α), tr BA ( i ϕ) = K i tr BA (ϕ), tr BA ([+ J α]ϕ) = [A + J α , a 1 ]tr BA (ϕ),
where

A + J α = (A, (R A i ) i∈Ag , pre, post
) is the action model (see Figure 2) such that:

• A = {a 1 , a 2 , a 3 }; • ∀i ∈ J, R A i = {(a 1 , a 2 ), (a 2 , a 3 ), (a 3 , a 3 )}; • ∀i ∈ Ag \ J, R A i = {(a 1 , a 3 ), (a 2 , a 3 ), (a 3 , a 3 )}; • pre(a 1 ) = , pre(a 2 ) = tr BA (α), pre(a 3 ) = ; • post(a 2 , p) = post(a 3 , p) = p for all p ∈ Ap, post(a 1 , p) = p for all p ∈ i∈J {p i α }, and post(a 1 , p i α ) = if i ∈ J.
The following is the core result of this section highlighting the correctness of our polynomial embedding.

Theorem 1 Let ϕ ∈ L B . Then, ϕ is satisfiable for the class M if and only if tr BA (ϕ) is DEL-satisfiable.

PROOF.

The proof relies on the fact the language L B can be equivalently interpreted relative to models of the form Ω = (S, B, (⇒ i ) i∈Ag , τ ) where S is a non-empty set of states, B : Ag × S -→ 2 L0 is a belief base function, τ : Ap -→ 2 S is valuation function, ⇒ i ⊆ S × S is agent i's epistemic accessibility relation. L B -formulas are interpreted with respect to pointed models (Ω, s) with s ∈ S as follows (boolean cases are omitted for simplicity):

(i) (Ω, s) |= p iff s ∈ τ (p), (ii) (Ω, s) |= i α iff α ∈ B(i, s), (iii) (Ω, s) |= i ϕ iff ∀s ∈ S, if s ⇒ i s then (Ω, s ) |= ϕ, (iv) (Ω, s) |= [+ J α]ϕ iff (Ω + J α , s * ) |= ϕ, where Ω + J α = (S + J α , B + J α , (⇒ + J α i ) i∈Ag , τ + J α ) with s * ∈ S and: • S + J α = S ∪ {s * }, • B + J α (i, s ) = B(i, s) if s = s * , • B + J α (i, s * ) = B(i, s) if i ∈ J, • B + J α (i, s * ) = B(i, s) ∪ {α} if i ∈ J, • ⇒ + J α i = ⇒ i ∪{(s * , s ) | s ⇒ i s } if i ∈ J, • ⇒ + J α i = ⇒ i ∪{(s * , s ) | s ⇒ i s and (Ω, s ) |= α} if i ∈ J, • τ + J α (p) = τ (p) ∪ {s * } if s ∈ τ (p), • τ + J α (p) = τ (p) if s ∈ τ (p).
In particular, for every ϕ ∈ L B , it is shown in [START_REF] Lorini | Rethinking epistemic logic with belief bases[END_REF]) that ϕ is satisfiable for the class of multi-agent belief models M iff ϕ is satisfiable for the subclass of the previous models such that, for every i ∈ Ag and for every s ∈ S, ⇒ i (s) ⊆ α∈B(i,s) ||α|| (Ω,s) with ||α|| (Ω,s) = {s ∈ S | (Ω, s ) |= α}. Such a subclass is called the class of "notional" models and is noted N. Given the previous semantic equivalence between the two classes M and N wrt language L B , in the rest of the proof we simply need work with the latter class.

We just need to show that ϕ is satisfiable for the class N if and only if tr BA (ϕ) is DEL-satisfiable. The left-to-right direction of the proof is by induction on the structure of ϕ and relies on a two-step transformation. The non-trivial case is ψ = [+ J α]ϕ. Given a pointed notional model (Ω, s) satisfying ϕ we transform it into an isomorphic pointed epistemic model M, w. Then, we compute the pointed notional model (Ω + J α , s * ) on the one hand, and the epistemic model M × A + J α , (w, a 1 ) on the other hand. We transform (Ω + J α , s * ) into an isomorphic pointed epistemic model (M , w ). We show that M , w and M × A + J α , (w, a 1 ) are bisimilar. The right-to-left direction is proved in an analogous way.

From Action Models to Belief Base Operations

In this section, we show how to polynomially translate a DEL fragment into L B . We focus on the DEL fragment, based on so-called ruby action models, that encompasses 2 a :

pre : post : the action models A + J α used in the translation tr BA (see Figure 2).

Ruby Action Models

Definition 10 A ruby action model A, a is a pointed action model of the form given in Figure 3 where β 1 , . . . , β n are propositional formulas (that is, boolean combinations of propositional variables).

A ruby action model corresponds to the concurrent private announcement of β 1 to agent 1, . . . , of β n to agent n. Interestingly, the action model A + J α of Figure 2 in which we ignore the postcondition is equivalent to the ruby action model with β i = tr BA (α) for all i ∈ J and β i = for all i ∈ J.

In the definition of ruby action models, we suppose that the root a 0 is linked to a unique action a i for agent i. This definition can be leveraged also if this uniqueness is not assumed: several i-successors can be replaced by a single isuccessor whose precondition is the disjunction of the preconditions. While, the absence of i-successors can be replaced by a single i-successor whose precondition is ⊥. The obtained ruby action model is equivalent to the original action model3 .

Note that ruby action models commute (M × A × A = M × A × A when A and A are ruby action models).

Translation We define a translation tr AB from the DEL language with only ruby action models into the language L B as follows:

tr AB (p) := p tr AB (¬ϕ) := ¬tr AB (ϕ) tr AB (ϕ ∧ ψ) := tr AB (ϕ) ∧ tr AB (ψ) tr AB (K i ϕ) := i ϕ tr AB ([A, a]ϕ) := [tr Act (A, a)]tr AB (ϕ)
where tr Act (A, a) is the sequence of operations that [+ 1 β 1 ; . . . ; + n β n ]; when A, a is the action model depicted in Figure 3. Note that tr AB is computable in polynomial time.

Theorem 2 Let ϕ be a DEL formula only containing ruby action models. Then: ϕ is DEL satisfiable iff tr AB (ϕ) is satisfiable for the class M.

Before proving Theoreom 2, we need to make a detour and explain how to encode tree-like epistemic models. For the rest of the section, we suppose that Ap is finite and contains the atomic propositions in ϕ.

Tree-like Epistemic Models When a DEL formula is satisfiable, it is in a tree. This comes from the fact that epistemic logic has the tree-like model property (see Th. 5.2 in [START_REF] Vardi | Why is modal logic so robustly decidable?[END_REF], or Proposition 2.15 in [START_REF] Blackburn | Modal Logic[END_REF]), and that any DEL formula is equivalent to a formula in standard epistemic logic (van Ditmarsch, van der Hoek, and Kooi 2007a). We consider here tree-like epistemic models, in which the actual world is the root of the tree. Given a tree-like epistemic model τ with root w and a ruby action model (A, a), we call τ × (A, a) the connected component of τ × A containing the world (w, a). This component is still tree-like, and (τ × (A, a), (w, a)) is bisimilar to (τ × A, (w, a)).

We first give a function describing any tree-like epistemic model. We define desc(τ ) by induction on τ :

desc( w ) =Prop(w) ∧ i∈Ag i ⊥ desc( w τ 1 τ m . . . J 1 J m ) =Prop(w) ∧ i∈Ag i k|i∈J k desc(τ k ) ∧ i,k|i∈J k ♦ i desc(τ k ) where Prop(w) is p∈Ap|p is true in w p ∧
p∈Ap|p is false in w ¬p (we restrict to relevant atomic propositions, so Ap is finite here). The formula desc( w ) intuitively describes the valuation in w and that there are no successors in w. The formula desc(

w τ 1 τ m . . . J 1 J m )
intuitively describes the valuation w, says that successors are among trees τ 1 , . . . , τ k and finally says that each tree τ 1 , . . . , τ k is present as a successor.

It is known that the DEL version of this function, which for any τ gives the formula tr BA (desc(τ )), describes epistemic models up to depth(τ )-bisimulation [START_REF] Moss | Finite models constructed from canonical formulas[END_REF]. We first show that this description function similarly describes belief bases up to bisimulation. In order to generalize this correspondence to formulas which do include dynamic operators, we must first establish that the translation of ruby action models effectively simulates the effects of these actions in belief bases. More precisely, the formula tr BA (desc( τ PROOF.

We show this by induction on the height on τ . Consider a ruby-like action model such as the one in Figure 3.

If τ = w , then τ × (A, a) is equal to τ and desc(τ ,a) and B agree on all propositional variables, we have B trAct (A,a) , Cxt |= Prop(w). Moreover, suppose that B trAct (A,a) R i B for some agent i and some B in Cxt: then

) = desc(τ × (A, a)) = Prop(w) ∧ i∈Ag i ⊥. Consider then B and Cxt such that B, Cxt |= desc(τ ). As B trAct (A
BR i B as B i ⊆ B trAct (A,a) i
. But B, Cxt |= i ⊥, so there can be no such state B , and therefore B trAct (A,a) , Cxt |= i ⊥ for all i. Hence B trAct (A,a) , Cxt |= desc(τ ).

Suppose now that τ is of the form w τ 1 τ m . . . J 1 J m , and call

w k the root of τ k for 1 k m. Then τ × (A, a) is bisimilar to w τ 1 τ m . . . G 1 G m where for every k m, G k = {i ∈ J k | τ, w k |= β i }; and desc(τ × (A, a)) =Prop(w) ∧ i∈Ag i k|i∈G k desc(τ k ) ∧ i,k|i∈G k ♦ i desc(τ k ).
Consider ,a) , it must be the case that B , Cxt |= β i . Then by Proposition 1, we have ,a) , Cxt |= i k|i∈G k desc(τ k ) for all i in Ag.

β i ∈ B trAct (A
τ k , w k |= β i , hence τ, w k |= β i and i ∈ G k . Therefore B trAct (A
It remains to show that B trAct (A,a) , Cxt |= ♦ i desc(τ k ) for all i and k such that i ∈ G k . Consider such a i and k: then i ∈ J k and τ, w k |= β i . As i ∈ J k and B, Cxt |= desc(τ ), we know that B, Cxt |= ♦ i desc(τ k ), that is, there exists some B such that BR i B and B , Cxt |= desc(τ k ). But then by Proposition 1, B , Cxt |= β i as τ, w k |= β i . Hence B trAct (A,a) R i B , and B trAct (A,a) , Cxt |= ♦ i desc(τ k ). We conclude that B trAct (A,a) , Cxt |= desc(τ × (A, a)).

We can now extend the result of Proposition 1 to any formulas of the DEL language in which all actions are ruby action models.

Proposition 3 For any formula ϕ of the DEL language in which all action models are ruby action models, for any state B, context Cxt and tree-like epistemic model τ with root w, if B, Cxt |= desc(τ ) then B, Cxt |= tr AB (ϕ) iff τ, w |= ϕ.

PROOF.

We show this by induction on ϕ. We have shown all cases in the proof of Proposition 1 except for that of dynamic operators.

If B, Cxt |= desc(τ ), then:

B, Cxt |= [tr Act (A, a)]tr AB (ϕ) iff B trAct (A,a) , Cxt |= tr AB (ϕ) iff τ × A, (w, a) |= ϕ (by IH with Proposition 2) iff τ, w |= [A, a]ϕ.
This ends the proof.

Correctness of the Translation

We are now ready to prove Theorem 2. We show both directions of the equivalence separately.

Proposition 4 For any formula ϕ containing only ruby action models, if ϕ is DEL-satisfiable then tr AB (ϕ) is Bsatisfiable.

PROOF.

Suppose that ϕ is DEL-satisfiable. Then by the treelike model property, ϕ is satisfied in a tree-like epistemic model τ = (W, (R i ) i∈Ag , V ) with root w: τ, w |= ϕ. For each v ∈ W , consider a fresh variable p v (appearing neither in ϕ not in V (w) for w ∈ W ) and define the model

τ = (W, (R i ) i∈Ag , V ) such that for all v ∈ W , V (v) = V (v) ∪ {p v }. Then we also have τ , w |= ϕ. Now define the state B v = (B v i ) i∈Ag , V v as follows: B v i = { u|vRiu p u } V v = V (v) Consider the context Cxt W = {B v | v ∈ W }. Then B w , Cxt W |= desc(τ ). Hence B w , Cxt W |= tr AB (ϕ)?
by Proposition 3.

Proposition 5 For any formula ϕ containing only ruby action models, if tr AB (ϕ) is B-satisfiable then ϕ is DEL satisfiable.

PROOF.

For any state B and context Cxt, define the epistemic model M B,Cxt = (W B,Cxt , (R B,Cxt i ) i∈Ag , V B,Cxt ), where:

W B,Cxt ={B} ∪ Cxt; R B,Cxt i ={(B, B ) | B ∈ Cxt and BR i B } ∪ {(B , B ) | B , B ∈ Cxt and B R i B } for all i ∈ Ag; V B,Cxt (B ) ={p | B |= p} for all B ∈ W.
An induction on ϕ proves that for any formula ϕ of the DEL language containing no dynamic operators, for any Consider now a formula ϕ of the DEL language only containing ruby action models, and suppose that B, Cxt |= ϕ for some B and Cxt. We know that M B,Cxt , B is md(ϕ)bisimilar to some tree-like epistemic model (τ, w) (consider the unravelling of M B,Cxt , B up to depth md(ϕ)), where w is the root of τ . In particular, M B,Cxt , B |= tr BA (desc(τ )), hence B, Cxt |= desc(τ ) as tr BA (desc(τ )) contains no dynamic operators. By Proposition 3, this implies that for any formula ψ of the DEL language only containing ruby action models, B, Cxt |= tr AB (ψ) iff τ, w |= ψ. In particular, τ, w |= ϕ, hence ϕ is DEL satisfiable.

Introspective Variant

In this section, we explain the changes to make to keep translations from L B into L DEL , and from L DEL into L B wrt. to the introspective variant. The latter translation will however require some technical restrictions. The underlying epistemic logic is K45: the epistemic relations are transitive and Euclidean (see Def. 2.13 of (Van Ditmarsch, van Der Hoek, and Kooi 2007b)).

Definition

In [START_REF] Lorini | Rethinking epistemic logic with belief bases[END_REF] an alternative definition of the epistemic indistinguishability relation is given. It works for introspective agents that have perfect knowledge of their belief bases. Specifically, for a state to be considered possible by an introspective agent, (i) it must satisfy all information in the agent's actual belief base, and (ii) the agent should have the same belief base in the actual state and in the epistemically accessible state. Definition 11 (Epistemic alternatives) The introspective variant of the epistemic indistinguishability relation is the binary relation

R i ⊆ S × S such that, for all B = (B i ) i∈Ag , V , B = (B i ) i∈Ag , V ∈ S: BR Int i B if and only if (i)∀α ∈ B i : B |= α, (ii)B i = B i .
In other words, the previous definition adds to Definition 3 the introspection condition (ii). It is easy to verify that the relation R Int i so defined is transitive and Euclidean.

a :

pre : post : p i α ← for all i ∈ J a i1 :

pre : tr BA (α) post : / a i k :

pre : tr BA (α) post : / . . .

a t : pre : post : / i 1 i k Ag\{i 1 } Ag\{i k } i 1 i k Ag Ag \ J
pre : post : / 1 n A g \ { 1 } A g \ { n } 1 n

Ag

Figure 5: Introspective ruby action model.

From Belief Base Operations to Action Models

When considering introspective agents (i.e., when the epistemic indistinguishability relations are defined according to Definition 11), we need to slightly redefine the translation from the language L B to the language L DEL given in Section 4.1. In particular, we need to define a new translation tr intr BA whose lines 1-5 are identical to lines 1-5 of translation tr BA and whose line 6 is:

tr intr BA ([+ J α]ϕ) = [A intr + J α
, a]tr intr BA (ϕ), where A intr + J α is the action model for introspective agents depicted in Figure 4. The following theorem is the analog of Theorem 1 for introspective agents. We omit its proof since it is similar to that of Theorem 1. Theorem 3 Let the epistemic indistinguishability relation R i be defined according to Definition 11 and let ϕ ∈ L B . Then, ϕ is satisfiable for the class M if and only if tr intr BA (ϕ) is DEL-satisfiable.

From Action Models to Belief Base Operations

In this section, we discuss the natural generalisation of tr AB to the introspective case.

Introspective Ruby Action Models Definition 12 An introspective ruby action model A, a is a pointed action model of the form given in Figure 5 where β 1 , . . . , β n are propositional formulas.

The translation tr intr AB differs from tr AB given in Subsection 4.2 in two aspects. First the language is the DEL one with introspective ruby action models. Second, we need to add a special variable for each action model A. We define:

tr intr AB ([A, a]ϕ) := [tr Act (A, a)]tr intr AB (ϕ)
where tr Act (A, a) is the sequence of operations that [+ 1 β 1 ; . . . ; + n β n ; + 1 p A ; . . . ; + n p A ; + 1 p 1 A ; . . . ; + n p n A ]; when A, a is the action model depicted in Figure 5, and where p A and all p i A are fresh atomic propositions. Note that tr intr AB is still computable in polynomial time. Just like the ruby action models of Definition 10, introspective ruby action models commute. Thus, the sole information of action model A being executed is sufficient; the order of the executions is not important. In other words, the propositions p A are sufficient to represent the list of already executed actions. Also, as the variables p A are fresh, they are supposed to be initially false.

Theorem 4 Let ϕ be a DEL formula only containing introspective ruby action models. Then: ϕ is DEL K45satisfiable implies tr intr AB (ϕ) is satisfiable for the class M with the introspective relations.

The proof of Theorem 4 is involved. We start by giving some definitions and notations surrounding introspective ruby action models. For any formula ϕ containing only introspective ruby action models, we call Seq(ϕ) the set of sequences of pointed action models corresponding to sequences of actions found in ϕ, defined inductively as:

Seq(p) = { } ( is the empty sequence) Seq(ϕ ∧ ψ) = Seq(ϕ) ∪ Seq(ϕ) Seq(¬ϕ) = Seq(ϕ) Seq([A, a]ϕ) = {(A, e)σ | σ ∈ Seq(ϕ)
and e is an event in A} If w is a world in an epistemic model M and σ is a sequence (A 1 , e 1 ) . . . (A m , e m ) of pointed action models, we abbreviate the model (M × A 1 ) × • • • × A m to M × σ and the world ((w, e 1 ), . . . , e m ) in M × σ to (w, σ), when that world is defined in M × σ.

As ruby action models commute, the order of the ruby action models of a sequence σ does not matter when constructing M × σ; we will therefore be identifying any sequence σ of actions with the set of actions found in σ. In particular, two sequences σ and σ are considered equal when they consist of the same actions.

Finally, given a sequence σ of introspective ruby action models, we call Add i (σ) the set of formulas β such that + i β appears in tr Act (A, a) for some action (A, a) or (A, a i ) in σ. Formally:

Add i (σ) = {β | ∃(A, e) ∈ σ, e ∈ {a, a i } and pre A (a i ) = β}
From Epistemic Models to Contexts Remark that introspective ruby action models preserve the K45 properties: the product of a K45 epistemic model with an introspective ruby action model is again a K45 epistemic model. We now define contexts simulating the effects of actions in an epistemic model. Given a formula ϕ containing only introspective ruby action models and a K45 epistemic model (M, r) with M = (W, (R i ) i∈Ag , V ), we take for every action model A in ϕ three fresh variable p A , p i A and p t A appearing neither in ϕ nor in V (w) for any w ∈ W , and we similarly take for every w ∈ W a fresh variable p w . We define the following context:

Cxt ϕ,M = {B w,σ |σ ∈ Seq(ϕ) and (w, σ) is a world of M × σ}
where for all w ∈ W and σ ∈ Seq(ϕ), B w,σ = (B w,σ i ) i∈Ag, , V w,σ with:

B w,σ i ={ v|wRiv p v } ∪ {p A | (A, e) ∈ σ} ∪ Add i (σ) ∪ {p i A | (A, e) ∈ σ, e ∈ {a, a i }} ∪ {p t A | (A, e) ∈ σ, e ∈ {a, a i }} V w,σ =V (w) ∪ {p w } ∪ {p A | (A, e) ∈ σ} ∪ {p i A | i ∈ Ag, (A, a i ) ∈ σ} ∪ {p t A | (A, a t ) ∈ σ}.
Intuitively, we want every state B w,σ to correspond to the world (w, σ). The fresh variables identify the worlds of w, the actions models in σ, as well as instances of events a i and a t . The constraints in B w,σ i correspond to the following properties:

• from the definition of product updates, it follows that is (w, σ)R i (w , σ ) then wR i w ;

• the inclusion of p A for all (A, e) ∈ σ identify the world as being part of the model M × σ; • Add i (σ) corresponds to the new beliefs introduced for i by σ in (w, σ); • when e∈{a, a i } for some (A, e) ∈ σ then eR i e iff e =a i ;

• when e ∈{a, a i } for some (A, e) ∈ σ then eR i e iff e =a t .

We are now going to show that this context corresponds to the collection of all models M × σ for σ ∈ Seq(ϕ), which will then allow us to simulate model checking of ϕ in M.

Proposition 6 Let ϕ be a DEL formula containing only introspective ruby action models. Let (M, r) be a K45 epistemic model , and Cxt ϕ,M be the corresponding context. Then M, r |= ϕ iff B r , Cxt ϕ,M |= tr intr AB (ϕ). Proof of Theorem 4 Suppose that ϕ is K45-satisfiable. Then ϕ is satisfiable in a K45 epistemic model. Consider a K45 epistemic model (M, r) satisfying ϕ. By Proposition 6, we have B r, , Cxt ϕ,M |= tr intr AB (ϕ). Other Direction Suppose that tr intr AB (ϕ) is satisfiable. To adapt the proof of Proposition 5, we need to construct a K45 epistemic model for ϕ from the unravelling of the belief base inferred epistemic model for tr intr AB (ϕ). The issue will be that the application of a i must exactly correspond to the addition of the β i in the belief bases. And the obtained state must be in the context. To cope with that, we simply suppose that the context is the set S of all states.

Theorem 5 tr intr AB (ϕ) is satisfiable for the class of states whose context is S implies ϕ is DEL K45-satisfiable.

Related Work

Let us mention syntactic approaches for describing epistemic actions. Action models of DEL are captured up to n-bisimilarity by a language of programs introduced in [START_REF] French | A composable language for action models[END_REF]. A logical language for capturing specific kinds of private/public announcements in which propositions expressing which agents are listening to the source is introduced in [START_REF] Bolander | Announcements to attentive agents[END_REF]. However in all of these approaches only actions are syntactic, while the underlying models remain Kripke structures.

As emphasized in the introduction, the multi-agent semantics for epistemic logic using belief bases was first introduced in [START_REF] Lorini | In praise of belief bases: Doing epistemic logic without possible worlds[END_REF][START_REF] Lorini | Rethinking epistemic logic with belief bases[END_REF]. It allows us to represent both agents' beliefs about propositional facts and their higherorder beliefs (i.e., an agent i's belief about an agent j's belief). 4 It was applied to the formalization of a large variety of epistemic concepts and of aspects of both individual and collective epistemic reasoning including graded belief (Lorini and Schwarzentruber 2021a), distributed belief [START_REF] Herzig | A logic of explicit and implicit distributed belief[END_REF], common belief [START_REF] Lorini | Logical theories of collective attitudes and the belief base perspective[END_REF], multiagent belief revision and planning (Lorini and Schwarzentruber 2021b;[START_REF] Davila | A simple framework for cognitive planning[END_REF]. A preliminary comparison of this semantics and the DEL update semantics was given in [START_REF] Lorini | Rethinking epistemic logic with belief bases[END_REF] in which it was shown that private belief base expansion corresponds to a specific kind of DEL private announcement represented through arrow models [START_REF] Kooi | Arrow update logic[END_REF]. In particular, it was shown that the epistemic model obtained via private belief expansion is bisimilar to the epistemic model obtained via private announcement relative to the language of standard of epistemic logic.

The semantics for epistemic logic using belief bases can be seen as a way of defining a single 'canonical' model from the description of a state. This point was explored in [START_REF] Lorini | Exploiting belief bases for building rich epistemic structures[END_REF] in the static setting of epistemic logic. In this paper, we extend the analysis to the dynamic setting (Theorem 2).

In the belief base approach, as seen in Definition 1, the state contains a set of formulas for each agent, and then a 'canonical' model is inferred by the relations of Definition 3. There are other approaches that define a 'canonical' model based on what agents see. The description of what agents see is either given by positions of agents in a geometrical environment [START_REF] Balbiani | Agents that look at one another[END_REF][START_REF] Gasquet | Big brother logic: visual-epistemic reasoning in stationary multi-agent systems[END_REF], or given by an abstract specification [START_REF] Cooper | A lightweight epistemic logic and its application to planning[END_REF]. A similar idea is at the center of symbolic model checking for epistemic logic [START_REF] Van Benthem | Symbolic model checking for dynamic epistemic logic[END_REF]. Techniques described in Section 4 and 5 may also be applied to the comparison of these frameworks with action models.

Conclusion

This paper is about a seminal bridge between two opposite approaches for dynamic knowledge change: the syntactic one with belief base operations, and the semantic one with DEL action models (see Theorems 1 and 2). We also discussed the introspective case (see Theorems 3,4,5).

For going from belief bases to action models, we proposed action models for simulating the operation of privately expanding belief bases of some agents by an input α (see Figures 2 and4). For the other direction, we restricted to so-called ruby action models (see Figures 3 and5). For going from (ruby) action models to belief base operations, we adopted two different proof techniques. For the nonintrospective case, we simply add each formula β j to the base of agent j for simulating a ruby action model. We were also able to fully describe (tree-like) epistemic models obtained by product as a formula of L B . For the introspective case, the simulation of a ruby action model is more involved and requires the use of fresh atomic propositions.

Our work leaves many open questions for future work. Modal Formulas. Currently, formulas in ruby action models are Boolean. We plan to generalize to any modal formulas. This is challenging because formulas in action models can only be about implicit beliefs, and the translation from implicit to explicit beliefs is not straightforward.

Belief Revision. Belief bases form an elegant formalism for bringing belief revision in a multi-agent context (Lorini and Schwarzentruber 2021b). We aim at studying the link with the plausibility models for multi-agent belief revision [START_REF] Baltag | Dynamic belief revision over multi-agent plausibility models[END_REF].

Epistemic Planning. On the one hand, multi-agent epistemic planning with belief bases is decidable, even with belief revision (Lorini and Schwarzentruber 2021b). On the other hand, epistemic planning in DEL when models are trees and propositional is also decidable [START_REF] Bolander | Complexity results in epistemic planning[END_REF]. Future work will be devoted to investigate the connection between the two approaches to epistemic planning.

Public Announcement. Recently, the belief base approach was extended to capture a notion of public announcement [START_REF] Lorini | Logical theories of collective attitudes and the belief base perspective[END_REF]. This paper paves the way for investigating the connection between DEL-notions of public and semi-private announcement and corresponding notions expressed the belief base semantics.

Figure 2 :

 2 Figure 2: Action model A + J α that simulates adding α in bases of agents in J.

Figure 3 :

 3 Figure 3: Pointed ruby action model (A, a).

Proposition 1

 1 For any DEL formula ϕ in which there are no dynamic operators, for any state B, context Cxt and treelike epistemic model τ with root w, if B, Cxt |= desc(τ ) then B, Cxt |= tr AB (ϕ) iff τ, w |= ϕ. PROOF. This is shown by induction on ϕ. Suppose τ is of the form propositional variables is straightforward as B, Cxt |= Prop(w). The cases of boolean operators are standard. If B, Cxt |= i tr AB (ϕ), consider w such that wR i w . Then w is the root of τ k for some k such that i ∈ J k . As B, Cxt |= desc(τ ), we have in particular B, Cxt |= ♦ i desc(τ k ). Hence there exists B such that BR i B and B , Cxt |= desc(τ k ). But then by the induction hypothesis, B , Cxt |= tr AB (ϕ) iff τ k , w |= ϕ, and B , Cxt |= tr AB (ϕ) as BR i B . Hence τ, w |= ϕ and τ, w |= K i ϕ. If τ, w |= K i ϕ, consider B such that BR i B . Then there exists k such that i ∈ J k and B , Cxt |= desc(τ k ). Call w the root of τ k in τ . Then wR i w so τ, w |= ϕ, and by the induction hypothesis B , Cxt |= tr AB (ϕ) iff τ k , w |= ϕ, the latter being equivalent to τ, w |= ϕ. Hence B , Cxt |= tr AB (ϕ) and B, Cxt |= i tr AB (ϕ).

  )) → [(A, a)]tr BA (desc(τ × (A, a))) is valid in DEL for any tree-like epistemic model τ and pointed ruby action model (A, a); we show a corresponding property for belief bases. Proposition 2 For any tree-like epistemic model τ , for any ruby action model (A, a), state B and context Cxt, if B |= desc(τ ) then B tr(A,a) , Cxt |= desc(τ × (A, a)).

  B and Cxt, B, Cxt |= tr AB (ϕ) iff M B,Cxt , B |= ϕ. (For the proof, remark that given B, Cxt, and any B ∈ Cxt, (M B,Cxt , B ) and (M B ,Cxt , B ) are equal when B ∈ Cxt and bisimilar when B ∈ Cxt.)

Figure 4 :

 4 Figure 4: Action model A intr+ J α that simulates adding α in bases of agents in J = {i 1 , . . . , i k } in the introspective variant.

Example 4

 4 For example, given two pointed action models (A, a) and (A , a ), we have Seq([A, a]p ∧ [A , a ](q ∧ ¬[A, a]p)) = {(A, e), (A , e ), (A , e )(A, e) | e is an event of A and e is an event of A }.

  B and Cxt such that B, Cxt |= desc(τ ). Once again, it is clear that B trAct (A,a) , Cxt |= Prop(w). Consider now a state B ∈ Cxt such that B trAct (A,a) R i B . Then as before, BR i B ,and therefore B , Cxt |= k|i∈J k desc(τ k ), that is, there exists a k m such that i ∈ J

k and B , Cxt |= desc(τ k ). Moreover, as

Proceedings of the 19th International Conference on Principles of Knowledge Representation and ReasoningMain Track

if we ignore postconditions Proceedings of the 19th International Conference on Principles of Knowledge Representation and Reasoning Main Track

This is true because there is an emulation (see[START_REF] Van Eijck | Action emulation[END_REF]) between the original action model and the obtained ruby action model.

Related work on the connection between the syntactic representation of preferences based on priority graphs and Kripke models for preference representation can be found in (de Jongh[START_REF] De Jongh | Preference, priorities and belief[END_REF][START_REF] Liu | Reasoning about preference dynamics[END_REF][START_REF] Souza | Belief base change as priority change: A study based on dynamic epistemic logic[END_REF]. It was extended todeontic logic in (van Benthem, Grossi, and Liu 2014). This work is focused on the single-agent case and does not consider reasoning about higher-order beliefs or preferences.

Acknowledgements

This work is supported by the ANR projects CoPains ("Cognitive Planning in Persuasive Multimodal Communication") and BE4musIA ("Belief Change for better Multi-Source Information Analysis").