Loic Constantin

Nada Kraiem

Zhipeng Wu

Bai Cui

Jean-Luc Battaglia

Christian Garnier

Jean-François Silvain

Yong Feng Lu

Manufacturing of Complex Diamond-Based Composite Structures via Laser Powder-Bed Fusion

Keywords: additive manufacturing, copper, diamond, selective laser melting, metal matrix composites

Therefore, pursuing research into nonconventional LPBF

Abstract: Manufacturing complex metal matrix composite (MMC) structures by laser powderbed fusion (LPBF) could unleash their full potential but is difficult due to the presence of reinforcement. Unmelted particles negatively affect the pool dynamics, cause critical spatter ejections, and form printing defects. In this work, by taking copper (Cu) / diamond (D) composite as an example for its prospective thermal management applications and machinability limitations; we discovered that adding steps to LPBF enables the fabrication of high-quality materials and structures. We demonstrated that adding a recoating step improves the composite quality compared to structures manufactured by conventional LPBF. Adding a remelting step enabled further improvement by limiting the generation of spatter and printing defects, leading to 3D laser print dense (97%), highly thermally conductive (349 W/m.K) and complex Cu/5 vol.% D structures.

Introduction

Laser powder-bed fusion (LPBF) provides a unique way of fabricating intricate designs that are not possible by conventional manufacturing methods, thereby revolutionizing the biomedical, defense, and aerospace industries [START_REF] Hojjatzadeh | Pore elimination mechanisms during 3D printing of metals[END_REF]. On the other hand, metal matrix composites (MMCs) are among the most promising materials for advanced applications. The combination of a metal matrix with a reinforcement enables the formation of materials with enhanced and tailored mechanical, thermal, and electrical properties [START_REF] Chawla | Metal Matrix Composites[END_REF]. To date, MMCs are manufactured by solid (i.e., powder metallurgy) and liquid state methods (i.e., stir casting, squeeze casting), producing simple designs, thus restricting their full capabilities [START_REF] Sharma | Manufacturing of metal matrix composites: A state of review[END_REF]. Copper (Cu)/diamond (D) composite is an excellent example of an MMC, where adding D particles tailors the Cu's coefficient of thermal expansion (CTE) while enhancing its thermal conductivity (TC) [START_REF] Silvain | A review of processing of Cu/C base plate composites for interfacial control and improved properties[END_REF]. This MMC is one of the most advanced materials for thermal management applications, yet its shape remains simple due to complexities of machining D-based materials. Therefore, merging the abilities of LPBF to create sophisticated designs with the outstanding properties of MMCs can unleash the full capability of the Cu/D materials and remodel their applications [START_REF] Neugebauer | Additive manufacturing boosts efficiency of heat transfer components[END_REF][START_REF] Constantin | Additive Manufacturing of Copper/Diamond Composites for Thermal Management Applications[END_REF]. Nonetheless, the LPBF of Cu/D composite remains challenging and less explored due to the intricacy of laser three-dimensional (3D) printing of micro composites, the high thermal conductivity of Cu and D, and the low optical absorption of Cu in the infrared (IR) domain.

The difficulties encountered in the 3D printing of micro composites arise from the significant size of the solid reinforcement, which plays a critical role in the densification behavior. L. Xi et al. showed that finer reinforcements allow manufacturing denser composites compared to a coarser reinforcement [START_REF] Xi | Effect of ceramic particle size on densification behavior, microstructure formation, and performance of TiB2-reinforced Al-based composites prepared by selective laser melting[END_REF]. Micrometric particles significantly alter the transit molten pool dynamics, rheology, and flowability, leading to porous parts' formation [START_REF] Simchi | Direct laser sintering of iron-graphite powder mixture[END_REF][START_REF] Ma | On the study of tailorable interface structure in a diamond/Al12Si composite processed by selective laser melting[END_REF][START_REF] Li | Selective laser melting W-10 wt.% Cu composite powders[END_REF]. Consequently, different approaches were developed to 3D laser print MMCs via LPBF. The first approach involves creating a composite powder containing a metal matrix and a nano reinforcement to form a homogenous mixture at the microscale, allowing 3D printing of dense MMCs [START_REF] Aversa | Microstructural and Mechanical Characterization of Aluminum Matrix Composites Produced by Laser Powder Bed Fusion[END_REF][START_REF] Gu | Nanocrystalline TiC reinforced Ti matrix bulk-form nanocomposites by Selective Laser Melting (SLM): Densification, growth mechanism and wear behavior[END_REF][START_REF] Gao | Selective laser melting of TiN nanoparticle-reinforced AlSi10Mg composite: Microstructural, interfacial, and mechanical properties[END_REF][START_REF] Gu | Densification behavior, microstructure evolution, and wear property of TiC nanoparticle reinforced AlSi10Mg bulk-form nanocomposites prepared by selective laser melting[END_REF]. However, this method limits the type of MMCs to nanocomposites and cannot be generalized for Cu/D materials. The second approach consists of the in-situ formation of the reinforcement [START_REF] Leong | In-situ formation of copper matrix composites by laser sintering[END_REF][START_REF] Slocombe | Selective laser sintering of TiC-Al2O3 composite with self-propagating hightemperature synthesis[END_REF][START_REF] Kumar | Composites by rapid prototyping technology[END_REF][START_REF] Almangour | In-situ formation of novel TiC-particle-reinforced 316L stainless steel bulk-form composites by selective laser melting[END_REF]. This technique enables the manufacturing of dense MMCs with micro/nano-sized reinforcement; but once again, a limited number of reinforcements can be formed during LPBF.

Furthermore, Cu is one of the most challenging metals for 3D laser printing due to its low optical absorption in the near-IR and its high TC. In other words, the laser energy is either reflected or quickly dissipated when striking the Cu powder [START_REF] Lindström | Laser Powder Bed Fusion of Metal Coated Copper Powders[END_REF]. SD. Jadlhav, et al. reported the LPBF of dense (99%) and highly thermally conductive Cu parts (336 W/m.K) using a high-power laser (800 W) [START_REF] Jadhav | Influence of selective laser melting process parameters on texture evolution in pure copper[END_REF]. However, they also noticed damage to the laser optics caused by the high IR reflectivity of Cu. Later, they showed that adding carbon (C) nanoparticles into the Cu matrix enhanced the laser absorption; but the TC of the Cu/C composite was severely degraded (164 W/m.K) [START_REF] Jadhav | Influence of Carbon Nanoparticle Addition (and Impurities) on Selective Laser Melting of Pure Copper[END_REF].

Alternatively, the LBPF of pure Cu with low laser power (200 to 600 W) results in decreased part density (83 to 96%), as reported by others [START_REF] Ikeshoji | Selective Laser Melting of Pure Copper[END_REF][START_REF] Constantin | Laser 3D printing of complex copper structures[END_REF][START_REF] Silbernagel | Electrical resistivity of pure copper processed by medium-powered laser powder bed fusion additive manufacturing for use in electromagnetic applications[END_REF]. For these reasons, extensive research has focused on Cu alloys, which have much higher absorption in the IR domain. Among them, Cuchromium (Cr) appears to be one of the most promising alloys because of its high TC (340 W/m.K) and improved laser absorption compared to Cu [START_REF] Jadhav | Surface Modified Copper Alloy Powder for Reliable Laser-based Additive Manufacturing[END_REF]. Nevertheless, the TC of the as-printed Cu-Cr alloys is low (<115 W/m.K) due to the dissolution of the Cr phase in the Cu matrix [START_REF] Uchida | Microstructures and electrical and mechanical properties of Cu-Cr alloys fabricated by selective laser melting[END_REF][START_REF] Zhang | Microstructure and properties of high strength and high conductivity Cu-Cr alloy components fabricated by high power selective laser melting[END_REF].

Here, we overcame these challenges and successfully 3D print dense, complex, and highly conductive Cu/5 vol.% D composites by implementing different printing strategies (i.e., recoating and remelting). The printing parameters (e.g., laser power, scan speed, hatch distance, etc.) are well known to play a vital role in the quality and properties of the LPBF part'. Analogously, the printing strategy has an influence on the material characteristics but has been less explored. In other words, optimizing the printing parameters, along with the strategy, could open new ways to additively manufacture a broader range of MMCs. In our approach, the relative density of the Cu/D composite is increased from 88 to 92% by first adjusting the printing parameters and subsequently adding a recoating step, allowing the composite's TC to be increased from 236 to 327 W/m.K.

Next, the relative density was further improved to 97% by implementing a remelting step, leading to the manufacturing of highly conductive composites with a TC of 349 W/m.K. Finally, the fabrication of sophisticated Cu/D designs was demonstrated with potential thermal management applications.

Material and methods

Preparation of titanium (Ti)-based interphase on diamond particles.

Diamond particles (Eastwind Diamond Abrasives, mean diameter 25 µm) were coated using a graded TiO2-TiC molten salt process. For that, D particles were mixed with metallic Ti powders (99.5%, mean diameter 40 µm) and potassium chloride (KCl) salt (Sigma-Aldrich 99%) at a molar ratio of KCl:C:Ti =0.64:1:0.04. The mixture was then placed in an alumina crucible and heated in a tube oven filled with argon (Ar) at 850 ℃ for 4 h. After cooling down, the crucible was immersed in boiling distilled water to dissolve the salt. The coated D particles were then retrieved by filtration and air-dried in an oven at 60 ºC for 12 h. SEM micrographs of the original and coated -D are presented in Fig. S1ab. The coating thickness is estimated to be about 600 to 700 nm based on a previous study [START_REF] Constantin | Additive Manufacturing of Copper/Diamond Composites for Thermal Management Applications[END_REF]. More details of the molten salt coating can be found elsewhere [START_REF] Constantin | Design of tailored oxide-carbide coating on carbon fibers for a robust copper/carbon interphase[END_REF].

Laser powder-bed fusion of Cu/D composites.

Coated -D particles were mixed with spherical Cu powder (US Nano, Cu 99.5%, mean diameter 45 µm) at a volume percentage of 5% (Fig. S1cf). The laser powder-bed fusion was carried out using the SLM ® 125 3D printer from SLM Solutions. The printer is equipped with a continuous wave (CW) fiber laser (IPG photonics, YLM, Yb: YAG, λ = 1070 nm) with a maximum output power of 400 W. The laser spot was focused to a diameter of 70 μm on the powder-bed surface.

The printing chamber was filled with Ar (99.9%, Matheson) to reduce the oxygen level to about 0.10%. A constant pressure of 18 mbar was maintained in the chamber during all experiments. A stripe laser scan strategy was used for all prints. The laser power and layer thickness were kept constant at 400 W and 0.03 mm throughout the study. First, the single-melting strategy was investigated for scan speeds and hatches ranging from 100 to 600 mm/s and 0.03 to 0.20 mm, respectively. Next, the recoating and remelting strategies were employed at scan speeds and hatch distances ranging from 200 to 450 mm/s and 0.06 to 0.12 mm, respectively. The energy density (J/mm 3 ) was calculated with the following equation:

𝐸 = 𝑃 𝐿𝑇×ℎ×𝑠 , ( 1
)
where P is the laser power (W), LT is the layer thickness (mm), h is the hatch distance (mm), and s is the scan speed (mm/s). The structures designed using SOLIDWORKS ® were uploaded to the Materialise Magics software for parameter assignment and print job generations. The parts were then printed on a stainless-steel building plate (stainless steel, 123 × 123 × 25 mm 3 ). The LPBF was monitored using a layer control system installed in the SLM ® 125 printer. Pictures of the ongoing printing were captured using a 12-megapixel CCD camera.

Characterization of Cu/D composites.

The Archimedes method (Mettler Toledo AT201) was used to measure the real density of the samples. The relative density was calculated by dividing the real density by the theoretical one (D Theoretical = 8.63 g/cm 3 ). The apparent density was measured using the open-source ImageJ software on SEM micrographs [29]. The microstructure of printed parts was observed using an SEM (FEI Quanta 200 ESEM ™ ). The cross-sectional view of the samples was analyzed by cutting and polishing the Cu/D composites. The average surface roughness was measured using an optical surface profiler (Zygo NewView 8000). The sample's thermal diffusivity (TD) was measured using an advanced photothermal experiment described elsewhere [START_REF] Ruffio | Signal noise ratio improvement technique for bulk thermal diffusivity measurement[END_REF]. The TD was carried out by measuring the temperature change at the rear surface when the front face was submitted to a heat flux with a Dirac comb waveform. The photothermal method did not require the sample surfaces to be coated with graphite layers to enhance the absorptivity and emissivity and thus led to more accurate thermal conductivity measurement. To estimate the in-plane and out-of-plane thermal properties, the experiment was performed within the samples' two directions. The TD was converted into TC using the following equation :

𝑘(𝑇) = 𝑎(𝑇) × 𝐶 𝑝 (𝑇) × 𝜌(𝑇), (2)
Where k is the thermal conductivity (W/m.K), a is the thermal diffusivity (mm 2 /s), ρ is the density, and Cp is the heat capacity of the sample (J/Kg.K). The effective Cp is calculated from that of D, TiC, and Cu that were 630, 190, and 392 J/Kg.K, respectively [START_REF] Azina | Effect of titanium and zirconium carbide interphases on the thermal conductivity and interfacial heat transfers in copper/diamond composite materials[END_REF]. More information on the measurement can be found in supplementary information.

Results and discussion

Conventional laser powder bed fusion of Cu/D composites.

Copper and D are known as a nonreactive system with a lack of chemical affinity [START_REF] López | The solubility of C in solid Cu[END_REF][START_REF] Kong | Enhanced wetting and properties of carbon/carbon-Cu composites with Cr 3 C 2 coatings by Cr-solution immersion method[END_REF]. Several studies have indicated that the Cu-D interfacial resistance is the main parameter that governs the resulting TC [START_REF] Azina | Effect of titanium and zirconium carbide interphases on the thermal conductivity and interfacial heat transfers in copper/diamond composite materials[END_REF]. Therefore, creating interphase between both components can allow a proper thermal transfer from the matrix to the reinforcement [START_REF] Silvain | A review of processing of Cu/C base plate composites for interfacial control and improved properties[END_REF][START_REF] Zhang | Enhanced thermal conductivity in copper matrix composites reinforced with titanium-coated diamond particles[END_REF][START_REF] Schubert | Interfacial characterization of Cu/diamond composites prepared by powder metallurgy for heat sink applications[END_REF]. Hence, in this study, D particles were coated with a graded titanium dioxidetitanium carbide (TiO2-TiC) coating to enhance Cu's wettability on D (Fig. S1a -b) [START_REF] Constantin | Design of tailored oxide-carbide coating on carbon fibers for a robust copper/carbon interphase[END_REF]. A 5 vol.% coated -D was then mixed with a pure Cu powder, as shown in Fig. S1 cf. In conventional LPBF, a 3D object is printed layer upon layer by repeating a three-step process, as illustrated in Fig. S2: 1) deposition of a thin powder layer, 2) laser melting, and 3) lowering the build plate. The three-step process is referred to as the singlemelting strategy. The typical parameters to print Cu as reported in the literature, are a laser power (P) ranging from 200 to 1000 W, a scan speed (s) between 200 and 1000 mm/s, a layer thickness (LT) from 0.03 to 0.05 mm, and a hatch distance (h) from 0.05 to 0.12 mm [START_REF] Jadhav | Influence of selective laser melting process parameters on texture evolution in pure copper[END_REF][START_REF] Ikeshoji | Selective Laser Melting of Pure Copper[END_REF][START_REF] Constantin | Laser 3D printing of complex copper structures[END_REF][START_REF] Colopi | Selective laser melting of pure Cu with a 1 kW single mode fiber laser[END_REF]. Initially, the following parameters were investigated to print Cu/D composites: s = 100 to 600 mm/s and h = 0.03 to 0.12 mm. The laser power and layer thickness were fixed at 400 W and 0.03 mm, respectively, for all experiments.

The ongoing printing of Cu/D composites using a single-melting strategy is presented in Fig. 1a.

In this image, a significant number of spatter ejections appeared during the laser/Cu+D powder interactions (Fig. 1b), accompanied by a dense metal vapor. Spatter ejection is a common phenomenon in LPBF and is often categorized into two types. When the laser strikes the powder bed, a spheroidal molten pool is created within a few microseconds [START_REF] Gusarov | Model of Radiation and Heat Transfer in Laser-Powder Interaction Zone at Selective Laser Melting[END_REF]. Quickly, a metal vapor emerges above the molten pool, inducing a recoil pressure onto the pool surface. Meanwhile, inside the liquid, a thermal gradient rapidly develops, producing a flow of molten metal from the hot area to the coldest one, known as the Marangoni effect. Both effects caused particle ejection, called recoil-induced ejection, as illustrated in Fig. 1c [START_REF] Yin | High-power laser-matter interaction during laser powder bed fusion[END_REF]. In the meantime, a denudation zone was created on the molten track's sides, where powders can be swept under the flow of the metal vapor and the gas shield. This type of particle ejections is called entrainment particle, as illustrated in Fig. 1c [START_REF] Ly | Metal vapor micro-jet controls material redistribution in laser powder bed fusion additive manufacturing[END_REF]. Intuitively, micrometric particles altered the molten pool rheology, surface tension, and metal vapor density, as has been reported [START_REF] Xi | Effect of ceramic particle size on densification behavior, microstructure formation, and performance of TiB2-reinforced Al-based composites prepared by selective laser melting[END_REF][START_REF] Sitek | Analysis of Microstructure and Properties of a Ti-AlN Composite Produced by Selective Laser Melting[END_REF][START_REF] Li | Effect of Trace Addition of Ceramic on Microstructure Development and Mechanical Properties of Selective Laser Melted AlSi10Mg Alloy[END_REF]. Also, a pronounced spattering phenomenon was observed when 5 vol.% coated -D was mixed with Cu compared to a pure Cu bed situation (Fig. S3a), thus confirming the negative effect of the coated -D on the molten pool. illustration of the spatter effect on the printing part.

After ejection, the hot particles fall back into the powder bed, cool down, and partially fuse with the solid material (Fig. 1d). Then, during the deposition of a subsequent layer, the solidified particles create bumps, pores, and cavities on the part. Figs. 2ac display the surface of Cu/D composites printed with low, moderate, and high energy densities of 278, 555, and 1111 J/mm 3 , respectively. As can be observed, the samples appear porous with large cavities and bumps.

Several studies showed that laser power and scan speed are two key parameters that affect the number and size of spatters at constant layer thickness, pressure, and gas flow [START_REF] Yin | High-power laser-matter interaction during laser powder bed fusion[END_REF]. It is commonly accepted that a fast scan speed lowers the molten pool temperature, vapor pressure, and decreases the denudation zone, limiting the spatter ejection. However, all samples printed with a scan speed between 100 to 450 mm/s, were porous, as can be noted on the two-dimensional (2D) map in Fig.

2d -e and Fig. S4 with a relative density ranging from 67 to 90%, depending on the energy density (Fig. 2f).

Furthermore, the energy required to 3D print Cu/D composites appears more substantial than that for pure Cu, where an energy density of 275 J/mm 3 led to a dense material (95%), as shown in Fig.

2f [START_REF] Constantin | Laser 3D printing of complex copper structures[END_REF]. The higher energy density required to print Cu/D composites compared to pure Cu is caused by solid coated -D into the molten Cu pool. The solid particles increased the molten metal viscosity and limited its ability to flow and fuse [START_REF] Gu | Nanocrystalline TiC reinforced Ti matrix bulk-form nanocomposites by Selective Laser Melting (SLM): Densification, growth mechanism and wear behavior[END_REF]. Y. Ma et al. reported the same phenomenon when they attempted to print aluminum-silicon /diamond (Al12Si/D) composites [START_REF] Ma | On the study of tailorable interface structure in a diamond/Al12Si composite processed by selective laser melting[END_REF]. Alongside, the high energy density required to 3D laser print Cu/D generates critical back reflection of the laser beam because of the low Cu's optical absorption in the near-IR domain (i.e., λ = 1070 nm) [START_REF] Hess | Continuous Wave Laser Welding of Copper with Combined Beams at Wavelengths of 1030nm and of 515nm[END_REF]. It was noticed that when setting an energy density above 1000 J/mm 3 (i.e., low scan speed < 200 mm/s), more than 10 W is back-reflected into the laser, thus shutting it down. Consequently, a scan speed limit of 200 mm/s must be set to prevent damage to the laser optics and to the laser itself. Hence, the maximum relative density which can be reached using a single-melting is about 88%. As a result, the LPBF of Cu/D composites raises several issues. The micrometric coated -D induces spattering, and Cu reflectivity in the IR domain limits the scan speed to 200 mm/s.

Effect of printing strategies on density, quality, and microstructure of Cu/D materials.

To overcome these limitations, a recoating and remelting strategies were implemented in the LPBF. The recoating strategy aims to fill the pores by repassing the recoater on the same layer and shining the laser a second time. It involved a five-step process, as illustrated in Fig. S5: 1) deposition of a thin powder layer, 2) laser melting, 3) redeposition of thin powder layer, 4) laser melting, and 5) lowering the build plate. On the contrary, the remelting strategy has for objective to suppress printing defects, fill pores, and smoothens the surface by shining the laser twice on the same layer. This approach included a four-step process, as shown in Fig. S6: 1) deposition of a thin powder layer, 2) laser melting, 3) laser remelting, and 4) lowering the build plate. Metal matrix composites were then printed with different strategies and printing parameters. They were qualitatively categorized into porous, dense, and cracked, represented by the black, blue, and red circles in Fig. 2d. Photographs of the composites printed are included in Fig. S7 andS8.

The process map using the recoating strategy is shown in Fig 3a . As can be noted, fast scan speeds and large hatch distances led to porous surfaces. In contrast, slower speeds and smaller hatch distances (i.e., high energy density) generated cracked samples as a result of critical thermal gradients [START_REF] Zaeh | Investigations on residual stresses and deformations in selective laser melting[END_REF]. The same tendency was observed when the remelting strategy was used to print Cu/D composite materials; but a small process window was found for scan speeds of 200, 250, 300, and 350 mm/s with hatch distances of 0.12, 0.1, 0.8, and 0.6 mm, respectively (Fig. 3b).

The relative density of the Cu/D composite as a function of the energy density is shown in Fig. 3c.

The remelted MMC with an energy density ranging from 500 to 700 J/mm 3 were the densest with a relative density of 97%. Next, the recoated Cu/D composite had a relative density of 92%, followed by the single melted sample with 88%. In addition, Fig. 3d displays the Cu/D composites'

relative density printed at a fixed hatch distance and using different printing strategies and scan speeds. As can be noted, the sample relative density decreased with faster speeds, a consequence of an insufficient flow and fusing of Cu. These observations imply that the scan speed must be sufficiently slow to have an adequate flow of the molten metal but must be fast enough to avoid critical back reflection (< 200 mm/s).

The high relative density achieved using the remelting strategies can be explained by different phenomena during the printing. Layer thickness is a crucial parameter in spatter generation, as demonstrated by C. Qiu et al. [START_REF] Qiu | On the role of melt flow into the surface structure and porosity development during selective laser melting[END_REF]. They showed that when the layer thickness increased from 20 to 100 μm, the spatter and metal vapor increased and induced large pores, discontinuous laser tracks, and cavities on sample surfaces. Surface maps and average surface roughness (Ra) of printed Cu-D composites with different printing strategies are presented in Fig. S9. It is noticed that the Cu/D composites printed with a single melting and recoating strategies have numerous defects and are significantly rough with a Ra of 130 and 85 μm, respectively (Fig. S9a-b).

Naturally, holes, pores, and bumps alter the layer thickness, affecting the pool dynamic, thus promoting spatter ejection. Also, a rough surface induces a laser focus shift and inhibits its ability to melt the powder accurately [START_REF] Bean | Effect of laser focus shift on surface quality and density of Inconel 718 parts produced via selective laser melting[END_REF]. Both effects combined or separately, can explain the formation of porous microstructures with the single-melting or recoating strategy, as illustrated in Fig. 3ef.

In contrast, the remelting strategy produces smooth (Ra = 25 μm) and dense samples in the process window, as shown in Fig. 3g and Fig. S9 c-d. Several studies have demonstrated that the remelting of each layer allows the suppression of printing defects, filling pores, and smoothing the surface, as shown in Fig. 3g. W. Yu et al. show that the remelting of printed AlSiMg alloys allows decreasing the surface roughness for 21 to 10 μm [START_REF]Influence of re-melting on surface roughness and porosity of AlSi10Mg parts fabricated by selective laser melting[END_REF]. E. Yasa et al. reported that shining the laser twice on a stainless steel layer improves the final material's surface finish from 12 to 1.5 μm and reduces its porosity from 0.8% down to 0.04% [START_REF] Yasa | Manufacturing by combining Selective Laser Melting and Selective Laser Erosion/laser re-melting[END_REF]. R. Li. et al. demonstrated that a remelting approach allows to melt and incorporate printing defects into the printed layer [START_REF] Li | Balling behavior of stainless steel and nickel powder during selective laser melting process[END_REF]. Hence, the remelting appears to be beneficial to print metallic parts and certainly to print MMCs. (e) illustration of Cu/D composites surfaces when using the single-melting, (f) recoating, and (g) remelting strategies.

Furthermore, to get more insight into the influence of the printing strategies on the powder bed and composite qualities, in-situ monitoring of the LPBF was carried out using the layer control system (Fig. 4 a). Note that the images allowed observing only significant particles representing a small amount of the spatter ejection but were the most likely to have influenced the printing quality (Fig. S3). S. Ly et al. reported that the size of the recoil-induced particle ejections (i.e., molten pool) was larger than the raw powders, while entrainment particles had a similar size [START_REF] Ly | Metal vapor micro-jet controls material redistribution in laser powder bed fusion additive manufacturing[END_REF]. Hence, the large particles observed here may have originated from an instability of the molten pool induced by the coated -D. As can be seen in Fig. S10, the molten pools have a comparable size with the D particles (Fig. S1b) for all printing strategies. Note that the scan speed and hatch distance were fixed to 250 mm/s and 0.10 mm, respectively, for the rest of the study.

The powder-bed pictures after the coating of a new layer at different stages of the printing using the single-melting strategy are presented in Fig. 4b. As can be seen at the beginning of the print (Layer 10), the powder bed was flat, uniform, and without defects. Next, when the number of layers increased (Layer 60), large white particles appeared on the bed surface. Later, they disappeared (Layer 110), and the printed part emerged, with particle-like features at the surface.

The fading away of the large particles after 110 layers was due to their absorption inside the bed and incorporation into the printed part, causing the part's roughness (Ra ~ 130 μm). At the end of the print, the Cu/D composite materials appear rough and porous (d= 88 %) as shown Fig. 4e.

When using the recoating strategy, the bed was uniform at the beginning of the print; then, significant spatter appears when the LPBF went on, as displayed in Fig. 4c. The recoating filled some pores by redepositing powders and applying the laser a second time. Nonetheless, a more considerable amount of powder was deposited on each layer, leading to an increased number of spatter ejections. Finally, the part showed an improved surface (Ra ~ 85 μm); but the pores remained as observed in Fig. 4e (d= 92 %). In contrast, only a few spatter particles were found on the powder bed with the remelting strategy (Fig. Next, the microstructures of the previously printed samples (s = 250 mm/s, h= 0.10 mm) were studied along the building direction to further comprehend the role of the printing strategy on the layer-upon-layer manufacturing of Cu/D composites (Fig. 5a). The microstructure of the Cu/D composite printed using the single-melting strategy is presented in Fig. 5b. The sample shows pores, cavities with sizes up to 500 μm, and partially fused powder. The presence of these printing defects was expected, considering the previous observations. Undoubtedly, adding a new layer on a high defect Cu/D surface led to uneven powder dispersion. Thus, the defects remained during the printing, contributing to the low-density microstructure formation with an apparent density of about 80 %. Fig. 5c shows the microstructure of the sample printed using the recoating strategy.

The sample appeared denser compared to the single-melting approach with an apparent density of about 88%, but pores remained with size up to 200 μm. As mentioned previously, the recoating approach allowed pores and cavities to be filled during the LPBF by adding a new powder layer and shining the laser a second time. Nevertheless, spatter ejection continued, adding defects to the composite material. Thus, a partially dense sample was obtained, as shown in the SEM micrographs. Fig. 5d displays the microstructure of the sample printed using the remelting strategy. The sample cross-section and surface are dense with an apparent density of about 98 %, and only a few pores can be observed with size up to 50 μm. These observations confirmed the advantages of using the remelting printing strategy to manufacture dense microcomposites.

3D printing of complex Cu/D composites for thermal management applications.

The TC of the as-built Cu/D composites printed with the different printing strategies (s = 250 mm/s and h= 0.10 mm) was measured using the photothermal method and is presented in Fig. 6a. More information on the TC measurement can be found in Fig S9 . The single melted Cu/D composite had a low TC of 236 W/m.K. This result was expected considering its low relative density. Next, the samples printed using the recoating strategy showed improved thermal performance with a TC of 327 W/m.K. Finally, the MMC printed using the remelting strategy showed a high TC of 349 W/m.K. These measurements match the previous density and microstructural analyses. In addition, it is worth mentioning that the TC of the as-built remelted Cu/D composite was higher than the one reported for as-built pure Cu (336 W/m.K) and the as-built Cu-Cr alloy (112 W/m.K), both by LPBF [START_REF] Jadhav | Influence of selective laser melting process parameters on texture evolution in pure copper[END_REF][START_REF] Jadhav | Surface Modified Copper Alloy Powder for Reliable Laser-based Additive Manufacturing[END_REF].

Copper/D composite materials are commonly referred to as the next generation of thermal management materials for microelectronic applications (Fig. 6b). Heat sinks require a high surface area to enhance heat dissipation, to be realized by intricate designs. However, sophisticated designs are challenging to achieve on D-based materials due to the presence of D particles [START_REF] Neugebauer | Additive manufacturing boosts efficiency of heat transfer components[END_REF]. On the other hand, Cu has a high density (8.93 g/cm 3 ), limiting its thermal management applications as compared to aluminum (Al, 2.7 g/cm 3 ), which is often preferred for its lightweight. Nevertheless, LPBF offers the possibility of fabricating efficient and lightweight Cu/D heat sinks by forming intricate and architectural objects. In a recent report, M. Wong et al. showed that the heat dissipation of a D-structure made of Al alloys was 60 % more efficient than an array heat sink while being lightweight and strong [START_REF] Wong | Pressure Loss and Heat Transfer Through Heat Sinks Produced by Selective Laser Melting[END_REF][START_REF] Liu | Mechanical Properties of Optimized Diamond Lattice Structure for Bone Scaffolds Fabricated via Selective Laser Melting[END_REF][START_REF] Leary | Selective laser melting (SLM) of AlSi12Mg lattice structures[END_REF]. In the same manner, honeycomb structures are wellestablished as effective and lightweight heat dissipators [START_REF] Sahoo | Application of TCE-PCM based heat sinks for cooling of electronic components: A review[END_REF]. Therefore, combining these shapes with the excellent thermal properties of Cu/D materials could improve their thermal performance by increasing the surface area while enhancing their attractiveness by lowering the weight. Fig. 6c shows a photograph and SEM micrographs of a dense and highly complex D-structure with dimensions of about 3.5 × 3.5 × 1.5 cm 3 manufactured using the optimized printing parameters (i.e., remelting, s = 250 mm/s, h = 0.1 mm). Note that the structure was sandblasted to remove the residual particles on sidewalls. Next, three honeycomb objects were additively manufactured with filling percentages ranging from 40 to 60%, as presented in Fig. 6d.

Conclusion

In summary, by adding new printing steps in LPBF (i.e., remelting), we demonstrated the 3D laser printing of dense (97%), complex, and high TC (349 W/m.K) Cu/D composites. Studying the effect of the single-melting, recoating, and remelting strategies on the powder-bed quality and composite characteristics shows the prospective benefits of adding new steps to 3D laser printing of high-quality microcomposites. We expect that developing new printing strategies for LPBF could open new avenues to laser 3D print a broad range of MMCs that is not possible today.

Thereby, unleashing their full potential, as demonstrated on a Cu/D composite, for thermal management applications. In particular, the ability to design complex Cu/D parts as the most efficient thermal structures could significantly enhance their thermal performance, lower their density, and thus increase their attractiveness for advanced thermal management applications.

Fig. 1 .

 1 Fig. 1. Spatter ejection phenomenon during the LBPF of Cu/D composite materials. (a) Optical

Fig. 2 .

 2 Fig. 2. Microstructure, quality, and relative density of Cu/D composite materials printed using the

Fig. 3 .

 3 Fig. 3. Comparative study of Cu/D composites quality and relative densities printed using the

  4c), leading to 3D laser printing of smoother (Ra ~ 25 μm) and denser composite structures (d= 97 %) as presented in Fig 4e.

Fig. 4 .

 4 Fig. 4. Effect of the printing strategy on the powder-bed and composites qualities. (a) Illustration of the layer control system in the SLM ® 125 printer. (b) Images of the powder bed after coating a

Fig. 5 .

 5 Fig. 5. Microstructural study of Cu/D composites printed with different printing strategies. (a)

Fig. 6 .

 6 Fig. 6. Thermal performance and utility of complex Cu/D structures. (a) Thermal conductivity of

Acknowledgments

The authors are grateful for the financial support provided by the National Science Foundation (CMMI 1825608) and the Nebraska Center for Energy Sciences Research (NCESR). The authors would like to thank Jamie Eske for her help on the English editing.

Data Availability

Data generated during or analyzed during the current study are included in this published article and its Supplementary Information file.

AUTHOR INFORMATION

Corresponding Authors

*Jean-François Silvain : jean-francois.silvain@icmcb.cnrs.fr; Yong Feng Lu :ylu2@unl.edu.

Author Contributions ‡ Dr. Loic Constantin, Mrs. Nada Kraiem, and Mr. Zhipeng Wu contributed equally to this work.

All authors approved the final version.


Jan Broersen

Andreas Herzig

A Normal Simulation of Coalition Logic and an Epistemic Extension

In this paper we show how coalition logic can be reduced to the fusion of a normal modal STIT logic for agency and a standard normal temporal logic for discrete time, and how this multi-modal system can be suitably extended with an epistemic modality. Both systems are complete, and we provide a new axiomatization for the STIT-fragment. The epistemic extension enables us to express that agents see to something under uncertainty about the present state or uncertainty about which action is being taken. In accordance with established terminology in the planning community, we call this version of STIT the 'conformant STIT'. The conformant STIT enables us to express that agents are able to perform a uniform strategy. As a final word of recommendation for this paper we want to point out that its subject is at the junction of four academic fields, viz. modal logic, philosophy, game-theory and AI-planning.

Introduction

Coalition Logic, CL for short, was proposed by Pauly in [START_REF] Pauly | Logic for Social Software[END_REF] as a logic for reasoning about social procedures characterized by complex strategic interactions between agents, individuals or groups. Examples of such procedures are fairdivision algorithms or voting processes. CL facilitates reasoning about abilities of coalitions in games by extending classical logic with operators [J] ϕ for groups of agents J, reading: "the coalition J has a joint strategy to ensure that ϕ". 1 In this paper we show how CL is naturally embedded in a variant of STIT theory [START_REF] Belnap | Facing the future: agents and choices in our indeterminist world[END_REF]. STIT theory is the most prominent account of agency in the philosophy of action. It is the logic of constructions of the form "agent i sees to it that ϕ holds". Our variant of STIT theory is the fusion of a multi-agent version of the so-called Chellas STIT operator 2 with a temporal next-time operator. We thus assume no interaction between the STIT operator and the next-time operator. We acknowledge that any reasonable logic for time and agency cannot present the action and time dimensions as logically independent, but our point in this paper is that we do not need the interaction to give a correct embedding of Coalition Logic. For easy reference, we adopt the name NCL ('Normal Simulation of Coalition Logic') for the fusion of our multi-agent STIT operator and the next-time operator. The embedding of CL in NCL is an interesting result since it shows how to extend CL with capabilities of reasoning about what a coalition is actually doing (as opposed to what it could do).

1 Note that we use [J] ϕ as an alternative notation for Pauly's non-normal operator [J]ϕ. We introduce this alternative syntax for two reasons: (1) the new syntax evokes the quantifier combination ∃ -∀ underlying the semantics, and (2) we use Pauly's original syntax [J]ϕ to denote the STIT operator, thereby emphasizing that this is a normal modal necessity operator.

2 The logic of Chellas STIT together with the logic of the deliberative STIT form what is called the deliberative STIT theories and have been axiomatized in [START_REF] Belnap | Facing the future: agents and choices in our indeterminist world[END_REF]Chap. 17].

In social choice theory, in particular since Harsanyi, the interaction between ability models and epistemic models has been a main focus of research. It has been realized that intentionality of action presupposes awareness or knowledge of the means by which effects are ensured. Philosophers refer to this ability of agents as having the power to ensure a condition. So, in order to say that an agent 'can' or 'has the power to' ensure a condition, there should not only be an action in the agent's repertoire that ensures the condition, the agent should also know how to choose the action.

More recently the issue of 'knowing how to act' has come up in the logic ATEL [START_REF] Wiebe Van Der Hoek | Tractable multiagent planning for epistemic goals[END_REF] which is the epistemic extension of the logic of strategic ability ATL [START_REF] Alur | Alternatingtemporal logic[END_REF]. The problem is often referred to as the problem of uniform strategies. In particular, ATEL does not allow to distinguish the situations where:

1. the agent a knows it has a particular action/choice in its repertoire that ensures ϕ, possibly without knowing which choice to make to ensure ϕ.

2. the agent a 'knows how to' / 'can' / 'has the power to' ensure ϕ.

In this paper we do not reason about series of choices, alias strategies, which is why our starting point is CL instead of ATL. We extend NCL with an S5 modal operator for knowledge and show that the resulting complete logic, that we refer to as ENCL, solves the problem of uniform strategies. Furthermore, the epistemic extension enables us to define a notion of 'seeing to it under uncertainty'. In accordance with established terminology in the planning literature, we call this version of STIT, the 'conformant STIT'. Finally, we give an argument for our view that ATEL-based approaches to the problem of uniform strategies are not likely to succeed.

Coalition Logic

Let AGT be a finite set of agents and P rop a countable set of atomic formulas. The syntax of Coalition Logic is defined as follows:

ϕ ::= p | ¬ϕ | ϕ ∨ ϕ | [J] ϕ
where p ranges over P rop and J ranges over the subsets of AGT . The other boolean connectives are defined as usual.

Coalition model semantics

Definition 1 (effectivity function). Given a finite set of agents AGT and a set of states S, an effectivity function is a function E : 2 AGT -→ 2 2 S . An effectivity function is said to be:

• J-maximal iff for all X ⊆ S, if S \X ∈ E(J) then X ∈ E(J).
• outcome monotonic iff for all X ⊆ X ⊆ S and for all J ⊆ AGT , if

X ∈ E(J) then X ∈ E(J). • superadditive iff for all X 1 , X 2 , J 1 , J 2 such that J 1 ∩ J 2 = ∅, X 1 ∈ E(J 1 ) and X 2 ∈ E(J 2 ) imply that X 1 ∩ X 2 ∈ E(J 1 ∪ J 2 ).
E intuitively associates every coalition J to a set of X ⊆ S (a set of possible outcomes) for which J is effective. That is, J can force the world to be in some state of X at the next step. • S is a nonempty set of states;

• E : S -→ (2 AGT -→ 2 2 S
) is a playable effectivity structure;

• V : S -→ 2 P rop is a valuation function.

The mapping E associates every state s to a playable effectivity function E(s). We will write E s (J) instead of E(s)(J).

Truth conditions are standard for classical formulas. We evaluate the coalitional operators against a coalition model M and a state s according to:

M, s |= [J] ϕ iff {s | M, s |= ϕ} ∈ E s (J).
Note that Coalition Logic is a non-normal weak modal logic, and that the semantics given above is a so-called 'neighborhood semantics'. For instance, the outcome monotonicity property can be reformulated as reachability of neighborhoods being closed under supersets.

Game semantics

In It appears that there is a strong link between a coalition model (whose effectivity structure is playable by definition) and a strategic game.

Definition 5. Given a strategic game G, the effectivity function

E G : 2 AGT -→ 2 2 S of G is defined as X ∈ E G (C) iff there is σ C ∈ i∈C Σ i such that for every σ C ∈ i∈C Σ i we have o(σ C × σ C ) ∈ X.
Pauly then gives the following characterization:

Theorem 1 ([Pau02]
).

E : 2 AGT × S -→ 2 2 S such that µ E (J, s) = {Y | Y is a minimal effectivity outcome at s for J }.
The outcome of a strategic game is completely determined when every agent has made its choice.

Proposition 1. µ E (AGT , s) is a set of singletons.

Proof. With Definition 5 this is a corollary of Theorem 1.

Axiomatization

The set of formulas that are valid in coalition models is completely axiomatized by the following principles [START_REF] Pauly | A modal logic for coalitional power in games[END_REF].

(ProTau) any sufficient set of propositional logic schemas

(⊥) ¬ [J] ⊥ ( ) [J] (N ) ¬ [∅] ¬ϕ → [AGT ] ϕ (M ) [J] (ϕ ∧ ψ) → [J] ψ (S) [J 1 ] ϕ∧ [J 2 ] ψ → [J 1 ∪J 2 ] (ϕ∧ψ) if J 1 ∩ J 2 = ∅ (M P ) from ϕ and ϕ → ψ infer ψ (RE) from ϕ ↔ ψ infer [J] ϕ ↔ [J] ψ Theorem 2 ([Pau02]
). The principles (ProTau), (⊥), ( ), (N ), (M ), (S), (M P ) and (RE) are complete with respect to the class of all coalition models.

Note that the (N ) axiom corresponds to AGTmaximality of the effectivity structures. It says that if a formula is not settled true, the coalition of all agents (AGT ) can always coordinate their choices to make its negation true. The axiom (S) corresponds to superadditivity and says that two disjoint coalitions can combine their efforts to ensure a conjunction of properties. Note that from (S) and (⊥) it follows that

[J 1 ] ϕ ∧ [J 2 ] ¬ϕ
is not satisfiable for disjoint J 1 and J 2 . So, two disjoint coalitions cannot ensure opposed facts.

Theoremhood and consistency are defined as usual.

A Normal Simulation of Coalition Logic

Let AGT = {0, . . . , n -1} be a finite set of n ≥ 1 agents and P rop a countable set of atomic formulas. The logic NCL has the following syntax, where p ranges over elements of P rop and J ranges over the set of subsets of AGT :

ϕ ::= p | ¬ϕ | ϕ ∨ ϕ | Xϕ | [J]ϕ
Other boolean connectives are defined by abbreviations, as usual. Furthermore, we define

J ϕ = def ¬[J]¬ϕ.
The intended meaning for the operator X is 'next time', and for the operator [J] 'agents J see to it that'.

Axiomatization We give the following axiom schemas for NCL.

(ProTau) any sufficient set of propositional logic schemas

S5([J]) any sufficient set of S5- schemas, for every [J] (Mon) [J 1 ]ϕ → [J 1 ∪ J 2 ]ϕ Elim([∅]) ∅ ϕ → J J ϕ Triv([AGT ]) ϕ → [AGT ]ϕ K(X) X(ϕ → ψ) → (Xϕ → Xψ) D(X) Xϕ → ¬X¬ϕ Det(X) ¬X¬ϕ → Xϕ
We also assume the standard inference rules of modus ponens, and necessitation for X and [∅].

From the latter necessitation for every [J] follows by the inclusion axiom (Mon). J denotes the complement of J w.r.t. AGT . The axiom Elim([∅]) will be central in this section. Note that the converses of Elim(∅) and Triv(AGT ) can be proved from (Mon), S5(∅) and S5(AGT ). Hence, we have

∅ ϕ ↔ J J ϕ and ϕ ↔ [AGT ]ϕ Semantics An NCL-model is a tuple M = (W, R, F X , π)
where:

• W is a set of worlds (alias contexts);

• R is a collection of equivalence relations R J (one for every coalition J ⊆ AGT ) such that:

-

R J 1 ∪J 2 ⊆ R J 1 -R ∅ ⊆ R J • R J -R AGT = Id; • F X : W -→ W is a total function; • π : W -→ 2 P rop is a valuation function.
The truth conditions are:

• M, w |= p iff p ∈ π(w) • M, w |= Xϕ iff M, F X (w) |= ϕ • M, w |= [J]ϕ iff for all u ∈ R J (w), M, u |= ϕ
together with the usual definitions for the other operators. Validity and satisfiability are also defined as usual.

Theorem 3. NCL is determined by the class of NCL-models.

Proof. Soundness is obtained by a routine argument and completeness is immediate from Sahlqvist's theorem, cf. [START_REF] Blackburn | Modal Logic[END_REF].

In the rest of the section we prove two lemmas that are useful in the sequel.

Lemma 1. ∅ ϕ → J 1 J 2 ϕ if J 1 ∩ J 2 = ∅.
Proof. By Elim(∅) we have ∅ ϕ → J 1 J 1 ϕ. Then, by hypothesis, J 1 ∩ J 2 = ∅, or equivalently J 2 ⊆ J 1 . Thus, by (Mon), J 1 ϕ → J 2 ϕ. We obtain

J 1 J 1 ϕ → J 1 J 2 ϕ by standard modal principles for [J 1 ]. We conclude that ∅ ϕ → J 1 J 2 ϕ.
In [BPX01, Chap. 17] the authors provide an axiomatization of the theories of deliberative STIT in terms of a family of axiom schemas (AIA k ). These capture a central idea of multi-agent STIT theories saying that agents' choices are independent. We now give a theorem of NCL that generalizes (AIA 1 ) from individuals to coalitions, and that will be instrumental later in the proof of superadditivity in Theorem 4.

Lemma 2. ∅ [J 0 ]ϕ 0 ∧ ∅ [J 1 ]ϕ 1 → ∅ ([J 0 ]ϕ 0 ∧ [J 1 ]ϕ 1 ) for J 0 ∩ J 1 = ∅.
Proof. Suppose J 0 ∩ J 1 = ∅. We establish the following deduction:

1. ∅ [J 0 ]ϕ 0 → J 1 J 0 [J 0 ]ϕ 0 by Lemma 1 2. ∅ [J 0 ]ϕ 0 → J 1 [J 0 ]ϕ 0 from 1 by S5([J 0 ]) 3. ∅ [J 0 ]ϕ 0 ∧ [J 1 ]ϕ 1 → J 1 [J 0 ]ϕ 0 ∧ [J 1 ][J 1 ]ϕ 1 from 2 by S5([J 1 ]) 4. ∅ [J 0 ]ϕ 0 ∧ [J 1 ]ϕ 1 → J 1 ([J 0 ]ϕ 0 ∧ [J 1 ]ϕ 1 ) from 3 by S5([J 1 ]) 5. ∅ ( ∅ [J 0 ]ϕ 0 ∧ [J 1 ]ϕ 1 ) → ∅ J 1 ([J 0 ]ϕ 0 ∧ [J 1 ]ϕ 1 )
from 4 by standard modal principles

6. ∅ [J 0 ]ϕ 0 ∧ ∅ [J 1 ]ϕ 1 → ∅ J 1 ([J 0 ]ϕ 0 ∧ [J 1 ]ϕ 1 ) from 5 by S5([∅]) 7. ∅ [J 0 ]ϕ 0 ∧ ∅ [J 1 ]ϕ 1 → ∅ ([J 0 ]ϕ 0 ∧ [J 1 ]ϕ 1 )
from 6 by (Mon) and S5([∅])

In [[Sch07, BGH + 07]] it is shown that it follows that the problem of deciding the satisfiability of a formula of NCL is NEXPTIME-complete.

Translating Coalition Logic

We give the following translation from Coalition Logic to NCL.

tr(p) = p tr( [J] ϕ) = ∅ [J]Xtr(ϕ)
and homomorphic for the other connectives.

Theorem 4. If ϕ is a theorem of CL then tr(ϕ) is a theorem of NCL.

Proof. First, the translations of the CL axiom schemas are theorems of NCL. The only non trivial cases are superadditivity and AGTmaximality. We start with the latter:

tr(¬ [∅] ¬ϕ → [AGT ] ϕ) = ¬ ∅ [∅]X¬tr(ϕ) → ∅ [AGT ]Xtr(ϕ).
Since [AGT ]ψ ↔ ψ by Triv(AGT ), and

∅ [∅]ψ ↔ [∅]ψ by S5([∅]), the translation of (N ) is equivalent to ¬[∅]X¬tr(ϕ) → ∅ Xtr(ϕ)
. This is equivalent to ∅ ¬X¬tr(ϕ) → ∅ Xtr(ϕ) which is proved a theorem from applying [∅]-necessitation to Det(X) and applying a variant of the K-axiom.

For superadditivity we have:

tr( [J 1 ] ϕ ∧ [J 2 ] ψ → [J 1 ∪ J 2 ] (ϕ ∧ ψ)) = ∅ [J 1 ]Xtr(ϕ) ∧ ∅ [J 2 ]Xtr(ψ) → ∅ [J 1 ∪ J 2 ]X(tr(ϕ) ∧ tr(ψ)) 1. ∅ [J 1 ]Xtr(ϕ) ∧ ∅ [J 2 ]Xtr(ψ) → ∅ ([J 1 ]Xtr(ϕ) ∧ [J 2 ]Xtr(ψ)). by Lemma 2 2. [J 1 ]Xtr(ϕ)∧[J 2 ]Xtr(ψ) → [J 1 ∪ J 2 ]Xtr(ϕ)∧ [J 1 ∪ J 2 ]Xtr(ψ). by (Mon) 3. ∅ ([J 1 ]Xtr(ϕ) ∧ [J 2 ]Xtr(ψ)) → ∅ ([J 1 ∪ J 2 ](Xtr(ϕ) ∧ Xtr(ψ))
from previous line by standard modal principles

4. ∅ [J 1 ]Xtr(ϕ) ∧ ∅ [J 2 ]Xtr(ψ) → ∅ [J 1 ∪ J 2 ]X(tr(ϕ) ∧ tr(ψ))
from lines 1 and 3 by standard modal principles for X.

Second, clearly the translation of modus ponens preserves validity. To prove that the translation of CL's (RE) preserves validity suppose tr(ϕ ↔ ψ) = tr(ϕ) ↔ tr(ψ) is a theorem of NCL. We have to prove that tr(

[J] ϕ ↔ [J] ψ) = ∅ [J]tr(ϕ) ↔ ∅ [J]tr(ψ) is a theorem of NCL.
This follows from the theoremhood of tr(ϕ) → tr(ψ) by standard modal principles.

Lemma 3. Let M = ((S, E), V ) a coalition model and selec : S -→ S some mapping such that if selec(s) = s then {s } ∈ µ E (AGT , s). 3Let M = (W, R, F X , π) be constructed as follows:

• W = { s, s | s ∈ S, {s } ∈ µ E (AGT , s)} • R J = {( s, s 1 , s, s 2 ) | ∃Y ∈ µ E (J, s), s 1 , s 2 ∈ Y } • F X ( s, s ) = s , selec(s ) • π( s, s ) = V (s)
Then M is an NCL-model.

Proof. The proof consists in checking that the constructed model satisfies every constraint on NCL models. Everything is almost immediate.

The main point is that we are permitted to define F X this way because of Proposition 1.

A peculiarity of the models thus constructed is that next moments are chosen using a selection function. The reason this is possible is that we have chosen not to impose a logical relation between the temporal dimension and the action (/ choice) dimension in NCL. Now we can turn to the proof of satisfiability preservation of the translation and the model construction.

Theorem 5. If ϕ is CL-satisfiable then tr(ϕ) is NCL-satisfiable.

Proof. Given a coalition model M = ((S, E), V ) we construct an NCL-model M N CL = (W, R, F X , π) for some mapping selec as in Lemma 3. We prove by structural induction that M, s |= ϕ iff there is a s, s ∈ W s.t.

M N CL , s, s |= tr(ϕ).

The cases of atoms and classical connectives are straightforward, so we just consider the case of ϕ = [J] ψ.

1. Suppose, M, s |= [J] ψ. Then, there is Z ∈ E s (J) such that for all t ∈ Z , M, t |= ψ.
Then there is a minimal effectivity outcome The other direction of the induction hypothesis is verified by reverse arguments.

Z ∈ µ E (J,
Corollary 1. ϕ is a theorem of CL iff tr(ϕ) is a theorem of NCL.

Proof. The right-to-left direction is Theorem 4. The left-to-right direction follows from Pauly's completeness result for Coalition Logic and Theorem 5.

Seeing to it under imperfect knowledge: an epistemic extension

In this section we extend NCL with an S5 knowledge operator. This enables us to express that an agent sees to something although it is uncertain about the present state or the action being taken. In the planning community this kind of actions are called conformant [START_REF] Goldman | Expressive planning and explicit knowledge[END_REF]; they ensure a property ('the goal') in spite of uncertainty about the present state. The logic presented here enables us to express this as K i [{i}]ϕ for "agent i knows that it sees to it that ϕ, without necessarily knowing the present state". In accordance with established terminology in the planning community we call this combination of the knowledge operator and the STIT operator the 'Conformant STIT'.

The idea of combining a logic for multi-agency with a logic for knowledge naturally stems from game theory [START_REF] Osborne | A Course in Game Theory[END_REF]. In game theory, conformant plans are called 'uniform strategies'.

In ATEL [START_REF] Wiebe Van Der Hoek | Tractable multiagent planning for epistemic goals[END_REF], the epistemic extension of Alternating-time Temporal Logic (ATL), which in turn extends coalition logic by allowing coalitions to perform series of choices to ensure a certain condition, the issue of how to express existence of uniform strategies has drawn considerable attention [START_REF] Jamroga | Agents that know how to play[END_REF][START_REF] Jamroga | Constructive knowledge: What agents can achieve under incomplete information[END_REF]. The problem concerns the disambiguation of the notion of knowing a strategy: ATEL is not expressive enough to distinguish the sentence "for all epistemically indistinguishable states, there exists a strategy of J that leads to ϕ". from there exists a strategy σ of the coalition J such that for all states epistemically indistinguishable for J, σ leads to ϕ."

The former is a ∀ -∃ schema of "knowing a strategy", in philosophy referred to as the de dicto reading. It is opposed to the de re reading exemplified by the latter sentence, which is a ∃ -∀ schema.

In [START_REF] Broersen | A STIT-extension of ATL[END_REF] we sketched how the problem can be solved in a STIT-extension of ATL we called ATL-STIT. In the present setting we are only concerned with one step choices. We show how, as an extension of NCL, we can easily obtain a complete system whose semantics distinguishes between uniform and non-uniform strategies. The logic system we present here does not have the restricted syntax of the first presented proposal in [START_REF] Herzig | Knowing How to Play: Uniform Choices in Logics of Agency[END_REF] and, in addition, has a complete and straightforward axiomatization as an extension of NCL. The problem of 'uniform strategies' already arises with individual knowledge. For purpose of simplicity we thus do not consider group knowledge.

ENCL has the following syntactic form, where p ranges over P rop, J ranges over 2 AGT and i over AGT :

ϕ ::= p | ¬ϕ | ϕ ∨ ϕ | Xϕ | [J]ϕ | K i ϕ
Axiomatization The logic is obtained by adding to NCL the principles of the standard epistemic logic S5 for every individual agent i.

Semantics ENCL-models are tuples M = (W, R, F X , ∼, π) where:

• (W, R, F X , π) is a model of NCL.
• ∼ is a collection of equivalence relations ∼ i (one for every agent i ∈ AGT ).

Theorem 6. ENCL is determined by the class of models of ENCL.

To explain how this logic solves the problem of uniform strategies, we consider two scenarios.

Example 1. Ann is in a room. She is blind and cannot distinguish a world where the light is off from a world where the light is on. The light in the room is controlled by a button that activates a timer. When the button is pushed the light bulb will shine for a determinate time. When the light is on, there is no way to switch it off. Ann can also do nothing (skip). In the actual situation the light is off and Ann is pushing the button. The above picture represents the example, and we now explain how the picture can be seen as an ENCL-model. The worlds of the semantics of NCL and ENCL are here state-action pairs. The states are positions before and after execution of an action. In the picture there are 6 of these positions. For this example this results in 8 ENCL worlds. We thus have the following ENCL-model M 1 = W, R, F X , ∼, π : (we abbreviate 'push' to p, and 'skip' to s)

• W = {(1, p), ( 1, s), (2, s), (2, p), (3, s), (4, s), (5, s), (6, s)
} • R ∅ = { (1, p), ( 1, s) , (2, s), (2, p) , (3, s), (3, s) , (4, s), (4, s) , (5, s), (5, s) , (6, s), (6, s)
} • R Ann = { w, w | w ∈ W } • F X is defined by F X ((1, p)) = (3, s), F X ((1, s)) = (4, s), F X ((2, s)) = (5, s), F X ((2, p)) = (6, s), F X ((3, s)) = (3, s), F X ((4, s)) = (4, s), F X ((5, s)) = (5, s), F X ((6, s)) = (6, s) • ∼ Ann = { (1, p), (2, p) , (1, s), (2, s) } • π is defined by π((2, p)) = π((2, s)) = π((3, s)) = π((5, s)) = π((6, s)) = 'on', and π((1, p)) = π((1, s)) = π((4, s)) = 'off'
where is a reflexive, symmetric and transitive closure. It is not difficult to check that M 1 is a genuine ENCL-model, satisfying also all the constraints we defined for the NCL-sub-models.

The reader may have noticed that the model adds detail to the example. In particular, Ann is given the choice between pushing and skipping only once, and "determinate time" is interpreted as forever. Of course, the model is a very simple one, with only one agent in the system: AGT = {Ann}. Ann's actions thus coincide with system actions, and all her choices are deterministic.

The four basic properties we consider are: ϕ 1 = "One of Ann's choices ensures the light will be on" = ∅ [{Ann}]X on ϕ 2 = "Ann knows one of her choices ensures the light will be on" = K Ann ∅ [{Ann}]X on ϕ 3 = "Ann knows she has the power to ensure the light is on"

= ∅ K Ann [{Ann}]X on ϕ 4 = "Ann conformantly sees to it that the light is on" = K Ann [{Ann}]X on
It is easy to verify that in M 1 the first three formulas are true in the first four possible ENCL worlds:

M 1 , w |= ϕ 1 ∧ ϕ 2 ∧ ϕ 3 for all w ∈ {(1, p),
} • R Ann = { w, w | w ∈ W } • F X is defined by F X ((1, t)) = (3, s), F X ((1, s)) = (4, s), F X ((2, s)) = (5, s), F X ((2, t)) = (6, s), F X ((3, s)) = (3, s), F X ((4, s)) = (4, s), F X ((5, s)) = (5, s), F X ((6, s)) = (6, s) • ∼ Ann = { (1, t), (2, t) , (1, s), (2, s) } • π is defined by π((2, t)) = π((2, s)) = π((3, s)) = π((5, s)) = 'on', and π((1, t)) = π((1, s)) = π((4, s)) = π((6, s)) = 'off'
Now, in the actual world where the light is off and Ann toggles, the light will actually be on, so the formula Xon holds. Yet, Ann does not conformantly see to it that the light is on, since she does not know that the light is off at the present moment. So, the fourth of the above properties does not hold: M 2 , (1, t) |= ϕ 4 . Also, she does not have a uniform strategy, and indeed the third of the above properties does not hold either: M 2 , (1, t) |= ϕ 3 . The first and the second property do hold in the actual world, since in each state Ann indeed has an action that ensures the light is on and she knows that: But her problem is that the decision which one to take depends on the state she is in, which is something she does not know: M 2 , w |= ϕ 1 ∧ ϕ 2 for all w ∈ {(1, t), (1, s), (2, s), (2, t)}.

Let us compare our approach with the situation in ATEL. For representing uncertainty in ATEL a family of equivalence relations among states (one for each agent) is assumed, interpreting a standard normal S5 operator K i in the language. Since our uncertainty relations are among stateaction pairs, our knowledge operator is more expressive.

We will now argue that the known approaches to the problem of uniform strategies in the literature are unlikely to succeed. Note first that in example 2 above we might have given different names to the actions. And there is no reason why this renaming should be uniform. In particular, the left toggle action can be called 'put the light on' and the right toggle action 'put the light off '. Obviously, non-uniform renaming of actions should not influence Ann's basic capabilities or her knowledge concerning her capabilities. Our theory satisfies this consideratum, since changing the names of the actions in the way described, does not in any way change the evaluation of ENCL formulas. In particular, Ann still does not have a uniform strategy: using the new terminology provided by the new action names she now 'cannot distinguish between putting the light on when it is off and putting the light off when it is on'. However, all ATEL-based approaches in the literature do not satisfy the consideratum. In these variants and extension of ATEL (see e.g. [START_REF] Schobbens | Alternating-time logic with imperfect recall[END_REF]) the following condition is imposed on the models: if one state is indistinguishable from another, then any action name appearing for a choice in the first state also appears as an action name for a choice in the second state. It is clear right away that under this restriction, a non-uniform renaming of actions as we discussed above, may result in uncertainty relations being eliminated, and thus in a gain in knowledge. In particular, in the renamed version of example 2 above, Ann would always be able to distinguish the two states, and there would be no uncertainty left at all, which directly contradicts the requirement having to express that Ann does not know a uniform strategy in this situation.

Concluding remarks

We have some brief concluding remarks. The establishment of complete axiomatizations for NCL and ENCL opens up interesting perspectives on the use of (semi)-automatic theorem provers for reasoning about properties of games. Such theorem provers could then also be used for conformant planning, through the established link between planning and satisfiability checking [START_REF] Kautz | Planning as satisfiability[END_REF].

A natural investigation concerns the introduction of group knowledge in the present picture. In particular the integration of common knowledge is a worth challenge. But, while it is straightforward to import the principles of common knowledge in NCL, completeness of the resulting logic does not follow immediately, as with standard epistemic logic.

As a third future perspective we want to point out the relation with product update [START_REF] Baltag | Logics for epistemic programs[END_REF]. In the models after a product update, uncertainty relations also range over state-action pairs. And it is actually quite easy to describe our examples of the previous section as updates of epistemic models with suitable epistemic action models. The difference with product update as described by Baltag is that in our product models, we should not take the intersection of the original uncertainty relations but the union. This is because in the present setting actions are not 'suspected observations' like in the work of Baltag. In our setting we assume 'no learning' and uncertainty may either come from performing a known action in an unknown state, or an unknown action in a known state, which is why in product models we have to take the union of the uncertainty relations. However, this is not the place to discuss this in more detail, and we leave the issue for future research.

Some more detailed issues remain unresolved.

For instance, if we would replace the axiom schema Triv([AGT ]) of NCL, that is, ϕ → [AGT ]ϕ, by the weaker ∅ Xϕ → ∅ [AGT ]Xϕ we would still be able to prove that we can embed CL (see the proof of the translation of AGTmaximality in Theorem 4, which is the only place where Triv([AGT ]) is used). This seems to suggest that R AGT need not be a 'normal' relation.

Another issue is the extension with the schema [J]Xϕ → X[J]ϕ. This property seems essential for STIT-semantics, since it adds an interaction between time and action for which it is hard to find intuitive counter-examples. The reason that it was not studied before in STIT-semantics is that discrete time itself is never assumed. We plan to consider these and more issues in an extended version of this paper.

Last but not least, a clear objective is to extend the axiomatizations we gave to the setting with extensive form games. We already studied the semantics for this extension in [START_REF] Broersen | A STIT-extension of ATL[END_REF]. The most notable feature of the generalization of the semantics to extensive form games is that evaluation should be defined with respect to statestrategy pairs.

  (1, s), (2, s), (2, p)}. In particular, in the actual world (1, p) the third property holds, saying that Ann has a uniform strategy to ensure the light is on. In the actual world also the fourth property holds (M 1 , (1, p) |= ϕ 4 ), while in the two worlds where Ann skips, it does not (M 1 , (1, s) |= ϕ 4 and M 1 , (2, s) |= ϕ 4 ).

								(5, s), (6, s)}
								• R ∅ = { (1, t), (1, s) , (2, s), (2, t) ,
								(3, s), (3, s) , (4, s), (4, s) , (5, s), (5, s) ,
								(6, s), (6, s)
	Example 2. Ann is in a room. She is blind and
	cannot distinguish a world where the light is off
	from a world where the light is on. The light in
	the room is controlled by a switch. In her reper-
	toire of actions, Ann can toggle or remain passive
	(skip), which correspond to switching the state of
	the light and maintaining the state of the light,
	respectively. In the actual situation the light is
	off and Ann toggles.
		off	1			on	2
	toggle		skip		skip	toggle
	on	3	off	4	on	5	off	6
	skip		skip	skip		skip
	This example is encoded by the following ENCL-
	model M 2 = W, R, F X , ∼, π : (we abbreviate
	'toggle' to t, and 'skip' to s)
	• W = {(1, t), (1, s), (2, s), (2, t), (3, s), (4, s),

Such a function exists by the axiom of choice.

Acknowledgement

We would like to thank Philippe Balbiani, Olivier Gasquet and François Schwarzentruber for useful discussions concerning parts of the paper.