Stefan W Hell

Steffen J Sahl

Mark Bates

Xiaowei Zhuang

Rainer Heintzmann

Martin J Booth

Joerg Bewersdorf

Gleb Shtengel

Harald Hess

Philipp Tinnefeld

The 2015 super-resolution microscopy roadmap

come   L'archive ouverte pluridisciplinaire

Status

The remarkably efficient functioning of the brain largely mirrors its multiscale complexity. In particular, it became clear since the emergence of modern neuroscience that basic investigations of brain organization at the subcellular scales are key for the understanding of brain processes. Since then, our knowledge of the cellular mechanisms involved in neuronal communication and their evolution during life has literally exploded. Most of our comprehension comes from the discovery of molecules, genes and signaling cascades that play central roles in the maintenance and plasticity of the neuronal communication. Notably, the identification of the synapse has attracted much attention. It contains key molecules required to induce plastic changes and mediates a large fraction of fast neuronal communication and long-term adaptations. The dimensions of brain synapses are however small, at the submicron scale and therefore they cannot be resolved with conventional optical microscopes. Electron microscopy has been providing ultra-structural information of synapse architectures and to some extent, knowledge about the content and local molecular densities in these structures. However, this knowledge is out-of-reach in living cells and in intact tissues with this imaging modality. In this context, light microscopy constitutes the imaging modality of choice to study synapses in live cell. Continuous developments and refinements of optical imaging techniques delivered an impressive amount of information about synaptic molecule organizations at the nanoscale. A first decisive achievement came from the possibility to detect single molecules in living cells. It not only allows localizing molecules with nanometer accuracies, far below the optical resolution, but also provides the ability to probe the intimate variety of molecule dynamic environments from nano-to micro-scales levels. In the early 2000's the presence of mobile receptors for neurotransmitters was revealed for the first time 2,3 in the postsynaptic membrane of dissociated neuronal cultures and the role of neurotransmitter receptor diffusion in fast synaptic transmission was demonstrated 4 . The advent of several super-resolution methods that followed these early achievements raised great expectations in neurosciences by providing optical images of neuronal structures with unprecedented resolutions.

Current and Future Challenges

Super-resolution microscopy methods mostly rely on the control of the number of emitting molecules in specific imaging volumes. Two major types of techniques found major applications in neurosciences: those based on STED where highly localized fluorescence emission volumes are optically shaped and those based on single-molecule localizations ((f)PALM, (d)STORM or (u)PAINT) 5 . Single-molecule localization based methods have proven to be powerful to study nanoscale molecular organizations such as that of postsynaptic receptors and scaffold proteins (e.g., 1,6- 8 ) but also that of actin and actin binding proteins in axons 9 . On the other hand, STED microscopy allows visualizing dendritic spine shapes in living neurons with remarkable resolution and revealed that spine neck plasticity regulates compartmentalization of synapses 10 and further deciphered the dynamic organization of actin. As exemplified in the aforementioned achievements, imaging synaptic structures and their molecular contents with nanometer resolutions constitute already a breakthrough for the understanding of synaptic functioning. Because these demonstrations have predominantly been performed on dissociated neuronal cultures or thin fixed brain slices, one of the main challenge will be to transfer such cutting edge technologies to more integrated and ultimately intact living sample. This will allow major fields of neurosciences, such as development, aging and neurodegenerative disorders, to benefit from the unprecedented degree of details provided by these microscopies.

Advances in Science and Technology to Meet Challenges

The possibility to perform super-resolution imaging in intact living brain tissues to address major neurosciences questions is conditioned to several requirements. Because both light scattering and absorption limit the penetration depth in thick biological samples in the visible domain, superresolution imaging in tissues is only emerging. Scattering is particularly detrimental for STED microscopy, since the realisation of a zero intensity region for depletion is mandatory. Two-photon microscopy, which uses near-infrared laser excitations, is a common strategy to excite molecules deep in brain tissues. However, due to the high intensities required, photobleaching rates are drastically enhanced and single molecule based super-resolution with two-photon excitation remains difficult. To fully exploit the biological transparency window, near-infrared probes with favourable photophysics still need to be developed, as current red-shifted dyes are not strong emitters.

Another key development will consist in achieving fast wide field super-resolved imaging at rates compatible with the inherent movements of living samples in 3D. Current super-resolution methods are indeed restricted to rather small imaging areas (sub-millimetric), which is constitutive to the novel opportunity to obtain images at superior resolutions. Indeed, by increasing the image resolution by typically a factor 10 to 50 per dimension, the size of a 3D image will be increased by a factor 10 3 up to ~10 5 . This imposes constrains on the imaging speed as well as data handling. A promising strategy would consist in performing multi-scale imaging where only "relevant" brain sub-areas are imaged at the nanometer scales while the other regions are imaged at lower resolutions. The definition of imaging "relevance" within the brain might depend on time and/or the physio-pathological local state of the sample as well as the nature of the information provided by the low resolution imaging modality. In this context, correlative imaging with other modalities, which do not have to be limited to optical techniques, are interesting routes.

Combining multiple super-resolution approaches can also provide brand novel opportunities. Finally, a full understanding of the brain function cannot be obtained solely by images. Manipulation of the brain physio-pathological state will be needed using methodologies that are compatible with superresolution imaging. Optogenetic methods provide promising tools in this context.

Concluding Remarks

For widespread application of super-resolution imaging in neuroscience, the main challenge will be to link subcellular information gathered at the molecular scale (e.g. about synaptic processes) to the global organ function obtained at a macroscopic scales, in a welldefined functional state. To this aim, a multidisciplinary effort will be needed where physicist, chemists, computer scientist, neurophysiologists and neuropathologists will have to work together. The task is vertiginous, but is at the level of the complexity of the brain.

  in on synapses -Laurent COGNET & Brahim LOUNIS, Univ. Bordeaux, Institut d'Optique & CNRS, Talence, France

Figure 1 -

 1 Figure 1 -High-density single-molecule trajectories (top) and super-resolution imaging (bottom) of endogenous glutamate receptors measured on a live neuron with the uPAINT method The circle highlights the presence of a synapse (from 1 ). Scale bar = 500 nm.

Figure 2 -

 2 Figure 2 -Effect of long term potentiation on the morphology of synaptic spines and heads revealed by STED microscopy (bottom panels are zoom of top ones. (from 10 ). Scale bar = 500 nm.


Dr Friederike A Ernst

Prof Zhenghong L Gao

Raul Arenal

Prof Timm R Heek

Antonio Setaro

Rodrigo Fernandez-Pacheco

Rainer Haag

Laurent Cognet

Dr Stephanie Reich

Raúl Arenal

Prof S Setaro

Reich

LP2N Cognet

Dr R Arenal

Haag

Noncovalent Stable Functionalization Makes Carbon Nanotubes Hydrophilic and Biocompatible

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Single wall carbon nanotubes are becoming popular for biomedical applications for several reasons: They are small, chemically stable, hollow and fillable, have a large absorption cross section, and lend themselves to chemical functionalization. [START_REF] Liu | Carbon Nanotubes in Biology and Medicine: In vitro and in vivo Detection, Imaging and Drug Delivery[END_REF][START_REF] Kruss | Carbon nanotubes as optical biomedical sensors[END_REF] Consequently, they can be used as cargo vessels to release drugs in a precise location in a cell or in the body, [START_REF] Bartholomeusz | In Vivo Therapeutic Silencing of Hypoxia-Inducible Factor 1 Alpha (HIF-1α) Using Single-Walled Carbon Nanotubes Noncovalently Coated with siRNA[END_REF][START_REF] Bianco | Applications of carbon nanotubes in drug delivery[END_REF] to locally convert electromagnetic radiation into heat in photothermal therapy, [START_REF] Chakravarty | Thermal ablation of tumor cells with antibody-functionalized single-walled carbon nanotubes[END_REF][START_REF] Kam | Carbon nanotubes as multifunctional biological transporters and near-infrared agents for selective cancer cell destruction[END_REF] and can be functionalized with antibodies and a plethora of other attachments to be used as sensors. [START_REF] Barone | Near-infrared optical sensors based on single-walled carbon nanotubes[END_REF][START_REF] Chen | Protein microarrays with carbon nanotubes as multicolor Raman labels[END_REF] Carbon nanotubes also possess optoelectronic properties that make them a highly desirable candidate for biological imaging. [START_REF] Welsher | Selective probing and imaging of cells with single walled carbon nanotubes as near-infrared fluorescent molecules[END_REF][START_REF] Godin | Single-nanotube tracking reveals the nanoscale organization of the extracellular space in the live brain[END_REF] The intrinsic luminescence emission of single wall carbon nanotubes lies in the near infrared region from 900nm -1400nm, the second optical window of tissue. Biological tissues are relatively transparent between 650nm and 1350nm. However, in practice, the first optical window between 650nm and 950nm for which there are well established biocompatible fluorophores is obfuscated by tissue autofluorescence. [START_REF] Smith | Bioimaging: second window for in vivo imaging[END_REF] The second optical window between 1000nm and 1350nm has been demonstrated to exhibit 100-fold improved signal-to-noise ratios when imaging whole animals as compared to the first window, making carbon nanotube based systems with an intrinsic luminescence in the NIR highly desirable for biomedical imaging. [START_REF] Lim | Selection of Quantum Dot Wavelengths for Biomedical Assays and Imaging[END_REF][START_REF] Welsher | A route to brightly fluorescent carbon nanotubes for near-infrared imaging in mice[END_REF][START_REF] Welsher | Deep-tissue anatomical imaging of mice using carbon nanotube fluorophores in the second near-infrared window[END_REF] Carbon nanotubes are intrinsically hydrophobic and have a propensity for bundling. For applications in biomedical imaging nanotubes require solubilization through non-covalent functionalization with amphiphilic surfactants.

Using small molecules in carbon nanotube solubilization has the distinct advantage that the molecule's design is flexible and can easily be modified to incorporate additional functionalities. The biocompatibility of small surfactantnanotube hybrids faces three challenges.

(i)

The surfactant has to suspend nanotubes in a wide range of pHs, reflecting the variable environments encountered in biological tissues.

(ii)

The surfactant ought not interact with the surrounding cellular environment itself and preclude interaction between nanotube and environment as well.

(iii)

The dynamic equilibrium between adsorbed and free surfactant must be strongly in favor of the adsorbed state. This stability is important not only for the sake of continued suspension, but also because the surfactant shields the environment against the nanotube.

We recently introduced an amphiphilic molecule for functionalizing carbon nanotubes, PerPG.

PerPG forms tightly bound complexes with carbon nanotubes, in effect serving as a surfactant in solution. Instead of a charged headgroup, PerPG possesses a hydrophilic polyglycerol (PG) dendron. [START_REF] Ernst | Energy Transfer in Nanotube-Perylene Complexes[END_REF] PG dendrons are known to interact with proteins very weakly and are consequently popular as a protein repellent coating, for instance in anti-fouling surfaces. [START_REF] Frey | Dendritic polyglycerol: a new versatile biocompatible material[END_REF][START_REF] Wyszogrodzka | A convergent approach to biocompatible polyglycerol "click" dendrons for the synthesis of modular core-shell architectures and their transport behavior[END_REF][START_REF] Wyszogrodzka | Study of single protein adsorption onto monoamino oligoglycerol derivatives: a structure-activity relationship[END_REF][START_REF] Wyszogrodzka | Synthesis and characterization of glycerol dendrons, self-assembled monolayers on gold: a detailed study of their protein resistance[END_REF] We combine the PG dendron with two parallel alkyl chains for nanotube individualization and a perylene diimide dye. The perylene-derived dye forms ultra-efficient energy transfer complexes with the nanotube. [START_REF] Ernst | Energy Transfer in Nanotube-Perylene Complexes[END_REF][START_REF] Ernst | Excitation characteristics of different energy transfer in nanotube-perylene complexes[END_REF] This energy transfer functionality enables the excitation of all nanotube chiralities at one single wavelength, possibly potentiating their use in biomedical sensing, imaging and photothermal therapy applications.

Here, we present high-resolution transmission electron microscopy (HRTEM) and electron energy-loss spectroscopy (EELS) data confirming that. The nanotube-PerPG complexes are highly stable and robust against change of medium and drying. We assess PerPG's biocompatibility by performing a cell viability study and find that at relevant concentrations the small designer molecules are as biocompatible as conventionally used biocompatible poloxamers such as pluronic. 21

Results and Discussion

Imaging of nanotube-PerPG hybrids

PerPG, shown in Fig. 1, solubilizes nanotubes efficiently in aqueous solution and forms energy transfer complexes with the carbon nanotubes. [START_REF] Ernst | Energy Transfer in Nanotube-Perylene Complexes[END_REF] Its design is modular in nature, comprising alkyl chains for nanotube individualization (green), a pH independent PG dendron for water solubility (blue) and a functional perylene unit (red), which is the donor unit in the energy transfer complexes formed with the nanotubes. The combination of these three building blocks was previously systematically varied and optimized for both energy transfer and nanotube suspension capabilities. [START_REF] Ernst | Functional Surfactants for Carbon Nanotubes: Effects of Design[END_REF] The suspension was further optimized for negligible free surfactant and stability through absorption and photoluminescence measurements. [START_REF] Ernst | Energy Transfer in Nanotube-Perylene Complexes[END_REF] The energy transfer is FRET-like with an efficiency on the order of 99.99%. In conjunction with the spectral overlap between donor emission and acceptor absorption we calculate the distance between nanotube and dye in a point-dipole approximation to be 0.7nm. [START_REF] Ernst | Excitation characteristics of different energy transfer in nanotube-perylene complexes[END_REF] However, the point dipole approximation overestimates the distance between donor and acceptor in this case, as an exciton in the nanotube is not well described as a point-dipole. Instead, the dipole is oblong, decreasing the interaction with the point-dipole of the dye. Consequently, for a given interaction strength, the point-dipole approximation overestimates the distance between dye and nanotube. [START_REF] Wong | Ideal dipole approximation fails to predict electronic coupling and energy transfer between semiconducting single-wall carbon nanotubes[END_REF] Based on this, the upper limit for the distance between nanotube and dye unit is 0.7nm, only slightly more than for van der Waals bound systems. Hence we assumed that the extended π-orbitals of the carbon nanotube interact with the extended π-orbitals of the perylene derivative dye, resulting in an adsorption of the dye on the tube wall through a π-π stacking mechanism. [START_REF] Ernst | Excitation characteristics of different energy transfer in nanotube-perylene complexes[END_REF][START_REF] Wong | Ideal dipole approximation fails to predict electronic coupling and energy transfer between semiconducting single-wall carbon nanotubes[END_REF][START_REF] Wu | Resonance Energy Transfer: Methods and Applications[END_REF] The coverage rate depends slightly on the nanotube chirality and was previously estimated to be on the order of 0.1 through a detailed analysis of absorption, photoluminescence and photoluminescence excitation measurements of the carbon nanotube-PerPG complexes. [START_REF] Ernst | Excitation characteristics of different energy transfer in nanotube-perylene complexes[END_REF] To confirm the adsorption of PerPG on the nanotube surface, we perform a high resolution TEM study. HRTEM micrographs show microcrystalline regions on top of the nanotubes, Fig. 2.

These microcrystalline structures are shown to be PerPG in subsequent EELS measurements, Fig. 3. Spatially resolved EEL spectroscopy is an ideal technique to gain insight into the morphology of the nanotube-PerPG complexes, letting us directly visualize the organic biomolecules on top of the nanotube through the evaluation of the local chemical composition. [START_REF] Setaro | Preserving πconjugation in covalently functionalized carbon nanotubes for optoelectronic applications[END_REF][START_REF] Alvarez | One-Dimensional Molecular Crystal of Phthalocyanine Confined into Single-Walled Carbon Nanotubes[END_REF] Particular attention was paid to reducing the beam dose used to optimize the acquisition conditions, see Experimental Section. Figure 3B shows EEL spectrum-lines recorded on a bundle of functionalized carbon nanotubes. The locations of the spectral acquisitions are given in Fig. 3A, the corresponding high angular annular dark field (HAADF) scanning transmission electron microscopy (STEM) image. In the HAADF image we see a bright area of ~5 x 1.5 nm 2 in the middle of the nanotube bundle. Other regions like the one indicated are also visible. These brighter appearing moieties are localized at the surface of the nanotubes as we have observed in the HRTEM micrographs, Fig. 2. The EEL spectrum line was acquired following the green line marked in Fig. 3A, which crosses this bright region. The sum of 5 EEL spectra was extracted from the two areas marked in the HAADF image, Fig. 3B. Both EEL spectra display the C-K edge with the characteristic peaks of sp 2 materials at 285 eV and 292 eV corresponding to the π* and σ* contributions, respectively. [START_REF] Setaro | Preserving πconjugation in covalently functionalized carbon nanotubes for optoelectronic applications[END_REF][START_REF] Alvarez | One-Dimensional Molecular Crystal of Phthalocyanine Confined into Single-Walled Carbon Nanotubes[END_REF][START_REF] Ayala | The physical and chemical properties of heteronanotubes[END_REF][START_REF] Arenal | Boron-nitride and boron-carbonitride nanotubes: synthesis, characterization and theory[END_REF][START_REF] Alvarez | Charge Transfer Evidence between Carbon Nanotubes and Encapsulated Conjugated Oligomers[END_REF][START_REF] Arenal | Spatially-resolved EELS analysis of antibody distribution on biofunctionalized magnetic nanoparticles[END_REF] In the spectrum acquired over the bright region in the HAADF micrograph we additionally detect oxygen, see Fig. 3B: The O-K edge is clearly visible, as designated on the graph. The amount of oxygen in this region, obtained from the analysis of this EEL spectrum, is ~14%. The combination of the HRTEM images and local analytical findings confirm that the perylene unit of PerPG π-π stacks on the nanotube wall, resulting in extremely tight binding between nanotube and the small amphiphilic molecule.

We would like to draw special attention to the fact that the nanotube-PerPG complexes were created in aqueous solution, resulting in well-individualized nanotubes as attested to by strong PL emission. [START_REF] Ernst | Energy Transfer in Nanotube-Perylene Complexes[END_REF] The aqueous sample was subsequently dried and then re-dispersed in ethanol for these measurements, Fig. 4. This means the complexes are bound tightly enough to remain bundled throughout the drying process. Furthermore they are stable in air and highly robust with respect to change of the environment, indicating that this PerPG-nanotube complex may be bound tightly enough to satisfy challenges (i) and (iii) of achieving biocompatibility for small surfactant-nanotube hybrids.

Suitability for biomedical applications

We designed PerPG such that it is tailored for use in biomedical applications.

Regarding the three challenges (i)-(iii) of biocompatibility discussed in the introduction, challenge (i) pH independence and (ii) weak interaction with surrounding tissue, are met by the polyglycerol dendron as hydrophilic unit. The PG dendron's hydrophilicity is a result of its multiple oxygen bonds, not of the presence of a charged salt. All bonds are fully saturated and much less susceptible to deprotonation upon pH variation. PG dendrons have previously reported to not interact with proteins as necessary to meet challenge (ii). [START_REF] Frey | Dendritic polyglycerol: a new versatile biocompatible material[END_REF][START_REF] Wyszogrodzka | A convergent approach to biocompatible polyglycerol "click" dendrons for the synthesis of modular core-shell architectures and their transport behavior[END_REF][START_REF] Wyszogrodzka | Study of single protein adsorption onto monoamino oligoglycerol derivatives: a structure-activity relationship[END_REF][START_REF] Wyszogrodzka | Synthesis and characterization of glycerol dendrons, self-assembled monolayers on gold: a detailed study of their protein resistance[END_REF] Challenge (iii), a strong preference towards the adsorbed over the free state, is tackled by the adsorption mechanism of PerPG as we verify from the absence of perylene luminescence. The emission of the perylene dye in the PerPG is quenched when it is in close contact with a nanotube. Residual dye emission is an excellent measure of free PerPG. The residual emission is quenched by a factor of 10 4 , corresponding to a ratio of adsorbed to free molecules of 10 000:1 or higher. [START_REF] Ernst | Energy Transfer in Nanotube-Perylene Complexes[END_REF] This remarkably high ratio of adsorbed to free PerPG is attributed to the strong π-π stacking exhibited between the sp 2 bonded carbon network of the nanotube and the organic perylene moiety. This agrees with the HRTEM and EELS studies that highlighted the stability of the complexes with respect to drying and re-dispersal in another medium.

We assess the biocompatibility of PerPG suspended nanotubes by a cell viability study.

To access cell viability we conduct an MTT assay after exposure to the nanotube-PerPG hybrids. [START_REF] Wilson | Cytotoxicity and viability[END_REF] It measures the activity of cellular enzymes in metabolically active cells. The enzymes reduce yellow tetrazolium dyes into its insoluble, purple formazan that can be solubilized and quantified by spectrophotometric means. The MTT assay is widely accepted as a sensitive and reliable tool in assessing cytotoxicity. [START_REF] Wilson | Cytotoxicity and viability[END_REF] Carbon nanotubes suspended in PerPG are added to COS-7 cell cultures with a final nanotube concentration of 0.1µg/ml, 0.1µg/ml being the highest concentration that may be used for applications in single nanotube imaging and tracking. PerPG is benchmarked against Pluronic as a standard surfactant to make nanotubes biocompatible [START_REF] Ghosh | Oxygen Doping Modifies Near-Infrared Band Gaps in Fluorescent Single-Walled Carbon Nanotubes[END_REF] The relative cell viability, Fig. 5A, demonstrated that PerPG is equally biocompatible as Pluronic. Figure 5B reports the cell viability of PerPG suspended nanotubes for different nanotube concentrations ranging from 1•10 - 3 µg/ml to 1 µg/ml. The biocompatibility of the suspended nanotubes increases drastically with lowered concentration reaching 90% viability at 10 -3 µg/ml. PerPG thus reached the same performance as the polymeric biocompatible surfactant Pluronic, demonstrating that it is well suited as a biocompatible surfactant for nanotubes.

Experimental

Sample preparation 0.11g/l of HiPco SWNTs (Unidym, batch SP0295) were dispersed with 6•10 -5 M of PerPG or Pluronic through tip sonication with a Bandelin SonoPlus HD 2070 with 30% of the maximum power of 70W and a 3mm microtip. The sample was subsequently centrifuged at 30 000g and 23° for 90 min. Only the supernatant was used in the experiments.

The synthesis of PerPG is published. [START_REF] Ernst | Energy Transfer in Nanotube-Perylene Complexes[END_REF] HRTEM and EELS The transmission electron microscopy (TEM) samples were prepared by re-dispersing the dried nanotube-PerPG complexes in ethanol. The dispersions were ultrasonicated and subsequently deposited on holey carbon 3 mm copper grids. High resolution TEM was performed employing an imaging-side aberration-corrected FEI Titan-Cube microscope working at 80 keV, equipped with a Cs corrector (CETCOR from CEOS GmbH).

Spatially resolved electron energy loss spectroscopy (EELS) measurements were performed on probe-corrected scanning TEM (STEM) FEI Titan Low-Base 60-300 operating at 80 keV (fitted with a X-FEG® gun and Cs-probe corrector (CESCOR from CEOS GmbH)). EEL spectra were recorded using the spectrum-imaging (SPIM in 2D or spectrum-line (SPLI) in 1D) mode [START_REF] Jeanguillaume | Spectrum-image: The next step in EELS digital acquisition and processing[END_REF][START_REF] Arenal | Extending the analysis of EELS spectrum-imaging data, from elemental to bond mapping in complex nanostructures[END_REF]

Figure 1 .

 1 Figure 1. PerPG is a small amphiphilic molecule which serves as a biocompatible surfactant for

Figure 2 .

 2 Figure 2. High resolution transmission electron micrographs of nanotube-PerPG hybrids. The

Figure 3 .

 3 Figure 3. A HAADF image of a bundle of functionalized carbon nanotubes. EELS data were

Figure 4 .

 4 Figure 4. The carbon nanotube-PerPG complexes created in aqueous solution remain stable

Figure 5 .

 5 Figure 5. Relative cell viability as measured in an MTT assay for different surfactants and NT

  in a Gatan GIF Tridiem ESR 865 spectrometer. The convergent semi-angle was 25 mrad and the collection semi-angle 80 mrad with an energy resolution ~ 1.2 eV. Cell viability is demonstrated by an MTT assay. At concentrations relevant for biotechnological applications the nanotube-PerPG complexes are equally biocompatible as nanotubes suspended in the biocompatible standard surfactant, pluronic F108. Until now, research into nanotube surfactants with a low cytotoxicity has concentrated on polymers. The modularity of the design of the small molecule surfactant is inherently flexible and could accommodate many other functionalities besides the energy transfer exhibited by PerPG in this manuscript. Tailored small functional molecules that efficiently act as surfactants have the potential to play a significant role in the solubilization of carbon nanotubes for biomedical applications.

	Cell viability study COS-7 cells are seeded in a 96 wells plate with 100µl of COS-7 solution
	(2.5•10 4 cells/ml) in each well and cultured at 37°C and 5% CO 2 for 24h. 11µl of SWNT
	suspension (1µg/ml) is added to each well, resulting in a final concentration of 0.1µg/ml.
	Cultures are kept at 37°C and 5% CO 2 for further 24h. Subsequently, 10µl MTT kit is added to
	each well, followed by 3.5h incubation at 37°C in the dark for reaction. All medium is carefully
	removed and 150µl MTT solvent is added into each well, the plate is then wrapped with
	aluminum paper to isolate from light and put on a laboratory shaker (60 rpm) for thoroughly
	dissolving MTT reduction. Absorptions were measured with a plate reader (PLUOstar Omega) at
	570nm, the fingerprint region for viable cells, and background is subtracted. The control culture
	was treated the same without adding SWNT suspension. All data were normalized by the control
	average.
	Conclusions
	In this paper, we show that a small functional molecule with a modular design, PerPG, is
	capable of suspending carbon nanotubes in biological surroundings. HRTEM imaging and EEL
	spectroscopy demonstrate that PerPG adsorbs to the nanotube with its perylene unit, confirming
	deductions made from earlier photoluminescence measurements. The HRTEM data also confirm
	the physical robustness of the nanotube-PerPG complexes to drying and solvent exchange and
	prove the complexes are stable in air, a result of the exceptionally strong binding interaction
	between nanotube and PerPG's perylene moiety. A polyglycerol dendron known for low levels of
	interaction with proteins was chosen for the hydrophilic part of PerPG to achieve
	biocompatibility of the nanotube complexes.

AUTHOR INFORMATION

Corresponding Author

* Correspondence should be addressed to f.ernst@fu-berlin.de

Author Contributions

The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript.

ACKNOWLEDGMENT

We gratefully acknowledge the financial support of the European Research Council (grant number 210642) and the German Research Foundation (DFG via SFB 658, subprojects A6 and B7). ZG and LC thank J. Varuela and L. Groc for the use of the absorption reader for MTT assay and acknowledge financial support from CNRS, Agence Nationale de la Recherche (ANR-14-OHRI-0001-01), IdEx Bordeaux (ANR-10-IDEX-03-02) and the France-BioImaging national infrastructure (ANR-10-INBS-04-01). The microscopy works were conducted in the Laboratorio de Microscopias Avanzadas at the Instituto de Nanociencia de Aragon -Universidad de Zaragoza (Spain). R.A. gratefully acknowledges the support from the Spanish Ministry of TOC Ultra-stable functionalization of carbon nanotubes based on a pi-pi stacking mechanism yields nanotube-surfactant complexes that are fully biocompatible as well as resistant to change of medium and drying.


Jorge A Laval

Ludovic Leclercq

Nicolas Chiabaut

Minimal Parameter Formulations of the Dynamic User Equilibrium using Macroscopic Urban Models: Freeway vs City Streets Revisited

Keywords:

This paper investigates the dynamic user equilibrium (DUE) on a single origin-destination pair with two alternative routes, a freeway with a f xed capacity and the surrounding city-streets network, modeled with a network macroscopic fundamental diagram (NMFD). We f nd using suitable transformations that only a single network parameter is required to characterize the DUE solution, the freeway to NMFD capacity ratio. We also show that the stability and convergence properties of this system are captured by the constant demand case, which corresponds to an autonomous dynamical system that admits analytical solutions. This solution is characterized by two critical accumulation values that determine if the steady state is in free-f ow or gridlock, depending on the initial accumulation. Additionally, we also propose a continuum approximation to account for the spatial evolution of congestion, by including variable trip length and variable NMFD coverage area in the model. It is found that gridlock cannot happen and that the steady-state solution is independent of surface network parameters. These parameters do a ect the rate of convergence to the steady-state solution, but convergence rates appear virtually identical when time is expressed in units of the NMFD free-f ow travel time.

Introduction

The simplest models that incorporate the e ects of congestion on urban areas are reservoir models. They are based on vehicle conservation inside the reservoir with its outf ow given by a known function of the accumulation. This function, called the Network Macroscopic Fundamental Diagram (NMFD), was f rst introduced in [START_REF] Godfrey | The mechanism of road network[END_REF] and later used by [START_REF] Mahmassani | Investigation of network-level traffic flow relationships: Some simulation results[END_REF][START_REF] Mahmassani | Performances of urban traffic networks[END_REF], but only recently was shown to have strong empirical support suggesting a rather stable shape Daganzo (2007); [START_REF] Geroliminis | Macroscopic modeling of traffic in cities[END_REF]; Wang et al. (2015). Many control applications have been proposed since (e.g., [START_REF] Haddad | On the stability of traffic perimeter control in two-region urban cities[END_REF][START_REF] Aboudolas | Perimeter and boundary flow control in multi-reservoir heterogeneous networks[END_REF][START_REF] Ampountolas | Robust control of bi-modal multi-region urban networks: An lmi optimisation approach[END_REF][START_REF] Hajiahmadi | Optimal dynamic route guidance: A model predictive approach using the macroscopic fundamental diagram[END_REF][START_REF] Knoop | Network transmission model: a dynamic traffic model at network level[END_REF]Yildirimoglu et al., 2015;[START_REF] Kouvelas | Enhancing feedback perimeter controllers for urban networks by use of online learning and data-driven adaptive optimization[END_REF].

Apart from the shape of the NMFD, the main assumption of a reservoir model is that the travel time of a vehicle entering at time t is a function of the accumulation at the same time, which makes it applicable only when the inf ow varies slowly. Otherwise it may be subject to the "inf nite-wave-speed problem" due to the lack of a space dimension in the model. Thus, when fast inf ow variations occur, they have immediate repercussions on the outf ow meaning that information travel is inf nitely fast inside the reservoir. Improved reservoir models have been proposed that guaranty that all vehicles travel their trip length before exiting [START_REF] Arnott | A bathtub model of downtown traffic congestion[END_REF][START_REF] Fosgerau | Congestion in the bathtub[END_REF][START_REF] Lamotte | The morning commute in urban areas: Insights from theory and simulation[END_REF] 2017. The Authors. This manuscript version is made available under the CC-BY-NC-ND 4.0 licence. http://creativecommons.org/licenses/by-nc-nd/4.0/ [START_REF] Mariotte | Macroscopic urban dynamics: Analytical and numerical comparisons of existing models[END_REF] at the expense of mathematical tractability. But when demand varies slowly, the simple reservoir gives good approximations, with the big advantage of being analytical. This has allowed extensions such as congestion pricing [START_REF] Arnott | A bathtub model of downtown traffic congestion[END_REF][START_REF] Daganzo | Distance-dependent congestion pricing for downtown zones[END_REF]. This paper focuses on the simple reservoir model in the context of dynamic traffic assignment (DTA), in particular the dynamic user equilibrium (DUE) when there is the alternative of reaching the destination using a freeway of limited capacity. Although there have been efforts in the past to combine DTA and MFD (Yildirimoglu and Geroliminis, 2014a; [START_REF] Leclercq | Estimating mfds in simple networks with route choice[END_REF], existing methods are algorithmic or numerical. Apart from the numerical errors introduced, this approach makes it difficult to answer fundamental questions such as: (i) what are the parameters, or combinations thereof, that affect the solution? (ii) under what conditions is the simple reservoir model a good approximation? (iii) will the system converge to gridlock and how fast?

To answer these and other questions, here we establish the minimum set of parameters needed to fully characterize the DUE solution. Towards this end, section 2 considers the simplest problem of a single NMFD (no DUE) to show that the accumulation evolution is characterized by two critical accumulation values that determine if the steady state is in free-flow or gridlock, depending on the initial accumulation. It also shows that the stability and convergence properties of this system are captured by the constant demand case, which admits analytical solutions. Section 3 shows that only a single network parameter is required to characterize the DUE solution, the freeway to NMFD capacity ratio. Section 4 proposes a continuum approximation to account for the spatial evolution of congestion, by including variable trip length and variable NMFD coverage area, typically assumed constant in the literature (e.g., [START_REF] Haddad | On the stability of traffic perimeter control in two-region urban cities[END_REF][START_REF] Aboudolas | Perimeter and boundary flow control in multi-reservoir heterogeneous networks[END_REF][START_REF] Hajiahmadi | Optimal dynamic route guidance: A model predictive approach using the macroscopic fundamental diagram[END_REF][START_REF] Leclercq | Macroscopic traffic dynamics with heterogeneous route patterns[END_REF]Yildirimoglu and Geroliminis, 2014b). Finally section 5 presents a discussion.

Single NMFD loading

We start by considering the simplest problem of the dynamic loading of a surface network using a single NMFD; there is no freeware alternative, and therefore no DUE. It turns out that the solution to this problem is the building block for the following sections.

We assume that the surface network can be well described by an outflow-NMFD, f (n), with capacity µ ≡ max n f (n). This function gives the production, i.e. the number of trip completions per unit time, as a function of the number of vehicles in the network, n. Let λ(t) be the demand inflow into the network at time t. 1 The NMFD dynamics studied in this paper are given by the following ordinary differential equation (ODE):

n-ODE: n ′ (t) = λ(t) -f (n), (reservoir dynamics) n(0) = n 0 , (initial conditions) (1a) (1b)
where primes denote differentiation, λ(t) the inflow and n 0 specify the initial conditions. A key point is whether or not demand is restricted by the supply function of the NMFD, Ω; i.e.,

λ(t) ≤ Ω(n(t)). (supply constraint) (2)
Although this appears to be a standard assumption in the latest literature, we argue that this constraint negates the fact that distance traveled within the reservoir may increase with congestion. Therefore, we are interested here in the unconstrained system solution. We will show how solutions to the constrained system can be derived from the solutions presented here.

It is convenient to express system dynamics in terms of the dimensionless variables occupancy, k(t), and demand intensity ρ(t):

k(t) ≡ n(t)/(κL), 0 ≤ k(t) ≤ 1, (occupancy) (3a) ρ(t) ≡ λ(t)/µ, (demand intensity) ( 3b
)
where κ is the jam density for one lane, assumed identical for all lanes in the network, and L corresponds to the length of the network, such that and κL is the maximum number of bumper-to-bumper vehicles inside the NMFD coverage area. We assume a general speed-occupancy NMFD, V k (k), of the form:

V k (k) = ug(k), (MFD speed-occupancy) ( 4
)
where u is the NMFD free-flow speed (which is ≤ FD free-flow speed as it includes the effects of signals) and g(k) is an arbitrary function of the occupancy satisfying g(0) = 1, g(1) = 0 and g ′ < 0. The speed-accumulation NMFD, V(n), and the f (n) can be derived as follows (Daganzo, 2007):

V(n) = V k (n/(κL)), (MFD speed-accumulation) (5a) f (n) = V(n) n/ℓ = g(n/(κL))nu/ℓ, (outflow-MFD) ( 5b
)
where ℓ is the trip length, assumed identical for all commuters.

The key observation here is that ℓ and u can be eliminated from the formulation by noting that the free-flow travel time inside the NMFD coverage area, namely τ * ≡ ℓ/u, corresponds to the timescale of our problem, and that therefore it can be eliminated by rescaling time. To see this, (i) let t ≡ t/τ * be the rescaled time, so that demand becomes ρ(t) = ρ(t/τ * ), and (ii) express the NMFD capacity as µ ≡ κLc/τ * for some positive constant c. This constant is the single supply parameter needed in this formulation and defines the "shape" of the NMFD, e.g. c = 1/4 for parabolic and c = 1/2 for isosceles f (k). Dividing (1) by κL and using (3a) gives that in original time the occupancy ODE of the system can be expressed as:

k-ODE: ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ k ′ (t) = F(t/τ * , k)/τ * , ( reservoir dynamics)
F(t/τ * , k) ≡ cρ(t/τ * ) -kg(k), k(0) = k 0 , ( initial conditions) (6a) (6b) (6c)
where F(t/τ * , k) is an auxiliary function. It follows from the chain rule that in rescaled time (6a) becomes k ′ (t) = F(t, k), which is independent of τ * . Hence, hereafter we will analyze the solution of k-ODE:

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ k ′ (t) = F(t, k), ( reservoir dynamics)
F(t, k) ≡ cρ(t) -kg(k), k(0) = k 0 , (initial conditions) (7a) (7b) (7c)
where we have dropped the hat for clarity (and from now on). Notice that this rescaling eliminates τ * regardless of the shape of both g(k) and ρ(t), which are the only parameters involved in our problem. As shown next, however, an additional transformation can be applied to time and space to justify a linear parameter-less function for g(x).

A Greenshield approximation

It has been shown [START_REF] Laval | Stochastic approximations for the macroscopic fundamental diagram of urban networks[END_REF][START_REF] Daganzo | Traffic flow on pedestrianized streets[END_REF][START_REF] Laval | Symmetries in the kinematic wave model and a parameter-free representation of traffic flow[END_REF] that in the context of the Kinematic Wave model [START_REF] Lighthill | On kinematic waves. I Flow movement in long rivers. II A theory of traffic flow on long crowded roads[END_REF][START_REF] Richards | Shockwaves on the highway[END_REF] with triangular FD for road segments, under a set of linear transformations of flow, occupancy, space and time, individual delays (and other measures of performance) are invariant. This means that individual vehicular delays remain the same regardless of the parameters in the transformation. As argued in [START_REF] Laval | Stochastic approximations for the macroscopic fundamental diagram of urban networks[END_REF] this can be used to streamline calculations by choosing the FD that simplifies the problem the most. In particular, they showed that for an isoscele triangle FD the resulting network NMFD becomes smooth and symmetric with respect to the y-axis. This gives the strong indication that a parabolic NMFD should a good approximation, which here means that:

g(k) ≡ 1 -k. (Greenshield approximation) (8)
The NMFD capacity becomes µ = f (κL/2) = uκL/ℓ/4 = κL/(4τ * ), which means that c = 1/4. Model (8) will be assumed at some point in each section below. 12), which depends on the single parameter ρ. Recall that t is measured in units of τ *

Analytical solutions

The ODE (7) (or (1)) cannot be solved analytically for general ρ(t) and g(k). However, when demand is constant, i.e. ρ(t) = ρ, then F(t, k) = F(k) = cρkg(k). In this case ( 7) is said to be an autonomous system since its evolution depends only on the occupancy and not on other time-dependent functions. Autonomous dynamical systems are well understood (see e.g. Teschl, 2010), and despite their simplicity, they capture most of the dynamics under more general demand patterns in our case.

In the case of traffic flow, the solution of these systems is fully characterized by two critical occupancies that are the roots of F(k * ) = 0: one in free-flow, k * 1 , and another in congestion, k * 2 ; see Fig. 3

(inset). Since F ′ (k * 1 ) < 0 and F ′ (k *
2 ) < 02 it follows that k * 1 is stable (attractor) and k * 2 unstable (repellor). This means that initial conditions "close to" k * 1 will tend to k * 1 as t → ∞, and that initial conditions close to k * 2 will diverge away either to k * 1 or to gridlock. Under the Greenshield approximation we have:

k * 1 = 1 -1 -ρ /2, (attractor) (9a) k * 2 = 1 + 1 -ρ /2, (repellor) (9b)
Note that ρ is the single parameter that describes the system. The solution of autonomous dynamical systems can be obtained by the standard techniques for separable ODEs. In our case, we write (7) as dk/F(k) = dt and integrate to obtain t = T (k) -T (k 0 ), where T (k) = dk/F(k). This gives:

k(t) = T -1 (T (n 0 ) + t) ,
(autonomous case solution) (10)

It can be shown that in our case:

T (k) = 1 c 1 tanh -1 1/2 -k c 1 , (11)
with c 1 = 1ρ/2, and thus the solution (10) can be expressed as:

k(t) = 1/2 -c 1 tanh (T (k 0 ) + c 1 t) . (autonomous, Greenshield solution) (12)
Fig. 1 shows the occupancy k(t) for 4 initial values k 0 = k(0) and 2 values for ρ. We can see that (i) when ρ ≤ 1 then

k → k * 1 if k 0 ≤ k * 2 and to 1 (gridlock) if k 0 > k * 2 ,
as expected, (ii) the convergence to gridlock happens at an increasing rate, (iii) the relaxation time of the system is comparable to 5τ * in most cases, (iv) when ρ > 1, i.e. when the demand exceeds the MFD capacity, the system converges to gridlock for all initial occupancies, as expected because the inflow is unrestricted.

Rush-hour demand patterns

We have been able to find analytical solutions for a family of demand curves well suited to model rush-hour periods, but only under the Greenshield approximation. These include second degree polynomial, exponential, and logistic functions. In particular, consider the following demand functions:

λ(t) ≡ q 0 + [1 -exp(-t/t * )](q 1 -q 0 ), (exponential) (13a) λ(t) ≡ q 0 + [1 + exp(-t/t * )] -1 (q 1 -q 0 ), (logistic) (13b)
where q 0 and q 1 are the initial and final demand flows, respectively, and t * is the time scale, which regulates how fast this transition takes place and which could be measured by fitting the appropriate functional form to empirical data. Since e -4 ≈ 0 we can say that the relaxation time is ≈ 4t * for exponential and ≈ 8t * for logistic demands.

The analytical solutions (produced with mathematical software) are too lengthy in general to include here; a few simple examples are included in the appendix. The important point is that (i) these solutions can be evaluated with arbitrary precision, even under a time-varying demand, and (ii) in the case of the rush-hour patterns considered here, at most two dimensionless parameters are needed to evaluate k(t) if time is measured in units of τ * : ρ ≡ q 1 /µ, (steady-state intensity) (14a)

β ≡ t * /τ * , (time scale ratio) (14b)
The parameter ρ is now the steady-state intensity, which should be ≤ 1 if (2) is assumed. The significance of the β parameter is that it can be used to discern when model ( 1) is a good approximation. To see this, note that time scale of NMFD dynamics (or relaxation time) should be comparable to the average trip time, which should be a small multiple of τ * , and we have already noted that the demand time scale is proportional to t * . It follows that β is proportional to the ratio of demand and supply time scales. The constant of proportionality depends on the type of demand function, but in general large values of β indicate that the demand varies slowly compared to the speed at which the MFD converges to equilibrium. For these more general demand patterns we found that the solution is qualitatively very similar to the autonomous case where λ(t) = q 1 . This is as expected because in the steady-state, demands (13) are constant (and equal to q 1 ). The main difference is in the transition to the steady-state solution, in particular the initial value above which the system tends to gridlock, which equals k * 2 in the autonomous case, but here it depends on β; see Fig. 2. It is important to remember that when the supply constraint (2) is active, the solution changes such that gridlock is impossible: if k 0 > k * 2 then k(t) = k 0 , else k(t) tends to k * 1 exponentially.

Freeway bottleneck vs NMFD

In this section we add a freeway as an alternative to the city street network (CS) modeled as in the previous section. It is generally agreed that freeways cannot be well approximated with an MFD because of the infinite-wave-speed problem and hysteresis loops (Saberi and [START_REF] Mahmassani | Urban network gridlock: Theory, characteristics, and dynamics[END_REF][START_REF] Mahmassani | Urban network gridlock: Theory, characteristics, and dynamics[END_REF][START_REF] Geroliminis | Properties of a well-defined macroscopic fundamental diagram for urban traffic[END_REF][START_REF] Leclercq | Macroscopic fundamental diagrams: A cross-comparison of estimation methods[END_REF]. In this section we present the results for the UE between a freeway, modeled as a bottleneck of capacity µ 0 , and the CS, modeled with an NMFD, f (n), of capacity µ 1 and with constant network length, L, per the previous section; see Fig. 3. In general, the subindex i=0 will be used for freeway and i = 1 for the CS, except for CS-only variables, where it will be omitted.

Destination l( ) t l 0 ( ) t l 1 ( ) t m 0 Freeway k f k * 1 u l 1,ss m 1 0 1 k * 2 ½ City streets MFD
We consider here the following system dynamics:

τ 0 (t) = τ * 0 + w 0 (t), (freeway travel time) (15a) τ 1 (t) = n(t)/ f (n(t)), (CS travel time) (15b) n ′ (t) = λ 1 (t) -f (n(t)), ( reservoir dynamics) (15c)
λ(t) = λ 0 (t) + λ 1 (t), ( demand conservation) (15d)
where λ i (t) denotes the demand flow to each route, τ * 1 ≡ ℓ/u is the free-flow CS travel time, and w 0 (t) is the freeway queuing delay, which is given by w 0 (t) = A 0 (t)/µ 0t ≥ 0; see [START_REF] Laval | Graphical solution and continuum approximation for the single destination dynamic user equilibrium problem[END_REF]. As in this reference, we set t = 0 when travel times are identical on both alternatives; i.e., τ 0 (0) = τ 1 (0). This enables us to use the UE condition in differential form for t ≥ 0, which equalizes the rate of change in travel times: τ ′ 0 (t) = τ ′ 1 (t), with τ ′ 0 (t) = λ 0 (t)/µ 0 -1 and τ ′ 1 (t) = (1n f ′ / f )n ′ / f . This gives:

λ 0 (t) = (1 + τ ′ 1 (t))µ 0 , (differential UE condition) (16a) = (1 + (1 -n f ′ / f )n ′ / f )µ 0 . (16b)
Combining (15d), ( 15c) and ( 16) to eliminate λ 1 gives the ODE describing system dynamics:

UE-ODE: ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ n ′ (t) = (λ -µ 0 -f ) f (1 -n f ′ / f )µ 0 + f n(0) = n 0 , (17a) (17b)
which is an explicit nonlinear first-order ODE of degree 1, and is valid for t ≥ 0. Unfortunately, (17) does not admit an explicit analytical solution without specifying λ(t) and f (t). However, one can still extract valuable insight. In particular, one can see that:

(i) the equilibrium travel time τ(t) increases with n ′ . To see this, notice that τ

′ (t) = (1 -n f ′ / f )n ′ / f and that 1 -n f ′ / f ≥ 0, (18)
for concave NMFDs.

(ii) a steady-state accumulation is reached when n ′ (t) = 0, i.e. when λµ 0f = 0. It follows that, qualitatively, the steady-state solution in this section is similar to the single NMFD loading problem, which is characterized by two critical accumulations (one attractor and one repellor) that are the roots of f (n * ) = λµ 0 . (iii) the flow diverted to the CS network as function of n can be obtained combining (15c) and (17a), which gives:

λ 1 (t, n) = λ -µ 0 n f ′ / f 1 + (1 -n f ′ / f )µ 0 / f (19)
This expression indicates that: (a) diversion always takes place as both the numerator and denominator in ( 19) are always positive. This is a consequence of ( 18) and of λ > µ 0 (recall that congestion in the CS can only happen if the freeway is already congested), (b) at the CS capacity, f ′ = 0 and thus λ 1 = λµ 1 /(µ 0 + µ 1 ) is a constant. This diversion pattern was called diversion type-2 in [START_REF] Laval | Graphical solution and continuum approximation for the single destination dynamic user equilibrium problem[END_REF] who modeled the CS as a bottleneck: the total demand is split proportionally to each alternative's capacity, (d) the limit of λ 1 as n → Lκ is always zero, and the limit of λ 1 as n → 0 is always positive, but depends on the NMFD, e.g., a linear free-flow branch gives λµ 0 .

Analytical solutions

It turns out that the problem in this section can also be expressed as k

′ (t) = F(t, k)/τ * 1 .
The derivation is similar to that of (6); divide (17) by κL, use (3a) and ( 5). This means that the parameter τ * 1 can be eliminated by rescaling time, and that one may solve the simplified problem:

k-UE-ODE: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ k ′ (t) = F(t, k),
(reservoir dynamics)

F(t, k) ≡ cρ(t) -kg 1 -cmg ′ /g 2 , k(0) = k 0 , (initial conditions) (20a) (20b) ( 20c
)
where t is measured in units of τ * 1 , and the dimensionless parameters m and ρ are given by: m = µ 0 /µ 1 , (capacity ratio) (21a) Notice how as m → 0 (no freeway) then ODE (20) → ODE (7), and therefore the solution to the single MFD loading problem in section 2 is a special case of the solution in this section. Unfortunately, unlike section 2, we were not able to find analytical solutions for ODE (20) under any time-dependent demand ρ(t). The autonomous case ρ(t) = ρ, however, can still be solved analytically, and it turns out that the solution has a lot in common with the single NMFD case: we have the same two critical occupancies, k * 1 and k * 2 , with the same stability properties as given by ( 9). To see this, it is clear from (20b) that the solution of F(k * ) = 0 is still given by ( 9); for the stability properties, one can verify that F ′ (k * ) is proportional to -(k * 2 g ′ (k * )cρ), which can be written as k * (k * g(k * )) ′ . This expression is proportional to the wave speed (k * g(k * )) ′ and the conclusion follows.

ρ(t) ≡ (λ(t) -µ 0 )/µ 1 = λ(t)/µ 1 -m, (MFD demand intensity) (21b) Destination l( ) t l 0 ( ) t l 1 ( ) t m 0 x( ) t City streets MFD coverage area 0 x k f k * 1 u l 1,ss m 1 0 1 k * 2 ½ d Freeway
Therefore, the solution of (20) in the autonomous case ρ(t) = ρ is also given by ( 10), which depends upon the shape of the MFD. Under the Greenshield approximation we obtain F(k) = (ρ -4(1k)k)/(4 + m/(1k) 2 ), and T (k) = dk/F(k) is given by :

T (k) = 1 + (2 -ρ)m/ρ 2 T 1 (k) + m ρ 2 ρ 1 -k + 2 log ρ -4(1 -k)k (1 -k) 2 (22)
where T 1 (k) is the corresponding T -function for the single NMFD loading problem, given by (11). Again, as m → 0 (no freeway) then T → T 1 and therefore the solution to the single MFD loading problem is a special case of the solution in this section. Similarly, as m → ∞ (no CS) then T → ∞, which means that the CS will never be used. Fig. 4 shows the occupancy k(t) given by ( 10)-( 22) and the resulting CS inflow λ 1 (t)/µ 1 for 4 initial values k 0 = k(0), and for different ρ, m values. As mentioned earlier, we can see that the evolution of the system is very similar to single NMFD case in Fig. 1, and therefore the same conclusions apply here. The main differences are that (i) the convergence to gridlock happens at a decreasing (rather than increasing) rate, and (ii) the relaxation time of the system might be larger. Notice the demand patterns produced in the CS are not time-independent (as one might have expected from an autonomous case). In fact they appear to be exponential for free-flow initial conditions, and logistic or bell-shaped for congested initial conditions.

A continuum approximation for off-ramps

In this section we combine the formulation proposed in the previous sections with the continuum approximation (CA) proposed in [START_REF] Laval | Graphical solution and continuum approximation for the single destination dynamic user equilibrium problem[END_REF]. In this approximation, the set of discrete off-ramps is treated as a continuum, where vehicles can exit the freeway at any given location x ≥ 0 upstream of a freeway bottleneck located at x = 0. They show that when the CS speed is constant the UE solution is characterized by an "information wave", ξ(t), that marks the most upstream location where vehicles divert; i.e., vehicles divert in 0 ≤ x ≤ ξ(t). Assuming a constant lateral exit capacity, φ, in units of vehicles /(time×distance), the total lateral diverting flow at time t, λ 1 (t), can be expressed as:

λ 1 (t) = φξ(t). (23)
Here, the CS speed is given by the NMFD over a coverage area that increases (quadratically) with ξ(t).

To illustrate, let δ be the average distance between two parallel and consecutive streets ad in Fig. 5. The number of blocks inside a coverage area A is then A/δ 2 , each contributing 2δ to the network length, assuming that only southbound and eastbound links will be used. Taking an area given by a quarter-disk of radius ξ(t), gives that the network length can be expressed as:

L(ξ) = πξ 2 2δ , (24)
We assume that the trip length is proportional to the network length, i.e. that:

γ ≡ L/ℓ (25)
is a constant. With all, system dynamics are identical to the previous section, except that now there are two unknown functions, n(t) and ξ(t); i.e.:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ λ(t) = (τ ′ 1 (t) + 1)µ 0 + φξ(t) , n ′ (t) = φξ(t) -f (n(t), ξ(t)) (n(0), ξ(0)) = (n 0 , ξ 0 ), (26a) (26b) (26c)
where n 0 < κL(ξ 0 ) has to be imposed so that the initial accumulation does not exceed jam accumulation, and f (n, ξ) is given by ( 8) with L = L(ξ) and ℓ = ℓ(ξ), as prescribed in this section.

It can be verified that in steady-state we have the following equilibrium values:

λ 1,ss = λ ss -µ 0 , ξ ss = λ 1,ss φ , ℓ ss = πξ 2 ss 4γδ , L ss = ℓ ss γ µ 1,ss = 1 4 γκu 1 , τ * 1,ss = ℓ ss u 1 . ( 27
)
where the subscript "ss" means the steady-state of the variable, e.g. ξ ss ≡ ξ(t → ∞). The two critical occupancy values k * 1 and k * 2 are still given by ( 9) with ρ = λ 1,ss /µ 1,ss . Notice that the steady-state location of the information wave, ξ ss , is independent of the supply parameters on the CS, and it is only a function of the steady-state demand and the exit capacity. This means, for example, that the free-flow speed, or the number of lanes on the CS facilities does not affect the solution for large times. These parameters do affect the rate of convergence to the steady-state solution, as shown next.

Unfortunately, we were not able to find analytical solutions even for the Greenshield approximation under constant demand, and therefore we resort to numerical methods. Fig. 6 shows the evolution of various quantities of interest for problem in the case of a constant demand λ(t) ≡ q for 4 different free-flow CS speed u 1 , and for empty CS initial conditions ξ 0 = 0 + , n 0 = 0 (a very small value 0 + is needed since the NMFD has to cover a nonzero area).

The first column in the figure uses regular units, while the second uses the dimensionless forms from the last section to express the of convergence to steady-state values measuring time in units of τ * 1 . The main result of this figure is the low scatter observed for the solution n(t), ξ(t) in the dimensionless form. This means that the solution to our problem can be cast independently of the CS free-flow speed u 1 . Also, we have verified that k ss → k * 1 if k 0 ≤ 1/2, as opposed to k 0 ≤ k * 2 in the previous subsection. The congested initial conditions case k 0 > 1/2 is shown in Fig. 7, which indicates that the CS occupancy tends to gridlock shortly after t = 0 but then eases back to the equilibrium occupancy k * 2 . We have verified that the effect of the initial condition for ξ 0 (=1 km in the figure) is only transient, and the solution quickly converges to the solution when ξ 0 = 0 + .

Discussion

We analyzed two formulations of the dynamic user equilibrium on a freeway with a fixed capacity and the surrounding city-streets network, modeled with an NMFD. The traditional approach of constant network and trip lengths for the NMFD was amenable for analytical solution in the constant total demand case, which could be used as building block for more general demand patterns. When network and trip lengths are allowed to vary to reflect that off-ramp capacities are finite, analytical solutions become impossible, but numerical solutions reveal that the system shares many similarities the constant length model. The main difference being that in the varying length model gridlock does not happen, and that the steady-state solution is independent of surface network parameters. In all cases we find that convergence rates are very similar when time is expressed in units of the NMFD free-flow travel time.

Our results can be extended in the number of directions. It is expected that formulations for the trip length other than (25) might lead to different dynamics. Also, queue spillbacks are not considered here. As argued in [START_REF] Laval | Graphical solution and continuum approximation for the single destination dynamic user equilibrium problem[END_REF], these should not be a problem for the freeway queue because the information tends to travel faster than the shock wave, and since chances are that the freeway bottleneck capacity is higher than the off-ramp capacity, off-ramps located inside the freeway queue would not be starved, i.e. able to discharge at capacity. Spillbacks from off-ramps queues could be a problem if they starve the freeway bottleneck, however. These and other extensions are being studied by the authors.

Fig. 1 .

 1 Fig. 1. Single NMFD loading, constant demand case. Illustration of the analytical solution (12), which depends on the single parameter ρ. Recall that t is measured in units of τ *

Fig. 2 .

 2 Fig.2. Single NMFD loading, exponential demand case, given by (13a) with q 0 = 0, ρ = q 1 /µ = 0.9, for 4 initial values k 0 = {0, .3, .8, .94}, and for 2 different β values. Notice that for k 0 = 0.94 the steady-state solution is different for different values of β.

Fig. 3 .

 3 Fig. 3. Freeway vs CS network: constant network length model.

Fig. 4 .

 4 Fig. 4. Freeway vs MFD, constant network length model. Occupancy k(t) and the resulting CS inflow λ 1 (t) for 4 initial values k 0 = k(0), and for different ρ, m values.

Fig. 5 .

 5 Fig. 5. Freeway vs CS network: variable network length model.

Fig. 6 .

 6 Fig.6. Freeway vs NMFD, variable network length model. Evolution of various quantities in the case of a constant demand λ(t) ≡ q for 4 different free-flow CS speeds u 1 , and for empty CS initial conditions. First column: regular units; second column: dimensionless. We took: number of freeway lanes = 3, Q 0 = 2500 veh/hr/ln, κ = 150 veh/km/ln, γ = 4, δ = .1 km, φ = 1500 veh/hr/km.

Fig. 7 .

 7 Fig. 7. Freeway vs MFD, variable network length model. Similar to Fig. 6 but with congested initial conditions.

The word "demand" is used in the traffic flow context to distinguish the willingness to traverse a bottleneck (demand) from the ability to do so (flow); it is not used in the economics context to reflect cost elasticity. Here, the demand is an inelastic and exogenous time-dependent function.

Since F ′ (k * ) = (k * g(k *)) ′ corresponds to the wave speed of the k-MFD.

Acknowledgments.

This study has received funding from NSF research project # 1562536, and from the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation program (grant agreement No 646592 MAGnUM project). Saberi, M., Mahmassani, H., 2013. Empirical characterization and interpretation of hysteresis and capacity drop phenomena in freeway networks. Transportation Research Record 2391, 44-55. Teschl, G., 2010. Differential equations and dynamical systems. Graduate Studies in Mathematics. American Mathematical Society, Providence, Rhode Island, autre tirage : 2006. Wang, P., Wada, K., Akamatsu, T., Hara, Y., 2015. An empirical analysis of macroscopic fundamental diagrams for sendai road networks. Interdisciplinary Information Sciences 21 (1), 49-61. Yildirimoglu, M., Geroliminis, N., 2014a. Approximating dynamic equilibrium conditions with macroscopic fundamental diagrams. Transportation Research Part B: Methodological 70, 186 -200. Yildirimoglu, M., Geroliminis, N., 2014b. Approximating dynamic equilibrium conditions with macroscopic fundamental diagrams. In: Transportation Research Board 93rd Annual Meeting. 14-0709. Washington DC. Yildirimoglu, M., Ramezani, M., Geroliminis, N., 2015. Equilibrium analysis and route guidance in large-scale networks with {MFD} dynamics. Transportation Research Part C: Emerging Technologies 59, 404-420. URL http://www.sciencedirect.com/science/article/pii/S0968090X15001813

Appendix A. Expression k(t) in section 2 for a few special cases.

Exponential demand function with β = 1/2, q 1 = 0 and arbitrary k 0 , q 0 :

where t is measured in units of 2t * , ρ = q 0 /µ and a = (1e -t ) √ ρβ.

Exponential demand function with β = 3/2, q 1 = 0 and arbitrary k 0 , q 0 :

2 3 √ ρ cos(a) (4k 0 (e t -1) + 3ρ) -2 sin(a) (9k 0 ρ + e t (4k 0 -3ρ))

t is measured in units of 2t * and ρ, a same as above.

Exponential demand function with β = 1/(2 (1ρ)), q 0 = 0 and arbitrary k 0 , q 1 :

where t is measured in units of 2t * , ρ = q 1 /µ and a = ρ 1e -t /(2 (1ρ)ρ).

Logistic demand function with q 0 = 0 and arbitrary k 0 , q 1 :

where t is measured in units of τ * , ρ = q 1 /µ and:

], and W[•] is the Hypergeometric 2F1 function.


Franz Baader
email: franz.baader@tu-dresden.de

Pierre Ludmann
email: pierre.ludmann@ens-cachan.fr

The Unification Type of ACUI w.r.t. the Unrestricted Instantiation Preorder is not Finitary

The unification type of an equational theory is defined using a preorder on substitutions, called the instantiation preorder, whose scope is either restricted to the variables occurring in the unification problem, or unrestricted such that all variables are considered. It is known that the unification type of an equational theory may vary, depending on which instantiation preorder is used. More precisely, it was shown that the theory ACUI of an associative, commutative, and idempotent binary function symbol with a unit is unitary w.r.t. the restricted instantiation preorder, but not unitary w.r.t. the unrestricted one. Here, we improve on this result, by showing that, w.r.t. the unrestricted instantiation preorder, ACUI is not even finitary.

Introduction

The first preorder introduced to deal with unification [START_REF] Alan | A machine-oriented logic based on the resolution principle[END_REF] was in fact the unrestricted instantiation preorder. It was designed for syntactic unification and it worked pretty well considering that it gave a unitary unification type. Yet, when it comes to equational unification, researchers usually employ the restricted instantiation preorder, though the reason for this change was not explained in early papers.

We use the following notation to distinguish between these two preorders:

σ ≤ X E τ iff ∃λ∀x ∈ X. λ(σ(x)) = E τ (x) and σ ≤ ∞ E τ iff ∃λ∀x ∈ V. λ(σ(x)) = E τ (x) Here, V is the countably infinite set of all variables, whereas X is a (usually finite) subset of it. From now on, we will use ≤ E to denote the restricted preorder ≤ Var (Γ) E when the unification problem Γ is clear from the context.

Note that ≤ ∞ E ⊆ ≤ E , which has as an easy consequence that the unification type in the restricted case can never be worse than the type in the unrestricted case. In particular, syntactic unification is also unitary w.r.t. ≤ E . Hence, for the empty theory it makes no difference which instantiation preorder is used. For this reason, the distinction between these two orders was not always rigorously made clear, though it was known that the unrestricted preorder could lead to unpleasant results in weak unification [START_REF] Eder | Properties of substitutions and unifications[END_REF].

It took until 1991 before the first example of an equational theory for which the unrestricted and restricted unification types are different was published. That particular theory is ACUI, the theory of idempotent abelian monoids:

ACUI := {x + 0 = x, x + (y + z) = (x + y) + z, x + y = y + x, x + x = x}
From [START_REF] Baader | Unification in commutative idempotent monoids[END_REF] it was known that ACUI is unitary in the restricted case; in [START_REF] Baader | Unification, weak unification, upper bound, lower bound, and generalization problems[END_REF], it was shown that its unrestricted type is not unitary. This paper thus showed that the choice of the instantiation preorder makes a difference for equational unification. However, it did not show how big that difference actually is: the unrestricted type of ACUI might be finitary, which is still a quite pleasant type from the application point of view. Here we show that this is not the case: ACUI is at least infinitary. However, the precise unrestricted unification type of ACUI is still an open problem: it is either infinitary or of type zero.

Auxiliary Results

In the following, Var (s) denotes the set of variables occurring in the term s, Dom(σ) = {x ∈ V | σ(x) = x} the domain of the substitution σ, and VRan(σ) = x∈Dom(σ) Var (σ(x)) the variable range of σ.

The proof of our main result (Theorem 2) is based on the fact that ACUI is a theory in which no variable is going to pop -in or out -of a term, i.e. the set of variables of a term is stable under equality modulo the theory. As a matter of fact, this property is called regularity in unification theory.

Definition 1. An identity s = t is regular if Var (s) = Var (t). A set of identities E is regular if all elements of E are regular.

Regularity of the defining set of identities of an equational theory implies regularity of the whole theory.

Lemma 1 ([Yel85]). E is regular iff = E is regular.
Obviously, all the identities of ACUI are regular, which by the above lemma yields that s = ACUI t implies Var (s) = Var (t). Thus, the following lemma applies to ACUI.

Lemma 2. Let E be a regular theory and

Γ = s = t an E-unification problem s.t. Var (s) ∩ Var (t) = ∅. Then the set U of all unifiers σ of Γ satisfying ∀y ∈ VRan(σ).∃x, x ∈ V s.t. x = x and y ∈ Var (σ(x)) ∩ Var (σ(x )) is complete w.r.t. ≤ ∞ E . Proof.
Let σ be a unifier of Γ that does not belong to U, i.e., there is y 0 ∈ VRan(σ) s.t. there exists a unique x 0 verifying y 0 ∈ Var (σ(x 0 )) (note that x 0 = y 0 is possible). Let τ = {x 0 → y 0 , y 0 → x 0 } be the substitution that exchanges x 0 with y 0 (and is the identity if x 0 = y 0 ). Then define the new substitution σ as σ (x) := τ (σ(x)) for x ∈ V \ {x 0 } and σ (x 0 ) := x 0 . Note that VRan(σ )

VRan(σ): in fact, if x 0 ∈ VRan(σ), then x 0 ∈ VRan(σ ); otherwise, y 0 ∈ VRan(σ ). Using the fact that, in any case, x 0 ∈ VRan(σ ), we can show that σ is an (unrestricted) instance of σ :

∀x ∈ V \ {x 0 } : (τ • {x 0 → τ • σ(x 0 )} • σ )(x) = (τ • {x 0 → τ • σ(x 0 )})(σ (x)) = τ (σ (x)) = τ • τ • σ(x) = σ(x)
Additionally, we have for the variable x 0 :

(τ • {x 0 → τ • σ(x 0 )} • σ )(x 0 ) = (τ • {x 0 → τ • σ(x 0 )})(x 0 ) = τ (τ • σ(x 0 )) = σ(x 0 )
This shows σ ≤ ∞ E σ. However, we need to show that σ is a unifier of Γ in order to make the comparison useful. Because E is a regular theory, σ(s) = E σ(t) implies Var (σ(s)) = Var (σ(t)). Then, if x 0 occurs in one of the terms s, t, it also has to occur in the other one since it is the unique producer of y 0 . This contradicts our assumption on Γ. Since, on the variables different from x 0 , σ is an instance of σ, we thus know that σ is a unifier of Γ.

Consequently, we can replace σ with σ and so shrink VRan(σ). Since VRan(σ) is finite, repeating this process will end with a unifier σ * that satisfies the required condition and is more general w.r.t. ≤ ∞ E than the original unifier σ.

A second auxiliary result that will be employed in the proof of our main theorem follows easily from the order-theoretic point of view on unification types [START_REF] Baader | Characterizations of unification type zero[END_REF][START_REF] Baader | Unification theory[END_REF]. Given any preorder on the set U of E-unifiers of Γ, we denote with ∼ its induced equivalence relation, i.e., σ ∼ τ iff σ τ and τ σ. We denote the ∼-equivalence class of a unifier σ as [σ] and the set of all equivalence classes of unifiers as [U ]. The partial order induced by on equivalence classes is defined as usual, i.e., [σ]

≤ [τ ] iff σ τ . We say that M ⊆ [U ] is complete w.r.t. ≤ if every element of [U ] is above (w.r.t. ≤) some element of M . Theorem 1 ([BS01]). Let M be the set of ≤-minimal elements of [U ]. If C is a minimal complete set of E-unifiers of Γ w.r.t. , then M = {[σ] | σ ∈ C}. Conversely, if M is complete in U , then any set of representatives of M is a minimal complete set of E-unifiers of Γ.
The following easy consequence of this result holds w.r.t. any preorder on unifiers, and thus in particular both for the restricted and the unrestricted instantiation preorder.

Lemma 3. Let C be a complete set of E-unifiers of an E-unification problem Γ. Then Γ has a minimal complete set of E-unifiers iff C contains a minimal complete set of E-unifiers of Γ.

The Unrestricted Type of ACUI is at Least Infinitary

We will more generally show the result for regular theories satisfying certain properties, and then show that ACUI satisfies these properties. From now on we assume that • E is a regular theory,

• Γ = s = t is an E-unification problem s.t. Var (s) ∩ Var (t) = ∅,
• there is a ≤ ∞ E -minimal unifier σ of Γ that uses fresh variables, i.e., VRan(σ) \ X = ∅ where X = Var (s) ∪ Var (t), and

• this unifier σ belongs to the set U defined in the formulation of Lemma 2.

We will prove that in such a configuration, Γ, and so E, is at least infinitary w.r.t. unrestricted instantiation.

Let x 0 ∈ VRan(σ) \ X and consider the following construction of substitutions:

σ z := σ • (x 0 z) where (x 0 z) := {x 0 → z, z → x 0 } and z ∈ V.
We will show that, under certain conditions on z, such substitutions σ z are ≤ ∞ E -minimal unifiers that are incomparable to each other w.r.t. ≤ ∞ E . By Theorem 1, this implies that Γ cannot have a finite minimal complete set of unifiers w.r.t. ≤ ∞ E since there are infinitely many variables z satisfying these conditions.

Lemma 4. For any z ∈ X, σ z is a minimal unifier of Γ w.r.t. ≤ ∞ E . Proof. Note that σ z is a unifier of Γ because x 0 , z ∈ X. Moreover, let θ be a unifier of Γ s.t. θ ≤ ∞ E σ z , i.e.
, there is a substitution λ s.t.

∀x ∈ V. σ z (x) = E λ • θ(x)
Consequently, if we "multiply" from the right with (x 0 z), we obtain

∀x ∈ V. σ(x) = E λ • (θ • (x 0 z))(x). Because (θ•(x 0 z)) is a unifier of Γ with (θ•(x 0 z)) ≤ ∞ E σ, minimality of σ yields σ ≤ ∞ E (θ•(x 0 z)), i.e., there exists µ s.t. ∀x ∈ V. (θ • (x 0 z))(x) = E µ • σ(x).
Consequently, "multiplying" again from the right with (x 0 z) yields

∀x ∈ V. θ(x) = E µ • σ z (x).
Thus, we have shown that θ ≤ ∞ E σ z for a unifier θ of Γ implies σ z ≤ ∞ E θ, which proves that σ z is a minimal unifier of Γ w.r.t. ≤ ∞ E .

Lemma 5. For any two different variables z, z ∈ Dom(σ) ∪ VRan(σ), σ z and σ z are incomparable w.r.t. ≤ ∞ E .

Proof. Assume there exists λ s.t. ∀x ∈ V. σ z (x) = E λ • σ z (x), and let t 0 := σ(x 0 ).

Case 1:

Var (t 0 ) = ∅ Then σ z (z) = E λ • σ z (z) implies z = E λ(t 0
). This definitely contradicts the existence of λ since it cannot create a variable from a ground term.

Case 2: Var (t 0 ) = ∅ Then there exists y 0 ∈ Var (t 0 ). Note that, independent of whether y 0 = x 0 or y 0 = x 0 , we have y 0 ∈ VRan(σ). Since σ ∈ U by our assumptions on σ, there is

x 1 ∈ V different from x 0 s.t. σ(x 1 ) = E t 1 and y 0 ∈ Var (t 1 ). Again, σ z (z) = E λ • σ z (z) implies z = E λ(t 0 ). Because E is regular, this implies Var (λ(y 0 )) ⊆ {z}. Since z, z ∈ Dom(σ) ∪ VRan(σ), x 1 = z and x 1 = z ; so σ z (x 1 ) = E λ • σ z (x 1 ) implies t 1 = E λ(t 1 ) Again, regularity yields Var (λ(y 0 )) ⊆ Var (t 1 ).
Then Var (λ(y 0 )) = ∅ as z ∈ VRan(σ) and Var (t 1 ) ⊆ VRan(σ). In fact, if x 1 ∈ Dom(σ), then Var (t 1 ) is contained in VRan(σ) by the definition of VRan. Otherwise, we must have x 1 = σ(x 1 ) = E t 1 . Since y 0 ∈ Var (t 1 ), regularity of E yields y 0 = x 1 and Var (t 1 ) = {y 0 }. Thus, y 0 ∈ VRan(σ) yields Var (t 1 ) ⊆ VRan(σ).

However, since z = E λ(t 0 ), the variable z is produced by λ from at least one variable y that occurs in t 0 , i.e., there is a variable y ∈ Var (t 0 ) \ {y 0 } such that Var (λ(y)) = {z}.1 Since y ∈ VRan(σ) and σ ∈ U, there exists x = x 0 s.t. y ∈ Var (σ(x)). Yet again, as z, z ∈ Dom(σ) ∪ VRan(σ), we know x = z and x = z , and thus σ z (x) = E λ • σ z (x) implies σ(x) = E λ • σ(x). Since we know that z ∈ Var (λ • σ(x)), regularity yields z ∈ Var (σ(x)). However, this is absurd since x = z and z ∈ VRan(σ).

To sum up, we have shown that σ z ≤ ∞ E σ z does not hold. A symmetric argument yields that σ z ≤ ∞ E σ z also does not hold.

Since the complement of the finite set X ∪ Dom(σ) ∪ VRan(σ) in the countably infinite set V of all variables is infinite, the set of unifiers of Γ must have infinitely many minimal elements. By Theorem 1, this shows that Γ cannot have a finite minimal complete set of unifiers. Lemma 6. The unification problem Γ does not have a finite minimal complete set of E-unifiers w.r.t unrestricted instantiation, and thus E is at least infinitary w.r.t. ≤ ∞ E .

We are now ready to apply this result to ACUI.

Theorem 2. ACUI is at least infinitary w.r.t. ≤ ∞ ACUI .

Proof. Since ACUI is regular, it is sufficient to show that there is an ACUI-unification problem Γ and a minimal unifier σ of Γ satisfying the conditions stated at the beginning of this section.

According to Corollary 3.6 in [START_REF] Baader | Unification in commutative idempotent monoids[END_REF], any most general unifier (w.r.t. restricted instantiation) of the ACUI-unification problem Γ = x + y + z = u + v must use a fresh variable. Let θ be such an mgu.

If Γ does not have a minimal complete set of ACUI-unifiers w.r.t. unrestricted instantiation, then we are done. Thus, assume that Γ has a minimal complete set M w.r.t. unrestricted instantiation. By Lemma 2 and Lemma 3, we can assume without loss of generality that M ⊆ U, and by Theorem 1 we know that the elements of M are ≤ ∞ ACUI -minimal. Since θ is an ACUI-unifier of Γ, there is a σ ∈ M such that σ ≤ ∞ ACUI θ. Since ≤ ∞ ACUI ⊆ ≤ X ACUI , this implies that σ is also an mgu of Γ w.r.t. restricted instantiation, and thus it introduces a fresh variable.

Consequently, we have shown that all prerequisites for applying Lemma 6 are satisfied, which proves the theorem.

Conclusion

In this paper we have shown that the gap between the unification types of equational theories w.r.t restricted and unrestricted instantiation is wider than previously known. In fact, for ACUI, which is unitary w.r.t. restricted instantiation, it was only known that the unrestricted type is at least finitary [START_REF] Baader | Unification, weak unification, upper bound, lower bound, and generalization problems[END_REF]. Now we know that it is at least infinitary, which makes using unrestricted instantiation in this setting even less desirable.

Regarding future work, it would of course be good if we could determine the exact unification type of ACUI w.r.t. unrestricted instantiation (infinitary or type zero), but we have not been able to achieve this yet.

Note that, if there is no such additional variable in t 0 , then this already contradicts the existence of λ, making the point.


Franz Baader
email: franz.baader@tu-dresden.de

Pierre Ludmann
email: pierre.ludmann@ens-cachan.fr

The Exact Unification Type of Commutative Theories

The exact unification type of an equational theory is based on a new preorder on substitutions, called the exactness preorder, which is tailored towards transferring decidability results for unification to disunification. We show that two important results regarding the unification type of commutative theories hold not only for the usual instantiation preorder, but also for the exactness preorder: w.r.t. elementary unification, commutative theories are of type unary or nullary, and the theory ACUIh of Abelian idempotent monoids with a homomorphism is nullary.

Introduction

It is well-known that deciding solvability of a disunification problem modulo an equational theory E [START_REF] Wray | On solving equations and disequations[END_REF][START_REF] Comon | Disunification: A survey[END_REF] can be reduced to E-unification in case the theory E is effectively finitary, i.e., for every E-unification problem a finite complete set of E-unifiers always exists and can be effectively computed. Given a disunification problem Γ, one first computes a finite complete set of E-unifiers for the equations in Γ, and then checks whether this set contains a substitution that also satisfies all the disequations of Γ.

In order to extend this approach to non-finitary equational theories, Cabrer and Metcalfe [START_REF] Cabrer | From admissibility to a new hierarchy of unification types[END_REF] introduced a new preorder on substitutions that contains, but is usually larger than, the instantiation preorder:

σ X E τ iff σ(s) = E σ(t) ⇒ τ (s) = E τ (t) for all terms s, t built using only variables from X.

Using this preorder in place of the instantiation preorder (where X is the set of variables occurring in the unification problem), one can now define (minimal) complete sets of unifiers, and thus obtain a new classification of equational theories w.r.t. their type, which we call exact unification type. The definition of the exactness preorder is tailored towards ensuring that the above reduction from disunification to unification still works. The advantage of using the exactness preorder rather than the instantiation preorder is that the former is larger, and thus it may be the case that a theory that is non-finitary w.r.t. the instantiation preorder is finitary w.r.t. the exactness preorder. For instance the theories of Idempotent Semigroups and of Distributive Lattices are nullary w.r.t. the instantiation preorder, yet their exact types are respectively finitary and unitary (see [START_REF] Cabrer | From admissibility to a new hierarchy of unification types[END_REF] for these and more examples). In the following, we will denote the instantiation preorder as ≤ E and the exactness preorder as E , where the set of variables X on which the substitutions are compared is always assumed to be the set of variables occurring in the unification problem under consideration.

Commutative theories were introduced in [Baa89] in order to generalize approaches and results for unification in the theories axiomatizing Abelian Monoids (ACU), idempotent Abelian Monoids (ACUI), and Abelian groups (AG). One important result in [START_REF] Baader | Unification in commutative theories[END_REF] is the fact that, w.r.t. elementary unification, commutative theories are of unification type either unitary or nullary. Recall that a unification problem is elementary if it does not contain free constant or function symbols. The three theories mentioned above are all unitary. To provide an example for a non-unitary (and thus nullary) equational theory, it is shown in [START_REF] Baader | Unification in commutative theories[END_REF] that the theory ACUIh of idempotent Abelian monoids with a homomorphism is nullary:

ACUIh := {x + 0 = x, x + (y + z) = (x + y) + z, x + y = y + x, x + x = x, h(x + y) = h(x) + h(y), h(0) = 0}
The mentioned results use the instantiation preorder to compare substitutions. We will show below that these results also hold if the exactness preorder is used in place of the instantiation preorder.

The Dichotomy Result for Exact Types

The class of commutative theories is defined in [START_REF] Baader | Unification in commutative theories[END_REF] using the category of finitely generated free algebras.1 To be more precise, let E be an equational theory and V a denumerable set (of variables). Then the category C(E) is defined as follows:

• The objects of C(E) are the algebras F E (X) for finite subsets X of V , where F E (X) is the free algebra generated by X in the variety defined by E.

• The morphisms of C(E) are the homomorphisms between these objects, where the composition of morphisms is the usual composition of mappings.

An equational theory E is commutative if the associated category C(E) is semiadditive. The exact definition of semiadditive categories can be found in [START_REF] Herrlich | Category Theory[END_REF][START_REF] Baader | Unification in commutative theories[END_REF]. For the purpose of this paper it is sufficient to know that in such a category the coproduct is also a product. In [START_REF] Baader | Unification in commutative theories[END_REF] there is also a more algebraic characterization of commutative theories, which shows that the theories ACU, ACUI, AG, and ACUIh are commutative.

The following theorem generalizes Theorem 6.3 in [START_REF] Baader | Unification in commutative theories[END_REF] from the instantiation preorder to all preorders extending the instantiation preorder. It thus applies to the exactness preorder, but also to the preorder on substitutions introduced in [START_REF] Hoche | Essential unifiers[END_REF].

Theorem 1. For elementary unification, commutative theories are either unitary or nullary w.r.t. any preorder containing the instantiation preorder.

Proof. A theory is either at most finitary or at least infinitary. In the first case, use the first lemma below to show that it is actually unitary. In the second case, the condition of the second lemma is satisfied, which shows that the theory is actually nullary.

For the remainder of this section, we assume that is a preorder that contains the instantiation preorder. To align our proofs of the two lemmata with the corresponding ones given in [START_REF] Baader | Unification in commutative theories[END_REF] for the instantiation preorder, we use the categorical formulation of unification introduced there. To obtain this formulation, a unification problem

s i = t i | 1 ≤ i ≤ n is translated into a pair of morphisms σ = τ , where Dom(σ) = Dom(τ ) = I = {x 1 , . . . , x n } and σ(x i ) = s i ∧ τ (x i ) = t i for all i, 1 ≤ i ≤ n.
In this setting, a unifier is a morphism γ such that σγ = τ γ.

Lemma 1. Let E be a commutative theory and Γ = σ = τ be an elementary E-unification problem, and let {γ 1 , . . . , γ n } be a finite complete set of E-unifiers of Γ w.r.t. . Then there exists a E-unifier γ of Γ s.t. the singleton set {γ} is a complete set of E-unifiers of Γ w.r.t. . Proof. Let σ, τ : F E (I) → F E (X) and γ i :

F E (X) → F E (Y i ). Let Y = Y 1 • • • Y n is the disjoint union of the generating sets Y i .
Then F E (Y ) is the coproduct and the product of the objects F E (Y i ). Let p 1 , . . . , p n be the projections for the product. Then there exists a unique morphism γ : F E (X) → F E (Y ) such that γp i = γ i for all i, 1 ≤ i ≤ n. Consequently, γ ≤ E γ i and thus also γ γ i . The morphism γ is an E-unifier of Γ since σγp i = σγ i = τ γ i = τ γp i for i = 1, . . . , n implies σγ = τ γ (by the definition of product).

It remains to show that {γ} is complete w.r.t. . Thus, let δ be an E-unifier of Γ. Since {γ 1 , . . . , γ n } is complete w.r.t. , there is an index i such that γ i δ. But now transitivity of the preorder yields γ δ, which shows that {γ} is indeed complete.

Lemma 2. Let E be a commutative theory and Γ = σ = τ be an elementary E-unification problem, and let {γ 1 , γ 2 , γ 3 , . . . } be an infinite set of E-unifiers of Γ such that there is no Eunifier α of Γ with α γ i for all i ≥ 1. Then there does not exist a minimal complete set of E-unifiers of Γ.

Proof. Let σ, τ : F E (I) → F E (X) and γ n : F E (X) → F E (Y n ), and let U E (Γ) denote the set of E-unifiers of Γ.
The morphisms δ n are inductively defined as follows: δ 1 is just γ 1 . Assuming that δ n :

F E (X) → F E (Z n ) is already defined, we consider the product F E (Z n+1 ) with the projections p 1 , p 2 of F E (Y n+1 ) and F E (Z n ). Then δ n+1 : F E (X) → F E (Z n+1
) is defined to be the unique morphism such that δ n+1 p 1 = γ n+1 and δ n+1 p 2 = δ n .

As in the proof of the previous lemma, it is easy to see that the morphisms δ n are E-unifiers of Γ, and that δ n+1 ≤ E δ n and thus δ n+1 δ n . The condition imposed on {γ 1 , γ 2 , γ 3 , . . . } implies that the decreasing chain δ 1 δ 2 δ 3 . . . has no lower bound in U E (Γ). As a matter of fact, if such a bound α existed, then we would have for all n ≥ 1 that α δ n γ n , which contradicts our assumption regarding the set {γ 1 , γ 2 , γ 3 , . . . }.

We now assume that there exists a minimal complete set M of E-unifiers of Γ w.r.t. ≺. Since M is complete, there is θ ∈ M such that θ δ 1 . The fact that δ 1 δ 2 δ 3 . . . has no lower bound in U E (Γ) yields an n ≥ 1 satisfying θ δ n , but θ δ n+1 . Let θ : F E (X) → F E (Y ), and let F E (Z) with the projections q 1 , q 2 be the product of F E (Z n+1 ) and F E (Y ). The morphism θ : F E (X) → F E (Z) is defined to be the unique morphism such that θq 1 = δ n+1 and θq 2 = θ.

Again, it is easy to see that θ is an E-unifier of Γ, θ δ n+1 , and θ θ. Since M is complete there is θ ∈ M such that θ θ. Now θ θ for θ, θ ∈ M yields θ = θ by minimality of M . But then θ = θ θ δ n+1 is a contradiction to our choice of n.

A Commutative Theory whose Exact Type is Nullary

The following unification problem was used in [START_REF] Baader | Unification in commutative theories[END_REF] to show that the theory ACUIh is nullary (w.r.t. the instantiation preorder):

Γ = h(x 1 ) + h(x 2 ) = x 2 + h 2 (x 3 )
In [START_REF] Baader | Combination problems for commutative/monoidal theories or how algebra can help in equational unification[END_REF] the same problem was used to extend this result to a large subclass of the class of commutative theories. The proofs use the fact that for all n ≥ 0 the substitution

θ n := {x 1 → y, x 2 → h(y) + h 2 (y) + • • • + h n+1 (y), x 3 → h n (y)}
is a unifier of Γ. Below, we use the same problem and the same sequence of unifiers to show that an appropriate variant of Lemma 8.1 in [START_REF] Baader | Unification in commutative theories[END_REF] also holds if the instantiation preorder is replaced by the exactness preorder. However, the proof is a bit more involved.

Lemma 3. Let γ be an ACUIh-unifier of Γ such that γ ACUIh θ n . Then there is a variable z and a number k ≥ n such that x 3 γ contains h k (z) as a summand.

Proof. For simplicity, we denote ACUIh as E in this proof. First, note that x 1 γ = E 0γ = 0 since γ E θ n and x 1 θ n = y = E 0 = 0θ n . Consequently, x 1 γ is of the form x 1 γ = p i=1 h ni (y i ) where p ≥ 1 and n i ≥ 0 and y i is a variable (possibly, but not necessarily, distinct from the variables x 1 , x 2 , x 3 ) for all 1 ≤ i ≤ p. Now, assume that h m (α) is a summand of (h(x 1 ) + h(x 2 ))γ, where α is a variable. Since (h(x 1 ) + h(x 2 ))γ is finite, there is a nonnegative integer k s.t.

• h m+k (α) is a summand of (h(x 1 ) + h(x 2 ))γ, but

• h m+k+1 (α) is not a summand of (h(x 1 ) + h(x 2 ))γ.
Consequently, h m+k (α) must be a summand of h 2 (x 3 )γ, which implies that m + k ≥ 2 and h m+k-2 (α) is a summand of x 3 γ.

Because the terms h ni+1 (y i ) are all summands of (h(x 1 ) + h(x 2 ))γ, we obtain nonnegative integers k 1 , . . . , k p s.t. h ni+ki-1 (y i ) is a summand of x 3 γ for all i, 1 ≤ i ≤ n. Without loss of generality we can assume that

k 1 ≥ k 2 ≥ • • • ≥ k p . Now, consider t = h k1-1 (x 1 ) + x 3 + p i=2 h k1-ki (x 3 ) and u = x 3 + p i=2 h k1-ki (x 3 ). A simple calculation shows that (h k1-1 (x 1 ))γ = h n1+k1-1 (y 1 ) + p i=2 h ni+k1-1 (y i ) = h n1+k1-1 (y 1 ) + p i=2 h k1-ki (h ni+ki-1 (y i )).
Given what we know about the summands of x 3 γ, this implies tγ = E uγ, which in turn yields tθ n = E uθ n , i.e.,

h k1-1 (y) + h n (y) + p i=2 h n+k1-ki (y) = E h n (y) + p i=2 h n+k1-ki (y). Thus, k 1 -1 = n or there is an i, 1 ≤ i ≤ n, with k 1 -1 = n + k 1 -k i .
In the first case, h n1+n (y 1 ) is a summand of x 3 γ, and in the second, h n1+n (y i ) is a summand of x 3 γ.

Obviously, this lemma implies that the E-unification problem Γ satisfies the prerequisite of Lemma 2 for = E . Thus, Γ does not have a minimal complete set of E-unifiers for E = ACUIh.

Theorem 2. For elementary unification, ACUIh is nullary w.r.t. the exactness preorder.

Conclusion

The theory ACUIh actually axiomatizes equivalence in the Description Logic FL 0 [START_REF] Baader | Unification of concept terms in description logics[END_REF], and thus ACUIh-unification corresponds to unification in FL 0 . As shown in [START_REF] Baader | Unification of concept terms in description logics[END_REF], deciding solvability of such unification problems is ExpTime-complete (for unification with constants). Since ACUIh is of exact type nullary, disunification in FL 0 (modulo ACUIh) cannot be reduced to unification using the method outlined in the introduction. Nevertheless, solvability of disunification problems (with constants) was shown to be decidable (more precisely, ExpTime-complete) in [START_REF] Baader | Solving language equations and disequations with applications to disunification in description logics and monadic set constraints[END_REF] using an extension of the automata-based decision procedure for FL 0 -unification.

Another Description Logic for which unification has been investigated in detail is the Description Logic EL. Equivalence in this logic can be axiomatized by the equational theory SLMO of semilattices with monotone operator, which is not a commutative theory. As shown in [START_REF] Baader | Unification in the description logic EL[END_REF], deciding solvability of EL-unification problems is NP-complete and the unification type (w.r.t. the instantiation preorder) is again nullary. Whether EL (and thus SLMO) also has exact unification type nullary is still an open problem.

See[START_REF] Nutt | Unification in monoidal theories[END_REF] for an algebraic definition of the closely related class of monoidal theories.