Susanne Fuchs
email: fuchs@zas.gwz-berlin.de

Bernd Pompino-Marschall
email: bernd.pompino-marschall@rz.hu-berlin.de

Pascal Perrier
email: perrier@icp.inpg.fr

IS THERE A BIOLOGICAL GROUNDING OF PHONOLOGY? DETERMINING FACTORS, OPTIMIZATION, AND COMMUNICATIVE USAGE

Keywords: physics, speech, biology of speech

This special session discusses the issue if there is a biological grounding of phonology. By reviewing current and past work from different speech research disciplines we suggest that (1) biological factors provide the limits, the frame of reference for phonology, (2) phonology is shaped by optimization processes taking into account the nonlinear relations of different representations of speech (acoustics, articulation, speech perception), and (3) sociolinguistic factors and communicative usage affect, for instance, speech acquisition and sound change. The first of the three suggestions is biological in nature whereas the last represents the non-biological nature of speech.

INTRODUCTION

"[In trying to understand] how human beings communicate by means of language, it is impossible for us to discount physical considerations, the facts of physics and physiology." (Halle 1954, cited in Ohala 1978, [START_REF] Ohala | Production of Tone[END_REF] p.5). It is hard to deny the biological foundation of speech since the speech signals we are able to realize and perceive stem from bio-physical systems, (1) the vocal apparatus and (2) the eyes and ears both controlled by the neural system. However, if all normal and fully developed human beings have more or less the same basic bio-physical system, why do we find such a variety of languages (at least 3000-6000 extant) with a tremendous amount of speech sounds and their combinations? How does the biological foundation of speech shape phonology?

Classical and recent approaches in phonology [START_REF] Trubetzkoy | Grundzüge der Phonologie[END_REF][START_REF] Chomsky | The Sound Patterns of English[END_REF][START_REF] Clements | The geometry of phonological features[END_REF] have mainly worked on the definition of terminologies and methods to allow description and characterization of languages, including typological classification among them, and prediction of their potential diachronic evolution. However, this work consists in abstract formalizations of experimental observations of the languages' properties and variations. It does not provide a way to understand the origins of the different speech units, why some combinations of sounds do not exist, and why the possibilities of diachronic evolution are limited.

During the ICPhS conference 1983 in Utrecht a comparable issue was addressed in a session organized by Peter Ladefoged. Ladefoged [START_REF] Ladefoged | Out of the chaos comes order; Physical, biological, and structural patterns in phonetics[END_REF] suggested that "sound patterns are the result of languages being a self-organizing social institution" (p.91). "Evolutionists teach us that such things are properties of a culture, and not of an individual's physiology" (p.94). Contrary to this view, Lindblom [START_REF] Lindblom | Economy of speech gestures[END_REF][START_REF] Lindblom | Can the models of evolutionary biology be applied to phonetic problems?[END_REF] in the same session proposed that fundamental speech units and processes can be derived deductively from independent premises anchored in physiological and physical realities.

Since then, experimental techniques for studying speech production and perception have tremendously improved (among others Electromagnetic Articulography, Magnetic Resonance Imaging, Ultrasound, and Brain Imaging techniques). The increase of computer capabilities and the development of powerful software packages permitted the elaboration of complex realistic physiological models in order to study the underlying control mechanisms in the production and perception of sounds. Recently, a number of approaches have taken into account the characteristics of the speech production and perception apparatus and provided interesting suggestions that not only describe, but also explain some important aspects of the morphogenesis of language units [START_REF] Macneilage | The frame/content theory of evolution of speech production[END_REF][START_REF] Schwartz | Phonology in a theory of perception-for-actioncontrol[END_REF].

In phonology, on the other hand, e.g. constraint based models were developed during the last decade or so that ask for a natural grounding of universal principles [START_REF] Clements | The geometry of phonological features[END_REF][START_REF] Prince | Optimality Theory: Constraint Interaction in Generative Grammar[END_REF]. The lesson we should have learned from OT and related (phonological) models is that "universal", i.e. natural, constraints may shape phonological structure. Moreover, the interface model of Articulatory Phonology [START_REF] Browman | Articulatory gestures as phonological units[END_REF] lets us see the question of biological grounding of phonology in a new light. Consequently, we have decided to update this discussion in the light of recent findings and propose the special session entitled "Is there a biological grounding of phonology?". Our thoughts will be organized using three keywords: determining factors, optimization, and communicative usage.

DETERMINING FACTORS

We suggest that the biological foundation of speech provides the determining factors for the development of phonology. Biology determines the limits, the frame of reference for phonology and specifies the course that languages can take in their development. Hereafter we will provide some examples of biological origin constraining the speech production process.

On the basis of tongue shapes in the midsagittal plane, Harshman et al. [START_REF] Harshman | Factor analysis of tongue shapes[END_REF] have shown for English speaking subjects by a statistical procedure called PARAFAC, that two factors can explain most of the variance in the data. The 'front raising' factor is associated with movements along a highfront to low-back axis, and the 'back raising' factor is associated with movements along a low-front to high-back axis. Similar results were also reported in the literature for other languages (Icelandic, German, French) leading to the assumption that it could be a universal principle. But what would be the reason for its emergence?

Using a 2 D biomechanical model of the tongue for the analysis of a large amount of simulations, Fuchs & Perrier [START_REF] Fuchs | On the complex nature of speech kinematics[END_REF] proposed that these two factors would be a consequence of muscle anatomy and the internal structure of the tongue. They would not be speech specific. From this perspective, the front raising and back raising factors would be totally independent of any linguistic specification. Contrariwise, they seem to correspond to a biological constraint that influences the production of speech sounds, their variability, and their potential evolution.

In recent work Winkler et al. [START_REF] Winkler | The relation between differences in vocal tract geometry and articulatory control strategies in the production of French vowels: Evidence from MRI and modelling[END_REF] we considered different vocal tract shapes as a biologically determining factor for individual articulatory control strategies aiming towards acoustic goal regions. Human adult vocal tracts differ particularly with respect to the length of the pharynx between males and females Fitch & Giedd [START_REF] Fitch | Morphology and development of the human vocal tract: A study using magnetic resonance imaging[END_REF]. Figure 1 displays the outer surface of different vocal tracts from 4 speaker's mid-sagittal MRI data. The plots on top exhibit larger horizontal proportions (schematically depicted as a dashed line) than vertical proportions (schematically depicted as a bold line) and the lower pair shows the reverse. As a starting point we suppose that speakers should have more space to move vertically in a tract with a large vertical space whereas they are more constrained (in terms of articulatory precision) in the horizontal direction. The opposite should be true for speakers with a larger horizontal than vertical vocal tract geometry. Consequently, to reach the same acoustic goal regions, speakers with different vocal tract shapes also need to posture their tongue differently, which in turn requires different muscle activation. Preliminary results of our experimental and modeling work provide evidence that in the production of French /i/ speakers with a large vertical vocal tract proportion show more upward than forward tongue movement. In our biomechanical simulations this movement was attributed to a higher activation of the Styloglossus. Similarly, Brunner et al. [START_REF] Brunner | On the relation between the palate shape and articulatory behaviour[END_REF] discussed the impact of the individual palate shape in the coronal plane as a determining factor for articulatory precision. Speakers with a more flat palate shape need to place their tongue more precisely than speakers with dome-shaped palates, since small articulatory changes in tongue height position have a larger impact on the area function and hence on the acoustics. By means of articulatory and acoustic data from 32 speakers we found that speakers with a flat palate shape realized a reduced token-totoken variability. Speakers with dome shaped palates were less constrained. Some had a large and some had a small degree of variability. Additionally, the actual effect of differences in the coronal palate shape was tested by means of a model. The influence of the palate shape on articulatory precision is generally strong for high vowels. However, this determining factor may be particularly relevant for language with a dense vowel inventory, but of less importance for the ones with a 3-vowel system.

OPTIMIZATION

Optimization processes are at work, taking into account the multimodal representations of speech (Perrier 2005). There are a number of concepts in the literature focusing on the optimization of the nonlinear relations of these representations.

Optimization between acoustic, perception and articulation

According to Stevens' [START_REF] Stevens | The quantal nature of speech: Evidence from articulatory-acoustic data[END_REF][START_REF] Stevens | On the quantal nature of speech[END_REF] 'Quantal nature of speech', spoken languages prefer those sounds or regions where articulatory changes have only little impact on the acoustic consequences. Following this perspective, articulatory regions of acoustic stability shape phonology.

Functional Phonology [START_REF] Boersma | Functional Phonology: Formalizing the interaction between articulatory and perceptual drives[END_REF] and Hyper-& Hypospeech [START_REF] Lindblom | Economy of speech gestures[END_REF] take into account the oppositional principles of articulatory economy and perceptual comprehension. According to them, the speech production systems in common with all motor systems, is organized to minimize effort, resulting in articulatory reduction, weakening or even deletion of movements. Minimization of effort, however, goes against the minimization of perceptual confusion within the communicative process. Functional Phonology as well as H&H theory focuses on the optimization between articulatory effort and perceptual recognition.

Another study discussing the relation between articulation and perception is reported in

Liljencrant & Lindblom [START_REF] Liljencrants | Numerical simulation of vowel quality systems: the role of perceptual contrast[END_REF]. They hypothesized that a maximal perceptual distinctiveness principle underlies the distribution of the vowels. On this basis the authors could predict why /i/, /u/, and /a/ are present in almost all vowel systems.

The Dispersion-Focalization Theory (DFT) proposed by Schwartz et al. [START_REF] Schwartz | The Dispersion-Focalization Theory of vowel systems[END_REF] also relies on an optimality criterion. It involves a maximal perceptual distinctiveness principle, but combines it with the concept of 'good perceptual objects', inspired from the 'Gestalt' theory. Schwartz et al. consider that 'focalized' frequency characteristics, i.e. spectra depicting a close proximity and even a merging of different maxima of energy, are perceptually more salient and more stable. Indeed, the DFT predicts very well the distribution of the vowels in the world languages as described in the UPSID data base [START_REF] Maddieson | Updating UPSID[END_REF].

Optimization at the motor control level

Comparing the production of alveolar stops with fricatives, Fuchs et al. [START_REF] Fuchs | What role does the palate play in speech motor control? Insights from tongue kinematics for German alveolar obstruents[END_REF] hypothesized differences in the control of these sounds: alveolar stops have a target above the contact location resulting in a collision of the tongue tip at the palate as opposed to a precise positioning of the tongue at the lateral margins of the palate for alveolar fricatives. The first assumption was based on Löfqvist and Gracco [START_REF] Löfqvist | Lip and jaw kinematics in bilabial stop consonant production[END_REF] who found for bilabial stops that articulatory gestures could be directed toward a target that is beyond the actual contact location. Thus, in the production of coronal stops the palate is used as a reference, which automatically blocks the tongue's movement at the required location in the vocal tract and guarantees the acoustic silent closure. In terms of stability and simplicity, such a control strategy seems to be extremely efficient in comparison to the control of a fine positioning. By means of kinematic data Fuchs et al. found strong support for the different control strategies. They concluded that the control strategy used in stop production would be more simple and stable than the one in fricative production. This difference in the complexity of control could be part of the explanation for the fact that most languages have more plosives than fricatives (Vallée et al.[26]).

COMMUNICATIVE USAGE

The question of the extent to which biological factors influence speech during the communicative usage is a very challenging one, since it raises many methodological problems, especially with disentangling linguistic, sociolinguistic, communicative, biological and frequency factors -to name just a few.

As we all know for a long time, in speech communication we not only transport meaning, but also signal our attitudes, emotions, values etc. [START_REF] Trubetzkoy | Grundzüge der Phonologie[END_REF]. And of course, spoken language in general would not exist without social communication. This point has been taken up recently by Exemplar Theory [START_REF] Pierrehumbert | The next toolkit[END_REF] modeling for instance speech acquisition as the development of a self-assembling system from variable environmental input.

Sociolinguistic variation in itself is one of the most important non-biological factors structuring spoken language as pointed out by Scobbie in this session.

OUTLOOK

So far the question of the biological grounding of phonology was mainly expressed as the question of how the internal structure of different phoneme inventories can be explained (cf. the above discussion of Dispersion Theory and DFT). And indeed, evidence for quite different perceptual factors determining these segmental inventories have been found. Some of the newer observations (cf. the paper of Flemming to this session) point in the direction that we should look also more closely at the perceptual cues for individual segments in the acoustics of connected speech -and not only from a purely concatenatively oriented point of view (e.g. sonority sequencing). Sonority sequencing itself is surely grounded in our productive and perceptual abilities. Aren't all speech signals prosodic at their core? The contributions of Slifka and Goldstein et al. to this session already point in this direction. In this same vein we might extend the H&H Theory to a model of continuous modulation within individual utterances. At the hierarchically structured prosodic domains we might look for specific hypothesised hyper-to-hypo contours giving rise to boundary markers. For example, downdrift and final lengthening might be considered biologically natural phenomena. Can we trace back phonological lengthening rules in phrasing (e.g. Chichewa penultimate lengthening) to such mechanisms?

Another question worth pursuing from the biological, i.e. anatomical, perspective seems to be the question of inter-speaker variability. Recently growing data on anatomical differences between individual vocal tracts in connection with the interspeaker variability in articulatory movements observed for long times provoke some new questions. Do different vocal tracts require different neural motor control structures to yield the same categorical output? Or are their articuloacoustic results taken care of by their listener as exemplars around a prototype? Until now, the discussion has centred around artificial prototypes (e.g. an acoustically defined 7-step height pattern of vowels in DFT). What would the picture be if we took into account productive constraints and anatomically induced exemplar variability? What can we learn from 'compensatory' articulation aiming at the same auditory target (cf. the paper by Honda et al in this session)?

Figure 1 :

 1 Figure 1: Different outer vocal tract shapes from 4 speakers (labeled on MRI data)

ACKNOWLEDGEMENTS

This work was supported by grants from the German Research Council (DFG) GWZ 4/8-1 P1, PO 334/4-1 and by the P2R program of the French Foreign Office. Thanks to the members of the French-German network POPAART for reviewing several papers of this special session and Jim Scobbie for proof reading and helpful discussions.


J S Perkell

D H Klatt

J. VAJ1SSIERE in INVARIANCE AND VARIABILITY lN SPEECH PROCESSES

Sucll ."\ sep."\ratioll has 1I0t proverl to tIc very prodllctivc in the PI\St, anrl there is lIothing t() suggcst thut it wollid 11(, any more so today. lt is probably 1\ solllld prillciplc ill !'cicncc to nvoid making shl\rp dichotomies. ln his pnpcr, Abbs makes a clistillctioll betwcen the abstract goals of the spe."\ker, bis/lIer phollological intentiolls, and their articwl\tory realizatiolls. lt is obviously attractive to view spcech-motor control as u process where a COilllllullicativc act successively unfolds itself into greater and greater articulatory detail. On the other band, such a view requires a very complex set of operatiolls transforming the abstract intentions into artiçulatory and acoustic results. 1 am particularly concerned that sooller or later we will have to worry about these transformations in order to provide a more explicit account.

Jacqueline Vaissière: Comment Abbs has noted that syllables were observed to be more variable in their movement patterns when isolated than when they were placed in a carrier phrase. l would like to comment on the lack of invariance in isolated CVCs.

The observed variance may be explained, first, by assuming the superimposition of gestures unrelated to the linguistic and phonetic aspects of speech. Figure 10b.1 represents the time function of the vertical position of two pellets, one located on a speaker's velum and the other on the lower lip, for the pronunciation of a series of isolated syllables. The data have beell obtained with the x-ray microbeam system at the University of Tokyo (Fujimura, Kiritani, & Ishida, 1973) and analyzed using the facilities developed at Bell Laboratories.

The tracings illustrate two points. Firsi, the velùm is rising prior to alld at Borne points during the realization of the consonant lm]. This gesture is antagonistic io the nrticulntory requirement of the nasal fenture. Articulatory patterns in speech cannot th us be explàined in terme of segmentaI fe."\tures onJy, but they have to be integmted into a global view of speech activity. Second, the onset of elevation of the velum (designated by the let ter T on the figure) is not fixed relative to the segmentaI events, but highly contextdependent as exemplified on ihese three tracings. The height or the velum pellet at ihe bcginning of voicing for the lm] (dèsignnted by the letter H on the figure) is depcndent on such onset timillgj the e.vlier the onsei, the higher the velum peak. The fnct thnt the iniiinl consonmlt is a nnsnl deternlines an upper limit for such an elevation, to keep the velophnryngenl port open. Thc tcIIsing or muscles illvolvcd ill speech productioll prior to speech i!l n common observntioll. This s()-cnllcd "spc(~h-rc11dy gcsture," is highly vnrinblc in nnture. Its cfTects cnn dis.,ppcûr bcrorc the first phoncmc ill ml uttcrnncc, or Clln bc maillulincù du ring the cntire bcgillllilig or the IIttcrnllcc (or thc first CVCs in û list). Silililarly, a "spc(~h-rcll\Xation gcsture" (a lowcring or the velum in this cuBe) bcrore the cnd or the uttcrmlce or artcr iLs complction lIIay bc obscrvcd on tllc 11\.'\t CVCs just bcrorc a piluse, nlld its onset is IIOt rcaùily prcdictnble. llcsidcs the spccchrc.'ldy gcsturc .'lnd the sp(,cch-rclnxntion gcsturc, dRUl orteIl illustr.'ltc n tClldclICY ror llrticlllatory pûttt'rlls or illtCrlllCdiiltC CVCs to rcnect n dL'Crcilsillg rUndnlllclltnl rrcqucllcy (rclntcd to the so-cnllcd dcclillntioll Iillc-in scntcIICt'S), a lower 11lld lowcr vclnm positioll, nnd ùccreilsillg IlIagllitudc of tllC jilW excursions. Tht~sC c/Tccts reprcscnt a gcncral rclûxntion of thc ilTticulators wit.h tilllc. Such a tendcllcy CûD be countcractcd in a repct.ition or the s.'lmc list. The spcech-rcady gesture, the relaxntion gesture, nnd declination c/Tcct colllplicate t.he interpretat.ion of the data, silice such phenolllena are not prcdictable from the linguistic colltent of speech, and thcy obscure the relationship betwecn the segmentaI features and articulatory and EMG patterns.

The speaker's "semantic" interpretation of the speech mnterinl that is read can also cause observed articulatory variability. It is weil known that in conversation al speech, speakers tend to focus on the elements of speech (a feature, a syllable, a word) that they feel nre important. Such a tendcncy is valid Dot only for meaningfui speech but also for nonsense speech, in isolation or in carrier sentences. ln other words, in a series like [ted, ted, tenJ, speakers tend to overnrticulate the final consonants to contrast them optimally, while in a series like [tcd, ned, medJ, they will tend to place more emphasis on the syllable-initial consonant. As a consequence, the way of articulating the same syllable [tedJ may be slightly different depending on the context and its position in the list. ln summary, the lack of invariance in movements patterns form one repetition of an isolated CVC to another may be partly explained by the intcrference of nonlinguistic factors, and by context-dcpendent articulatory "weights" assigned by the speaker to different parts of the syllable.

  lindillgH cau!l(~d SOIllC colIstcrlll\tioll, sillcC it was gcncrally bclievccl thnt the Hillile solllld wowd IIc articlllatcd in tlle saille way irrespcctivc of tlle conl.cxtail I\SSlIlllptioll tllat scenls to rcappCl\r I\t certain intcrvals. ln Imrticnll\r, a serious discIIssion starterl ."\bollt tllc rcu\tionship bctwccn phonctics 1\lId pllonology. AIthollgh the callsallinks ill tlll\t rliscIIssionare IIOt cntircly cl~"\r, many people argned for a rl\dical separation betwcell pllonetics l\nrll~lIonology. Phonology w."\8 tliIIS sccn as a mcntl\list sllbjcct, wllcrcl\S pholletics W/L.. a physic."\l scicncc. Part of tItis rlcbate occllrrcd rlllriJlg tllC Sccollrlll1terlll\tiollal COIlgrcHs of Phonctic Sciences (.Jolies and Fry, 19:JCi); otllcr cOlltrilllltions I\rc reprintccj ill Zwirllcr allrl J';ZI\WI\ (19Ci8).

  FIG. IOb.l. Time Iunction of the vcrtical position of the vclum pcllct, thc lowcr lip pcllet, ;lnd cnvclopc of thc spcc(,h signm for thr<:c"'yllables wilh thc initial conson:lnt lm] and vowcJ IE:I. T indicatcs thc onsct of the clcvalion of lilc VC!lIIll, and Il t.hc 11(,ig!lt of thc VCIUIII pcllct at the voicing ()II""t of thc Inl!.


Yunfan Lai
email: khroskyabs@gmail.com

Causativisation in Wobzi and other Khroskyabs dialects

Keywords:

Mainly basing on first hand data, this paper deals with the causative constructions in the Khroskyabs language (Rgyalrongic, Sino-Tibetan), focusing on the Wobzi dialect. Causative prefixes, anti-causative, analytic causative and labile verbs are described. In addition, comparative analyses within Rgyalrongic as well as the Sino-Tibetan family are made.

Background

Khroskyabs 1 is a Rgyalrongic language 2 spoken in the Rngaba Prefecture in Sichuan Province, China (prc). Previous studies on this language include Huang (2007, Guanyinqiao dialect), Lai (2013a, 2015b, Wobzi dialect), etc. The majority of Khyroskyabs speakers live in an area that extends from Thugschen (endonym grəmde, 觀音橋鎮 Guānyīnqiáo Zhèn), Northwest of Chuchen County, towards 'Brongrdzong (mbrôŋrdzoŋ 木爾宗鄉 Mù'ěrzōng Xiāng), Southern part of 'Barkhams County (mbaerkʰaem 馬爾康縣 Mǎ'ěrkāng Xiàn), as well as Phosul (pʰosé 蒲西鄉 Púxī Xiāng) in Eastern Dzamthang County (ndzaemtʰʊŋ, 壤塘縣 Rǎngtáng Xiàn). The total Khroskyabs speaking population is estimated to be less than 10,000 people [START_REF] Huang | Lāwùróngyǔ yánjiū (拉塢戎語研究) [Study on the Lavrung language[END_REF]. The black circle in Figure 1 shows the approximate location of the Khroskyabs speaking area in Sichuan.

In this paper, I will present new data from one of the Khroskyabs dialects, Wobzi (spoken in Ere Township ʁôvzi 俄熱鄉 Érè Xiāng in Chuchen County). I additionally make reference to firsthand data from the Siyuewu dialect (fsəjó=vu 1 Local pronunciation kʰrəscaev. The native name of this language is bósraei, literally meaning "Tibetan language". It is previously known as "Lavrung", which was erroneously considered an endonym of the native speakers. In fact, the term "Lavrung" is unknown to most of the inhabitants. 2 Rgyalrongic is a sub-branch of the Sino-Tibetan family (since no evidence of any innovation common to all languages beside Chinese has been discovered, I refrain from using the term "Tibeto-Burman"), first proposed by [START_REF] Sun | Stem alternations in Puxi verb inflection: Toward validating the rGyalrongic subgroup in Qiangic[END_REF], comprising four Rgyalrong languages, Japhug, Situ, Zbu (Showu by [START_REF] Sun | Verb-stem variations in Showu rGyalrong[END_REF], and Tshobdun [START_REF] Sun | Stem alternations in Puxi verb inflection: Toward validating the rGyalrongic subgroup in Qiangic[END_REF][START_REF] Jacques | Phonologie et morphologie du japhug (rGyalrong)[END_REF]), and at least three Horpa languages, Stau, Stodsde, and Dgebshes (Jacques et al. forthcoming).

cahiers de linguistique asie orientale manuscript clao-1076

October 23, 2016 3

sharpening.stone=loc 'under the sharpening stone'), spoken in Siyuewu village 斯躍武村 Sīyuèwǔ Cūn in 'Dzamthang County) and the 'Brongrdzong dialect (spoken in 'Barkhams County), as well as data published by other scholars. Except for the Njorogs dialect [START_REF] Yin | Yèlóng lāwùróngyǔ yánjiū (业隆拉坞戎语研究) [Study on Njorogs Lavrung[END_REF]), which might be better treated as a distinct language, Khroskyabs dialects share a relatively high degree of intelligibility.

1.1 Syllable structure Presenting the syllable structure of the Khroskyabs language gives us better access to the understanding of its causative morphophonology. Each core Khroskyabs dialect has a similar syllable structure. I herewith present the syllable structure of the Wobzi dialect. The syllable in Wobzi Khroskyabs is comprised of an onset that is only absent in several affixes and ideophones, an obligatory nucleus, and a coda. In Wobzi, the coda is always mono-consonantal, while in Siyuewu and other dialects, complex codas with two consonants are attested.

Onsets can be comprised of a single consonant or a consonant cluster (hereafter a complex onset):

(1)

(C)(C)(C)(C)(C)V(C)
A complex onset is divided in three parts, an obligatory initial, one or several optional preinitial(s), and an optional medial. Any consonantal phoneme can be initial, while only sonorants and fricatives can be preinitials, and only non-nasal sonorants can be medials. In order to recognise these three onset positions, the consonantal reduplication test can be applied (Lai 2013c):

(2) a.

C p C i C m VC → C p C i C m VC~(C p )C i (C m )ɑ/u b. mbraê'rice' i. → mbraêmbru ii. → mbraêbru iii. → mbraêmbu iv. → mbraêbu
The process is done by reduplicating the onset and replacing the rhyme with -ɑ or -u. Some elements in the onset can be omitted in the reduplicated syllable, but one of them must be present. The initial of the onset is always reduplicated, while the consonants that precede (i.e. preinitials) and follow (i.e. the medial) it can either be preserved or deleted in the reduplicated syllable, as in the four possibilities in (2).

Elements in the preinitial part of an onset strictly follow the order indicated in Example (3) below (Lai 2013b (3)

ʁ-> j-> nasals > v-> r-, l-> s-, z-> initial > medial 1.2 Tone
Khroskyabs is a tonal language. In Wobzi Khroskyabs, there are two distinctive tonemes attested, a high tone (σ) and a high-falling tone (σ). Only one tone bearing syllable is allowed in a single phonological word, of which the position is not predictable. The surface tones of toneless syllables are derivable according to the tone of the tone-bearing syllable (Lai 2015a). Minimal pairs in Wobzi Khroskyabs are listed in Table 1.

The verb

A verb in Khroskyabs can have one, two, or three stems: a non-past stem (stem 1), a past stem (stem 2), and an irrealis stem (stem 3) [START_REF] Yi | Stem alternation in Khroskyabs[END_REF][START_REF] Sun | Stem alternations in Puxi verb inflection: Toward validating the rGyalrongic subgroup in Qiangic[END_REF].

Most of the verbs have stem 1 and stem 2 only, and stem 1 covers the functions of stem 3 in these cases. One-stem verbs are formally invariable in all contexts. Like other Rgyalrongic languages (Jacques 2013: 195), Khroskyabs exhibits templatic morphology with a long prefixal chain. Again, the verbal template is more or less the same within the core dialects. The Wobzi verbal template is shown in Table 2, and the affix list is shown in Table 3. 4. 3 In the intransitive conjugation, person markers are coreferent with S. 4 The transitive conjugation follows a hierarchical alignment (Lai 2015b), person markers follow the empathy hierarchy (1>2>3, [START_REF] Silverstein | Hierarchy of features and ergativity[END_REF], and the inverse direction (u-in Wobzi, əin Guanyinqiao) is also indexed on the verb. 5 The intransitive and transitive paradigms in Wobzi Khroskyabs is illustrated in Tables 5 and6. Transitivity is morphologically marked in an unambiguous way in Khroskyabs. Inverse constructions have the A marked by the ergative =ɣə and the There are two types of alignment in ditransitive constructions: indirective and secundative (Lai 2015b[START_REF] Haspelmath | Argument marking in ditransitive alignment types[END_REF][START_REF] Malchukov | Ditransitive constructions: A typological overview[END_REF]. From the perspective of person marking, in the secundative alignment, the recipient 6 Speech-act participant, referring to first or second person. cahiers de linguistique asie orientale manuscript clao-1076 October 23, 2016 is treated as P (recipient = P), while in the indirective alignment, the theme is treated as P (theme = P). Secundative verbs in Wobzi include ldzê 'teach' , bə' give (food) ' and sŋí 'lend' , etc.; indirective verbs include kʰɑ̂'give' , rŋî 'borrow' , fsé 'lead' and raé'say' , etc. ( [START_REF] Comrie | The syntax of causative constructions: Cross-language similarities and divergences[END_REF] who proposes the term "paradigm case". This principle is based on the case hierarchy:

-11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 ae-u-mə- -ŋ sə-nae-ɑ-tə-/tʰae-zə-ʁ-N-v-s-ʁjae-Noun Verb -j -Cɑ/u etc. ə-/
(6) Subject > Direct object > Indirect object > Oblique object

Once the causer occupies the subject position, the causee must be demoted to the highest unoccupied position. Therefore, from a case-marking point of view, in a causative construction, the causee stands in the direct object position if the base verb is intransitive, in the indirect object position if the base is monotransitive, and in the oblique object position if the base is ditransitive.

Example (7) shows the causativisation of the Wobzi intransitive verb gɑɣ 'to bow' , in which its S, the causee, ʁû 'head' , occupies the direct object position in the causative construction. Causative derivation turns a monotransitive verb into a ditransitive one. As mentioned in Section 1.4, indirective and secundative alignments are attested in ditransitive constructions. The derived causative verb is always secundative (recipient = P), that is, the causee is regarded as the recipient. In Wobzi, the causee can be left unmarked, or marked with the dative =kʰe. Causativisation of ditransitive verbs is rarer in Khroskyabs. The causative verb takes four participants, the causer, the causee, the theme, and the recipient. The alignment is still similar to a secundative one: when the causee and the recipient are both sap arguments, the causee is treated as P; while in the presence of a non-sap argument and an sap argument (no matter which one appears as the causee), the sap argument is treated as P.

As in Example (10a), the causee nû (2sg) and the recipient marked by dative ŋɑ=kʰe (1sg=dat) are both sap arguments. In this case, the causee nû (2sg) is treated as P, with which the main verb agrees: n-u-s-kʰɑ-n (pfv-inv-caus-give-2). While in Example (10b), only the recipient ŋɑ=kʰe (1sg=dat) is an sap argument, the verb must therefore agree with it, despite its dative marking: n-u-skʰɑ-ŋ (pfv-inv-vaus-give-1sg).

S-causative

It is common among Sino-Tibetan languages to find a causative morpheme with the form s-, see Sun (1999: 184-185) for a general discussion on Rgyalrong and Dulong, Sun (2007: 214-224) for Horpa, and [START_REF] Sagart | Reconstructing the *s-prefix in Old Chinese[END_REF] for Old Chinese. Similarly, in Khroskyabs dialects, we find a sibilant prefix on verbs used to mark a causative. In this section, I focus on the morphophonology of this prefix in Wobzi Khroskyabs.

The causative prefix s-in Wobzi exhibits richer morphophonological properties than the other prefixes in the verbal template. There are six categories of phonological processes: voicing assimilation, lateral dissimilation, lateral assimilation, affrication, metathesis, and cluster reduction.

Voicing assimilation

When followed by a voiced stop (b-, d-, ɟ-g-, the causative prefix s-undergoes assimilation and becomes a voiced z-(for affricates and fricatives, see Section 3.1.2). However, voiced sonorant consonants (ʁ-, v-, j-, r-, l-, and nasals) do not trigger the assimilation process. Example (11) illustrates the phenomenon. Dissimilated s-causative also undergoes voicing assimilation (see examples in Section 3.1.2).

(11) S-causative voicing assimilation a. s-+ qʰrɑ́'to be big' → s-qʰrɑ́'to cause to be big' b. s-+ gí 'to put on (clothes') → z-gí 'to cause to put on (clothes)' c. s-+ raé'to write' → s-raé'to cause to write'

Lateral dissimilation

The causative prefix s-is lateralised into l-or ɬ-depending on the voicing of the initial consonant, when preceding coronal affricates and fricatives s-, z-, ts-, tsʰ-, dz-, ɕ-, ʑ-, tɕ-, tɕʰ-, dʑ-, tʂ-, tʂʰ-, and dʐ-. Similar to the fricative variant present above, it is voiceless when preceding voiceless consonants and voiced when preceding voiced ones. See the examples in Example ( 12). ( 12) S-causative lateral dissimilation a. s-+ tsʰə̂'to hit' → ɬ-tsʰə̂'to cause to hit' b. s-+ tɕʰərɑ 'to stand up' → ɬ-tɕʰərɑ 'to cause to stand up' c. s-+ tʂəlé 'to fold' → ɬ-tʂəlé 'to cause to fold' d. s-+ dzaev 'to bark' → l-dzaev 'to cause to bark' e. s-+ dʑé 'to hold' → l-dʑé 'to cause to hold' f. s-+ dʐaeń 'to recall' → l-dʐaeń 'to cause to recall'

Affrication

An epenthetic stop consonant (t-or d-) is inserted between the causative prefix, dissimilated into l-or ɬ-, and the initial consonant of the base verb. Affrication is not restricted to word-internal sandhi, it also occurs with some enclitics, such as the inferential marker =si (→ =tsʰi) and the yes-no question marker =ɕəɣ (→ =tɕʰəɣ) when preceded by a coronal nasal. While affrication must be present with such enclitics, it is unproductive in morphological causatives in Wobzi.

Only one example is found in our Wobzi data, shown in Example ( 13).

(13) s-rʑə̂(caus-wash) → l-d-ʑə̂→ l-dʑə̂'cause to wash'7

Other verbs with initial coronal fricatives do not undergo this process: s-sɑ( caus-to kill) → ɬ-sɑ̂'cause to kill' . As far as I know, in Siyuewu Khroskyabs, there are more examples of affrication, see Section 3.1.9.

Metathesis

Two types of metathesis are presented in this section: preinitial ordering metathesis together with cluster reduction, and vCVr metathesis.

First, preinitial ordering metathesis and cluster reduction is related to the ordering of the preinitials presented in (3), repeated in Example ( 14).

(14) ʁ-> j-> nasals > v-> r-, l-> s-, z-> initial > medial
The prefixation of causative s-is restricted to the preinitial domain of the syllable onset. Therefore, causative s-cannot occur on the right-hand side of the initial or the medial. When the verb contains a complex onset, the placement of the causative prefix must not violate the pre-determined ordering. Automatic metathesis prevents any violation of this ordering, as shown in Example (15). As ʁ-, j-, and v-rank higher in the hierarchy, s-must be placed after them.

(15) Preinitial ordering metathesis a. s-+ ʁbɑɣ 'to explode' → ʁ<z>bɑɣ 'to cause to explode' b. s-+ jdə̂'to buy' → j<z>də̂'to cause to buy' c. s-+ vɟê 'to be true' → v<z>ɟê 'to correct'

When a verb takes both the causative s-and the reflexive ʁjae-, s-is inserted between ʁ-and jae-: ʁjaeŝɑ 'to commit suicide' → ʁ<s>jaeŝɑ [ sjaeŝɑ] 'to cause to commit suicide' .

Except for ʁ-, the nasal preinitials, in Wobzi r-and l-obligatorily drop after metathesis. This process may be called the cluster reduction, shown in ( 16).

(16) a. s-ntʰəmrɑ́(caus-paralysed) → *n<s>tʰəmrɑ́→ s-tʰəmrɑ́'to cause to be paralysed' b. s-mkʰaê(caus-intelligent) → *m<s>kʰaê→ s-kʰaê'to cause to be intelligent' c. s-rlaé(caus-peel) → *r<s>laé→ s-laé'to cause to peel' d. s-ɬqʰaeĺ (caus-dirty) → *ɬ<s>qʰaeĺ → s-qʰaeĺ 'to cause to be dirty'

The cluster reduction rule is optionally applied to preinitials j-and v-. In the variety of Wobzi spoken by younger speakers, j<z>də̂'cause to buy' and v<z>ɟê 'cause to be slow' are more often simplified to z-də̂and z-ɟê, respectively. However, when a syllabic prefix (e.g. directional-tam prefixes, inverse, etc.) is present, more often than not, j-and v-are still produced, as is shown in (17). The other type of metathesis is vCVr metathesis. When a base has v-as preinitial and -r as coda, the use of s-causative yields vCVr metathesis, as is shown in (18). ( 18 The base vCVr first metathesizes its preinitial and coda, and the new base rCVv undergoes normal s-causative processes presented above. See ( 19) for the steps of vCVr metathesis. This process is a Wobzi-specific innovation. Consider the cognates in other Khroskyabs dialects in Table 7. The verbs vdaer 'to open (one's eye)' and vdɑŕ 'to be flat' do not seem to have a cognate in Rgyalrong dialects, while vzɑŕ 'spicy' corresponds to Japhug Rgyalrong mɤ-rtsaβ 'to be spicy' . We can therefore conclude that *rCVv should be closer to the proto language (although the irregularity of -r vs. ɣ-in vdɑŕ :: ɣdav⁵⁵ :: ɣdap⁵⁵ is yet to be explained), and that syllables with a structure *rCVv have undergone a metathesis to vCVr. The causative derivation of the ancestor of Wobzi, z-dɑv 'to flatten' and l-zɑv 'to cause to be spicy' , predates and bleeds this metathesis. Another verb, vʑaeŕ 'shave' , seems to fit in this category. However, its causative form is attested as l-ʑaeŕ 'cause to shave' . This is probably because it is a recent borrowing from Tibetan (cf. bzhar 'to shave'), which took place after the *rCVv → vCVr sound change.

Lateral assimilation

Lateral assimilation happens in the causative form of verbs with -ər rhyme when the initial consonant is a coronal fricative or affricate (a consonant that triggers lateral dissimilation). The causativisation process replaces the coda -r with -l. cahiers de linguistique asie orientale manuscript clao-1076 October 23, 2016 13 Examples in (20) illustrate the process. The initial consonants of each base verb are s-(çsər 'to include'), tɕʰ-( tɕʰər 'to be sour'), dʑ-(jdʑər 'to mill'), tsʰ-(tsʰər 'to milk') and z-(ʁzər 'to crack'), respectively. Note that this process disregards the presence or absence of preinitials, as well as their place of articulation.

(20) Lateral assimilation a. s-+ çsər 'to include' → çsə-l 'to cause to include' b. s-+ tɕʰər 'to be sour' → tɕʰə-l 'to cause to be sour' s-+ jdʑər 'to mill' → jdʑə-l 'to cause to mill' d. s-+ tsʰər 'to milk' → tsʰə-l 'to cause to mikl' e. s-+ ʁzər 'to crack' → ʁzə-l 'to cause to crack' On the surface, the only difference between the base and the causative form is that of the coda, -r in the base and -l in the causative from. This is the outcome of no less than five distinct sound changes, as illustrated in Example ( 21 The lateral variant of causative s-assimilates the coda -r into -l before disappearing, then the coda -l becomes the causative marker in these cases. As more and more speakers merge r and l in preinitial and coda positions, lateral dissimilation is becoming increasingly rare.

Tonal alternation

Generally, the causative prefix s-is added to the base verb without any suprasegmental changes. However, there are s-causative forms where tones are different from those of their bases. Tonal alternation is unproductive and uni-directional with a high-falling toned base and a high toned causative form. To illustrate the alternation, corresponding examples from Wobzi and Siyuewu are shown in Table 8.

Tonal alternation is not always uniform among dialects. Some causative forms exhibiting tonal alternation in one dialect might be regular causative forms (without tone change) in another. For instance, the Siyuewu causativisation exhibits an alternation s-mpʰjaer (caus-be.beautiful) → ɬ-s-pʰjaeŕ 'to beautify' , which has a cognate in Wobzi without alternation: s-mpʰjaeŕ → s-pʰjaeŕ. 8 8 For an analysis of the origin of the tonal alternation in Khroksyabs causatives, see [START_REF] Lai | The denominal origin of voice derivation in Rgyalrongic: Passive and causative[END_REF] This irregular coda might be related to the stem 3 -m suffix attested in Rgyalrong languages [START_REF] Jacques | Phonologie et morphologie du japhug (rGyalrong)[END_REF]: 353, Sun 2004: 280).

The suffix -m is preserved in only a limited number of stem 3 verbs in Rgyalrong languages, and according to Sun (2004: 280), it must be one of the oldest markers of stem 3 in Rgyalrongic. The form s-tə-m 'to shorten' is the only example attested with the suffix -m in Wobzi, and it is not found elsewhere among the Khroskyabs dialects.

Additional remarks

Our data show that Wobzi s-causative used to be quite productive at some points in the history of the language, as it can be prefixed to Tibetan loan words: s-dʐaeń (caus-recall) → l-dʐǽn 'cause to recall' (Tibetan dran), s-ndʑǽm (causbe.soft) → l-dʑǽm 'to cause to be soft' (Tibetan 'jam). In Rgyalrong language, such as Japhug Rgyalrong, the causative prefix sɯ-is fully productive and can be added to verbs of Tibetan and even Chinese origin, for instance, sɯ-sci (causbe.born) 'to give birth to' is derived from a Tibetan loanwoard, skye 'to be born, cahiers de linguistique asie orientale manuscript clao-1076 October 23, 2016 to grow' . (Jacques 2014b: 264).

It is generally assumed that derivational morphology is far less productive than inflectional morphology [START_REF] Stump | Inflection[END_REF][START_REF] Haspelmath | Understanding morphology[END_REF]. S-causative derivation is also restricted and its productivity is visibly decreasing today, as younger people use less s-causative forms than their elders, a particular causative form may be judged acceptable by one consultant and rejected by another, in this case, they would prefer the analytic strategy (Section 6). For example, speakers prefer not to use s-causative when the base has already an spreinitial: for instance, the causative form ɬ-srí of the verb srí 'to look' , is never attested in narratives or daily production. The form is only judged marginally possible when elicited. In Siyuewu Khroskyabs, as will be shown in Section 3.1.9, this constraint is less strict, allowing sequences like r<ɬ>s-.

Some s-causative forms are related to different interpretations of syllable structure depending on individual speakers. For instance, the onset structure of the verb base vɣəm 'to rest head on' is ambiguous. It can be either an initialmedial sequence, or a preinitial-initial one. Both interpretations are attested with different consultants through the reduplication test: apart from the fully reduplicated vɣəm-vɣɑ, we have vɣəm-vɑ if it is analysed as initial-medial and vɣəm-ɣɑ if preinitial-initial. Since s-causative can only interact with preinitials, speakers with the initial-medial interpretation will find it impossible to add the s-causative to the base, as the phonotactics of Wobzi does not allow two sonorants or fricatives behind s-; on the other hand, speakers with the preinitial-initial interpretation will find the causative form s-ɣəm legitimate. 9Many of the s-causative forms are lexicalised and have idiomatic meanings. For example, the action that z-bré (caus-loud) denotes is actually that of playing a wind instrument, rather than causing any object to sound loud;10 the most frequent meaning of s-lé (caus-be.slow) is to delay, hence its reflexive form ʁjaesle (rfl-delay) means to spend some time somewhere (耽擱 dān gē in Western variants of Sichuanese Mandarin); s-cʰaé(caus-immense) is used with the meaning of holding someone in esteem. [START_REF] Kulikov | The "second causative": A typological sketch[END_REF] introduced the terms first causative and second causative to discuss languages having at least two different causative strategies. The first causative is the formally simpler causativisation strategy, often showing idiomatic meanings and referring to more natural and typical activities or processes than the standard causative meanings [START_REF] Kulikov | Remarks on double causatives in Tuvan and other Turkic languages[END_REF]. In this sense, we can assume that s-causative is the first causative derivation in Wobzi. Wobzi's second causative under this framework should be analytic causative, presented in Section 6.

S-causative in other Khroskyabs dialects

In other Khroskyabs dialects, the s-causative also undergoes complex morphophonological processes, including voicing assimilation, lateral dissimilation, and preinitial ordering metathesis which we saw earlier.

In Siyuewu Khroskyabs,11 the morphophonogical processes are similar to Wobzi, but there is no cluster reduction (23a). The preinitial N-is denasalised into l-or ɬ-, depending on the voicing of the following consonant (23b, 23c), and the preinitial m-(not to be confused with the bilabial variant of N-before bilabial consonants) is fricativised into v-, then devoiced into f-by the following voiceless consonant (23d), after metathesis.

(23) a. s-rsvɑɣ (caus-coarse) → r-ɬ-svɑɣ 'to cause to be coarse' b. s-ɲcô (caus-throw) → ɬ-s-cô 'to cause to throw' c. s-mpjaepjû (caus-suck.in.the.mouth) → ɬ-s-pjaepjû 'to cause to suck' d. s-mqlóɣ (caus-swallow) → f-s-qlóɣ 'to cause to swallow'

We find more examples of affrication in Siyuewu Khroskyabs than in Wobzi Khroskyabs, such as l-dzé 'to cause to be small/young' , from zê 'to be small/ young' and l-dzóɣ 'to cause to be sharp' from zóɣ 'to be sharp ' , etc. There is also an isolated example of metathesis attested: s-təĺ 'caus-hit' → ɬtə-z 'cause to hit' . The causative prefix s-is metathesised with the original coda -l, and automatically becomes -z due to the phonotactics of the dialect.

The 'Brongrdzong dialect12 has a syllabic causative morpheme sə-preceding the base, which seems to be largely productive.

(24) a. sə-bə̂(caus-give.food) 'to cause to give food' b. sə-vzê (caus-teach) 'to cause to teach' c. sə-ʁɴjə́(caus-be.thin) 'to cause to be thin'

The original s-prefix seems only to occur in lexicalised forms: sməmô 'to move (tr.)' (c.f. Wobzi s-mumû (caus-move.intr) 'to move (tr.)'), sjú 'to finish' (c.f. Wobzi s-jəɣ (caus-be.finished) 'to finish'). Sometimes sə-and s-co-occur, the causative form of ŋkaekaê'to be separated' is sə-s-kaekaê'to separate' .

The Guanyinqiao variant described in Huang ( 2007) is similar to 'Brongrdzong, with a productive sə-prefix and unpredictable co-ocurrence of sə-and s-: sə-xsû (caus-be.alive) 'to cause to be alive' , sə-s-lû (caus-caus-be.wet) 'cause to be wet' .

Njorogs [START_REF] Yin | Yèlóng lāwùróngyǔ yánjiū (业隆拉坞戎语研究) [Study on Njorogs Lavrung[END_REF]) is a phonologically more conservative dialect as it preserves plosive codas, which have undergone lenition in all core Khroskyabs dialects. As in 'Brongrdzong, the preservation of s-causative in Njorogs is restricted to lexicalised examples, and the productive prefix is sə-. In Example ( 25), there are a few examples taken from Yin (2007: 177):

(25) a. sə³³-ndzoʔs⁵⁵ (caus-learn) 'to cause to learn' b. sə³³-raʔt⁵⁵ (caus-write) 'to cause to write' c. sə³³-ɣdoʔ⁵⁵ (caus-hit) 'to cause to be hit' d. sə³³-dzəʔ⁵⁵ (caus-eat) 'to cause to be eat'

V-causative

The causative prefix v-in Wobzi is cognate to Situ Rgyalrong wɐ-, Zbu Rgyalrong vɐ- [START_REF] Sun | Jiāróngyǔ dòngcí de pàishēng xíngtài (嘉戎語動詞的派生形 態) [Derivational morphology in the Rgyalrong verb)[END_REF]) and Japhug Rgyalrong ɣɤ- [START_REF] Jacques | Jiāróngyǔ yánjiū (嘉絨語研究) [Study on the Rgyalrong language[END_REF]. It is believed to be exclusively used with stative verbs (Sun 2006: 11). In Wobzi this prefix is not productive, only one confirmed example is found in our vocabulary: f-tsʰû 'to boil' , from the stative verb tsʰû 'to be boiled' (v-devoiced into f-due to voicing assimilation). Minimal pairs with s-causative are also found with the stem: ɬ-tsʰû 'cause to be boiled'; the difference between v-and s-causatives can be illustrated in Example [START_REF] Jacques | The origin of causative prefix in Rgyalrong languages and its implication for Proto-Sino-Tibetan reconstruction[END_REF].

(

) ŋô 1sg jdə=tə water=def ae-f-tsʰú-ŋ pfv-caus v -boiled-1sg 26
aenɑ but ae-mae-ɬ-tsʰú-ŋ pfv-neg-caus s -boiled-1sg 'I tried to boil the water but I didn't make it boil.' Huang (2007: 80) states that in Guanyinqiao, v-seems to be a variant of the syllabic prefix sə-, but does not discuss the conditioning environment.

In Siyuewu, v-appears only lexically with verbs whose first consonant is a dental fricative or an affricate, and it is not restricted to stative verbs.

(27) a. f-skí (caus-be.hot) 'to heat' b. f-stʰəź (caus-wake) 'to cause to wake' c. f-tsʰû (caus-be.boiled) 'to boil'

The original difference between v-causative and s-causative remains opaque in Khroskyabs with the data available to us; although [START_REF] Sun | Jiāróngyǔ dòngcí de pàishēng xíngtài (嘉戎語動詞的派生形 態) [Derivational morphology in the Rgyalrong verb)[END_REF][START_REF] Sun | Typology of generic-person marking in Tshobdun Rgyalrong[END_REF] reports cahiers de linguistique asie orientale manuscript clao-1076 October 23, 2016 a difference in Tshobdun, the semantic contrast between the two prefixes appears to have been lost also in Japhug, except in examples like sɯ-mto 'to show' vs. ɣɤ-mto 'cause to recover (sight)' [START_REF] Jacques | The origin of causative prefix in Rgyalrong languages and its implication for Proto-Sino-Tibetan reconstruction[END_REF].

Anticausative

Anticausative verbs are derived through voicing alternation from their causative counterparts, of which the initials are voiceless consonants. As anticausative examples are more or less identical among Khroskyabs dialects, Table 10 only provides an exhaustive list of anticausative forms in the Wobzi data: The direction of the anticausative derivation is undoubtedly from voiceless base verbs to voiced anticausative verbs. Voicing alternation is widely attested in Sino-Tibetan languages. In closely related Rgyalrong dialects, the anticausative forms are characterised by a prenasalised voiced initial, as in Japhug Rgyalrong prɤt 'to cut' vs. mbrɤt 'to be cut' , and the process is still productive (Jacques 2012: 214): Japhug Rgyalrong derived ʁndɤr 'to be dispersed' from tɤr 'to disperse' , which is borrowed from Tibetan gtor 'to disperse, to toss' . Even though Khroskyabs anticausatives do not exhibit prenasalised consonants, there is evidence indicating that at least some voiced onsets are actually from prenasalised consonants. For instance, Wobzi Khroskyabs bró 'horse' is cahiers de linguistique asie orientale manuscript clao-1076 October 23, 2016 related to Japhug Rgyalrong mbro 'horse' < Proto-Rgyalrong *mraŋ. The voiced bilabial stop -b-is clearly epenthetic, and a prenasalised stage was to be posited for this word. The onset of Wobzi bró must have come from *mbr-< *mr-.

Most of the causative verbs listed in Table 10 begin with voiceless aspirated consonants, only tɕərə 'to tear' has an unaspirated onset. Note the aspiration neutralisation tɕ-→ dʑ-, tɕʰ-→ dʑ-which confirms that the diachronicality of the derivation was from transitive to intransitive. Other related languages also exhibit anticausative voicing alternation. In Stodsde, a Horpa language (Sun 2007: 214), for example: prə 'to startle' vs. brə 'to be startled' , kʰʌ 'to cause to snap' vs. gʌ 'to snap'; in G.yurong Horpa (personal investigation in 2014): prɯ 'to loosen' vs. brɯ 'to become loose' , kʰwae 'to cut' vs. gwae 'to be cut' .

In Wobzi, the preinitial f-in the base verb drops in the anticausative form, instead of becoming a voiced v-, as in ftɕʰə̂'to melt (tr.)' → dʑə̂'to melt (intr.)' , ftsʰər 'to uprear' → dzər 'to stand' . The same case is found in Japhug Rgyalrong, ftʂi 'to melt (tr.)' → ndʑi 'to melt (intr.)' (Jacques 2014a: 246).

In ftɕʰəm → ʁdʑəm, we observe the loss of preinitial f-and also an addition of preinitial ʁ-. This should be a redundant passive prefix. 13 The Wobzi alternation seems to be restricted to stops and affricates, unlike Khang.gsar Stau səla 'to let fall' vs. zəla 'fall' (Jacques et al. forthcoming). Uvular stops are not attested in the alternation in Wobzi since the voiced uvular stop ɢ-does not belong to the consonantal inventory. In G.yurong Horpa this phonological restriction is absent, as ɢ-is attested in voicing alternation: qətsɐ' to break (branch) (tr.)' vs. ɢətsɐ́'(branch) to break (intr.)' .

Semantically, the anticausative is similar to passive constructions, but without the presence of an agentive participant (Ottosson 2013: 330); while passive usually entails an agent that causes the event, compare cʰô 'to open (tr.)' , -cʰô (pass-open) 'to be opened ' and ʁɟô 'to open (intr.) Sometimes the passive form conveys the same meaning as the anticausative. Therefore -pʰɑɣlɑɣ (pass-lay.down) 'to be laid down' and bɑɣlɑɣ 'to be laid down' have no apparent difference in meaning, as is illustrated in (29). nae--pʰɑɣləɣ pfv-pass-be.laid.down 'The stick was lying on the ground.'

Labile verbs

Labile verbs can be both causative and non-causative verbs without morphological transformation. The causative reading is sometimes termed lexical causative [START_REF] Song | Linguistic typology: Morphology and syntax[END_REF]: 260, Kulikov 2001: 887). There are two types of labile verbs: A-preserving labile verbs (S = A) and P-preserving labile verbs (S = O) (Dixon 1994: 230). Several P-preserving labile verbs are attested in Wobzi Khroskyabs: çsər 'to include (tr.), to be included' , and zbî 'to dry up (tr.), to be dry (intr.)' , skí 'to heat, to be hot' and səɣ 'to tie (tr.), to be tight' . See examples (32), ( 33 k-u-səɣ=si pfv-inv-tie=infr 'He tied the things up with a rope.' All four of the labile verbs have cognates in Japhug Rgyalrong, illustrated in Table 11. While they are labile in Wobzi Khroskyabs, none of them are labile in Japhug Rgyalrong, causative prefixes sɯ-/sɤ-/ɣɤ-and the passive prefix a-must be added to derive transitive or intransitive counterparts.

Actually, P-preserving labile verbs are absent in Japhug Rgyalrong, all labile verbs are A-preserving [START_REF] Jacques | Argument demotion in Japhug Rgyalrong[END_REF]. As all of the P-preserving labile verbs in Wobzi are with a coronal sibilant somewhere in the onset (çs-, zb-, sk-and s-), this kind of lability may be due to the fact that the causative s-is lost without trace before a cluster whose first element is a sibilant fricative, and does not undergo lateralisation: The direction from intransitive to transitive may be confirmed by səɣ 'to be tight': the preinitial -is a variant of ʁ-which can be either the passive prefix or a denominal marker deriving exclusively intransitive verbs (Lai 2013a) (c.f. its Japhug cognate asɯɣ 'to be tight'). Therefore, səɣ 'to be tight' is formally an intransitive base, and its transitive meaning must be derived from some process, probably through the merger of the proto *s-and the following coronal sibilant. This evidence might give us a clue about the origin of lability in Khroskyabs.

6 Analytic causative 6.1 Formation Comrie (1989: 167) describes the prototypical analytic causative as a clause in which there are separate predicates expressing the notion of causation and cahiers de linguistique asie orientale manuscript clao-1076 October 23, 2016 the predicate of the effect. Thus, the English phrase cause John to go fits the prototype, while the French causative phrase faire courir Jean (make run John) does not. In this sense, Wobzi Khroskyabs exhibits a prototypical analytic causative with the nominaliser =spi and the causation predicate vî 'to do' . See Example (38), taken from a natural narrative. In (38), the effect clause nominalised by =spi is a sentence with the causee, the semantic agent of the effect transitive predicate dʑé 'to get' , marked with ergative. A more canonical expression can be elicited as: As with morphological causative, the causee of the transitive predicate is either unmarked or marked with the dative =kʰe.

Analytic causative constructions with an intransitive effect predicate do not mark the causee, illustrated in (40).

(40) nêɲɟi 2pl snɑɣlî moon nae-mae-ɲcʰaeĺ=spi ipfv-neg-play=nmlz v-ɑŋ do-1sg 'I will not let you play under the moon.' Analytic causatives are fully productive in Wobzi. However, when the causer and causee are both human, the verb raé'say' with a subordinate imperative clause is often used with a causative meaning, as is shown in (41). Although the sentence in (41) literally means 'He said to him, go practice Buddhism' , it does not necessarily imply that the causer actually uttered any words. Rather, it simply implies passive causation, best translated by the English let or Mandarin Chinese 讓 ràng. Note also that such constructions can also be used even when the causer is hearing/speaking-impaired. While the canonical analytic causative with =spi denotes the causative sensu stricto 'cause to do' , the quasi-causative construction with the verb raé'to say' indicates a command or a suggestion.

Comparative remarks

The Horpa languages have similar nominalising patterns using enclitics. Some of those enclitics developed independently of Khroskyabs. In Stodsde, the analytic causative construction is characterised by the nominliser -ldo, as attested by Sun (2007: 225)

Conclusion

This paper described lexical, morphological, and analytic causative constructions in Khroskyabs, mainly focusing on the Wobzi variant.

Lexical causatives are not common in Wobzi, with only a handful of labile lexical causative verbs attested. There are two morphological causative prefixes in Wobzi, and s-, v-. The former exhibits rich morphophonological alternations, including voicing assimilation, lateral assimilation, lateral dissimilation, two types of metathesis, affrication and cluster reduction. These processes are similar to those of the Stodsde causative, described by [START_REF] Sun | Morphological causative formation in Shangzhai Horpa[END_REF]. However, Wobzi does not possess a syllabic causative prefix sə-which is fully productive in other Khroskyabs dialects, such as Guanyinqiao, 'Brongrdzong, and Njorogs. Although the syllabic sə-looks more archaic, the absence of morphophonological alternations betrays the fact that it is a recently generalised allomorph.

A distinct phonomenon is the anticausative derivation whereby an intransitive verb is derived from a transitive one by voicing of the onset. Like the other Rgyalrongic and Sino-Tibetan languages, Khroskyabs exhibits a limited number of voiceless causative vs. voiced anticausative pairs, which originated from prenasalisation.

Khroskyabs also has a productive analytic causative construction that uses the nominaliser =spi (or =spə in 'Brongrdzong, =spe in Njorogs) and the causation verb vî 'to do' . With human causees, the analytic causative with =spi is less applied; instead, the verb raé'say' with an imperative subordinate clause is used to conveys a suggestion or a demand.

Figure 1

 1 Figure 1 Location of Khroskyabs in Sichuan, China

(

(

  (17) a. n-u-( j)<z>dəṕ fv-inv<caus>buy 'He caused to buy.' b. n-u-(v)<z>ɟê pfv-inv<caus>be.correct 'He corrected.'

  (19) a. *s-vzɑŕ (caus-to be spicy) b. → *s-rzɑv (vCVr metathesis) c. → *r-s-zɑv (preinitial ordering metathesis) d. → *s-zɑv (cluster reduction) e. → l-zɑv (lateral dissimilation, voicing assimilation) 'cause to be spicy'

  ): (21) a. *s-jdʑər (caus-mill) b. → *j<s>dʑər (preinitial ordering metathesis) c. → *j<l>dʑər (lateral dissimilation) d. → *j<l>dʑəl (lateral assimilation) e. → jdʑə-l (loss of the former l) 'cause to mill'

  crossed the river.' b. Mǒo-lu people.def-pl tee-ta cross-pf.pos.intr 'The people crossed.' (31) P-preserving labile verb in English a. He breaks the window. b. The window breaks.

(

  37) a. Prefixation of *s-to an intransitive verb beginning with *s-or *z-, deriving a transitive causative verb. b. Merger of *s-and *s-/*z-of the base verb, causing the derived causative verb to be homophonous with the original intransitive verb. c. Labile verbs were created due to homophony.

  with the faster horse was throned.'

  with the faster horse was throned.'

Table 1

 1 Wobzi tones

	H	Gloss	HL		Gloss
	jlé	'rabbit'	≠	jlê	'flute'
	ʁbɑɣ 'to explode' ≠	ʁbɑɣ 'to be many'
	vlé	'to be slow' ≠	vlê	'to care for (child)'
	zɑɣ	'to be sharp' ≠	zɑɣ	'potentilla anserina'

Table 2

 2 Wobzi verbal template

Table 3

 3 Affixes in the verbal template

	ɕə-			-n
	Inflection	Derivation	Stem	Infl. Redup.
	cahiers de linguistique asie orientale

1.4 Argument flagging and indexing

Case markers in Khroskyabs are listed in Table

Table 4

 4 Wobzi Khroskyabs case markers

	Functions	Markers
	ergative	=ɣə
	dative/ablative	=kʰe
	possessive/dative =ji
	locative	=tʰɑ, =gə, etc.

Table 5

 5 Wobzi intransitive paradigm

	Person endings Pronouns
	1sg stem-ŋ	ŋô
	1du stem-j	ŋgəne
	1pl stem-j	ŋgəɲɟi
	2sg stem-n	nû
	2du stem-n	nêne
	2pl stem-n	nêɲɟi
	3sg stem	aetə3
	du stem	aetəne
	3pl stem	aetəɟi

Table 6

 6 Wobzi transitive paradigm

							A
					1sg	1pl	2	3
			1sg			stem-n	stem-ŋ
		P	1pl 2	stem-ŋ	u-stem-j	stem-n	stem-j stem-n
			3
	(4)	a. Direct
		nû	cə3	nae-sɑ-n
		2sg	sg	pfv-kill-2
		'You killed him.'
		b. Inverse
		nû	ɣə	ŋô		n-u-sɑ-ŋ
		2sg	erg	1sg	pfv-inv-kill-1sg
		'You killed me.'

u-stem-ŋ u-stem-j u-stem-n (u-)stem verb indexed by the inverse prefix u-. As for direct constructions, the transitive verb agrees with A in sap 6 →3 situations, otherwise it agrees with P.

Table 7

 7 The verb vdaer 'to open (one's eye)' does not seem to have undergone the rCVv → vCVr sound change, since it is related to the more conservative Siyuewu Khroskyabs vdaer 'to open (the eyes)' . The causativsation into z-daev 'to cause to open eye' is therefore analogical.

				vCVr cognates
	Wobzi Guanyinqiao Siyuewu Njorogs Gloss
	vdaer	-	vdaer	-	'to open (one's eye)'
	vdɑŕ	ɣdáv	ɣdɑv	ɣdap⁵⁵ 'to be flat'
	vzɑŕ	rzɑv	rdzɑv	rzaʔp⁵⁵ 'to be spicy'

Table 8

 8 . Tonal alternation in causative derivations in Khroskyabs

	cahiers de linguistique asie orientale

3.1.7 Irregular -m coda An irregular causative form with an additional -m coda is attested. It concerns the stative verb çtə̂'to be short' : (22) s-çtə̂(caus-be.short) → s-tə-m 'to shorten'

Table 9

 9 Stem 3 in Rgyalrong languages

	Language	Stem 1 Stem 3	Gloss
	Japhug kɯmɲɯ	mto	mtɤ-m	'to look'
	Japhug Gsar.rdzong sɤndu sɤndɯ-m 'to exchange'
	Zbu	rzû	rzə-m	'to keep, to raise'

Table 10

 10 Anticausative in Wobzi Khroskyabs

	Base verb Gloss	Anticausative Gloss
	cʰô	'to open (tr.)'	ʁɟô	'to open (intr.)'
	ftɕʰə̂'to melt (tr.)'	dʑə̂'to melt (intr.)'
	ftɕʰəm	'to gather (tr.)'	ʁdʑəm	'to gather (intr.)'
	kʰlaé'to perish'	glaé'to die out'
	ntɕʰətɕʰɑv 'to trip'	ndʑədʑɑv	'to tumble'
	ftsʰər	'to uprear'	dzər	'to stand'
	ntsʰɑɣ	'to wear'	dzɑɣ	'to be there (attached)'
	pʰɑɣlɑɣ 'to lay down'	bɑɣlɑɣ	'to be laid down'
	pʰjá	'to destroy (financially)' bjá	'to go bankrupt'
	pʰrɑɣ	'to tie'	brɑɣ	'to be tied'
	pʰrə̂'to loosen'	brə̂'to become loose'
	tɕʰɑlɑɣ	'to scatter'	dʑɑlɑɣ	'to be scattered'
	tɕʰaev	'to break (tr.)'	dʑaev	'to break (intr.)'
	tɕərə	'to tear'	dʑərə	'to be torn (intr.)'

  Following this hypothesis, the transitive interpretation of the labile verb is derived from the original intransitive verb:

	manuscript clao-1076	October 23, 2016	22
	Table 11 Japhug Rgyalrong cognates of Wobzi Khroskyabs labile verbs
	Wobzi	Gloss		Japhug	Gloss
	Khroskyabs			Rgyalrong
	çsər	'to include/to be included'	a sɤr sɤr	'to be counted' 'to count'
	zbî	'to dry (tr.)/to be dry'	zbaʁ sɯzbaʁ/ɣɤzbaʁ 'to dry' 'to be dry'
	skí	'to heat/to be hot'	ɕke sɯɕke	'to be hot' 'to burn'
	səɣ	'to tighten/to be tight'	asɯɣ sɤsɯɣ	'to be tight' 'to tighten'
	(36) Proto-Khroskyabs
	a.
	cahiers de linguistique asie orientale

*s-sC-> *sCb. *s-zC-> *zC-

  in (42).

	(42) ɬʌmu=ɣə	lŋae	ɟʌló=ldo	ɣə-vzo
	Lhamo=erg	child	go.to.bed-nmlz	pfv-make
	'Lhamo made the child go to bed.'

In Wobzi Khroskyabs, there is a set of locative markers, expressing different spatial relations: =tʰɑ 'on the surface of' , =gə 'inside' , =vi 'under, at the bottom of' , =ʁɑ 'in an unspecified space' , etc.

A: transitive agentive, P: transitive patientive, S: intransitive subjective, T: ditransitive theme, R: ditransitive recipient

2 → 1, 3 → 1, 3 → 2, and all 3 → 3 directions; c.f.[START_REF] Zúñiga | Deixis and alignment: Inverse systems in indigenous languages of the Americas[END_REF] cahiers de linguistique asie orientale manuscript clao-1076 October 23, 2016

For the deletion of the preinitial r-see Section 3.1.4. cahiers de linguistique asie orientale manuscript clao-1076 October 23, 2016

The forms were elicited with four different speakers in the summer of 2012. Language consultants: Rig.'dus Lha.mo, Mtsho.skye Sgrolma.ma, Shakyaa Tshe.ring, Snyan.grags.

10 In Japhug Rgyalrong, the cognate ʑmbri 'to play (an instrument)' has the same idiomatic meaning, suggesting that this causative form was already lexicalised in Proto-Rgyalrongic. cahiers de linguistique asie orientale manuscript clao-1076October 23, 2016

Personal investigation since 2014. Language consultant: G.yu Lha.

Referring to the Siniao (斯鳥) variant. Personal investigation in 2012. Language consultant: Dngosgrub Lhamo. cahiers de linguistique asie orientale manuscript clao-1076 October 23, 2016

Acknowledgments

This research was funded by the HimalCo project (ANR-12-CORP-0006), the Humanities and Social Sciences Research Foundation of Ministry of Education of China for Young Scholar (Project No. 16 yjc740058), the Key Project of Humanities and Social Sciences Research of Chongqing Education Commission (Project No. 16skgh084), the Key Project of Humanities and Social Sciences Research of Chongqing Education Commission (Project No. 16skgh083) and is related to the research strand LR-2.5 of the Labex EFL (funded by the ANR/ CGI). I would like to thank Guillaume Jacques, Thomas Pellard, Amos Teo and the three anonymous reviewers who provided useful comments. I would also like to express my sincere gratitude to my main language consultants, Rig 'dus Lha Mo and G.yu Lha.


Steve Huntsman
email: sch213@nyu.edu

DISCRETE TOPOLOGICAL METHODS FOR CYBERSECURITY, NETWORK SCIENCE, AND MACHINE LEARNING

Keywords: topological data analysis, Dowker homology, path homology, magnitude homology AMS subject classifications. 55N31, 55N35, 55U05

This is an advertisement for (or a rant about) discrete topological methods-fairly new even by the standards of topological data analysis, and certainly underutilized at present-that are naturally suited to analyze (hyper)graphical data prevalent in cybersecurity applications, network science, and tasks in machine learning. As outstanding exemplars in this vein, we briefly discuss path homology and magnitude homology before paying lip service to sheaves, curvatures, and geodesics in discrete settings. Throughout, we mention new or plausible near-term applications.

1. Introduction. The field of topological data analysis (TDA) arguably originated with the introduction of topological persistence in 2000 [START_REF] Edelsbrunner | Topological persistence and simplification[END_REF]. Since then, it has blossomed into a vast enterprise (see, e.g. [START_REF] Zomorodian | Topology for Computing[END_REF][START_REF] Edelsbrunner | Computational Topology: An Introduction[END_REF][START_REF] Ghrist | Elementary Applied Topology[END_REF][START_REF] Oudot | Persistence Theory: From Quiver Representations to Data Analysis[END_REF][START_REF] Dey | Computational Topology for Data Analysis[END_REF]), albeit one that has been mostly focused on computing topological properties of structures that in some sense embody or approximate continuous objects such as spaces, probability distributions, etc. The fundamental data structure here, familiar to every student of topology, is an abstract simplicial complex. Meanwhile, the basic idea of mainstream TDA is that algebraic bookkeeping enables the construction of a filtration of abstract simplicial complexes (e.g., the computationally tractable Vietoris-Rips complexes) that adequately represents topological properties of a finite metric space across scales. Here "adequately" means that the constructions are sufficient to reconstruct many of the features of the original data, that they are resilient to noise, with quantitative error (bottleneck distance) bounds, etc.

A more peripheral branch in applied topology has an earlier origin but also ties into recent foundational developments. This branch deals with structures in which simplicial complexes are optional technical devices (if they appear at all) and are not embodiments or approximations (but are perhaps alternative representations or models) of a structure that is the direct subject of investigation. For example, in [START_REF] Dowker | Homology groups of relations[END_REF], Dowker showed that a relation naturally gives rise to a pair of homotopy equivalent simplicial complexes. Meanwhile, the homology over F 2 of these complexes can be readily computed directly from the relation itself, without ever constructing either simplicial complex. That is, relations give rise directly to F 2 -(co)chain complexes [START_REF] Ghrist | Elementary Applied Topology[END_REF]. For applications (besides the implicit connection to witness complexes [START_REF] Silva | Topological estimation using witness complexes[END_REF]), see, e.g. [START_REF] Atkin | Mathematical Structure in Human Affairs[END_REF][START_REF] Ghrist | Topological landmarkbased navigation and mapping[END_REF][START_REF] Giusti | Two's company, three (or more) is a simplex[END_REF][START_REF] Erdmann | Topology of privacy: lattice structures and information bubbles for inference and obfuscation[END_REF][START_REF] Speranzon | On sensor network localization exploiting topological constraints[END_REF][START_REF] Huntsman | Topology[END_REF]. More generally, bipartite graphs can be identified with a relation up to duality, and so Dowker homology could be applied to, e.g. user/computer authentication data such as [START_REF] Hagberg | Credential hopping in authentication graphs[END_REF][START_REF] Kent | User-computer authentication associations in time[END_REF], process/file interactions, bipartite network data from [START_REF] Kunegis | Konect: the Koblenz network collection[END_REF],1 etc. (these are exercises for the reader). 2Discrete structures such as finite topological spaces (e.g., the four-point pseudocircle) [START_REF] Barmak | Algebraic Topology of Finite Topological Spaces and Applications[END_REF] and posets [START_REF] Wachs | Poset topology: tools and applications[END_REF] have long been analyzed using classical methods of algebraic topology and the classical structure of a simplicial (or in some cases CW-) complex. More recently, hypergraphs [START_REF] Joslyn | Hypernetwork science: from multidimensional networks to computational topology[END_REF] have been analyzed via the abstract simplicial complexes they generate [START_REF] Firoz | IEEE International Parallel and Distributed Processing Symposium Workshops (IPDPSW)[END_REF]. Meanwhile, discrete versions of classical methods such as Morse theory [START_REF] Scoville | Discrete Morse Theory[END_REF] are seeing increasing use. Still, there is a sense in which the distinctly recent and computationally convenient topological techniques of path and magnitude (co)homology generalize classical techniques (though we do not specifically elaborate on the generalization aspects here) while also permitting applications to new types of data, most notably weighted digraphs and even directed hypergraphs. Because these techniques-each of which we briefly discuss in turn below-construct chain complexes directly from the data structures that they apply to, without any need to construct an intervening simplicial complex, they are particularly suited for novel applications that can give rise to new capabilities.

Path homology.

Here we sketch the basic elements of path homology as treated in [START_REF] Grigor'yan | Homologies of path complexes and digraphs[END_REF][START_REF] Chowdhury | Path homologies of motifs and temporal network representations[END_REF]. 3 Let R be a ring and consider the free modules R X ∼ = R |X| with standard basis {e x : x ∈ X}; also, set R ∅ := {0}. Next, let D = (V, A) be a loopless digraph, and consider the non-regular boundary operator ∂ [k] : R V k+1 → R V k defined as the linear map whose action on the standard basis is

∂ [k] e (v0,...,v k ) = k j=0 (-1) j e ∇j (v0,...,v k ) , where ∇ j (v 0 , . . . , v k ) := (v 0 , . . . , v j-1 , v j+1 , . . . , v k ) is the result of deleting v j from (v 0 , . . . , v k ). A few lines of algebra focused on index bookkeeping shows that ∂ [k-1] • ∂ [k] ≡ 0, i.e. (R V k+1 , ∂ [k] ) is a chain complex.
Path homology arises indirectly from another chain complex that is derived from

(R V k+1 , ∂ [k] ). Consider the set A k (D) := {(v 0 , . . . , v k ) ∈ V k+1 : (v j-1 , v j ) ∈ A, 1 ≤ j ≤ k} of allowed k-paths, with A 0 := V , V 0 ≡ A -1 := {0} and V -1 ≡ A -2 := ∅. Write Ω k := ω ∈ R A k : ∂ [k] ω ∈ R A k-1 , Ω -1 := R {0} ∼ = R, and Ω -2 := R ∅ = {0}. Now ∂ [k] Ω k ⊆ R A k-1 , so ∂ [k-1] ∂ [k] Ω k = 0 ∈ R A k-2 and ∂ [k] Ω k ⊆ Ω k-1 .
The (non-regular) path complex of D is accordingly defined to be the chain complex (Ω k , ∂ k ), where 4 The (non-regular) path homology of D is just the homology of the path complex (Ω k , ∂ k ). 5 A simple example illustrating the mechanics of path homology is provided by the digraphs D 1 and D 2 in Fig. 1. A 1 (D 1 ) and A 1 (D 2 ) are given by the digraph arcs, A 2 (D 1 ) = ∅, and A 2 (D 2 ) = {(w, x, z), (w, y, z)}. We have ∂ [START_REF] Alur | Principles of Cyber-Physical Systems[END_REF] (e (i,j) -e (i,k) -e (l,j) + 3 See also a series of papers including [START_REF] Grigor'yan | Graphs associated with simplicial complexes[END_REF][START_REF] Grigor'yan | Homologies of digraphs and Künneth formulas[END_REF][START_REF] Grigor'yan | On the path homology theory of digraphs and Eilenberg-Steenrod axioms[END_REF]. There are associated notions of cohomology [START_REF] Dimakis | Differential calculus and gauge theory on finite sets[END_REF][START_REF] Dimakis | Discrete differential calculus: Graphs, topologies, and gauge theory[END_REF][START_REF] Grigor'yan | Cohomology of digraphs and (undirected) graphs[END_REF][START_REF] Grigor'yan | On a cohomology of digraphs and Hochschild cohomology[END_REF] and homotopy [START_REF] Grigor'yan | Homotopy theory for digraphs[END_REF][START_REF] Grigor'yan | Fundamental groupoids of digraphs and graphs[END_REF]; the constructions of [START_REF] Babson | Homotopy theory of graphs[END_REF][START_REF] Barcelo | Discrete homology theory for metric spaces[END_REF] are closely related for the case of undirected graphs. Persistent path homology is introduced in [START_REF] Chowdhury | Persistent path homology of directed networks[END_REF] and an algorithm for computing it in dimension one is in [START_REF] Dey | An efficient algorithm for 1-dimensional (persistent) path homology[END_REF]. There are extensions of path homology to (among other structures) hypergraphs [START_REF] Grigor'yan | Homology of path complexes and hypergraphs[END_REF] and directed hypergraphs [START_REF] Muranov | Path homology of directed hypergraphs[END_REF], as well as an associated discrete Morse theory [START_REF] Lin | Discrete Morse theory on digraphs[END_REF]. 4 The implied regular path complex prevents a directed 2-cycle from having nontrivial 1-homology. In our view non-regular path homology is simpler, richer, and more likely useful in applications. As a practical matter, our rationale amounts to the convention that a directed 2-cycle counts as a "hole." 5 For convenience, we generally replace the path complex (Ω k , ∂ k ) with its reduction . . .

∂ k := ∂ [k] | Ω k .
Ω k+1 ∂ k+1 -→ Ω k ∂ k -→ Ω k-1 ∂ k-1 -→ . . . ∂ 1 -→ Ω 0 ∂0 -→ R -→ 0.
Assuming the original complex is nondegenerate, this has the minor effect H0 ⊕ R ∼ = H 0 , while Hk ∼ = H k for k > 0.

e (l,k) ) = 0, and similarly ∂ [START_REF] Alur | Principles of Cyber-Physical Systems[END_REF] (e (x,z) + e (w,x) -e (y,z) -e (w,y) ) = 0. Now ∂ [START_REF] Alvarez-Melis | Gromov-Wasserstein alignment of word embedding spaces[END_REF] e (w,x,z) = e (x,z) -e (w,z) + e (w,x) ∈ R A1(D2) and ∂ [START_REF] Alvarez-Melis | Gromov-Wasserstein alignment of word embedding spaces[END_REF] e (w,y,z) = e (y,z) -e (w,z) + e (w,y) ∈ R A1(D2) (because the arc w → z is not present), but ∂ [START_REF] Alvarez-Melis | Gromov-Wasserstein alignment of word embedding spaces[END_REF] (e (w,x,z) -e (w,y,z) ) = e (x,z) -e (w,z) + e (w,x) -e (y,z) + e (w,z) -e (w,y) = e (x,z) + e (w,x) -e (y,z) -e (w,y) ∈ R A1(D2) .

It follows that H 1 (D 1 ) = R and H 1 (D 2 ) = {0}, i.e., the Betti numbers are different: 1. The digraph D 2 has trivial path homology but the digraph D 1 does not. The nontrivial path homology of D 1 can be interpreted as a consistent and robust unidirectional flow. The trivial path homology of D 2 can be attributed to w and z appearing as "bottlenecks" that prevent robustness of the flow. This intuition is formalized by results in [START_REF] Chowdhury | Path homologies of deep feedforward networks[END_REF].

β 1 (D 1 ) = 1 and β 1 (D 2 ) = 0. D 1 i j k l i j k l = D 2 w x y z w x y z = Fig.
We note that there is good evidence that path homology provides a robust characterization of neural network architectures [START_REF] Chowdhury | Path homologies of deep feedforward networks[END_REF], a stronger variant of cyclomatic complexity [START_REF] Huntsman | Path homology as a stronger analogue of cyclomatic complexity[END_REF], and that it can identify salient motifs and behavior in temporal networks [START_REF] Chowdhury | Path homologies of motifs and temporal network representations[END_REF]. Path homology is also phenomenologically rich, exhibiting torsion and nonvanishing homology in dimension three for various digraphs with just six vertices.

3. Magnitude homology. We follow the concise formulation of [START_REF] Hepworth | Magnitude cohomology[END_REF]. Let (X, d) be a Lawvere metric space, i.e., d is an extended quasipseudometric. 6 A k-simplex in X is an ordered tuple x (k) := (x 0 , . . . , x k ) ∈ X k+1 such that adjacent entries are distinct; its length is λ(x (k) ) := k j=1 d(x j-1 , x j ). The R-graded magnitude chain complex has k-chains (where as before R is a coefficient ring)

M C k,L (X) := R x (k) a k-simplex in X : λ(x (k) ) = L and differential ∂ k : M C k,L (X) → M C k-1,L (X) given by ∂ k := k-1 j=1 (-1) j ∂ (j) with ∂ (j) (x (k) ) := ∇ j x (k) if d(x j-1 , x j+1 ) = d(x j-1 , x j ) + d(x j , x j+1 ) 0 otherwise,
and where as before ∇ j acts on a tuple by deleting the corresponding entry. The appropriate notion of chain map is induced by distance-nonincreasing maps.

The magnitude homology of X is the homology of the magnitude chain complex. Conveniently for a digraph D, there is a direct sum decomposition of the form

M C •,L (D) = s,t∈V (D) M C (s,t) •,L (D)
where the direct summands on the right are subcomplexes generated by simplices with initial and terminal entries s and t, respectively [START_REF] Asao | Geometric approach to graph magnitude homology[END_REF]. This provides a very specific algebraic structure and a reduction in computational complexity that makes magnitude homology particularly convenient for unweighted (or Z + -weighted) digraphs. 7 The dual theory of magnitude cohomology has an analogue of the cup product [START_REF] Hepworth | Magnitude cohomology[END_REF]. This additional ring structure allows magnitude cohomology to completely determine its underlying space up to isometry for finite extended quasi-metric spaces, such as digraphs and finite metric spaces. By comparison, trees on the same number of vertices have isomorphic magnitude (co)homology groups.

These constructs all generalize to the setting of enriched categories, and we sketch a slightly nontrivial construction likely to be of interest and use. There is an enriched category of sub-flow graphs of a given flow graph in the sense of [START_REF] Huntsman | The multiresolution analysis of flow graphs[END_REF], and it can be shown that the topological entropy (roughly, the logarithmic rate of growth of the number of possible paths as a function of length [START_REF] Kitchens | Symbolic dynamics: one-sided, two-sided and countable state Markov shifts[END_REF]) of such a flow graph with its (unique) entry and exit identified is a suitable "size function" over the max-plus semiring. This setup dovetails nicely with the direct sum decomposition above, and a categorification of these observations would presumably yield an interesting and efficiently computable instance of magnitude (co)homology, with representatives that encode detailed structural information. Meanwhile, there are effective normalization procedures to represent control flow graphs of the sort compilers actually produce in this way, so the theory has sufficient scope for genuine applications to be possible.

Besides the notional application sketched above, another interesting situation ripe for investigation/application is a graph with probability distributions on the vertices. By computing Bregman (e.g., Kullback-Leibler) divergences between adjacent distributions, we obtain an asymmetric weighted graph suited for analysis via (blurred) magnitude homology (or even just the decategorification into magnitude [START_REF] Leinster | The magnitude of a metric space: from category theory to geometric measure theory[END_REF][START_REF] Leinster | Entropy and Diversity: the Axiomatic Approach[END_REF]).

Finally, it seems likely that magnitude homology will also be useful for characterizing DAGs, and an interesting problem is to develop a formula akin to that in [START_REF] Chowdhury | Path homologies of deep feedforward networks[END_REF] for the path homology Betti numbers of DAGs that correspond to densely connected multilayer perceptrons. For example, the DAG underlying a densely connected multilayer perceptron with consecutive layers of 5, 4, 3, and 2 neurons has only a single nontrivial Betti number in path homology, viz. β 3 = 24, whereas a computer calculation yields the more complicated magnitude homology Betti numbers β 1,1 = 38, β 2,2 = 61, and β 3,3 = 60 (recall that the second index is for the length grading): all other Betti numbers are zero except for β 0,0 , which is just the number of vertices.

4. Proto-topological and geometrical tools. Sheaves [START_REF] Ghrist | Elementary Applied Topology[END_REF][START_REF] Robinson | Topological Signal Processing[END_REF], (positive) Ricci curvatures and geodesics [START_REF] Myers | Riemannian manifolds with positive mean curvature[END_REF][START_REF] Sakai | Riemannian Geometry[END_REF] each control topological properties of spaces they are associated with. These objects also have analogues in the discrete setting that inherit properties familiar from traditional continuous or smooth contexts, while also being amenable to computational implementation. 8 As such, they are likely to 7 Notably, magnitude homology is related to Vietoris-Rips homology via a "blurring" procedure that essentially replaces equalities of the form λ = L in the definition of the chain complex with inequalities λ ≤ L [START_REF] Otter | Magnitude meets persistence: homology theories for filtered simplicial sets[END_REF][START_REF] Cho | Quantales, persistence, and magnitude homology[END_REF][START_REF] Govc | Persistent magnitude[END_REF], and it is also closely related to path homology [START_REF] Asao | Magnitude homology and path homology[END_REF]. 8 See, e.g. https://github.com/kb1dds/pysheaf.

find progressively more useful applications adjacent to topological invariants.

4.1. Sheaves. For heterogeneous data, sheaves are the representational framework par excellence [START_REF] Purvine | A topological approach to representational data models[END_REF]. Without belaboring definitions, sheaves provide the mathematical framework assembling global data from local data. In this context it is natural that they should be used to model communications networks [START_REF] Robinson | Understanding networks and their behaviors using sheaf theory[END_REF][START_REF] Robinson | Analyzing wireless communication network vulnerability with homological invariants[END_REF][START_REF] Huntsman | Topology[END_REF] and the process of data fusion [START_REF] Robinson | Sheaves are the canonical data structure for sensor integration[END_REF][START_REF] Robinson | Dynamic sensor fusion using local topology[END_REF][START_REF] Robinson | Hunting for foxes with sheaves[END_REF][START_REF] Joslyn | A sheaf theoretical approach to uncertainty quantification of heterogeneous geolocation information[END_REF][START_REF] Robinson | Assignments to sheaves of pseudometric spaces[END_REF].

This data fusion process, predicated on a consistency radius, requires a reasonable (pseudo)metric on observations. While for many applications to cybersecurity this can be a challenging if not prohibitive obstacle, it is probably less of one in the important case of networked cyber-physical systems, where models can inform the construction of a metric on observable data [START_REF] Alur | Principles of Cyber-Physical Systems[END_REF][START_REF] Lee | Fundamental limits of cyber-physical systems modeling[END_REF][START_REF] Bakirtzis | Compositional cyber-physical systems modeling[END_REF][START_REF] Taha | Cyber-Physical Systems: A Model-Based Approach[END_REF]. In particular, hierarchical classifications and their corresponding ultrametrics are readily constructible in practice, and these have the advantage of being realizable in Euclidean space via a Cholesky decomposition of a (finite) distance matrix. Such a construction (which we note can avoid the need for continuous models/metrics in a way analogous to how the homology theories of preceding sections avoid constructing simplicial complexes) can be lifted to sequences of observed data that could then be meaningfully fused using sheaf-theoretical techniques.

Curvatures.

For a reasonably up-to-date overview of discrete notions of curvature, see [START_REF] Najman | Modern Approaches to Discrete Curvature[END_REF]. Various such notions exist, e.g. the Ollivier-Ricci curvature for metric spaces [START_REF] Ollivier | Ricci curvature of metric spaces[END_REF][START_REF] Ollivier | A survey of Ricci curvature for metric spaces and Markov chains[END_REF] and directed hypergraphs [START_REF] Eidi | Ollivier-Ricci curvature of directed hypergraphs[END_REF]; and Forman-Ricci curvature for directed graphs, simplicial complexes [START_REF] Saucan | Discrete Ricci curvatures for directed networks[END_REF] and hypergraphs [START_REF] Leal | Forman-Ricci curvature for hypergraphs[END_REF]. These yield complementary tools for network science applications [START_REF] Leal | Ricci curvature of random and empirical directed hypernetworks[END_REF] and jointly provide features that can statistically distinguish between different model network ensembles. Other variants of Ricci curvature have also been constructed for {weighted, directed, hy-per}graphs: see, e.g. [START_REF] Saucan | A simple differential geometry for complex networks[END_REF].

Applications of such discrete curvatures include analysis of neural network architectures [START_REF] Saucan | A simple differential geometry for complex networks[END_REF]; graph matching [START_REF] Ni | Network alignment by discrete Ollivier-Ricci flow[END_REF] and community detection [START_REF] Ni | Community detection on networks with Ricci flow[END_REF][START_REF] Sia | Ollivier-Ricci curvature-based method to community detection in complex networks[END_REF]; and addressing congestion in communication [START_REF] Wang | Interference constrained network control based on curvature[END_REF] and power networks [START_REF] Jonckheere | Curvature, entropy, congestion management and the power grid[END_REF]. Coming full circle, discrete curvatures also inform the construction of persistent topological invariants [START_REF] Saucan | Discrete Morse theory, persistent homology and Forman-Ricci curvature[END_REF].

4.3. Geodesics. Gromov-Wasserstein distance [START_REF] Mémoli | Gromov-Wasserstein distances and the metric approach to object matching[END_REF][START_REF] Sturm | The space of spaces: curvature bounds and gradient flows on the space of metric measure spaces[END_REF] is a Riemannian geodesic distance on the space of (isomorphism classes of) metric measure spaces. It can be computed using optimal transport methods, and algorithms based on this insight have enabled state-of-the-art performance for matching-and meaningfully interpolating between-discrete structures such as (hyper)graphs endowed with measures [START_REF] Xu | Gromov-Wasserstein learning for graph matching and node embedding[END_REF][START_REF] Xu | Scalable Gromov-Wasserstein learning for graph partitioning and matching[END_REF][START_REF] Chowdhury | Quantized Gromov-Wasserstein[END_REF][START_REF] Chowdhury | Generalized spectral clustering via Gromov-Wasserstein learning[END_REF][START_REF] Chowdhury | Hypergraph co-optimal transport: Metric and categorical properties[END_REF] and (via word embeddings) unsupervised translation [START_REF] Alvarez-Melis | Gromov-Wasserstein alignment of word embedding spaces[END_REF].

Conclusion.

A recurring theme that we have sought to illustrate is that while topological and adjacent methods may have been (and likely often still are) thought of as implicitly inappropriate for fundamentally discrete data and problems, the opposite is actually true. Recently developed topological invariants and adjacent geometrical constructions are amenable to computation in silico and can enable new capabilities for discrete data that are ubiquitous in cybersecurity, network science, and machine learning.

This collection of data is presently at http://konect.cc/.

Dowker complexes (and for that matter, cosheaves) also feature prominently in an effort to characterize consistency of program variants' behavior on inputs[START_REF] Ambrose | Topological differential testing[END_REF][START_REF] Ewing | Metric comparisons of relations[END_REF][START_REF] Robinson | Cosheaf representations of relations and Dowker complexes[END_REF][START_REF] Robinson | Statistical detection of format dialects using the weighted Dowker complex[END_REF]. While topological invariants per se are relevant to this effort, they feature only peripherally in such work to date.

Here "extended" means d : X × X → [0, ∞]; "quasi" means d need not be symmetric, and "pseudo" means that d(x, x) need not be zero. A Lawvere metric space is best thought of here as a category enriched over the poset ([0, ∞], ≥), not least because the theory of magnitude (co)homology applies to enriched categories with a suitable notion of "size."

Acknowledgments. I thank Samir Chowdhury and Michael Robinson for conversations that informed portions of this advertisement.