Anne Boud'hors

Sofia Torallas Tovar

PHILOLOGICAL WORK ON THE SAHIDIC GOSPEL OF MARK IN THE 9TH CENTURY: THE CASE OF SA 123

In our research for the critical edition of the Sahidic variants of the Gospel of Mark, the manuscript sa123 stands out as a very special example: it is likely to be a fragmentary witness of philological work in the Fayyum well after the Arabic conquest. The manuscript features extensive corrections performed by a second hand and based on a different type of text than the first copy. In this article we present the edition of one of its pages with the purpose of exploring the process and philological activity attested by it.

Establishing a new critical edition of the Sahidic text of the Gospel of Mark Gospel is the ultimate goal of the work we have been pursuing for many years, at a more or less regular pace. Our starting hypothesis is the existence of three types of texts: 1 sa 1 = text of the main witness of the ancient translation (P.Palau 182, ed. Quecke), also attested by other fragmentary manuscripts.

sa II = the revised translation, which is not witnessed by a unique manuscript, but by several fragmentary manuscripts that have the same textual type. It is the text most frequently giventhough not always, by Horner's edition.

sa 9 = text of M 569 (ed. Aranda Perez), which occupies an intermediary or mixed position.

In recent years, the development of digitization of manuscript images, easier access to these reproductions, and collaboration with INTF Münster, has allowed us to examine many witnesses and to make progress in several aspects: determining the existence of additional witnesses for type sa 1, 2 clarifying the text of sa II, and confirming that sa 9 is really a type of text, and not the text of an isolated manuscript, since at least three other manuscripts are obviously dependent on it (sa 125, sa 152, sa 156). In addition, it is now very likely that sa 9 is a kind of mixed text, the

result of a compilation work between sa 1 and sa II, carried out in Fayyum, perhaps at the monastery of Saint Michael of Phantoou3 . The purpose of this article is to present the case of another highly illustrative manuscript, which shows how such a compilation work could have been done. This is the manuscript sa 123. Horner's edition uses this manuscript (No. 73), with various acronyms indicating that it contains many corrections, but it is impossible to rely on his edition to understand the meaning of those corrections. As we will see, these corrections are intended, at least in the Gospel of Mark, to conform, to an extent that needs to be assessed, a text originally related to sa 1 to the text of sa II. Moreover, this manuscript poses two kinds of difficulties for a publisher: 1) the edition of the corrections in a clear manner and 2) the use of this manuscript in the critical apparatus of the edition of the Gospel of Mark.

To try to illustrate both these difficulties and the interest of the manuscript sa 123, we have chosen a page, whose edition we present below, as well as a detailed analysis of the corrections for the episode of the Transfiguration (Mk 9:2-13). They are preceded by a description of the manuscript based on all the surviving sheets and a brief characterization of the various other witnesses to the passage.

Description of sa 123

Identification: SMR (Münster) sa 123 = Biblia Coptica (Schüssler) sa 531 Provenance: White Monastery ? ultimately Fayyum ? Contents: Four gospels State of conservation: remains of 28 folia, of which 9 contain Mk: 1:10-13, [22][23][33][34]45 Paris Layout: Two column pages, at ca 36 lines in the column. There is pagination and headers (ⲡⲡⲡⲡⲡ in verso pages and ⲡⲡⲡⲡⲡⲡ in recto ones), as well as paragraphoi and polychrome coronides and drawings in the intercolumnar spaces.

Paleography: the hand is bimodular, a so-called Alexandrian majuscule, which is not far from the hand of sa 9. It also presents common features with the Fayyumic codices A and B4 , with less regularity. Apart from its standard features, the supralinear stroke is used on top of autosyllabic vowels (also the case of sa 9), which can be characterised as a quite frequent use in Sahidic manuscripts copied in Fayyum. It also appears occasionally on ⲡ before consonant (e.g. Mk 9:5,12:22,12:35,12:42). The supralinear stroke is more or less long and more or less centered on the letter or slightly shifted to the left or right; in the sequence ⲡⲡ, the stroke is not distinguished from the extension of the horizontal bar of ⲡ on the left. The current title (ⲡⲡⲡⲡⲡ on even pages, ⲡⲡⲡⲡⲡⲡ on odd pages) was adopted by the first copyist.

There seems to be two correcting hands, one using the same kind of brown ink as that of the main copyist, the other writing in black ink (the latter is much less frequent, it is found mainly in the London sheets, to the point that one may wonder whether it is really a different hand). These two hands are quite coarse, it is difficult to date them and to say if they are contemporary with each other, but their corrections seem to go in the same direction. These corrections consist of deletions -sometimes crossing out some words to rewrite others -, adding words or letters in the margin, more rarely in interlinear spaces (see below for a detailed presentation of the corrections on the edited page). Since the erasures are never total, we often manage to read what was originally there. It is possible that the correcting hand was also responsible for the pagination.

Date: based on the comparison with sa 9 (dated between 820 and 915 approximately), one could propose the 9th or early 10th century for the hand of the first copyist.

The Fayyumic origin of the manuscript can still be detected in some spelling features: ⲡⲡⲡⲡⲡⲡ (12,2); ⲡⲡⲡⲡⲡⲡⲡⲡⲡ (12,42): see the spelling of the fa version ⲡⲡⲡⲡⲡⲡⲡⲡⲡ vs. sa ⲡⲡⲡⲡⲡⲡⲡⲡⲡ; ⲡⲡⲡⲡⲡⲡⲡⲡⲡⲡⲡⲡ (12,41), which is once found in the fa version. An influence may also be reflected in some syntactic (9,4,9,5) or lexical (8,35) features: see the details of the edition. It is well known that the monastery of St. Michael in Fayyum and its scriptorium in Touton had relations with the White Monastery in the 10th century: either manuscripts copied in Fayyum were sent to the White Monastery or, perhaps, fayyumizing copyists worked at the White Monastery.

Other testimonies

The following fragments have been collated either on the originals (Paris, Oxford, Berlin, Leipzig), or on photographs (Moscow and Rome). Horner has not used all of these fragments, although he included the manuscript numbers in his edition.

-8 sa II -------- -------- Leipzig, inv. 3000 8 , f. 7v- 8 Fayyum 9: 1-8, 11b-13a sa 1
A Fayyumic version (fa) is also available for this passage: it is a folio from a White Monastery codex kept at the French Institute of Oriental Archaeology in Cairo (IFAO, Coptic 122), which gives the text of Mk 8:24b to 9:12. This is manuscript "B" mentioned above (n. 3). As we will see in detail below, the text of sa 123 is very close to this version on a few occasions.

Edition of London BL 3579 B19 f.36R

(Fig. 1)

The page we edit below is page 89 (ⲡⲡ). For the sake of clarity, given the large amount of text contained in a single page, we edit the two columns successively. They contain verses 8:31b to 9:13, but only the text of 9:2-13 is detailed afterwards.

The supralinear stroke, which adopts various placements and dimensions, is edited by convention and placed on the second letter when it is connective. Occurrences of ektheseis are in col. 1, l. 12 (8:34), l. 33 (9:1), l. 39 (9:2).

As mentioned in the description, it seems that two correcting hands can be distinguished; one (column 1) uses a blacker ink and is coarser than the other (col. 1 and col. 2), which is itself not very refined. Both hands make corrections in the same direction. But the "black" hand seems to have 'overcorrected', for instance when it has crossed out the two ⲡⲡⲡ which had already been deleted (col. 1, l. 23-24) The correction methods are often crossing out of words with a line, erasure produced by scratching or abrasion with or without second text superposed, and added text in the interlinear spaces or the margins to the left and right of the text. We indicate with double brackets (⟦ ⟧ ) the deleted words or passages, and in italics the words or passages added or rewritten on the deleted passages by one of the two correcting hands. Some details on the position or nature of the corrections are given in the apparatus or commentaries. Most of them modify the text copied by the first hand, but some only repair errors or omissions. sa 123a ⲡⲡⲡ ⲡⲡϥϩⲡⲡⲡⲡ ⲡⲡⲡⲡⲡϩ ⲡ ⲡⲡⲡⲡⲡⲡϣ ⲡ ⲡⲡⲡⲡ• ⲡ ⲡⲡϣ ⲡⲡⲡⲡϩ<ⲡ> ⲡ ⲡⲡⲡⲡ ⲡⲡⲡⲡⲡⲡⲡⲡⲡ ϩⲡ ⲡⲡⲡ ϩⲡϫⲡ ⲡⲡⲡϩ => The corrector deleted ⲡⲡⲡϩ and replaced it with ⲡⲡⲡⲡ9 ; he deleted the ⲡ from ⲡ ⲡⲡⲡⲡ (which is also in sa 112) and mistakenly replaced it with an ⲡ (not an ⲡ); he replaced ⲡⲡⲡϣ with ⲡⲡⲡϣ, probably added ⲡ in ⲡⲡϩⲡ, and deleted ⲡ in front of ⲡⲡⲡⲡⲡⲡⲡⲡ (this ⲡ is also in sa 291 L ); he corrected the final ⲡ from ⲡⲡⲡⲡ to ⲡ, which is again an error (this spelling is also in sa 112); but he did not delete ⲡ in front of ⲡⲡⲡⲡⲡⲡϣ and did not change the order of the words at the end of the sentence. Note the form ⲡⲡⲡⲡⲡ (found in two witness of sa II).

Col

Verse 4 sa ⲡⲡⲡ ⲡϥⲡⲡⲡⲡϩ ⲡⲡⲡⲡⲡ ⲡϭⲡ ϩⲡⲡⲡⲡⲡ ⲡⲡ ⲡⲡⲡⲡⲡⲡ ⲡⲡϣⲡϫⲡ ⲡⲡ ⲡⲡ • Some manuscripts reverse Elijah and Moses : sa 9, sa 122, sa 291 L , sa 134 => sa 123a gave ⲡⲡⲡⲡⲡⲡϩⲡⲡ, which is also the lesson of fa; this lesson has been corrected into ⲡϥⲡⲡⲡⲡϩ in sa 123b.

Verse 5 sa 1 ⲡϥⲡⲡⲡϣⲡ ⲡⲡ ⲡ ϭⲡ ⲡⲡⲡⲡⲡⲡ ⲡϥϫⲡ ⲡ ⲡⲡⲡ ⲡⲡ ⲡ ϫⲡ ϩⲡⲡⲡⲡⲡⲡ ⲡⲡⲡⲡⲡⲡ ⲡⲡⲡ ⲡⲡⲡⲡⲡϭⲡ ⲡ ⲡⲡⲡ ⲡⲡ• ⲡⲡⲡⲡ ⲡⲡⲡⲡⲡ ϭⲡ ⲡ ϣⲡⲡⲡⲡ ⲡ ⲡⲡⲡⲡⲡ ⲡⲡⲡⲡ ⲡⲡⲡ ⲡⲡⲡ ⲡⲡⲡⲡ ⲡ ⲡⲡⲡⲡⲡⲡ ⲡⲡⲡ ⲡⲡⲡⲡ ⲡ ⲡⲡⲡⲡⲡⲡ• sa 9 ⲡϥⲡⲡⲡϣⲡ ⲡⲡ ⲡϭⲡ ⲡⲡⲡⲡⲡⲡ ⲡϥϫⲡ ⲡⲡⲡⲡ ⲡⲡ ⲡ ϫⲡ ϩⲡⲡⲡⲡⲡⲡ ⲡⲡⲡⲡⲡⲡ ⲡⲡⲡ ⲡⲡⲡⲡⲡϭⲡ ⲡⲡⲡⲡⲡⲡ• ⲡⲡⲡⲡⲡⲡⲡⲡⲡⲡ ⲡϣⲡⲡⲡⲡ ⲡⲡⲡⲡⲡⲡ. ⲡⲡⲡⲡ ⲡⲡⲡ ⲡⲡⲡⲡ ⲡⲡⲡⲡⲡⲡⲡ ⲡⲡⲡ ⲡⲡⲡⲡ ⲡϩⲡⲡⲡⲡⲡⲡ• sa II ⲡϥⲡⲡⲡϣⲡ ⲡ ϭⲡ ⲡⲡⲡⲡⲡⲡ ⲡϥϫⲡ ⲡ ⲡⲡⲡ ⲡⲡ ⲡ ϫⲡ ϩⲡⲡⲡⲡⲡⲡ ⲡⲡⲡⲡⲡⲡ ⲡⲡⲡ ⲡⲡⲡⲡⲡϭⲡ ⲡ ⲡⲡⲡⲡⲡ• ⲡⲡⲡⲡ ⲡⲡⲡⲡⲡ ⲡ ϣⲡⲡⲡⲡ ⲡ ⲡⲡⲡⲡⲡ. ⲡⲡⲡⲡ ⲡⲡⲡ ⲡⲡⲡ ⲡⲡⲡⲡ ⲡ ⲡⲡⲡⲡⲡⲡ ⲡⲡⲡ ⲡⲡⲡⲡ ⲡ ϩⲡⲡⲡⲡⲡ• => sa 123 does not have ⲡⲡ (this omission of a particle may have been independent of the influence of sa II); ⲡⲡⲡ had been omitted and was added by the corrector; one reads ⲡϭⲡ and not ⲡⲡⲡⲡⲡϭⲡ; ϭⲡ is present after ⲡⲡⲡⲡⲡⲡⲡⲡⲡ; the manuscript has neither of the two ⲡⲡⲡ, which is also the reading of fa (and probably that of sa 134). It therefore has a fairly independent text for this passage, even though these variations are not of great importance. => ϩⲡ ⲡⲡⲡⲡⲡ probably is the reading of sa II (and fa): it is attested only by sa 112, whereas sa122 and sa 134 are lacunar here, but the size of the lacunas is in favor of ϩⲡ; ⲡⲡⲡ had probably been omitted (added by the corrector); the latter also deleted the first two letters of ⲡⲡϣⲡⲡ and added ⲡⲡ at the beginning of the following line, thus conforming the text to that of sa II. The rest has not been modified.

Verse 10 sa1 /sa 9 : omitted! sa II ⲡⲡⲡ ⲡϣⲡϫⲡ ⲡⲡⲡⲡⲡϩⲡⲡ ⲡ ⲡⲡϥ ϩⲡⲡⲡ ⲡ ϩⲡⲡⲡⲡ ⲡⲡⲡⲡⲡⲡⲡⲡⲡ. ϫⲡ ⲡⲡ ⲡⲡ ⲡⲡⲡⲡⲡⲡ ⲡⲡⲡⲡ ϩⲡ ⲡⲡⲡⲡⲡⲡⲡⲡ sa 123b ⲡⲡⲡ ⲡϣⲡϫⲡ ⲡⲡⲡⲡ ⲡ ϩ ⲡ ⲡ ⲡ ⲡⲡϥ ϩⲡⲡⲡ ⲡ ϩⲡⲡⲡⲡ ⲡⲡⲡ ⲡ ⲡ ⲡⲡⲡⲡⲡ ϫⲡ ⲡⲡ ⲡⲡ ⲡ ⲡ ⲡ ⲡ ⲡ ⲡ ⲡ ⲡ ⲡ ⲡ ϩ ⲡ ⲡ ⲡ ⲡ ⲡⲡⲡⲡⲡ => The initial ⲡⲡⲡ was the one of the following verse, which also starts with ⲡⲡⲡ in sa 1/sa 9; then a line was deleted to insert verse 10 according to the text of sa II (attested in sa 122 and sa 112): it was too important an omission not to be noticed. The beginning of verse 11 was also rewritten by the corrector. Note that the homoioteleuton that probably caused this omission does not exist in any Greek witness, but that D, W, f 1 and f 13 have a situation (different from that of the other uncials) that could have caused the same phenomenon: repetition of οταν εκ νεκρων αναστη (i. e. the loss of a Greek model for this could be assumed).

Verse 11 sa 1 ⲡⲡⲡ ⲡⲡϫⲡⲡⲡϥ ⲡⲡϫⲡ ⲡ ⲡⲡⲡ ϫⲡ ⲡⲡⲡⲡ ⲡⲡ ⲡⲡⲡⲡⲡⲡⲡⲡⲡⲡⲡⲡ ⲡⲡϫⲡ ⲡ ⲡⲡⲡ ϫⲡ ϩⲡⲡⲡⲡⲡⲡ ϥⲡⲡϫⲡⲡ ⲡⲡ ⲡ ϣⲡⲡⲡ sa 9 ⲡⲡⲡ ⲡⲡϫⲡⲡⲡϥ ⲡⲡϫⲡ ⲡⲡⲡⲡ ϫⲡ ⲡⲡⲡⲡ ⲡⲡ ⲡⲡⲡⲡⲡⲡⲡⲡⲡⲡⲡⲡ ϫⲡ ⲡⲡⲡⲡ ϫⲡ ϩⲡⲡⲡⲡⲡ ϥⲡⲡϫⲡⲡⲡ ⲡⲡ ⲡϣⲡⲡⲡ. sa II ⲡⲡϫⲡⲡⲡϥ ⲡⲡ ⲡⲡϫⲡ ⲡⲡⲡⲡ ϫⲡ ⲡⲡⲡ ⲡϩⲡⲡⲡⲡ ϭⲡ ⲡⲡⲡⲡⲡⲡⲡⲡⲡⲡⲡⲡ ϫⲡ ⲡⲡⲡⲡ ϫⲡ ϩⲡⲡⲡⲡⲡ ⲡⲡϫⲡⲡ ⲡⲡ ⲡϣⲡⲡⲡ (text of sa 122; sa 112 omits ⲡⲡⲡ and has ⲡⲡⲡⲡⲡⲡⲡⲡⲡⲡⲡⲡ ⲡⲡϫⲡ instead of ⲡⲡⲡⲡⲡⲡⲡⲡⲡⲡⲡⲡ ϫⲡ) sa 123b ⲡ ⲡ ϫ ⲡ ⲡ ⲡ ϥ ⲡ ⲡ ⲡⲡϫⲡ ⲡ ⲡⲡⲡ ϫ ⲡ ⲡ ⲡ ⲡ ⲡϩⲡ ⲡⲡ? ⲡⲡⲡⲡⲡⲡⲡⲡⲡⲡⲡⲡ ⲡⲡϫⲡ ⲡ ⲡⲡⲡ ϫⲡ ϩⲡⲡⲡⲡⲡ ⲡⲡϫⲡ ⲡ ⲡⲡ => After the insertion of v. 10, the corrector of sa 123 joins the original text of sa 123a at (ϩⲡ)ⲡⲡⲡⲡ, so sa 123a seems to have omitted ⲡϣⲡⲡⲡ; moreover, the corrector deleted only one line to rewrite three, so the original text of v. 11 was shorter than that of the other sa witnesses, but we no longer see any trace of it.

Verse 12 sa 1/sa 9 ⲡⲡⲡϥ ⲡⲡ ⲡⲡϫⲡϥ ⲡⲡⲡ ϫⲡ ϩⲡⲡⲡⲡⲡ ⲡⲡⲡ ⲡϥϣⲡⲡⲡⲡ ϥⲡⲡϫⲡⲡ ϩⲡⲡ ⲡⲡⲡ ⲡⲡⲡⲡ• ⲡⲡⲡ ⲡⲡⲡ ⲡⲡⲡⲡⲡϩⲡⲡ ⲡϫⲡ ⲡϣⲡⲡⲡ ⲡⲡⲡⲡⲡⲡ ϫⲡⲡⲡⲡ ⲡϥⲡϣⲡⲡ ϩⲡϩ ⲡϩⲡⲡⲡ ⲡⲡⲡⲡⲡϣϥ• sa II is not easy to determine because the manuscripts are quite different, but sa122 is the best candidate: sa 134 12 ⲡⲡⲡϥ ⲡⲡ ⲡⲡϫⲡϥ ⲡⲡⲡ ϫⲡ ϩⲡⲡⲡⲡⲡ ⲡⲡⲡ ⲡϥϣ ⲡ ⲡ ⲡ ⲡ ϥⲡⲡϫⲡⲡ ϩⲡⲡ ⲡⲡⲡ ⲡⲡⲡⲡ ⲡⲡⲡ ⲡ ⲡ ⲡ ⲡⲡⲡⲡⲡϩⲡⲡ ϫ ⲡ ⲡϣⲡⲡⲡ ⲡⲡⲡⲡⲡ ⲡ ⲡⲡϣⲡ ϩⲡϩ ⲡϩⲡ ⲡⲡ ⲡⲡ ⲡ ⲡ ⲡϣϥ (This particular reading can be compared with fa for the use of the Future 1: ⲡⲡⲡⲡ ⲡϣⲡⲡⲡ ⲡⲡⲡⲡⲡⲡ ϫⲡ ϥⲡⲡϫⲡ ⲡⲡⲡⲡⲡ ⲡϩⲡⲡⲡ). sa 123b ⲡⲡⲡϥ ⲡⲡ ⲡⲡϫⲡϥ ⲡⲡⲡ ϫⲡ ϩⲡⲡⲡⲡⲡ ⲡⲡⲡ ⲡϥϣⲡⲡⲡⲡ ϥⲡⲡϫⲡⲡ ϩⲡⲡ ⲡⲡⲡ ⲡⲡⲡⲡ ⲡⲡⲡ ⲡⲡⲡ ⲡⲡⲡⲡⲡϩⲡⲡϥ ⲡϫⲡ ⲡϣⲡⲡⲡ ⲡⲡⲡⲡⲡⲡ ϫⲡⲡⲡⲡ ⲡϥⲡϣⲡ ϩⲡϩ ⲡϩⲡⲡⲡ ⲡⲡⲡⲡⲡϣϥ => The text is in conformity with that of sa 1/sa 9, with the exception of ⲡϩⲡⲡϥ, where ϥ was added above the line by the corrector; ⲡⲡⲡⲡⲡⲡ ϫⲡⲡⲡⲡ was rewritten by the corrector on a few erased letters, perhaps the first scribe had written ϫⲡⲡⲡⲡ there and forgotten to write ⲡⲡⲡⲡⲡⲡ.

Verse 13 sa ⲡⲡⲡⲡ ϯϫⲡ ⲡⲡⲡⲡ ⲡⲡⲡⲡ ϫⲡ ⲡⲡⲡϩⲡⲡⲡⲡⲡⲡ ⲡϥⲡⲡ ⲡⲡⲡ ⲡⲡⲡⲡⲡⲡ ⲡⲡϥ ⲡⲡⲡⲡⲡⲡⲡⲡⲡⲡϣⲡⲡ ⲡⲡⲡⲡ ⲡⲡ ⲡⲡⲡⲡⲡϩⲡⲡ ⲡⲡⲡⲡⲡⲡϥ No variant in sa 123.

Results of the analysis

The text of sa 123a is a witness to the old translation (version sa I). However, it is not totally dependent on or in agreement with the manuscript sa1, and its text is already somewhat mixed or contaminated:

-with some exclusive features of sa II (the passage analyzed here does not show any, but there are in 2:7, 2:21, 11:32, 12:14, 12:29, 12:36);

-with apparently very few exclusive features of sa 9 (2:6, 11:30, 12:26, which have to be checked carefully); -with some readings of its own that could betray an influence of Fayyumic; however, it does not seem to have any remarkable affinity with the Leipzig manuscript (sa 92), another witness of sa1 certainly copied in Fayyum, perhaps around the 7th-8th centuries (this point will be specified once the edition of sa 123 is completed).

Therefore, in general, sa 123a can be used in the apparatus as a witness for sa I.

The manuscript sa 123 has subsequently undergone a number of corrections in order to be harmonized with the text of sa II. Therefore, sa 123b can be used, to a certain extent, in the apparatus as a witness of saII. This revision is however not systematic and, at the current stage of our research, not very predictable: what were the criteria chosen by the corrector to intervene? Did he have a witness of sa II at his disposal, and if so, which one? Since we have very few witnesses of sa II from Fayyum, should we assume that this revision was done at the White Monastery? Should we even assume that sa 123 was copied at the White Monastery by a 'fayyumising' scribe?

It seems that the revision work of sa 123 is comparable (but not similar) to the one that led to the manuscript sa 9, where we observe the final result of what is not a systematic harmonization, but rather a compilation work between the two translations sa I and sa II. In addition, sa 9 appears to be slightly earlier than sa 123, but does not appear to have played a role in correcting sa 123a. Instructive in this respect is the case of verse 10: this verse was omitted by sa 1, sa 9 and sa 123a; it was reinstated in sa 123b, in accordance with sa II (which has here two witnesses : sa 112, which also betrays some Fayyumic influences; sa 122, copied in a very classical biblical majuscule); sa 134, often close to sa 9, does not have this verse; the Leipzig papyrus (sa 92) probably did not have it either; but the fa version does, which reduces the scope of its influence on sa 123a.

There are still questions to be addressed. For example, in sa 123, is there a difference in the corrector's interventions according to the chapters? Are there also traces of a revision work in the other gospels of the manuscript, and in what direction do they go? Finally, one can also wonder whether sa 123 was a copy that was corrected in order to become the apograph for further manuscripts. It looks very untidy to be used in liturgy, but as exemplar in the scriptorium, it presented a collated text, ready to be copied further in new manuscripts.

In conclusion, the existence of these parallel processes attested by sa 9 and sa 123 testifies to the interest shown in the text of Mark's gospel at that time. Is it related to an attempt to make the text attributed to the evangelizer of Egypt reliable? Can this attempt be situated in the 9th century, at a time when Christianity in Egypt was beginning to weaken? This is matter for further research.

Insérer Fig. 1 Fig. 1. London BL 3579 B19 f.36R. © British Library

  sa 122 12 ⲡ ⲡⲡϥ ⲡⲡ ⲡⲡϫⲡϥ ⲡⲡⲡ ϫⲡ ϩⲡⲡⲡⲡⲡ ⲡⲡⲡ ⲡϥϣⲡⲡⲡⲡ ⲡ ϣⲡⲡⲡ ϥⲡⲡϫⲡⲡ ϩⲡⲡ ⲡⲡⲡ ⲡⲡⲡⲡ ⲡⲡⲡ ⲡⲡⲡ ⲡ ⲡⲡⲡⲡϩⲡⲡϥ ⲡϫⲡ ⲡϣⲡⲡⲡ ⲡ ⲡⲡⲡⲡⲡ ϫⲡⲡⲡⲡ ⲡϥⲡⲡϣⲡ ϩⲡϩ ⲡ ϩⲡⲡⲡ ⲡⲡⲡ ⲡ ⲡⲡⲡⲡϣϥ • sa 112 12 ⲡ ⲡⲡϥ ⲡⲡ ⲡⲡϫⲡϥ ⲡⲡⲡ ϫⲡ ϩⲡⲡⲡⲡⲡ ⲡⲡⲡ ⲡϥϣⲡⲡⲡⲡ ⲡ ϣⲡⲡⲡ ϥⲡⲡϫⲡⲡ ϩⲡⲡ ⲡⲡⲡ ⲡⲡⲡⲡ ⲡⲡⲡ ⲡⲡⲡ ⲡ ⲡⲡⲡⲡϩⲡⲡⲡⲡϥ ⲡϫⲡⲡ ⲡϣⲡⲡⲡ ⲡ ⲡⲡⲡⲡⲡ ϫⲡⲡⲡⲡ ⲡϥⲡϣⲡⲡ ϩⲡϩ ⲡ ϩⲡⲡⲡ ⲡⲡⲡ ⲡ ⲡⲡⲡⲡϣϥ • sa 249 12 ***]ⲡ ϩⲡⲡ ⲡⲡⲡ [ⲡⲡⲡⲡ] ⲡⲡⲡ ⲡⲡⲡ ⲡ [ⲡⲡⲡⲡⲡϩ]ϥ ⲡϫⲡ ⲡϣⲡ[ⲡⲡ ⲡ ⲡⲡ]ⲡⲡⲡ ϫⲡⲡⲡⲡ [ⲡϥⲡϣⲡ]ⲡ ϩⲡϩ ⲡ ϩⲡⲡⲡ ⲡⲡⲡ [ⲡ ⲡⲡⲡⲡ]ϣϥ • (Text as edited by Satzinger, but ⲡ [ⲡⲡⲡⲡϩⲡⲡ]ϥ and [ⲡϥⲡⲡϣ]ⲡ are equally possible).

For a detailed explanation of the first steps in this inquiry, see Boud'hors, "L'évangile de Marc en coptesahidique", "The Gospel of Mark in Sahidic Coptic", Boud'hors and Torallas Tovar, "Mc 1,1-11 et 1,40-45: La tradition manuscrite copte". The sigla used for the manuscripts and versions are based on the list established by Schmitz & Mink, G., Liste der Koptischen Handschriften (SMR, Münster: http://intf.uni-muenster.de/smr/), and Schüssler, Biblia Coptica.

 2 We have identified fragments of at least five manuscripts: sa 143, sa 162, sa 188 (?), sa 92 (papyrus), sa 99487 (= PM 601); the last two come from Fayyum.

On this monastery and its library, see Depuydt, Catalogue CIII-CXVI.

Cf. Chassinat, "Fragments de manuscrits coptes"; Boud'hors, "Réflexions supplémentaires". These manuscripts come from the White Monastery.

Unpublished. Cf. Munier, Manuscrits coptes 9-10.

The case of sa 134 is somewhat complicated: the attestation of the same passage of Mk 16 by two sheets (one of the BnF [161 4 f. 26h], and the other of the Bodleian [Copt b 11 f. 4]) which had been gathered under the same acronym sa 134 in SMR, makes it clear that they are in fact two rather similar manuscripts. Horner may have been conscious of it, since he had assigned two numbers, 103 and 108, which were grouped under sa 134. K. Schüssler had also perceived the problem, since he isolated the BnF sheet under the acronym sa 0857, grouping all the others under the acronym sa 0587. This is not the place for a complete demonstration, but it is quite clear that, copied by scribes from the same school, these two manuscripts (between which the surviving fragments must now be divided) present different types of texts: the one we will temporarily call "sa134 bis" is apparently a witness to sa II, while sa 134 has a more floating text: many proximities with sa 1 and sa 9, but occasionally also with sa II, as well as some peculiar readings. The fragment we are interested in here belongs to sa 134. As both manuscripts include Oxford fragments, it is plausible that they come from the White Monastery (the rating of the BnF fragments is not entirely discriminatory in this respect, since these fragments are part of the disparate lots purchased by Seymour de Ricci). Yet their materiality clearly links them to the scriptorium in Touton, Fayyum. Were they copied there and sent to the White Monastery, like many other manuscripts during this period? If so, they are additional clues to textual activity on the gospel of Mark, such as sa 9 and sa 123.

Published by Elanskaya, The Literary Coptic manuscripts 431-440.

Edition Boud'hors forthcoming. See Boud'hors, "The Gospel of Mark in Sahidic Coptic".

In sa II ⲡⲡⲡⲡ is probably a harmonization with Lk 9:28. Note that it is this reading that is cited in Shenoute's Canon 8 (manuscript YE [the quotation is omitted in XO], p. 135): seeBoud'hors, Le Canon 8 vol.1, 190, vol. 2, 411 n. 369.


Rahma Chaabouni
email: chaabounirahma@gmail.com

Eugene Kharitonov

Emmanuel Dupoux

Marco Baroni

Melanie Mitchell

Michael S Gazzaniga

Communicating artificial neural networks develop

published or not. The documents may come

W ords partition our world into semantic categories. Con- verging evidence indicates that, while these categories differ widely across languages, they are shaped by universal constraints [START_REF] Goldenweiser | The principle of limited possibilities in the development of culture[END_REF][START_REF] Brown | Human universals, human nature & human culture[END_REF][START_REF] Hauser | The possibility of impossible cultures[END_REF]. In particular, it has been suggested that semantic categorization evolves to support efficient communication [START_REF] Regier | Word meanings across languages support efficient communication[END_REF]. Humans develop naming systems to talk about their experience under two competing pressures: "accuracy maximization" (words should encode precise information about their referents) and "complexity avoidance" (preventing unwieldy languages). At an extreme, a maximally accurate system would have a different term for each perceptual or mental experience. At the other, a maximally simple system would use only one term to refer to all experiences, completely hindering communication.

Actual human naming systems are efficient in the sense that they optimize the accuracy/complexity trade-off. More generally, since the foundational work of Zipf [START_REF] Zipf | Human Behavior and the Principle of Least Effort[END_REF], a similar trade-off between precision and simplicity has been observed in many areas of language [START_REF] Gibson | How efficiency shapes human language[END_REF].

Zaslavsky et al. [START_REF] Zaslavsky | Efficient compression in color naming and its evolution[END_REF] formalized the measurement of namingsystem efficiency within the general information-theoretic framework of the Information Bottleneck (IB) [START_REF] Tishby | The information bottleneck method[END_REF] (see also the closely related rate-distortion theory framework in ref. 9). A system is deemed efficient if it reaches the maximum possible accuracy for a given complexity. In the IB framework, both accuracy and complexity are computed in a communication model where an idealized Speaker aims to communicate a meaning to an idealized Listener. Accuracy is then inversely related to the cost of a misinterpreted meaning, while complexity measures the quantity of information needed to convey the meaning. The IB efficiency of a system is effectively visualized in plots (see Fig. 3).

The black curve in Fig. 3 represents the theoretical limit: no system of a certain complexity (horizontal axis) can have accuracy (vertical axis) above the curve. Hence, according to IB, a system is optimal if it lies on the curve. Equipped with this framework, Zaslavsky et al. [START_REF] Zaslavsky | Efficient compression in color naming and its evolution[END_REF] demonstrated that color-naming systems [START_REF] Regier | Word meanings across languages support efficient communication[END_REF][START_REF] Kay | Color appearance and the emergence and evolution of basic color lexicons[END_REF][START_REF] Berlin | Basic Color Terms: Their Universality and Evolution[END_REF] are notably close to the theoretical limit and hence efficient in a quantifiable way.

IB theory is agnostic about where on the theoretical-limit curve a system should lie. Degenerate systems lying at the extremes of the curve, and expressing each referent with a different term or all referents with a unique term, are also efficient according to this theory. However, such systems are not attested. Instead, real color-naming systems approximate a small range of possible optimal solutions, avoiding the extremes, and in particular highcomplexity trade-offs [START_REF] Zaslavsky | Efficient compression in color naming and its evolution[END_REF]. This avoidance of complexity extremes has been observed more broadly in studies of categorization and naming across many semantic domains [START_REF] Regier | Word meanings across languages support efficient communication[END_REF][START_REF] Rosch | Natural categories[END_REF][START_REF] Rosch | Basic objects in natural categories[END_REF][START_REF] Fan | Pragmatic inference and visual abstraction enable contextual flexibility during visual communication[END_REF].

We study the efficiency of color naming from a different perspective. We compare natural language systems with those emerging from the interaction of modern neural networks (NNs) faced with a color-communication task. Artificial NNs trained with deep-learning methods [START_REF] Lecun | Deep learning[END_REF] have recently been used to study human (neuro)cognition in many fields (e.g., refs. [START_REF] Yamins | Performance-optimized hierarchical models predict neural responses in higher visual cortex[END_REF][START_REF] Kell | A task-optimized neural network replicates human auditory behavior, predicts brain responses, and reveals a cortical processing hierarchy[END_REF][START_REF] Futrell | Neural language models as psycholinguistic subjects: Representations of syntactic state[END_REF][START_REF] Manning | Emergent linguistic structure in artificial neural networks trained by self-supervision[END_REF], including color naming [START_REF] Steinert-Threlkeld | Ease of learning explains semantic universals[END_REF][START_REF] Emil | A reinforcement-learning approach to efficient communication[END_REF]. Traditional simulations in cognitive science are specifically designed to assess how certain factors of interest affect system behavior by developing ad hoc models, an approach illustrated by Baronchelli et al. [START_REF] Baronchelli | Modeling the emergence of universality in color naming patterns[END_REF] and Loreto et al. [START_REF] Loreto | On the origin of the hierarchy of color names[END_REF], in the domain of color naming, and

Significance

Color names in human languages are organized into efficient systems optimizing an accuracy/complexity trade-off. We show that artificial neural networks trained with generic deep-learning methods to play a color-discrimination game develop color-naming systems whose distribution on the accuracy/complexity plane is strikingly similar to that of human languages. We proceed to show that efficiency and narrow complexity crucially depend on the discrete nature of communication, acting as an information bottleneck on the emergent code. This suggests that efficient categorization of colors (and possibly other semantic domains) in natural languages does not depend on specific biological constraints of humans, but it is instead a general property of discrete communication systems. applied by Carr et al. [START_REF] Carr | Simplicity and informativeness in semantic category systems[END_REF] to the study of complexity/accuracy trade-offs in semantic categorization. Deep networks, however, are high-performance general-purpose learners, independently developed for engineering purposes, with no claims of cognitive plausibility concerning their architecture or learning process. In this respect, they might be best seen as complex "animal models" [START_REF] Mccloskey | Networks and theories: The place of connectionism in cognitive science[END_REF][START_REF] Scholte | Fantastic DNimals and where to find them[END_REF]. The main interest lies in whether the emergent behavior of these powerful mechanisms mirrors nontrivial properties of human behavior [START_REF] Cichy | Deep neural networks as scientific models[END_REF]. If it does, we can entertain the intriguing hypothesis that the specific converging human and deep-network patterns we observe have common roots. We can moreover directly intervene on the artificial organisms (more easily so than we can on humans), in order to causally assess how different components affect their emergent behavior.

Specifically, we show that, when two deep learning-trained NNs play a simple color discrimination game, they develop naming systems that closely match the distribution of human languages on the IB plane, showing both efficiency maximization and complexity control (Fig. 3). The use of human-like artificial systems emerges without imposing ad hoc constraints favoring efficiency or limiting complexity on the training procedure. Having observed the systematic emergence of efficiency and complexity reduction in the NN systems, we proceed to test the hypothesis that these properties crucially depend on the bottleneck imposed by the discrete communication channel. Indeed, as we let NNs exchange messages that are increasingly more continuous, their naming systems become more complex, and, eventually, no longer efficient. Varying the degree of color-discrimination granularity required to play the game affects the complexity of the emergent systems, but not efficiency, and only within the range of attested human variation. NN capacity only affects the complexity of the system in function of discreteness of communication.

The emergence of efficient and reasonably simple semantic categorization is not specific to human language but might generally arise in cognitive devices exchanging discrete messages about their world. Discreteness of communication plays a central role in the emergence of efficient and low-complexity naming systems among our artificial agents, raising intriguing questions about the role of discreteness in human language.

Color-Naming Task Stimuli. Following prior work [START_REF] Regier | Word meanings across languages support efficient communication[END_REF][START_REF] Zaslavsky | Efficient compression in color naming and its evolution[END_REF][START_REF] Regier | Color naming reflects optimal partitions of color space[END_REF], we use the World Color Survey (WCS). The WCS contains the names of 330 color chips (Fig. 1) in 110 languages of nonindustrialized societies [START_REF] Kay | The World Color Survey[END_REF]. We represent each color stimulus as a three-dimensional vector in CIELAB space (a color space designed to approximate human vision). In particular, we measure color similarity based on Euclidean distance in CIELAB, as it correlates with human perceptual sensitivity [START_REF] Zaslavsky | Efficient compression in color naming and its evolution[END_REF].

Discrimination Game. We implement a classic discrimination game [START_REF] Lewis | Convention: A Philosophical Study[END_REF] played by 2 NN agents, Speaker and Listener. Speaker receives a target color ct from the palette and sends one word w from its vocabulary V to Listener. Speaker chooses the word from a fixed vocabulary of size |V | = 1,024. As |V | is larger than the number of colors (330), it is always possible, in principle, for Speaker to use a unique word to denote each distinct color. Given w and two distinct colors, ct and a distractor c d , Listener must predict the target. The agents succeed if Listener guesses the correct target (as in Fig. 2).

As in previous work [START_REF] Regier | Word meanings across languages support efficient communication[END_REF][START_REF] Regier | Color naming reflects optimal partitions of color space[END_REF], we assume a uniform prior distribution p(c) over target colors. In SI Appendix, Supporting Information Text, 3. Salience-Weighted Source Distribution, we test an alternative nonuniform prior [START_REF] Gibson | Color naming across languages reflects color use[END_REF], and the results still hold.

The game is implemented in EGG [START_REF] Kharitonov | EGG: A toolkit for research on emergence of language in games[END_REF]. Further details are in Materials and Methods.

Discriminative Need. Despite the presence of universal tendencies [START_REF] Kay | Color appearance and the emergence and evolution of basic color lexicons[END_REF][START_REF] Ratliff | On the psychophysiological bases of universal color terms[END_REF][START_REF] Kay | Resolving the question of color naming universals[END_REF], color-naming variance is also observed [START_REF] Roberson | Color categories are not universal: Replications and new evidence from a stone-age culture[END_REF][START_REF] Roberson | Color categories: Evidence for the cultural relativity hypothesis[END_REF]. Prior studies hypothesized that such variance depends on distinct frequencies of occurrence of colors across communities [START_REF] Gibson | Color naming across languages reflects color use[END_REF][START_REF] Zaslavsky | Color naming reflects both perceptual structure and communicative need[END_REF]. In lack of data capturing these differences, we explore a complementary source of variation, that is easier to model computationally. We hypothesize that different cultures have different discriminative needs. Intuitively, highly industrialized societies might need to distinguish between subtly different color shades characterizing different goods, whereas nonindustrialized societies can rely on coarser distinctions. As indirect evidence, Gibson et al. [START_REF] Gibson | Color naming across languages reflects color use[END_REF] reported that, in the nonindustrialized Tsimané community, color terms are "only used to discriminate between familiar artificial objects." Since in a nonindustrialized community, there is relatively low variety of artificial objects, discrimination need will be low. In English, instead, speakers systematically use color terms to discriminate between objects of all kinds [START_REF] Gibson | Color naming across languages reflects color use[END_REF]. † Concretely, we define discriminative need as the minimum allowed Euclidean distance between targets and distractors in CIELAB space. Agents trained with small minimum targetdistractor distance, distmin, simulate communities with high discriminative need; the ones trained with large distmin represent communities with low discriminative need. We quantify distmin in terms of the nth percentile in the list of pairwise distances between the 330 distinct color chips. For example, with percentile = 50, for a given target color ci , a distractor cj is sampled uniformly among candidate colors such that

dist(ci , cj ) ≥ med ({dist(c k , c l ); k , l ∈ {1..330}, k > l }), [1]
where med is the median function and dist is the Euclidean distance in CIELAB space. Note that larger percentiles correspond to games requiring less granular discrimination. We provide examples in SI Appendix, Supporting Information Text, 1. Example of Nearest Target-Distractors for Different Percentiles. ‡ Speaker Word Distributions. Just like in natural language, we allow fuzzy naming: the same color might be, in different occasions, Fig. 2. A successful round of the discrimination game. A chip c is drawn from a uniform distribution and fed to Speaker. Speaker outputs a probability distribution p(W|c) over its vocabulary of size |V|. Here, a probability is mapped to a color according to a gray gradient (with darker colors representing higher probabilities). A word w is sampled from p(W|c) and fed to Listener. Finally, Listener-given w, the target chip (in position 1 in this illustration), and a distractor chip (in position 2 in this illustration)-assigns a probability to both positions, representing its guess about the position of the target (in this illustration, Listener correctly assigns a higher probability to the target position). ¶ denoted by different words. To estimate the probability distribution P (w |c) associated to a color chip c, we sample 25 words with replacement from Speaker after convergence. § For instance, since Speaker's outputs form a categorical distribution over V , if this distribution is a Dirac, the resulting set of 25 samples will correspond to a unique word. At the other extreme, if Speaker has no confidence about c's category, we might get 25 distinct words equiprobably naming c.

Evaluating the Accuracy/Complexity Trade-Off. To compareNN and human naming systems, we adopt the communication model of Zaslavsky et al. [START_REF] Zaslavsky | Efficient compression in color naming and its evolution[END_REF], keeping the same notation. U represents the set of world's objects, in our case, the set of colors; W represents the set of words; and M represents the set of Speaker's meanings. We assume that a NN Speaker, similarly to what is conjectured for humans (4, 7), internally represents each target color chip c as a Gaussian distribution m ∈ M over U centered at c and defined upon CIELAB color similarity. That is, for a given target chip c, Speaker constructs an internal representation m(c) reflecting its belief about the color chip it wishes to communicate to Listener. The Gaussian m(c), of mean c, is then only parameterized by variance σ 2 , that informs about the Speaker's (un)certainty about its belief. Concretely, an m(c) with low variance, reflecting a certain belief, would only cover c and few neighboring chips according to the CIELAB space (e.g., slightly darker and lighter chips). Similarly to Zaslavsky et al. [START_REF] Zaslavsky | Efficient compression in color naming and its evolution[END_REF], we set σ 2 = 64 for all target chips. Note that M is only introduced to compute the accuracy and complexity measures below, and it plays no direct role in the discrimination game.

In the framework by Zaslavsky et al. [START_REF] Zaslavsky | Efficient compression in color naming and its evolution[END_REF], the complexity of a naming system is quantified by the number of bits of information required for expressing the intended meanings. As shown by Zaslavsky et al. [START_REF] Zaslavsky | Efficient compression in color naming and its evolution[END_REF], this is measured by the mutual information, I (M ; W ) between M and W .

Also following Zaslavsky et al. [START_REF] Zaslavsky | Efficient compression in color naming and its evolution[END_REF], we use I (U ; W ) to measure the accuracy of a naming system. The latter measure is inversely related to the Kullback-Leibler divergence between Speaker and Listener meanings. That is, the better Listener is at reconstructing Speaker's meaning, the larger I (U ; W ) is. § A majority of WCS languages contains names elicited from 25 speakers, leading to comparable a sample size for P(w|c) estimation. ¶ Target and distractor positions are randomly shuffled at each round to prevent Listener from relying on position to succeed at the game.

The theoretically optimal trade-offs between complexity and accuracies are approximated by minimizing the IB objective function:

I (M ; W ) -βI (U ; W ) s.t. β ≥ 1, [2]
where β is the trade-off parameter determining the relative weight a system will attribute to complexity avoidance vs. accuracy maximization. Both complexity and accuracy are quantified by mutual information terms. However, the IB objective minimizes the first term (lowering complexity) and maximizes the second (increasing accuracy; note the minus sign preceding the second term in Eq. 2), two constraints that will be in contrast.

To minimize Eq. 2 for a fixed β, we look for the set {P (wi |cj )}i,j , where j ∈ [1, 330] and i ∈ [1, K ], with K a variable to optimize. To get the theoretical-limit curve shown in Fig. 3, we repeat this procedure for each β, as described in Materials and Methods. # The farther a system is to the theoretical-limit curve, the less efficient it is. To quantify the inefficiency of a system s, characterized as a point on the accuracy/complexity plane (Fig. 3), we introduce the Inef score:

Inef (s) = min β { s -s * β 2 s.t. β ≥ 1}, [3]
where s * β are the coordinates of the optimal naming system (on the theoretical-limit curve) for a fixed β.

Experiments and Results

Human vs. NN Naming Systems. To simulate communities with needs, we run the discrimination game varying minimum target-distractor distance, defined in terms of percentile of nearest distractor (see Discriminative Need). With percentile < 20, target-distractor pairs are too close, and agents fail to converge. Above percentile = 80, there are no distractors sufficiently distant to any given target. We hence played the game with percentile ∈ {20, 30, 40, 50, 60, 70, 80}, resulting in 60 successful games in total. Control experiments are in SI Appendix, Supplementary Information Text, 5. Encouraging the Emergence of a Two-Word System during Training.

Looking at how human and NN naming systems spread along the IB line in Fig. 3, we can make two striking observations. First, NN systems lie near the theoretical IB limit just like human languages do. SI Appendix, Supplementary Information Text, 4. Efficiency: Comparing Human vs. NN Systems, and Actual vs. Rotated Systems shows that NN system inefficiency (Eq. 3) falls within the human range. Second, both human and NN systems lie on a narrow segment of the curve. While this segment does not include the minimal values, it is still clearly tilted toward the low-complexity end of the curve. Note that minimum complexity would be achieved by a system with a single color term. As this makes no sense, we do not expect minimum-complexity systems to emerge. Intriguingly, neither WCS nor NN systems include two-word codes (which are exceedingly rare in natural languages in general) [START_REF] Hardin | Basic color terms and basic color categories[END_REF]. SI Appendix, Supplementary Information Text, 5. Encouraging the Emergence of a Two-Word System during Training shows that, even when we manipulate the game so that agents could achieve perfect discrimination with two words only, they minimally converge to a three-word system. [START_REF] Zaslavsky | Efficient compression in color naming and its evolution[END_REF]. The IB curve (black line) defines the theoretical limit on accuracy given complexity. All color-naming systems achieve near-optimal efficiency.

We have no explanation for why two-word systems are avoided. Still, both WCS and NN systems are clearly coming much closer to the lower end of the complexity scale than to the upper bound.

In sum, standard NNs trained on the discrimination game develop systems that support efficient communication (i.e., are close to the IB curve) while preferring low complexity, similarly to human color-naming systems. Our focus here is on the IB trade-off. However, the way in which NN systems accomplish this trade-off is not radically different from that of human languages. SI Appendix, Supplementary Information Text, 10. Direct Comparison of Color Space Partitions presents a detailed comparison of color partitioning in human and NN naming systems, highlighting partial differences but also important commonalities, in particular, in terms of the convexity of regions corresponding to distinct color names (see also SI Appendix, Fig. S12 for qualitative comparison between both systems).

Effect of discriminative need. Fig. 3 shows the NN systems resulting from exploring the full range of possible percentile values (the parameter controlling discriminative need). While all systems are efficient, we observe some variability in complexity (within the [0.84, 2.8] range), that might be due to different discriminative needs. This is confirmed by Fig. 4, which shows NN naming system complexity in function of percentile. Smaller percentile values (requiring more granular discrimination) make systems more complex. Still, this trend is gradual with no significant pairwise differences, suggesting the need for distant discriminative needs to observe a significant difference in systems' complexity. Furthermore, NN systems' complexity remains within human-range complexity when exploring the full range of percentile values. Interestingly, Fan et al. [START_REF] Fan | Pragmatic inference and visual abstraction enable contextual flexibility during visual communication[END_REF] showed, in the context of visual communication, that humans are also sensitive to discriminative need and adapt the complexity of their communicative system accordingly.

Thus, discriminative need (or related environmental/societal pressures to make more/less granular distinctions) could account for the range of complexity variation we observe in NN and human naming systems (and that might be somewhat underesti-NN systems are tilted toward the top of the human complexity range. This is probably an artifact of WCS' focus on nonindustrialized societies. English, the only industrializedsociety language in Fig. 3, is more complex than any NN language.

mated by the WCS sample). However, alone, it does not explain why the range of observed systems is so narrow.

Preference for low complexity. Both human and NN systems show much lower complexity than what could be found in an optimal naming system by systematically varying the trade-off parameter β ∈ [1, +∞[. * *

The attested systems all occur within a small segment corresponding to β ∈]1, 1.14].

One might conjecture that more complex codes do not evolve simply because the attested ones are sufficiently granular to support all required discriminations. For our NN agents at least, this is not the case, as they systematically fail to achieve 100% success in the discrimination game, which would instead be possible with more complex systems. To illustrate the latter, we generate additional naming systems by partitioning the color space using the "fuzzy c-means" (FCM) soft clustering algorithm [START_REF] Bezdek | Pattern Recognition with Fuzzy Objective Function Algorithms[END_REF], treating cluster labels as color names. We obtain different systems by varying the number-of-clusters hyperparameter. We then play the discrimination game with Speakers and Bayesian Listeners that use p(w |c) distributions derived from the soft clustering solutions.

The FCM-based agents can reach 100% communication success at all percentiles. However, this comes at the cost of higher complexity. Table 1 compares, for each percentile, the 100% successful FCM system with lowest complexity to the NN system with highest success rate. In all cases, NNs came up with systems that are considerably less complex but that also fail to reach perfect discrimination success. † † We conclude that the low complexity of NN systems cannot be explained by lack of sufficient communicative pressure toward more complex solutions. We explore next other possible sources of low-complexitypreference.

Roots of Efficiency and Complexity Avoidance. Building on recent work (41), we explore the idea that the discrete nature of the communication channel acts as bottleneck on the amount of information that the agents are able to transmit, leading them to establish efficient and low-complexity naming systems. Another natural bottleneck could be agents' capacity. Perhaps, the "neural power" of our NNs does not suffice to develop more complex languages. We show next that channel discreteness plays a fundamental role in complexity reduction, whereas NN capacity only matters insofar as it allows the agents to further simplify the code in presence of a discrete channel.

Effect of channel discreteness. We fix percentile = 50 and explore different training regimes ranging from a fully discrete setup to a virtually continuous one, relying on two commonly used methods to train deep networks in language emergence scenarios (e.g., refs. 42 and 43; also see Materials and Methods). The REINFORCE (RF) algorithm uses fully discrete symbol transmission during both training and evaluation. The Gumbel-Softmax (GS) method is fully discrete at evaluation time, but it estimates symbol probabilities through a smooth approximation training. At training time, discrete symbols are approximated by continuous vectors with most of the mass concentrated around a single value. The "peakiness" (and thus discreteness) of this approximation is controlled by the temperature parameter τ . The lower the τ , the peakier the vector (practically converging to a discrete "1-hot" encoding for low τ s). We ** In practice, distinct optimal systems are only obtained for β ∈ [1, 2 13 ], as all optimal systems with β > 2 13 are identical and assign a unique word to each color. † † In the few cases in which FCM systems converged to success rates comparable to those of NN systems, FCM systems were on average less complex, suggesting that the relatively high FCM complexity we observe in Table 1 is not due to an inherent tendency of the latter to converge to high-complexity solutions.

Fig. 4. Complexity distributions of NN systems across different discriminative needs (human distribution included for comparison).

There is a decreasing trend in complexity when increasing percentile (P = 0.004; Kruskal-Wallis). Pairwise differences are not significant when evaluated with Bonferroni-corrected Mann-Whitney-Wilcoxon.

explore τ ∈ {1, 5, 10}, corresponding to increasingly smoother communication channels. Settings with less smooth channels, and in particular fully discrete RF, are harder to train. Hence, we launch 60 runs for each GS setting and 180 for RF. In SI Appendix, Supplementary Information Text, 6. Discreteness and Success Rate, we discuss the relation between channel smoothness and successful convergence, arguing that the high failure rate of more discrete settings is due to a higher complexity-reduction pressure.

Fig. 5A shows that agents trained with RF (thus, in the completely discrete setting) develop significantly less complex systems compared to the ones trained with GS. Within GS, lower τ (more discreteness) leads to simpler codes. With more complexity, smoother systems also become less efficient, an effect that is clear with the highest τ = 10 (Fig. 5B). ‡ ‡ In SI Appendix, Supplementary Information Text, 9. How Are Color-Naming Systems (In)efficient?, we study one concrete way in which these systems are inefficient, comparing them with complex but still efficient NN systems resulting from high discriminative need.

Effect of agent capacity. Only Speaker capacity has a significant impact on complexity and only with a discrete communication channel. Interestingly, larger Speakers further reduce the complexity of the emerging naming system (SI Appendix, Fig. S9). As further discussed in SI Appendix, Supplementary Information Text, 8. Impact of Agent Capacity, a reasonable interpretation for this pattern is that, when the channel is discrete, transmitting information is difficult. Consequently, a "smarter" Speaker will use its extra computational power to come up with an encoding that allows it to transmit even less bits through the channel while maintaining reasonable accuracy. Thus, the agents' capacity experiments further confirm the importance of the discrete-channel bottleneck for complexity minimization.

Discussion

We have shown that NNs trained to play a color discrimination game develop naming systems whose distribution on the accuracy/complexity trade-off plane strikingly resembles that of human languages. We obtained this result using game success as the sole training signal, without imposing any constraint on ‡ ‡ The trend toward higher complexity and lower efficiency continues with larger τ . However, above τ = 10, agents rarely succeed at the game, making the interpretation of results difficult. FCM success rate is always 100%. For FCM, we report minimal complexity among fully successful solutions. For NN, we report complexity and success rate of the system achieving highest success rate. the emergent code, except that it had to consist of single discrete symbols. A very recent study by Kågebäck et al. [START_REF] Emil | A reinforcement-learning approach to efficient communication[END_REF] reports that deep NN agents trained with generic techniques to play a color-naming game strike a similarly human-like complexity/accuracy trade-off, despite important differences between their game and ours (in their setup, the Listener receives only the message as input, and it has to reconstruct the color chip seen by the Speaker), different methods to derive a discrete protocol, and different factors modulating the trade-off (the complexity cline, in their experiments, depends on different amounts of noise added to the communication channel). This constitutes important converging evidence that deep-network communication tends to naturally optimize the accuracy/complexity trade-off, independently of the specifics of the simulations.

We observed, in particular, that the networks developed "lowcomplexity" systems, again in accordance with natural language data. We then looked for the source of this low-complexity pressure in NN systems. Building up on a recent study reporting similar results in artificial tasks (41), we showed that the presence of a discrete communication bottleneck plays a crucial role. As we relax discreteness, the emergent naming systems become complex beyond what is attested in human language, and, eventually, significantly inefficient. § §

In the last few years, much evidence for the efficiency of human languages in general [START_REF] Gibson | How efficiency shapes human language[END_REF] and semantic categorization in particular (4) has been accumulated. Yet, we still lack a full scientific understanding of "why" language is efficient. Our results provide two contributions relevant to this question. First, since efficiency and complexity avoidance also characterize the code evolved by communicating NNs, these factors cannot be explained away by least-effort factors specific to biological agents. Second, the fact that NNs exchange a discrete signal is crucial. Discreteness is a striking, possibly unique characteristic of human language [START_REF] Hockett | The origin of speech[END_REF][START_REF] Studdert-Kennedy | Launching language: The gestural origin of discrete infinity[END_REF], often adduced as a precondition for the combinatorial infinity of expression that characterizes it [START_REF] Chomsky | New Horizons in the Study of Language and Mind[END_REF]. Our finding suggests that it might also be responsible for the efficient nature of semantic categorization (and possibly language at large). We do not have direct evidence on how the language of our ancestors became discrete and on how this affected the structure of categorization. However, our results pave the way for experiments with contemporary humans, exploring how a continuous/discrete transition in communication § § Intriguingly, while K ågeb äck et al. [START_REF] Emil | A reinforcement-learning approach to efficient communication[END_REF] do not explicitly discuss it, their results also point to a correlation between complexity reduction and discreteness, despite the fact that they control discreteness in a different way, that is, by injecting noise into a continuous channel. systems affects the nature of information exchange. With human subjects, we might not have a direct equivalent of the GS temperature parameter. We can however build on a strong tradition of experimental semiotics studies using continuous signals, such as drawings, whistles, and nonconventionalized gestures, and sometimes reporting a tendency to discretize the signals as systematic communication strategies emerge [START_REF] Fan | Pragmatic inference and visual abstraction enable contextual flexibility during visual communication[END_REF]47,[START_REF] Namboodiripad | Measuring conventionalization in the manual modality[END_REF]. By using this framework, we should be able to design experiments that probe a causal relation between discreteness and communicative efficiency, ultimately strengthening our understanding of the roots of efficiency in language.

Chaabouni

Materials and Methods

Human Languages. We used the WCS database (www1.icsi.berkeley.edu/ wcs/). Two languages with extremely sparse information (judgments from 1 speaker only for at least some chips) were removed, resulting in 108 analyzed languages. English, which is not in WCS, was approximated based on the relevant figures from the study by Zaslavsky et al. [START_REF] Zaslavsky | Efficient compression in color naming and its evolution[END_REF].

Agent Architecture and Training. Both agents are feed-forward NNs. Speaker contains 3 hidden layers, each of size 1,000 and with leaky-ReLU (rectified linear unit) activations. For each color, the Speaker's output layer defines a Categorical distribution over its vocabulary V. Listener is modeled as a linear layer of hidden size 5. The impact of agents' capacity on results is discussed in SI Appendix, Supplementary Information Text, 8. Impact of Agent Capacity.

Training NNs to communicate through a discrete channel is nontrivial, as we cannot backpropagate into the Speaker through this bottleneck. We two methods commonly employed in the deep agent language emergence literature: 1) GS relaxation (e.g., ref. [START_REF] Havrylov | Emergence of language with multi-agent games: Learning to communicate with sequences of symbols[END_REF]) and 2) REINFORCE (e.g., ref. 42)) (in both cases, Listeners' gradients are obtained with standard backpropagation). We plug the obtained gradient estimates into Adam [START_REF] Kingma | A method for stochastic optimization[END_REF].

GS. Samples from the GS distribution (50, 51) approximate those from a Categorical distribution through a reparameterization trick, thus enabling gradient-based training. Let us denote σ : R n → R n the standard softmax function. To get a sample that approximates an n-dimensional categorical distribution with probability p, we draw g = [g 1 , . . . , gn], where for each i, g i ∼ Gumbel(0,1) and use it to calculate y such that:

y = σ g + log p τ , [ 4
]
where τ is the temperature hyperparameter. As τ tends to 0, the samples get closer to one-hot, making communication more discrete; as τ → +∞, the samples tend to uniform, resulting in smooth communication

where ia are agent's inputs with a = s if agent is Speaker and a = l if it is Listener. o denotes Listener's prediction, t denotes the ground-truth, and L denotes the cross-entropy loss function; sg refers to the "stopgradient" operation. We use the standard running mean baseline b [START_REF] Greensmith | Variance reduction techniques for gradient estimates in reinforcement learning[END_REF][START_REF] Williams | Simple statistical gradient-following algorithms for connectionist reinforcement learning[END_REF] to reduce estimate variance. To achieve more robust convergence, we also adopt the common trick to add an entropy maximization term (55, 56) on Speaker's words. This could favor higher code complexity, which makes our low-complexity result even more striking. not stated otherwise, results are based on GS training with temperature τ = 1. Training consists in letting the agents play the game until their performance converge (this happens, on average, after about 6 million interactions). For each considered setting, we repeat experiments with 20 different random initializations and only focus the analysis on the successful runs. We consider a run successful if, after convergence, the agents have at least a 95% success rate. Following standard practice, success rates are computed in games in which the most likely word is deterministically sampled from the Speaker distribution.

IB Curve. We use the Agglomerative IB method (57) with β init = 2 13 . At each step of the annealing process, we evaluate the IB solution, i.e., P(w|c) for each (w, c) using Iterative IB [START_REF] Slonim | Communicating artificial neural networks develop efficient color-naming systems[END_REF]. The latter is an iterative method that alternates between evaluating P(w|c) and m(c) (Speaker's meaning for each c) until convergence. We refer readers to Zaslavsky et al. (SI Appendix, section 1.4 in ref. [START_REF] Zaslavsky | Efficient compression in color naming and its evolution[END_REF] for more details about this two-step process. When annealing β according to Agglomerative IB, the IB solution is initialized with the one found with the previous value of β. Optimization ends when β = 1.

Data Availability. The models reported in this paper have been deposited in GitHub (https://github.com/rahmacha/EGG).

Fig. 1 .

 1 Fig. 1. The 330 WCS color chips. Rows correspond to equally spaced lightness values and columns to equally spaced Munsell hues. Each stimulus is at the maximum available saturation for that hue/lightness combination.

  Refer to Zaslavsky et al. (in particular, Bounds on Semantic Efficiency in the main text of ref. 7 and SI Appendix, section S1.3 in ref. 7) for more details about definitions and derivations.

#Fig. 3 .

 3 Fig. 3. Human (blue circles) and NN (orange circles) color-naming systems on the information plane. English (light blue circle) is not in WCS, but it is approximated relying on Zaslavsky et al. (SI Appendix, figure S7 in ref.[START_REF] Zaslavsky | Efficient compression in color naming and its evolution[END_REF]. The IB curve (black line) defines the theoretical limit on accuracy given complexity. All color-naming systems achieve near-optimal efficiency.

Fig. 5 .

 5 Fig. 5. Complexity and inefficiency of NN color-naming systems trained with REINFORCE or GS with different τ s. Pairwise differences evaluated with Bonferroni-corrected Mann-Whitney-Wilcoxon. * P < 0.05; * * P < 0.01; * * * P < 0.001; * * * * P < 0.0001. Differences that are not significant are not marked.

Table 1 . Complexity and success rate (game accuracy after training) of FCM-based and NN systems in function of the game percentile parameter

 1

		min complexity	Complexity	Success rate
	Percentile	FCM	Best NN	Best NN
	20	5.39	2.50	95.45%
	30	4.34	2.28	96.97%
	40	4.01	2.23	95.76%
	50	3.75	2.68	98.79%
	60	3.44	2.17	96.97%
	70	3.39	2.30	97.56%
	80	3.12	2.24	98.78%

  . At training time only, we use the relaxed samples as messages from Speaker, making the entire Speaker/Listener setup differentiable. We look at the impact of τ on Speakers' output distribution in SI Appendix, Supplementary Information Text, 7. Effect of More/Less Discrete Training on Speakers' Output Distribution.

Reinforce

Following Schulman et al.

[START_REF] Schulman | Gradient estimation using stochastic computation graphs[END_REF]

, we sample Speaker's words and estimate its gradients as follows:

E is ,i l E w∼S(is ) [L(o; t) + sg (L(o; t) -b)log P θ (w)],

† Humans can also handle varying contextual discrimination needs by producing longer or shorter phrasal descriptions[START_REF] Monroe Hawkins | Colors in context: A pragmatic neural model for grounded language understanding[END_REF], a strategy we are not modeling here. ‡ SI Appendix, Supplementary Information Text, 2. Random Sampling of Distractors demonstrates that highly efficient systems also emerge when no percentile is imposed, although the latter never reach our threshold for minimum game accuracy (95%). 2 of 7 | PNAS https://doi.org/10.1073/pnas.2016569118 Chaabouni et al. Communicating artificial neural networks develop efficient color-naming systems Downloaded by guest on August 30, 2021

of 7 | PNAS https://doi.org/10.1073/pnas.2016569118 Chaabouni et al. Communicating artificial neural networks develop efficient color-naming systems Downloaded by guest on August 30, 2021

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ACKNOWLEDGMENTS.

We thank Emmanuel Chemla, Thomas Brochhagen, Roger Levy, Louise McNally, Rachid Riad, Noga Zaslavsky, the PNAS reviewers, and, especially, Gemma Boleda and Diane Bouchacourt for feedback. Ted Gibson and Bevil Conway generously shared their data with us. This research was funding by Agence Nationale pour la Recherche Grants ANR-17-EURE-0017 Frontcog, ANR-10-IDEX-0001-02 PSL * , and ANR-19-P3IA-0001 PRAIRIE 3IA.