Disambiguating Numeral Sequences to Decipher Ancient Accounting Corpora (2023)

Abstract: A numeration system encodes abstract numeric quantities as concrete strings of written characters. The numeration systems used by modern scripts tend to be precise and unambiguous, but this was not so for the ancient and partially-deciphered proto-Elamite (PE) script, where written numerals can have up to four distinct readings depending on the system that is used to read them. We consider the task of disambiguating between these readings in order to determine the values of the numeric quantities recorded in this corpus. We algorithmically extract a list of possible readings for each PE numeral notation, and contribute two disambiguation techniques based on structural properties of the original documents and classifiers learned with the bootstrapping algorithm. We also contribute a test set for evaluating disambiguation techniques, as well as a novel approach to cautious rule selection for bootstrapped classifiers. Our analysis confirms existing intuitions about this script and reveals previously-unknown correlations between tablet content and numeral magnitude. This work is crucial to understanding and deciphering PE, as the corpus is heavily accounting-focused and contains many more numeric tokens than tokens of text.

## introduction proto-elamite (pe) is a partially-deciphered script unearthed at early 3rd millennium bce sites across the iranian plateau. this script was exclusively used to record spreadsheet-style administrative accounts, and well over half of the attested glyphs are known to be digits. despite this abundance of numeric notations, no large-scale quantitative analysis of pe numerals has ever been undertaken, and most prior work has focused on the adjoining text. this is likely because the script employs multiple distinct number systems, which use partially-overlapping sets of digits and occasionally assign distinct values to identical-looking sign shapes (figure 1 ). many numerals can be read according to two or more of these systems, and represent different values depending on the system used. some pe signs have been given tentative readings in prior work, and on the basis of these readings there appears to be a regular relationship between the kind of object recorded and the number system used to count it (in a manner not entirely dissimilar to the measure words found across east asian languages). knowing which system is in use for a given numeral therefore increases the possibility of understanding what category of object is recorded in the adjoining text, and thus opens new avenues for the ongoing decipherment of this script. in this work, we consider the task of disambiguating which systems are used in ambiguous pe numeral notations in order that the values of these numerals may be determined. we describe a simple rule-based technique to extract lists of possible readings from pe numeral notations, which allows us to give the first large-scale survey of pe numerals since friberg 1978 (whose manual analysis occurred at a time when fewer texts were known). we then propose two disambiguation techniques, one based on the subset-sum problem and another which uses a bootstrap classifier @xcite . we describe the construction of a test set for evaluating pe numeral disambiguation models, and propose a novel approach to cautious rule selection which significantly improves the performance of a bootstrap classifier on our data. our analysis shows how these techniques lead to a deeper understanding of this ancient and undeciphered writing system. ## data & background we base our analysis off the transliterated pe corpus hosted by the cdli. 1 each text in this corpus contains an optional header, followed by a series of "entries" written one per line of the transliterated file. each entry contains a (possibly empty) span of text, a comma delimiter, and a (possibly empty) numeral notation. the transliterations use a work-in-progress signlist that reflects experts' current understanding of the texts, but which may not exactly match the true character inventory of the underlying script. in this signlist, characters that are believed to represent text are transliterated with labels beginning in "m" (e.g. m001, after meriggi who pioneered the study of this script), and those representing digits are labeled with "n" (e.g. n01). the notation n(n00) means that the digit n00 is written down n times. figure 2 shows an example of a tablet alongside its transliteration. most proto-elamite numerals are written using one of four 2 number systems, which are called decimal (d), sexagesimal (s), bisexagesimal (b), and capacity (c). in spite of their names, all of these systems use mixed radices. figure 1 shows the relative values of the digits in each of these systems, as derived through prior manual analyses @xcite @xcite . unlike hindu-arabic notation, where the value of a digit depends on its position, the values 1 cuneiform digital library initiative, https://cdli. mpiwg-berlin.mpg.de ; corpus downloaded 3 oct 2022. since n01 occurs as part of every number system, we use this sign as a standard unit and report values as multiples of n01 whenever we convert to hindu-arabic notation. the s, d, and b systems are understood to represent unitless cardinal numbers, while c records unitful measures of volume. when it is necessary to emphasize that these systems are not commensurable, we add a superscript to denote the system in use: e.g. 12 × n01 c is a measure of volume which is not equivalent to 12 × n01 d , despite having identical magnitude. we summarize the above points with an illustrative example. the following notation (read from right-to-left) would be transliterated as 1(n45) 2(n14) 7(n01). this numeral must use either the s or c system, as the large circle n45 only occurs in these systems (figure 1 ). using the readings from the s system, this notation encodes a value of 3627 × @xmath0, it instead encodes of these, we know that the s reading must be the correct one, since 7(n01) should never occur in the c system (every 6 n01 would be bundled into an n14, so the actual notation for 79 × n01 c would be 1(n45) 3(n14) 1(n01)). it is not clear whether these notations would have been considered ambiguous at the time they were written. it is very plausible that the original scribes would have been able to infer the correct reading for a numeral based on contextual information which is not salient to the modern reader (for example, by knowing that certain items are consistently counted with a particular number system, a possibility discussed by @xcite . there is equally the possibility that these notations were ambiguous even to their authors, but that this ambiguity did not interfere with their intended use, for example if the pe texts were intended for shortterm use when the scribes would still recall which system they intended at the time of writing. ## analysis in this section, we investigate how the features from our bootstrap classifier relate to known or hypothesized properties of the script. a number of signs have been suggested as indicating types of livestock (sheep, goats, etc.); these include m346, m362, m367, and m417 (dahl, 5 here we assume that the absolute value of n01 is the same across all three systems, and not just its value relative to the other digits. current understandings suggest that this is the case, but the undeciphered nature of the script means this is technically not certain. 2005). in our analysis, all of these signs also predict the decimal system, which suggests that they were typically used to count flock sizes. m376 has been suggested as either a "high-status human" (dahl, 2005, p. 95) or livestock @xcite ; it is strongly predicative of the sexigesimal system in our analysis. other objects associated with this system include m056∼f (a plow), m219, m371 and m296 (very speculatively, syllables used to write personal names), m059, m145, and m365 ("owners", possibly persons or institutions to whom these accounts belonged), and m269∼c (a vessel?). this may cast some doubt on the livestock reading for m376, in favor of the high-status human reading which is a more natural fit among the owner signs and possible personal names in this collection. we note one text, p008212, which may exhibit a consistent ratio related to m376. p008212 alternates between entries ending in m288 and entries ending in m376 or m367∼i. the magnitude of the numeral in an m288 entry is always exactly 4 times as large as the magnitude of an adjacent m376 entry, or 2 times as large as an adjacent m367∼i entry. this pattern is very unlikely to be due to chance, as it holds across 44 total entries. whatever the meaning of m376, on the basis of this text we can assert that it is associated with amounts of m288 that are exactly twice as large as those associated with m367∼i. to our knowledge this ratio has not yet been noted in previous publications. after disambiguating every numeral in the corpus to the most likely system according to our bootstrap classifier, we measure the average magnitude of counts associated with each feature. we observe that certain features accompany significantly higher or lower counts than others. entries ending in m288 have the largest capacity magnitudes on average, while those ending in m263 are among the smallest. both signs have been speculated to represent containers; from our results one might further speculate that m263 is a container of smaller dimension, or one that was never dealt with in bulk quantities. entries ending in m297∼b also accompany unusually large capacity measures, though the visually-similar m297 does not stand out as unusually large or small. this may point towards these signs having distinct meanings or uses despite sharing a visual resemblance. the numerical systems of proto-elamite have been proposed to have functional uses relating to cultural practices in 3rd millennium south-western iran. for instance, the capacity system (c) is suggested to be used for counting rations disbursed to households or workers @xcite . among the recipients of these rations are m388 and m124, parallel "worker categories" which may represent the heads of work teams. we find that entries beginning in m388 accompany significantly larger capacity measures on average than those in m124, which may point towards m388 individuals heading teams of larger sizes or comprising workers of higher status. while introducing a benchmark to test lm numeracy, @xcite note that numeral representations can vary across scripts; however, they assume a setting where the conversion to hindu-arabic notation is straightforward, and do not discuss ambiguities which may result from this conversion. one approach to handling numeric values in word problems is to replace them with variables v 1 , v 2 , ..., generate the solution as an equation in terms of these variables, and substitute the original values back to obtain a concrete solution. @xcite note that the choice of equation can sometimes depend on whether the original quantities were absolute values or percentages, and therefore this replacement can introduce ambiguities which make some problems unsolvable. they introduce a magnitude-aware encoding for digit sequences, and describe a numerical properties prediction mechanism which estimates whether a numeral is an integer, fraction, percentage, etc. this mirrors our attempt to predict an underlying number system. berg-kirkpatrick and spokoyny (2020) investigate the task of predicting a numeric value given surrounding text as context. they find that models which implicitly separate a value's mantissa from its exponent achieve better results than those which predict the value directly, and that context from large pretrained text encoders is useful even when the pretraining task was not focused on promoting numeracy. as we are dealing with an undeciphered corpus, our models are unfortunately unable to rely on pretrained embeddings for context. @xcite is responsible for early analyses of proto-elamite and proto-cuneiform (a related script which is also partially deciphered) which helped to establish the relative values of the digits in these corpora. @xcite perform what is possibly the earliest computer-assisted analysis of bookkeeping practices in proto-cuneiform, while @xcite and @xcite discuss accounting practices in proto-cuneiform and proto-elamite and the relationships between the two. ## conclusion we have automated the process of extracting candidate readings for ancient proto-elamite numeral notations, and have described ambiguities in the original script which make this extraction challenging. we present two approaches for disambiguating these ambiguous notations: one exploits a common structural property of proto-elamite accounts to look for unambiguous summations, and the other exploits the few unambiguous notations to train a bootstrap classifier. we create a test set for the disambiguation task by manually disambiguating a subset of the corpus, and we describe a novel variant of cautious rule selection which significantly improves bootstrapping performance on this test set. as a result of this work, we are able to assign confident values to a majority of the numeral notations in proto-elamite, to identify and correct a number of transliteration errors in the proto-elamite corpus, and to shed new light on existing hypotheses about the meanings of some signs in this partially-deciphered script. as the proto-elamite script was fundamentally an accounting technology, we believe that this work represents a crucial step towards deepening our understanding of this ancient corpus.