Reconsidering Ordered Inventories
Reconsidering Ordered Inventories An information-science approach to correspondence, order and exploratory analysis.
This page and website more broadly are meant as an explainer of the test care of the following independent research project: Ordered Set Correspondence (2026 Preprint) by Hugo Cartwright.
The latest official preprint for the project can be found at the following address:
https://zenodo.org/records/21959747
The project repository is on Gitlab:
https://gitlab.com/hugo.deploys/abjads
Introduction
It’s helpful to know that I’m a software engineer who grew up in the Netherlands and Borneo, and spent 7+ years in the Levant before becoming an amateur epigraphist after getting hooked on ancient inscriptions in the deserts surrounding Damascus. Did I just make all that up? Don't take this sites' word for it, see if you can spot my first or last name anywhere on this site (thanks again, Jürg). Now enough about me.
It may seem surprising but I actually make sure the project has as little epigraphy as possible seep into it to keep it relevant to my software engineer experience. When reading the PDF, seeing many epigraphy or language terms may seem confusing to computer people and that's why I created this website. Hopefully, it will also serve as an introduction to epigraphists and to the public to what I'm working on and is much more than "looking at the alphabet". But looking at "the alphabet" is precisely what we're going start with first.
DISCLAIMER: The rest of this page is still very much under construction
The project test case asks the following question: can we gather data from the way things are ordered, even before we know exactly what those things are or how they are related? I’ll explore this question using ancient alphabets and other writing systems as a case study because it is both intuitive and difficult. The goal of the epigraphy example we will discuss below is not to prove where alphabets came from or whether hypotheses made are 100% true, but to ask a more basic question: What patterns appear when we compare the orders of alphabets?
Why isolate one attribute?
Writing systems have many attributes such as:
- graphical form
- order
- writing direction
- number of signs
- names
- pronunciation
- language
- chronology
- geography
- scribal conventions
- materials written on
- writing tools used
Principle Separate attributes first. Recombine them afterwards. This does not assume that the attributes are independent.
An alphabet has an order, but it also has many other attributes.
We can compare its letters by their graphical form, their names and pronunciations, the language they represent, the direction in which they are written, the number of signs, their chronology, their geographical distribution, and many other properties.
All of these attributes can potentially tell us something — but considering them all at once makes the problem much harder when it comes to ancient or undeciphered scripts.
For example, attributing many attributes to a script if one of the attributes is present.
So I’m going to start with just one attribute: the order of the elements.
This does not mean that order is independent of the other attributes. It is simply a deliberate way of reducing the problem: first explore what order alone can tell us, and then bring the other information back when interpreting the results.
What is an ordered inventory?
An ordered inventory is more than a collection of elements
Alphabet
A B G D …
Ordered inventory
element → position
Key idea: we can compare inventories through correspondence between their elements.
Possible examples:
- alphabets / abecedaries
- syllabaries
- other ordered collections
Let me start with a very simple statement: an alphabet is not just a collection of symbols.
It is also an ordered inventory: a set of elements occupying positions in a particular order.
And that order is what I want to focus on.
Letters are a particularly useful and intuitive example, but the idea is not fundamentally about alphabets. It applies whenever we have a collection of elements whose positions can be compared.
If the elements of two such inventories can be put into correspondence, we can then ask how their orders relate to one another.
So, in this work, alphabets are an example of a more general object: the ordered inventory.
How do we compare orders?
Correspondence tables
| Inventory A | Inventory B | Position |
|---|---|---|
| A | 𐤀 | 1 |
| B | 𐤁 | 2 |
| G | 𐤂 | 3 |
| D | 𐤃 | 4 |
Correspondence → positional comparison → order comparison
The basic tool for doing this is a correspondence table.
Suppose we have two different inventories. Their elements may look completely different, so we first establish which elements we want to treat as corresponding.
For alphabets, one simple way of doing this is to transliterate both into a common representation.
We might therefore turn two different sequences of symbols into something like: A — B — G — D — H — ...
Once we have done that, we can ignore the symbols themselves and simply compare their positions.
The important point is that the correspondence tells us what we are comparing, while the order tells us what we measure.
This lets us deliberately focus on one attribute — order — without first having to decide whether the systems share a language, graphical origin, geographical origin, or historical family.
We can therefore ask a much narrower question: Given these correspondences, how similar are their orders?
Where does language fit?
Script ≠ language ≠ inventory
One language → multiple scripts
One script → multiple languages
Similar transliterations → potentially misleading comparisons
Language as an auxiliary signal
Character-frequency distributions → possible proxy for language identification
So far, I’ve deliberately treated the elements as things that can be put into correspondence without making language the starting point. But when we work with writing systems, language is obviously relevant.
A script is not a language, and a language is not a script. The same language can be written in different scripts, and the same script can be used for different languages.
There is also a practical problem with ancient material: once different inscriptions are transliterated into the same system of letters, things that originally looked quite different can appear much more similar.
So language should not be the main object of the analysis, but it should eventually become another attribute that we can evaluate.
One possibility we are exploring is character frequency: whether the frequency of corresponding characters can provide an additional signal for distinguishing languages.
In this way, we don't need to solve the language question perfectly before comparing orders. Language can become another dimension that we bring back when evaluating the results.
What does the historical data actually look like?
Ancient writing rarely gives us a complete dataset
- fragmentary inscriptions
- small corpora
- uncertain dating
- incomplete sign inventories
- uncertain reading direction
- uncertain original ordering
- reconstructed correspondences
Observed evidence ≠ reconstructed model
And the spoken text: This becomes particularly important when we move to ancient writing.
We rarely have a complete dataset. We may have fragmentary inscriptions, small corpora, uncertain dates, incomplete sign inventories, uncertain reading directions, and sometimes no direct evidence for the original ordering. We may also group scripts in uncertain ways.
There is therefore an important distinction between what is actually preserved and what we reconstruct from it. A modern table of an ancient alphabet is already a model of the evidence.
And that gives us an interesting opportunity: rather than treating a reconstruction as the final answer, we can also use it as data and ask what patterns emerge from it.
When many assumptions become one theory
One model can connect many uncertain attributes
Origin -> Script -> Order -> Language -> Chronology -> Graphical development
Risk
Assumptions in one dimension constrain interpretations in another.
Question
Have we reached the limits of what the current model can safely infer?
The difficulty becomes greater when several uncertain attributes are combined into one historical explanation. A theory might connect origin, script, order, language, chronology and graphical development into a single account. That can be extremely useful but it can also create assumptions that turn into circular interpretation. For example, an assumed chronology might affect which graphical similarities seem possible. Similarly, chronology can be used to reject a correspondence, while the proposed correspondence may itself have influenced how something was dated. Even the terminology we use can introduce assumptions. The word ‘alphabet’, for example, can encourage us to imagine a familiar kind of ordered inventory, with a particular sequence and particular letter names and that this is the only possible arrangment for any alphabet.
I don't think the useful question is simply, "which existing theory is wrong?"
The question is more basic: have we reached the limits of what these models can safely infer from the available evidence?
When multiple uncertain attributes are used to constrain one another, it becomes difficult to know how much evidential independence remains between them.
That is where I think an attribute-by-attribute approach can be useful.
When many assumptions become one theory
A deliberately difficult test case
- Proto-Canaanite
- Phoenician / Paleo-Hebrew
- Ugaritic
- Safaitic and related traditions
- Byblos
- Important unknowns
- Proto-Sinaitic
- incomplete corpus
- uncertain sign inventory
- uncertain original order
- interpretation of glyphs remains debated
- small corpus size
This is where the epigraphic case study comes in.
There are a number of early alphabetic and alphabet-like writing traditions that are potentially relevant here, including Proto-Sinaitic, Proto-Canaanite, Phoenician and Paleo-Hebrew traditions, Ugaritic, Safaitic and related traditions, and the material associated with Byblos. I am particularly interested in this material because it contains exactly the kind of uncertainty that makes exploratory analysis interesting. For example, with Proto-Sinaitic, we don't simply have a complete and ordered ancient inscription of the alphabet sitting in front of us.
There are questions about the corpus, what the complete sign inventory is, and, importantly for this project, the original ordering of the signs.
So when a modern reconstruction presents an apparently coherent alphabetic order, we need to distinguish between what was actually preserved and what has been reconstructed.
These are precisely the kinds of cases where I think it is useful to separate what we can observe about an inventory from the historical story we eventually want to tell about it. There is also a danger of allowing our uncertainty about a writing system to become an assumption about the people who produced it.
Why not just assume the established genealogy?
Byblos Syllabary as a second difficult case
- distinct and poorly understood corpus
- assumed to be a syllabary by specialists but undeciphered
- many graphical comparisons with phoenician are possible
- historical relationship with other scripts remains uncertain
- the underlying language may be pretty well understood if it is a form of Phoenician.
- The kings of Byblos when the Byblos syllabary was in use had Phoenician names.
- most inscriptions unveiled at Byblos, where the phoenician alphabet is supposed to have been developped and spread around the globe in various forms.
- phoenician alphabet palimpsest inscriptions have been found with Byblos syllabary.
Key distinction
Observable correspondence ≠ proven historical descent
The Byblos material provides another useful example. We have a substantial-looking corpus of signs, apparent regularities, and evidence of skilled production, but we don't possess a generally accepted decipherment.
It is another corpus where we can observe signs and relationships between signs without having a completely established interpretation of the system.
And this illustrates an important distinction for the project.
A correspondence between two inventories does not automatically establish historical descent.
If we find that two orders are surprisingly similar, that is a result about order.
It is not automatically a result about origin.
This distinction is really central to what I am trying to do.
Rather than beginning with a historical tree and asking where each inventory belongs on that tree, we can first ask what relationships appear when we compare particular attributes directly.
Transforming the data in ways it might have been
Transforming observed material into comparable ordered inventories
- transliteration / normalization
- correspondence mapping
- reconstructed sequences
- 1D order representations
- 2D layouts where available or reconstructed
- alternative reading directions
And importantly: We are not generating arbitrary grids first.
Instead: We can take existing ordered layouts and ask how they behave under plausible transformations.
So how do we actually turn this kind of material into something that we can compare?
The first step is to put the elements into a common representation. In the case of writing systems, this can involve transliteration, normalization and correspondence mapping.
We can then represent an inventory as an ordered sequence — but potentially also as a two-dimensional layout when the historical material gives us reason to do so.
And this is important because the spatial arrangement of an existing abecedary contains information that is lost if we immediately flatten it into a single line.
Rather than generating arbitrary two-dimensional grids and searching through all of them, we can start from existing layouts and examine plausible transformations of those layouts.
These can include different reading directions, such as left-to-right, right-to-left and boustrophedon-style readings, as well as other transformations that are justified by the writing system being studied.
The goal is therefore not to invent an enormous number of possible alphabets. It is to transform the data we actually have into a controlled set of comparable representations.
How surprising is a result?
The search-space problem
If we test:
- many permutations
- many correspondences
- many layouts
- many reading directions
- many inventories
- many comparisons
then some strong-looking matches are expected by chance.
Therefore:
How surprising is the result given the size of the search space?
Raw similarity ≠ evidential strength
This brings us to one of the biggest methodological issues in the project. It is not enough to find an interesting correspondence.
If we search a huge number of possible correspondences, layouts, permutations and reading directions, then eventually we are going to find things that look surprisingly good.
So the important question is not simply, "How good is this match?"
It is: How surprising is this match given everything we searched?
This is slightly different from simply talking about false discovery rates.
The fundamental issue is the relationship between the result and the size and structure of the search space.
A result that looks extraordinary after searching twenty plausible alternatives is not equivalent to the same result after searching millions of arbitrary possibilities.
So we need to make the search space explicit and, ideally, make it substantially smaller.
Reducing the search space
From arbitrary grids → existing 2D layouts
Instead of:
Generate every possible grid
use:
Existing abecedaries / inscriptions
↓
Test plausible readings:
- LTR
- RTL
- boustrophedon
- vertical directions
- rotations / reflections where justified
- documented unusual reading practices
Principle
Constrain the search using structures already present in the evidence.
This gives us a very practical improvement.
One reason we initially had to generate large numbers of grids was that the historical data are generally not represented as two-dimensional datasets.
An inscription may be presented as a sequence or transcription rather than as a spatial arrangement that we can directly analyse. So we could generate all possible grids.
But that creates exactly the search-space problem I just described.
A better approach is to start with the two-dimensional layouts that actually exist, where they are available, and ask how those layouts can plausibly be read.
That can include left-to-right, right-to-left, boustrophedon and other directions or transformations where there is evidence that they are meaningful.
The unusual ways in which some Libyco-Berber inscriptions have been read are interesting here because they remind us that the reading process itself can have more possibilities than a simple left-to-right assumption.
The principle is therefore: don't invent an enormous space of arbitrary possibilities when the historical evidence can give us constraints.
What comes out of the comparison?
The output is not automatically a family tree
Instead:
Correspondences -> Order similarities -> Clusters / outliers -> Candidate patterns -> Hypotheses for further evaluation
Example: MISS.A 2 Potentially interesting dimensions:
- order
- visual grouping
- character frequency
- Semitic correspondences
This changes what we expect the analysis to produce.
We aren't necessarily trying to produce another family tree.
The output can instead be a collection of candidate correspondences, similarities, clusters, outliers and unexpected relationships.
Those become hypotheses that can then be evaluated using the other dimensions we temporarily set aside.
MISS.A 2 is an interesting example from the current results.
It may be worth considering whether it represents a category or grouping that isn't adequately captured by existing classifications.
Its order is interesting, but so is the apparent visual grouping of its characters.
There may also be an interesting relationship with character-frequency distributions across languages, including potentially several Semitic languages.
But I want to be very clear about the status of this:
these are hypotheses generated by the analysis, not conclusions established by it.
Rebundling the inventories
From one genealogy → multiple attribute-based groupings
Instead of asking:
“Which script descended from which?”
we can ask:
- Which orders correspond?
- Which graphical forms correspond?
- Which inventories have similar frequency profiles?
- Which systems share reading conventions?
- Which relationships survive chronological constraints?
Possible research directions:
- MISS.A 2
- Ugaritic
- South Arabian traditions
- Safaitic
- later Arabic / related graphical developments
- Byblos
This suggests a different way of thinking about classification.
Instead of assuming that every attribute has to produce the same historical tree, we can potentially rebundle the inventories according to individual attributes.
One group might be particularly interesting because of order. Another because of graphical form. Another because of language-frequency characteristics. Another because of writing direction.
MISS.A 2 is interesting in this context because it may sit at an intersection of several of these observations.
That could eventually lead to questions about Ugaritic, South Arabian traditions, Safaitic, and even later graphical developments.
It could also provide new ways of approaching difficult corpora such as the Byblos Syllabary.
But again, the important point is that these are research questions generated by the method.
The method isn't telling us that one script is the ancestor of another.
It is telling us where relationships may be worth investigating.
What is the broader contribution?
An exploratory method for structured uncertainty
Ordered inventories -> Correspondence -> Attribute isolation -> Constrained search -> Search-space-aware evaluation -> Candidate patterns -> Hypotheses -> Reintroduce other attributes
Beyond epigraphy
The case study is an application — not the boundary of the method.
And this is where I want to return to the information-science aspect of the project.
The contribution is not necessarily a new way of determining new relationships between data.
It is a way of exploring structured data when the relationships between the objects are uncertain and when several attributes could potentially explain those relationships.
We establish correspondences, isolate an attribute, constrain the search space, evaluate the resulting patterns in relation to that search space, and then generate hypotheses.
Only after that do we bring any other attribute back into the analysis. In our case study, these are things like chronology, geography, language, graphical form, etc.
In a sense, this is also where the idea of temporarily setting aside time becomes useful. We aren't saying that time doesn't matter.
We're methodology, the treatment of the search space, and how this kind of attribute-based comparison might be applied beyond the particular case study asking what happens if we suspend chronology as the first organizing principle, examine another dimension such as order, and then restore time afterwards as an evaluative constraint.
And because the underlying object is an ordered inventory rather than necessarily an alphabet, the approach could potentially be applied far beyond this particular epigraphic case study.
The method does not eliminate historical assumptions. It makes them explicit inputs that can be varied, compared and tested.
More information
I'd be very interested in your questions, particularly about the methodology, the treatment of the search space, and how this kind of attribute-based comparison might be applied beyond the particular case study I've presented. Feel free to contribute or support the project by going to abjads.org.