An open problem in computational stemmatology -  a model for contamination
10.6092/issn.2532-8816/8555
In questo contributo viene preso in esame il problema della contaminazione in computational stemmatology. Il termine contaminazione si riferisce ad un insieme di fenomeni collegati alla variantistica e la presenza di più di un manoscritto durante la copia di un testo. Tale problema è tra I più sentiti in stemmatologia, essendo foriero di un approccio completamente diverso alla ricostruzione della storia dei testi di una tradizione. Maas (1937) afferma non esistere alcun rimedio a tale contaminazione. Pasquali e Pieraccioni (1952) hanno coniato i termini recensione 'aperta' o 'chiusa' per distinguere I fenomeni di contaminazione. In questo articolo presentiamo un modello teorico a grafo per la rappresentazione formale di tradizioni che presentano un grado di contaminazione, mantenendo l'ordine temporale e fornendo una formula per la deduzione di possibili scenari.
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Armin Hoenen
Umanistica Digitale
counting stemmata
An open problem in computational stemmatology -  a model for contamination
Text.Serial.Journal
https://umanisticadigitale.unibo.it/
Articles
2019-05-27
2018-09-26
2019-05-27
2020-05-19
Um. Dig.
Armin Hoenen
2019/05/27
10.6092/issn.2532-8816/8555
Goethe University Frankfurt, Academy of Sciences Göttingen
2532-8816
5
counting stemmata
en
An open problem in computational stemmatology -  a model for contamination
https://umanisticadigitale.unibo.it/article/view/8555/9188
    Umanistica Digitale - ISSN:2532-8816 - n.5, 2019
A. Hoenen – An open problem in computational stemmatology – a model for contamination
DOI: https://doi.org/10.6092/issn.2532-8816/8555


 An open problem in computational stemmatology - a
              model for contamination

                                            Armin Hoenen

                                Goethe-Universität Frankfurt am Main
                                  hoenen@em.uni-frankfurt.de




       Abstract. In this contribution, an open problem in computational stemmatology is being
       considered: contamination. Contamination is used as an umbrella term referring to all
       phenomena of admixture of text variants resulting from scribes considering more than
       one manuscript or even memory when copying a text. This problem is one of the biggest
       to date in stemmatology since it implies an entirely different formal approach to the
       reconstruction of the copy history of a tradition and in turn to the reconstruction of an
       urtext. Mass 36. famously stated that there is no remedy against contamination and
       Pasquali and Pieraccioni 40. coined the terms 'open' vs. 'closed' recensions to distinguish
       contaminated from uncontaminated. We present a graph theoretical model which
       formally accommodates traditions with any degree of contamination while maintaining a
       temporal ordering and give combinatorial numbers and formula on the implication for
       numbers of possible scenarios.
       In questo contributo viene preso in esame il problema della contaminazione in
       computational stemmatology. Il termine contaminazione si riferisce ad un insieme di
       fenomeni collegati alla variantistica e la presenza di più di un manoscritto durante la
       copia di un testo. Tale problema è tra I più sentiti in stemmatologia, essendo foriero di un
       approccio completamente diverso alla ricostruzione della storia dei testi di una tradizione.
       Maas 36. afferma non esistere alcun rimedio a tale contaminazione. Pasquali e Pieraccioni
       40. hanno coniato i termini recensione 'aperta' o 'chiusa' per distinguere I fenomeni di
       contaminazione. In questo articolo presentiamo un modello teorico a grafo per la
       rappresentazione formale di tradizioni che presentano un grado di contaminazione,
       mantenendo l'ordine temporale e fornendo una formula per la deduzione di possibili
       scenari.



       Introduction

      This contribution presents a significantly enhanced and reworked version of an abstract
presented at the AIUCD at Sapienza University, Rome in 2017 27.. The field this contribution
is centered in is stemmatology or more precisely computational and theoretical stemmatology.
Stemmatology itself is the science of reconstructing the genealogy of text versions belonging to


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the same tradition (that is work) based on surviving text versions in order to obtain a text as
close as possible in form to how the authorial original might have looked (compare 45., 9..
Stemmatology is thus a philological sub-discipline primarily concerned with works of ancient
and medieval authors that had been transmitted in hand writing. Graphically and theoretically,
the main framework with which stemmata are being modelled and described is graph theory,
more specifically, nodes (or vertices) symbolize text versions while edges symbolize genealogical
relations, that is copy processes. Before briefly introducing the history of the field with a focus
on computation and graphics in order to subsequently describe the two main target problems
tackled in this paper, the reader may arm herself/himself with an understanding of the some
very recurrent basic stemmatological terms which will be used otherwise unexplained
throughout the rest of the article, for all of them compare also 45.:
      • stemma or stemma codicum: literally a genealogical tree of the codices (45., 190) is a
          reconstuction of the copy history of one and the same work (for instance of all
          versions of Caesar's “De Bello Gallico”). In Figure 1, such a stemma is shown. Note,
          that it contains two different sets of nodes: such which are mapped to an extant text
          and such for which the text versions need to be reconstructed bottom-up so as to
          ultimately arrive at the text of root (should this not be present in one of the extant
          witnesses).
      • tradition (or textual tradition): other than the common usage of the term, in
          stemmatology tradition refers to all versions of a text in which it has been transmitted.
      • textual witness: A (textual) witness is one concrete version (or manifestation) of a text
          of a tradition. For stemmata, witnesses are what a node symbolizes, not manuscripts
          (since those are rather the physical embodiment of the witness and since they can
          carry other texts alongside the witness of the actual tradition).
      • reading: a reading is a “short piece of text” varying between manuscripts (45., 162),
          where one can think of the tradition as an alignment with one column per witness
          and a row holding the aligned corresponding readings.
      • vorlage: Vorlage is a loanword from German which refers to the model of a copy, thus
          if one imagine a scribe copying the text of one manuscript A into a new manuscript
          B, then A is the vorlage of B. The German plural is vorlagen, the English one can be
          vorlages.
      • archetype: an archetype of a tradition is usually the witness that coincides with the
          root of a stemma or a further reconstruction based on it. It needs to be differentiated
          from the original, which is assumed to be lost for most actual cases of application of
          stemmatology. With the original lost, the reconstruction closest to the original is that
          text which can be reconstructed on the basis of all extant versions and their stemma
          (and if one will an additional step of changing obvious fallacies etc.). Often it is graph
          technically the latest common ancestor of all surviving witnesses (sometimes one node
          above). Its text is the archetype and as such, if to be reconstructed, the best-we-can-do
          reconstruction.
      • edition: an edition is essentially a print version of a work. Firstly, version here entails a
          reworking rather than variation through copy edits and errors. Secondly, there is a
          difference between born-printed texts and born-handwritten texts when it comes to


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    A. Hoenen – An open problem in computational stemmatology – a model for contamination

           editions. A print edition of a born-handwritten text can contain for instance a) one
           retyped (transcribed) version of the text from one particular manuscript or b) a
           reconstructed archetypical text or c) multiple versions of the text and others more,
           which born-printed editions do not map to. Especially for c) there are many different
           scholarly edition types that have come into existence and with them different methods
           to derive them, the most famous of which is probably a so-called critical edition often
           featuring a base text to be read with superscript anchors relating to a (often heavy)
           footnote apparatus noting all variants.


         Brief History of Stemmatology and its Assets in Computation

      The beginnings of stemmatology depend on its definition. For the author, the occurrence
of modern stemmatology is bound to a key event: the invention of the printing press. This
ground-breaking invention brought with it a fundamental change to the way (and of course
numbers) in which texts were transmitted. Namely, a printed book features the exact same text
as many times as it was printed. 1 A manually copied work, due to the imperfection and
variation of human concentration and copy skill (but also philological activity) differed in each
exemplar. Furthermore, with years of transmission smaller copy errors, idiosyncratic
translations and small edits (for instance modernizing a text) could accumulate and lead to
quite different versions in different regions. Thus, only with the invention of print, a key
question gradually came into focus: given a variety of versions in different manuscripts, which of
these should be printed? For the author, this question is the birth-place of modern stemmatology.
The discipline investigating variation in textual transmission is much older (although
presumably itself a product of the introduction of writing) and is called textual criticism. One of
the first textual critics could have been Zenodotus of Alexandria 13., 10., 14. who investigated
the differences between the versions of Homeric works already roughly 2300 years ago.
      While prima facie the early printers tended to choose manuscripts in their vicinity 42., 9.,
philologists gradually started discussing ever more the genealogy of manuscripts as a tool to
identify good extant texts and (for instance in case that text is far removed from the prospective
original) a reasonable way of reconstructing bottom-up an archetypical text. In 1737 Bengel
proclaimed “a perfect edition of the New Testament would propose a classification of the
codices for their genealogical relations” (40., 9) and not even one hundred years later, in 1827,
what has been identified as the first stemma in a modern sense by (51., 62) is published for
Swedish law texts by Carl Johan Schlyter, see Figure 1.




1    To be fair, printing blocks wore down, the ink spread a little differently on each page and other factors
     more lead to a certain diversity especially among early prints. However, those can be judged minimal
     in confrontation with average manual transmission especially on the graphemic and lexical levels.


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      The next most important two
events are the summarizing formulation
of a method of deriving a stemma
mechanically from the surviving
witnesses attributed to Karl Lachmann
(see also 33.) and the fundamental
criticism at the method by Joseph
Bédier in 1928 5. who proposed an
editing paradigm which would only
determine a best manuscript by some
reasoning but not construct or draw
graphical equivalents of the copy
history. Bédiers criticism was existential
and had large effects, 2 one of which was
the reaction of Paul Maas in 1937 36.
who claimed that despite not sharing
Bédiers preoccupations3 on the
genealogical method of Lachmann in
general, there was a phenomenon
which           was          irremediable:
contamination.4           In       section
Contamination, this will be explained
in detail. Some years later, Pasquali and
Pieraccioni 40. talk about 'vertical' and Figure 1: First stemma in a modern sense, 1827,
'horizontal' transmission distinguishing identified as such by (Timpanaro 2005, p. 62).
contamination from inheritance. The
phenomenon has a slight implication
that the Lachmannian method of stemmatology 5 be more appropriate for the latter. 8. recently
argued that precisely such factors should be clearly reflected when applying certain digital
methodology.
      Already in the early days of computation, stemmatological tasks have been conducted by
computer programs 16.. While in the beginning the computer was thought to be capable of

2   Today, philology or more precisely editorship of ancient works is to some extent divided between
    Lachmannianists and Bedierists, which are both frequent terms used to refer to the concurrent style of
    editing consistent with one of both methods.
3   In fact, 36. tries to show why Bédier was wrong for the material he had used. Many publications
    outline and discuss the core arguments of this debate, please consider 11., 22., 25., 54., 24., 30. and
    sources therein.
4   A quote from (38., 294) famous among editors is “Gegen die Kontamination ist kein Kraut
    gewachsen” which freely translated means “Against contamination, no herb has grown.” meaning no
    medicine is available against contamination.
5   According to 45. stemmatology or stemmatics in some usage refers exclusively to the genealogical
    method as proposed in its entirety by Lachmann although he is not to be held the sole inventor as he
    built upon previous insights and prinicples from other philologists.


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    A. Hoenen – An open problem in computational stemmatology – a model for contamination

helping only with some more unsubtle subtasks (55., 72), continuously growing hardware
capacities finally led to the complete automation of the stemmatological task. With the 1990ies
an influx of bio-informatic software was noticable and dominated the field until the time of
publication of this article, but genuinely stemmatological algorithms and procedures
                                                              independent of the use of bio-
                                                              informatical     software    have
                                                              appeared lately, for instance 46.,
                                                              44., 28..




                                                                       Transfer     of    software
                                                                 generated for one field to
                                                                 another one is often correlated
                                                                 with a cumbersome process of
                                                                 adaptation and accommodation.
                                                                 As for stemmatology, this shall
                                                                 be exemplified by the properties
                                                                 of the graphical representation of
                                                                 the stemma since it is very
                                                                 closely correlated with the
                                                                 formal model one can apply to
                                                                 it, overtly or covertly. Figure 2
                                                                 shows some classical stemma as
                                                                 they appear in the prefaces of
                                                                 editions. As (2., 509) mention,
                                                                 the preferred graph theoretical
Figure 2: Example of a (typical) philolocigal stemma, here by
                                                                 model used for stemmata in the
(Lundström 1989) reprinted in (Silvas, Anna M. 2013).
                                                                 literature adhering rather to the
                                                                 uncontaminated tradition type is
a Directed Acyclic Graph (DAG) or in more common terms tree. As one can immediately see
from the previous figure, this is not the case for some classical stemmata, where philologists
included various nodes with an indegree > 1 in order to indicate that contamination happened.
Bio-informatically generated stemmata further are marked by a graphical focus on the leafs
(which are the only labelled nodes) and by being unrooted and thus undirected. Hence, they,
too are no DAG, but an unrooted tree. This implies the question what model one can apply to
a stemma satisfying the needs of classical philology without dissecting it into open and closed
or imposing a threshold on how much assumed contamination must be present to keep a
stemmatic hypothesis satisfactory. Since the focus of this discipline is on the most uncorrupted 6
and thereby authentic text version, hence on the root (compare 9.), we can conclude that
temporal order of the tree or directedness and rootedness are utterly important. Yet at the same
time, some way has to be found to maintain the possibility of indegrees >1. The result is not

6    By variation introduced during transmission


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only a graph, but a specialized type of graph which can be modelled in different ways one of
which is the subject of the next sections.



        Contamination

      (45., 47) define contamination as “The confluence of readings from more than one
exemplar”, (50., 15) explains contamination as “The entering into a manuscript of readings
that derive from a source other than its exemplar”. As mentioned in the abstract, the term is an
umbrella term for different phenomena. To begin with, 45. mention three types of
contamination, extra-stemmatic or extra-archetypical (52., 134) contamination refers to
variants in extant manuscripts which come from lost exemplars (or such lost exemplars from
branches between the original and the archetype). (52., 108) estimates that for the average
tradition no more than 27% of the witnesses have presumably come down to us. Extra-
stemmatic contamination must be one of the most prevalent types of contamination given
these large numbers of witness loss (see 3. for a general account of causes of manuscript loss).
The second contamination type is simultaneous contamination where simultaneously a scribe
had two or more vorlagen and chose between variants of them for instance, this was one main
vorlage and a secondary one which she/he consulted in case the main one was hardly readable,
had a lacuna or was damaged. The third type, successive contamination refers to the type of
text mixture where for instance the first half of a text is copied from one vorlage and the second
half from another also known as exemplar shift. However, there are even more types of
contamination, for instance contamination from memorized text. More importantly, medieval
manuscripts may contain some text in the margins or between the lines, which partly note or
discuss alternative variants.


        Towards a universal stemmatic model

      Few explicit models other than DAGs have been outlined for stemmata. 7 None of them
maintains two philological requirements at the same time. Firstly, the temporal order of a
stemma graph, secondly, the requirement of having two types of nodes, extant ones and (at
least partly) reconstructed ones. However, in philological stemmata, we do see that even though
their underlying models are hardly ever formulated mathematically, the existing models and
especially the DAG could only be taken as a basis for a fraction of them. Thus, we try to
generalize the DAG in a way that would allow the new model(s) to be postulated as implicit
model underlying any philological stemmatic depiction. One advantage would be that if this
succeeded, any philological stemma would become more readily accessible as stemma
comparationis for computation.


7   Among them 19. presents so-called “Greg graphs” (the name refers to the philologist W.W. Greg,
    comp. 23.) having some relation to the model of 41.and 18. referring back to 37. had introduced
    Greg trees.


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    A. Hoenen – An open problem in computational stemmatology – a model for contamination

     The main idea of the current proposal is inspired by existing graph types which are more
complex in their definition than simple graph types such as DAGs. They require some
additional elements or constraints. Such graphs are for instance multiple edge class graphs as
defined by 43. or multi layer graphs, as defined in 6..
     The simplest way of modelling what could lie behind the graphics of classical
stemmatology maintains the time-ordered DAG defined through a set {V, E, r} and expands it
by a second set of edges E’ the members of which may not belong to E. Those edges would be
termed contamination edges. The model would be applicable to all philological stemmata
which use two different line types (e.g. normal and dashed) such as the stemma of Lundström
above, see Figure 2.
     Let S be a stemma graph with S={V, E, E', r}
     with V the vertex (or node) set entailing the vertices vi ∈ V and with E a set of pairs
from V: E⊆{{u,v}: u,v ∈ V, u≠v}
     and with E' another set of pairs from V: E'⊆{{u,v}: u,v ∈ V , u ≠ v, {uv , }∉ E}
     and r is a distinguished node root.
     The logic behind S is that for any copy c (which would be labeled vi), we assume there to
be one main vorlage vm and any number of secondary ones v s1, vs2, ... Secondary vorlagen must
be older than c and different from v m, there is no necessary hierarchy between them. From this
follows that an edge {vm , c} ∈ E and for each secondary vorlage an edge {vsi, c} ∈ E' exists.
Empirically it appears conclusive that a scribe most of the cases chose a main exemplar since
switching too often could be impractical. This model can hold well for simultaneous or
successive contamination. For extra-stemmatic and extra-archetypical contamination, in this
model, we would assume additional vertices which have at least one secondary (contaminatory)
edge but no primary (main vorlage) edge. Thus, we would include them into the stemma as
hypothetical nodes (for which only the contamination inducing portions might [have to] be
reconstructed). Consequently, the term extra-stemmatic would loose some of its adequacy, but
in general, since philological stemmata often tend to represent those vertices, rather something
like extra-reconstructional or extra-full-reconstructional than literally extra-stemmatic may
describe the phenomenon. Any other type of contamination could be accommodated as well by
simply typing the edges and allowing multiple types for one edge.


         Getting more Complex – Back to the (Oral) Roots

     Furthermore, in case we deal with an oral origin of a work, 8 traditional stemmata are
complicated by the fact that variation that has sprung from oral transmission is partly different
from written-copy variation compare 34., 20.. While this can be mapped somehow on the edge
type, one of its consequences needs another model-theoretical intervention: roots. The text of
an orally transmitted epic is different every time it is being performed and any first written
manifestation springs from a dictation or writing-down event 34., 39.. As one can imagine,
popular plays such as the Odyssey can have manifested in written multiple times independently

8    Oral origin means that the work has been composed orally that is presumably by a bard. In early
     times, bards could not write, see discussions on Homer for instance in 34..


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and thus all bare a slightly different (oral variation type) text. The work of Zenodotus in
comparing Homeric versions was thus surely somehow different from that of philologists of the
post-print age and the model of transmission he could have had in mind presumably somehow
different from a modern stemma by this necessity. For such a work, basically from the different
first manifestations, copies may have sprung in the usual way, one could thus assume two or
more independent DAGs or stemma graphs, but due to contamination these would mix with
each other ever more, the further down one proceeded in the tree. For consolidating this with
the model, either one could allow multiple roots or one could assume one ultimate oral root
from which then in the second generation all written manifestations (from the oral) sprang.
Even a later back and forth of written and oral transmission as attested or supposed variously,
(see for instance 1.) could be accommodated by such an extension where necessarily some
nodes would be of an oral type. Lastly, the phenomenon of strata in a manuscript as alluded to
for instance by 32. could be taken into account for a general stemmatic model by defining a
node as a sequence of successive states and requiring edges to “dock” onto node states. 9 This is a
kind of maximally general model which could be more fit to express the high complexity of
stemmatology in many respects and yet graphically it would still look strikingly similar to a
simple DAG. Leaving this as narrated theoretical allusion, coming back to the stemma graph S,
we realize that this was consistent only with philological graphical stemmata which have no
more than two linetypes but there are several, which have three.


        A Refined Model for Contamination

      In case two main types of contamination would be distinguished, as is presumably
intended by graphical stemmata through displaying three line types (e.g. normal, dashed and
dotted), things get more complicated. If we deal with extra-stemmatic contamination in the
same way as above and assume that successive and simultaneous contamination are the two
types to be displayed differently, then one could deal with successive contamination by
assuming that there is still one witness contributing the largest part to any copy c. Only in case
of truly the same number of contributions could one allow multiple main edges belonging to E.
This however would be a model theoretical problem in the sense the graph constituted only by
E would no longer be a DAG which could mean loosing some analytical benefits. For that
reason, here we introduce an assumption which relaxes that situation albeit at the cost of
loosing comparability between different traditions through a differently set parameter. The
reasoning why one can presumably always find only one single main vorlage builds on a
structural observation. Witness size can be counted in different ways and different granularities:
by pages, by chapters, by words, by letters. The level(s) on which we count is our additional
parameter. In case more than one witness contribute exactly the same to a copy in a completely
unambiguous case, they would have to contribute the same number of pages, chapters, words
and letters. This however is highly unlikely if there is free independent text in all parts of the

9   Despite being encodable in the standard of the Text Encoding Initiative, theoretical or computational
    (which often follow theoretical) approaches dealing with strata are to date not available to the best
    knowledge of the author.


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  A. Hoenen – An open problem in computational stemmatology – a model for contamination

witness. In other words, given different options for counting the size of the contribution to a
copy, one could theoretically always find quantitative arguments to support intuitive
prioritization of one vorlage if only slightly (by more letters or more words). But this would be,
so my guess, relevant only in few cases anyway.
        If we thus define one vorlage as global main vorlage vgm, then in any successive or previous
part, where it is not the principal vorlage, it can still be used for simultaneous contamination,
just not for both at the same time and likewise all partial main vorlagen vpm cannot be secondary
vorlagen vps for the portion where they are the principal vorlage. Thus, we can define two edge
sets E'sim and E'succ for simultaneous and succesive contamination. For each copy c, we define
possible ancestry to be composed of one main (successive) vorlage vgm, entailing an edge {vgm, c}
∈ E. Additionally there can be edges indicating successive contamination for each vorlage of
another section of c ( vgm can be vorlage to discontinuous sections) {vpm, c} with vpm ≠ vgm, all ∈
E'succ. Finally, we can have edges indicating simultaneous contamination. These are of three
qualities. First, there may be witnesses not serving as a main vorlage to any part of c. Those
simply form edges {vi,c}∈ E'sim .
        Secondly, the global main vorlage was used for simultaneous contamination in any of the
further sections, then we add another edge {vgm, c} to E'sim . And lastly, if any of the other main
vorlagen for the sections were used for simultaneous contamination, this constitutes an edge
{vpm, c} ∈ E'sim. Thus we allow multiple edges from one to another node. And in fact, we do
find philological stemmata or partial stemmata with this number of edge types (see for instance
the stemma in 17., 48). Moreover, the classical stemmata where we find two edge types seem
most prevalent, the ones where a third edge type is present exist, but more than three edge
types, I conjecture to be rare. Not always do two additional edge types map to successive and
simultaneous contamination. Finally, some graphical depictions draw the same edge type for
contamination and main edges pointing to and allowing uncertainty – as outlined above, we
assume that if enough evidence on the text was available one could resolve such uncertainty
and therefore not further operate with so-modelled graphical stemmata. It must likewise be said
that two other types of stemmata which are often given are not touched upon by our
considerations: stemmata variantii (55., 52) for words and partial stemmata.
        There is an important restriction on S, whether we define it with one or two additional
edge sets: E must constitute a DAG, that is no cycles are allowed in E, any vertex vi has an
indegree(Ev  , i)=1 and there is a certain number of nodes with outdegree( Ev, i)=0. The graph is of
course directed. In order to maintain this, we need a second assumption for the model, namely
that ultimately no two nodes are generated at the exact same time, but that even among
descendents of the same main ancestor, the time of completion differs.


       Counting Contamination Scenarios

     The statements that stemmatics is not useful if contamination is present in its most
extreme fashion is one which we can now theoretically and quantitatively assess. In order to do
that, we count and confront the number of possible scenarios of stemmata involving



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     Umanistica Digitale - ISSN:2532-8816 - n.5, 2019

contamination with those which do not and try to align the result as quantitative argument
with the philological discussion.
      If we model a stemma tree as S with one additional edge set, we can ask, for a tree of n
nodes, how many contamination edges are maximally possible. 10 The number of edges for any
tree is (n–1). Thinking of the possible place to insert the first contamination edge of E', we have
apriori n possible sources and (n–1) possible target positions if selfloops are disallowed, which
they are.11 By differentiating source and target, we imply direction. But, younger nodes can just
be contaminated by older ones, which is why we have to divide n (n–1) by 2. This is so
because when taking all n possible sources combining them with each of the ( n–1) possible
targets, then every undirected edge is generated twice, once with ni being the source and nj, i≠
j being target and once the opposite where nj is source and ni target. If we assume a different
point in time of generation for all nodes, even siblings, then only one of the two generated
edges can be a valid one, namely that where source is the node with the earlier point in time of
generation whether this be ni or nj.12 (n(n-1))/2 would be the number of insertible edges. But, we
would still have to subtract the existing (n–1) edges of E arriving at a formula for all places, a
first contamination edge could be inserted as: (n (n–1)/2) – (n–1). Now, for any number z of
contamination edges to be inserted, we seek the formula for all possible contamination
scenarios. We know that wherever of the (n (n–1)/2)–(n–1) possible places we place the first
one, the second one can be placed at exactly one place less so as to not coincide with the first
one. More generally, (i–1) places are already occupied if i is the ordinal index of the actual
contamination edge to be inserted, for the second edge i=2, for the third 3 etc. Thus (n (n–
1)/2) – (n–1) – (i–1) positions are possible at each step. Now the question is how to count all
different contamination scenarios and an obvious assumption would be multiplication. For
each scenario of insertion of the first contamination edge, we can insert a second one at (n (n–
1)/2) – (n–1) – 1 places. But, if we would leave it at that and multiply (n (n–1)/2) – (n–1) –
(i–1)) with (n (n–1)/2) – (n–1) – 1), we would overgenerate since the sequence of
contamination edges is unmeaningful in our context. So while (n (n–1)/2)–(n–1)) gives us all
possible places for a first contamination edge and (n (n–1)/2) – (n–1) – 1) gives us the places
for a second one, when spelling out all scenarios the case that we have chosen any specific
10 For our goal, it is sufficient to consider trees (more precisely rooted labelled trees) in general, not to
   incoporate an explicit distinction between labelled and unlabelled nodes. We know, considering OEIS
   sequences A000169 for rooted (labelled) trees 12. and A005264 of rooted Greg trees (mixed labelled
   and unlabelled) 35., 25., 18. that the numbers of rooted Greg trees are, apart from n=1, always larger
   than for rooted labelled trees. Labels in our case do not refer to only extant witnesses but are necessary
   to model the distinctness of the witness texts associated with a node, be the witness extinct or
   surviving.
11 Yet again, stemmatology reveals its utter complexity for modelling. Theoretically, a manuscript which
   has repeated sections can contain such sections which are copied from one place of the same
   manuscript to another instead of being copied twice from the vorlage. Philologically interesting as it
   allows a comparison of the two near same sections, allowing an assessment of which range of variation
   might be expectable in general, model theoretically one could use self-loops to depict this.
12 Another interesting side effect of this model is that root and the first copy of root can never be
   contaminated.


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  A. Hoenen – An open problem in computational stemmatology – a model for contamination

possible first contamination edge e i and with it a second one e j will essentially be equal to the
scenario where we have first chosen e j and then ei . So in this case we would have to divide the
result by 2 but more generally we would for each sequence e i ,ej ,ek … generate all possible
permutations and thus we have to divide by z! (which for 2 is 2). Then, we can answer the
question for how many contamination scenarios C can we have for a tree with n nodes if we
want to insert exactly z contamination edges by the formula:
       (1) Cz( )=(Πi=1..z (nn  ( –1)/2 – (n-1) – (i–1))/z!
       The maximum number of z is of course, as outlined above nn          ( –1)/2 – (n-1) which implies
exactly one scenario13 – a scenario of extremely “promiscuous witnesses” where any possible
contamination edge is realized –, warranting that Πi=1..(n(n–1)/2 – (n-1)) (n(n–1)/2 – (n-1) – (i–1)) =
z.! If we look more closely, we see that very regularly the exact middle between 1 and (n(n–1)/2
– (n-1) is the place with the maximal number of possible contamination scenarios, which could
be interesting when talking about large amounts of contamination being plausible in philology.
If these amounts would be so large that virtually all manuscripts are contaminated by all
possible contamination sources, then the contamination hypothesis would simply be as
probable as the one without contamination.1415
       Now in order to know how many scenarios are possible for contamination at all, when we
don't know z to confront the full complexity of contaminated versus uncontaminated
traditions, we can sum all contamination scenarios of all possible values of z. Then, the
maximum possible number of differing contamination scenarios Cmax with a tree of n nodes
under the model is:
       (2) Cmax =Σz=1..(n(n–1)/2 – (n-1)) C(z)
       Ultimately, we will also know how many trees including contamination are possible at all
by multiplying with the number of all possible trees for n nodes:16
       (3) n(n–1) * Cmax
       The problem is one based on graphs and many equivalent problems have been identified
already in the literature, for instance the number of possible labeled planar graphs, compare the
OEIS sequence number A288266 and sources therein, for instance 7.. Ultimately, the numbers
we generate are intimately related with Pascal's triangle by reproducing its rows for each n
when incrementing z (but with the restriction that we do not allow z to be 0 and thus loose the
first number). Abstracting towards a more easily expressed but more abstract formula, we see
that Cz ( ) equals C(C(n-1,2),z) and that Cmax equals 2((n(n–1))/2) – 1, compare OEIS sequences
A084546 and A006125 (A126883).

13 Would z surpass the maximum, factors and the whole product will become 0.
14 Of course, complexity would come with pointing out where in the manuscripts contamination would
   have happened, for which many possibilities would exist, which a closed type tradition scenario
   would not have to care for.
15 As (21., 228) blatantly conjectures, oral variation may lead the modern editor to assuming more
   variation than actually was present in written works with oral influence and given the minimal spread
   of literacy in antiquity and the middle ages and the role of memorization for instance in monasteries,
   it seems plausible that many works could carry evidence of some effect of oral transmission.
16 Here, we do not add the number of non contaminated trees (or in other words we do not allow z to
   equal zero) since they are exactly what we have to confront.


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     Umanistica Digitale - ISSN:2532-8816 - n.5, 2019

     Turning to an assessment of the philological debate, we now confront the number of
scenarios for contaminated traditions (according to (3)) with the number of scenarios for
uncontaminated traditions (as n(n–1)) for any tree of size |V|=n up to 10 in a table. The table
shows the factor with which the number for uncontaminated tradition scenarios would have to
be multiplied to equal the one for open tradition scenarios, which equals Cmax, the number of all
possible contamination scenarios per single tree since n(n–1) is a factor in (3).


Tree        No                   Contaminated Tradition                  Factor
size        Contamination
1           1                    1                                       1
2           2                    14                                      7
3           9                    567                                     63
4           64                   65472                                   1023
5           625                  20479375                                32767
6           7776                 16307446176                             2097151
7           117649               31581162845295                          268435455
8           2097152              144115188073758720                      68719476735
9           43046721             1514571848868095273151                  35184372088831
10          1000000000           36028797018963967000000000              36028797018963967
Table 1: Scenarios and factors



      As one can see from the values, the numbers for contamination scenarios are much larger
and in addition grow quicker which in terms of the philological debate would imply that
contamination being present makes a specific stemmatic hypothesis mathematically a much
weaker hypothesis.
      At the same time at very low numbers of overall nodes, contamination implies moderately
more complexity. For n=2, the contaminated traditions are only 7 times as complex as the
uncontaminated ones where at 8 nodes they are already more than roughly 69 billion times as
complex. Looking at the mere numbers, one can also read the table in a relativizing way: a
contaminated tradition of a backbone tree with only 5 nodes is roughly half as complex as an
uncontaminated one with 9. Then, rejecting stemmatics on the basis of complexity of
contamination would be more consistent if one would also reject it for uncontaminated
traditions of certain sizes.
      One could argue that for different traditions, different amounts of contamination may
seem likely and that only contamination below a certain amount could be dealt with. We
tabulate anew this time with the tentative threshold for z = (n–1)/3 rounded up to the next



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    A. Hoenen – An open problem in computational stemmatology – a model for contamination


integer. Thus threshold t gives the maximally allowed number of contamination edges (0 still
excluded).

Tree size Uncontaminated              t   Contaminated Traditions (t)      Factor
1           1                         0 0                                  0
2           2                         0 0                                  0
3           9                         1 9                                  1
4           64                        1 192                                3
5           625                       1 3750                               6
6           7776                      2 427680                             55
7           117649                    2 14117880                           120
8           2097152                   2 484442112                          231
9           43046721                  3 158498026722                       3682
10          1000000000                3 7806000000000                      7806
Table 2: Scenarios and factors


      Such an assumption heavily influences numbers and the added complexity of
contaminated traditions is no more as astonishing as in the unconditioned case though still
large at larger numbers of n.
      In summary, this small and very theoretical counting exercise using a simple model that
could lie behind many a philological stemma graph implies conditionals on a statement like
“contamination precludes stemmatology” if one were to believe in it in this extreme form.
Firstly, it depends on the estimated degree of contamination (as we have seen above where
fewer scenarios were possible both at very mild and very large degrees of contamination),
secondly on the overall size of the tree constituted by E, where smaller trees do not imply as
severe a difference in complexity between contaminated and uncontaminated tradition
scenarios as larger ones. To understand the true meaning of such statements or opinions as
“contamination precludes stemmatics” one would have to investigate philological stemmata in
more detail observing for up to how many witnesses do philologists empirically publish
stemmata and how many nodes do these have. How many of them have 2 and how many 3
edge types. How are they judged by the community. Are factors such as time pressure and
payment important in a decision on whether to edit in a Bedierian or Lachmannian fashion. Is
there any empirical consensus among the community and is the community aware explicitly.
And especially, how large is the amount of contamination that philologists include in their
stemmata. For this purpose a database of stemmata where scholars could input a published
stemma, the tradition, the graph encoded in some way distinguishing contamination and the
publication would be utterly useful.



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     Umanistica Digitale - ISSN:2532-8816 - n.5, 2019

     It is self-evident that contamination complicates stemma building in the first place and
entails a localization of contaminatory processes increasing its complexity so that our
quantitative argument reflects only an aspect of the implication of contamination. Hopefully it
could however be shown nonetheless that unconditioned statements can entail contradictory
implications which should be resolved as much as possible.
     A final word to a computational realization of contamination models. Well, of course,
using a model like the one outlined, a fallback to tree-generating software (such as bio-
informatic one) without some adaptation would hardly be possible. There is a way of automatic
exemplar shift detection 15. and consequently, stemmata for different parts of a work can be
produced, but simultaneous contamination is currently possible to assess automatically only if
one abandons the temporal ordering altogether and generates for instance phylogenetic
networks. However, here no real difference between uncontaminated and contaminated
inheritance can be spotted. The here-outlined model for contamination allows to generate and
maintain a backbone tree constituting E and then add contamination for instance by some
post-processing scenario.
     However the reception of all this, there is still hidden complexity, for instance in
modelling different scribes (hands), redactions, different proportions of lost material in any one
witness and developing even more complex contamination models. Doing this will enable
rational comparisons and a larger throughput and capacity of evaluation of otherwise easily
unreflectedly overread implications of different assumptions for those who are willing and able
to use such tools and to adapt them if necessary.


       Visualisation of contamination via an excursus on multiple
       languages

     A problem of a similar magnitude as contamination and correlated with it is translation.
Translation of a renowned work has some benefits, for instance a certainly less heavy creative
workload. Translation is important in many textual traditions. By using evidence from
manuscripts of the same textual tradition in other languages but also by introducing changes to
the text required by the different grammatical set-up of languages contamination and
translation are interrelated and both produce variation. We will exemplify visualisations which
can be used to display either contamination or translation. In order to show how they can be
applicable to translation as well, we use this phenomenon here although our toy tradition to be
introduced is not contaminated.
     Practically, various philological stemmata, for instance 53. incorporate translation (using
yet again different edge types). If one wanted to use a distance matrix based method to induce a
genealogical hypothesis, translated texts would be a problem in the sense that words could be
no more compared as simple strings to derive some similarity because even if a corresponding
translation of a word was used, the surface form in both languages would differ and a simple
metric such as Hamming asking if two items are the same would need a hidden meaning layer
to operate. Another possibility would be the use of language independent structural elements
such as the chapter sequence as used monolingualy by 49. or the sequence of first appearances
of named entities or other rather structural (for instance narrative) elements.

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  A. Hoenen – An open problem in computational stemmatology – a model for contamination

      In order to know and compare with the true tree, we produce a toy tradition by
introducing a root chapter sequence where each chapter is symbolized by one letter and then
induce copying. Upon each copy slight changes can happen. Both sequences and copy
processes are recorded. First, a randomization decides between one and four copies per
generation and then proceeds for each copy. Translations outnumber again by randomization
monolingual copies twice. Randomly the copy is assigned an ancestor from a previous
generation and the copying starts. We suppose that translation, since it is a freer process than
monolingual copying introduces more deletions, insertions, expansions (simply using two
chapters for where the original sequence has one) and contractions (simply contracting two
adjacent chapters into one new chapter) as well as transpositions on the level of chapters. We
set the probability for all but transposition to 1% impressionistically and for transpositions to
2% in the monolingual copy case whereas we choose twice that probability for the cross-lingual
case. The process ends when the size of the tree has grown to 20 and produces a tree, see Figure
3. Finally, we produce a second flavour of the tradition and loose 4 chapter sequences, among
them root.




       Figure 3: The true tree generated by our chapter sequence copy process. Root is 0-A.
       Letters mark languages.

     This toy tradition is sufficient as a case that reflects data which probably has never but
could be structured in the way it is (possibly through other processes and possibly not as
historical tradition but as Wikipedia article versions or homeworks concerning similar topics
copied from internet precedents etc.). In the following, we want to look at possibilities of
visualisation and analysis for this toy tradition exemplifying what can be done with at least
comparable data in the humanities.




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     Umanistica Digitale - ISSN:2532-8816 - n.5, 2019




       Figure 4: NJ tree for the entire toy tradition, generated with R, package ape.



      First, two methods from the realm of
bio-informatics are being presented. The
Neighbor Joining algorithm 47. generates
an unrooted phylogenetic tree and can be
seen in Figure 4. As one can see, it is
retracing many of the relationships in the
true tree albeit in the typical bifid manner.
The root however is not distinguishable.
The second visualisation, see Figure 5 is a
so-called      NeighbourNet       4..    This
visualisation is applicable to displaying
which sequence (or text) is close to which
other and can thus be used to display
contamination although a genealogical Figure 5: NeighbourNet generated with R package
hypothesis is not immediately derivable. phangorn.
Again, the true relationships are captured
by the general outline of the net. The third
visualisation goes back to Minimum Spanning Trees (MST). MSTs depict only the surviving
nodes and connect them to a tree which spans all nodes of the graph and on which the sum of
edge weights (corresponding to the corresponding distance matrix field) is minimal given all
such possible trees. Now, whenever there is one value in the matrix, which is the same in more
than one field, theoretically more than one equally likely MST can exist 57.. 29. uses an
implementation of method 1 as described there to generate all MSTs which we also use. In one
further step, the algorithm counts all edges in all MSTs and creates a consensus graph with the
edge weights corresponding to the number of MSTs that the edge was present in. Because of a


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  A. Hoenen – An open problem in computational stemmatology – a model for contamination

quick combinatorial increase, already two or three instances of duplicate values in the
underlying distance matrix can quickly lead to larger numbers of MSTs and we found a
number of 1536 MSTs. 56. has proved that all those MSTs must constitute a chain where
neighbours differ only in one edge which is why despite this enormous number, the consensus
MST features only 33 edges in all, where a tree would have 19 (n-1) and the weights show that
the enormous number is truly a result of combination, see Figure 6. This consensus MST could
be used as a way to display contamination visually. In the case of translations it may help us
find out which witnesses belong to the same redactions in case there have been multiple
translations from and into the same language. Again the general typology reflects the true
                                           givens to a certain extent.
                                                 Lastly, an approach similar to the monolingual
                                           approach in 28. produces editable LaTeX code as in
                                           (Hoenen 2016) and can be characterized as iterative
                                           clustering. Finding a way of representing
                                           contamination in this method would come even
                                           closer to philological habit, which is why we shortly
                                           introduce the method here. Instead of a bifid
                                           clustering, starting from the pairwise matrix of
                                           distances, all witnesses cluster which are below a
                                           certain threshold of distance (hence where they are
                                           very similar). There can be incompatible clusters 17
                                           which can be resolved merging them, as for
                                           phylogenetic networks in 31.. The next step is to
                                           apply philological or linguistic knowledge by rules.
 Figure 6: The network of edges present in For each tradition, the user can edit and compose
 1536 MSTs, visualized with gephi. Edges   these rules herself/himself. Here, we only
 by degree of thickness correspond to      simplistically sketch one that is based on our prior
 presence in all (1536), half (second      knowledge of A to be the source language:
thickest), a third or a quarter of the MSTs.
                                                   1. If in a clustered group, there is only one
           manuscript of language A, it will be considered cluster root otherwise a lost A
           exemplar will be hypothesized.
      Extant manuscripts end up either as internodes or cluster root with this procedure. The
next step consists of updating the matrix, which is a step similar to the concurrent steps in NJ
or its predecessors. All clusters/groups are identified, named and a new empty pairwise distance
matrix between them is set up. Every cluster will be represented by its root. Unchanged
pairwise distances (in case either a cluster had only one member or in case root had been a
present witness according to the rules) are then retrieved from the old matrix. The empty fields
are filled by the average of distances of all with all group members of the two clusters. When all
fields of the new distance matrix are filled, another round of clustering is executed. This
happens iteratively until after a terminal clusterstep either there is only one group left uniting

17 This happens whenever one witness A is member of more than one cluster. When A differs from B
   less than the threshold and also from C, but B and C differ more, two such groups would result.


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     Umanistica Digitale - ISSN:2532-8816 - n.5, 2019

all previous roots or when only groups remain, which have one single member. Ultimate root is
again generated according to the rules above. Since by this logic, an idiosyncratic text would get
placed very high up the tree per se, since furthermore, the nodes closest to root in the actual
stemma end up rather removed from the root, the outcome should be yet interpreted as
unrooted (see Figure 7). While the other visualizations were either entirely bifid or produced no
hypothetical nodes, this tree has both multifurcations and hypothetical nodes and offers
manipulation through external rules. However, the accuracy seems inferior to the other
methods, an update step closer to the NJ procedure taking into account different evolution
rates could improve results. Groups such as (4,6,1,8) have come up and nodes close in the true
tree appear mostly close here, too. Contamination could now be introduced in various ways in
the latter procedure, for instance by detecting similarities among the textual versions across
branches of the final outcome and introducing contamination edges where these are the
strongest (until a certain threshold). This however should only be conducted if the results of
the method are more reliable, which has to be further tested and developed. Be it this method
or another to be developed, the technical possibility to produce graphs which are non-bifid and
include hypothetical nodes exists. Furthermore, such a graph can be rooted and enriched by the
display of contamination. All four visualisations display witness relations very differently; two
of them are per se applicable to display contamination, all 4 to display translation.




           Figure 7: Iterative cluster tree on the toy tradition with missing texts 3, 7, 14, 18
           and 0, unrooted.




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  A. Hoenen – An open problem in computational stemmatology – a model for contamination


       Conclusion

      We have dealt with an important problem in stemmatology: contamination. We
formulated explicit models which could underly philological stemmatological depictions and
performed a small combinatorial counting exercise from which we concluded that assuming
undifferentiatedly a too large complexity for the application of stemmatics because (much)
contamination is present, is too simplistic an argument. It misses factors such as the absolute
size of the tradition at hand, which as our table shows makes a small heavily contaminated
tradition not more (or less) complex than a large uncontaminated one. Further, the estimated
amount of contamination is important since at very high rates of contamination, the number
of possible scenarios gets smaller.
      In a second part, by example of translation, we presented visualizations, two of which
are/can already be used as a primer for the display of contamination. All these use distance
matrices. For generating automatically non-bifid stemmatic hypotheses with hypothetical and
extant nodes, a final method has been presented, which needs further development in general
and in particular as to the incorporation of contamination, which is technically possible, but
must be checked if it is feasible.
      NeighbourNets and consensus-MSTs can display manuscript networks. Whereas
NeighbourNets are visualizations which a user must learn to read, consensus-MSTs are simple
graphs with weighted edges. Both these automatically computable visualizations do not entail
an evolutionary perspective and could thus be used side-by-side with an automatically or
manually created “backbone stemma” which displays no contamination in order to (manually)
insert contamination edges (of course evaluating this given the witness texts) into the latter
where the former shows strong relationships not captured by the latter. More sophisticated
models and methods (which often follow models) still await birth for stemmata with
contamination, but by outlining and spelling out some possibilities I hope to have shown that
these are at least not completely implausible.


       References

     1. Amodio, Mark. 2004. Writing the Oral Tradition. Poetics of Orality and Literacy. Notre
        Dame, IN.: University of Notre Dame Press.
        http://www.loc.gov/catdir/toc/ecip052/2004023679.html.
     2. Andrews, Tara L., and Caroline Macé. 2013. “Beyond the Tree of Texts: Building an
        Empirical Model of Scribal Variation through Graph Analysis of Texts and Stemmas.”
        Literary and Linguistic Computing 28 (4): 504–521.
        https://doi.org/10.1093/llc/fqt032.
     3. Baez, Fernando. 2008. A Universal History of the Destruction of Books: From Ancient
        Sumer to Modern Iraq. New York: Atlas & Co.
     4. Bandelt, Hans-Jürgen, and Andreas W. M. Dress. 1992. “Split Decomposition: A
        New and Useful Approach to Phylogenetic Analysis of Distance Data.” Molecular


                                              53
Umanistica Digitale - ISSN:2532-8816 - n.5, 2019

   Phylogenetics and Evolution 1 (3): 242–52.
5. Bédier, Joseph. 1928. “La Tradition Manuscrite Du ‘Lai de l’Ombre’: Réflexions Sur
   l’Art d’Éditer Les Anciens Textes.” Romania 54 (214, 215-216): 161–96, 321–56.
6. Boden, Brigitte, Stephan Günnemann, Holger Hoffmann, and Thomas Seidl. 2012.
   “Mining Coherent Subgraphs in Multi-Layer Graphs with Edge Labels.” In
   Proceedings of the 18th ACM SIGKDD International Conference on Knowledge
   Discovery and Data Mining, 1258–1266. New York: ACM.
7. Bodirsky, Manuel, Clemens Gröpl, and Mihyun Kang. 2007. “Generating Labeled
   Planar Graphs Uniformly at Random.” Theoretical Computer Science 379 (3): 377–86.
8. Buzzoni, Marina, Eugenio Burgio, Martina Modena, and Samuela Simion. 2016.
   “Open versus Closed Recensions (Pasquali): Pros and Cons of Some Methods for
   Computer-Assisted Stemmatology.” Digital Scholarship in the Humanities 31 (3): 652–
   69.
9. Cameron, H. Don. 1987. “The Upside-down Cladogram: Problems in Manuscript
   Affiliation.” In Biological Metaphor and Cladistic Classification: An Interdisciplinary
   Approach, edited by H. M. Hoenigswald and L. F. Wiener, 227–42. Philadelphia,
   University of Pennsylvania Press.
10. Casson, Lionel. 2002. Bibliotheken in Der Antike. Düsseldorf: Artemis & Winkler.
11. Castellani, Arrigo E. 1957. Bédier Avait-Il Raison?: La Méthode de Lachmann Dans Les
    Éditions de Textes Du Moyen Age: Leçon Inaugurale Donnée à l’Université de Fribourg Le
    2 Juin 1954. Fribourg: Éditions universitaires.
12. Cayley, Arthur. 1889. “A Theorem on Trees.” Quarterly Journal of Mathematics 23:
    376–78.
13. Cisne, John L., Robert M. Ziomkowski, and Steven J Schwager. 2010. “Mathematical
    Philology: Entropy Information in Refining Classical Texts’ Reconstruction, and Early
    Philologists’ Anticipation of Information Theory.” PloS One 5 (1).
    https://doi.org/10.1371/journal.pone.0008661
14. Dearing, Vinton A. 1974. Principles and Practice of Textual Analysis. Oakland:
    University of California Press.
15. den Hollander, August. 2004. “How Shock Waves Revealed Successive
    Contamination: A Cardiogram of Early Sixteenth-Century Printed Dutch Bibles.” In
    Studies in Stemmatology II, edited by P. van Reenen, A. den Hollander and M. van
    Mulken, 99–112. Amsterdam: John Benjamins.
16. Ellison, John W. 1957. The Use of Electronic Computers in the Study of the Greek New
    Testament Text. Cambridge, MA: Harvard University.
17. Fazzo, Silvia. 2017. “Lo Stemma Codicum Della Metafisica Di Aristotele.” Revue
    d’Histoire Des Textes 12: 35–58.


                                          54
A. Hoenen – An open problem in computational stemmatology – a model for contamination

  18. Flight, Colin. 1990. “How Many Stemmata?” Manuscripta 34 (2): 122–28.
  19. Flight, Colin. 1992. “Stemmatic Theory and the Analysis of Complicated Traditions.”
      Manuscripta 36 (1): 37–52.
  20. Foley, John M. 2002. How to Read an Oral Poem. Urbana, IL: University of Illinois
      Press. http://www.oraltradition.org/hrop/.
  21. Foley, John M. 2012. Oral Tradition and the Internet: Pathways of the Mind. Urbana,
      IL: University of Illinois Press. http://www.pathwaysproject.org/.
  22. Fourquet, Jean. 1946. “Le Paradoxe de Bédier.” Mélanges 1945 (II): 1–46.
  23. Greg, Walter W. 1927. The Calculus of Variants: An Essay on Textual Criticism. Oxford:
      Clarendon Press.
  24. Haugen, Odd E. 2015. “The Silva Portentosa of Stemmatology Bifurcation in the
      Recension of Old Norse Manuscripts.” Digital Scholarship in the Humanities 31 (3):
      594–610.
  25. Hering, Wolfgang. 1967. “Zweispaltige Stemmata.” Philologus-Zeitschrift Für Antike
      Literatur Und Ihre Rezeption 111 (1–2): 170–85.
  26. Hoenen, Armin. 2016. “Das Erste Dynamische Stemma, Pionier Des Digitalen
      Zeitalters?” In DHd 2016 Konferenzabstracts. http://dhd2016.de/boa.pdf
  27. Hoenen, Armin. 2017. “Beyond the Tree - a Theoretical Model of Contamination
      and a Software to Generate Multilingual Stemmata.” In AIUCD, Book of Abstracts,
      155–59. DOI 10.6092/unibo/amsacta/5885. http://amsacta.unibo.it/5885/#
  28. Hoenen, Armin. 2018a. “From Manuscripts to Archetypes through Iterative
      Clustering.” In Proceedings of the 11th International Conference on Language Resources
      and Evaluation. LREC 2018. Miyazaki (Japan). http://www.lrec-
      conf.org/proceedings/lrec2018/summaries/314.html
  29. Hoenen, Armin. 2018b. “Tools, Evaluation and Preprocessing for Stemmatology.”
      PhD diss., Johann Wolfgang Goethe-Universität, Frankfurt am Main.
  30. Hoenen, Armin, Steffen Eger, and Ralf Gehrke. 2017. “How Many Stemmata with
      Root Degree K?” In Proceedings of the 15th Meeting on the Mathematics of Language,
      11–21. https://www.aclweb.org/anthology/papers/W/W17/W17-3402/
  31. Huson, Daniel H., Regula Rupp, and Celine Scornavacca. 2010. Phylogenetic
      Networks. Cambridge: Cambridge University Press.
  32. Irigoin, Jean. 1954. “Stemmas Bifides et États de Manuscrits.” Revue de Philologie, de
      Littérature et d’Histoire Anciennes 28: 211.
  33. Lachmann, Karl. 1853. In T. Lucretii Cari De Rerum Natura Libros Commentarius:
      Index. Berlin: Georg Reimer.
  34. Lord, Albert B. 1960. The Singer of Tales. Cambridge, MA: Harvard University Press.


                                           55
Umanistica Digitale - ISSN:2532-8816 - n.5, 2019

35. Lundström, Sven. 1989. Die Überlieferung Der Lateinischen Basiliusregel. Uppsala:
    Academia Ubsaliensis; Stockholm (distributor Almqvist & Wiksell International).
36. Maas, Paul. 1937. “Leitfehler Und Stemmatische Typen.” Byzantinische Zeitschrift 37
    (2): 289–94.
37. Maas, Paul. 1958. Textual Criticism. Oxford: Clarendon Press.
38. Maas, Paul. (1927) 1960. Textkritik. 4. Auflage. Leipzig: Teubner.
39. Parry, Millman, and Adam Parry. 1987. The Making of Homeric Verse: The Collected
    Papers of Milman Parry. Oxford: Oxford University Press.
    http://books.google.de/books?id=cbvyswUgSnEC.
40. Pasquali, Giorgio, and Dino Pieraccioni. 1952. Storia della tradizione e critica del testo.
    Firenze: Le Monnier.
41. Quentin, Henri. 1926. Essais de Critique Textuelle:(Ecdotique). Paris: Picard.
42. Reynolds, Leighton D., and Nigel G. Wilson. 2013. Scribes and Scholars, A Guide to
    the Transmission of Greek & Roman Literatures. Oxford: Oxford University Press.
43. Rocklin, Matthew, and Ali Pinar. 2013. “On Clustering on Graphs with Multiple
    Edge Types.” Internet Mathematics 9 (1): 82–112.
44. Roelli, Philipp, and Dieter Bachmann. 2010. “Towards Generating a Stemma of
    Complicated Manuscript Traditions: Petrus Alfonsi’s Dialogus.” Revue d’histoire Des
    Textes 5 (4): 307–21.
45. Roelli, Philipp, and Caroline Macé. 2015. Parvum Lexicon Stemmatologicum. A Brief
    Lexicon of Stemmatology, Helsinki: Helsinki University Homepage; Zurich (distributor
    Zurich Open Repository and Archive). https://www.zora.uzh.ch/id/eprint/121539/
46. Roos, Teemu, and Tuomas Heikkilä. 2009. “Evaluating Methods for Computer-
    Assisted Stemmatology Using Artificial Benchmark Data Sets.” Literary and Linguistic
    Computing 24: 417–33.
47. Saitou, Naruya, and Masatoshi Nei. 1987. “The Neighbor-Joining Method: A New
    Method for Reconstructing Phylogenetic Trees.” Molecular Biology and Evolution 4
    (4): 406–25.
48. Silvas, Anna M. (transl.). 2013. The Rule of St Basil in Latin and English: A Revised
    Critical Edition. Collegeville, MN: Liturgical Press.
49. Spencer, Matthew, Barbara Bordalejo, Li-San Wang, Adrian C. Barbrook, Linne R.
    Mooney, Peter Robinson, Tandy Warnow, and Christopher J. Howe. 2003. “Analyzing
    the Order of Items in Manuscripts of The Canterbury Tales.” Computers and the
    Humanities 37 (1): 97–109. https://doi.org/10.1023/A:1021818600001.
50. Tarrant, Richard J. 2016. Texts, Editors, and Readers: Methods and Problems in Latin
    Textual Criticism. Roman Literature and Its Contexts. Cambridge: Cambridge


                                           56
A. Hoenen – An open problem in computational stemmatology – a model for contamination

      University Press.
  51. Timpanaro, Sebastiano. (1963) 2005. The Genesis of Lachmann’s Method. Edited and
      translated by G. W. Most. Chicago: University of Chicago Press.
  52. Trovato, Paolo. 2014. Everything You Always Wanted to Know about Lachmann’s
      Method, A Non-Standard Handbook of Genealogical Textual Criticism in the Age of Post-
      Structuralism, Cladistics, and Copy-Text. Padova: Libreria Universitaria.
  53. Vancamp, Bruno. 2010. Untersuchungen Zur Handschriftlichen Überlieferung von
      Platons" Menon". Stuttgart: Franz Steiner Verlag.
  54. Weitzman, Michael P. 1987. “The Evolution of Manuscript Traditions.” Journal of the
      Royal Statistical Society. Series A (General) 150 (4): 287–308.
  55. West, Martin L. 1973. Textual Criticism and Editorial Technique: Applicable to Greek
      and Latin Texts. Stuttgart: Teubner.
  56. Wright, Perrin. 1997. “Counting and Constructing Minimal Spanning Trees.”
      Bulletin of the Institute of Combinatorics and Its Applications 21: 65–76.
  57. Yamada, Takeo, Seiji Kataoka, and Kohtaro Watanabe. 2010. “Listing All the
      Minimum Spanning Trees in an Undirected Graph.” International Journal of
      Computer Mathematics 87 (14): 3175–85.




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