Michael Gasser 11institutetext: Indiana University, Bloomington IN USA,
11email<EMAIL_ADDRESS>
WWW home page: http://homes.soic.indiana.edu/gasser/

# Minimal Dependency Translation:
a Framework for Computer-Assisted Translation
for Under-Resourced Languages

Michael Gasser

###### Abstract

This paper introduces Minimal Dependency Translation (MDT), an ongoing project
to develop a rule-based framework for the creation of rudimentary bilingual
lexicon-grammars for machine translation and computer-assisted translation
into and out of under-resourced languages as well as initial steps towards an
implementation of MDT for English-to-Amharic translation. The basic units in
MDT, called groups, are headed multi-item sequences. In addition to wordforms,
groups may contain lexemes, syntactic-semantic categories, and grammatical
features. Each group is associated with one or more translations, each of
which is a group in a target language. During translation, constraint
satisfaction is used to select a set of source-language groups for the input
sentence and to sequence the words in the associated target-language groups.

##  1 Introduction

For the majority of the world's languages we lack adequate resources to make
use of the machine learning techniques that have become the standard for
modern computational linguistics. For machine translation (MT) and computer-
assisted translation (CAT), the lack is even more serious because what is
required for machine learning is sentence-aligned translations, which are even
less common than monolingual corpora. However, linguistic descriptions and
sizable communities of native speakers do exist for many under-resourced
languages, including Asian languages such as Telugu and Burmese, African
languages such as Amharic and Hausa, and indigenous American languages such as
Quechua and Guarani. There is thus a need for frameworks that facilitate the
rapid creation of computational grammars and lexica by people and their
automatic extension through the limited corpora that are available.

The lack of computational linguistic resources for a language usually
correlates with a lack of written material in the language, an even more
serious disadvantage for the community of speakers. The gap in available
material is easily seen on Wikipedia, where the Amharic edition currently has
13,767 articles and the Hausa edition 1,504 articles. Compare these numbers
with the editions for more privileged languages with comparable numbers of
speakers: 1,909,454 articles for Dutch and 1,237,519 for Polish.wikipedia  One
way to alleviate this gap between the more privileged and less privileged
languages is to accelerate the translation of documents into the under-
resourced languages.

For this reason, this project focuses on MT, and especially CAT, into
languages like Amharic. The long-term goal is a system that allows users with
little or no linguistic experience to write bilingual lexicon-grammars for
low-resource languages that can also be updated on the basis of corpora, when
these are available, and that can be easily integrated into a CAT system,
where they are also updated on the basis of feedback from users.

This paper describes Minimal Dependency Translation (MDT), a lexical-
grammatical framework for MT and CAT. The core of MDT is a lexicon of phrasal
units called groups. A group's entry specifies translations to groups in one
or more other languages. Our focus to date has been on the language pairs
Spanish-Guarani and English-Amharic. Examples and implementation details
discussed in this paper are from the English-Amharic system, called Mit'mit'a.

##  2 Lexica and grammars

###  2.1 Phrasal lexica

The idea of treating phrases rather than individual words as the basic units
of a language goes back at least to the proposal of a Phrasal Lexicon by
Becker becker . In recent years, the idea has gained currency within the
related frameworks of Construction Grammar steels  and Frame Semantics
fillmoreFS  as well as in phrase-based statistical machine translation
(PBSMT). Arguments in favor of phrasal units are often framed in terms of the
ubiquity of idiomaticity, that is, departure from strict compositionality.
Seen another way, phrasal units address the ubiquity of lexical ambiguity. If
a verb's interpretation depends on its object or subject, then it may make
more sense to treat the combination of the verb and particular objects or
subjects as units in their own right.

Arguments based on idiomaticity and ambiguity are semantic, but they extend
naturally to translation. If the meaning of a source-language phrase fails to
be the strict combination of the meanings of the words in the phrase, then it
is unlikely that the translation of the phrase will be the combination of the
translations of the source-language words. Adding lexical context to an
ambiguous word may permit an MT system to select the appropriate translation.

###  2.2 A simple phrasal lexicon

The basic lexical entries of MDT are multi-word units called groups. Each
group represents a catena osborneetal12 . Catenae go beyond constituents,
including all combinations of elements that are continuous in the vertical
dimension within a dependency tree. For example, in the sentence I gave her a
piece of my mind, {I,gave} and {gave,her,piece} are catenae but not
constituents of the sentence.

A catena has a head, and each MDT group must also have a head, which indexes
the group within the lexicon. The other elements in the group are dependents
of the head, but the group has no further structure; it is thus a minimal
dependency structure. A group's entry also specifies translations to groups in
one or more other languages. For each translation, the group's entry gives an
alignment, representing inter-group correspondences between elements, as in
the phrase tables of PBSMT. Entry 1 shows a simple group entry of this sort.
The English group $<$one way or the other$>$ with head way111In the figures,
heads are enclosed in brackets. has as its Amharic translation the group
$<$bazihm hona baziyA$>$ (bEzzih1m honE bEzziya), which has its own entry in
the Amharic lexicon. In the alignment, three of the words in the English group
are associated with positions in the Amharic group; the others (indicated with
``0'') correspond to no word in the Amharic group.

$<$one [way] or the other$>$

  * $\rightarrow$amh bazihm hona baziyA

    * align:[0,1,2,0,3]

Entry 1 Group entry for one way or the other and its Amharic translation

###  2.3 The lexicon-grammar tradeoff

A rudimentary lexicon with entries of this sort is simple in two senses: given
an appropriate interface, a user with no formal knowledge of linguistics can
add entries in a straightforward manner, and the resulting entries are easily
understood. Such a lexicon permits the translation of sentences that are
combinations of the wordforms in the group entries, as long as group order is
preserved across the languages and there are no constraints between groups
that would affect the form of the target-language words. However, such a
lexicon permits no generalization to combinations of wordforms that are not
explicit in the lexicon. It would require a group entry for every reasonably
possible combination of wordforms.

At the other extreme from this simple lexicon is a full-blown grammar that is
driven by the traditional linguistic concern with parsimony: every possible
generalization must be ``captured''. Although such a grammar has the advantage
of compactness and of reflecting general principles of linguistic structure,
it is difficult to write, to debug, and to understand, requiring significant
knowledge of linguistics.

In the MDT project, the goal is a range of possibilities along the continuum
from purely lexical (and phrasal) to syntactic/grammatical, with the emphasis
on ease of entry creation and interpretation.

###  2.4 Lexemes

We can achieve significant generalization over simple groups consisting of
wordforms by permitting lexemes in groups. As an example, consider the English
group $<$lose_v hope$>$, where lose_v is the verb lexeme lose. In order to
make such a group usable, the system requires knowledge of verb morphology,
either in the form of a morphological analyzer or a list of wordforms
associated with each lexeme in the lexicon. For example, the system needs to
be able to recognize that loses is the third person singular present tense
form of the lexeme lose_v. MDT assumes such a resource for the source language
and in addition a part-of-speech tagger to reduce the syntactic and
morphological ambiguity that can result when words are analyzed in isolation.

Because a source-language lexeme will normally be translated as a lexeme
rather than a wordform, the system also requires knowledge of target-language
morphology, specifically either a morphological generator or a list of
wordform associated with each combination of lexeme and grammatical features.
For example, the system needs to know that for the Amharic verb .tafA
(t’Effa)222For simplification, Amharic verb lexemes are given in their usual
citation form, the third person singular masculine perfect form. In fact they
are represented internally in terms of their abstract roots, the sequence of
consonants that characterize words in Semitic languages., one translation of
lose, the forms corresponding to loses are y.tafAl (y1t’Efal) and t.tafAla^c
(t1t’EfalEčč), for masculine and feminine respectively. Entry 2 shows the
entry for the expression lose hope and its Amharic translation tasfA qora.ta
(tEsfa k’orrEt’E), literally `cut hope'.

$<$[lose_v] hope$>$

  * $\rightarrow$amh tasfA qora.ta

    * align:[2,1] ; agr:[([2,1],(tam:tam,sb:sb))]

Entry 2 Group entry for lose hope and its Amharic translation

Because this entry accommodates multiple sequences of English wordforms, we
need to map these onto appropriate target-language sequences. This is
accomplished through pairs of agreement features for the lexeme, constraining
the corresponding target language form to agree with the source form on those
features. In the example, the head lose_v and its translation in the Amharic
group agree on the tense-aspect-modality (tam) and subject (sb) features. For
example, if this group is selected in the translation of the sentence John
loses hope, the head of the corresponding Amharic group will be constrained to
be third person singular present tense: ^gon tasfA yqor.tAl (ǰon tEsfa
y1k’ort’al).333In fact the system would also generate the (incorrect) feminine
of the verb in this case since the group does not include the subject itself:
tqor.talA^c. For more about where the features of source-language words come
from, see the section on morphosyntactic transformations below.

###  2.5 Lexical/grammatical categories

Another straightforward way to generalize across groups is to introduce
syntactic or semantic categories. Consider the English expression make fun of
somebody. We can generalize across specific word sequences such as made fun of
her and made fun of the mayor by replacing the specific wordforms in position
4 in the group with a category that includes the wordforms that can fill that
position. This requires a dictionary of category labels for wordforms. Entry 3
shows how this appears in the lexicon. Category names are preceded by $.

$<$[make_v] fun of $sbd$>$

  * $\rightarrow$ amh $sbd[+acc] 'a^sofa

    * align:[2,0,0,1] ; agr:[([2,1],(tns:tns,sb:sb)),([4,1],(num:num))]

Entry 3 Group entry for make fun of somebody and its Amharic translation

Because group positions that are filled by categories do not specify a surface
form, during translation they must be merged with other groups that match the
category and do specify a form. For example, in the translation of the
sequence made fun of the mayor, position 4 in the group $<$make_v fun of
$sbd$>$ may be filled by the head of the group $<$the mayor$>$. This node
merging process is illustrated in Figure 1.

Figure 1: Merging of two groups in make fun of the mayor

Finally, the Amharic group that is the translation of make fun of $sbd
includes the constraint that whatever word fills the role of $sbd in the
Amharic sentence must be accusative since it is the direct object of the verb
'a^sofa.

###  2.6 Morphosyntactic transformations

For languages pairs, such as English and Amharic, that differ greatly in their
syntax and morphology, a further elaboration of the framework permits many
generalizations that save on the number of groups required. Amharic verb
morphology is extremely complex, including tense-aspect-modality, subject and
sometimes object agreement, as well as morphemes indicating whether the verb
is negative and the main verb of a relative clause. In English almost all of
these features are indicated syntactically rather than morphologically.
Consider the sentence they do not know her and its Amharic translation
'ayAwquAtm (ayawkw’at1m). With a different subject, different object, or an
affirmative rather than a negative verb, the single Amharic verb translating
this sentence would differ. Clearly without including the subject, the object,
or the word not in groups, there is no way the Amharic morphology can end up
correct. However, doing this entails a significant combinatorial explosion:
for each English verb there would need to be hundreds of groups to cover all
the different combinations of subject, object, polarity, and tense-aspect-
modality.

We can deal with some aspects of this problem by incorporating a pre-
processing phase that modifies the English input to make it more like Amharic.
Following morphological analysis of the input sentence, it is matched against
a set of morphosyntactic transformation rules, each of which may delete words
and/or modify the features of words. One of these is shown in Entry 4. This
rule matches sequences in input sentences consisting of the words they do not
followed by a verb. It modifies the features of the verb to make its subject
third person plural, its aspect imperfective (corresponding roughly to English
present and future), and its polarity negative, and it deletes the first three
words of the sequence.

they do not $v[sb$=$3p,tam$=$impf,$+$neg] ; del 1, 2, 3

Entry 4 Morphosyntactic transformation rule for they do not $v

Getting all of this morphology right is a challenge, and the MDT approach only
handles some of the cases. Since the morphosyntactic transformation phase does
not actually involve a syntactic parse of the input sentence, gaps present a
problem. For example, adverbs can intervene between not and the main verb in
English, in which case a rule such as that in Entry 4 would not apply.

##  3 Constraint satisfaction and translation

The steps in MDT translation are illustrated in Figure 2 for the input
sentence she made fun of the mayor.

(1) She made fun of the mayor.
(2) she make_v[tns=pst] fun_n of the mayor_n
(3) make_v[tns=pst,tam=prf,sb=3psf] fun_n of mayor_n[+def]
(4) $<$make_v fun of $sbd$><$mayor_n$>$
(5) $<$make_v[tns=pst,tam=prf,sb=3psf] fun of $<$mayor_n[+def]$>>$
(6) $<<$kantibA_n[+def,+acc]$>$ 'a^sofa_v[tam=prf,sb=3psf]$>$
(7) kantibAwn 'a^sofa^c

Figure 2: Steps in the translation of she made fun of the mayor

Following tokenization of the input sentence, the wordforms in the sentence
are tagged for part-of-speech and analyzed morphologically (2). Next, the
sequence of analyzed words is matched against the morphosyntactic
transformation rules (3). In the example sentence two rules match, one for she
followed by a past tense verb, one for the followed by a noun. The first rule
assigns perfective aspect to the verb and deletes she. The second assigns
definiteness to the noun and deletes the (definiteness is expressed by a
suffix in Amharic). Next the words or lexemes resulting from this first pass
are used to look up candidate groups in the groups dictionary (4).

To complete sentence analysis, the system assigns a set of groups to the input
sentence. A successful group assignment associates as many words in the
sentence as possible with a group, and no word to more than one group, unless
that word represents a node merging (see below). Longer groups over sequences
of shorter groups. Node merging takes place during this phase; in the example,
the $sbd node in the instantiation of the group $<$make_v fun of $sbd$>$ is
merged with an instantiation of the group $<$mayor_n$>$ (5).

Group selection is implemented in the form of constraint satisfaction, making
use of insights from the Extensive Dependency Grammar framework (XDG)
debusmann . Although considerable source-sentence ambiguity is eliminated
because groups incorporate context, ambiguity is still possible, particularly
for figurative expressions that also have a literal interpretation. In this
case, the constraint satisfaction process undertakes a search through the
space of possible group assignments, creating an analysis for each successful
assignment.

During the transfer phase, a source-language group assignment is converted to
an assignment of the crorresponding target-language groups (6). In this
process some target-language items are assigned grammatical features on the
basis of cross-language agreement constraints from the source group's entry.
In the example sentence, the Amharic verb gets its tam and sb feature values
from the English verb, and the noun gets its def feature value from the
English noun. A source-language group may have more than one translation;
unless specified otherwise, the transfer phase returns all of these.

During the realization phase, surface forms are generated for each target-
language group assignment, based on the lexemes and grammatical features that
resulted from the transfer phase (7).

##  4 Related work

Our goals are similar to those of the Apertium apertium  project. As with
Apertium, we are developing open-source, rule-based systems for MT, and we
work within the framework of relatively shallow, chunking grammars. We differ
mainly in our willingness to sacrifice linguistic coverage to achieve our
goals of flexibility, robustness, and transparency. We accommodate a range of
lexical-grammatical possibilities, from the completely lexical on the one
extreme to phrasal units consisting of a single lexeme and one or more
syntactic/semantic categories on the other, and we are not so concerned that
MDT grammars will accept many ungrammatical source-language sentences or even
output ungrammatical (along with grammatical) translations. Because MDT
focuses on the translation of phrases and outputs usually outputs multiple
translations rather than complete sentences, it is more appropriate for CAT
than for full-scale MT.

In terms of long-term goals, MDT also resembles the Expedition project
mcshane+nirenburg , which makes use of knowledge acquisition techniques and
naive monolingual informants to develop MT systems that translate low-resource
languages into English. Our project differs first, in assuming bilingual
informants and second, in aiming to develop systems that are unrestricted with
respect to target language. In fact we are more interested in MT systems with
low-resource languages as target languages because of the lack of documents in
such languages.

Although MDT is not intended as a linguistic theory, it is worth mentioning
which theories it has the most in common with. Like Construction Grammar
steels  and Frame Semantics fillmoreFS , it treats linguistic knowledge as
essentially phrasal. Like synchronous context-free grammar (SCFG) chiang , it
associates multi-word units in two languages, aligning the elements of the
units and representing word order within each. MDT differs from SCFG in having
nothing like rewrite rules or non-terminals. MDT belongs to the family of
dependency grammar (DG) theories because the heads of its phrasal units are
words or lexemes rather than non-terminals. However, it remains an extremely
primitive form of DG, permitting only flat structures with unlabeled arcs and
no relations between groups other than through the merge operation described
in Section 2.5. This means that complex grammatical phenomena such as long-
distance dependencies and word-order variability can only be captured through
specific groups.

##  5 English-Amharic implementation

We are in the process of creating an English–Amharic implementation of MDT,
called Mit'mit'a (mi.tmi.tA). In doing so, we have relied on the tokenizer and
POS tagger from spaCyspacy , the Amharic-English dictionary of Amsalu
Akliluamsalu , the Amharic morphological generator within the HornMorpho
system for morphological processinggasser , the extensive grammatical
descriptions of Amharic, and the author's own knowledge of the grammars of
English and Amharic. Although far from finalized, the implementation already
contains approximately 7000 groups and 500 morphosyntactic transformation
rules. The MDT framework offers a range of possibilities with respect to how
many grammatical generalizations are captured through the use of
morphosyntactic transformations and category nodes in groups, and Mit'mit'a
falls on the heavily grammatical end of the spectrum. Thus there are
transformation rules accommodating combinations of pronoun subjects with all
English tenses and modal verbs in both affirmative and negative forms. The
result is that the system often, though by no means always, gets Amharic
morphology right.

Evaluation of an MDT implementation should be of two types: for accuracy of
the translations and for usability of the system with CAT. Since Mit'mit'a is
still under development, we have not undertaken a systematic evaluation of
either of these sorts. However, we have begun to informally compare the
system's accuracy with that of existing statistical MT systems.

There are several commercial English–Amharic machine translation systems,
including Google Translate google  and Abyssinica Translator ethiocloud ,
developed by EthioCloud. We can highlight the strengths of Mit'mit'a by
examining the grammaticality of the output of Google Translate fpr the
grammatical patterns that Mit'mit'a is designed to capture. Mit'mit'a knows
how to translate all patterns consisting of a pronoun subject followed by a
negative or affirmative verb in any of eight possible English tenses/aspects
and combinations of modal and main verbs, as well as combinations of
transitive verbs with personal pronoun objects. Given the roughly 2,700
English verbs that the sytem has Amharic translations for, the result is
hundreds of thousands of translatable patterns such as she is about to break
it. Randomly selecting from the possible verb patterns, pronouns, and six
common verbs, we get an idea of how well Google Translate performs on such
combinations. Of the 54 resulting sentences, Google Translate outputs only one
grammatically correct verb. Mit'mit'a, on the other hand, makes only one minor
mistake on three sentences, treating him in write him as a direct rather than
an indirect object. Mit'mit'a also has the advantage of returning multiple
translations when there is ambiguity, for example, translating English you in
three ways (feminine singular, masculine singular, plural). Needless to say,
this is not really a fair comparison since it is based on examples that
Mit'mit'a is designed to handle, but it does give an idea of what sorts of
advantages this rule-based system can have over a statistical system for a
morphologically complex language in the context of limited training data.

##  6 Status of project, ongoing and future work

MDT code, including implementations for Spanish–Guarani and English–Amharic,
is available at https://github.com/hltdi/mainumby and
https://github.com/hltdi/mitmita under the GPL license.

In order to develop more complete lexicon-grammars for English–Amharic and
Spanish–Guarani, we are working on methods for automatically extracting groups
from the limited bilingual corpora that are available. We are also
implementing an interface for the use of MDT implementations for CAT; working
versions for Spanish-Guarani and English-Amharic can be found at
https://plogs.soic.indiana.edu/mainumby/ and
https://plogs.soic.indiana.edu/mitmita/. Here it will be important to evaluate
to what extent translators find their task simplified through the use of the
system. Finally, since the interface records the user's translations whether
or not they make use of the suggestions from MDT, there is the opportunity to
update the system's lexicon-grammar on the basis of those translations.

##  7 Conclusions

Relatively sophisticated computational grammars, parsers, and/or generators
exist for perhaps a dozen languages, and usable MT systems exist for at most
dozens of pairs of languages. This leaves the great majority of languages and
the communities who speak them even more disadvantaged than they were before
the digital revolution. What is called for are methods that can be quickly and
easily deployed to begin to record the grammars and lexica of these languages
and to use these tools for the benefit of the linguistic communities. The MDT
project is designed with these needs in mind. Though far from achieving our
ultimate goals, we have developed a simple, flexible, and robust framework for
bilingual lexicon-grammars and MT/CAT that we hope will be a starting point
for a large number of under-resourced languages.

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(ASIAA), and Princeton University.

This paper is based (in part) on data collected at the Subaru Telescope and
retrieved from the HSC data archive system, which is operated by Subaru
Telescope and Astronomy Data Center (ADC) at National Astronomical Observatory
of Japan. Data analysis was in part carried out with the cooperation of Center
for Computational Astrophysics (CfCA), National Astronomical Observatory of
Japan.

Software: NumPy (van der Walt et al., 2011; Harris et al., 2020), SciPy
(Virtanen et al., 2020), Astropy (Astropy Collaboration et al., 2013, 2018,
2022), Photutils (Bradley et al., 2022), PetroFit, Scikit-learn (Pedregosa et
al., 2011), Matplotlib (Hunter, 2007),
GalfitPyWrap111111https://github.com/Grillard/GalfitPyWrap, SExtractor (Bertin
& Arnouts, 1996), PSFEx (Bertin, 2011), Galfit (Peng et al., 2002, 2010a),
Dynesty (Speagle, 2020).

## Data Availability

Hyper Suprime-Cam Subaru Strategic Program (HSC-SSP) Public Data are available
at https://hsc.mtk.nao.ac.jp/ssp. CLAUDS u-band images of Deep and UltraDeep
layers of the HSC-SSP and a catalogue that includes the parameters of the
best-fit 2D Sérsic models we produced for this paper as well as matched
COSMOS2020 photometry, stellar masses, and SFRs are available at
https://www.clauds.net.

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##  Appendix A Robustness of the Fitting Pipeline and Sérsic parameters

Figure 25: Heat maps showing the simulation success rate (Equation 4) as a
function of input magnitude and input size. Galaxy sizes do not affect the
success rate significantly in a given magnitude bin as indicated by the
(almost) vertical colour stripes. Figure 26: Heat maps showing the simulation
success rate (Equation 4) as a function of input magnitude and input Sérsic
index. Sérsic indices of galaxies do not affect the success rate significantly
in a given magnitude bin as indicated by the vertical colour stripes. Figure
27: Heat maps showing the simulation success rate (Equation 4) as a function
of input magnitude and input axis ratio. Axis ratios of galaxies do not affect
the success rate significantly in a given magnitude bin as indicated by the
vertical colour stripes.

In Section 3.1, we mention the magnitude limits that we adopt in each band
while fitting the light profiles of galaxies. These magnitude limits are as
follows: u$<26$; g$<25.5$; r, i, z$<25$. We introduce these limits based on
the robustness analysis of our fitting pipeline through the simulations we
perform (Section 3.2). We adopt the simulation success rate described in
Equation 4 as a proxy for the robustness of the pipeline. This rate tells us
what fraction of galaxies with a certain properties we can fit successfully.

Through simulations we investigate how this success rate changes with
intrinsic Sérsic parameters of the simulated galaxies. In Figure 7, we show
how success rates in each CLAUDS+HSC bands change with the band magnitude of
galaxies. It is clear from the figure that the success rate is sensitive to
the brightness of a galaxy beyond $24^{\text{th}}$ magnitude in the bands used
in this study.

In Figures 25, 26 and 27, we investigate how the success rate depend on other
Sérsic parameters - size ($R_{e}$), Sérsic index ($n$) and axis ratio ($b/a$)
respectively through heat maps. These figures show the success rate as
function of both the input/intrinsic values of these Sérsic parameters and
magnitude in a given band. We find that the colours in the heat map in every
panel in each figure change horizontally but remain more or less constant
vertically. This observed colour pattern in the heat map shows that the
success rate and thereby the robustness of the fitting algorithm does not
depend on the intrinsic shape of the galaxy ($R_{e}$, $n$ and $b/a$) but the
brightness alone. Therefore, we do not adopt any cuts on the data based on
$R_{e}$, $n$ and $b/a$.

##  Appendix B Size-Mass Relation Fitting in Rest-frame UV Light

Figure 28: Fitting smoothly broken double power law to the QGs and single
power law to the SFGs at rest-frame $3000$ Å in $5$ redshift bins. The
individual SFGs are shown as blue points and QGs in red. The median
uncertainties in their measurements are given in the top left corner of each
panel. The best-fitting power law relations for SFGs and QGs are shown in navy
blue and maroon respectively with their uncertainties from Bayesian posterior
plotted as shaded region.

In Figure 28, we show the size-mass relation (SMR) for star-forming galaxies
(SFGs; blue) and quiescent galaxies (QGs; red) in rest-frame UV light ($3000$
Å). We fit a single power-law for SFGs but a double power-law for QGs
initially followed by a single power-law fit for QGs above the pivot mass
($M_{p}$; mass at pivot point) in the double power-law SMR (see Section 4.1).
We provide the best-fit parameters of double power-law SMR in Table 4 and
single-power law SMR in Table 4.

As we see in Figure 28, the SFGs are on average more extended than QGs in
rest-frame UV light in every redshift bin at $0.1<z<0.9$. The SFGs and low
mass QGs ($M<M_{p}$) have shallow SMR with an exponent around $0.2$ but the
high mass QGs have a steep exponent ($\sim 0.6$). These differences in SFG and
QG SMRs are similar to those we find in rest-frame $5000$ Å (see Figure 11).

##  Appendix C Removing Progenitor Bias from QG Population

Figure 29: Removing progenitor bias in rest-frame $3000$ Å shown in size-mass
plane. We describe the removal process in Section 5.2.2 and Appendix C. The
QGs removed (newcomers) are shown in red and the remaining QGs are in black.
The percentage of QGs removed are given in each panel. Figure 30: Removing
progenitor bias in rest-frame $5000$ Å shown in size-mass plane. As in Figure
29, removed newcomers are in red and other QGs are in black.

Newly quenched galaxies introduces a bias (progenitor bias) when we analyze
the size evolution of the quiescent population. Hence, we remove progenitor
bias from our QG population as described in Section 5.2.2. In short, we assume
that the newly quenched galaxies are more extended than the galaxies those
quenched earlier in a given stellar mass bin. We also assume that the
quenching efficiency increases with mass and thus, the fraction of newly
quenched galaxy in our QG sample varies in stellar mass bins. We estimate this
fraction using Equation 14. We normalize this equation so that we remove
approximately $10\%$ of QGs from the parent population in each redshift bin.

In Figures 29 and 30 we show the QGs removed in the size-mass plane in rest-
frames $3000$ Å and $5000$ Å respectively. In both wavelengths, we remove
$8-13\%$ of QGs in each redshift bin and this percentage is given in each
panel. The removed QGs (red) are more extended than the remaining QGs in their
respective mass bin and their percentage increases towards higher masses by
design.

##  Appendix D Central Stellar Mass Surface Density

Figure 31:  The best-fitting $\log\Sigma_{1}-\log M$ relation for SFGs (blue)
and QGs (red) with mass $\log M>9.5$ in $5$ redshift bins. The best-fit single
power-law parameters are given in each panel.  Figure 32:  The best-fitting
$\log\Sigma_{1}-\log M$ relation for blue and red SFGs (blue and purple
colours respectively) with mass $\log M>9.5$ in $5$ redshift bins. The best-
fit single power-law parameters are given in each panel.

Following Cheung et al. (2012) and Ji & Giavalisco (2022), we calculate the
projected central stellar mass surface density as

$\Sigma_{1}\left[\frac{\hbox{M${}_{\odot}$}}{\text{kpc}^{2}}\right]=\frac{M_{1\text{kpc}}}{\pi\times
1\text{kpc}^{2}}=\frac{M}{\pi\times
1\text{kpc}^{2}}\frac{\gamma(2n,x)}{\Gamma(2n)},$ (15)

where $M_{1\text{kpc}}$ is the stellar mass within $1$ kpc radius in M⊙, $M$
is the total stellar mass in M⊙, $\gamma$ and $\Gamma$ are incomplete and
complete gamma functions, and
$x=b_{n}\left(\frac{1\text{kpc}}{R_{e}}\right)^{(1/n)}$. We calculate $b_{n}$
by solving the equation $\Gamma(2n)=2\gamma(2n,b_{n})$ numerically (Graham &
Driver, 2005). The caveat here is that we assume a uniform mass-to-light (M/L)
ratio within each galaxy. Although this assumption is reasonable for QGs, we
acknowledge that the radial variation in $M/L$ ratio is significantly higher
for SFG (e.g., Tortora et al., 2011). We attempt to minimize the effect of
$M/L$ variation by analyzing the $\Sigma_{1}$ only at the rest-frame $5000$ Å
which has more contribution from bulk of the stars (old low-mass stars) than
at $3000$ Å. However, even at $5000$ Å, radial variation of $M/L$ ratio can be
significant (at least) in SFGs where the overall profile is dominated by the
light from young stars in the disks. Hence, the $\Sigma_{1}$ that we estimate
are lower limits, especially for SFGs with steep negative colour gradients.

To explore how the characteristic $\Sigma_{1}$ value for SFGs and QGs evolves
in Section 5.3, we fit a single power-law to the $\Sigma_{1}$-stellar mass
relation in log-log space for SFGs and QGs with mass $\log M>9.5$ in $5$
redshift bins (Figure 31). Across the redshift bins, we do not find any signs
of significant evolution in the power-law parameters, in congruence with
previous works (e.g., Fang et al. (2013) at $z\sim 0.1$ and Barro et al.
(2017) at $0.5<z<3$). The only parameter that we observe changing with time is
the zero point for QGs. The $\Sigma_{1}$ of QGs with a characteristic mass
$5\times 10^{10}$ M⊙ increases by $\sim 20\%$ from the lowest to the highest
redshift bin.

The $\Sigma_{1}-$stellar mass relation is tighter for QGs than for SFGs
(Figure 31). One of the possible reasons for this difference is the strong
variation in $M/L$ ratios in SFG radial profiles compared to QG profiles
(Tortora et al., 2011). QGs in general have higher $\Sigma_{1}$ than SFGs in
any given mass bin and this $\Sigma_{1}$ value is very similar for all QGs
with similar stellar mass due to the very tight $\Sigma_{1}-$stellar mass
relation. Thus, there exists a threshold $\Sigma_{1}$ value required for a
galaxy to be quiescent and this threshold value increases with galaxy stellar
mass. It means that forming a dense core is a prerequisite for quenching (or a
consequence of quenching) as argued by Cheung et al. (2012), Fang et al.
(2013), van Dokkum et al. (2014), Barro et al. (2017), Ji & Giavalisco (2023)
and others. However, since some SFGs also have $\Sigma_{1}$ close to the
threshold value, having a dense core alone is insufficient for a galaxy to be
quiescent.

A lack of evolution in the slopes indicates that the galaxies build up their
inner core while they grow in stellar mass in a similar manner across the
redshift range. However, $\Sigma_{1}-$stellar mass relation has $15-20\%$
steeper slope for SFGs. This difference in the slopes is at least partially
due to the differences in how galaxies evolve before and after quenching. SFGs
build up mass more uniformly across the galactocentric radii than QGs.
However, we note that the scaling relations for QGs are also affected by
recently quenched galaxies (newcomers) through the so-called progenitor bias
(Section 5.2.2; Ji & Giavalisco, 2022, 2023).

To further probe the evolution in characteristic $\Sigma_{1}$ for blue and red
SGFs separately in Section 5.3, we fit the $\Sigma 1-$stellar mass relations
for these two SFG subpopulations in Figure 32. Because the majority of SFGs
within our redshift interval is red (purple points in 32; see also Figure 19),
the $\Sigma_{1}$ values of the overall SFGs reflect those of red SFGs. The
blue SFGs (blue points in 32) generally have lower $\Sigma_{1}$ than red SFGs
at a given stellar mass. This suggests that, as blue SFGs turn red, they also
build up their central regions.

However, we acknowledge that the parameters of the $\Sigma_{1}-$mass relation
for blue SFGs are quite uncertain because of the small sample that covers a
narrow dynamic range in stellar masses. At $0.75<z<0.9$, around $45\%$ of SFGs
are blue but this percentage drops down to below $15\%$ at $0.1<z<0.3$. Most
of these blue SFGs have stellar mass in a narrow $9.5<\log M<10$ interval.
Hence, the parameters of the fit for blue SFGs at $z<0.45$, where the fraction
of blue galaxies in the star-forming population is below $20\%$, are not well
constrained.