# The Greedy and Recursive Search for Morphological Productivity

Caleb Belth1 (cbelth@umich.edu), Sarah Payne2 (paynesa@sas.upenn.edu), Deniz
Beser3 (beser@isi.edu),
Jordan Kodner4,5 (jordan.kodner@stonybrook.edu), Charles Yang2,6
<EMAIL_ADDRESS>
1Department of Computer Science and Engineering, University of Michigan
2Department of Linguistics and Department of Computer and Information Science,
University of Pennsylvania
3Information Sciences Institute, University of Southern California
4Department of Linguistics, Stony Brook University
5Institute for Advanced Computational Science, Stony Brook University
6Department of Psychology, University of Pennsylvania

###### Abstract

As children acquire the knowledge of their language’s morphology, they
invariably discover the productive processes that can generalize to new words.
Morphological learning is made challenging by the fact that even fully
productive rules have exceptions, as in the well-known case of English past
tense verbs, which features the -ed rule against the irregular verbs. The
Tolerance Principle is a recent proposal that provides a precise threshold of
exceptions that a productive rule can withstand. Its empirical application so
far, however, requires the researcher to fully specify rules defined over a
set of words. We propose a greedy search model that automatically hypothesizes
rules and evaluates their productivity over a vocabulary. When the search for
broader productivity fails, the model recursively subdivides the vocabulary
and continues the search for productivity over narrower rules. Trained on
psychologically realistic data from child-directed input, our model displays
developmental patterns observed in child morphology acquisition, including the
notoriously complex case of German noun pluralization. It also produces
responses to nonce words that, despite receiving only a fraction of the
training data, are more similar to those of human subjects than current neural
network models’ responses are.

Keywords: linguistics; language acquisition; morphology; computational
modeling

## Introduction

The acquisition of English past tense is one of the most extensively studied
problems in cognitive science (?, ?, ?). Yet its simplicity—a single, and
numerically dominant, rule (-ed) along with a list of irregular verb
exceptions—is hardly representative of the complexity of the world’s
morphological systems (?, ?). In comparison, German noun pluralization is a
more challenging test for morphological learning theories. The plural is
formed by five suffixes, but even the _least_ frequent, -s, is productive and
applies to novel nouns (e.g., iPhones), while the other four suffixes are also
productive for sub-categories of nouns characterized by their gender and
phonological properties (?, ?). Nevertheless, children exhibit remarkable
proficiency of their native language’s morphology at a very early age; see (?,
?) for a cross-linguistic review.

Children’s discovery of productive rules is also attested experimentally when
presenting young children with nonce words in a Wug test (?, ?) and when new
words (e.g., _google-googled_) enter a language. As a result, traditional
linguistic approaches to language acquisition often make use of productive
rules (?, ?). ? (?) proposed an impactful neural-network (NN) model,
suggesting that NNs might be capable of exhibiting rule-like behavior despite
having no explicit representation of rules. Ensuing critiques of the model by
? (?) and of connectionist models of cognition more generally by ? (?) sparked
what came to be known as the “Past Tense Debate.” Due to the extensive
advancements in NN architecture in natural language processing and machine
learning, the debate has recently been revived by ? (?) with some initially
promising improvements over early models (in particular, more realistic
accuracy). However, these accomplishments have been challenged, especially
when compared against human behavioral results on Wug tests in English past
tense (?, ?) and German plurals (?, ?).

The Tolerance Principle (TP) may provide another solution to some of these
challenges (?, ?). It provides a tipping point for rule productivity based on
the level of rote-memorized exceptions that the rule needs to withstand. A
mathematical consequence of the TP is that rules defined over smaller
vocabularies can tolerate a larger fraction of exceptions. This property is
attractive. First, it makes the TP a promising candidate for modeling early
language development, when children’s vocabulary size is quite limited.
Second, if the learner fails to acquire a broad rule defined over a
vocabulary, the vocabulary can be partitioned into subsets, within which
narrower rules may be identified by recursive application of the TP. In this
work, we propose an abductive learning model of morphology, which we call
_Abduction of Tolerable Productivity_ (ATP). ATP is a search procedure that
recursively hypothesizes rules and evaluates productivity, until all words in
the vocabulary are either accounted for by productive rules or listed as
exceptions.

## Background

### Psychological and Developmental Considerations

Computational models of cognition, which inevitably make simplifying
assumptions, must still operate within the boundaries established by empirical
research. We review several important lines of results from child language
acquisition, which serve as the design specifications for computational models
of morphological learning.

#### Data size

One of the most remarkable characteristics of child language acquisition is
how small the vocabulary on which they learn is. All English-learning children
acquire the productive use of past tense by age three (?, ?), some even before
age two (?, ?). Likewise, German-learners show over-regularization of suffixes
such as -(e)n and -e before or around age two (?, ?, ?). At such an early
stage of development, children have a very modest vocabulary: a two-year-old
English learner’s total vocabulary is 500 at most, and a three-year-old’s
vocabulary has an upper limit of just over 1000; most children’s vocabulary
size is considerably smaller (?, ?, ?). See (?, ?, ?) for similar findings in
other languages. Thus, a developmentally plausible cognitive model should be
able to learn the core morphology of inflection from datasets containing only
a few hundred words.

#### Productivity and Wug Tests

The classic study by ? (?) was the first systematic demonstration that young
children apply productive rules to form noun plurals (wug-wugs) and past tense
verbs (rick-ricked). Berko also found that children nearly categorically
resist the analogical use of irregular forms, even for words very similar to
existing irregulars. For example, only 1 of the 86 children in the study
produced glang for the novel verb gling. The categorical status of
productivity is also strongly confirmed in naturalistic production. One of the
first quantitative longitudinal studies (?, ?) noted that regular forms are
rarely analogized into irregular forms, while irregular forms are often
regularized. More recent quantitative studies estimate the rate of past tense
over-regularization at 8-10% of all past tense forms (?, ?, ?). By contrast,
children almost never over-extend an irregular form (e.g., bite-bote from
write-wrote, fry-frew from fly-flew): the most comprehensive study places the
error rate at 0.2% (?, ?).

### The Tolerance Principle

The Tolerance Principle (TP) is a cognitively-motivated, theoretical tipping
point that makes quantifiable predictions about when a linguistic process is
used productively. It is inspired by studies of lexical processing and the
hypothesis that children use a morphological process productively when it is
computationally more efficient to do so. The TP only depends on two
quantities: N—the number of words in the rule’s scope—and e, the number of
exceptions. For instance, the -s process for English pluralization applies to
singular nouns and has exceptions like _children, sheep, fish_ , etc. In our
model, rules are of the form $r:A\implies C$ where $A$ is the antecedent and
$C$ the consequent; N measures how many times $A$ applies and e measures the
number of times $C$ fails to follow from $A$. Given a rule $r$ with a scope of
size N and e exceptions, the TP states that

$r\text{ is productive iff
}\emph{e}\leq\theta_{\emph{N}}\triangleq\frac{\emph{N}}{\ln{\emph{N}}}.$ (1)

The TP has consistently made accurate predictions on when children accept
rules as productive and when they do not, as confirmed in artificial language
learning experiments (?, ?, ?) with precisely controlled conditions.
Productivity under the TP is categorical, which mirrors children’s
morphological use reviewed above. It is also parameter-free: the two values N
and e are word counts directly from the training data that require neither
parameter tuning nor statistical fitting.

Critically for complex morphological systems, the TP can be applied
recursively: if a rule $r$ is unproductive over its current scope, its scope
can be recursively narrowed. For example, none of the five German noun plural
suffixes covers a sufficiently large number of nouns to tolerate the rest as
exceptions: the learner will attempt to organize nouns into
subcategories—defined by morphological gender and the phonological form of the
noun—to recursively search for productivity within. As we will see, this
divide-and-conquer strategy is fully automated in our search procedure.

## Model

### Abductive Search for Productivity

The recursive application of the TP lends itself to a Peircean abductive
learning procedure. Given data in pairs, such a procedure hypothesizes rules
that map one set to the other (e.g., lemmas to their inflected forms). If no
such rule is productive via the TP, then it subdivides the words according to
some feature—that is, it refines the hypothesis over more narrow scopes—and
recursively tries again.

We propose just such a procedure: _Abduction of Tolerable Productivity_ (ATP).
Its input is a set $\mathcal{X}$ of instances in the form
($\ell,\mathcal{F},\textsc{i}$), where $\ell$ is a lemma,
$\mathcal{F}\subseteq\Omega$ is a set of features from the feature space
$\Omega$, and i is the inflected form corresponding to lemma $\ell$ and
features $\mathcal{F}$; for instance (walk, $\\{$3,SG,PST$\\}$, walked), where
the example features 3,SG,PST carry the information that the inflection is 3rd
person, singular, and past tense.

ATP recursively grows a decision tree, where each instance’s “label” is the
morphological change that produces the inflection; the resulting tree thus
encodes a map from lemma and features to inflection. In principle ATP could
model any type of inflectional morphology, but the inflections modeled in this
paper involve only suffixation, in which case an instance’s “label” is the
suffix that is concatenated to the lemma. For clarity, we describe ATP in
terms of suffixation.

At each recursive level, the decision of which feature to split on is selected
to maximize consistency: the relative frequency of the most frequent suffix
that the instances with that feature take. That is, it splits on
$\hat{s}=\operatorname*{arg\,max}_{s\in\Omega}=\frac{f_{\sigma_{\max}(\mathcal{X}_{s})}}{|\mathcal{X}_{s}|}$,
where
$\mathcal{X}_{s}=\\{(\ell,\mathcal{F},\textsc{i})\in\mathcal{X}:s\in\mathcal{F}\\}$
is the set of instances with feature $s$ and
$f_{\sigma_{\max}(\mathcal{X}_{s})}$ is the frequency of the most frequent
suffix in $\mathcal{X}_{s}$. The split is formed by recursing separately on
those instances with feature $s$ (i.e., $\mathcal{X}_{s}$) and those without
it (i.e., $\mathcal{X}\setminus\mathcal{X}_{s}$).

If _productivity_ is defined as the frequency of the most frequent suffix in
$\mathcal{X}_{s}$, then this is one possible formulation of ? (?)’s proposed
_Principle of Maximize Productivity_ , which he described as “Pursue rules
that maximize productivity.” Furthermore, it is motivated by findings that
consistent patterns promote generalization and category formation in learning
(?, ?, ?).

#### Features

The set of features provided as input is expanded to include regularities in
the ending of lemmas. For instance, the null suffix for German nouns is
predominantly used for non-feminine lemmas ending with a schwa followed by l,
r, or n (?, ?). Regularities of this sort are extracted at each recursive
level as the shortest lemma endings that productively predict a suffix in
$\mathcal{X}$ (the training instances at that level).

To do so, ATP considers suffixes one-by-one and, for each suffix, it considers
the endings of lemmas that take that suffix, starting with the shortest
ending. It caches any ending where the number of lemmas that do have the
ending but do _not_ take the suffix is tolerably low; effectively, _ending_
$\implies\ $ _suffix_ passes the TP. For example, enough English verb lemmas
ending in [k] take the [-t] suffix to pass the TP; similarly for [p], [], and
other voiceless segments. We denote the lemmas that take any such endings as
$\mathcal{E}$ (e.g., $\mathcal{E}=$ {lemma : lemma ends in [k] or [p] or … or
[]} in the English verb example). If $\mathcal{S}$ consists of the lemmas
whose inflections take the suffix being considered, then ATP tests the TP for
both
$\emph{N}_{1}=|\mathcal{E}|,\emph{e}_{1}=|\mathcal{E}\setminus\mathcal{S}|$
and
$\emph{N}_{2}=|\mathcal{S}|,\emph{e}_{2}=|\mathcal{S}\setminus\mathcal{E}|$.
If both tests pass the TP, the endings contributing to $\mathcal{E}$ are added
to the feature space $\Omega$. The purpose of these tests is to establish that
there is a productive relationship between the endings and the suffix. In the
running example, all words that end in a voiceless segment other than [t] take
the [-t] suffix, and all words that take the [-t] suffix end in a voiceless
segment, so both tests pass. Further examples of the result can be seen in
Figs. 3-4, where sets of lemma endings appear in brackets, separated by “|”
for _logical or_. This has the interpretation of picking out the lemmas that
end in any such ending (i.e., $\mathcal{E}$).

Thus, at the end of the lemma ending extraction, the feature space $\Omega$
consists of whatever features were provided as input, together with the newly
added sets of endings. ATP is agnostic to the nature of a feature, such as
whether it has to do with content or form. It considers them all equally,
seeking to maximize consistency, as described above.

Moreover, alternative approaches to incorporating or discovering features
could be used without changing the fundamental recursive, abductive search
procedure. Such adaptions could be useful in morphological processes beyond
suffixation, where ATP could make use of features relevant to processes such
as infixation and stem change within the same abductive search procedure.

#### Recursive and Base Cases

Once a split is performed, the node’s children are formed recursively on the
partitioned set. Features that are equivalent across all instances in
$\mathcal{X}$ are ignored since they are completely uninformative. The base
case of this recursion is reached when the most frequent suffix at the node
passes the TP or when there are no more features to split on. The path to a
node with a productive suffix is the rule (e.g., Figs. 3-4), and all
exceptions to the rule are memorized by storing them at the corresponding
node. If a node is reached where no suffix is productive and there are no more
features to split on, the node’s instances are memorized.

### Inflection Production

Once trained, ATP can use its acquired knowledge to produce the inflected form
corresponding to a lemma and its features. It does so by traversing the
decision tree to a leaf. If the leaf has a productive suffix, ATP produces the
inflected form via the rule. If there is no productive suffix at the leaf, ATP
makes an analogical guess by retrieving the memorized lemma at that node with
the smallest character-level Hamming distance, padding 0’s if necessary to
make the two strings the same length. ATP uses the learned inflected form for
the nearest-neighbor lemma to produce the inflection for the target lemma. In
some cases—such as when nonce words are presented—some features may be
unknown. In this case, an inflection is produced by traversing all logically
compatible paths and using the most specific (i.e., deepest in the tree)
compatible rule. A path is _logically compatible_ if none of the features it
specifies are contradicted by the word’s known features. For instance, if the
gender of a nonce word is unknown, and ATP encounters a branch point in the
tree that is determined by gender, it takes both branches.

It is worth pointing out that analogical guessing forces an answer in order to
quantitatively evaluate our model on test data. However, in many cases of
language use, speakers may refuse to produce any form when they do not have an
applicable productive rule, as in the well-known case of morphological gaps
(?, ?).

#### Code

We make the code for ATP, along with instructions for using it on new data,
available online.111https://github.com/cbelth/ATP-morphology.

## Evaluation

We evaluate the developmental plausibility of ATP with respect to the order in
which rules are acquired and its quantitative performance on realistic data.
We further evaluate how the regularities ATP discovers and its productions on
Wug tests correspond to children’s discoveries and productions.

### Data and Setup

We briefly discuss the data that we use in our experiments, and name each
dataset for reference.

CHILDES-DE contains 442 German nominative singular/plural pairs of child-
directed speech from the Leo (?, ?) CHILDES corpus. Features encode gender,
one of $\Omega=$ {feminine, masculine, neuter}, and frequency of words is
known. We removed umlauts, as these follow a separate morphological process
from suffixation. For developmental experiments, we sampled subsets of size
400, weighted by frequency, each sample modeling a child’s 400 word
vocabulary. The findings (reviewed earlier) that age-two children have
productive plural morphology suggests that suffixes are learnable on
vocabularies of this size or even smaller. While the CHILDES-derived words are
used for training, we test on CELEX words. Morphological knowledge is
generally acquired during childhood but must be able to generalize to other
words in the lexicon.

CHILDES-EN was constructed from 6,539 word-inflection pairs extracted from
child-directed English (?, ?). There are 3,321 plural nouns (23 irregular),
1,494 past tense verbs (120 irregular), and 1,724 progressive verbs (the
exceptionless -ing). As reviewed earlier, children’s vocabulary size during
morphological learning is very modest. We thus log-binned words into 20 bins
based on frequency and sampled 50 from each, simulating a child’s vocabulary
growth from 50 to 1K words. We repeated 100 times with different random seeds
to simulate 100 children. Features are one of $\Omega=$ {progressive, past,
plural}.

CELEX-EN contains just the stem/past tense pairs from CHILDES-EN, intersected
with CELEX (?, ?) for word frequency. For cross-validation, we formed 10
random 1000/100/200 train/dev/test splits, and for developmental experiments,
we test on the top $n$ items, $n\in\\{100,200,400,600,800,1000\\}$. To
simulate children, we added random jitters between 0 and 5 to the frequencies.
This has the Zipfian effect of scrambling which low-frequency items appear in
each learner’s training data.

#### Setup

Throughout the experiments, we use orthography for German and IPA
transcriptions for English, following ? (?)’s use of orthography and ? (?)’s
use of phonological transcriptions. Token frequencies are used to construct
realistic datasets, but only type frequencies are used in learning and
evaluation. We use paired $t$-tests at a 0.95 confidence level for statistical
analysis. When testing ? (?)’s ED model, we follow their setup, using an
identical RNN implementation, trained for 100 epochs, with batch size 20. Both
encoder and decoder RNNs are bidirectional LSTMs (?, ?) with two layers, 100
hidden units, and a vector size of 300.

### Developmental Results

#### Order of Acquisition

Figure 1: Fraction of “children” (ATP runs on different data) who have learned
each major suffixation rule. The order of acquisition closely follows child
development. Legend order and shade of gray match acquisition order. All
suffixes are consistently acquired, except German -s ($\approx 20$%).

We study the order in which ATP discovers productive processes in English and
German to assess its consistency with children’s developmental patterns on
both English pluralization, verb past tense, and verb present participle
(CHILDES-EN) and German pluralization (CHILDES-DE).

Figure 1 shows that ATP acquires the -s for pluralization (e.g., book, books)
and -ing for the verb present participle (e.g., walk, walking) on the smallest
dataset (50 pairs). In contrast, the exception-laden -ed of the past tense
takes longer to overcome the irregulars, but is consistently acquired by
around 500 words. These patterns align well with the order of morpheme
acquisition by English-learning children (?, ?).

On German pluralization, which has five primary suffixes (-(e)n, -e,
-$\emptyset$, -er, -s; we separate -n and -en), ATP acquires -n immediately
(on 50 words), which closely follows “A" in ? (?)’s diary study, who learned
-e(n) words most quickly. By 100 words, -$\emptyset$ and -_e_ have been
acquired; again matching “A," where the rate of learning of -$\emptyset$ and
-e words is virtually identical. Other longitudinal studies have also shown
the very early acquisition of these suffixes (?, ?). The -er suffix is learned
by 95% of simulated children by 400 words. In ? (?), -s and -er words were
learned at similar rates, while other studies (?, ?, ?, ?) also find that -s,
while productive, generally emerges later. In our data, -er is attested 15
times, while -s only 8 times; ? (?) reports that A’s _s_ -over-regularizations
jump at 2;1, when her vocabulary contains around 50 -_s_ words, which may
explain the difference.

Figure 2: Accuracy on English and German vs. ED. ATP outperforms ED by a stat.
sig. amount on English and the first four training sizes for German. English
accuracy on irregulars in the training data (dotted line) demonstrates the
classic U-shape. The dashed line on German is the performance when the models
are presented lemmas without gender.

#### Performance

We evaluate quantitative performance on CELEX-EN and CHILDES-DE in terms of
how accurately the model generates inflections for held-out (lemma, features)
pairs as the training data is progressively grown, simulating vocabulary
growth. We compare to ? (?)’s ED model. Results are in Fig. 2.

On English past tense, ATP outperforms ED by a statistically significant
amount on all training sizes. Furthermore, when evaluated on irregulars that
it has seen during training, ATP over-applies learned rules before correcting,
exhibiting the developmental regression (?, ?, ?) that is a hallmark of the
English past tense acquisition (dotted line). Before the model learns the
productivity of -ed, all training verbs are memorized (irregulars and regulars
alike). Once the productive rule is learned, the model erases the memory of
rule-following verbs, which no longer require storage. The two drops in
irregular training accuracy correspond with the two largest jumps in test
accuracy, demonstrating the acquisition of a rule (likely /-t/ and /-d/; see
Fig. 3). Throughout learning, all test errors were over-regularizations of
rules (unless guessing occurred due to no rule having yet been acquired),
matching the regularization vs. irregularization contrast in child acquisition
(?, ?, ?, ?).

On German pluralization, ATP outperforms ED by a statistically significant
amount on the first four datasets (60-240 words) and is statistically
indistinguishable after that point. Though the training data contains gender,
the models can also be tested with novel nouns without gender information.
Plural suffixation is not only conditioned on gender but also the phonological
properties of nouns (?, ?, ?). ATP always outperforms ED without gender,
indicating that it is more capable of extracting the phonological regularities
in the German system.

In addition to its accuracy, ATP runs in seconds on all dataset sizes,
compared to minutes for ED.

We note that the setup of this evaluation differs in certain respects from
that used in the neural modeling literature. In particular, each number in
Fig. 2 is the average _test_ accuracy of _fully-trained_ models on each of the
training sets of that size (with the exception of the dotted U-shape line, as
discussed above). In contrast, learning curves in works like ? (?) report
_training_ accuracy of _partially-trained_ models by epoch. Moreover, in such
works, the model’s final test-accuracy is an average only over different
random initializations of the model, holding the training data constant. We
chose our setup because our focus is on modeling development, and evaluating
periodically on held-out test data is a more faithful measure of a model’s
developmental trajectory.

### Acquired Knowledge

#### Discovered Rules

Figure 3: ATP decision tree for CHILDES-EN past tense. IPA symbols in brackets
are separated by “|” to indicate that a lemma that ends in any of the listed
endings follows (or does not follow, if preceded by “$\lnot$”) the branch.
Figure 4: ATP decision tree for CHILDES-DE. “M” indicates nouns with masculine
gender.

ATP learns rules explicitly, represented in a decision tree. The output trees
in Figs. 3-4 have transparent linguistic interpretations.

The English tree (Fig. 3) correctly characterizes the -ed rule for English
past tense: the first branch to the [-t] node splits on voiceless stem
endings, the next branch to [-d] splits on voiced stem endings, and [-d]
captures the remaining instances.

The German tree (Fig. 4) captures all five primary suffixes (-(e)n, -e,
-$\emptyset$, -er, -s). The -s suffix—well-known to be the most
idiosyncratic—is picked up at the deepest point in the tree, conforming to its
role as the rule of last resort (?, ?).

This particular English tree was learned from 1000 words (one simulated child)
of CELEX-EN, and the German tree was learned from the 442 German nouns in
CHILDES-DE.

#### Wug Test Production

Table 1: Correlations with human production results (bold $=$ stat. sig.).
Training ATP on just 400 words of child-directed speech yields higher
correlation with human productions than training a NN on 22x times the data
(?, ?).

|  | Neuter |  |  | Unknown |
---|---|---|---|---|---|---
| %R | %NR | $\rho$ | %R | %NR | $\rho$
-(e)n | 0.17 | 0.04 | -0.26 | 0.19 | 0.23 | 0.43
-e | 0.27 | 0.35 | -0.14 | 0.45 | 0.62 | 0.01
-$\emptyset$ | 0.11 | 0.0 | 0.55 | 0.07 | 0.00 | 0.55
-er | 0.44 | 0.17 | 0.53 | 0.29 | 0.0 | 0.46
-s | 0.01 | 0.44 | 0.3 | 0.01 | 0.15 | 0.64
other | 0.00 | 0.00 |  | 0.00 | 0.00 |

The Wug test can be used to assess the morphological knowledge acquired by
both humans and computational models. ? (?) carried out a Wug test study with
24 nonce words, divided into Rhyme (rhymes with familiar German words) and
Non-Rhyme (unfamiliar) nonce words. The same stimuli were used by ? (?) and ?
(?). ? (?) compared the predictions of ? (?)’s ED neural network model to
human productions. We passed the same 24 nonce words to ATP trained on
CHILDES-DE. This experiment follows ? (?): it computes (a) the fraction of
productions for each suffix, divided into Rhyme (R) and Non-Rhyme (NR) and (b)
Spearman’s rank correlation ($\rho$) between the production probabilities of
human and ATP productions. To simulate multiple people, we ran ATP on 500
samples of size 400 (by frequency) from CHILDES-DE. As in ? (?), we treated
each run as a model of a human, and computed the production probabilities for
a particular word-suffix pair as the fraction of models that produced that
suffix for that word.

When presenting ATP nonce words with unknown gender (right columns of Tab. 1),
ATP correlates with human productions statistically significantly for all
suffixes except -e. This is in contrast to the ED model, which shows no
correlation for any suffix (?, ?). Moreover, ATP was trained on a realistic
400 words of child-directed speech—over which the core inflectional morphology
is learned—while ED was trained on 8.7K words, or 22x the data. Furthermore,
both ? (?) and ? (?) found that -(e)n and -s are used more frequently for non-
rhyme words than rhyme words. ATP again matches this behavior, and ED did not
(?, ?).

When presenting nonce words with unknown gender to ATP, the correlation with
human performance is higher than when presenting with neuter gender (left
columns). The participants in ? (?) were presented nonce words with the neuter
determiner _das_. However, the impact of this is unknown. As noted earlier (?,
?), the suffix choice is conditioned on both gender and phonology, and a
conflict generally arises only in non-feminine gender (nearly all feminine
nouns add -(e)n). When they conflict, as in the test stimuli, human subjects
may persist with a gender-conditioned rule, or they may eschew gender
altogether and rely on phonological similarity to existing words (?, ?). Our
model makes a testable prediction regarding this open question that can be
pursued in future research.

## Discussion

ATP is not limited to morphology acquisition. Further research could
investigate its use in learning phonology, syntax, or anything where
linguistic generalizations are learned. Children’s adeptness at language
acquisition is a constant reminder of how much knowledge can be learned from
tiny amounts of evidence. The decision trees that ATP learns leave a step-by-
step trace of what was learned and how. They thus provide explicit places to
look for the steps children take in acquiring language.

## Acknowledgments

This work was supported by an NSF GRFP to CB. We thank the participants of the
2020 distributional learning seminar at the University of Pennsylvania for
helpful discussion.

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# Direct photons $\sim$basis for characterizing heavy ion collisions$\sim$

Takao Sakaguchi  Brookhaven National Laboratory, Physics Department, Upton, NY
11973, U.S.A<EMAIL_ADDRESS>
###### Abstract

After years of experimental and theoretical efforts, direct photons become a
strong and reliable tool to establish the basic characteristics of a hot and
dense matter produced in heavy ion collisions. The recent direct photon
measurements are reviewed and a future prospect is given.

##  1 Introduction

Direct photons are an excellent probe for extracting thermodynamical
information of a matter produced in nucleus-nucleus collisions, as they are
emitted from all the stages of collisions, and don’t interact strongly with
medium once produced. They are produced through a Compton scattering of quarks
and gluons ($qg\rightarrow q\gamma$) or an annihilation of quarks and anti-
quarks ($q\overline{q}\rightarrow g\gamma$) as leading order processes, and
the next leading order (NLO) process is dominated by bremsstrahlung (fragment)
($qg\rightarrow qg\gamma$) as depicted in Fig. 1(a). Theoretical studies show
that a part of fragment processes arises as leading order [1]. The source of
radiations are from various processes and manifest as a function of transverse
momentum ($p_{T}$) [2](Fig. 1(b)).

Figure 1: (a)Production processes of direct photons (left), and (b) their
manifestation as a function $p_{T}$ (right).

Photons with high $p_{T}$ are primarily produced in the initial hard
scattering, and often called as ”hard photons”. Under the formation of hot and
dense medium, in addition to the hard photons, a calculation predicts that the
photon contribution from a quark gluon plasma (QGP) state dominates lower
transverse momentum ($p_{T}$) region in heavy ion collisions (1$<p_{T}<$3
GeV/$c$ in Au+Au collisions at $\sqrt{s_{NN}}$=200 GeV [3]). The signal from a
hadron rescattering process dominates even lower $p_{T}$. Compton scattering
of hard-scattered partons and partons in the medium (jet-photon conversion),
or bremsstrahlung of the hard scattered partons in the medium will also arise
[4]. Photons from the processes would become an another measure of the parton
density of the medium since they are produced through an interaction of hard-
scattered partons and the medium. All the contributions are overwhelmed by
huge photonic background from known hadron sources such as $\pi^{0}$’s or
$\eta$’s, except for high $p_{T}$.

##  2 High $p_{T}$ direct photons $\sim$how well are they calibrated?$\sim$

###  2.1 $p+p$ collisions: precision test

Hard direct photon production in p+p collisions is extensively studied both
experimentally and theoretically. Figure 2 shows the ratios of direct photon
cross-sections to NLO pQCD calculation measured by various experiments [5].

Figure 2: (a)Data over NLO pQCD from various experiments in p+p collisions,
and (b)fragment photons measured through $h-\gamma_{dir}$ $\Delta\phi$
correlation by PHENIX.

The data are explained by NLO pQCD calculations within $\sim$20 %.

As depicted in Fig. 1(a), photons are produced in the fragment process as
well. Several experiments have measured the prompt to all hard photons by
applying an isolation cut. The same cut is applied to a NLO pQCD calculation
to compare with the measurement. The PHENIX experiment at RHIC has used a
similar technique and confirmed that the calculation is consistent with the
result [6]. There is a new attempt of measuring fragment contribution more
directly. The PHENIX experiment has recently measured photons associated with
the same side of trigger high $p_{T}$ hadrons [7]. These photons are
considered to be from fragment processes. Fig. 2(b) shows the associated
photon yield $\Delta\phi$ distributions from the analysis. The ratio of near-
side-associated fragment to inclusive photons is also measured, and show a
consistency with the previous PHENIX measurement [6]. A detailed study of PID
efficiency important in the analysis because possible mis-identification of
photons/hadrons would produce a trigger bias.

###  2.2 Hard photons suppressed in Au+Au?

The yield of high $p_{T}$ direct photons are well-scaled by a nuclear overlap
function ($T_{AB}$) in heavy ion collisions. The first measurement of such
photons at RHIC confirmed that the high $p_{T}$ hadron suppression is a
consequence of an energy loss of hard-scattered partons in the hot and dense
medium. The latest high statistics data from PHENIX showed a trend of
decreasing at high $p_{T}$ ($p_{T}>$14 GeV/$c$) (Fig. 3(a)). The decrease of
the yield in Au+Au starts at $\sim$12 GeV/$c$ ($x_{T}$=0.12) and drops by
$\sim$30 % at 18 GeV/$c$ ($x_{T}$=0.18) [8]. Parton distribution functions
(PDFs) do not change by 30 % between the two $x_{T}$ regions [9].

Figure 3: (a) $R_{AA}$ for direct photons, $\pi^{0}$ and $\eta$ in Au+Au
collisions at $\sqrt{s_{NN}}$=200GeV (left), (b) expected $R_{AA}$ from NLO
pQCD calculation at 200 and 62 GeV (middle), and (c) $R_{AA}$ for direct
photons in Au+Au collisions at 62 GeV (right) by PHENIX.

Isospin effect has been proposed to explain the suppression [10]. The photon
production cross-section is proportional to $\alpha\alpha_{s}\Sigma
e_{q}^{2}$. Therefore, the yield of photons will be different between p+p, p+n
and n+n collisions. It results in the deviation of $R_{AA}$ from unity at high
$p_{T}$ in Au+Au collisions, where the contribution of valence quarks become
prominent. There is a $\sim$15 % drop at 18 GeV/$c$ at $\sqrt{s_{NN}}$=200 GeV
expected from the effect (Fig. 3(b)). Combining PDF effect with the isospin
effect would explain the data. As shown in Fig. 3(b), the effect will manifest
in lower $p_{T}$ region at $\sqrt{s_{NN}}$=62.4 GeV because the effect scales
with $x_{T}$. The PHENIX experiment has measured photons in Au+Au collisions
at $\sqrt{s_{NN}}$=62.4 GeV [11] and divided them by NLO pQCD instead of p+p
yield (Fig. 3(c)), since there is no p+p data from the experiment. The
difference between p+p yield and NLO pQCD calculation measured at 200 GeV is
scaled to 62.4 GeV and shown as a dot-dashed line in 62.4 GeV. Assuming this
is the baseline, we may have confirmed the isospin effect (i.e., suppressed at
$p_{T}>$5 GeV/$c$, corresponding to 16 GeV/$c$ at 200 GeV), also at 62.4 GeV.
Combining the Au+Au data with the ones from future high statistics d+Au data
would disentangle the PDF and isospin effect.

##  3 Application of well-calibrated probe $\sim$$\gamma$-jet analysis$\sim$

Well calibrated high $p_{T}$ photons are ideal as measure of the initial
momenta of back-scattered partons. The idea was first proposed a decade ago
[12], but the measurement has not become realized until recent. The PHENIX has
measured an associated away-side hadron yield when triggered by a hard photon,
both p+p and Au+Au collisions at $\sqrt{s_{NN}}$=200 GeV as shown in Fig.
4(a).

Figure 4: Away-side associated hadron yield measured in $\gamma_{dir}-h$
correlation in p+p and Au+Au collisions at (a) PHENIX (left) and (b) STAR
(right).

The result shows that the away-side per-trigger hadron yield is reduced in
Au+Au collisions compared to that expected from p+p collisions [7]. This is
qualitatively consistent with single particle measurement [8]. In this
analysis, all the hadron contribution associated with photons from hadron
decay are subtracted on statistical basis to obtain $\gamma_{dir}-h$
correlation as:

$(\gamma_{dir}-h)=(\gamma_{incl.}-h)-(\gamma_{dec}-h)$

There is a new result from the STAR experiment showing $I_{AA}$ in
$\gamma_{dir}-h$ correlation [13]. In this analysis, the correlation is
obtained by:

$(\gamma_{dir}-h)=(Clus_{\gamma-en}-h)-\alpha\times(\pi^{0}-h)$

where $Clus_{\gamma-en}$ stands for $\gamma$-enriched clusters by a shower
shape cut. $\alpha$ is determined such that the near side associated hadron
yield be zero. This procedure will be justified under the assumption that the
$\eta$-triggered hadron yield and fragment-photon-triggered hadron yield are
as same as the $\pi^{0}$-triggered hadron yield. The cross-section measurement
of direct photons would help justifying the procedure.

##  4 Thermal photons from CERN to RHIC

The measurement of thermal photons delivers the temperature of the system.
Combining the temperature with an entropy derived from a particle multiplicity
measurement will deduce the degree of the freedom of the system [14, 15].
There is a direct photon measurement in thermal region in Pb+Pb collisions,
made by WA98 experiments at CERN [16]. However, the lack of p+p measurement at
the same energy made it difficult to understand whether or not there is a
thermal emission [3]. WA98 has recently analyzed p+Pb and p+C collision data
to measure hard photons with a nuclear effect ($k_{T}$ smearing). Taking the
ratio of the yield in Pb+Pb to p+Pb should be able to quantify the pure non-
hard photon component. However, the error is too large to make a conclusion.
The experiment is now making an effort to minimize the systematic errors [17].

Figure 5: (a) Direct photon measurement in thermal region by WA98 at CERN and
theoretical interpretation (left), and (b) comparison of direct photon yield
in p+Pb and Pb+Pb (right).

The thermal photon contribution is believed to be $\sim$10 % at RHIC energy,
and might need a measurement with an error of $<$5 %. Measurement of the
internal conversion of direct photons ($\gamma\rightarrow\gamma^{*}\rightarrow
e^{+}e^{-}$) opened up a possibility to significantly reduce the systematic
errors. The PHENIX experiment has applied the technique of measuring low
$p_{T}$ and low mass di-electrons to high $p_{T}$ and low mass di-electrons.
The measured yield is converted into a direct photon yield using Kroll-Wada
formula [18, 19]. If $M_{ee}\ll p_{T}$, and $M_{ee}<2M_{\pi}$, there is little
contribution from $q\overline{q}\rightarrow\gamma^{*}$, and thus the
conversion is straightforward. The yield of direct photons are found to be
higher than the ones expected from p+p collisions scaled by the number of
binary collisions, suggesting there are additional sources of photons in Au+Au
system (Fig. 6(a)).

Figure 6: (a) Low $p_{T}$ direct photon spectra (left) and (b) direct photon
elliptic flow (right) in Au+Au collisions measured by the PHENIX experiment at
RHIC

The average of simple exponential fits to the low $p_{T}$ regions gives a
temperature of 220$\pm$23$\pm$8 MeV. However, the possible contribution from a
nuclear effect ($k_{T}$ smearing) to the $p_{T}$ region still remains [20].
The internal conversion technique would help precisely determining the
contributions in d+Au collisions. It should be noted that there are a number
of theoretical analyses for extracting thermodynamical quantities [15, 21].

##  5 Decomposition of photon sources –photon elliptic flow–

It is predicted that the elliptic flow ($v_{2}$) of photons show the different
sign and/or magnitude, depending on the production processes of photons [22].
The observable is powerful to disentangle the contributions from various
photon sources in the $p_{T}$ region where they intermixes. The photons from
hadron-gas interaction and thermal radiation follow the collective expansion
of a system, and would give a positive $v_{2}$. The amount of photons produced
by jet-photon conversion or in-medium bremsstrahlung increases as the medium
to traverse increases. Therefore these photons show a negative $v_{2}$. The
intrinsic fragment or bremsstrahlung photons will give positive $v_{2}$ since
larger energy loss of jets is expected in out-plane.

PHENIX has measured the $v_{2}$ of direct photons by subtracting the $v_{2}$
of hadron decay photons off from that of the inclusive photons, following the
formula below:

${v_{2}}^{dir.}=(R\times{v_{2}}^{incl.}-{v_{2}}^{bkgd.})/(R-1),\ \ \
R=(\gamma/\pi^{0})_{meas}/(\gamma/\pi^{0})_{bkgd}$

The result is shown in Fig. 6(b). Although the systematic error is very large,
the $v_{2}$ of direct photons tends to be positive in 3-6 GeV/$c$ independent
of centrality [23]. It naively implies that the contribution from intrinsic
fragment or bremsstrahlung photons are dominant over that from jet-photon
conversion process. It could happen if the energy loss is very large and most
of the hadrons observed are produced near surface of the system; hard
scattered partons are absorbed before making enough Compton scattering to
produce additional photons. In any case, minimizing the systematic error is
desired before making a conclusion.

##  6 What would be the next measurement?

###  6.1 LHC

At LHC energies, the cross-section of hard photons increases drastically, and
therefore the primary target will be to measure the energy loss of hard
scattered partons with a trigger of prompt photons; the measurement of
$\gamma$-jet correlation [24].

Figure 7: (a) ALICE simulation of tagging efficiency of an optimized cut for
photons (triangle), and rejection power to hadrons (circle) (left), (b) CMS
simulation of fragmentation function reconstruction (middle), and (c) possible
scenario of photon contributions at low to mid $p_{T}$ region at LHC (right).

The experiments planning the measurement at LHC have already started
feasibility studies on the measurement using realistic simulations. Figure
7(a) shows the tagging efficiency of prompt photons and rejection power to
hadrons with an optimized cut in the ALICE detector [25]. At high $p_{T}$, the
tagged samples are shown to be mostly photons. Figure 7(b) shows the
reconstructed fragmentation function in CMS detector [26]. The function is in
good agreement with the input fragmentation function within systematic errors
in hand. These studies show that measurement of energy loss of partons in the
medium is promising at LHC.

Turing eyes into the low to mid $p_{T}$ region, thermal photon emission would
be of great interest as are at CERN and RHIC. Here, we can estimate how well
such photons are resolved. The temperature and the degree of freedom can be
associated with energy density. Using the fact and following the idea
presented in [3], the yield of thermal photons can simply be written as:

$\sigma\sim N_{part}\times(\tau_{freeze}-\tau_{0})\times(s^{1/2})^{1/4\times
2}$

At LHC, the c.m.s. energy will increase by a factor of 70 and the $N_{part}$
by a little, therefore the yield may increase by a factor of $\sim$9\. On the
other hand, photons related to hard-scattered partons would increase
drastically in LHC, because the jet cross-section becomes exponentially larger
as a function of c.m.s. energy. The jet-photon conversion yield would be
proportional to the multiple of jet cross-section and QGP volume, resulting
in:

$\sigma\sim N_{part}\times N_{coll}\times(s^{1/2})^{n}\times g(x_{T})$

where the last term represents hard-scattering cross-section, $n$ is the
$x_{T}$-scaling power, and $\sim 5-8$. Therefore, the jet-photon conversion
would overwhelm thermal photon production. A rough schematics is shown in Fig.
7(c). From these consideration, it would be hard to observe thermal photons,
instead, the medium can be investigated by observing photons from the jet-
photon conversion process, together with $v_{2}$ measurement.

###  6.2 Forward measurement

Comparison of the hadron production at mid and forward rapidities has deduced
particle production mechanisms, such as CGC. There is an interesting
prediction on photon production at mid and forward rapidity, which can
discriminate system expansion scenarios as shown in Fig. 8 [27].

Figure 8: Direct photon cross-section at (a)y=0 (left) and (b)y=2 (right),
under different system expansion scenario.

Landau and Björken expansion of the system would differ the ratio of the yield
in mid and forward rapidity. At RHIC, STAR has photon detector [28] and has
already measured photons at a forward rapidity. PHENIX also has a photon
detector at the rapidity [29]. The detectors would provide interesting results
on the system expansion.

##  7 Summary

The recent direct photon measurements were reviewed and a future prospect was
given. Direct photons would establish a status as one of the most fundamental
measurement in heavy ion collisions in the future.

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