Building robust natural language understanding systems will require a clear characterization of whether and how various linguistic meaning representations complement each other. To perform a systematic comparative analysis, we evaluate the mapping between meaning representations from different frameworks using two complementary methods: (i) a rule-based converter, and (ii) a supervised delexicalized parser that parses to one framework using only information from the other as features. We apply these methods to convert the STREUSLE corpus (with syntactic and lexical semantic annotations) to UCCA (a graph-structured full-sentence meaning representation).  Both methods yield surprisingly accurate target representations, close to fully supervised UCCA parser quality—indicating that UCCA annotations are partially redundant with STREUSLE annotations.  Despite this substantial convergence between frameworks, we find several important areas of divergence.

§ COMPARING MEANING REPRESENTATIONS

This work is licensed under a Creative Commons
Attribution 4.0 International License.
License details:

Several symbolic meaning representations (MRs) support human annotation of text with broad coverage <cit.>.
To date, it is still not completely clear, for all frameworks, what linguistic semantic phenomena they encode, and how it compares to the content represented by the others.
It therefore behooves us to develop a firm linguistic understanding of MRs.
In particular:
are they merely a coarsening and rearranging of syntactic information, such as is encoded in Universal Dependencies <cit.>?
To what extent do they take lexical semantic properties into account?
What does this suggest about the potential for exploiting simpler or better-resourced linguistic representations for improved MR parsing?
Intuitively, we ask whether:

\begin{align*}
\text{sentence-level MR} \stackrel{?}{=} %\\
\text{syntax} + \text{lexical semantics}
\end{align*}

To address this question, we examine UCCA, a document-level MR often used for sentence-level semantics (see <ref>).
hershcovich2019content began to examine the relation of UCCA to syntax, contributing a corpus with gold standard UD and UCCA parses, heuristically aligning them, and quantifying the correlations between syntactic and semantic labels.

Conversely, hershcovich2018multitask
provided some initial evidence that other MRs can be brought to bear on the UCCA parsing task via multitask learning,
but left the details of the relationship between representations to latent (and opaque) parameters of neural models.

In this paper, we aim to close the gap between the two previous investigations by (1) building an interpretable rule-based system to convert from shallower representations
(syntax and lexical semantic units/tags)
into UCCA, forcing us to be linguistically precise about
what UCCA captures and how it “decomposes”;
and (2) training top-performing supervised parsers in a delexicalized setting with
only syntactic and lexical semantic features, as a data-driven mapping
corroborating the rule-based approach.

We perform our analysis on the Reviews section of the English Web Treebank <cit.>,
which has been manually annotated with UD and UCCA; and STREUSLE for lexical semantics (<ref>).
Although at present we only have the necessary evaluation data for English,
the linguistic representations we examine have been applied to multiple languages (<ref>).

Our approach can thus be applied cross-linguistically with minimal adaptation.
Our contributions are:

* Delexicalized rule-based and supervised UCCA parsers, based only on syntax and lexical semantics.

* A linguistically motivated analysis of similarities and differences between the frameworks.

§ REPRESENTATIONS UNDER CONSIDERATION

Example sentence from the Reviews training set
(, “We took our vehicle in for a repair to the air conditioning”),
with UCCA, STREUSLE, and UD annotations.

$\diamond$ UCCA abbreviations: H = parallel scene, L = scene linker, P = process (dynamic event), S = state, A = scene participant, D = scene adverbial, E = non-scene elaborator, C = center (non-scene head), R = relator, F = functional element. The STREUSLE and UD part is adapted from liu2020lexical.

The increasing interest in semantic representation and parsing, and the partial overlap in content between the different frameworks <cit.>, is a main motivation for our inquiry into content differences between UD and STREUSLE, and UCCA. We expect our inquiry to be relevant to other schemes, both in developing a general methodology, and in the insights gathered. For example, besides STREUSLE, UD also serves as the backbone of the DeComp scheme <cit.>, and so information as to its semantic content is important there as well. Argument structural phenomena are at the heart of many MRs, which provide further motivation for empirical studies to the extent lexical semantics and syntax can encode them.

§.§ Universal Conceptual Cognitive Annotation

Universal Conceptual Cognitive Annotation <cit.> targets a level of semantic granularity that abstracts away from syntactic paraphrases in a typologically-motivated, cross-linguistic fashion <cit.>, building on Basic Linguistic Theory <cit.>, an influential framework for linguistic description.
The scheme does not rely on language-specific resources, and sets a low threshold for annotator training. Beyond syntactic paraphrases, UCCA encodes lexical semantic properties such as the aspectual distinction between states and processes (whether an event evolves in time or not).

UCCA has been applied to text simplification <cit.>
and evaluation of text-to-text generation <cit.>.
UCCA corpora are available for English, French and German,
and pilot studies have been conducted on additional languages.

Here we summarize the principles and main distinctions in UCCA.[For further details, see the extensive UCCA annotation manual:

In UCCA, an analysis of a text passage is a DAG (directed acyclic graph)
over semantic elements called units.
A unit corresponds to (is anchored by) one or more tokens,
labeled with one or more semantic categories in relation to a parent unit.[UCCA also supports implicit units which do not correspond to any tokens <cit.>,
but these are excluded from parsing evaluation and we ignore them for purposes of this paper.]
The principal kind of unit is a scene denoting a situation mentioned in the sentence, typically involving a scene-evoking predicate, participants, and (perhaps) modifiers.
Each predicate is labeled as either State (S) or Process (P).
<Ref> contains three scenes: one anchored by the Process took; one anchored by the Process a repair; and one anchored by the possessive pronoun our, which indicates a stative possession relation.
A Participant (A) of a scene is typically an entity or location involved.
Adverbials (D) modify scenes with respect to properties like negation, modality, causativity, direction, manner, etc., which do not constitute an independent situation or entity.
Temporal modifiers are labeled Time (T).

Scenes in UCCA can relate to one another in one of three ways.
A Scene can serve as a Participant within a larger scene;
a Scene can serve to elaborate on a Participant within a Scene (typically relative clauses);
or scenes can be related by parallel linkage in a unit that consists of Parallel Scenes (H) and possibly Linkers (L) describing how they are related. This is seen at the top level of <ref>, where the taking and repair scenes are parallel and the purposive for is a linker.

Other categories only apply to units with no predicate: a semantic head—the Center (C); modifiers of Quantity (Q); and other modifiers, called Elaborators (E).
An Elaborator may itself be a scene, as in our vehicle, where the scene of possession elaborates on the vehicle entity. Similarly, blue vehicles would be analyzed with a stative scene of blueness that elaborates on the vehicles.

Apart from the main semantic content of scenes and participants, UCCA provides the categories: Relator (R) for grammatical markers expressing how a unit relates to its parent unit—in English, these are mainly prepositions and the possessive 's; Function (F) for other grammatical markers with minimal semantic content, such as tense auxiliaries, light verbs, and articles.
Other categories are used for expressing coordination and for expressions expressing speaker perspective outside the propositional structure of the sentence.

Semantically opaque multi-word expressions (e.g., air conditioning in <ref>) are called unanalyzable units,
and are not analyzed internally.
UCCA distinguishes primary edges that always form a tree, and remote edges, which express reentrancies, such as the dotted edge from the possession scene unit to vehicle.

§.§ Universal Dependencies

UD is a syntactic dependency scheme used in many languages,
aiming for cross-linguistically consistent and coarse-grained treebank
annotation. Formally, UD uses bi-lexical trees, with edge labels
representing syntactic relations.
An example UD tree appears at the bottom of <ref>.

§.§ STREUSLE

STREUSLE (Supersense-Tagged Repository of English with a Unified Semantics for Lexical Expressions)
is a corpus annotated comprehensively for several forms of lexical semantics <cit.>.
All kinds of multi-word expressions (MWEs) are annotated, giving each sentence a lexical semantic segmentation.[STREUSLE distinguishes strong MWEs, which are opaque (noncompositional) or idiosyncratic in meaning, and weak MWEs, which represent looser collocations that are nevertheless semantically compositional, like “highly recommended”.]
Syntactic and semantic tags are then applied
to individual units (single- and multi-word).
The semantic tags are supersenses for noun, verb, and prepositionalpossessive units.
Preposition supersenses include two tiers of annotation: scene role labels represent the semantic role of the prepositional phrase marked by the preposition, and function labels represent the lexical contribution of the preposition in itself. The two labels are drawn from the same supersense inventory and are identical for many tokens.

The lexcat annotations (syntactic category of lexical unit) is a slight extension to the Universal POS tagset, adding categories for certain MWE subtypes, such as light verb constructions, following walsh-18 and idiomatic PPs; it also distinguishes possessive pronouns, the possessive clitic 's, and discourse expressions.[STREUSLE tagset documentation:
<Ref> illustrates the MWE, lexcat, and supersense layers.

STREUSLE itself is limited to English, but many of its component annotations have been applied to other languages: verbal multi-word expressions <cit.>, noun and verb supersenses <cit.>, and preposition supersenses <cit.>.

liu2020lexical presented a comprehensive lexical semantic tagger for STREUSLE, which predicts the comprehensive lexical semantic analysis from text, and is freely available.
prange2019made proposed several procedures for integrating STREUSLE supersenses directly into UCCA, refining its coarse-grained categories with preposition supersenses. Enriching a supervised UCCA parser with preposition supersense features from STREUSLE—and, even more so, training a parser to predict supersenses jointly with UCCA—improved parsing performance, revealing the two frameworks to be overlapping but complementary.

§.§ Related Representations

The above annotation schemes define finite inventories of coarse-grained categories to avoid depending on language-specific lexical resources, and thus can in principle be applied to any language.
This fact distinguishes UCCA and STREUSLE from finer-grained sentence-structural representations like FrameNet <cit.> and the Abstract Meaning Representation <cit.>, which relies on PropBank <cit.>. The Prague Dependency Treebank tectogrammatical layer <cit.> uses few lexicon-free roles, but its semantics is determined by a valency lexicon.

The Parallel Meaning Bank <cit.> uses lexicon-free[While the lexical items comprising a linguistic utterance
are naturally essential to its meaning, and therefore influence its semantic representation,
by lexicon-free we mean the ontology and label set of the representation
are not tied to a lexicon or a particular language.] VerbNet <cit.> semantic roles. The STREUSLE tagset for preposition supersenses generalizes VerbNet's role set
to cover non-core arguments/adjuncts of verbs, as well as prepositional complements of nouns and adjectives.
Universal Decompositional Semantics (DeComp) defines semantic roles as bundles of lexicon-free features.

Cross-linguistic applicability in this case is delegated to the parser, which parses sentences in other languages to their corresponding English semantics <cit.>.

[->,level distance=12mm,line width=1pt,
level 2/.style=sibling distance=25mm,
level 3/.style=sibling distance=8mm,
every circle node/.append style=fill=black,
every node/.append style=text height=.6ex,text depth=0]
word = [font=,color=black]
(1_1) [circle]

child node (1_2) [circle]

child node
child node (1_4) [word] Blue cross  edge from parent [draw=none]
edge from parent [draw=none]
child node
child node (1_5) [word] has  edge from parent [draw=none]
edge from parent [draw=none]
child node (1_6) [circle]

child node (1_11) [word] no  edge from parent node[midway, fill=white]  E
child node (1_12) [word] record  edge from parent node[midway, fill=white]  C
edge from parent node[midway, fill=white]  D
child [draw=Pcolor] node (1_7) [circle]

child [draw=black] node (1_8) [word] of  edge from parent node[midway, fill=white]  R
child [draw=black] node (1_9) [word] aa  edge from parent node[midway, fill=white]  F
child [draw=black] node (1_10) [word] reversal  edge from parent node[midway, fill=white]  C
child [draw=black] node (1_13) [word] .  edge from parent node[midway, fill=white]  U
edge from parent node[midway, fill=white]  P
edge from parent node[midway, fill=white]  H
[bend right,draw=Acolor] (1_2) to node[midway, fill=white] A (1_4);
(1_2) to node[midway, fill=white] F (1_5);

n.GROUP           v.stative           n.COM-     p.Topic        n.EVENT

[edge below]

Blue  & cross & has  & no  & record & of  & aa  & reversal & .

propn & propn & verb & det & noun   & adp & det & noun     & punct

[edge unit distance=1em]39punct

[->,level distance=13mm,line width=1pt,
level 2/.style=sibling distance=29mm,
level 3/.style=sibling distance=15mm,
level 4/.style=sibling distance=7mm,
every circle node/.append style=fill=black,
every node/.append style=text height=.6ex,text depth=0]
word = [font=,color=black]
(1_1) [circle]

child node (1_2) [circle]

child node
child node
child node (1_4) [word] Blue cross  edge from parent [draw=none]
edge from parent [draw=none]
edge from parent [draw=none]
child node
child node
child node (1_3) [word] has  edge from parent [draw=none]
edge from parent [draw=none]
edge from parent [draw=none]
child [draw=Acolor] node (1_5) [circle]

child node
child node (1_8) [word] no  edge from parent [draw=none]
edge from parent [draw=none]
child node
child node (1_6) [word] record  edge from parent [draw=none]
edge from parent [draw=none]
child [draw=black] node (1_9) [circle]

child node (1_11) [word] of  edge from parent node[midway, fill=white]  R
child node (1_12) [word] aa a  edge from parent node[midway, fill=white]  F
child [draw=Pcolor] node (1_10) [word] reversal
edge from parent node[midway, fill=white]  E
edge from parent node[midway, fill=white]  A
child node
child node
child node (1_7) [word] .  edge from parent [draw=none]
edge from parent [draw=none]
edge from parent [draw=none]
edge from parent node[midway, fill=white]  H
[bend right,draw=Acolor] (1_2) to node[midway, fill=white] A (1_4);
[draw=Scolor] (1_2) to node[midway, fill=white] S (1_3);
[draw=Pcolor] (1_9) to node[midway, fill=white] P (1_10);
[bend left] (1_2) to node[midway, fill=white] U (1_7);
(1_5) to node[midway, fill=white] D (1_8);
(1_5) to node[midway, fill=white] C (1_6);

Example sentence (, reading “Blue cross has no record of aa[sic] reversal”) with gold-standard UCCA graph;
STREUSLE MWEs
and supersenses; and UD coarse-grained POS tags and relations (left);
and UCCA graph output by the rule-based converter (right).

§ FIRST CONVERSION APPROACH: RULE-BASED UCCA PARSING FROM SYNTAX AND LEXICAL SEMANTICS

Here we describe a system to produce UCCA analyses for text,
given UD syntactic graphs and STREUSLE lexical semantic annotation.

This is a rule-based converter that inspects the UD structure in tandem with STREUSLE annotations to build an UCCA parse.[The new conversion and analysis code is available at <https://github.com/danielhers/streusle/tree/streusle2ucca>]
An analysis of the converter's successes and failures (<ref>)
will, in turn, reveal the similarities and differences between the schemes. What follows is an overview of the algorithm; futher details about the rules are given in <ref>.[Before writing the new converter from the ground up, we tried modifying the hershcovich2019content UD-only conversion code to use STREUSLE information. Details and analysis of that system appear in <ref>.]

MWEs and unanalyzable units.
Based on STREUSLE MWE annotation,
we group together text tokens into single unanalyzable semantic units when
they are annotated as strong MWEs (we do not use weak MWEs, as UCCA does not encode them),
except for light verbs and annotations that would lead to cycles.
MWE-internal dependency edges are discarded so they will not be
processed later.
Punctuation is marked .
Mappings between units and dependency nodes are maintained.
For example, in

This is one of the worst places I have stayed, we cut out[sic] stay short and went to the Mulberry. (),

cut…short is an unanalyzable unit according to the gold UCCA annotation,
but the syntactic relation between
cut and short, namely ,
does not indicate that; only few
s, in general, correspond to UCCA unanalyzable units
In STREUSLE, however, cut…short is annotated as a strong MWE,
whose lexcat is V.VID (idiomatic verb).
Our rules create an unanalyzable unit covering this phrase,
which matches the gold UCCA annotation in this case.
The same is true for Blue cross[Short for Blue Cross Blue Shield, a well-known health insurance organization in the United States.] in <ref>.

STREUSLE supersenses and UCCA scene-evoking phrases.
In a top-down traversal of the dependency parse, we visit each word's
lexical unit and decide whether it evokes a main relation (scene-evoking phrase), using rules based on
syntactic and lexical semantic features:
copular be and stative have,
adjectives (excluding a small list of quantity adjectives),
existential there,
non-discourse adverbs with a copula dependent,
predicative prepositions and
copulas introducing a predicate nominal,
as well as common nouns supersense-tagged as n.attribute, n.feeling, or n.state
are labeled S;
thanks, thank you and common nouns supersense-tagged as n.act, n.phenomenon, n.process, or n.event (with the exception
of nouns denoting a part of the day) are labeled P.
For example, the noun reversal in <ref>
is marked as P,
since its supersense of n.event informs us that this is
a scene-evoking noun.
Relational nouns, supersense-tagged as n.person or n.group that also
match kinship/occupation lists or suffixes, are labeled P+A.

STREUSLE lexical categories and UCCA edge categories.
After identifying main relations and non-scene units, the exact category for some units still needs to be determined. The main relations themselves are labeled
P or S
depending on lexical categories and supersenses.
Determiners, auxiliaries, and copulas are generally labeled F; vocatives
and interjections, G. Exceptions include modal auxiliaries
(D), and demonstrative/quantifier determiners modifying a non-scene unit
(E/Q respectively).
Verbal arguments and modifiers of scene-evoking non-verbal phrases are labeled
Possessive clitics and prepositions are attached as R, unless
a possessive clitic marks canonical possession, in which case it is

Secondary verb constructions.
These constructions are structured differently in UCCA and UD:
in “members who won't stop talking”,
the verb “stop” is the UD head and “talking” is a dependent.
In UCCA “stop” is D and “talking” is the main
relation, labeled P.
To normalize the treatment of these constructions,
they are marked and eventually restructured such that the syntactic head is labeled
D and the syntactic dependent is labeled as the main relation.

Coordination and lexical heads.
Traversing the graph top-down again, coordination between
scene units is labeled L (Linker), and between
non-scene units, N (Connector).
Scene units are labeled H,
and non-scene unit heads, C.
heads of units are labeled C, P, or S, and
scene units as H where necessary;
in “X of Y” constructions involving
quantities/species, Y is identified as the C.

§ SECOND CONVERSION APPROACH: DELEXICALIZED SUPERVISED UCCA PARSING

Previous work tackled the UCCA parsing task using supervised learning.
In order to complement and validate the analysis of the rule-based converter, we  compare its findings to a delexicalized supervised parser, that can be seen as inducing a converter from data.

By removing all word and lemma features from these parsers, and instead adding features based on gold UD and STREUSLE annotations, we obtain supervised “converters”, which can be used for data-driven analysis and complement the rules.

This UCCA parser <cit.>
is based on a transition-based algorithm with a neural network
transition classifier, using a BiLSTM for encoding input representation,
with word, lemma, and syntactic features embedded as real-valued features.

We add the supersense and lexcat from STREUSLE as
embedding inputs to the TUPA BiLSTM
(concatenated with existing inputs).
For prepositions, we add both the scene role and function
(see <ref>).[Our code to enrich UCCA data with STREUSLE features is available at <https://github.com/jakpra/ucca-streusle>]

HIT-SCIR Parser.
This is a transition-based parser for several MR frameworks, including UCCA <cit.>.
It achieved the highest average score in the CoNLL 2019 shared task <cit.>.

While che-etal-2019-hit fine-tuned BERT <cit.> for contextualized word representation,
our delexicalized version replaces it with UD and STREUSLE features: POS tag, dependency relation, supersenses (scene role and function; see <ref>), the lexical category of the word or the MWE that the word is part of, and the BIO tag. These are concatenated to form word representations.[Our modified code for the HIT-SCIR parser is available at <https://github.com/danielhers/hit-scir-ucca-parser>]

Training     Development

# Tokens     44,804     5,394

# Sentences     2,723     554

EWT Reviews data statistics. We use only the training and development splits for analysis.

§ EXPERIMENTS

We use the Reviews section from UD 2.6 English_EWT <cit.>,
with lexical semantic annotations from STREUSLE 4.4

and with UCCA graphs from UCCA_English-EWT v1.0.1
We use the standard train/development split for this dataset,
and do not use the test split to avoid over-analyzing it,
although all datasets contain annotations for it too.

The data statistics are listed in <ref>.

Rule-based converters.

We evaluate the rule-based converter (<ref>),
as well as the syntax-based converter from
hershcovich2019content, which uses the UD tree and a majority-based
category mapping (based on the most common UCCA category in the training set for each UD relation).
This converter is oblivious to lexical semantics.

We train TUPA v1.3
and the HIT-SCIR parser
with gold-standard features from UD and STREUSLE, for equal conditions
with the converters,
using default hyperparameters.
Categorical features are added as 20-dimensional embeddings.

Scores are averaged over 3 models with different random seeds.
For TUPA, we ablate UD- or STREUSLE-based features to quantify the contribution of each.

We use standard UCCA parsing evaluation, matching edges by the terminal yields of their endpoint units.[The terminal yield of a unit is defined based on the
graph's primary edges only, as standard in UCCA evaluation.]
Labeled precision, recall and F1-score consider the edge categories when matching
Where an edge has multiple categories,
each of them is considered separately.

Primary F1     Remote F1

Syntax-based converter with gold UD <cit.>     56.6     28.0

Our rule-based converter with gold UD + STREUSLE     71.7     44.2

TUPA, delexicalized with gold UD + STREUSLE features           69.5     46.4

UD features only           64.4     35.9

STREUSLE features only     62.4     27.5

HIT-SCIR, delexicalized with gold UD + STREUSLE features      67.9     41.6

TUPA with gold UD features + GloVe <cit.>     71.7     47.0

HIT-SCIR (BERT-Large)      71.9     41.8

HIT-SCIR (GloVe)      67.0     42.4

with gold UD + STREUSLE features      72.2     46.9

Labeled F1 (in %) for primary and
remote edges on the UCCA EWT Reviews dev set,
for rule-based systems (top),
delexicalized supervised parsers with gold UD+STREUSLE (middle),
and supervised parsers with word features (bottom).

§ RESULTS

<Ref> shows the EWT Reviews dev scores.
For comparison with parsers that have access to words,
we also show the TUPA dev results from
hershcovich2019content, who used syntactic features from
the gold UD annotation and GloVe <cit.>;[Parsing from gold features is by no means a realistic scenario, but we give the scores as a reference for the converters.]
and the HIT-SCIR parser with
BERT/GloVe, and with UD+STREUSLE features.

Rules with gold UD and STREUSLE close the gap
between the syntax-based converter and parsers with word information,
reaching the same primary labeled F1 as TUPA with word features.
This is surprising (since supervised parsers are known to usually outperform rule-based ones),
and suggests that the training data (see <ref>) was insufficient for the parser
to learn a mapping as accurate as the complex conversion rules (described in <ref>).
Enhancing GloVe-based HIT-SCIR with UD and STREUSLE yields similar results.
However, many errors remain in both approaches, indicating that UCCA and STREUSLE are
far from equivalent.
We analyze these errors in <ref> to investigate the frameworks
and the relationship between them.

Noticeable drops in the ablations (TUPA with UDSTREUSLE only) show that both UD-provided structure and relationentity types from STREUSLE supersenses are needed to make up for the missing lexical information, but also that lacking UD hurts more.
This is expected, as the parser resorts to guessing when it lacks sufficiently informative input, and the chance of errors when guessing the UCCA structure (for which UD is informative) is much larger than for assigning edge labels (for which STREUSLE provides more fine-grained cues).
The ablated TUPA models still outperform the syntax-based converter, indicating that there are indeed structural signals in STREUSLE and semantic signals in UD, which TUPA can salvage.

5@lPredicted Category     15cGold Category

1cA   1cA|G   1cA|P   1cA|S   1cC   1cD   1cD|T   1cE   1cF   1cG   1cH   1cL   1cN   1cP   1cQ   1cR   1cS   1cT   1c$\emptyset$

A   758   4   7   12   17   11      9   4   1   6   1      14   1   1   19      150

A|P            1   1

A|S            8   2

C   50      7   12   457   27      11   1   1   12   3      31   2   5   12   1   48

D   10            12   280      40   8   12   2   2      6   4   1   7   18   20

E   48   1         20   42   1   294   3   1   17         3   7   1   24   4   49

F   3                        613               1   1      3      1

G      2                     2   6   2                  2      4

H   40   2      1   29   6      13   1      450   4      22      2   8      265

L                  7      1   19   1      221   14   1      27         5

N               1   1      1            10   31      1            2

P   3            16   15   1   2   13   12   1   1      345      2   29      32

Q               8   5      1                     40            1

R   3            6                     13      1      211   14      3

S   6            48   49      4   26      6         10      1   251      5

T   2            4   2   3                        1         45   5

$\emptyset$   148   1   3   6   136   60      100   32   1   124   9   2   65   12   34   23   6

dev set confusion matrix for the rule-based converter.
The last column (row)
shows the number of predicted (gold-standard) edges of each category
that do not match any gold-standard (predicted) unit.

§ ANALYSIS

<Ref> presents the EWT Reviews dev confusion matrix for
the converter's output and gold UCCA.
The delexicalized parsers' confusion matrix (in <ref>) is similar.[For multiple UCCA units with the same terminal yield (i.e., units with a single non-remote child),
we take the top category only, to avoid double-counting.]
Note that we consulted the training set iteratively while developing the rules, addressing many recurring issues that would show up as prominent confusions.[A full report
of dev set outputs is included in <https://github.com/danielhers/streusle/blob/streusle2ucca/uccareport.dev.tsv>.]

We proceed with an extensive error analysis of the converter,
to point out similarities and delineate remaining
divergences, which we stipulate  constitute content differences between UCCA and the combination of syntax and lexical semantics from UD and STREUSLE.
<Ref> shows gold annotation and the converter's predictions.

STREUSLE Annotation     Predicted UCCA Annotation     Gold UCCA Annotation

Noun compounds

$\underset{\sst{n.substance}}{\text{tap\_water}}$     tap water (unanalyzable)     [E tap] [C water]

$\underset{\sst{n.event}}{\text{road\_construction}}$     [P road construction]     [A road] [P construction]

Adverbs and linkage

$\underset{\sst{v.vid}}{\text{Gets\_busy}}$ so $\underset{\sst{v.motion}}{\text{come}}$ early     [H [P Gets busy] ] [L so] [H [P come] [T early] ]     [H [D Gets] [S busy] ] [L so] [H [P come] [T early] ]

so easy to $\underset{\sst{v.motion}}{\text{load}}$     [L so] [H [S easy] [A [F to] [P load] ] ]      [D [E so] [C easy] ] [F to] [P load]

Scene-evoking nouns

a $\underset{\sst{n.food}}{\text{meal}}$ $\underset{\sst{p.locus}}{\text{on}}$ the $\underset{\sst{n.communication}}{\text{menu}}$     [F a] [C meal] [E [R on] [F the] [C menu] ] ]     [F a] [C meal] [E [R on] [F the] [C menu] ] ]

$\underset{\sst{v.communication}}{\text{answered}}$ all $\underset{\sst{p.gestalt}}{\underset{\sst{p.originator}}{\text{my}}}$ $\underset{\sst{n.communication}}{\text{questions}}$     [P answered] [A [Q all] [A my] [C questions] ]     [P answered] [A [D all] [A my] [P questions] ]
Examples for cases where the rule-based converter produced the correct UCCA annotation due to converging analyses,
as well as cases where it produced a wrong annotation due to a divergence.

§.§ High Match—Converging Analyses

As are recovered with high precision and recall.
This is generally expected as most syntactic subjects and objects, as well as some obliques and even clauses, signify scene participants.
Where syntax and semantics diverge, STREUSLE supersenses can rule out unlikely candidates.
The most common sources of missed As are structural errors, i.e., incorrect scene structures, overly flat units containing more than the referential words, or misinterpreted noun compounds (see <ref> below).

Function words.

As evident in <ref>, Function words (F) are
accurately predicted. The distinction between words that
contribute to the semantic meaning and those that do not
is preserved between STREUSLE and UCCA, except for some
cases—mainly infinitive “to”.

Linkers (L) are relatively easy: they are prototypically instantiated by syntactic co- and subordinators. To the extent that these are considered adpositional by STREUSLE, their supersense helps disambiguate between inter-scene linkage and Connectors (N) of non-scenes.

§.§ Partial Match—Inferrable by Combining Syntax and Lexical Semantics

Time (T) and Quantifier (Q) expressions frequently coincide with certain syntactic categories such as adverbs and prepositions, and can typically be identified from corresponding supersenses, if available.
The converter tends to err on the conservative side, falling back to Adverbials (D) and Elaborators (E) when it cannot find sufficient explicit semantic evidence.

§.§ Low Match—Divergences or Insufficient Information

Noun compound interpretation.
Lexical composition in noun compounds evokes various forms of event structures,
which are underspecified by  the meaning of the constituent words
While often compounding is used for Elaboration, as in
[E tap] [C water],
it is not necessarily always the case. For example, in [C sea] [C bottom] both “sea" and “bottom" are Center, since they reflect part-whole relations.
The modifier may also be a Participant in the scene evoked by the head,
as in [A road] [P construction].
This is partially encoded in STREUSLE, as the fact that the MWE “road construction”
has the n.event supersense indicates that it is scene-evoking,
but it still does not reveal the relationship between the constituent words.

Adverbs and linkage.
While many syntactic adverbs are semantically Linkers (“well”, “though”),
neither UD nor STREUSLE distinguish them from Adverbials
(“really”, “possibly”).
Some adverbs, like “so”, can serve either role (see <ref>),
a distinction that is only made in UCCA.

C is often unaligned due to the different notions of
multi-word expressions in STREUSLE and UCCA:
“tap water” is considered a strong MWE in STREUSLE,
but is internally analyzed (with “water” being the Center) in UCCA
(see <ref>), leading to an unmatched C.

§.§.§ Scene-evokers

While the concept of scenes is central to UCCA,  correctly identifying scene-evoking words
is one of the more difficult tasks for our converter.
“Scene-ness” clearly goes beyond syntax (not all verbs evoke scenes and scenes can be evoked by a wide range of POS) and STREUSLE supersenses in isolation are often too coarse to resolve the question whether a given word evokes a scene and, if so, whether it is a Process (P) or a State (S).
The former decision is generally somewhat easier for the converter (Recall of scene-evokers: 71.3%) than distinguishing between P (Recall: 69.1%) and S (64.0%).
Below we examine a few recurring phenomena involving scenes.

Scene-evoking nouns.
STREUSLE underspecifies whether nouns evoke scenes.
For example,
“menu” and “question” both have the n.communication supersense
in STREUSLE, but “menu” does not evoke a scene, while “question” might.
Other similarly broad supersenses include n.cognition (“decision”: potential scene, “fact”: non-scene) and n.possession (“purchase”: potential scene, “money”: non-scene).

Relational nouns.
This is a special case of scene-evoking nouns <cit.>, both refering to an entity and evoking a scene in which the entity generally or habitually participates.[E.g., a teacher is a person who teaches, and a friend is a person in a friendship relation with another person.]
These units have two categories in UCCA, either A|P or A|S.
The converter relies here on a combination of n.person or n.group supersenses and lexical lists.
However, these nouns' scene-ness is often not recognized and they are confused with regular A or C.

Scene-evoking adjectives.
Inspecting the high-frequency confusions, adjectives stand out as persistent error inducers. Different classes of adjectives are handled differently in UCCA: e.g., while most adjectives are scene-evoking, pertainyms (academic), inherent-composition modifiers (sugary), and quantity modifiers (many) are not. Some adjectives are ambiguous: a legal practice may refer to a behavior that is legal as opposed to illegal, in which case it should be scene-evoking, or to a law office, in which case it should not. Enriching STREUSLE with supersenses for adjectives <cit.> might be fruitful for such distinctions. Even with lexical disambiguation, the scene attachment of the adjective may be ambiguous: e.g. a good chef probably means a chef who cooks well, so good should be an Adverbial in the scene evoked by chef—in contrast with a tall chef, where tall is not part of the cooking scene and instead should evoke a State.
Predicative adjectives, and adjective modifiers in predicative NPs, are another source of difficulty, especially when they occur in fragments:
sometimes the adjective is annotated as evoking the main scene, and sometimes not. Determining this requires making various semantic distinctions, which are not fully represented in STREUSLE.

§ CONCLUSION

We have presented an extensive analysis of the similarities and differences between STREUSLE and UCCA on the EWT Reviews corpus,
assisted by two complementary methods:
manual rule-based conversion, and delexicalized parsing.
Both approaches arrived at similar results, showing that the conversion
between the frameworks can be moderately accurate, while also revealing
important divergences, namely distinctions made in UCCA but not in STREUSLE:
semantic relation between nouns in compounds,
adverbial and linkage usage of adverbs,
and the scene-evoking status of nouns, possessives and adjectives,
among others.

Enriching supervised parsers with lexical semantic features improves parsing performance when using gold input. While this paper focuses on analysis, future work will investigate using predicted features with a parser/tagger <cit.>.
This approach is expected to improve parsing performance and robustness, demonstrating the utility of linguistically-informed approaches in complementing general supervised semantic parsers.

§ ACKNOWLEDGEMENTS

This research was supported in part by grant 2016375 from the United States–Israel Binational Science Foundation (BSF), Jerusalem, Israel. ML is funded by a Google Focused Research Award. We acknowledge the computational resources provided by CSC in Helsinki and Sigma2 in Oslo through NeIC-NLPL (www.nlpl.eu).

§ DETAILS OF RULE-BASED CONVERTER

The following is a detailed description of the rules used in the
rule-based parser (<Ref>).
This gives a step-by-step overview of the algorithm for constructing an UCCA semantic graph using STREUSLE/UD annotations.
It is not a full specification and omits many details of the criteria and operations, but should be helpful for understanding the full code, available in <https://github.com/danielhers/streusle/blob/streusle2ucca/conllulex2ucca.py>.
A running example is given at each step for the sentence:

There's plenty of parking, and I've never had an issue with audience
members who won't stop talking or answering their cellphones.

It has the following gold annotation:

[H [F There] [F 's] [D plenty]
[P [R of] [C parking] ] ] [U ,] [L and]
[H [A I] [F 've] [DT never] [F had]
[P [F an] [C issue] ] [A [R with] [C [A
audience] [P members] ] [E [R who] [H
[A* members] [F wo] [D n't] [D stop] [P talking] ]
[L or] [H [A* members] [P answering] [A [E
[SA their] [A* cellphones] ] [C cellphones] [U
.] ] ] ] ] ]

§.§ Step 0.1: Transform the UD dependency parse

Split the final preposition off from V.IAV MWEs like
take care of, as it is usually not treated as part of the
verbal unit in UCCA. If the remaining part is still an MWE, it is
labeled V.LVC.full or V.VPC.full depending on its

§.§ Step 0.2: Initialize lexical units under the UCCA

* Each strong lexical expression (single-word or MWE) in STREUSLE is
treated as an UCCA unit, with the following exceptions:

* An MWE with lexcat V.LVC.cause is broken into two units:
for the light verb modifies the main predicate in a

* An MWE with lexcat V.LVC.full is broken into two units:
for the light verb modifies the main predicate in a

* An MWE annotation is discarded if it would lead to a cycle in the
dependencies such that the highest token of the MWE is dependent on
a token outside of the MWE, which is dependent on another token of
the MWE.

* MWE-internal dependency edges are discarded so they will not be
processed later.

* Punctuation is marked .

* Mappings between units and dependency nodes are maintained.

§.§ Step 0.3: Identify which lexical units are main relations

In a top-down traversal of the dependency parse, visit each word's
lexical unit and decide whether it evokes a state (), process
(), undetermined between state or process (), or
does not evoke a scene ():

* If already labeled , , or , do nothing.

* If an adjective not from a small list of quantity adjectives, label

* If existential there, label . If a be verb in
an existential construction, label  and swap the positions
of be and there in the dependency parse so there
is the head.

* If an adverb not attached as discourse and it has a copula, label

* thanks and thank you are .

* In most cases, predicative prepositions are .

* A copula introducing a predicate nominal (non-PP) is labeled
and promoted to the head of the dependency parse, unless
the nominal is scene-evoking. (The top-down traversal order ensures
the nominal is reached first.)

* If a common noun, mark as

*  if supersense-tagged as attribute, feeling, or state

*  if act, phenomenon, process, or event (with the exception
of nouns denoting a part of the day)

* a relational noun if person or group and
matching kinship/occupation lists or suffixes

*  otherwise

* If a verb or copula not handled above, label

* Else label

In the notation, “UNA” means “lexical” (it originally meant

[DUMMYROOT [S [UNA There] ] [- [UNA 's] ] [- [UNA
plenty] ] [- [UNA of] ] [S [UNA parking] ] [U ,]
[- [UNA and] ] [- [UNA I] ] [- [UNA 've] ] [-
[UNA never] ] [+ [F had] … [UNA issue] ] [-
[UNA an] ] … [- [UNA with] ] [- [UNA
audience] ] [- [UNAAP [UNA members] ]
] [- [UNA who] ] [- [UNA wo] ] [- [UNA n't] ]
[+ [UNA stop] ] [+ [UNA talking] ] [- [UNA or] ]
[+ [UNA answering] ] [- [UNA their] ] [- [UNA
cellphones] ] [U .] ]

§.§ Step 1: Attach functional and discourse modifier

Determiners, auxiliaries, copulas are generally ; vocatives
and interjections, . Exceptions include modal auxiliaries
(), demonstrative determiners modifying a non-scene unit
(), quantifier determiners modifying a non-scene unit

Omitting the root:

[S [UNA There] [F [UNA 's] ] ] [- [UNA
plenty] ] [- [UNA of] ] [S [UNA parking] ] [U ,]
[- [UNA and] ] [- [UNA I] ] [+ [F [UNA
've] ] … [F had] [F [UNA an] ]
[UNA issue] ] [- [UNA never] ] … [- [UNA
with] ] [- [UNA audience] ] [- [UNAAP
[UNA members] ] ] [- [UNA who] ] [+ [F [UNA wo]
] … [UNA stop] ] [- [UNA n't] ] … [+
[UNA talking] ] [- [UNA or] ] [+ [UNA answering] ]
[- [UNA their] ] [- [UNA cellphones] ] [U .]

§.§ Step 2: Attach other modifiers: adverbial,
adjectival, numeric, compound, possessive, predicative-PP, adnominal-PP;
as well as possessive clitic and preposition (as , unless
possessive clitic marks canonical possession in which case it is

[S [UNA There] [F [UNA 's] ] ] [- [UNA plenty]
[E [S [R [UNA of] ] [UNA parking] ]
] ] [U ,] [- [UNA and] ] [- [UNA I] ] [+ [F
[UNA 've] ] [T [UNA never] ] [F had] [F
[UNA an] ] [UNA issue] [A [R [UNA with] ]
[E [UNA audience] ] [UNAAP [UNA
members] ] … [E [+ [A* members]
[F [UNA wo] ] [D [UNA n't] ] [UNA
stop] ] ] ] ] [- [UNA who] ] … [+ [UNA
talking] ] [- [UNA or] ] [+ [UNA answering] ] [-
[E [AS [UNA their] ] [A*
cellphones] ] [UNA cellphones] ] [U .]

§.§ Step 3: Process verbal argument structure relations:
subjects, objects, obliques, clausal complements; flag secondary
(non-auxiliary) verb

[S [UNA There] [F [UNA 's] ] [A [UNA
plenty] [E [S [R [UNA of] ] [UNA parking] ] ] ]
] [U ,] [- [UNA and] ] [+ [A [UNA I] ]
[F [UNA 've] ] [T [UNA never] ] [F had] [F [UNA
an] ] [UNA issue] [A [R [UNA with] ] [E [UNA
audience] ] [UNAAP [UNA members] ] [E
[+ [A* members] [R [UNA who] ] [F [UNA wo] ] [D
[UNA n't] ] [UNA stop] [ [+ [UNA talking]
] ] ] ] ] ] [- [UNA or] ] [+ [UNA answering]
[A [E [AS [UNA their] ] [A*
cellphones] ] [UNA cellphones] ] ] [U .]

§.§ Step 4: Coordination

Traversing the graph top-down: for each coordinate construction with
conjuncts' units' categories X and Y, create a ternary-branching
structure  if X is scene-evoking
(, , or ) and

There's plenty…and I've never had an issue with…

[S(COORD) [S [UNA There] [F [UNA 's] ] [A
[UNA plenty] [E [S [R [UNA of] ] [UNA parking] ]
] ] ] … [L [UNA and] ] [+ [A [UNA
I] ] [F [UNA 've] ] [T [UNA never] ] [F had]
[F [UNA an] ] [UNA issue] [A [R [UNA with] ] [E
[UNA audience] ] [UNAAP [UNA members]
] [E [+ [A* members] [R [UNA who] ] [F [UNA wo]
] [D [UNA n't] ] [UNA stop] [ [+ [UNA talking]
] ] ] ] ] ] ] [U ,] … [- [UNA or] ]
[+ [UNA answering] [A [E [AS [UNA their] ]
[A* cellphones] ] [UNA cellphones] ] ] [U .]

…won't stop talking or answering…

[S(COORD) [S [UNA There] [F [UNA 's] ] [A [UNA
plenty] [E [S [R [UNA of] ] [UNA parking] ] ] ]
] … [L [UNA and] ] [P [A [UNA I] ]
[F [UNA 've] ] [T [UNA never] ] [F had] [F [UNA
an] ] [UNA issue] [A [R [UNA with] ] [E [UNA
audience] ] [UNAAP [UNA members] ] [E
[P [A* members] [R [UNA who] ] [F [UNA wo]
] [D [UNA n't] ] [UNA stop] [ [+(COORD)
[P [UNA talking] ] [L [UNA or] ] [P
[UNA answering] [A [E [AS [UNA their] ] [A*
cellphones] ] [UNA cellphones] ] ] ] ]] ] ] ]
] [U ,] … [U .]

§.§ Step 5: Decide  or  for

Copula be and stative have are ; other verbs, as
well as nouns tagged as ACT, PHENOMENON, PROCESS, or EVENT, are

[S(COORD) [S [UNA There] [F [UNA 's] ] [A [UNA
plenty] [E [S [R [UNA of] ] [UNA parking] ] ] ]
] … [L [UNA and] ] [P [A [UNA I] ]
[F [UNA 've] ] [T [UNA never] ] [F had] [F [UNA
an] ] [UNA issue] [A [R [UNA with] ] [E [UNA
audience] ] [UNAAP [UNA members] ] [E
[P [A* members] [R [UNA who] ] [F [UNA wo]
] [D [UNA n't] ] [UNA stop] [ [+(COORD)
[P [UNA talking] ] [L [UNA or] ] [P
[UNA answering] [A [E [AS [UNA their] ] [A*
cellphones] ] [UNA cellphones] ] ] ] ]] ] ] ]
] [U ,] … [U .]

§.§ Step 6.1: Restructure for secondary verbs

[S(COORD) [S [UNA There] [F [UNA 's] ] [A [UNA
plenty] [E [S [R [UNA of] ] [UNA parking] ] ] ]
] … [L [UNA and] ] [P [A [UNA I] ] [F
[UNA 've] ] [T [UNA never] ] [F had] [F [UNA an]
] [UNA issue] [A [R [UNA with] ] [E [UNA audience]
] [UNAAP [UNA members] ] [E [P [A*
members] [R [UNA who] ] [F [UNA wo] ] [D [UNA
n't] ] [D [UNA stop] ] [+(COORD) [P
[UNA talking] ] [L [UNA or] ] [P [UNA answering]
[A [E [AS [UNA their] ] [A* cellphones] ]
[UNA cellphones] ] ] ]] ] ] ] ] [U ,] …
[U .]

§.§ Step 6.2: Articulation—marking lexical
heads of units as , , or , and renaming
scene units as  where necessary; determination of
involves “X of Y” constructions involving

[S(COORD) [H(S) [S [UNA There] ] [F [UNA 's]
] [A [Q [UNA plenty] ] [E [H(S) [R
[UNA of] ] [S [UNA parking] ] ] ] ] ]
… [L [UNA and] ] [H(P) [A [UNA I] ]
[F [UNA 've] ] [T [UNA never] ] [F had] [F [UNA
an] ] [P [UNA issue] ] [A [R [UNA with] ]
[E [UNA audience] ]
[AP [UNA members] ] ] [E [H
[A* members] [R [UNA who] ] [F [UNA wo] ] [D
[UNA n't] ] [D [UNA stop] ] [+(COORD) [H(P)
[P [UNA talking] ] ] [L [UNA or] ] [H(P)
[P [UNA answering] ] [A [E
[H(AS)S [AS [UNA their]
] ] [A* cellphones] ] [C [UNA cellphones] ]
] ] ] ] ] ] ] ] [U ,] … [U .]

§.§ Steps 7+: Cleanup

Remove temporary decorations on  units from articulation; move
units for punctuation to more convenient attachment points;
convert remaining  and  labels; wrap stray
and  units with  scenes; remove
and other temporary designations in the graph

[S(COORD) [H [S [UNA There] ] [F [UNA 's] ]
[A [Q [UNA plenty] ] [E [H [R [UNA of] ]
[S [UNA parking] ] ] ] ] ] … [L [UNA and]
] [H [A [UNA I] ] [F [UNA 've] ] [T
[UNA never] ] [F had] [F [UNA an] ] [P [UNA
issue] ] [A [R [UNA with] ] [E [UNA audience] ]
[C [AP [UNA members] ] ] [E [H [A*
members] [R [UNA who] ] [F [UNA wo] ] [D [UNA
n't] ] [D [UNA stop] ] [+(COORD) [H [P [UNA
talking] ] ] [L [UNA or] ] [H [P [UNA
answering] ] [A [E [C [AS [UNA their]
] ] [A* cellphones] ] [C [UNA cellphones] ] ] ]
] ] ] ] ] ] [U ,] … [U .]

[S(COORD) [H [S [UNA There] ] [F [UNA 's] ] [A
[Q [UNA plenty] ] [E [H [R [UNA of] ] [S [UNA
parking] ] ] ] ] ] [U ,] [L [UNA and] ]
[H [A [UNA I] ] [F [UNA 've] ] [T [UNA never]
] [F had] [F [UNA an] ] [P [UNA issue] ] [A [R
[UNA with] ] [E [UNA audience] ] [C [AP
[UNA members] ] ] [E [H [A* members] [R [UNA who]
] [F [UNA wo] ] [D [UNA n't] ] [D [UNA stop] ]
[+(COORD) [H [P [UNA talking] ] ] [L [UNA or] ]
[H [P [UNA answering] ] [A [E [C [AS [UNA
their] ] ] [A* cellphones] ] [C [UNA cellphones] ]
] ] ] ] ] ] ] [U .] ]

[H [H [S [UNA There] ] [F [UNA 's] ] [A
[Q [UNA plenty] ] [E [H [R [UNA of] ] [S [UNA
parking] ] ] ] ] ] [U ,] [L [UNA and] ] [H
[A [UNA I] ] [F [UNA 've] ] [T [UNA never] ]
[F had] [F [UNA an] ] [P [UNA issue] ] [A [R
[UNA with] ] [E [UNA audience] ] [C [AP
[UNA members] ] ] [E [H [A* members] [R [UNA who]
] [F [UNA wo] ] [D [UNA n't] ] [D [UNA stop] ]
[H [H [P [UNA talking] ] ] [L [UNA or] ]
[H [P [UNA answering] ] [A [E [C [AS [UNA
their] ] ] [A* cellphones] ] [C [UNA cellphones] ]
] ] ] ] ] ] ] [U .] ]

[H [H [S There] [F 's] [A [Q plenty] [E [R of]
[S parking] ] ] ] [U ,] [L and] [H [A I] [F
've] [T never] [F had] [F an] [P issue] [A [R
with] [E audience] [C [AP members] ] [E [A*
members] [R who] [F wo] [D n't] [D stop] [H [H
[P talking] ] [L or] [H [P answering] [A [E [C
[AS their] ] [A* cellphones] ] [C cellphones]
] ] ] ] ] ] [U .] ]

§ AN ALTERNATIVE CONVERTER

Prior to the converter described in this paper, we had developed a converter based on
the UD-to-UCCA converter of hershcovich2019content.
However, after rewriting the converter from scratch in a more modular fashion,
yielding the current converter (<ref>), we found that it obtains better performance, and therefore
abandoned the previous, alternative converter.
For completeness, we describe the previous converter here. The source code is available online.[<https://github.com/danielhers/streusle/blob/streusle2ucca-2019/conllulex2ucca.py>]

§.§ Dependency Transformations

apply several pre-conversion dependency transformation,
to match UCCA's flat linkage structure:
dependents and  dependents of
are promoted to be siblings of their heads, as in UCCA they are often L
between scenes.
, ,  and
are also promoted. We implement the same transformations using

§.§ Label Mapping

We assign UCCA categories to graph edges,
translating each UD relation to the
most commonly co-occurring UCCA category in the training set.

We augment this simple majority-based UD-to-UCCA category mapping
with rules based on lexical semantics, whose synopsis follows.
The rules are based on the concept of scene-evoking words and phrases, which
is not directly encoded in either UD or STREUSLE.
However, by defining a set of heuristics on the lexical semantic categories,
we obtain a good approximation of the UCCA scene/non-scene
These rules first apply
heuristics based on STREUSLE noun and verb supersenses—determined by examining the training data—and then take into account POS tags and word lists for the more complex cases.

Scene-evoking or not?
If a noun has one of the supersenses n.act, n.event, n.phenomenon, n.process, n.state, n.attribute, or n.feeling, it is identified as scene-evoking.
Proper nouns (UPOS=PROPN) are classified as non-scene-evoking.
Other nouns with the supersense n.person are first matched against a hand-crafted list of relational suffixes[-er, -ess, -or, -ant, -ent, -ee, -ian, -ist] and then against a list of relational nouns based on AMR heuristic lists and WordNet synsets.[From <http://amr.isi.edu/download.html> we obtained ,
which lists 39 types of government officials; and
which includes 85 kinship terms and a handful of person-to-person relations like boss, client, and roommate
(the `MAYBE' entries in these lists, which contain ambiguous words, are disregarded).
The WordNet list consists of 1,487 occupations given by the single-word lemmas in the synsets , , , and their hyponyms, minus the words man and woman.
The heuristics assign the State category to kinship terms like `father' and
Process to occupation nouns.]

Copulas with UPOS=AUX and supersense v.stative
are classified as non–scene-evoking.
v.change verbs are matched against a hand-crafted list of aspectual verbs,[start, stop, begin, end, finish, complete, continue, resume, get, become, quit, keep] which are non–scene-evoking (D) in UCCA.
Nouns and verbs that do not satisfy any of these criteria are canonically classified as non–scene-evoking and scene-evoking, respectively.

Scene-evoking verbs and nouns are labeled P
instead of C by default
(and their modifiers D instead of E).
However, they are S under the following conditions:
nouns with the supersenses n.state, n.attribute, and n.feeling, and adjectives modifying non-scene-evoking nouns.
Predicative nouns with no subject are labeled as A,
and their modifiers as S (e.g. “Great food!”).
Possessives with p.SocialRel or p.OrgRole supersenses
are E scenes with the possessive labeled
both S and A, and their head as a A in the scene they evoke (see example in <Ref>).

Special cases of verbs.
Verbs with the v.stative supersense are labeled F if they have
scene-evoking objects (e.g., “they have great customer service”),
as S if their lemma is “be”“have”, and as P otherwise.
Verbs in full light verb constructions or verb idiomatic expressions
(V.LVC.full, V.VID) are labeled F
(e.g., “pay attention”).

Further lexical decisions.
Expressions with the n.time supersense are labeled T.
Locative pro-adverbs from a pre-defined lexicon (“here”, “there”, etc.)
are labeled as A.
Expressions with the NUM lexcat are labeled Q.
If they are not labeled as R or D by the majority-based mapping, words from a pre-defined lexicon or having the DISC lexcat or the p.purpose supersense are labeled L.
Adverbs with the p.approximator supersense are E instead of D (e.g. “about 30%”).

Further structural decisions.
While  (conjunct, e.g., in coordination) most often correspond to
H, if the head is a non-scene then the dependent is labeled as C instead, as its corresponding unit is likely non-scene too.
Compound or possessive modifiers of scene nouns are labeled as A.
dependents are labeled as both A and G.

§.§ UCCA Postprocessing

UCCA enforces a number of well-formedness restrictions, in terms of which categories may be siblings or children of which.
These are sometimes violated by applying the rules described so far.

We apply the category replacements listed in <Ref> to enforce meeting them.
Additionally, H or L inside a scene are promoted to be a sibling
of the scene,
and remote H, N, L are removed.

C of unit with A    $\to$    P

P, S or D in unit with C and N    $\to$

P, S or D in unit with C and without N    $\to$

N in unit with H    $\to$    L

L in unit without H, starting a scene    $\to$    R

L in unit without H, not starting a scene    $\to$    N

Top-level, category not in
{L,H,F,G,U}    $\to$    H

UCCA postprocessing category replacements in the alternative converter.

§.§ Comparison with the Primary Converter

In the following head-to-head comparison, we refer to the primary system presented in the main part of the paper as “system A” and the secondary version presented here as “system B”.

Easy for both:

* both systems perform well on As

* both systems are good at recalling Fs (system A: 84.9, system B: 89.9), but system A (in contrast to system B) has almost perfect precision (98.6 vs 82.5)

Difficult for both:

* both systems perform okay on Cs; system B tends to confuse Cs for As and Ps more than system A, which tends to fail at predicting units matching gold Cs entirely

* Ds are difficult for both systems; system A underpredicts Ds more than system B, but it is also more precise

* Es are difficult for both systems; Es often get confused (by both systems and in both directions) with Ds, As and Ss

* relational nouns (A|S, A|P) are very difficult for both systems; system B doesn't predict them at all, and system A predicts a few A|Ss which are mostly correct, but still misses 3/4 of them (and all A|Ps)

* Gs are very difficult for both systems; system B doesn't predict them at all, and system A predicts a few but with low precision and recall

* system A is better at recalling Qs

* system A is better at recall (64.0 vs 53.6) and precision (61.2 vs 48.4) on Ss, but also confuses some gold Fs for Ss

* system A is better at recalling Ts; both systems tend to confuse Ts for Ds, but system B does it more than half of the time whereas system A only a quarter

* system A is more eager to predict Ls, thus has higher recall (83.7 vs 61.0) but lower precision (74.7 vs 87.5) here than system B; system A confuses some gold Fs and Rs for Ls, system B confuses some gold Ls for Cs, Ns and Rs

* system B is more eager to predict Ns, thus has higher recall (76.6 vs 66.0) but lower precision (37.1 vs 66.0) here than system A; system B confuses some gold Ls for Ns

* system B is more eager to predict Ps, thus has higher recall (78.6 vs 69.1) but lower precision (56.3 vs 73.1) here than system A; system B confuses some gold Cs, Ds and Fs for Ps, system A confuses some gold Ps for Hs

* system B is more eager to predict Rs, thus has higher recall (88.8 vs 74.0) but lower precision (65.5 vs 84.1) here than system A; system B confuses some gold Fs and Ls for Rs, system A confuses some gold Rs for Ls

Given the small differences between the converters in terms of performance, we decided to use system A for the main analysis in the paper, as it is more modular and interpretable.

§ DELEXICALIZED PARSER CONFUSION MATRIX

<Ref> shows the confusion matrix for the
delexicalized HIT-SCIR parser on the EWT reviews development set.

5lPredicted Category     15cGold Category

1cA   1cA|G   1cA|P   1cA|S   1cC   1cD   1cD|T   1cE   1cF   1cG   1cH   1cL   1cN   1cP   1cQ   1cR   1cS   1cT   1c$\emptyset$

A   741   8   3   14   52   14      40   3   1   15   5      16   1   5   6   4   199

A|C   14   1      2   77         1      1            4   1      1      6

A|P         4   1   1                           2

A|Q                                 1

A|S   1         6   6                           1               1

C   31      5   4   406   5      17   4   1   18      1   39   2   1   23   1   69

D   18            18   275   2   43   15   8   7   6      5   2   1   28   24   25

D|S                                                   1

D|T               1   2   2                                 6

E   49      2      24   36      169   20   1   9   3      3   8   2   9   4   147

F   2            3   42      16   626      1   2      7   12   6   19      13

G               1         2      1   1   1         1

H   11   1         10         6   1   1   578   1      4         3      264

H|R                                                         1

L   3            5   6   1   2   10   6   1   187   16      1   16   2   5   6

N                  1                  14   29         5   1      1

P   12      1   5   37   20      9   10   5   13   1      357   2   1   53      43

Q               6   5      4                     31            1

R   10            8   1      1   15         34      1      225   11      12

S   7      1      33   37      67   10   5   20         17   1   4   224      14

T   1            1   6      1      1   1   2               1   21   7

$\emptyset$   173      1   7   65   55      98   8   4   273   8   1   38   7   19   11   9

Development set confusion matrix for the delexicalized HIT-SCIR parser.
The last column (row), labeled $\emptyset$,
shows the number of predicted (gold-standard) edges of each category
that do not match any gold-standard (predicted) unit.


# Non-Standard Version of Egorov Algebra of Generalized Functions

Todor D. Todorov  Professor Emeritus of California Polytechnic State
University, San Luis Obispo, CA-93407, USA<EMAIL_ADDRESS>https://math.calpoly.edu/todor-todorov

###### Abstract.

We consider a non-standard version of Egorov’s algebra of generalized
functions, with improved properties of the generalized scalars and embedding
of the Schwartz distributions compared with the original standard Egorov’s
version. The embedding of distributions is similar to, but different from
author’s works in the past and independently done by Hans Vernaeve.

###### Key words and phrases:

Schwartz distributions, generalized functions, Colombeau algebra, Egorov
algebra, multiplication of distributions, sheaf of differential functional
spaces, partial differential equations, non-standard analysis, infinitesimals,
non-standard real numbers, non-standard complex numbers, transfer principle,
saturation principle, underflow principle

###### 2020 Mathematics Subject Classification:

Primary: 46F30; Secondary: 46F05, 46F10, 46S10, 46S20, 03H05, 03C50 35D05,
35D10, 35R05, 54B40

##  1\. Introduction

Egorov’s article (Egorov [2]) on algebra $\mathcal{G}(\Omega)$ on generalized
functions was published relatively soon after the arrival of Colombeau theory
of generalized functions (Colombeau [1]) and from the very beginning it was
treated from the mathematical community in close comparison with Colombeau
theory. One striking difference in this comparison is the _simplicity of
Egorov construction_ : Unlike in Colombeau construction, all representatives
of the generalized functions are _moderate_ and the _ideal is relatively
simple_ (even _trivial_ in the case of generalized scalars). This simplicity
of Egorov’s construction is particularly advantageous when one is trying to
define _composition between generalized functions_ or _generalized functions
on a manifold_. In addition to the general theory, numerous interesting
applications of the theory to partial differential equations appear in (Egorov
[2]). In spite of all of these, Egorov’s theory was mostly ignored from
mathematical community dealing with non-linear theories of generalized
functions (standard and non-standard alike) - for one and one reason only: the
embedding of Schwartz distribution into Egorov algebra $\mathcal{G}(\Omega)$
is not of _Colombeau type_ in the sense that the product on
$\mathcal{G}(\Omega)$, if restricted to smooth functions in
$\mathcal{E}(\Omega)$, reduces to classical product only for constant
functions. We summarize this in
${\mathbb{C}}\subset\mathcal{D}^{\prime}(\Omega)\subset\mathcal{G}(\Omega)$
(compared with
$\mathcal{E}(\Omega)\subset\mathcal{D}^{\prime}(\Omega)\subset\mathcal{G}(\Omega)$
in Colombeau theory). The purpose of this article is to improve the properties
of the generalized scalars and the embedding of the distributions as much as
possible, while preserving the rest of the attractiveness features of Egorov
approach including its simplicity.

The non-standard version of Egorov algebra
$\widehat{{}^{*}\mathcal{E}}(\Omega)$ of generalized functions had been
studied in the past under the notation $\mathcal{A}(\Omega)$ in (Todorov [11],
p. 680-684) and under the notation
${{}^{*}\mathcal{C}}^{\infty}(\Omega)|{{\rm{ns}}}(^{*}\Omega)$ in (Vernaeve
[13]). For convenience of the reader we give an independent presentation in
Section 3. One reason to involve non-standard analysis into non-linear theory
of generalized functions (Egorov or Colombeau theories alike) is to improve
the properties of generalized scalars (defined as generalized functions with
zero gradient). The sets of generalized scalars in both Egorov and Colombeau
algebras are rings with zero divisors. In contrast, the sets of scalars,
${}^{*}\mathbb{R}$ and ${}^{*}\mathbb{C}$, of the algebra
$\widehat{{}^{*}\mathcal{E}}(\Omega)$ are fields, real closed and
algebraically closed, respectively.

In Section 4 we discuss the existence of a particular non-standard delta-
function in the space ${}^{*}\mathcal{D}(\mathbb{R}^{d})$, slightly modifying
some results in Todorov [9]-[10]. Here ${}^{*}\mathcal{D}(\mathbb{R}^{d})$
stands for the non-standard extension of the space of test functions
$\mathcal{D}(\mathbb{R}^{d})$.

In Section 5 we define a particular embedding
$\iota_{\Omega}:\mathcal{D}^{\prime}(\Omega)\to\widehat{{}^{*}\mathcal{E}}(\Omega)$
of the space of Schwartz distributions. The embedding $\iota_{\Omega}$ is
similar to, but different the embedding of distributions in (Vernaeve [13]);
we make a short comparison below. The properties of $\iota_{\Omega}$ are
described in Theorem 5.2, but the differences with the previous works are best
visible in Corollary 5.4. In short, the product in
$\widehat{{}^{*}\mathcal{E}}(\Omega)$ reduces to the classical (pointwise)
product on the ring $\mathbb{C}[\Omega]$ of polynomials (not on the whole
$\mathcal{E}(\Omega)$) and in a weaker sense on the ring $\mathcal{C}(\Omega)$
of continuous functions. This is an improvement relative to Egorov theory
(Egorov [2]), where the product in Egorov’s algebra $\mathcal{G}(\Omega)$
reduces to the classical (pointwise) product only on $\mathbb{C}$ (if complex
numbers are treated as constant functions in $\mathcal{E}(\Omega)$). We should
mention that in both $\mathcal{G}(\Omega)$ and
$\widehat{{}^{*}\mathcal{E}}(\Omega)$ there is one more embedding (in addition
to what we discussed above) of $\mathcal{E}(\Omega)$ as a differential
subalgebra; in our text it appears under notation $\sigma$ (see the end of
Definition 3.1 and Corollary 5.4).

In Section 6 we introduce a differential subalgebra
$\widehat{\mathcal{R}}_{\rho}(\Omega)$ of
$\widehat{{}^{*}\mathcal{E}}(\Omega)$, which is similar to, but different from
the algebra $\mathcal{G}^{\infty}(\Omega)$ of regular generalized functions
introduced and study in (Oberguggenberger [6], [8]) within (standard)
Colombeau theory (Colombeau [1]). The algebra $\widehat{\mathcal{R}}_{\rho}$
does not contain the counterexample constructed in Vernaeve [15]) and in that
way we remove the obstacles to developing a regularity methods in non-standard
setting.

We shortly compare our approach based on the algebra
$\widehat{{}^{*}\mathcal{E}}(\Omega)$ with Hans Vernaeve’s work on his algebra
${{}^{*}\mathcal{E}}(\Omega)|{{\rm{ns}}}(^{*}\Omega)$ in (Vernaeve [13]). We
should mention that Vernaeve translated his theory also in standard setting
(Vernaeve [14]).

  * •

The algebra $\widehat{{}^{*}\mathcal{E}}(\Omega)$ defined here and Vernaeve’s
algebra ${{}^{*}\mathcal{E}}(\Omega)|{{\rm{ns}}}(^{*}\Omega)$ (Vernaeve [13])
are the same:
$\widehat{{}^{*}\mathcal{E}}(\Omega)={{}^{*}\mathcal{E}}(\Omega)|{{\rm{ns}}}(^{*}\Omega)$
and $\mu(\Omega)={{\rm{ns}}}(^{*}\Omega)$. So, the scalars, ${}^{*}\mathbb{C}$
and ${}^{*}\mathbb{R}$, are also the same. _The difference is only in the
embeddings_ of the space of distributions.

  * •

Vernaeve’s embedding is of _Colombeau type_ in the sense that, the product on
${{}^{*}\mathcal{E}}(\Omega)|{{\rm{ns}}}(^{*}\Omega)$ _reduces on
$\mathcal{E}(\Omega)$ to the usual classical product between smooth functions_
\- very much like in Colombeau theory (Colombeau [1]) as well as in its non-
standard versions (Oberguggenberger &Todorov [7] and Todorov &Vernaeve [12]).
Unfortunately, Vernaeve’s embedding is defined _for convex open sets $\Omega$
only_. Hence, the family of spaces of distributions
$\\{{\mathcal{D}^{\prime}}(\Omega)\\}_{\Omega\in\mathcal{T}^{d}}$ fails to be
a subsheaf of
$\\{{{}^{*}\mathcal{E}}(\Omega)|{{\rm{ns}}}(^{*}\Omega)\\}_{\Omega\in\mathcal{T}^{d}}$,
where $\mathcal{T}^{d}$ stands for the usual topology on $\mathbb{R}^{d}$.
Consequently, the embedding in Vernaeve’s approach does not, in general,
preserve the supports of distributions.

  * •

In contrast to the above, our embedding $\iota_{\Omega}$ is _not of Colombeau
type_ \- the product in $\widehat{{}^{*}\mathcal{E}}(\Omega)$ _generalizes the
classical product on the ring of polynomials $\mathbb{C}[\Omega]$ only_, not
on the whole space $\mathcal{E}(\Omega)$ (and also on the ring of continuous
functions $\mathcal{C}(\Omega)$ in a weaker sense - after the restriction of
the functions to $\Omega$). However, our embedding is well-defined for _any
open set $\Omega$ of_ $\mathbb{R}^{d}$, the family
$\\{{\mathcal{D}^{\prime}}(\Omega)\\}_{\Omega\in\mathcal{T}^{d}}$ is a
subsheaf of
$\\{\widehat{{}^{*}\mathcal{E}}(\Omega)\\}_{\Omega\in\mathcal{T}^{d}}$.
Consequently, $\iota_{\Omega}$ preserves the support of distributions (very
much like in Egorov and Colombeau algebras). Whether the “trade-off” is worth
doing, remains to be seen.

  * •

_The question_ for defining a _Colombeau type of embedding_ of Schwartz’s
distributions into $\widehat{{}^{*}\mathcal{E}}(\Omega)$ or
${{}^{*}\mathcal{E}}(\Omega)|{{\rm{ns}}}(^{*}\Omega)$ _for every open set_
$\Omega$ of $\mathbb{R}^{d}$, which preserves the support of distributions,
_remains open_ (with or without the requirement on the generalized scalars to
be fields).

##  2\. Notations and Set-Theoretical Framework

  * •

If $\Omega$ is an open subset of $\mathbb{R}^{d}$, we denote by
$\mathcal{C}(\Omega)$ the space continuous functions from $\Omega$ to
$\mathbb{C}$. Similarly, we write
$\mathcal{E}(\Omega)=\mathcal{C}^{\infty}(\Omega)$,
$\mathcal{D}(\Omega)=\mathcal{C}_{0}^{\infty}(\Omega)$,
$\mathcal{L}_{loc}(\Omega)$, $\mathcal{D}^{\prime}(\Omega)$ and
$\mathcal{E}^{\prime}(\Omega)$ for the popular classes of functions and
distributions. The _Schwartz embedding_
$\mathcal{S}_{\Omega}\\!:\mathcal{L}_{loc}(\Omega)\mapsto\mathcal{D}^{\prime}(\Omega)$
is defined by
$\langle\mathcal{S}_{\Omega}(f),\varphi\big{\rangle}=\int_{\Omega}f(x)\varphi(x)\,dx$
for all $\varphi\in\mathcal{D}(\Omega)$ (Vladimirov [16]). In addition, we let
$\mathbb{C}[\Omega]=\mathbb{C}[x_{1},\dots,x_{d}]\,\\!\\!\downharpoonright\\!\Omega$,
where $\mathbb{C}[x_{1},\dots,x_{d}]$ stands for the ring of polynomials in
$d$-many variables with coefficients in $\mathbb{C}$ and
$\;\\!\\!\downharpoonright\\!\;$ stands for the point-wise _restriction_.

  * •

Our framework is a $\mathfrak{c}_{+}$-saturated ultrapower non-standard model
with the set of individuals $\mathbb{R}$, where
$\mathfrak{c}={\rm{card}\,}\,\mathbb{R}$. For a presentation of the topic we
refer to (Lindstrøm [4], Loeb &Wolff [5]) and/or (the Appendix in Todorov
[11]). If $S$ is a set (in the superstructure of $\mathbb{R}$), we write
${}^{*}S$ for the non-standard extension of $S$. In particular,
${}^{*}\mathbb{N}$, ${}^{*}\mathbb{R}$, ${}^{*}\mathbb{C}$,
${}^{*}\mathbb{R}^{d}$, ${}^{*}\Omega$, ${}^{*}\mathcal{L}_{loc}(\Omega)$,
${}^{*}\mathcal{C}(\Omega)$,
${{}^{*}\mathcal{D}}(\Omega),{{}^{*}\mathcal{E}}(\Omega),{{}^{*}\mathcal{D}}^{\prime}(\Omega)$.
etc., are the non-standard extensions of $\mathbb{N}$, $\mathbb{R}$,
$\mathbb{C}$, $\mathbb{R}^{d}$, $\Omega$, $\mathcal{L}_{loc}(\Omega)$,
$\mathcal{C}(\Omega)$,
$\mathcal{D}(\Omega),\mathcal{E}(\Omega),\mathcal{D}^{\prime}(\Omega)$. etc.,
respectively. Recall that ${}^{*}\mathbb{R}$ is $\mathfrak{c}_{+}$–saturated
_real closed (non-Archimedean) field_ of cardinality $\mathfrak{c}_{+}$, which
contains $\mathbb{R}$ as a subfield. Also, ${}^{*}\mathbb{C}$ is an
_algebraically closed field_ containing $\mathbb{C}$ as a subfield and we have
the usual connection ${}^{*}\mathbb{C}={{}^{*}\mathbb{R}}(i)$. Notice that
${{}^{*}\mathcal{E}}(\Omega)$ and ${{}^{*}\mathcal{D}}(\Omega)$ are
_differentia algebras_ over the field ${}^{*}\mathbb{C}$. Also,
${}^{*}\mathbb{C}$ is a differential subring of ${}^{*}\mathcal{E}(\Omega)$
(if the elements of ${}^{*}\mathbb{C}$ are treated as _constant functions_).
We should mention that the functions in ${{}^{*}\mathcal{E}}(\Omega)$ are
mapping from ${}^{*}\Omega$ to ${}^{*}\mathbb{C}$ (not from $\Omega$ to
${}^{*}\mathbb{C}$) and similarly for the rest of the spaces.

  * •

If $X\subseteq\mathbb{R}^{d}$, we let $\mu(X)=\\{x+dx:x\in
X,\,dx\in{{}^{*}\mathbb{R}^{d}},dx\approx 0\\}$ for the set of _near-standard
points of_ ${}^{*}\\!X$. Here $dx\approx 0$ means that $||dx||$ is an
infinitesimal in ${}^{*}\mathbb{R}$. If $f\in{{}^{*}\mathcal{E}}(\Omega)$,
then the following are equivalent: (a)
$f\\!\\!\downharpoonright\\!\mu(\Omega)=0$; (b)
$f\\!\\!\downharpoonright\\!{{}^{*}K}=0$ for all $K\Subset\Omega$.

##  3\. Non-Standard Version of Egorov Algebra
$\widehat{{}^{*}\mathcal{E}}(\Omega)$

######  3.1 Definition (Non-Standard Version of Egorov Algebra).

Let $\Omega$ be an open set of $\mathbb{R}^{d}$.

  1. 1.

We let
$\widehat{{}^{*}\mathcal{E}}(\Omega)={{}^{*}\mathcal{E}(\Omega)}/\mathcal{N}(\Omega)$,
where
$\mathcal{N}(\Omega)=\\{f\in{{}^{*}\mathcal{E}(\Omega)}:f\\!\\!\downharpoonright\\!\mu(\Omega)=0\\}$.

  2. 2.

We supply $\widehat{{}^{*}\mathcal{E}}(\Omega)$ with the operations of a
differential algebra over the field ${}^{*}\mathbb{C}$ with the operations
inherited from ${}^{*}\mathcal{E}(\Omega)$.

  3. 3.

For every $f\in{{}^{*}\mathcal{E}}(\Omega)$ we let
$\widehat{f}=f+\mathcal{N}(\Omega)$ or
$\widehat{f}=f\\!\\!\downharpoonright\\!\mu(\Omega)$ and refer to
$\widehat{f}$ as a _generalized function on $\Omega$_. By exception, we shall
write simply $c$ instead of $\widehat{c}$ in the particular case
$c\in{{}^{*}\mathbb{C}}$ (if $c$ is treated as a constant function in
${}^{*}\mathcal{E}(\Omega)$). If
$\mathcal{S}\subseteq{{}^{*}\mathcal{E}(\Omega)}$, we let
$\widehat{\mathcal{S}}=\\{\widehat{f}:f\in S\\}$. In particular,
$\widehat{{}^{*}\mathcal{E}(\Omega)}=\widehat{{}^{*}\mathcal{E}}(\Omega)$,
$\widehat{\mathcal{N}(\Omega)}=\\{0\\}$,
$\widehat{{}^{*}\mathbb{C}}={{}^{*}\mathbb{C}}$ and
$\widehat{{}^{*}\mathbb{R}}={{}^{*}\mathbb{R}}$.

  4. 4.

For every $\widehat{f}\in\widehat{\mathcal{E}}(\Omega)$ we define
$\widehat{f}:\mu(\Omega)\mapsto{{}^{*}\mathbb{C}}$ by
$\widehat{f}(\xi)={{}^{*}\\!f}(\xi)$.

  5. 5.

Let $\mathcal{O}$ be an open subset of $\Omega$ and
$\widehat{f}\in\widehat{{}^{*}\mathcal{E}}(\Omega)$. We define the
_restriction_
$\widehat{f}\,\\!\\!\downharpoonright\\!\mathcal{O}\in\widehat{{}^{*}\mathcal{E}}(\mathcal{O})$
by
$\widehat{f}\,\\!\\!\downharpoonright\\!\mathcal{O}=\widehat{f\\!\\!\downharpoonright\\!{{}^{*}\mathcal{O}}}$.
We say that $\widehat{f}$ _vanishes on_ $\mathcal{O}$ if
$\widehat{f}\,\\!\\!\downharpoonright\\!\mathcal{O}=0$ in
$\widehat{{}^{*}\mathcal{E}}(\mathcal{O})$. The _support_
${\rm{supp}}(\widehat{f})$ of $\widehat{f}$ is the the complement to $\Omega$
of the largest open subset of $\Omega$, on which $\widehat{f}$ vanishes.

  6. 6.

Let $X$ be a Lebesgue measurable subset of $\mathbb{R}^{d}$ whose closure is a
compact subset of $\Omega$. We define a (Lebesgue) _integral of
$\widehat{f}\in\widehat{{}^{*}\mathcal{E}}(\Omega)$ over_ $X$ with values in
${}^{*}\mathbb{C}$ by the formula
$\int_{X}\widehat{f}(x)\,dx=\int_{{}^{*}X}f(\xi)\,d\xi$.

  7. 7.

We define the _pairing between_ $\widehat{{}^{*}\mathcal{E}}(\Omega)$ and
$\mathcal{D}(\Omega)$ by
$\langle\widehat{f},\varphi\rangle=\int_{{}^{*}\Omega}f(\xi)\;{{}^{*}\\!\varphi(\xi)}\,d\xi$
for all $\widehat{f}\in\widehat{{}^{*}\mathcal{E}}(\Omega)$ and all
$\varphi\in\mathcal{D}(\Omega)$, where ${}^{*}\\!\varphi$ stands for the non-
standard extension of $\varphi$.

  8. 8.

We say that two generalized functions
$\widehat{f},\;\widehat{g}\in\widehat{{}^{*}\mathcal{E}}(\Omega)$ are _weakly
equal_ or _associated_ , and write $\widehat{f}\;\cong\;\widehat{g}$, if
$\langle\widehat{f},\varphi\rangle=\langle\widehat{g},\varphi\rangle$ for all
$\varphi\in\mathcal{D}(\Omega)$.

  9. 9.

We define _standard embedding_
$\sigma:\mathcal{E}(\Omega)\to{{}^{*}\mathcal{E}(\Omega)}$ by
$\sigma(f)=\widehat{{}^{*}\\!f}$, where
$\widehat{{}^{*}\\!f}={{}^{*}\\!f}\\!\\!\downharpoonright\\!\mu(\Omega)$.

######  3.2 Theorem (Basic Properties of
$\widehat{{}^{*}\mathcal{E}}(\Omega)$).

  1. _missing_ fnum@T-enumi(i)(i)

$\widehat{{}^{*}\mathcal{E}}(\Omega)$ is a differential algebra over the field
${}^{*}\mathbb{C}$. Also, the mapping $f\,+\,\mathcal{N}(\Omega)\mapsto
f\\!\\!\downharpoonright\\!\mu(\Omega)$ from
${}^{*}\mathcal{E}(\Omega)/\mathcal{N}(\Omega)$ onto
$\\{f\\!\\!\downharpoonright\\!\mu(\Omega):f\in{{}^{*}\mathcal{E}(\Omega)}\\}$
is a differential algebra isomorphism (justifying the notation
$\widehat{f}=f\\!\\!\downharpoonright\\!\mu(\Omega)$ used in advance).

  2. _missing_ fnum@T-enumi(ii)(ii)

${}^{*}\mathbb{C}=\\{\widehat{f}\in\widehat{{}^{*}\mathcal{E}}(\Omega):\nabla\widehat{f}=0\\}$
for every open connected subset $\Omega$ of $\mathbb{R}^{d}$.

  3. _missing_ fnum@T-enumi(iii)(iii)

The family
$\\{\widehat{{}^{*}\mathcal{E}}(\Omega)\\}_{\Omega\in\mathcal{T}^{d}}$, is a
sheaf of differential algebras on $\mathbb{R}^{d}$, where $\mathcal{T}^{d}$
stands for the usual topology on $\mathbb{R}^{d}$.

  4. _missing_ fnum@T-enumi(iv)(iv)

$\sigma[\mathcal{E}(\Omega)]$ is a differential $\mathbb{C}$-subalgebra of
$\widehat{{}^{*}\mathcal{E}}(\Omega)$, isomorphic of $\mathcal{E}(\Omega)$.
Also, $\int_{X}\sigma(f)(v)\,dx=\int_{X}f(x)\,dx$ for all
$f\in\mathcal{E}(\Omega)$ and all Lebesgue measurable subset $X$ of
$\mathbb{R}^{d}$ with compact closure in $\Omega$. Moreover,
$\\{\sigma[\mathcal{E}(\Omega)]\\}_{\Omega\in\mathcal{T}^{d}}$ is a subsheaf
of $\\{\widehat{{}^{*}\mathcal{E}}(\Omega)\\}_{\Omega\in\mathcal{T}^{d}}$.

###### Proof.

For the proof we refer to (Todorov [11], §5) or/and (Vernaeve [13]). ∎

##  4\. Non-Standard Delta-Function

We discuss the existence of a particular non-standard delta-function $\Delta$
in the space ${}^{*}\mathcal{D}(\mathbb{R}^{d})$, the non-standard extension
$\mathcal{D}(\mathbb{R}^{d})$. In this section we slightly modify similar
results in Todorov [9]-[10].

######  4.1 Lemma (Non-Standard Delta-Function).

For every $d\in\mathbb{N}$ there exists (not necessarily unique)
$\Delta\in{{}^{*}\mathcal{D}}(\mathbb{R}^{d})$ such that $\Delta(\xi)=0$ for
all infinitely large and all finite, but non-infinitesimal
$\xi\in{{}^{*}\mathbb{R}^{d}}$ and such that
$\int_{{}^{*}\mathbb{R}^{d}}\Delta(\xi)\,{{}^{*}\\!\varphi}(\xi)\,d\xi=\int_{{}^{*}\mathbb{R}^{d}}\Delta(-\xi)\,{{}^{*}\\!\varphi}(\xi)\,d\xi=\varphi(0)$
for all continuous functions $\varphi\in\mathcal{C}(\mathbb{R}^{d})$. We let
$\rho={{}^{*}\\!\sup}\big{\\{}||\xi||:\xi\in{{}^{*}\mathbb{R}^{d}},\,\Delta(\xi)\not=0\big{\\}}$
for the radius of support of $\Delta$. Moreover, for each open
$\Omega\subseteq\mathbb{R}^{d}$ and each
$\varepsilon\in{{}^{*}\mathbb{R}_{+}}$ we let

$\Omega_{\varepsilon}=\\{\xi\in{{}^{*}\Omega}:{\rm
dist}(\xi,\partial\Omega)\geq\varepsilon\;\,\&\;\,{\rm dist}(\xi,0)\leq
1/\varepsilon\\},$

and define $\Pi_{\Omega}:{{}^{*}\mathbb{R}^{d}}\mapsto{{}^{*}\mathbb{C}}$ by
the formula $\Pi_{\Omega}(\xi)=\int_{\Omega_{3\rho}}\Delta(\xi-\eta)\,d\eta$.

######  4.2 Theorem (Regularization in ${}^{*}\mathcal{E}(\Omega)$).

Let $\Omega$ and $\Delta$ be as in Lemma 4.1. Then:

  1. _missing_ fnum@T-enumi(i)(i)

${{}^{*}\\!f}\star\Delta\in{{}^{*}\mathcal{E}}(\mathbb{R}^{d})$ for every
$f\in\mathcal{C}(\mathbb{R}^{d})$. Here
${{}^{*}\\!f}\star\Delta:{{}^{*}\mathbb{R}^{d}}\to{{}^{*}\mathbb{C}}$ is
defined by
$({{}^{*}\\!f}\star\Delta)(\xi)=\int_{{}^{*}\mathbb{R}^{d}}\,{{}^{*}\\!f(\eta)}\,\Delta(\xi-\eta)\,d\eta$.
Moreover, ${{}^{*}\\!f}\star\Delta$ is an extension of $f$ from
$\mathbb{R}^{d}$ to ${}^{*}\mathbb{R}^{d}$, in symbol,
$({{}^{*}\\!f}\star\Delta)\,\\!\\!\downharpoonright\\!\,\mathbb{R}^{d}=f$.

  2. _missing_ fnum@T-enumi(ii)(ii)

${}^{*}P\star\Delta={{}^{*}P}$ for all polynomials $P\in\mathbb{C}[\Omega]$.

  3. _missing_ fnum@T-enumi(iii)(iii)

$\rho$ is a positive infinitesimal in ${}^{*}\mathbb{R}$ and
$\mu(\Omega)\subseteq\Omega_{2\rho}\subset\Omega_{\rho}\subseteq{{}^{*}\Omega}$.

  4. _missing_ fnum@T-enumi(iv)(iv)

$\Pi_{\Omega}\in{{}^{*}\mathcal{D}(\Omega)}$,
$\Pi_{\Omega}\\!\\!\downharpoonright\\!\mu(\Omega)=1$. Moreover,
${\rm{supp}}(\Pi_{\Omega})=\Omega_{2\rho}$.

  5. _missing_ fnum@T-enumi(v)(v)

$\Pi_{\Omega}({{}^{*}T}\star\Delta)\in{{}^{*}\mathcal{D}(\Omega)}$ for all
$T\in\mathcal{D}^{\prime}(\Omega)$.

###### Proof.

For (i) and (ii) we refer to (Todorov [10], where the results are based on
infinite-dimensional linear algebra and saturation principle in non-standard
analysis (Lindstrøm [4] and/or Loeb &Wolff [5]). The fact that $\rho$ is an
infinitesimal follows by underflow principle (Lindstrøm [4]). For the standard
counterpart of the rest we refer to (Vladimirov [16], §4.6). ∎

##  5\. Embedding of Distributions into $\widehat{{}^{*}\mathcal{E}}(\Omega)$

Although the mapping $T\to\Pi_{\Omega}({{}^{*}T}\star\Delta)$ from
$\mathcal{D}^{\prime}(\Omega)$ to ${}^{*}\mathcal{E}(\Omega)$ (Theorem 4.2) is
injective, it does not commute with the partial derivatives
$\partial^{\alpha}$ in ${}^{*}\mathcal{E}(\Omega)$. Moreover, the family
$\\{{{}^{*}\mathcal{E}}(\Omega)\\}_{\Omega\in\mathcal{T}^{d}}$ is not a sheaf
on $\mathbb{R}^{d}$. Thus ${}^{*}\mathcal{E}(\Omega)$ cannot be treated as an
algebra of generalized functions on $\mathbb{R}^{d}$ we are looking for; we
return to the algebra $\widehat{{}^{*}\mathcal{E}}(\Omega)$ defined in Section
3.

######  5.1 Definition (Embedding of Distributions in
$\widehat{{}^{*}\mathcal{E}}(\Omega)$).

Let $\Omega$ and $\Delta$ be chosen (and fixed) as in Lemma 4.1. We define
$\iota_{\Omega}:\mathcal{D}^{\prime}(\Omega)\mapsto\widehat{{}^{*}\mathcal{E}}(\Omega)$
by $\iota_{\Omega}(T)=\widehat{\Pi_{\Omega}({{}^{*}T}\star\Delta)}$ or
equivalently, by
$\iota_{\Omega}(T)=\Pi_{\Omega}({{}^{*}T}\star\Delta)\\!\\!\downharpoonright\\!\mu(\Omega)$.
Here ${}^{*}T\star\Delta:{{}^{*}\Omega}\to{{}^{*}\mathbb{C}}$ is defined by
$(^{*}T\star\Delta)(\xi)=\langle{{}^{*}T}(\eta),\Delta(\xi-\eta)\rangle$ on
the ground of transfer principle (Lindstrøm [4] and/or Loeb &Wolff [5]).

######  5.2 Theorem (Properties of the Embedding).

  1. _missing_ fnum@T-enumi(i)(i)

$\iota_{\Omega}$ commutes with the partial derivatives on
$\mathcal{D}^{\prime}(\Omega)$. Moreover,
$\langle\iota_{\Omega}(T),\varphi\rangle=\langle T,\varphi\rangle$ for all
$\varphi\in\mathcal{D}(\Omega)$ (Definition 3.1). Consequently,
$\iota_{\Omega}$ is injective and
$\iota_{\Omega}[\mathcal{D}^{\prime}(\Omega)]$ is a differential
$\mathbb{C}$-vector subspace of $\widehat{{}^{*}\mathcal{E}}(\Omega)$.

  2. _missing_ fnum@T-enumi(ii)(ii)

$(\iota_{\Omega}\circ S_{\Omega})[\mathcal{D}(\Omega)]$, $(\iota_{\Omega}\circ
S_{\Omega})[\mathcal{E}(\Omega)]$, $(\iota_{\Omega}\circ
S_{\Omega})[\mathcal{C}(\Omega)]$ and $(\iota_{\Omega}\circ
S_{\Omega})[\mathcal{L}_{loc}(\Omega)]$ are $\mathbb{C}$-vector subspaces of
$\widehat{{}^{*}\mathcal{E}}(\Omega)$. Moreover, $(\iota_{\Omega}\circ
S_{\Omega})[\mathcal{D}(\Omega)]$ and $(\iota_{\Omega}\circ
S_{\Omega})[\mathcal{E}(\Omega)]$ are differential $\mathbb{C}$-vector
subspaces of $\widehat{{}^{*}\mathcal{E}}(\Omega)$. Also, we have
$(\iota_{\Omega}\circ S_{\Omega})(f)\cong\sigma(f)$ for all
$f\in\mathcal{E}(\Omega)$ (Definition 3.1).

  3. _missing_ fnum@T-enumi(iii)(iii)

$(\iota_{\Omega}\circ S_{\Omega})(P)=\sigma(P)$ for all polynomials
$P\in\mathbb{C}[\Omega]$. Consequently, $(\iota_{\Omega}\circ
S_{\Omega})\big{[}\mathbb{C}[\Omega]\big{]}$ is a differential subring (a
differential $\mathbb{C}$-subalgebra) of
$\widehat{{}^{*}\mathcal{E}}(\Omega)$, which is isomorphic to
$\mathbb{C}[\Omega]$. We summarize these in the chain of embeddings:
$\mathbb{C}[\Omega]\subset\mathcal{D}^{\prime}(\Omega)\subset\widehat{{}^{*}\mathcal{E}}(\Omega)$,
after dropping $\iota_{\Omega}$.

  4. _missing_ fnum@T-enumi(iv)(iv)

The family
$\big{\\{}\iota_{\Omega}[\mathcal{D}^{\prime}(\Omega)]\big{\\}}_{\Omega\in\mathcal{T}^{d}}$
is subsheaf of
$\\{\widehat{{}^{*}\mathcal{E}}(\Omega)\\}_{\Omega\in\mathcal{T}^{d}}$ on
$\mathbb{R}^{d}$. Consequently,
${\rm{supp}}(T)={\rm{supp}}(\iota_{\Omega}(T))$ for all
$T\in\mathcal{D}^{\prime}(\Omega)$.

  5. _missing_ fnum@T-enumi(v)(v)

Let $f\in\mathcal{C}(\Omega)$ be continuous function. Then
$(\iota_{\Omega}\circ S_{\Omega})(f)$ is an extension of $f$ (from $\Omega$ to
$\mu(\Omega$), i.e. $(\iota_{\Omega}\circ S_{\Omega})(f)(x)=f(x)$ for all
$x\in\Omega$. In particular, $\partial^{\alpha}\,(\iota_{\Omega}\circ
S_{\Omega})(f)(x)=\partial^{\alpha}f(x)$ for all $f\in\mathcal{E}(\Omega)$,
all $\alpha\in\mathbb{N}_{0}^{d}$ and all $x\in\Omega$.

  6. _missing_ fnum@T-enumi(vi)(vi)

Let $X$ and $Y$ be two open subsets of $\mathbb{R}^{d}$ and $\theta\in{\rm
Diff}(X,Y)$. Let the mapping $T\to T(\theta)$, from $\mathcal{D}^{\prime}(X)$
to $\mathcal{D}^{\prime}(Y)$, stands for the change of variables in the sense
of distribution theory (Hörmander [3], §6.3-§6.4) and (Vladimirov [16], p.26).
We define
$\theta_{*}:\widehat{{}^{*}\mathcal{E}}(X)\to\widehat{{}^{*}\mathcal{E}}(Y)$
by
$\theta_{*}(\widehat{f})=(f\circ{{}^{*}\theta^{-1}})\\!\\!\downharpoonright\\!\mu(Y)$,
where ${}^{*}\theta^{-1}$ stands for the non-standard extension of
$\theta^{-1}$. Then $\theta_{*}(\iota_{X}(T))\cong\iota_{Y}(T(\theta))$ for
all $T\in\mathcal{D}^{\prime}(X)$.

###### Proof.

Relatively straightforward consequences from Theorem 4.2. ∎

######  5.3 Examples (Some Particular Generalized Functions in
$\widehat{{}^{*}\mathcal{E}}(\Omega)$).

  1. 1.

$\iota_{\mathbb{R}^{d}}(\delta)=\widehat{\Delta}$ and more generally,
$\iota_{\mathbb{R}^{d}}(\partial^{\alpha}\delta)=\widehat{\partial^{\alpha}}\Delta$
for all $\alpha\in\mathbb{N}_{0}^{d}$. We write these more casually as
$\partial^{\alpha}\delta\in\widehat{{}^{*}\mathcal{E}}(\mathbb{R}^{d})$.

  2. 2.

$(\widehat{\Delta})^{n}\in\widehat{{}^{*}\mathcal{E}}(\mathbb{R}^{d})$ for all
$n\in\mathbb{N}$, since $\widehat{{}^{*}\mathcal{E}}(\mathbb{R}^{d})$ is an
algebra. We write this more casually as
$\delta^{n}\in\widehat{{}^{*}\mathcal{E}}(\mathbb{R}^{d})$.

  3. 3.

Let $f:\mathbb{C}\mapsto\mathbb{C}$ stand for $f(z)=e^{z}$. Clearly,
$e^{x+iy}\in\mathcal{E}(\mathbb{R}^{2})$. Thus
$e^{\Delta}\in{{}^{*}\mathcal{E}}(\mathbb{R}^{2})$ (we skip the asterisk in
front of ${}^{*}e^{z}$) and
$\widehat{e^{\Delta}}\in\widehat{{}^{*}\mathcal{E}}(\mathbb{R}^{2})$. We write
more causally, $e^{\delta}\in\widehat{{}^{*}\mathcal{E}}(\mathbb{R}^{2})$.
Notice that $e^{\delta}$ makes sense as well in (Egorov [2]), but not in
Colombeau algebra, since $e^{x+iy}$ is _non-moderate_ in the variable $x$
(Colombeau [1]).

  4. 4.

Let temporarily write $\Delta_{d}$ instead of $\Delta$ indicating that
$\Delta_{d}\in{{}^{*}\mathcal{D}}(\mathbb{R}^{d})$. Let
$(e_{1},e_{2},\dots,e_{d})$ be the standard basis for $\mathbb{R}^{d}$. Then
$\widehat{\Delta}_{d}\\!\\!\downharpoonright\\!{\rm
span}(e_{n})\cong\widehat{\Delta}_{1}$, where
$\widehat{\Delta}_{d}\\!\\!\downharpoonright\\!{\rm
span}(e_{n}):=\Delta_{d}\\!\\!\downharpoonright\\!\mu({\rm span}(e_{n}))$.
Notice that ${\rm span}(e_{n})$ is a _smooth submanifold_ (not open subset) of
$\mathbb{R}^{d}$.

Very much like in Colombeau and Egorov theories, Schwartz distribution can be
multiplied within $\widehat{{}^{*}\mathcal{E}}(\Omega)$, since the latter is a
differential (commutative and associative) algebra. How good (or bad) is this
product? In a lack of compelling applications to other branches of mathematics
or physics, we making our judgement mostly by applying this product to the
$\iota_{\Omega}$-images in $\widehat{{}^{*}\mathcal{E}}(\Omega)$ of the
classical functions. Here is our test:

######  5.4 Corollary (Multiplication of Classical Functions).

  1. _missing_ fnum@C-enumi(i)(i)

The product in the algebra $\widehat{{}^{*}\mathcal{E}}(\Omega)$, if
restricted to $\mathbb{C}[\Omega]$ (more precisely, on $(\iota_{\Omega}\circ
S_{\Omega})\big{[}\mathbb{C}[\Omega]\big{]}$), coincides with the usual
product between polynomials, i.e. for every $P,Q\in\mathbb{C}[\Omega]$ we have

$(\iota_{\Omega}\circ S_{\Omega})(PQ)=(\iota_{\Omega}\circ
S_{\Omega})(P)\cdot(\iota_{\Omega}\circ S_{\Omega})(Q).$

  2. _missing_ fnum@C-enumi(ii)(ii)

$(\iota_{\Omega}\circ
S_{\Omega})\big{[}\mathbb{C}[\Omega]\big{]}={\sigma}\big{[}\mathbb{C}[\Omega]\big{]}={\sigma}[\mathcal{E}(\Omega)]\cap(\iota_{\Omega}\circ
S_{\Omega})[\mathcal{E}(\Omega)]$.

  3. _missing_ fnum@C-enumi(iii)(iii)

For every two continuous functions $f,g\in\mathcal{C}(\Omega)$ and for all
(standard) $x\in\Omega$ we have $(\iota_{\Omega}\circ
S_{\Omega})(fg)(x)=(\iota_{\Omega}\circ
S_{\Omega})(f)(x)\cdot(\iota_{\Omega}\circ S_{\Omega})(g)(x)=f(x)g(x)$.

  4. _missing_ fnum@C-enumi(iv)(iv)

$\iota_{\Omega}(fT)\cong\sigma(f)\,\iota_{\Omega}(T)$ (Definition 3.1) for all
$f\in\mathcal{E}(\Omega)$ and all $T\in\mathcal{D}^{\prime}(\Omega)$, where
the product $fT$ is in the sense of distribution theory (Vladimirov [16],
§1.10). In particular, $\iota_{\Omega}(P\,T)\,\cong\,(\iota_{\Omega}\circ
S_{\Omega})(P)\;\iota_{\Omega}(T)$ for all polynomials
$P\in\mathbb{C}[\Omega]$ and all $T\in\mathcal{D}^{\prime}(\Omega)$.

##  6\. Regular Algebra

The algebra $\widehat{\mathcal{R}}_{\rho}(\Omega)$ defined below is similar,
but different from the algbra $\mathcal{G}^{\infty}(\Omega)$ introduced and
study in (Oberguggenberger [6], [8]). We should mention that
$\widehat{\mathcal{R}}_{\rho}$ does not contain the counterexample in Vernaeve
[15]).

######  6.1 Definition (Regular Algebra).

Let $\Omega$ and $\Delta$ be chosen (and fixed) as in Lemma 4.1. Let
${{}^{\sigma}\\!\mathcal{E}}(\Omega)=\\{^{*}\\!f:f\in\mathcal{E}(\Omega)\\}$
and $\mathcal{M}_{\rho}=\\{\xi\in{{}^{*}\mathbb{C}}:|\xi|\leq\rho^{-n}\text{
for some }n\in\mathbb{N}\\}$. Let $\mathcal{R}_{\rho}(\Omega)$ denote the
subring of ${}^{*}\mathcal{E}(\Omega)$ generated by
${{}^{\sigma}\\!\mathcal{E}}(\Omega)\cup\mathcal{M}_{\rho}$, in symbol,
$\mathcal{R}_{\rho}(\Omega)={{}^{\sigma}\\!\mathcal{E}}(\Omega)(\mathcal{M}_{\rho})$.
The _algebra of $\rho$-regular functions_ is defined by
$\widehat{\mathcal{R}}_{\rho}(\Omega)=\\{\widehat{f}:f\in\mathcal{R}_{\rho}(\Omega)\\}$
or equivalently, by
$\widehat{\mathcal{R}}_{\rho}(\Omega)=\\{f\\!\\!\downharpoonright\\!\mu(\Omega):f\in\mathcal{R}_{\rho}(\Omega)\\}$.

######  6.2 Theorem.

Under the assumption of the above definition we have:

  1. _missing_ fnum@T-enumi(i)(i)

$\widehat{\mathcal{R}}_{\rho}$ is a differential $\mathbb{C}$-subalgebra of
$\widehat{{}^{*}\\!\mathcal{E}}(\Omega)$.

  2. _missing_ fnum@T-enumi(ii)(ii)

If $\widehat{f}\in\widehat{\mathcal{R}}_{\rho}(\Omega)$, then
$(\forall\xi\in\mu(\Omega))(\exists
n\in\mathbb{N})(\forall\alpha\in\mathbb{N}_{0}^{d})(|\partial^{\alpha}\widehat{f}(\xi)|\leq\rho^{-n})$.

  3. _missing_ fnum@T-enumi(iii)(iii)

$\widehat{\mathcal{R}}_{\rho}(\Omega)\cap{\iota_{\Omega}}[\mathcal{D}^{\prime}(\Omega)]=({\iota_{\Omega}}\circ
S_{\Omega})[\mathbb{C}[\Omega]]$.

###### Proof.

We leave the verification to the reader. ∎

## References

  * [1] J. F. Colombeau, _New Generalized Functions and Multiplication of Distributions_ , Math. Studies 84, North-Holland-Amsterdam-New York-Oxford, 1984.
  * [2] Yu. V. Egorov, A contribution to the theory of generalized functions (Russian). Uspekhi Mat Nauk (Moskva) 45, No 5 (1990), 3-40.
_English translation_ in Russian Math. Surveys 45:5 (1990), p. 1-49.

  * [3] Lars Hörmander, The Analysis of Linear Partial Differential Operators I, volume 256 of Grundlehren der mathematischen Wissenschaften, Springer, Berlin, 1990.
  * [4] Tom Lindstrøm, An invitation to nonstandard analysis, in: Nonstandard Analysis and its Applications, N. Cutland (Ed), Cambridge U. Press, 1988, p. 1-105.
  * [5] Peter A. Loeb and Manfred Wolff (Eds.), _Nonstandard Analysis for the Working Mathematician_ , Kluwer Academic Publishers, Dordrecht/Boston/London, 2000.
  * [6] M. Oberguggenberger, _Multiplication of Distributions and Applications to Partial Differential Equations_ , Pitman Research Notes Math., 259, Longman, Harlow, 1992.
  * [7] M. Oberguggenberger and T. Todorov, _An Embedding of Schwartz Distributions in the Algebra of Asymptotic Functions_ , International J. Math. & Math. Sci., Vol. 21, No. 3 (1998), p.417-428.
  * [8] M. Oberguggenberger, _Regularity Theory in Colombeau Algebras_ , in Bulletin T. CXXXIII de l’Academie serbe des sciences et des arts, Classe des sciences mathématiques et naturelles, Sciences mathématique No 31, 2006, p. 147-162.
  * [9] Todor Todorov, A Nonstandard Delta Function, In: _Proceedings of the American Mathematical Society_ , Vol. 110, Number 4, 1990, p. 1143–1144.
  * [10] Todor Todorov, Pointwise Kernels of Schwartz Distributions, In: _Proceedings of the American Mathematical Society_ , Vol. 114, No 3, March 1992, p. 817–819.
  * [11] Todor Todorov, An Existence of Solutions for Linear PDE with $C^{\infty}$-Coefficients in an Algebra of Generalized Functions, In: _Transactions of the American Mathematical Society_ , Vol. 348, 2, Feb. 1996, p. 673–689 http://www.ams.org/journals/tran/1996-348-02.
  * [12] Todor Todorov and Hans Vernaeve, Full Algebra of Generalized Functions and Non-Standard Asymptotic Analysis, In: _Logic and Analysis_ , Springer, Vol. 1, Issue 3, 2008, pp. 205-234 (available at: (http://www.logicandanalysis.com/index.php/jla/article/view/193/79) and/or at: http://arxiv.org/abs/0712.2603).
  * [13] Hans Vernaeve, _Nonstandard Contributions to the Theory of Generalized Functions_ , Ph.D. Thesis, University of Gent, 2002.
  * [14] Hans Vernaeve, _Optimal Embeddings of Distributions into Algebras_ , Proceedings of the Edinburgh Mathematical Society (2003) 46, p. 373-378.
  * [15] Hans Vernaeve, _Regularity of nonlinear generalized functions: A Counterexample in the nonstandard setting_ , in Examples and Counterexamples 1 (2021) 100021.
  * [16] V. S. Vladimirov, Generalized Functions in Mathematical Physics, Mir-Publisher, Moscow, 1979.