# Scope-enhanced Compositional Semantic Parsing for DRT

Xiulin Yang
Georgetown University
<EMAIL_ADDRESS>
Jonas Groschwitz
University of Amsterdam
<EMAIL_ADDRESS>
Alexander Koller
Saarland University
<EMAIL_ADDRESS>
Johan Bos
University of Groningen
<EMAIL_ADDRESS>

###### Abstract

Discourse Representation Theory (DRT) distinguishes itself from other semantic
representation frameworks by its ability to model complex semantic and
discourse phenomena through structural nesting and variable binding. While
seq2seq models hold the state of the art on DRT parsing, their accuracy
degrades with the complexity of the sentence, and they sometimes struggle to
produce well-formed DRT representations. We introduce the AMS parser, a
compositional, neurosymbolic semantic parser for DRT. It rests on a novel
mechanism for predicting quantifier scope. We show that the AMS parser
reliably produces well-formed outputs and performs well on DRT parsing,
especially on complex sentences.

##  1 Introduction

Among current semantic representation formalisms used in NLP, Discourse
Representation Theory (DRT; Kamp and Reyle, 1993) stands out in its systematic
use of structural nesting and variable binding to represent meaning in detail.
Originating from linguistic theory, DRT has been designed to capture subtle
semantic and discourse phenomena such as anaphora, presupposition, and
discourse structure, as well as tense and aspect (see Fig. 1). This structural
and semantic richness distinguishes DRT from other popular frameworks in
semantic parsing, such as Abstract Meaning Representation (AMR; Banarescu et
al., 2013).

With the availability of the broad-coverage Parallel Meaning Bank (PMB;
Abzianidze et al., 2017), DRT has become an active target for the development
of semantic parsing methods. The current state of the art is held by purely
neural seq2seq models (Zhang et al., 2024). However, due to the structural
complexity of typical DRT representations, these models do not always generate
well-formed meaning representations. They also struggle on long sentences;
length generalization is a known challenge for transformers in semantic
parsing settings (Hupkes et al., 2020; Yao and Koller, 2022). Existing
compositional semantic parsers for DRT significantly lag behind the seq2seq
models in terms of parsing accuracy.

Figure 1: DRS (top) and DRG (bottom) for the sentence Every child misbehaves
occasionally; dashed lines represent scope assignments of connectives.

In this paper, we introduce the _AMS parser_ , an accurate compositional DRT
parser. The AMS parser extends the AM parser (Groschwitz et al., 2018), which
predicts meaning representations compositionally and has achieved high
accuracy across a range of sembanks (Lindemann et al., 2019; Weißenhorn et
al., 2022). The AM parser by itself struggles to predict structural nesting in
DRT. The key challenge is to predict _scope:_ how to assign each atomic
formula in Fig. 1 to one of the three boxes. Differences in scope assignment
affect the represented meaning significantly.

The technical contribution of this paper is to extend the AM parser with an
innovative mechanism for predicting scope. We train a dependency parser to
predict scope relations between word tokens and project this information into
the DRT representation using word-to-box alignments. We show that this
dependency mechanism can predict correct scope assignments at very high
accuracy. The overall parser always predicts well-formed DRT representations
(in contrast to all seq2seq models) and is almost on par with the best models
in parsing accuracy. On the PMB TestLong split, which contains particularly
long sentences, it outperforms all other DRT parsers that are trained on the
PMB gold dataset. Thus, the strength of the AMS parser is its ability to
remain accurate as sentences grow complex.

##  2 Background and Related Work

Discourse Representation Theory (DRT; Kamp and Reyle, 1993) is a well-
developed framework for dynamic semantics that aims to interpret meanings from
the context. It can model diverse linguistic phenomena ranging from anaphora
(Kamp, 1981; Haug, 2014) to rhetorical structures (Lascarides and Asher,
2007). In DRT, meanings are traditionally represented by Discourse
Representation Structures (DRS), which are composed of nested boxes that
contain discourse referents (the entities talked about in the discourse) and
propositions about these discourse referents. Fig. 1 (top) is an example of
DRS representing Every child misbehaves occasionally. The boxes act as logical
quantifiers that bind variables, and they can be connected with logical
operators such as implication.

Bos (2023) recently proposed an equivalent, variable-free notation for DRSs in
the form of directed acyclic graphs, called Discourse Representation Graphs
(DRGs; see Fig. 1, bottom). A DRG contains nodes representing boxes, predicate
symbols, and constants. Some edges (drawn solid in Fig. 1) connect predicates
to arguments with semantic roles. Others (drawn dashed) represent the
structural nesting of boxes and propositions: A dashed edge means that its
target node is inside the box from which the edge emanates. Universal
quantification, disjunction, and implication are represented in DRGs as
logically equivalent structures using only negation and conjunction.

The main resource for DRS and DRG is the Parallel Meaning Bank (PMB;
Abzianidze et al. (2017)), which is a multilingual parallel corpus comprising
sentences and texts paired with meaning representations. In this paper, we use
the latest version (PMB release 5.1.0, English) for evaluation. It includes
three distinct splits based on the quality and method of annotation: Gold
(manually verified), Silver (partially corrected), and Bronze (automatically
generated by Boxer). As our objective is to address challenges within a
limited data setting, our experiments specifically focus on utilizing gold-
annotated data.

###  2.1 DRS parsing

Deriving DRSs from sentences compositionally is a nontrivial challenge.
Efforts towards this goal include $\lambda$-DRT (Muskens, 1994; Kohlhase et
al., 1996, 1998), Compositional DRT (Muskens, 1996), and bottom-up DRT (Asher,
1993). All of these approaches use lambda calculus to compositionally combine
partial meaning representations, which is intractable in broad-coverage
semantic parsing (see e.g. the discussion by Artzi et al. (2015)).

To date, the most accurate broad-coverage DRT parsers are based on neural
sequence-to-sequence models (e.g., Liu et al., 2018; Fancellu et al., 2019;
Van Noord et al., 2018; van Noord et al., 2020). They achieve impressive
performances, especially when the models are trained on additional silver or
bronze training data (Wang et al., 2023a) or use additional features (van
Noord et al., 2019, 2020). However, due to the structure-unaware design of
these models, they sometimes struggle to generate well-formed DRT
representations (see Poelman et al. (2022)).

Existing compositional semantic parsers for DRT rely on syntactic dependency
parsers Le and Zuidema (2012); Poelman et al. (2022) or CCG parsers Bos (2008,
2015). These models reliably generate well-formed DRSs, but are not
competitive with seq2seq models in terms of parsing accuracy.

###  2.2 AM Parsing

The DRT parser we present here is based on the AM Parser (Groschwitz et al.,
2018), a neurosymbolic compositional semantic parser that has previously been
shown to be fast and accurate both on broad-coverage parsing, e.g. on AMR
(Lindemann et al., 2019), and in compositional generalization tasks
(Weißenhorn et al., 2022).

#### Apply and Modify

The AM parser uses a neural dependency parser and tagger to predict terms over
the AM algebra (Groschwitz et al., 2017), which combines graphs into bigger
graphs using the operations Apply and Modify. To this end, nodes of the graphs
can be decorated with _sources_ (Courcelle and Engelfriet (2012), marked in
blue), which assign names to nodes at which the graph can be combined with
other graphs. Every graph has a special source called root, drawn with a bold
outline, which is where the graph inserts into others when used as an
argument.

In the example of Fig. 2(a), the graph gwant has sources s and o indicating
where the arguments supplied by the subject and object should be inserted. It
also has a source m1 which allows it to attach to some other graph as a
modifier.

The Apply operation (app) models the combination of a complement (i.e.
argument) with its head. For example in Fig. 2(d), the appo operation combines
the head gwant with its argument gsleep, plugging the root of gsleep into the
o source of gwant (Fig. 2(d)). Because every graph may only contain one node
decorated with each source name, the s and m1 source nodes of gsleep and gwant
get merged. This allows the AM algebra to generate nontrivial graph
structures.

The Modify operation (mod) models the combination of a head with a modifier.
For example, the modm operation in our example attaches the adjunct glittle to
the root of its head gcat, using the adjunct’s m source (Fig. 2(b)). Again,
both graphs have an m1 source that gets merged.

(a) Graphs for the individual words

(b) modm(gcat, glittle)

(c) Complete DRG

(d) appo(gwant, gsleep)

[theme = simple] [column sep=.01cm, font=] start & The & little & cat & wanted
& to & sleep.
gstart & & glittle & gcat & gwanted & & gsleep
15MODm 54apps 57appo 43modm2

(e) AM dependency tree

Figure 2: Relevant graphs for sentence The little cat wanted to sleep.

#### AM dependency trees and AM parsing

The AM parser predicts a graph from a sentence by computing an _AM dependency
tree_ , as in Fig. 2(e). It uses a neural tagger to predict a _lexical graph_
for each word (drawn below the sentence) and a neural dependency parser to
predict app and mod edges. The AM dependency tree can be unraveled into a term
of app and mod operations over the AM algebra, which deterministically
evaluates into a graph; for instance, the AM dependency tree in Fig. 2(e)
evaluates to the graph in Fig. 2(c). Words that do not lexically contribute to
the meaning representation, such as the determiner the, are not assigned
incoming dependency edges and thus ignored in the construction of the graph.

In order to train the AM parser, one needs to construct an AM dependency tree
for every sentence-graph instance in the training data. _Decomposing_ the
graph into an AM dependency tree is a nontrivial task, which can fail:
Depending on the alignments between word tokens and nodes in the graph, an AM
dependency tree that evaluates to the given graph may not exist. We call such
training instances _non-decomposable_.

##  3 Scope in DRT is hard for the AM parser

We start with an attempt to directly apply the AM parser to DRT. As we will
see, the dashed scope edges in a DRG are difficult to handle with the AM
parser. We will solve this problem in the AMS parser, presented in Section 4.

###  3.1 A baseline AM parser for DRG

We construct AM dependency trees for the DRGs in the PMB using the approach
proposed by Groschwitz et al. (2021), which learns decomposition jointly with
training the neural parsing model. The learning algorithm represents the
latent space of possible AM dependency trees for each graph compactly,
allowing training on the whole latent space. This leads to the parser
converging on AM dependency trees that are consistent across the corpus.

This largely unsupervised method still requires two inputs beyond the graph.
First, node-token alignments (every node must be aligned), for which we use
the alignments given in PMB5.1. For the top box in each DRG, which is always
unaligned, we introduce a special START token to align it to (cf. Fig. 2(e)).

Second, each edge must be assigned to a graph constant, to fully partition the
DRG into lexical graphs for the individual words. This often makes the
difference between an Apply or Modify operation. For example in Fig. 2, the
Attribute edge between little and cat is grouped with the little node, making
little a modifier of cat, a linguistically plausible analysis. The edge could
also be grouped with the cat node, effectively making little an argument of
cat (the two would be combined with an app operation), an implausible
analysis. We follow the linguistically-informed principle to group edges
between a head and an argument with the head, and edges between a head and a
modifier (Lindemann et al., 2019); see Appendix C for our full heuristics. The
scope edges do not fall into these categories and provide a unique challenge,
see below.

All remaining aspects of the AM dependency tree, including the source names,
are then learned during training.

###  3.2 The Challenge of Scope Prediction

Figure 3: Failed combination of graphs for Fig 1

The scope edges of DRGs are not something that the Apply and Modify operations
were designed for. In particular, the scope edges do not fall
straightforwardly into the head/argument/modifier paradigm. The design of the
AM algebra forces us into an inconvenient choice: (1) include scope edges in
the lexical graph that contains the box and insert the contents of the box
with Apply operations; or (2) include scope edges in the lexical graphs of the
contents of the box and insert them into the box using Modify operations.

| NoPrep | CPT | SCPL
---|---|---|---
APP | 0.7 | 84.5 | 94.4
MOD | 76.7 | 77.7 | 78.0
Table 1: Decomposable graphs in PMB5 (%). APP: member edges grouped with the
box; resulting in Apply operations in the AM dependency tree. MOD: member
edges grouped with the content nodes, resulting in Modify operations.

The first approach fails completely, with only 0.7% of DRGs in PMB5 being
decomposable (see Table 1, NoPrep/APP; see also Appendix B). The second
approach works better, with 76% graphs being decomposable (Table 1,
NoPrep/MOD). For example, Fig. 2(e) shows a valid AM dependency tree for the
graph in Fig. 2(c) under this paradigm. However, this success is limited to
graphs with only a single box: only 30% of all multibox DRGs, i.e. DRGs that
contain more than one box node, can be decomposed into AM dependency trees.

To illustrate the challenge, consider the DRG in Fig. 1. Fig. 3 shows two
partial graphs in an attempt to build the full graph with the AM algebra, the
left representing child misbehaves occasionally, and the right representing
every. The lexical graph gevery introduces two boxes, and to obtain the DRG in
Fig. 1, we need to draw a scope edge from the upper box to the child node on
the left and, simultaneously, scope edges from the lower box to the misbehave,
time, and occasionally nodes. We can use a modm1 operation to unify the
M1-source of the left graph with the root of gevery (the upper box); but this
will put child into the wrong box. The problem is that both boxes are
introduced by the same lexical graph (a consequence of the alignments in the
PMB), and only one of them can receive outgoing edges through a single Modify
operation. Other attempts at decomposing the DRG in Fig. 1 fail in similar
ways.

##  4 Scope-enhanced AM Parsing

We will address this scope challenge through a two-step process. First, we
simplify the DRGs by removing scope edges, such that over 94% of DRGs can be
decomposed for training. Second, we recover the scope information at parsing
time through an independent scope prediction mechanism. The overall structure
of our parser is sketched in Fig. 4.

Figure 4: Overall structure of the AMS parser.

###  4.1 Simplifying DRGs

We identified two effective DRG simplification strategies: Compact DRG and
Scopeless DRG.

#### Compact DRG

The Compact DRG representation (CPT), inspired by Abzianidze et al. (2020),
makes use of the fact that many nodes share the same scope as their parent
node, i.e. are members of the same box. In this representation, we thus remove
all scope edges for nodes that are in the same scope as their parents (if
there are multiple parents, we only remove the scope edge if the node and
_all_ its parents are in the same box). This method removes around 70% of
scope edges, and the full scope information can be losslessly recovered with
the rule-based method in Section 4.2. The compact DRG for Fig. 1 is shown in
Fig. 5 with the removed edges marked in light blue.

Figure 5: Compact DRG for Fig 1.

#### Scopeless DRG

While Compact DRGs maintain at least one connection between a scope box and a
node within its scope, Scopeless DRGs (SCPL) remove all scope edges as long as
the graph remains connected. This results in graphs that are mostly reduced to
their predicate-argument structure, facilitating a more straightforward
decomposition with the AM Algebra, at the cost of losing some information. An
example is shown in Fig. 6. More complex examples are detailed in Appendix I.

Figure 6: Scopeless DRG for Fig 1.

Both Compact and Scopeless DRGs show much higher decomposability rates
compared to the full DRGs, see Table 1. This effect is particularly strong in
the setting where membership edges are grouped with the boxes (see row “APP”),
where Compact and Scopeless DRGs achieve decomposability rates of 84.5% and
94.4% respectively.

###  4.2 Scope Prediction

To recover the scope information, we designed two scope resolvers: one rule-
based, and the other relying on a dependency parser to predict the scope
edges.

#### Rule-based Scope Resolver

The rule-based scope resolver is the inverse of our Compact DRG simplification
method, but can also be applied to Scopeless DRG. This resolver traverses the
predicted graph top-down; if it encounters a node with no incoming scope edge,
it assigns the node the same scope as its parent. If a node has multiple
parents with conflicting scope, an arbitrary parent is chosen (this only
occurs with Scopeless DRG). For Compact DRG, this method recovers the full
scope information losslessly.

This rule-based approach is easy to implement, transparent and fully
explainable. However, it is imperfect for Scopeless DRG, and even for Compact
DRG it may propagate parsing errors into the recovered scope edges.

#### Dependency-based Scope Resolver

For the dependency-based scope resolver, we make use of the fact that an AM
dependency tree splits the graph into lexical graphs, each of which is linked
to a specific word token in the sentence. This induces an alignment relation
between nodes in the graph and tokens in the sentence: a node is aligned to
the token if it is part of the lexical graph for that token. We project the
scope edges in the DRG into edges between the word tokens by following this
alignment relation from the nodes to the tokens; this creates a _scope
dependency graph_ over the sentence (see Fig. 7). The scope dependency graph
is not necessarily a tree: it need not be connected, and a token might receive
multiple incoming edges if the aligned lexical graph contains multiple nodes
linked to different boxes (see Appendix F).

When the lexical graph for a token contains multiple nodes or boxes, we also
encounter a further challenge. In such a case, the scope dependency graph,
which connects only the two tokens, cannot fully specify which nodes in the
lexical graphs the scope edge connects. An example of this can be seen in Fig.
7. Here, the lexical graphs gchild and gmisbehaves are both children of gevery
in the scope dependency graph, but they should go in different boxes of
gevery.

To remove this ambiguity, we name the boxes in each lexical graph and encode
the box to which each child in the scope dependency graph connects in the
dependency edge label. For example, consider again the dependency graph in
Fig. 7. The relationship between the tokens every and child is annotated as
scope_b2, indicating that child goes into the upper box (b2111The labeling of
the boxes is decided by the hierarchy of the boxes in the whole graph: the
parent box is assigned by a smaller number than the children, the root box is
assigned with b1.). By contrast, the edge into misbehaves has the label
scope_b3, indicating that it goes into the lower box. We use a similar method
if different nodes from the same constant are members of different boxes, see
Appendix F. In this way, the labeled scope dependency graphs unambiguously
specify scope edges.

[theme = simple] [column sep=.01cm, font=] start & every & child & misbehaves
& ocasionally.
gstart & gevery& gchild & gmisbehaves & goccasionally
1root 23scope_b2 24scope_b3 25scope_b3

Figure 7: Scope dependency graph for Every child misbehaves occasionally.

This method allows us to use standard dependency parsing techniques for scope
prediction. We adopted the biaffine dependency graph parser of Dozat and
Manning (2018), which is simple and accurate. We use ordinary supervised
training, based on the annotated node-token alignments in the PMB5.
Hyperparameter details can be found in Appendix E.

Since the AM parser also predicts some scope edges (in particular for Compact
DRG, but also a bare minimum in Scopeless DRG), there can be conflicts between
the dependency-based scope predictions and scope edges already present in the
predicted simplified DRG. We use the following rules to resolve mismatches:
(1) We only use a dependency-based edge if its target has no scope edge in the
predicted simplified DRG; i.e. the AM parser predictions take precedence. (2)
Any remaining node without a scope edge inherits its scope from its parent (as
in the rule-based resolver).

##  5 Results & Discussion

###  5.1 Data & Evaluation

We evaluated on the latest Parallel Meaning Bank 5.1.0 Abzianidze et al.
(2017). Apart from the normal train, dev, and test split, the PMB 5.1.0 also
provides an extra TestLong set that contains 40 lengthy (average length: 39.7
tokens) sentences. Statistics can be found in Appendix A.

For the evaluation metric, following Poelman et al. (2022), we convert DRGs to
condensed Penman notations222Examples can be found in Appendix G. (Wang et
al., 2023b) and adopt the SMATCH F1 score (Cai and Knight, 2013) to assess
DRGs. We also report the percentage of test instances for which a parser
generated ill-formed DRGs.

###  5.2 Handling coreference

The PMB contains coreference annotations; these are non-compositional by
design and thus very tricky for a compositional system like the AM parser. We
reduce the impact of coreference on our evaluation through a simple pre- and
postprocessing method. We remove all edges indicating coreference in the DRG
and introduce a new tag p to the label of all coreferent nodes. In
postprocessing, we then simply add coreference edges between all nodes marked
as coreferent. This method further increases decomposability, up to 94% (see
Table 1). Details are in Appendix D.

This method has the advantage of only using information from the predicted
DRG, but it only really works when there is just one instance of coreference
in the graph. This is frequently the case in the PMB 5.1.0, but in a different
setting, more complex coreference resolution methods would likely be needed
(see e.g. Anikina et al. (2020)).

###  5.3 Experiment details

We use the implementation of Groschwitz et al. (2021) in all our AM Parser
experiments. Hyperparameter settings can be found in Appendix H.

We compare the AMS parser against the strongest published models for DRG
parsing listed in Zhang et al. (2024): byT5 (Xue et al., 2022), mT5 (Xue et
al., 2021) and mBART (Liu et al., 2020). All of these are sequence-to-sequence
models with no built-in awareness of semantic structure, compositionality, or
scope.

We also trained the AM Parser on the DRGs with the original scope annotations.
To make the root box easier to learn for the parser, we introduced a new token
START to the beginning of each input sentence. Finally, we fine-tuned T5-Base,
T5-Large (Raffel et al., 2020) as two further robust baselines.

###  5.4 Parsing Results

#### Scope Dependency Parsing

We first evaluate how accurately scope assignments can be predicted by
dependency parsing (cf. Section 4.2), using the usual UAS and LAS evaluation
measures for dependency parsing. Table 2 reveals high LAS and UAS of predicted
scope dependency graphs across both development and test sets, indicating
reliable scope prediction. This is remarkable, given the complexity of the
scope prediction task.

On the TestLong set, the accuracy dropped significantly, indicating the
difficulty of predicting scope as sentences grow in complexity. The much
larger drop in LAS compared to UAS indicates the difficulty of reliably making
scope assignment decisions within a lexical graph.

Dev | Test | TestLong
---|---|---
UAS | LAS | UAS | LAS | UAS | LAS
98.7 | 96.4 | 98.3 | 95.7 | 67.0 | 55.4
Table 2: Accuracy of scope dependency parsing.

#### DRG Parsing

For the task of DRG parsing itself, we compare the AMS parser to the baselines
in Table 3. Our focus is on models that are trained on the hand-annotated gold
dataset (G); we also include some models trained on gold and silver. The
suffix scpl denotes Scopeless DRGs, cpt refers to Compact DRGs, d indicates
the dependency-based scope resolver, and h signifies the heuristic scope
resolver. “Without scope resolution” groups together variants of the AMS
parser that directly predict compact or scopeless DRGs, without a mechanism
for reconstructing scope edges in postprocessing. The best results among the
gold-trained models are marked in bold. Three critical observations emerge
from the table.

First, the AMS parser, especially the scopeless (SCPL) version, excels against
the gold-data trained baselines. The only exception is byT5, which has a
token-free architecture that makes it particularly good at processing short
texts – a significant advantage given the very short average sentence length
of 6.7 tokens in the regular test set. The AMS parser also outperforms the
generic AM parser, indicating the effectiveness of our novel scope resolution
mechanism.

Second, in contrast to all seq2seq models, the AMS parser maintains a 0% error
rate, i.e. it never generated ill-formed DRGs. Furthermore, on the very long
sentences of the TestLong set, all variants of the AMS parser outperform the
gold-trained seq2seq baselines by a large margin, almost achieving parity with
the best model trained on silver data.

Finally, Scopeless DRGs perform better than Compact DRGs. This could be
attributed to the fact that Compact DRGs retain more scope edges, making the
graph more complex to learn. The higher decomposability rate of Scopeless DRGs
also means that we have more training data in that setting. The dependency-
based scope resolver outperforms its heuristic-based counterpart in accuracy
across in-domain development and test splits. This advantage makes sense given
the scope dependency parser’s high accuracy. It could also be that the
dependency resolver is better able to handle initial parsing inaccuracies
compared to the rule-based resolver, where AM Parsing errors can easily
propagate into more scope errors.

Models | Test | TestLong
---|---|---
| F1 | Err | F1 | Err
Baselines (gold only) |  |  |  |
ByT5(G) | 86.7 | 5.4 | 27.1 | 38.3
mT5(G) | 61.2 | 11.3 | 16.5 | 25.0
mBART(G) | 82.8 | 6.3 | 30.5 | 12.5
T5-base(G) | 76.4 | 20.0 | 13.9 | 77.5
T5-large(G) | 84.2 | 3.9 | 18.1 | 67.5
AM Parser(G) | 81.9 | 0.0 | 47.2 | 0.0
without scope resolution |  |  |  |
AMS Parserscpl(G) | 73.5 | 0.0 | 39.3 | 0.0
AMS Parsercmpt(G) | 73.1 | 0.0 | 37.3 | 0.0
with scope resolution |  |  |  |
AMS Parserscpl+h(G) | 84.6 | 0.0 | 48.4 | 0.0
AMS Parsercmpt+h(G) | 83.1 | 0.0 | 44.9 | 0.0
AMS Parserscpl+d(G) | 85.3 | 0.0 | 48.8 | 0.0
AMS Parsercmpt+d(G) | 83.3 | 0.0 | 46.0 | 0.0
Baselines (gold + silver) |  |  |  |
byT5(G+S) | 93.4 | 0.7 | 36.6 | 40.0
T5-base(G+S) | 86.0 | 1.6 | 44.3 | 37.5
mT5(G+S) | 93.1 | 0.8 | 55.8 | 15.0
mBART(G+S) | 86.2 | 4.4 | 7.8 | 12.5
Table 3: Accuracy and error rates for DRG parsing.

#### Scaling to complex DRGs

As we already saw in Section 3.2, scope prediction is easy when there are not
many boxes. Table 3 therefore splits the test instances by number of
boxes333For the TestLong split, we evaluate the models only on multi-box DRGs,
of which there are 33 out of 40.. For each of these classes, we report the
overall SMATCH score of our best model and the baselines, as well as the
SMATCH score when considering only scope edges. This allows us to explore how
the parsers scale to complex DRGs, and in particular how they maintain their
ability to predict scope edges when there are many boxes.

Compared to other models trained on gold data, the AMS parser excels at
maintaining its accuracy as the DRGs grow more complex. While the AM parser is
almost on par with the AMS parser on single-box DRGs, the gap widens
drastically with increasing complexity. For DRGs with four or more boxes, as
well as on the TestLong set, the AMS parser also decisively outperforms all
(gold) seq2seq baselines.

Models | Test |  |  |  | TestL
---|---|---|---|---|---
# Box | #1 |  #2 | #3 | #$\geq$4 | #$\geq$2
Count | 972 |  136 | 75 | 10 | 33
T5-base(G) | 86.2 |  17.5 | 17.5 | 16.7 | 9.0
| 92.9 |  3.0 | 5.0 | 14.6 | 10.3
mBART(G) | 84.8 |  81.5 | 83.1 | 76.6 | 17.2
| 91.7 |  80.7 | 84.8 | 80.2 | 17.2
mT5(G) | 65.6 |  61.6 | 55.8 | 49.9 | 13.5
| 76.7 |  65.4 | 63.3 | 59.3 | 17.7
T5-large(G) | 87.9 |  74.8 | 67.4 | 45.2 | 19.0
| 94.1 |  73.7 | 68.5 | 48.3 | 21.5
byT5(G) | 87.3 |  84.3 | 86.9 | 52.5 | 29.6
| 93.6 |  85.2 | 86.6 | 55.6 | 33.0
AM Parser(G) | 84.3 |  75.6 | 67.5 | 61.9 | 46.3
| 92.1 |  78.9 | 72.0 | 66.5 | 56.0
AMS Parserscpl+d(G) | 86.0 |  83.8 | 81.9 | 75.2 | 48.2
| 92.9 |  86.5 | 82.2 | 85.2 | 58.7
byT5(G+S) | 89.1 |  89.9 | 88.8 | 83.6 | 48.0
| 95.0 |  89.2 | 90.6 | 87.4 | 47.8
mT5(G+S) | 89.1 |  89.9 | 88.8 | 85.6 | 61.7
| 95.0 |  88.9 | 89.5 | 88.4 | 65.1
mBART(G+S) | 84.8 |  81.5 | 83.1 | 75.7 | 17.2
| 91.7 |  80.6 | 84.6 | 80.2 | 17.6
Table 4: SMATCH score for multi-box DRGs and corresponding scope score
(highlighted in gray)

At the same time, we observe that the AMS parser maintains a very high
accuracy on predicting scope edges even for complex DRGs. We observe that the
difference between the AMS Parser and the baseline AM parser is small on
single-box DRGs, but much larger on multi-box DRGs, showing that treating
scope prediction separately pays off.

###  5.5 What makes long texts so hard?

Anil et al. (2022) found that simple fine-tuning of transformer models does
not achieve length generalization, nor does scaling up the models. We
conducted a detailed error analysis and identified two factors that might
contribute to the limitations of the models in length generalization.

#### Structural Complexity

As shown in Table 4, all models show a decreasing trend as the number of boxes
increases. We find that a higher number of boxes generally results in longer
sequences, especially in the TestLong split - we assume the box complexity
brought by longer sequences could be a possible reason for length
generalization limitation.

Furthermore, byT5 tends to generate shorter sequences, averaging 70 roles and
relations in its predictions, in contrast to other models which average
approximately 100. This discrepancy underscores byT5’s limitation in handling
long texts.

#### Sense Generalization

Furthermore, longer sentences can introduce new word senses, which have to be
predicted as node labels. 25% senses in the TestLong split are absent in the
train split. All models show accuracies lower than 0.33 in predicting unseen
senses with the AMS Parser performing the best at this rate.

##  6 Conclusion and Future Work

In this work, we proposed a novel mechanism for predicting scope assignments
in DRT parsing. By combining it with the compositional AM parser, we obtain
the AMS parser, which outperforms existing DRT parsers trained on the same
dataset, especially for complex sentences. It also avoids the prediction of
ill-formed DRGs that plague other models. The prediction of scope information
has been a long-standing challenge in computational semantics; our dependency
parsing mechanism achieves very high accuracy on this task.

In the future, we plan to extend our work to tackle increasingly complex
meaning representation frameworks, such as Uniform Meaning Representation
(UMR) (Van Gysel et al., 2021). Since UMR-writer (Zhao et al., 2021), the UMR
annotation tool, provides node-token alignment automatically, no more manual
annotation is needed. Furthermore, our current system’s architecture, which
includes both the AM Parser and a dependency parser by Dozat and Manning
(2018), presents opportunities for optimization. We aim to streamline the
process by unifying these two models into a single framework that leverages
joint learning.

## Limitations

The AMS parser uses the AM parser to predict the predicate-argument relations
in the DRGs. The AM parser has not kept pace in accuracy with the development
of overall graph parsing models since it was published in 2019. This holds
back the accuracy of the AMS parser. If a more accurate sentence-to-graph
parser that induces node-token alignments became available, the AMS parser
could be combined with it for increased accuracy. Note, however, that the AM
parser shows strong performance with respect to the degradation of parsing
accuracy for long and complex sentences.

Furthermore, the treatment of coreference in the paper is quite shallow. One
might include the predictions of a coreference resolver into the parsing
process. On the relatively short coreference chains in the PMB test sets, this
would probably not make a significant impact on the evaluation.

## References

  * Abzianidze et al. (2017) Lasha Abzianidze, Johannes Bjerva, Kilian Evang, Hessel Haagsma, Rik van Noord, Pierre Ludmann, Duc-Duy Nguyen, and Johan Bos. 2017.  The Parallel Meaning Bank: Towards a multilingual corpus of translations annotated with compositional meaning representations.  In _Proceedings of the 15th Conference of the European Chapter of the Association for Computational Linguistics: Volume 2, Short Papers_ , pages 242–247, Valencia, Spain. Association for Computational Linguistics.
  * Abzianidze et al. (2020) Lasha Abzianidze, Johan Bos, and Stephan Oepen. 2020.  DRS at MRP 2020: Dressing up discourse representation structures as graphs.  In _Proceedings of the CoNLL 2020 Shared Task: Cross-Framework Meaning Representation Parsing_ , pages 23–32, Online. Association for Computational Linguistics.
  * Anikina et al. (2020) Tatiana Anikina, Alexander Koller, and Michael Roth. 2020.  Predicting coreference in Abstract Meaning Representations.  In _Proceedings of the Third Workshop on Computational Models of Reference, Anaphora and Coreference_ , pages 33–38, Barcelona, Spain (online). Association for Computational Linguistics.
  * Anil et al. (2022) Cem Anil, Yuhuai Wu, Anders Andreassen, Aitor Lewkowycz, Vedant Misra, Vinay Ramasesh, Ambrose Slone, Guy Gur-Ari, Ethan Dyer, and Behnam Neyshabur. 2022.  Exploring length generalization in large language models.  _Advances in Neural Information Processing Systems_ , 35:38546–38556.
  * Artzi et al. (2015) Yoav Artzi, Kenton Lee, and Luke Zettlemoyer. 2015.  Broad-coverage CCG semantic parsing with AMR.  In _Proceedings of the 2015 Conference on Empirical Methods in Natural Language Processing_ , pages 1699–1710, Lisbon, Portugal. Association for Computational Linguistics.
  * Asher (1993) Nicholas Asher. 1993.  _Reference to abstract objects in discourse_ , volume 50.  Springer Science & Business Media.
  * Banarescu et al. (2013) Laura Banarescu, Claire Bonial, Shu Cai, Madalina Georgescu, Kira Griffitt, Ulf Hermjakob, Kevin Knight, Philipp Koehn, Martha Palmer, and Nathan Schneider. 2013.  Abstract meaning representation for sembanking.  In _Proceedings of the 7th linguistic annotation workshop and interoperability with discourse_ , pages 178–186.
  * Bos (2008) Johan Bos. 2008.  Wide-coverage semantic analysis with Boxer.  In _Semantics in Text Processing. STEP 2008 Conference Proceedings_ , pages 277–286. College Publications.
  * Bos (2015) Johan Bos. 2015.  Open-domain semantic parsing with boxer.  In _Proceedings of the 20th nordic conference of computational linguistics (NODALIDA 2015)_ , pages 301–304.
  * Bos (2023) Johan Bos. 2023.  The sequence notation: Catching complex meanings in simple graphs.  In _Proceedings of the 15th International Conference on Computational Semantics_ , pages 195–208, Nancy, France. Association for Computational Linguistics.
  * Cai and Knight (2013) Shu Cai and Kevin Knight. 2013.  Smatch: an evaluation metric for semantic feature structures.  In _Proceedings of the 51st Annual Meeting of the Association for Computational Linguistics (Volume 2: Short Papers)_ , pages 748–752, Sofia, Bulgaria. Association for Computational Linguistics.
  * Courcelle and Engelfriet (2012) Bruno Courcelle and Joost Engelfriet. 2012.  _Graph structure and monadic second-order logic: a language-theoretic approach_ , volume 138.  Cambridge University Press.
  * Dozat and Manning (2018) Timothy Dozat and Christopher D. Manning. 2018.  Simpler but more accurate semantic dependency parsing.  In _Proceedings of the 56th Annual Meeting of the Association for Computational Linguistics (Volume 2: Short Papers)_ , pages 484–490, Melbourne, Australia. Association for Computational Linguistics.
  * Fancellu et al. (2019) Federico Fancellu, Sorcha Gilroy, Adam Lopez, and Mirella Lapata. 2019.  Semantic graph parsing with recurrent neural network DAG grammars.  In _Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing (EMNLP-IJCNLP)_ , pages 2769–2778, Hong Kong, China. Association for Computational Linguistics.
  * Groschwitz et al. (2017) Jonas Groschwitz, Meaghan Fowlie, Mark Johnson, and Alexander Koller. 2017.  A constrained graph algebra for semantic parsing with AMRs.  In _Proceedings of the 12th International Conference on Computational Semantics (IWCS) — Long papers_.
  * Groschwitz et al. (2021) Jonas Groschwitz, Meaghan Fowlie, and Alexander Koller. 2021.  Learning compositional structures for semantic graph parsing.  In _Proceedings of the 5th Workshop on Structured Prediction for NLP (SPNLP 2021)_ , pages 22–36, Online. Association for Computational Linguistics.
  * Groschwitz et al. (2018) Jonas Groschwitz, Matthias Lindemann, Meaghan Fowlie, Mark Johnson, and Alexander Koller. 2018.  AMR dependency parsing with a typed semantic algebra.  In _Proceedings of the 56th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers)_ , pages 1831–1841, Melbourne, Australia. Association for Computational Linguistics.
  * Haug (2014) Dag Trygve Truslew Haug. 2014.  Partial dynamic semantics for anaphora: Compositionality without syntactic coindexation.  _Journal of Semantics_ , 31(4):457–511.
  * Honnibal et al. (2020) Matthew Honnibal, Ines Montani, Sofie Van Landeghem, and Adriane Boyd. 2020.  spaCy: Industrial-strength natural language processing in python.
  * Hupkes et al. (2020) Dieuwke Hupkes, Verna Dankers, Mathijs Mul, and Elia Bruni. 2020.  Compositionality decomposed: How do neural networks generalise?  _Journal of Artificial Intelligence Research_ , 67:757–795.
  * Kamp (1981) Hans Kamp. 1981.  Evénements, représentations discursives et référence temporelle.  _Langages_ , (64):39–64.
  * Kamp and Reyle (1993) Hans Kamp and Uwe Reyle. 1993.  _From Discourse to Logic: An Introduction to Modeltheoretic Semantics of Natural Language, Formal Logic and DRT_.  Kluwer, Dordrecht.
  * Kohlhase et al. (1998) Michael Kohlhase, Susanna Kuschert, and Martin Müller. 1998.  Dynamic lambda calculus.  Manuscript.
  * Kohlhase et al. (1996) Michael Kohlhase, Susanna Kuschert, and Manfred Pinkal. 1996.  A type-theoretic semantics for-DRT.  In _Proceedings of the 10th Amsterdam Colloquium_ , pages 479–498.
  * Lascarides and Asher (2007) Alex Lascarides and Nicholas Asher. 2007.  Segmented Discourse Representation Theory: Dynamic semantics with discourse structure.  In _Computing meaning_ , pages 87–124. Springer.
  * Le and Zuidema (2012) Phong Le and Willem Zuidema. 2012.  Learning compositional semantics for open domain semantic parsing.  In _Proceedings of COLING 2012_ , pages 1535–1552, Mumbai, India. The COLING 2012 Organizing Committee.
  * Lindemann et al. (2019) Matthias Lindemann, Jonas Groschwitz, and Alexander Koller. 2019.  Compositional semantic parsing across graphbanks.  In _Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics_ , pages 4576–4585, Florence, Italy. Association for Computational Linguistics.
  * Liu et al. (2018) Jiangming Liu, Shay B. Cohen, and Mirella Lapata. 2018.  Discourse representation structure parsing.  In _Proceedings of the 56th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers)_ , pages 429–439, Melbourne, Australia. Association for Computational Linguistics.
  * Liu et al. (2020) Yinhan Liu, Jiatao Gu, Naman Goyal, Xian Li, Sergey Edunov, Marjan Ghazvininejad, Mike Lewis, and Luke Zettlemoyer. 2020.  Multilingual denoising pre-training for neural machine translation.  _Transactions of the Association for Computational Linguistics_ , 8:726–742.
  * Liu et al. (2019) Yinhan Liu, Myle Ott, Naman Goyal, Jingfei Du, Mandar Joshi, Danqi Chen, Omer Levy, Mike Lewis, Luke Zettlemoyer, and Veselin Stoyanov. 2019.  Roberta: A robustly optimized BERT pretraining approach.  _CoRR_ , abs/1907.11692.
  * Muskens (1994) R.A. Muskens. 1994.  A compositional Discourse Representation Theory.  In _Proceedings of the Ninth Amsterdam Colloquium_ , volume 9 of _Proceedings of the Amsterdam Colloquium_ , pages 467–486. ILLC/Department of Philosophy, University of Amsterdam.  Pagination: 20.
  * Muskens (1996) Reinhard Muskens. 1996.  Combining Montague semantics and discourse representation.  _Linguistics and philosophy_ , pages 143–186.
  * Poelman et al. (2022) Wessel Poelman, Rik van Noord, and Johan Bos. 2022.  Transparent semantic parsing with Universal Dependencies using graph transformations.  In _Proceedings of the 29th International Conference on Computational Linguistics_ , pages 4186–4192, Gyeongju, Republic of Korea. International Committee on Computational Linguistics.
  * Raffel et al. (2020) Colin Raffel, Noam Shazeer, Adam Roberts, Katherine Lee, Sharan Narang, Michael Matena, Yanqi Zhou, Wei Li, and Peter J Liu. 2020.  Exploring the limits of transfer learning with a unified text-to-text transformer.  _The Journal of Machine Learning Research_ , 21(1):5485–5551.
  * Van Gysel et al. (2021) Jens EL Van Gysel, Meagan Vigus, Jayeol Chun, Kenneth Lai, Sarah Moeller, Jiarui Yao, Tim O’Gorman, Andrew Cowell, William Croft, Chu-Ren Huang, et al. 2021.  Designing a uniform meaning representation for natural language processing.  _KI-Künstliche Intelligenz_ , 35(3-4):343–360.
  * Van Noord et al. (2018) Rik Van Noord, Lasha Abzianidze, Antonio Toral, and Johan Bos. 2018.  Exploring neural methods for parsing discourse representation structures.  _Transactions of the Association for Computational Linguistics_ , 6:619–633.
  * van Noord et al. (2019) Rik van Noord, Antonio Toral, and Johan Bos. 2019.  Linguistic information in neural semantic parsing with multiple encoders.  In _Proceedings of the 13th International Conference on Computational Semantics - Short Papers_ , pages 24–31, Gothenburg, Sweden. Association for Computational Linguistics.
  * van Noord et al. (2020) Rik van Noord, Antonio Toral, and Johan Bos. 2020.  Character-level representations improve DRS-based semantic parsing even in the age of BERT.  In _Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing (EMNLP)_ , pages 4587–4603, Online. Association for Computational Linguistics.
  * Wang et al. (2023a) Chunliu Wang, Huiyuan Lai, Malvina Nissim, and Johan Bos. 2023a.  Pre-trained language-meaning models for multilingual parsing and generation.  In _Findings of the Association for Computational Linguistics: ACL 2023_ , pages 5586–5600, Toronto, Canada. Association for Computational Linguistics.
  * Wang et al. (2023b) Chunliu Wang, Xiao Zhang, and Johan Bos. 2023b.  Discourse representation structure parsing for Chinese.  In _Proceedings of the 4th Natural Logic Meets Machine Learning Workshop_ , pages 62–74, Nancy, France. Association for Computational Linguistics.
  * Weißenhorn et al. (2022) Pia Weißenhorn, Lucia Donatelli, and Alexander Koller. 2022.  Compositional generalization with a broad-coverage semantic parser.  In _Proceedings of the 11th Joint Conference on Lexical and Computational Semantics_ , pages 44–54, Seattle, Washington. Association for Computational Linguistics.
  * Xue et al. (2022) Linting Xue, Aditya Barua, Noah Constant, Rami Al-Rfou, Sharan Narang, Mihir Kale, Adam Roberts, and Colin Raffel. 2022.  ByT5: Towards a token-free future with pre-trained byte-to-byte models.  _Transactions of the Association for Computational Linguistics_ , 10:291–306.
  * Xue et al. (2021) Linting Xue, Noah Constant, Adam Roberts, Mihir Kale, Rami Al-Rfou, Aditya Siddhant, Aditya Barua, and Colin Raffel. 2021.  mT5: A massively multilingual pre-trained text-to-text transformer.  In _Proceedings of the 2021 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies_ , pages 483–498, Online. Association for Computational Linguistics.
  * Yao and Koller (2022) Yuekun Yao and Alexander Koller. 2022.  Structural generalization is hard for sequence-to-sequence models.  In _Proceedings of the 2022 Conference on Empirical Methods in Natural Language Processing_ , pages 5048–5062, Abu Dhabi, United Arab Emirates. Association for Computational Linguistics.
  * Zhang et al. (2024) Xiao Zhang, Chunliu Wang, Rik van Noord, and Johan Bos. 2024.  Gaining more insight into neural semantic parsing with challenging benchmarks.  In _Proceedings of the Fifth International Workshop on Designing Meaning Representations @ LREC-COLING 2024_ , pages 162–175, Torino, Italia. ELRA and ICCL.
  * Zhao et al. (2021) Jin Zhao, Nianwen Xue, Jens Van Gysel, and Jinho D. Choi. 2021.  UMR-writer: A web application for annotating uniform meaning representations.  In _Proceedings of the 2021 Conference on Empirical Methods in Natural Language Processing: System Demonstrations_ , pages 160–167, Online and Punta Cana, Dominican Republic. Association for Computational Linguistics.

##  Appendix A Statistics of Parallel Meaning Bank Release 5.1.0

In our experiment, we excluded all ill-formed DRGs from the gold split of the
PMB5.1.0 dataset. Detailed statistics of the modified gold data as well as the
silver and bronze splits are presented below.

Gold | Silver | Bronze
---|---|---
Train | Dev | Test | TestLong |  |
9560 | 1195 | 1193 | 40 | 146,718 | 141,435
Table 5: Number of sentences across different splits

##  Appendix B Challenges Brought by Scope

In this section, we show that AM-Algebra struggles with even one-box DRG when
scope is taken as an argument of the root box, with sentence The little cat
wanted to sleep as an example.

The graphs corresponding to the sentences are illustrated in Fig. 8. As
depicted in Fig. 9, these graphs can be merged to form a scopeless lexical
graph. However, integrating this lexical graph with a box requiring four
arguments proves problematic for constructing the AM-tree. This is due to AM-
Algebra’s restriction against multiple apps (applications) between two sub-
graphs, a constraint that mirrors linguistic principles in English, where
different parts of one constituent cannot play unique roles relative to
another constituent.

Figure 8: graphs for DRGs with scope as argument

[theme = simple] [column sep=.01cm, font=] start & The & little & cat & wanted
& to & sleep.
gstart & $\bot$& glittle & gcat & gwanted & $\bot$ & gsleep
15xmodo1 54apps 57appo 43modm2

Figure 9: Failed combinations of graphs

##  Appendix C Heuristics on Edge Directions

The heuristics on edge directions can be found in Table 6.

Operation |  Edge Labels
---|---
APP |  Agent, Bearer, Participant, Creator, Proposition, Stimulus, Beneficiary, Co-Agent, Co-Patient, Co-Theme, Experiencer, Patient, Pivot, Product, Recipient, Theme, Owner, OF, User, Role, NEQ, APX, EQU, TPR
Mod |  Consumer, Topic, Result, member, Sub, Source, Destination, Goal, Product, ALTERNATION, ATTRIBUTION, CONDITION, CONSEQUENCE, CONTINUATION, CONTRAST, EXPLANATION, NECESSITY, NEGATION, POSSIBILITY, PRECONDITION, RESULT, SOURCE
Table 6: Mapping of operation types to edge labels in the DRG-to-graph
conversion process.

##  Appendix D Coreference Resolution

PMB5.1.0 explicitly marks coreference: two nodes that refer to the same entity
are connected with an ANA edge.

In our approach, we leverage the AM Parser’s supertagger for coreference
resolution. In PMB, node labels are annotated with lexical categories like n
(noun), a (adjective), r (adverb), and v (verb), such as female.n.02 in Fig.
10. To allow coreference resolution via supertagging, we introduce a new
category, denoted as p (pronoun). During preprocessing, this category is
assigned to nodes involved in coreference, identified by the ANA edge linking
them. For example, Fig. 10 shows the resulting penman notation after
preprocessing and postprocessing steps. The two nodes, s0 and s3 (bot labeled
female.n.01) are relabeled as female.p.01. This encodes the fact that the two
entities corefer is now encoded in the node labels, allowing us to remove the
ANA edge. While this is not always a lossless transformation when there are
multiple instances of coreference in the graph, we find it to work well in
practice (see Section 5). And crucially, this removes a reentrancy from the
DRG, making it more likely to be decomposable by the AM algebra. At training
time, the AM Parser’s supertagger can then learn to distinguish regular nouns
(i.e., n) and coreferent nouns (i.e., p).

Preprocessing:
(b0 / box :member (s1 / unscrew.v.01 :Agent (s0 / female.np.02 :Name (c0 /
"Mary")) :Time (s2 / time.n.08 :TPR (c1 / "now")) :Patient (s4 / lipstick.n.01
:User (s3 / female.np.02 :ANA s0))))
Postprocessing:
(b0 / box :member (s1 / unscrew.v.01 :Agent (s0 / female.n.02 :Name (c0 /
"Mary"))) :Time (s2 / time.n.08 :TPR (c1 / "now")) :Patient (s4 /
lipstick.n.01 :User (s3 / female.n.02 :ANA s0 ))))

Figure 10: An example of coreference after preprocessing and postprocessing
for the sentence Shei unscrewed heri lipstick.

At evaluation time, we reconstruct coreference information in a postprocessing
step. This step begins with identifying nodes marked as p in predicted DRGs.
However, if a DRG contains only one such p-tagged node, we do not treat it as
coreferent, since coreference involves multiple entities. In most cases, the
parser flags either one or two nodes as potential coreference candidates
within a single DRG. When two nodes are both tagged as p, we compare their
node concepts to see if they are identical. In our example (Fig. 10), since
both nodes are labeled female.p.02, indicating a match, we create an ANA edge
linking them. This edge is directed from node with a larger number on the node
label (like s3) to the one with a smaller node label (like s1). The final step
is to change the nodes’ categories from p back to n.

##  Appendix E Implementation details of the scope dependency parser

The original implementation of Dozat and Manning (2018) uses POS tags, lemma-,
and character-level word embeddings, processed through a BiLSTM and a Forward
Network (FNN), to predict if there is an edge between two tokens as well as
the corresponding edge label. Then a biaffian classifier is used to predict
the existence of an edge and the edge label.

In our experiment, we fine-tune roberta-large (Liu et al., 2019) and take POS
tags and characters as feature embeddings. All the linguistic information is
provided by spaCy444We use version 3.7.2 Honnibal et al. (2020). We keep all
other hyperparameters the same as the best model reported in their paper.

##  Appendix F Scope Annotation of a Complex Example

As discussed in Section 4, when a single token aligns with a lexical graph
that contains multiple nodes or boxes, it creates a complex scenario where
different nodes within the same lexical graph are linked to distinct boxes and
complicates the establishment of straightforward one-to-one dependency
relations between tokens. Our annotation method is straightforward: as long as
an aligned lexical graph contains multiple nodes or boxes, we make the scope
assignment of each node explicit in a top-down order.

We illustrate our method with two other possibilities when we build the
dependency edges between lexical graphs aligned with tokens.

(1) the two lexical graphs aligned with the token have multiple nodes and
boxes respectively, and each node is assigned a different scope box. An
example can be found in the scope assignment between the lexical graph aligned
with born ($G\textsubscript{born}$) and the lexical graph aligned with all
($G\textsubscript{all}$). We can see that the bottom node of
$G\textsubscript{born}$ receives the scope from the top box of
$G\textsubscript{all}$, while the top node of $G\textsubscript{born}$ receives
the scope from the bottom box of $G\textsubscript{all}$. In this case, the
dependency edge between born and all is scope_b3_b2.

(2) the two lexical graphs aligned with the token have multiple nodes and
boxes respectively, and each node is assigned the same scope box. This case
can be found in the scope assignment between the lexical graph aligned with
children ($G\textsubscript{children}$)and that with all. Although both nodes
of $G\textsubscript{children}$ receives the same scope, we still explicitly
annotating the scope for each node as shown in scope_b2_b2.

(a) Token Alignments: lexical graphs are color-coded to indicate alignment
with distinct tokens, denoted beneath each respective circle

[theme = simple] [column sep=.01cm, font=] start & All & of & their & children
& were & born & in & Malaysia.
gstart & gall & of & gtheir & gchildren & gwere & gborn& in & gMaylaysia
13no_scope 18no_scope 24scope_b2 25scope_b2_b2 26scope_b3 27scope_b3_b2
29scope_b3

(b) The converted dependency graph based on the scope information represented
in dashed lines

Figure 11: Complex DRG and dependency graph for All of their children were
born in Malaysia.

##  Appendix G Evaluation Format

In evaluation, we use a more compact format following Wang et al. (2023b).
This strict format integrates synset nodes’ information into a single entity
and eliminates variables representing constants, thereby avoiding inflated
scores. An example is shown in Fig. 12.

    (b0 / "box"
    :member (s0 / "synset"
        :lemma "person"
        :pos "n"
        :sense "01"
        :Name (c0 / "?"))
    :member (s1 / "synset"
        :lemma "time"
        :pos "n"
        :sense "08"
        :TPR (c1 / "now"))
    :member (s2 / "synset"
        :lemma "male"
        :pos "n"
        :sense "02"
        :Name (c2/"William W"))
    :member (s3 / "synset"
        :lemma "defeat"
        :pos "v"
        :sense "01"
        :Co-Agent s0
        :Time s1
        :Agent s2))

(a) Lenient Format used in Poelman et al. (2022)

    (b0 / box
    :member (s0 / person.n.01
        :Name "?")
    :member (s1 / time.n.08
        :TPR "now")
    :member (s2 / male.n.02
        :Name "William Wallace")
    :member (s3 / defeat.v.01
        :Co-Agent s0
        :Time s1
        :Agent s2))

(b) Strict Format

Figure 12: Comparison of DRG Representation in Lenient and Strict Formats for
the sentence Who did William Wallace defeat?

##  Appendix H Hyperparameters in AM Parser

The hyperparameters used in the experiments that show the best performance on
the scopeless SBN training data are summarized in Table 7.

Hyperparameter | Value
---|---
Activation function | tanh
Optimizer | Adam
Learning rate | 0.001
Epochs | 100
Early Stopping | 20
Dim of lemma embeddings | 64
Dim of POS embeddings | 32
Dim of NE embeddings | 16
Minimum lemma frequency | 7
Hidden layers in all MLPs | 1
Hidden units in LSTM (per direction) | 256
Hidden units in edge existence MLP | 256
Hidden units in edge label MLP | 256
Hidden units in supertagger MLP | 1024
Hidden units in lexical label tagger MLP | 1024
Layer dropout in LSTMs | 0.35
Recurrent dropout in LSTMs | 0.4
Input dropout | 0.35
Dropout in edge existence MLP | 0.0
Dropout in edge label MLP | 0.0
Dropout in supertagger MLP | 0.4
Dropout in lexical label tagger MLP | 0.4
Table 7: Common hyperparameters used in all experiments in AM Parser.

##  Appendix I More examples of Complex DRGs

In this section, we show examples of more complex scope assignments in Fig 13.

Figure 13: Examples of complete DRG (top), scopeless DRG(middle), and
simplified DRG (bottom) for the sentence You and he both are very kind.


#  Classical and quantum cosmology in $f(T)$-gravity theory: A Noether
symmetry approach

Roshni<EMAIL_ADDRESS>Sourav
<EMAIL_ADDRESS>Subenoy<EMAIL_ADDRESS>1Department of Mathematics, Jadavpur University, Kolkata-700032, West Bengal,
India
2Department of Mathematics, Dr. Meghnad Saha College, Itahar, Uttar
Dinajpur-733128, West Bengal, India.

###### Abstract

In the framework of $f(T)$-gravity theory, classical and quantum cosmology has
been studied in the present work for FLRW space-time model. The Noether
symmetry, a point-like symmetry of the Lagrangian is used to the physical
system and a specific functional form of $f(T)$ is determined. A point
transformation in the 2D augmented space restricts one of the variable to be
cyclic so that the Lagrangian as well as the field equations are simplified so
that they are solvable. Lastly for quantum cosmology, the WD equation is
constructed and possible solution has been evaluated.

Keywords: Noether Symmetry, f(T) Gravity, Quantum Cosmology.

##  I Introduction

In the context of recent series of observational evidences Riess:1998cb ;
Perlmutter:1998np ; Spergel:2003cb ; Tegmark:2003ud ; Eisenstein:2005su  which
predict that our Universe is going through an era of accelerated expansion, a
group of cosmologists are in favour of modifying Einstein gravity to
accommodate these predictions. There are several modified gravity theories in
the literature among which the popular one is the $f(R)$-gravity theory
Capozziello:2003tk ; Carroll:2003wy . In this gravity theory, the scalar
curvature $R$ in the Einstein-Hilbert action is replaced by an arbitrary
function $f(R)$. In recent years, another gravity theory gets much attention
and is known as teleparallel gravity. Here the gavitational interactions
Einstein/00 ; Einstein/01 ; Hayashi/00  are described by torsion (instead of
curvature). Such a gravity model was first proposed by Einstein with a view to
unify electromagnetism and gravity over Weitzenb$\ddot{\mbox{o}}$ck non-
Riemannian manifold. So the Levi-civita connection is replaced by
Weitzenb$\ddot{o}$ck connection in the underlying Riemann-cartan space-time.
As a result, pure geometric nature of the gravitational interaction is
violated and torsion behaves as force. Hence gravity may be considered as a
gauge theory of the translation group Arcos:2005ec . Although there are
conceptual differences between GR and teleparallel gravity theory, still at
classical level both of them have equivalent dynamics.

In analogy to $f(R)$-gravity theory, a generalization to teleparallel gravity
has been formulated Ferraro:2006jd ; Ferraro:2011ks ; Linder:2010py ; Finch/00
; capozziello/00 ; Basilakos/00 ; Bamba:2013jqa ; Li:2018ixg ; Basilakos/01 ;
Harko:2014sja  by replacing the torsion scalar $T$ by a generic function
$f(T)$. Linder Linder:2010py  termed this modified gravity as $f(T)$-gravity
theory. For a comparative study with $f(R)$-gravity theory, there are two
important differences namely (a) the field equations in $f(T)$-gravity theory
are second order while one has fourth order equations in $f(R)$-gravity
Bengochea:2008gz  (b) Although $f(R)$-gravity theory obey local lorentz
invariance but not by $f(T)$-gravity theory. As a result, in $f(T)$-gravity
theory all $16$ components of the vierbien are independent and a gauge choice
Li:2010cg  can not fix six of them. Further, the four linearly independent
vierbeins (i.e, tetrad) fields are the dynamical object in $f(T)$-gravity
theory. Also these tetrad fields form the orthogonal bases for the tangent
space at each point of space-time. The name “teleparallel” is justified as the
vierbeins are parallel vector fields (for a review see ref. Cai:2015emx ;
Darabi:2019qpz ; Golovnev/00 ; Aviles/00 ; Bamba/02 ; Mishra/00 ; Biswas/00 ).

The geometrical symmetries namely Lie point and Noether symmetries related to
space-time are usually very useful to solve/study physical systems. The
conserved charges in Noether symmetry are considered as a selection criterion
to discriminate similar physical processes Capozziello/100000 ;
Szydlowski/10000000 ; Capozziello:2007wc ; Capozziello/50000000 ;
Capozziello:2008ch ; Capozziello:2009te ; Vakili:2008ea ; Zang/10000 .
Mathematically a given system of differential equation can either be
simplified or to have a first integral (Norther integral) by imposing Noether
symmetry to the system. Further, it is possible to constrain or determine
physical parameters involved in a physical system by imposing Noether symmetry
to it Szydlowski/00 ; Dutta:2016jgj . In recent years, symmetry analysis has
been widely used to the physical systems in Riemannian spaces Tsamparlis/11 ;
Leach/11 ; Bluman/11 ; Aminova/11 ; Aminova/12 ; Feroze/11 ; Tsamparlis/111
and specially in the context of cosmology Dutta/00 ; Dutta/01 ; Dutta:2016exd
; Dutta:2016mbp ; Dutta:2018hyv ; Dutta:100 . The present work is an example
of it. Usually, evolution equations are simplified to a great extend by
determining a cyclic variable in the augmented space. As a consequence,
analytic solutions are possible with new variables (in the augmented space)
and are analyzed from cosmological context.

On the otherhand, Noether symmetry can also be used in quantum cosmology to
identify a typical subset of the general solution of the Wheeler-Dewitt (WD)
equation having oscillatory behavior Dutta:101 ; Dutta:2019ujc ;
Paliathanasis:2015gga ; Capozziello:2012hm ; Basilakos:2013rua;
Bajardi:2021tul. Also in Minisuperspace geometry, symmetry analysis identifies
equations of classical trajectories Capozziello:2013qha ; Halliwell/00 . Hence
classical observable universe can be related to the quantum cosmology through
the application of Noether symmetry analysis.

In the present work, both classical and quantum cosmology has been studied for
$f(T)$-gravity theory using the Noether symmetry analysis in the background of
homogeneous and isotropic flat FLRW space-time model. The paper is organized
as follows: a basic concept of Noether symmetry approach has been discussed in
Section II, Section III describes the brief review of conformal symmetry and
classical cosmology in f(T) gravity and Noether symmetry is presented in
Section IV, where as Section V presents a general description of quantum
cosmology: minisuperspace approach and paper ends with a summary in Section
VI.

##  II Basic Concept of Noether Symmetry approach

Noether’s first theorem states that every differentiable symmetry of the
action of a physical system with conservative forces has a corresponding
conservation law i.e, the lie derivative of the Lagrangian
Paliathanasis:2015gga , Basilakos:2011rx , Paliathanasis:2014zxa  of any
physical system, associated to some conserved quantities, will be invariant
along an appropriate vector field
($\mathcal{L}_{\overrightarrow{X}}f=\overrightarrow{X}(f)$). Using these one
can determine the conserved quantities (invariants) from the observed
symmetries of a physical system. Also these symmetry constraints are very much
useful to simplify the evolution equations of the physical system Stephani/00
.

If a point like canonical Lagrangian is of the form
$L[q^{\alpha}(x^{j}),\dot{q}^{\alpha}(x^{j})]$, then the Euler-Lagrange
equations take the form

$\partial_{i}\left(\frac{\partial
L}{\partial\partial_{i}q^{\alpha}}\right)=\frac{\partial L}{\partial
q^{\alpha}}$ (1)

Now contracting the equation (1) with some unknown functions
$\lambda^{\alpha}(q^{\beta})$ one can get

$\lambda^{\alpha}\bigg{[}\partial_{i}\left(\frac{\partial
L}{\partial\partial_{i}q^{\alpha}}\right)-\frac{\partial L}{\partial
q^{\alpha}}\bigg{]}=0$ (2) $\lambda^{\alpha}\frac{\partial L}{\partial
q^{\alpha}}+(\partial_{i}\lambda^{\alpha})\left(\frac{\partial
L}{\partial\partial_{i}q^{\alpha}}\right)=\partial_{i}\left(\lambda^{\alpha}\frac{\partial
L}{\partial\partial_{i}q^{\alpha}}\right)$

So, the Lie derivative of the Lagrangian i.e,
$\mathcal{L}_{\overrightarrow{X}}L$ takes the form,

$\mathcal{L}_{\overrightarrow{X}}L=\lambda^{\alpha}\frac{\partial L}{\partial
q^{\alpha}}+(\partial_{i}\lambda^{\alpha})\frac{\partial
L}{\partial\left(\partial_{i}q^{\alpha}\right)}=\partial_{i}\left(\lambda^{\alpha}\frac{\partial
L}{\partial\partial_{i}q^{\alpha}}\right)$ (3)

where $\overrightarrow{X}$, the infinitesimal generator, is of the form

$\overrightarrow{X}=\lambda^{\alpha}\frac{\partial}{\partial
q^{\alpha}}+\left(\partial_{i}q^{\alpha}\right)\frac{\partial}{\partial\left(\partial_{i}q^{\alpha}\right)}$
(4)

According to Noether theorem, the lie derivative of the Lagrangian $L$ along
the vector field $\overrightarrow{X}$ must vanish i.e,
$\mathcal{L}_{\overrightarrow{X}}L=0$. Further, from equation (3) we can say
that the Noether current or conserved current $Q^{i}$ is a constant of motion
of the system Capozziello:2010ih , where

$Q^{i}=\lambda^{\alpha}\frac{\partial
L}{\partial\left(\partial_{i}q^{\alpha}\right)}$ (5)

satisfying

$\partial_{i}Q^{i}=0$ (6)

Now the energy function associated with the system can be written as

$E=\dot{q}^{\alpha}\frac{\partial L}{\partial\dot{q}^{\alpha}}-L$ (7)

If there is no explicit time dependence in the Lagrangian, then the energy
function which is also known as Hamiltonian of the system is a constant of
motion Capozziello:2010ih . In section 4, we will show how Noether symmetry
analysis simplifies the present coupled cosmological model.

Hamiltonian formulation is very useful in the context of quantum cosmology.
The Noether symmetry condition can be rewritten as

$\mathcal{L}_{\overrightarrow{X}_{H}}H=0$ (8)

with

${\overrightarrow{X}_{H}}=\dot{q}\frac{\partial}{\partial
q}+\ddot{q}\frac{\partial}{\partial\dot{q}}$

The conserved canonically conjugate momenta due to Noether symmetry can be
written as

$\Pi_{l}=\frac{\partial L}{\partial q^{l}}={\sum}_{l}$ (9)

$l=1,2,...m$

where ’$m$’ is the no of symmetries. Now the operator version (i.e,
quantization) of equation (9) takes the form

$-i\partial_{q^{l}}\ket{\psi}={\sum}_{l}\ket{\psi}$ (10)

For real conserved quantity $\sum_{l}$, the equation (10) has oscillatory
solutionwhich is given by

$\ket{\psi}=\sum_{l=1}^{m}e^{i\sum_{l}q^{l}}\ket{\phi(q^{k})},k<n$ (11)

where ’$k$’ is the directions along which there is no symmetry and $n$ is the
dimension of the minisuperspace. This oscillatory part of the wave function
implies that the Noether symmetry exists and the conjugate momenta along the
symmetry directions should be conserved and vice-versa (Hartle Hartle/1000 ).
In fact Noether symmetry allows researchers to consider whole classes of
hypothetical Lagrangians with given invariants to describe a physical system.

##  III Conformal Symmetry: A brief review

In differential geometry, conformal invariance gives rich geometrical
structures. A vector field $\xi^{\alpha}$ is a Conformal Killing Vector (CKV)
of the metric $g_{ij}$ if

$\mathcal{L}_{\overrightarrow{\xi}}g_{ij}=\mu(x^{k})g_{ij}$ (12)

where $\mu$ is an arbitrary function of the space and notationally
$\mathcal{L}_{\overrightarrow{\xi}}$ indicates Lie derivative with respect to
the vector field $\overrightarrow{\xi}$. In particular if

(i) $\mu(x^{k})=\mu_{0}(\neq 0)$, a constant: $\xi^{\alpha}$ \- homothetic
vector field.

(ii) $\mu(x^{k})=0$: $\xi^{\alpha}$ \- killing vector field.

The above three class of vector fields individually form an algebra as
follows:

(a) The class of conformal killing vectors form an algebra, known as conformal
algebra (CA) of the metric Tsamparlis:2013aza .

(b) The class of homothetic vector fields form an algebra, known as homothetic
algebra (HA).

(c) The class of killing vector fields form an algebra, known as killing
algebra (KA).

These three algebras are related as

$KA\subseteq HA\subseteq CA$ (13)

Further, for a $n(>2)$ dimensional manifold of constant curvature, the
dimension of these three algebras are $\dfrac{(n+1)(n+2)}{2}$,
$\dfrac{n(n+1)}{2}+1$ and $\dfrac{n(n+1)}{2}$ respectively.

In a given space, two metrics $g$ and $g^{\prime}$ are said to be conformally
related if $\exists$ a function $\Pi(x^{k})$ so that

$g^{\prime}_{ij}=\Pi(x^{k})g_{ij}$ (14)

It is to be noted that two conformally related metrics have the same conformal
algebra but subalgebras are not necessarily same. In fact if
$\overrightarrow{\xi_{0}}$ is a conformal killing vector for the conformally
related metrics $g$ and $g^{\prime}$ then the corresponding conformal
functions $\mu(x^{k})$ and $\mu^{\prime}(x^{k})$ are related by the relation

$\mu^{\prime}(x^{k})=\mu(x^{k})+\mathcal{L}_{\overrightarrow{\xi_{0}}}(\ln\Pi)$
(15)

As Noether symmetries follow the homothetic algebra of the metric so two
conformally related physical systems are not identical.

In the context of conformal Lagrangian, Tsamparlis et al. Tsamparlis:2013aza
have shown that the equations of motion (i.e, Euler-Lagrange equations)
corresponding to two conformal Lagrangians transform covariantly under the
conformal transformation provided Hamiltonian (i.e, the total energy) is zero.
So systems with vanishing energy are conformally related and corresponding
equations of motion are conformally invariant. Further, in quantum cosmology,
due to the the Hamiltonian constraint the total energy of the system has be
zero, and consequently, one has conformally invariant systems with respect to
the equations of motion.

##  IV Classical Cosmology in $f(T)$-gravity and Noether Symmetry

In the background of flat FLRW space-time model, the point like Lagrangian in
$f(T)$ gravity theory takes the form

$L=a^{3}f(T)-a^{3}Tf_{T}(T)-6a\dot{a}^{2}f_{T}(T)-Da^{-3\omega}$ (16)

where $f(T)$ is a regular function of the torsion scalar $T$, $a$ is the scale
factor and $D$ is a constant of integration. Here matter field is chosen as
perfect fluid with $\omega=\dfrac{p}{\rho}$, the constant equation of state
parameter. The modified Friedmann equations are Bose:2020xdz

$H^{2}=\frac{1}{(2f_{T}+1)}\left[\frac{\rho}{3}-\frac{f}{6}\right]$ (17)

and

$2\dot{H}=\frac{(\rho+p)}{1+f_{T}+2Tf_{TT}}$ (18)

In the present Lagrangian system we have 2D configuration space $\\{a,T\\}$
and the momenta conjugate to configuration variables are

$\displaystyle p_{a}$ $\displaystyle=$ $\displaystyle\frac{\partial
L}{\partial\dot{a}}=-12a\dot{a}f_{T}(T)$ (19) $\displaystyle p_{T}$
$\displaystyle=$ $\displaystyle\frac{\partial L}{\partial\dot{T}}=0$ (20)

Using Legendre transformation, the Hamiltonian of the system is

$H=-\frac{1}{24}\frac{{p_{a}}^{2}}{af_{T}(T)}-a^{3}f(T)+a^{3}Tf_{T}(T)+Da^{-3\omega}$
(21)

So the Hamilton’s equations of motion are

$\displaystyle\dot{a}$ $\displaystyle=$
$\displaystyle\\{a,H\\}=-\frac{1}{12}\frac{p_{a}}{af_{T}(T)}$
$\displaystyle\dot{T}$ $\displaystyle=$ $\displaystyle\\{T,H\\}=0$
$\displaystyle\dot{p_{a}}$ $\displaystyle=$
$\displaystyle\\{p_{a},H\\}=-\frac{1}{24}\frac{{p_{a}}^{2}}{a^{2}f_{T}(T)}+3a^{2}f(T)-3a^{2}Tf_{T}(T)+3\omega
Da^{-3\omega-1}$ $\displaystyle\dot{p_{T}}$ $\displaystyle=$
$\displaystyle\\{p_{T},H\\}=6a\dot{a}^{2}f_{TT}(T)-a^{3}Tf_{TT}(T)$ (22)

We shall now impose Noether symmetry to the above physical system. According
to Noether’s theorem $\exists$ a vector field $\overrightarrow{X}$ about which
the Lie derivative of the Lagrangian should be zero i.e,

$\mathcal{L}_{\overrightarrow{X}}L=0$ (23)

where the infinitesimal generator $\overrightarrow{X}$ has the form

$\overrightarrow{X}=\alpha\frac{\partial}{\partial
a}+\beta\frac{\partial}{\partial
T}+\dot{\alpha}\frac{\partial}{\partial\dot{a}}+\dot{\beta}\frac{\partial}{\partial\dot{T}}$
(24)

Here $\alpha=\alpha(a,T)$ and $\beta=\beta(a,T)$ are the functions in the
configuration space with $\dot{\alpha}=\dfrac{\partial\alpha}{\partial
a}\dot{a}+\dfrac{\partial\alpha}{\partial T}\dot{T}$ and so on.

Now from the Noether symmetry condition (23) one obtains the following partial
differential equations:

$-6\alpha f^{\prime}(T)-6a\beta
f^{\prime\prime}(T)-12af^{\prime}(T)\frac{\partial\alpha}{\partial a}=0$ (25)
$-12af^{\prime}(T)\frac{\partial\alpha}{\partial T}=0$ (26)

and

$3\alpha a^{2}f(T)-3\alpha a^{2}Tf^{\prime}(T)+3\omega D\alpha
a^{-3\omega-1}-\beta a^{3}f^{\prime\prime}(T)T=0$ (27)

Using seperation of variables for the coefficients $\alpha$, $\beta$ of the
symmetry vector, the above set of partial differential equations are solvable
to give

$\displaystyle\alpha(a,T)$ $\displaystyle=$ $\displaystyle
ca^{1-\frac{3k}{2}}$ $\displaystyle\beta(a,T)$ $\displaystyle=$
$\displaystyle-3kca^{-\frac{3k}{2}}T$ (28)

and also $f(T)$ has the solution

$f(T)=f_{0}T^{\frac{1}{k}}$ (29)

with the equation of state parameter $\omega=0$.

In the solution, $c$, $k$ and $f_{0}$ are arbitrary integration constants.

In order to solve the modified Friedmann equations we make a transformation in
the configuration space $(a,T)\rightarrow(u,v)$ so that one of the transformed
variable (say $u$) becomes cyclic and consequently the transformed Lagrangian
becomes much simpler in form. Hence the evolution equations become very simple
to have analytic solutions. So the infinitesimal generator (i.e, the vector
field $\overrightarrow{X}$) due to this point transformation becomes

$\overrightarrow{X}_{T}=\left(i_{\overrightarrow{X}}du\right)\frac{\partial}{\partial
u}+\left(i_{\overrightarrow{X}}dv\right)\frac{\partial}{\partial
v}+\left\\{\frac{d}{dt}\left(i_{\overrightarrow{X}}du\right)\right\\}\frac{\partial}{\partial\dot{u}}+\left\\{\frac{d}{dt}\left(i_{\overrightarrow{X}}dv\right)\right\\}\frac{\partial}{\partial\dot{v}}$
(30)

Thus $\overrightarrow{X}_{T}$ may be considered as the lift of a vector field
defined on the augmented space. Now, without any loss of generality one may
restrict the above point transformation to be

$i_{\overrightarrow{X}}du=1$

and

$i_{\overrightarrow{X}}dv=0$ (31)

so that

$\overrightarrow{X}_{T}=\dfrac{\partial}{\partial u}$

with

$\dfrac{\partial L_{T}}{\partial u}=0$ (32)

Here $i_{\overrightarrow{X}}$ stands for the inner product with the vector
field $\overrightarrow{X}$. Usually with Noether symmetry there is an
associated conserved current (defined in the (5) of section 2). The time
component of it when integrated over spatial volume gives a conserved charge,
which in geometric notion can be defined as

$q=i_{\overrightarrow{X}}\theta_{L}$

where the cartan one form $\theta_{L}$ is defined as Dutta:2018hyv

$\theta_{L}=\frac{\partial L}{\partial a}da+\frac{\partial L}{\partial T}dT$
(33)

Now the first order partial differential equations corresponding to equation
(31) are

$\alpha\frac{\partial u}{\partial a}+\beta\frac{\partial u}{\partial T}=1$

and

$\alpha\frac{\partial v}{\partial a}+\beta\frac{\partial v}{\partial T}=0$
(34)

Using the solutions for $\alpha$ and $\beta$ from equation (28) the solution
for $u$ and $v$ gives

$u=\frac{2}{3kc}a^{\frac{3k}{2}}$ $v=\ln\left(aT^{\frac{1}{3k}}\right)$ (35)

and the transformed Lagrangian takes the form

$L=f_{0}\left(1-\frac{1}{k}\right)e^{3v}-\frac{6f_{0}}{k}c^{2}\dot{u}^{2}e^{\left(3-3k\right)v}-D$
(36)

The solution of the corresponding Euler-Lagrange equations take the form

$\displaystyle v$ $\displaystyle=$ $\displaystyle B$ $\displaystyle u$
$\displaystyle=$ $\displaystyle Ft+G$ (37)

with $B$, $F$ and $G$ are arbitrary constants.

Figure 1: $H$ vs $t$

Figure 2: $\frac{\ddot{a}}{a}$ vs $t$

So the classical cosmological solution in original variables can be written as

$\displaystyle a$ $\displaystyle=$
$\displaystyle\left\\{\frac{3kc}{2}\left(Ft+G\right)\right\\}^{\dfrac{2}{3k}}$
$\displaystyle T$ $\displaystyle=$
$\displaystyle\frac{4B_{0}}{9k^{2}c^{2}\left(Ft+G\right)^{2}}$ $\displaystyle
f(T)$ $\displaystyle=$ $\displaystyle
f_{0}\left\\{\frac{4B_{0}}{9k^{2}c^{2}\left(Ft+G\right)^{2}}\right\\}^{\dfrac{1}{k}}$
(38)

where $B_{0}$ is an arbitrary constant.

The above cosmological solution indicates power-law expansion of the Universe
with Hubble parameter decreases with respect to cosmic time as $\dfrac{1}{t}$
(see Figure 1) and the Universe is in an accelerating phase (see Figure 2)
with rate of acceleration decreases with the evolution.

##  V A general description of Quantum Cosmology: Minisuperspace Approach

In cosmology, homogeneous and isotropic metrics and matter fields are the
simplest and widely used minisuperspace models. In this model lapse function
$N(=N(t))$ is homogeneous and the shift function identically vanishes. Using
$(3+1)$-decomposition the metric in $4D$ manifold takes the form,

$ds^{2}=-N^{2}(t)dt^{2}+h_{ab}(x,t)dx^{a}dx^{b}$ (39)

and the Einstein-Hilbert action can be written as

$I(h_{ab},N)=\frac{{m^{2}}_{p}}{16\pi}\int
dtd^{3}xN\sqrt{h}\left[k_{ab}k^{ab}-k^{2}+(3)_{R}-2\Lambda\right]$ (40)

here $k_{ab}$ denotes the extrinsic curvature of the $3$ space; the trace of
the extrinsic curvature $k=k_{ab}h^{ab}$; $(3)_{R}$ represents the curvature
scalar of the three space and $\Lambda$ is a constant known as cosmological
constant.

The metric $h_{ab}$ can be characterized by a finite number of time functions
$q^{\alpha}(t)$, $\alpha=0,1,2,...,n-1$ due to homogeneity of the three space.
So, the above action takes the form of a relativistic point particle with self
interacting potential in $nD$ curved space time as

$I\left(q^{\alpha}(t),N(t)\right)=\int_{0}^{1}dtN\left[\frac{1}{2N^{2}}f_{\alpha\beta}(q)\dot{q}^{\alpha}\dot{q}^{\beta}-V(q)\right]$
(41)

So the equation of motion of the relativistic particle can be written as

$\frac{1}{N}\frac{d}{dt}\left(\frac{\dot{q}^{\alpha}}{N}\right)+\frac{1}{N^{2}}\Gamma^{\alpha}_{\mu\nu}\dot{q}^{\mu}\dot{q}^{\nu}+f^{\alpha\beta}\frac{\partial\nu}{\partial
q^{\beta}}=0$ (42)

where $\Gamma^{\alpha}_{\beta\gamma}$ is the Christoffel symbols in the
minisuperspace. Now we also have a constraint equation which is obtained by
variation with respect to the lapse function as

$\frac{1}{2N^{2}}f_{\alpha\beta}\dot{q}^{\alpha}\dot{q}^{\beta}+V(q)=0$ (43)

Now the momenta canonical to $q^{\alpha}$ can be written as

$p_{\alpha}=\frac{\partial L}{\partial
q^{\alpha}}=f_{\alpha\beta}\frac{\dot{q}^{\beta}}{N},$ (44)

so the Hamiltonian can be written as

$H=p_{\alpha}\dot{q}^{\alpha}-L=N\left[\frac{1}{2}f^{\alpha\beta}p_{\alpha}p_{\beta}+V(q)\right]=N\mathcal{H}$
(45)

where $f^{\alpha\beta}$ is the inverse metric. Using equation (43) and
equation (44) we get

$\mathcal{H}(q^{\alpha},p_{\alpha})\equiv\frac{1}{2}f^{\alpha\beta}p_{\alpha}p_{\beta}+V(q)=0$
(46)

Now, in quantization scheme, replacing $p_{\alpha}$ by
$-i\hbar\dfrac{\partial}{\partial q_{\alpha}}$ the operator version of
equation (46) on a time independent function one can obtain the Wheeler-Dewitt
(WD) equation in quantum cosmology as

$\mathcal{H}\left(q^{\alpha},-i\hbar\frac{\partial}{\partial
q^{\alpha}}\right)\psi(q^{\alpha})=0$ (47)

The above WD equation encounters the operator ordering problem because
generally the minisuperspace metric is dependent on $q^{\alpha}$. One can
resolve this problem by imposing the quantization in minisuperspace to be
covariant in nature. In the context of quantum cosmology for probability
measure, $\exists$ a conserved current for hyperbolic type of partial
differential equation as

$\overrightarrow{J}=\frac{i}{2}(\psi^{*}\nabla\psi-\psi\nabla\psi^{*})$ (48)

and $\overrightarrow{J}$ satisfies the relation
$\overrightarrow{\nabla}.\overrightarrow{J}=0$. Here $\psi$ represents the
solution of WD differential equation (hyperbolic type). So the probability
measure on the minisuperspace can be written as

$dp=|\psi(q^{\alpha})|^{2}dV$ (49)

where $dV$ denotes a volume element on minisuperspace.

In the present problem the Lagrangian of the system in the transformed
variables is given by (36). So the canonically conjugate momenta are

$\displaystyle p_{u}$ $\displaystyle=$ $\displaystyle\frac{\partial
L}{\partial\dot{u}}=-\frac{12f_{0}}{k}c^{2}\dot{u}e^{(3-3k)v}$ $\displaystyle
p_{v}$ $\displaystyle=$ $\displaystyle\frac{\partial L}{\partial\dot{v}}=0$
(50)

Hence the Hamiltonian of the system is

$H=p_{u}\dot{u}+p_{v}\dot{v}-L=-\frac{k}{24f_{0}c^{2}}e^{(3k-3)v}p_{u}^{2}-f_{0}\left(1-\frac{1}{k}\right)e^{3v}+D$
(51)

The above Hamiltonian is a very special type of Hamiltonian having only one
dynamical variable $u$ which is also cyclic in nature ($v$ can not be
considered as dynamical variable as $\dot{v}$ i.e, $p_{v}$ does not appear in
the Hamiltonian).

Hence the WD equation which is the operator version of the above Hamiltonian
takes the form

$\frac{d^{2}\phi}{du^{2}}+l\phi=0$ (52)

with
$l=\dfrac{f_{0}\left(1-\frac{1}{k}\right)e^{3v}-D}{\frac{k}{24f_{0}c^{2}}e^{(3k-3)v}}$,
a constant.

So the solution of WD equation can be written as

$\displaystyle\phi$ $\displaystyle=$ $\displaystyle
A_{1}e^{\sqrt{l}u}+A_{2}e^{-\sqrt{l}u},~{}~{}\mbox{when}~{}l>0$ (53)
$\displaystyle=$ $\displaystyle
B_{1}\cos\sqrt{-l}u+B_{2}\sin\sqrt{-l}u,~{}~{}\mbox{when}~{}l<0$
$\displaystyle=$ $\displaystyle C_{1}u+C_{2},~{}~{}\mbox{when}~{}l=0$

where $A_{i}$’s, $B_{i}$’s and $C_{i}$’s $(i=1,2)$ are the constants of
integration.

##  VI Summary

The present work deals with $f(T)$ cosmology from the point of view of
symmetry analysis. In particular Noether symmetry has been used both in
classical and quantum cosmology with $f(T)$ gravity theory. Using Noether
symmetry condition to the Lagrangian of the present model along the symmetry
vector, the coefficients of the symmetry vector are not only determined, it is
possible to determine the explicit form of the $f(T)$ function. Using a
transformation of variables in the configuration (satisfying condition
$(\ref{eq31})$) the Lagrangian simplifies to a great extend and it is possible
to have cosmological solution having power-law nature. In quantum cosmology,
the WD equation simplifies to a great extend due to only one dynamical
variable having cyclic nature. From the nature of the wave function one can
infer that big-bang singularity may be avoided quantum mechanically for the
present $f(T)$-cosmological model. So one may conclude that $f(T)$ cosmology
can avoid the big-bang singularity at the very early era of evolution of the
Universe.

## References

  * (1) Adam G. Riess et al.  Observational evidence from supernovae for an accelerating universe and a cosmological constant.  Astron. J., 116:1009–1038, 1998.
  * (2) S. Perlmutter et al.  Measurements of $\Omega$ and $\Lambda$ from 42 high redshift supernovae.  Astrophys. J., 517:565–586, 1999.
  * (3) D. N. Spergel et al.  First year Wilkinson Microwave Anisotropy Probe (WMAP) observations: Determination of cosmological parameters.  Astrophys. J. Suppl., 148:175–194, 2003.
  * (4) Max Tegmark et al.  Cosmological parameters from SDSS and WMAP.  Phys. Rev. D, 69:103501, 2004.
  * (5) Daniel J. Eisenstein et al.  Detection of the Baryon Acoustic Peak in the Large-Scale Correlation Function of SDSS Luminous Red Galaxies.  Astrophys. J., 633:560–574, 2005.
  * (6) Salvatore Capozziello, Sante Carloni, and Antonio Troisi.  Quintessence without scalar fields.  Recent Res. Dev. Astron. Astrophys., 1:625, 2003.
  * (7) Sean M. Carroll, Vikram Duvvuri, Mark Trodden, and Michael S. Turner.  Is cosmic speed - up due to new gravitational physics?  Phys. Rev. D, 70:043528, 2004.
  * (8) Preuss Akad Wiss A.Einstein, Sitzungsber.  Phys.Math.KI., 217:401, 1928.
  * (9) A.Einstein.  Math.Annal, 102:685, 1930.
  * (10) T.Shirafuji K.Hayashi.  Phys. Rev. D, 19:3524, 1979.  [Addendum: Phys.Rev.D 24, 3312 (1982)].
  * (11) H. I. Arcos and J. G. Pereira.  Torsion gravity: A Reappraisal.  Int. J. Mod. Phys. D, 13:2193–2240, 2004.
  * (12) Rafael Ferraro and Franco Fiorini.  Modified teleparallel gravity: Inflation without inflaton.  Phys. Rev. D, 75:084031, 2007.
  * (13) Rafael Ferraro and Franco Fiorini.  Spherically symmetric static spacetimes in vacuum f(T) gravity.  Phys. Rev. D, 84:083518, 2011.
  * (14) Eric V. Linder.  Einstein’s Other Gravity and the Acceleration of the Universe.  Phys. Rev. D, 81:127301, 2010.  [Erratum: Phys.Rev.D 82, 109902 (2010)].
  * (15) A.Finch J.L.Said.  Eur.Phys.J.C, 78:7560, 2018.
  * (16) O.Luongo S.Capozziello A.Ravanpak, R.Pincak.  Gen.Rel.Grav., 50:5,53, 2018.
  * (17) M.DeLaurents A.Paliathanasis M.Tsamparlis S.Basilakos, S.Capozziello.  Noether symmetries and analytic solutions in f(T)-cosmology: A complete study.  Phys. Rev. D, 88:103526, 2013.
  * (18) Kazuharu Bamba, Sergei D. Odintsov, and Diego Sáez-Gómez.  Conformal symmetry and accelerating cosmology in teleparallel gravity.  Phys. Rev. D, 88:084042, 2013.
  * (19) Chunlong Li, Yong Cai, Yi-Fu Cai, and Emmanuel N. Saridakis.  The effective field theory approach of teleparallel gravity, $f(T)$ gravity and beyond.  JCAP, 10:001, 2018.
  * (20) F.Anagnostopoulos E.Saridakis. S.Basilakos, S.Nesseris.  JCAP, 08:008, 2018.
  * (21) Tiberiu Harko, Francisco S. N. Lobo, G. Otalora, and Emmanuel N. Saridakis.  Nonminimal torsion-matter coupling extension of f(T) gravity.  Phys. Rev. D, 89:124036, 2014.
  * (22) Gabriel R. Bengochea and Rafael Ferraro.  Dark torsion as the cosmic speed-up.  Phys. Rev. D, 79:124019, 2009.
  * (23) Baojiu Li, Thomas P. Sotiriou, and John D. Barrow.  $f(T)$ gravity and local Lorentz invariance.  Phys. Rev. D, 83:064035, 2011.
  * (24) Yi-Fu Cai, Salvatore Capozziello, Mariafelicia De Laurentis, and Emmanuel N. Saridakis.  f(T) teleparallel gravity and cosmology.  Rept. Prog. Phys., 79(10):106901, 2016.
  * (25) F. Darabi and K. Atazadeh.  f(T) quantum cosmology.  Phys. Rev. D, 100(2):023546, 2019.
  * (26) T.Koivisto A.Golovnev.  JCAP, 012:1811, 2018.
  * (27) S.Capozziello O.Luongo A.Aviles, A.Bravetti.  Cosmographic reconstruction of f(T) cosmology.  Phys. Rev. D, 87(6):064025, 2013.
  * (28) D.Momeni R.Myrzakulov K.Bamba, M.Jamil.  Astrophys. Space Sci., 344:259–267, 2013.
  * (29) S.Mishra S.Chakraborty.  Eur.Phys.J.C, 79(4):328, 2019.
  * (30) S.K.Biswas S.Chakraborty.  Int.J.Mod Phys.D, 24(07):1550046, 2015.
  * (31) S.Capozziello, R.de.Ritis, C.Rubano, and P.Scudellaro.  Riv. Nuovo Cimento, 19:2, 1996.
  * (32) M.Szydlowski et al.  Gen. Relt. Grav., 38:795, 2006.
  * (33) S. Capozziello, A. Stabile, and A. Troisi.  Spherically symmetric solutions in f(R)-gravity via Noether Symmetry Approach.  Class. Quant. Grav., 24:2153–2166, 2007.
  * (34) S.Capozziello, S.Nesseris, and L.Perivolaropoulos.  JCPA, 12:009, 2007.
  * (35) Salvatore Capozziello and Antonio De Felice.  f(R) cosmology by Noether’s symmetry.  JCAP, 08:016, 2008.
  * (36) S. Capozziello, E. Piedipalumbo, C. Rubano, and P. Scudellaro.  Noether symmetry approach in phantom quintessence cosmology.  Phys. Rev. D, 80:104030, 2009.
  * (37) Babak Vakili.  Noether symmetry in f(R) cosmology.  Phys. Lett. B, 664:16–20, 2008.
  * (38) Y.Zang, Y.g.Gong, and Z.H.Zhu.  Phys.Lett.B, 688:13, 2010.
  * (39) M.Heller M.Szydlowski.  Acta.Phys.Pol.B, 14:571, 1983.
  * (40) Sourav Dutta and Subenoy Chakraborty.  A study of phantom scalar field cosmology using Lie and Noether symmetries.  Int. J. Mod. Phys. D, 25(05):1650051, 2016.
  * (41) M.Tsamparlis A.Paliathanasis.  J.Phys.A, 44:175202, 2011.
  * (42) P.G.L.Leach.  Australian Mathematical Society Lecture Series, 22:12, 2009.
  * (43) G.Bluman S.Kumei.  Symmetries and differential equations.  Springer-Verlar, N.Y., 1989.
  * (44) A.V.Aminova.  Mat.Sb., 186:1711, 1995.
  * (45) A.V.Aminova N.A.M.Aminov.  Tensor Newser, 62:65, 2000.
  * (46) A.Qadir T.Feroze, F.M.Mahomed.  Nonlinear Dynamics, 45:65, 2006.
  * (47) A.Paliathanasis M.Tsamparlis.  Gen.Relt.Grav., 42:2957, 2010.
  * (48) S.Chakraborty S.Dutta, M.M.Panja.  A scalar field dark energy model: Noether symmetry approach.
  * (49) S.Chakraborty S.Dutta, M.M.Panja.  A study of dynamical equations for nonminimally coupled scalar field using noether symmetry approach.
  * (50) Sourav Dutta, Muthusamy Lakshmanan, and Subenoy Chakraborty.  Quintom cosmological model and some possible solutions using Lie and Noether symmetries.  Int. J. Mod. Phys. D, 25(14):1650110, 2016.
  * (51) Sourav Dutta and Subenoy Chakraborty.  Answer to the Comments on the paper ”A study of phantom scalar field cosmology using Lie and Noether symmetries” by A. Paliathanasis, et al. arXiv:1605.05147.  7 2016.
  * (52) Sourav Dutta, Muthusamy Lakshmanan, and Subenoy Chakraborty.  Non-minimally coupled scalar field in Kantowski–Sachs model and symmetry analysis.  Annals Phys., 393:254–263, 2018.
  * (53) Santu Mondal, Sourav Dutta, Manjusha Tarafdar, and Subenoy Chakraborty.  A cosmological study of Einstein-Skyrme model in anisotropic Kantowski-Sachs spacetime using Lie and Noether symmetry.  Int.J.Geom.Meth.Mod.Phys., 15(06).
  * (54) Sourav Dutta, Muthusamy Lakshmanan, and Subenoy Chakraborty.  Quantum cosmology with symmetry analysis for quintom dark energy model.  Phys.Dark Univ., 32:100795, 2021.
  * (55) Sourav Dutta, Muthusamy Lakshmanan, and Subenoy Chakraborty.  Quantum cosmology for non-minimally coupled scalar field in FLRW space-time: A symmetry analysis.  Annals Phys., 407:1–14, 2019.
  * (56) Andronikos Paliathanasis, Michael Tsamparlis, Spyros Basilakos, and John D. Barrow.  Dynamical analysis in scalar field cosmology.  Phys. Rev. D, 91(12):123535, 2015.
  * (57) Salvatore Capozziello, Mariafelicia De Laurentis, and Sergei D. Odintsov.  Hamiltonian dynamics and Noether symmetries in Extended Gravity Cosmology.  Eur. Phys. J. C, 72:2068, 2012.
  * (58) Salvatore Capozziello and Mariafelicia De Laurentis.  Noether symmetries in extended gravity quantum cosmology.  Int. J. Geom. Meth. Mod. Phys., 11:1460004, 2014.
  * (59) J.J.Halliwell, eds. S.Coleman, J.B.Hartle, T.Piran, and S.Weinberg.  Quantum Cosmology and Baby Universes.  World Scientific, Singapore, 1991.
  * (60) Spyros Basilakos, Michael Tsamparlis, and Andronikos Paliathanasis.  Using the Noether symmetry approach to probe the nature of dark energy.  Phys. Rev. D, 83:103512, 2011.
  * (61) Andronikos Paliathanasis, Michael Tsamparlis, and Spyros Basilakos.  Dynamical symmetries and observational constraints in scalar field cosmology.  Phys. Rev. D, 90(10):103524, 2014.
  * (62) H.Stephani.  Differential Equations: Their Solutions Using Symmetry.  Camb.Univ.Press, Cambridge, England, 1989.
  * (63) Salvatore Capozziello, Farhad Darabi, and Daniele Vernieri.  Equivalence between Palatini and metric formalisms of f(R)-gravity by divergence free current.  Mod. Phys. Lett. A, 26:65–72, 2011.
  * (64) J.B.Hartle, eds. S.Carter, and J.B.Hartle.  ”Gravitation in Astrophysics” Cargese,1986.  Plenum, New York, 1986.
  * (65) Michael Tsamparlis, Andronikos Paliathanasis, Spyros Basilakos, and Salvatore Capozziello.  Conformally related metrics and Lagrangians and their physical interpretation in cosmology.  Gen. Rel. Grav., 45:2003–2022, 2013.
  * (66) Akash Bose and Subenoy Chakraborty.  Cosmic evolution in f(T) gravity theory.  Mod. Phys. Lett. A, 35(36):2050296, 2020.