# Evaluating Agent Interactions Through Episodic Knowledge Graphs

Selene Báez Santamaría  Piek Vossen  Thomas Baier
Vrije University Amsterdam
{s.baezsantamaria, p.t.j.m.vossen<EMAIL_ADDRESS>
###### Abstract

We present a new method based on episodic Knowledge Graphs (eKGs) for
evaluating (multimodal) conversational agents in open domains. This graph is
generated by interpreting raw signals during conversation and is able to
capture the accumulation of knowledge over time. We apply structural and
semantic analysis of the resulting graphs and translate the properties into
qualitative measures. We compare these measures with existing automatic and
manual evaluation metrics commonly used for conversational agents. Our results
show that our Knowledge-Graph-based evaluation provides more qualitative
insights into interaction and the agent’s behavior.

##  1 Introduction

In order to develop open-domain conversational agents, it is crucial to have
automatic and reproducible ways of evaluating the interaction and the agent’s
role. However, interaction with people is challenging to evaluate for several
reasons: 1) people behave differently in each interaction, 2) people
appreciate the interaction for different reasons and aspects, 3) different
goals and sub-goals may play a role simultaneously, and 4) personal
relationships and past experiences have an impact on every interaction. For
these reasons, most evaluations of interactive systems use human judges and
questionnaires in analogy to user-satisfaction methods.

In addition to these questionnaires, conversational systems are often
evaluated by comparing system responses to human responses on a turn-by-turn
basis, where the prompts and the gold responses are taken from human-human
conversations. Standard measures such as BLUE Papineni et al. (2002), METEOR
Banerjee and Lavie (2005), BERTscore Zhang* et al. (2020) test the similarity
between the system response and a gold response, whereas USR Mehri and
Eskenazi (2020) tests the coherence of the system response to the previous
prompt and context. However, these measures do not truly assess the quality of
the system’s interpretation and the relevance of the prompt, and they punish
systems for being creative and making responses personal. By detaching prompt-
response pairs from the whole conversation, these metrics evaluate the
reactivity of an agent, not its ability to engage in a coherent interaction.

Deriu et al. (2021) mention five general requirements for evaluation: 1)
automatic to reduce human labour and subjectivity, 2) repeatable when applied
to the same dialogue, 3) correlate with human judgments, 4) differentiated for
various strategies, and 5) explainable. None of the existing approaches
satisfy all these criteria. In this paper, we demonstrate that graph
properties can be used as an additional and independent evaluation of the
effectiveness of the communication. This evaluation is an automatic measure of
semantic quality that is also explainable and reproducible, meeting three of
the five previous requirements.

We present a novel evaluation method that qualifies conversations using an
episodic Knowledge Graph (eKG) Baez Santamaria et al. (2021). We calibrate
different groups of graph measures in relation to human evaluations and ground
truth independent measures. To test our selected metrics, we compare the
quality of types of conversations and show that the proposed evaluation
framework holds, regardless of differences in system design. Our contributions
are:

  1. 1.

We provide an reference-free and explainable method for evaluating the
interaction of conversational agents.

  2. 2.

We compare our method to other standard evaluation methods and show its
complementary value.

  3. 3.

We demonstrate that our method can be applied across multiple conversations
and different (types of) participants.

##  2 Related work

Dialogue systems have been studied for several decades and are further
developed within Conversational AI systems. In their survey, Deriu et al.
(2021) discuss different types of dialogue systems and how they are evaluated.
They conclude that evaluating open conversational agents is an open problem
due to the lack of a goal and variable structure. Therefore, evaluation
approaches focus on appropriateness and human likeness of responses or
specific linguistic properties such as variability, lexical complexity,
coherence, correctness, and relevance of system responses. Evaluations can
furthermore be done at a turn-level or conversation-level. There is a many-to-
many problem in both cases: multiple responses can be correct, multiple
dialogues can lead to the right/same result, and every interaction is unique.

Attempts to automate these notions often rely on metrics such as BLUE Papineni
et al. (2002), ROUGE Lin and Och (2004), METEOR Banerjee and Lavie (2005), and
BERTscore Zhang* et al. (2020) to measure the similarity of the agent response
to one or more ground-truth responses; or they borrow from information
retrieval measures when systems need to select the appropriate responses from
a set of possible alternatives. In contrast, USR Mehri and Eskenazi (2020) is
a new method created by fine-tuning RoBERTa-base Liu et al. (2019) on the
training set of Topical-Chat. Whereas METEOR and BERTscore compare a system
response to a ground-truth response, USR gives a quality evaluation score
without a ground-truth response (reference-free) by measuring the coherence of
the system response with the human prompt and the previous context.

Evaluation regimes with ground-truth responses limit agents’ "freedom and
creativity" to generate other responses that may also fit the purpose.
Therefore, it is unsurprising that evaluations often fall back on a posteriori
evaluation by human judges. However, human evaluations suffer from several
pitfalls: expensive, time-consuming, inconsistent across experiments,
difficult to reproduce, and challenging to scale. Researchers tried to
harmonize the evaluation criteria to address the inconsistency and lack of
coherence in terminology and methodology for human evaluations of open-domain
dialogue. Howcroft et al. (2020) survey of 165 papers with human evaluations
reports more than 200 quality criteria (such as Fluency, Accuracy, or
Readability.) that have been used in Natural Language Generation.
Independently, Fitrianie et al. (2019) analyzed the proceedings of the
conference of Intelligent Virtual
Agents111https://dl.acm.org/conference/iva/proceedings between 2013 and 2018.
They found 189 constructs from 89 questionnaires reported in 81 papers, which
they reduced to 19 measurement instruments. Measurements range from how
enjoyable, correct and useful to how fluent.

In an attempt to automate evaluations and make them more reproducible and
scalable, the Ninth Dialog System Technology Challenge, Track on Interactive
Evaluation of Dialog (DSTC9, Track 3)222http://dialog.speech.cs.cmu.edu:8003
carried out a variety of automatic and human evaluations on 33 systems
submissions to the Topical-Chat challenge Gopalakrishnan et al. (2019).
Topical-Chat consists of conversations between two Amazon Mechanical Turk
workers who were given prior knowledge or information to refer to during their
conversation. Systems need to respond to turns from these conversations,
replacing one worker. A human evaluation of system responses was done using
the questionnaire from the FED dataset Mehri and Eskenazi . An automatic
evaluation was done using the measures METEOR, BERTscore, and USR. Gunasekara
et al. (2020) report that the USR (0.3 Spearman) correlates better with human
judgments than METEOR (0.23 Spearman) and BERTscore (Spearman 0.22), although
they also admit that the correlation is not very high. It is still to be seen
how easy USR can be transferred to other dialogues and contexts, as it was
trained and tested on Topical-Chat.

We present a reference-free approach that is not based on coherence but
measures the interpretability of (multimodal) situations and accumulates these
over time. The basic idea is that effective interactions result in rich and
high-quality representations that can be measured in a Knowledge Graph.
Factors determining the communication effect are the agent’s response quality
and the collaboration between participants.

##  3 Problem formalization

We represent an interaction as a series of tuples [t, s, g, f, p], such that:

t

$\in$ T, a set of time points

s

$\in$ S, a set of situations

g

$\in$ G, a set of graphs

f

$\in$ F, a set of unknown features part of situation s

p

$\in$ P, a set of defined properties that can make up graph g

A graph g represents the interpretation of a sequence of situations s at time
point t. Each situation s can be modeled as a bundle of unknown features f and
each graph g as a set of properties p that are defined a priori. To quality
the conversations, we measure how many and which properties p are extracted
from each turn and the cumulative effect of adding these properties over time
to the graph g. The instrument’s effectiveness in measuring quality depends on
the ability to detect the properties p given the features f. The effectiveness
of the communication depends on the predefined properties p that are chosen
for the evaluation.

The properties used to define the quality of the conversation can be 1)
mathematical, e.g. measuring the average degree and sparseness, 2) semantic,
e.g. number and type of triples, 3) knowledge integrity, e.g. conflicts,
outliers, analogies, completeness, 4) subjective values, e.g. sentiment and
emotion, certainty, trust, and 5) dialogue properties, e.g. turn-property
ratio’s, utterance type distribution and density, style and quality of
expression. These measures depend on the capability to extract p from the
unknown features f -implicit in image, audio and text signals- and the
modelling of these properties in the knowledge graph g. An interaction will
result in a series of graphs over time. A cumulative graph can be seen as an
episodic Knowledge Graph (eKG) Baez Santamaria et al. (2021) for which the
qualitative evaluation over time can provide valuable additional insight.

In this paper, we implement the above formal model and propose a set of
properties p, defined as RDF triples, that correlate with human judgements and
can be used across different conversational setups and for different property
detection systems.

###  3.1 Model

Figure 1: Example of how an utterance is converted into an episodic graph with
source perspective values.

Figure 1 shows the interpretation of the statement "I like reading 1984" in an
eKG. The core triple reflects that "Nicole" "reads" something labelled as
"1984". The triple itself is a named graph representing a claim. The claim is
mentioned (denotedBy) by the speaker Nicole (attributedTo) and perspective
values are attributed to this mention, such as sentiment:positive,
polarity:positive, certainty:certain. The model can represent multiple
mentions of the same triple, with different perspective values and/or
attributed to different speakers.

Figure 2: Schematic representation of interaction process (in purple), golden
data (in yellow), EKG construction (in blue) and evaluation (in green). Arrows
flowing into the metrics determine the components needed for their
calculation.

##  4 Capturing interaction and episodic knowledge

In order to apply graph metrics, we need to generate properties from the
(multimodal) signals produced during the interaction. As we represent these
properties as RDF triples, we rely on a text-to-triple instrument to detect
these properties. This instrument’s effectiveness determines the evaluation’s
depth, precision and richness; therefore, the most precise and standardised
instrument is preferable. However, using the same instrument across
interactions, we can compare the interactions and draw conclusions about its
conversational variables.

Another factor for our evaluation is the graph properties modelled in the eKG.
Some properties are generic and can be measured in any graph, whereas other
properties depend on the semantics of the data model. In the following
subsections, we discuss conversational variables when comparing interactions,
the metrics that can be applied to the generated graphs and how they depend on
specific property types.

###  4.1 Conversational variables

Our framework is agnostic to an interaction setup. We can thus have various
combinations of prompt and response participants, and we can use different
triples extractors, as shown in Figure 2. Schematically, the interlocutors (P1
and P2) both produce conversational signals to which we can apply triple-
extraction, updating the eKG. Our evaluation framework is applied to this eKG,
which is reference-free and considers the interaction between both
interlocutors. This differs from other evaluation frameworks, such as USR,
which only evaluates P2 as a coherent response to P1, and BLUE, METEOR and
BERTscore, which evaluate P2 against a ground-truth response.

#### Triple extractor

Regardless of the interlocutors, we can apply triple extraction to the
utterances a-posteriori and derive an eKG from the communication. The
extracted triples represent factoid information and possibly the speakers’
perspectives. Some approaches to extract triples are: StandforOpenIE Angeli et
al. (2015), spaCy’s Dependency parser Vasiliev (2020), or tailored Context
Free Grammars (CFG).

#### Agents

As for the type of participants, we can take recorded human-human dialogues
and apply triple extraction to each prompt from an actor. Similarly, machine-
machine conversations can be generated where a chatbot mimics human input as a
prompt agent and another chatbot functions as the response agent.

Table 1: Statistics for 9 conversations with the number of turns, claims and perspectives. The conversation effect is measured by the average claim-triples (claim density) and perspective-triples (perspective density) per turn. Category | P1 | P2 | Triple Extractor | Turns | Claims | Claim density | Perspective | Perspective density
---|---|---|---|---|---|---|---|---
Human - Machine | Student | Leolani | CFG | 83 | 27 | 0.33 | 23 | 0.28
Human - Machine | Student | Leolani | CFG | 57 | 18 | 0.32 | 14 | 0.25
Human - Machine | Student | Leolani | CFG | 45 | 17 | 0.38 | 14 | 0.31
Human - Machine | Student | Leolani | CFG | 55 | 14 | 0.25 | 11 | 0.20
Human - Machine | Student | Leolani | CFG | 78 | 21 | 0.27 | 16 | 0.21
Human - Machine | Student | Leolani | CFG | 80 | 22 | 0.28 | 17 | 0.21
Machine - Machine | Blenderbot | Leolani | CFG-spacy | 298 | 6 | 0.02 | 4 | 0.01
Machine - Machine | Blenderbot | Eliza | CFG-spacy | 207 | 24 | 0.12 | 22 | 0.11
Human - Human | Monica | Chandler | Stanford-OIE | 243 | 109 | 0.44 | 0 | 0.00

###  4.2 Graph metrics

Our formal model allows us to evaluate conversations under different
frameworks: as a mathematical object (group A), as an RDF knowledge
representation tool (group B), and as an eKG hosting the accumulation of
interactions (group C). For groups A and B we used an implementation by
Pernisch et al. (2020) while for groups C we implemented the metrics using
rdflib.

After an exploratory analysis with 62 graph metrics (A-15, B-27, C-20), we
select a subset of 24, as many of these metrics are compositional and
therefore correlate highly and mostly reflect the same insights. Our selected
Group A metrics include volume (number of nodes and edges), centrality
(average node degree, degree centrality, and closeness), connectivity (average
degree connectivity and assortativity), clique (number of strong connected
components), entropy (centrality and closeness entropy) and density
(sparseness). Group B metrics include volume (number of axioms) and density
(average population). Finally, Group C metrics include density (ratios
comparing claims to triples, perspectives to triples, conflicts to triples,
perspectives to claims, mentions to claims, conflicts to claims), and
interaction (average perspectives per claim, mentions per claim, turns per
interaction, claims per source and perspectives per source).

The above measures can be applied to the evolving eKG during the interaction
or a posteriori. Although these measures were not originally intended as
quality measures for interaction, we hypothesize that some of these measures
can be used to characterize a conversation, and the resulting graphs can be
compared independently. For example, the average claims per source may signal
how much information the agent is getting per person it interacts with, the
ratio of mentions to claims signals how much a factoid has been repeated in
conversation, while the ratio of perspectives to claims signals how much
diversity of opinions or sentiment has been expressed on claimed factoids.

##  5 Experimental setup

Table 1 shows an overview of the conversations analyzed. Details on the
artificial agents and triple extractors can be found in the Appendix as well
as an in-depth analysis of the triples extracted. We calculate the number of
claims and the number of perspectives from each session. To measure the
overall effectiveness in interpreting the conversation, we derive the density
of claims and perspectives per turn333Note that the number of turns includes
Participant 2’s responses, while the extraction focuses on claims and
perspectives from P1. About half of the total number of turns are utterances
from the students, which makes densities of 0.25 and higher still effective.,
which can be seen as the first crude measure of quality.

#### Human - Machine conversations

Three groups of students had two conversations with Leolani Vossen et al.
(2018) during which they had to introduce themselves. Students were instructed
to converse for 5 to 10 minutes or 30-40 turns per conversation. They were
also instructed about the type of sentences and expressions from which the
agent could extract triples to make the conversations more successful.
Conversations will be more cumbersome in real open settings where users do not
know what the agent can understand.

#### Machine - Machine conversations

We set up dialogues between Blenderbot Roller et al. (2020) and Leolani, and
Blenderbot with Eliza Weizenbaum (1966), where we extract triples using a
tailored Context Free Grammar (CFG) and spaCy’s dependency parses.

#### Human - Human conversations

We take all dyadic dialogues in the Friends dataset Poria et al. (2018)
between Monica and Chandler and extract triples using StanfordOIE Angeli et
al. (2015).

##  6 Comparing within conversational variables

To measure the quality of multiple interactions across different participants
with the same agent, we collected human judgements of the human-machine
conversations and compared these with the graph metrics groups.

###  6.1 Human evaluations

The students evaluated the agent’s responses in their conversation using the
DSTC9 Track3 challenge evaluation metrics Mehri et al. (2022), which form
three submetric groups: 1) enjoyability (Interesting, Engaging, Specific,
Relevant), 2) semantic correctness (Correct, Semantically Appropriate,
Understandable), and 3) fluency. Each conversation got between two to four
evaluations where students score each turn for all submetrics and overall
score. Table 2 shows the aggregated results, showing the averaged scores for
all six conversations. Although Leolani has limited communication skills, most
ratings fall above mid-range.

Table 2: Human ratings of six conversations. Score range: 1 (very bad) to 5 (very good). | Conversations
---|---
| 1.1 | 1.2 | 2.1 | 2.2 | 4.1 | 4.2
Interesting | 2.65 | 2.97 | 1.56 | 1.60 | 2.98 | 1.91
Engaging | 2.79 | 3.12 | 1.92 | 2.69 | 2.94 | 2.22
Specific | 2.94 | 2.84 | 2.02 | 2.31 | 2.68 | 2.02
Relevant | 3.25 | 3.86 | 2.83 | 2.76 | 2.97 | 2.25
Correct | 3.15 | 3.93 | 2.63 | 2.55 | 3.01 | 2.20
Semantic appr. | 3.06 | 3.91 | 2.44 | 2.48 | 2.98 | 2.27
Understandable | 3.75 | 4.06 | 3.33 | 3.05 | 3.31 | 2.97
Fluent | 3.64 | 4.13 | 2.67 | 2.45 | 3.16 | 3.08
Average submetrics | 3.15 | 3.60 | 2.42 | 2.49 | 3.00 | 2.37
Overall hum. | 3.16 | 3.41 | 2.35 | 2.57 | 2.74 | 2.12

Table 3 shows the average overall score (2.73) and the average over the
submetrics (2.84), which hints that the submetrics comprehensively indicate
the overall appreciation. The submetrics vary across but are close to the
overall average. The enjoyable submetrics score lower than the semantic
correctness and fluency ones.

###  6.2 Automatic evaluations

Following DSTC9-track3, we scored the agent responses using the USR model
("adamlin/usr-topicalchat-roberta_ft"). We implemented a likelihood score (USR
LLH) that averages the masked-task prediction of the model for every token in
the agent response given the preceding utterances as context (up to 300
characters). For every token, we get the top 20 predictions to get the token’s
score, or a score of zero if it was not listed. We also average the likelihood
of the highest-scoring token (USR MAX) as the perfect response according to
the pre-trained model. We normalized the USR scores to a 5-point Likert scale
to match it with the human ratings and averaged over all responses and
conversations. The response of our agent scores significant lower (USR
LLH=1.68) than the Overall Human Rating of the conversations (2.73). The
maximal possible score by USR is close the to Overall Human Rating (USR
MAX=2.78).

Table 3: Average human ratings. Score range: 1 (very bad) to 5 (very good). Mean Squared Error (MSE) for submetric and USR metric averages against the overall human rating. |  | MSE against overall human score
---|---|---
| Avg | Human | USR LLH | USR MAX
Interesting | 2.28 | 0.05 | 0.15 | 0.16
Engaging | 2.61 | 0.06 | 0.13 | 0.19
Specific | 2.47 | 0.06 | 0.13 | 0.16
Relevant | 2.99 | 0.04 | 0.13 | 0.16
Correct | 2.91 | 0.03 | 0.14 | 0.13
Semantic appr. | 2.86 | 0.03 | 0.14 | 0.15
Understandable | 3.41 | 0.08 | 0.14 | 0.15
Fluent | 3.19 | 0.09 | 0.19 | 0.18
Average submetrics | 2.84 | 0.05 | 0.19 | 0.18
Overall hum. | 2.73 |  | 0.15 | 0.16

We measured the Mean Squared Error (MSE) by comparing the USR scores and the
human submetrics against the Overall Human Rating. The MSE scores for the
human submetrics below 1 point. The USR LLH and USR MAX scores are 2 to 3
times higher but remain below 2 points, which means they deviate less than
average from the human norm. Finally, our agent response (USR LLH) in most
cases correlates better than the most likely predicted tokens from the model
itself (USR MAX).

###  6.3 Episodic knowledge graph evaluations

The previous evaluations (human questionnaires and automatic USR) do not
evaluate the quality of the knowledge communicated. To that end, we apply the
graph measures described in Section 4.2 to the eKGs of the student
conversations. We compute the correlations between the graph metrics and the
human and automatic metrics for each student conversation (Appendix Figure 4).
Two patterns are visible: metric group A correlates more strongly with human
evaluations, while metric group B correlates with automatic evaluation. Seven
of the human metrics correlate the most with the average degree per node in
the eKG. The other two human evaluations, Overall Human Rating and Relevance,
correlate the most to sparseness.

##  7 Comparing across conversational variables

Three major factors determine the resulting graph: 1) the (human) participant,
2) the agent’s capability to understand the prompts, and 3) the agent’s
capability to respond adequately. Our eKG-based evaluation, therefore,
genuinely evaluates interaction from both ends. This makes it possible to
evaluate interaction across different (types of) people with the same and/or
different agents, resulting in different graphs due to the human input and/or
the agents’ capabilities.

###  7.1 Correlation with human judgements

Table 4 shows the values between human evaluations, automatic evaluations, and
the graph metric with highest correlation.444Chatbot conversations have not
been evaluated manually. For 6/9 human metrics, including Overall Human
Rating, two graph metrics correlate more strongly than the USR metrics. We
interpret this as evidence for these two graph evaluations to approximate
human evaluations instead of USR evaluations. Recall that Gunasekara et al.
(2020) reported a correlation of 0.3 for USR concerning the Topical-Chat
evaluation, which is higher than the score obtained for our conversations.
Since USR was fine-tuned on Topical-Chat training data, it is expected to
reflect stronger coherence relations on these conversations. Furthermore,
Topical-Chat consists of human-human conversations replaced by system
responses, whereas our conversations are naturally-born agent conversations
that partly come from the inner drives of the agent. The task to generate an
appropriate response is more challenging for our agent compared to fine-tuned
language models that mimic human responses.

Table 4: Human evaluation, automatic scores and the most correlated graph metric. |  Average degree |  Sparseness |  USR LLH
---|---|---|---
Interesting | 0.088 | 0.077 | 0.148
Engaging | 0.158 | 0.145 | 0.076
Specific | 0.124 | 0.067 | 0.072
Relevant | 0.055 | 0.062 | 0.091
Correct | 0.071 | 0.040 | 0.128
Semantically Appropriate | 0.124 | 0.076 | 0.053
Understandable | 0.119 | 0.050 | -0.013
Fluent | 0.184 | -0.061 | -0.039
Overall Human Rating | 0.120 | 0.194 | 0.088

#### Average degree

Average node degree reflects how many edges a node on the eKG has. Figure 3a
exposes a relation between average degree and fluency. Conversations 1.2 and
1.1 scored the highest on fluency, while conversation 1.1 shows the lowest
average degree curve. In contrast, conversation 2.2 scored the lowest on
fluency and showed an incremental behaviour for this metric. Both Blenderbot
conversations show a steep increase of the average degree as the conversation
proceeds. Manual inspection of these dialogues reveals that Blenderbot becomes
repetitive after several turns, resulting in an extreme increase of the
average degree. Responses from Leolani are more repetitive than responses from
Eliza, which is consistent as Blenderbot is trained with PersonaChat Zhang et
al. (2019), Empathetic Dialogues Rashkin et al. (2018) and Wikipedia topic
conversations Dinan et al. (2018), making Blenderbot responsive for Eliza’s
empathic prompts to talk about personal relations and emotions. The responses
from Leolani, on the other hand, are based on its drives which can be more
obscure and less "human", causing Blenderbot to fall back on standard
responses rapidly. Finally, the conversation between Monica and Chandler shows
a different pattern, where the degree drops linearly. The fast decreasing
curve can be explained by high fluency between (scripted) human-human
dialogues.

#### Sparseness

Sparseness reflects how well connected a graph is. Figure 3b shows that, for
all conversations, sparseness decreases as the conversation proceeds and the
Overall Human Rating get higher as the eKG gets less sparse. Conversation 4.2
has the slowest decaying curve, while conversation 1.1 decays the fastest.
Conversation 2.2 plateaus for a few turns, which is reflected by having the
lowest Overall Human Rating. Since all eKGs have the same sparseness starting
point, this suggests that conversations that fail to expand these initially
dense graphs might not be successful. The Blenderbot-Leolani conversation is
less steep, suggesting that it is less successful. The Blenderbot-Eliza
conversation appears to be very similar in decreasing sparseness to the
student-Leolani conversations, thus confirming that Blenderbot and Eliza are
well aligned. The most effective conversation is shown by Monica and Chandler,
having a curve that decays to the same level as conversation 1.1, but decays
further.

(a) Average node degree.

(b) Sparseness.

(c) Ratio of mentions to claims.

(d) Ratio of perspectives to claims.

Figure 3: Selected graph metrics progression over turns for all student
conversations

###  7.2 Complementing human judgements

While some metrics correlate with human judgments, other graph metrics
complement human evaluations. Evaluating conversations as eKGs allows to
observe how much knowledge has been accumulated so far, how much diversity of
opinions and conflicting information has been encountered, how often the same
factoids are mentioned, how long conversations are, and how much knowledge has
been acquired per source.

#### Ratio of mentions to claims

Figure 3c shows the ratio of mentions to claims, which relates to how often
the same topics are discussed in conversation. An increasing curve implies a
repetitive conversation. The blender-Leolani conversation has the steepest
curve, as the conversation stagnated with BlenderBot repeating the same
factoid ("I have a dog"). In contrast, the Blenderbot-Eliza conversation goes
well, almost like the student-Leolani conversations but still repetitive. Once
again, the highest quality conversation is the human-human, with the lowest
ratio.

#### Ratio of perspectives to claims

Figure 3d shows the ratio of perspectives to claims, where higher means
conversations contain more diverse views on the same topics. On the contrary,
a lower ratio represents a series of broad conversations on their topics
limited to the views of a few, if not only a single source. The lowest curve
belongs to the Chandler-Monica conversation; due to using StanfordOIE, which
does not extract perspectives.

##  8 Conclusion

We presented a new method for evaluating dyadic interactions that does not
require a ground-truth interaction or a human judgment a posteriori. Our
method analyses the episodic Knowledge Graph that results from interpreting
prompts. Like the USR score, our method is automatic and reference-free.
However, we provided evidence that our evaluation correlates better with human
judgments and gives deeper insight into the knowledge built up from the
conversation. We also conclude that the graph metrics provide nuanced
information about the growth of knowledge resulting from the interaction.
Although we cannot yet say anything about the absolute score of an
interaction, we can compare different interactions based on the resulting
graph and observe differences in graph properties that may or may not be
desirable.

Note that this line of work aims to approximate human evaluations in a cost-
free manner. Yet, human evaluations are generally highly subjective and not
reproducible to evaluate a conversation. Thus, even though Overall Human
Ratings highly correlate to eKG sparseness, outliers arise due to the impact
of individual judgments per student group.

In future work, we want to validate these metrics by evaluating benchmark
datasets, similar to the methodology by Li et al. (2019). We also want to
direct the conversations to aim for certain types of knowledge and
perspectives to validate that our metrics can detect such different
intentions. We want to demonstrate that we can associate graph structures with
functionality, e.g., having broad or deep knowledge of subjects, being able to
direct people to trustworthy informants, or directing people to shared
interests.

#### Supplemental Material Statement:

Code and data are publicly available from
https://github.com/selBaez/evaluating-conversations-as-ekg.git.

#### Acknowledgements

This research was funded by the Vrije Universiteit Amsterdam and the
Netherlands Organisation for Scientific Research (NWO) via the Spinoza grant
(SPI 63-260) awarded to Piek Vossen, the Hybrid Intelligence Centre via the
Zwaartekracht grant (024.004.022) and the research project (9406.D1.19.005)
Communicating with and Relating to Social Robots: Alice Meets Leolani.

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##  Appendix A Appendix

###  A.1 Leolani, the agent

The Leolani agent creates an eKG on the fly and formulates a response to a
prompt based on the changes in the graph. At the start, an empty eKG is
initialized with the Nice-to-Meet-You (N2MU) ontology Vossen et al. (2019)
that defines basic things in the world such as people, locations, and some
social properties, but without any knowledge about individual people. Triple
extraction during the conversation adds such knowledge.

In addition to passive processing of human prompts, our agent Leolani reflects
on the changing eKG during the conversation. After each change, a series of
semantically meaningful SPARQL queries is applied to the eKG to detect
specific graph properties that are deemed relevant to respond to. These
handmade query patterns detect phenomena like conflicts, uncertainty,
analogies, generalizations, knowledge gaps, trust in knowledge sources, the
novelty of factoids, and the relevance of factoids to interlocutors. We
consider these queries as "thoughts" that reflect on the current state of the
eKG, given the latest incoming information. Each type of thought is associated
to a broad higher level goal, which we call "drives", for example, to improve
the correctness or completeness of the agent’s current knowledge, or to
improve the engagement or social connection to the interlocutor. These drives
result in speech acts such as statements or questions that the agent directs
to the interlocutor, aiming of obtaining a relevant response. In the next
short dialog, the agent Leolani learned a new concept "reading" and tries to
generalise this to all people by asking a question to the human. The next
statement by the human triggers the agent to ask another follow-up question:

  * •

H: I like reading

  * •

A: I never heard about reading before! can you tell me, is reading something
people do?

  * •

H: Yes, people read books

  * •

A: I did not know that, what else do people read?

###  A.2 Context-Free-Grammar Triple Extractor

Triple extraction is performed by a dependency parser with a Context-Free
Grammar (CFG) consisting of 12 rules and a lexicon with mainly closed-class
words. It is adapted to resolve deictic references to the addressee or other
observed objects in a situation. As the conversation is open and can be about
any topic, part-speech-tagging and named entity-recognition is used to detect
patterns of open-class words as subjects, predicates, and objects for triples.

###  A.3 Sample student-Leolani conversations

⬇

Leolani: "What’s up? What is your name? Stranger?"

Student1: "Student1"

Leolani: "So your name is Student1?"

Student1: "yes"

Leolani: "Would you like to chat? I’ll do my best to keep up"

Student1: "I like chatting"

Leolani: "I would like to know. What types of Instance like chatting do person
or Instance usually like"

Student1: "Person likes convos"

Leolani: "I am curious. What types of noun or Instance like convos do tops or
Instance usually like"

Student1: "Convos are not people"

Leolani: "Let me ask you something. Has convos work at institution?"

Student1: "I work at institution"

Leolani: "Let me ask you something. Has thomas experience touch?"

Student1: "I have experience touch"

Leolani: "I am curious. Has thomas own object?"

Student1: "I own object"

Leolani: "Interesting! I am excited to get to know about you!"

Student1: "What do you want to know?"

Leolani: "I know agent usually want to verb.cognition, but I do not know this
case"

Student1: "I like sushi"

Leolani: "Exciting news! I did not know anything that Student1 like"

Student1: "I also like cats"

Leolani: "If you don’t mind me asking. What types of animal or Instance like
cats do person or Instance usually like"

Listing 1: Extract from conversation with Student 1

###  A.4 Analysis of student-Leolani conversations

In total, 81,631 triples were generated from the student conversations. Note
that not all triples count as factual knowledge. 23 predicates were extracted
of which "know" and "be" are most frequent (Table 5). The "know" predicates
are mostly derived from introducing oneself to the agent. The predicates
further show that the communication is open, although the semantics is not
further defined beyond the predicate itself, e.g. there is no relation between
"smell" and "can-smell". The triples further contained 31 unique subjects and
78 unique objects; most occurred only once in the communication (Table 6).

Table 5: Predicate labels and their frequency in student conversations. know | 43 | can-fly | 2 | favourite-animal-is | 1
---|---|---|---|---|---
be | 39 | like-to | 2 | favourite-cat-is | 1
like | 22 | work-at | 2 | fly | 1
live-in | 9 | be-in | 1 | hair-color-is | 1
sense | 7 | be-to | 1 | own | 1
have | 6 | can-learn | 1 | smell | 1
be-from | 3 | can-smell | 1 | wear | 1
love | 3 | could-help | 1 |  |
Table 6: Entity labels and their frequency in student conversations. Leolani |  23 |  a-dog |  1 |  go |  1
---|---|---|---|---|---
student1 |  11 |  a-girl |  1 |  great-that-i |  1
student2 |  10 |  a-man |  1 |  his-uncle |  1
student3 |  8 |  a-shame |  1 |  institution |  1
student4 |  8 |  a-student |  1 |  student10 |  1
student5 |  8 |  a-wise-man |  1 |  know |  1
student6 |  7 |  a-woman |  1 |  lasagne |  1
a-flamingo |  6 |  airplanes |  1 |  phd2-student5 |  1
student7 |  6 |  amstelveen |  1 |  phd2-name |  1
student8 |  5 |  an-animal |  1 |  Leolani-new-things |  1
amsterdam |  4 |  an-aunt |  1 |  my-daughter |  1
student9 |  4 |  an-emotion |  1 |  my-parents |  1
my |  3 |  an-uncle |  1 |  student5-friend |  1
orange |  3 |  brown |  1 |  now |  1
reading |  3 |  bulgaria |  1 |  object |  1
alkmaar |  2 |  business-class |  1 |  other-things |  1
cats |  2 |  candy |  1 |  parents |  1
chatting |  2 |  city |  1 |  people |  1
convos |  2 |  cook |  1 |  person |  1
garfield |  2 |  cook-by-myself |  1 |  pink |  1
japanese-food |  2 |  deloitte |  1 |  rotterdam |  1
love |  2 |  dogs |  1 |  shrimp |  1
None |  2 |  every-agent |  1 |  student |  1
phd1 |  2 |  experience-touch |  1 |  sushi |  1
10-fingers |  1 |  favorite-of-student6 |  1 |  tapas |  1
a-bird |  1 |  flamingo |  1 |  the-south-of-holland |  1
a-color |  1 |  food |  1 |  two-hands |  1
a-company |  1 |  front-camera |  1 |  what |  1
a-country |  1 |  garfield-favourite-food |  1 |  yes-candy |  1
a-daughter |  1 |  glasses |  1 |  |

###  A.5 Full list of metrics tested

The full list of 62 metrics used, sorted by group:

  * •

GROUP A

    * –

Total nodes

    * –

Total edges

    * –

Average degree

    * –

Average degree centrality

    * –

Average closeness

    * –

Average betweenness

    * –

Average degree connectivity

    * –

Average assortativity

    * –

Average node connectivity

    * –

Number of components

    * –

Number of strong components

    * –

Shortest path

    * –

Centrality entropy

    * –

Closeness entropy

    * –

Sparseness

  * •

GROUP B

    * –

Total classes

    * –

Total properties

    * –

Total instances

    * –

Total object properties

    * –

Total data properties

    * –

Total equivalent class properties

    * –

Total subclass properties

    * –

Total entities

    * –

Total inverse entities

    * –

Ratio of inverse relations

    * –

Property class ratio

    * –

Average population

    * –

Class property ratio

    * –

Attribute richness

    * –

Inheritance richness

    * –

Relationship richness

    * –

Object properties ratio

    * –

Datatype properties ratio

    * –

Total concept assertions

    * –

Total role assertions

    * –

Total general concept inclusions

    * –

Total domain axioms

    * –

Total range axioms

    * –

Total role inclusions

    * –

Total axioms

    * –

Total aBox axioms

    * –

Total tBox axioms

  * •

GROUP C

    * –

Total triples

    * –

Total world instances

    * –

Total claims

    * –

Total perspectives

    * –

Total mentions

    * –

Total conflicts

    * –

Total sources

    * –

Total interactions

    * –

Total utterances

    * –

Ratio claim to triples

    * –

Ratio perspectives to triples

    * –

Ratio conflict to triples

    * –

Ratio perspectives to claims

    * –

Ratio mentions to claims

    * –

Ratio conflicts to claims

    * –

Average perspectives per claim

    * –

Average mentions per claim

    * –

Average turns per interaction

    * –

Average claims per source

    * –

Average perspectives per source

###  A.6 Correlation matrix

Hereby we show the full correlation matrix between our proposed metrics and
the human and automatic evaluations. Metrics related to conflicts are not
informative since the short conversations did not produce conflicting
information. Metrics related to perspectives are hindered because the simple
triple extractor is limited in the range of perspectives it can extract.

Figure 4: Correlation matrix for evaluation metrics.


# Completely Invariant Escaping Set of Transcendental Semigroup

Bishnu Hari Subedi Central Department of Mathematics
Institute of Science and Technology
Tribhuvan University
Kirtipur, Kathmandu
Nepal<EMAIL_ADDRESS>/<EMAIL_ADDRESS>and  Ajaya Singh
Central Department of Mathematics, Institute of Science and Technology,
Tribhuvan University, Kirtipur, Kathmandu, Nepal<EMAIL_ADDRESS>/
<EMAIL_ADDRESS>
###### Abstract.

For a non-trivial transcendental semigroup, escaping set $I(S)$ is in general
S-forward invariant and it is S-completely invariant if semigroup $S$ is
abelian. In the contrary of this result, we investigate completely invariant
escaping set $K(S)$ in different way even if semigroup $S$ is not abelian and
we discuss some properties and structure of such type of escaping set. Also,
we establish some relations between completely invariant escaping set $K(S)$
and the general escaping set $I(S)$.

###### Key words and phrases:

Transcendental semigroup, escaping set, S-completely invariant escaping set,
completely invariant escaping set etc.

###### 2010 Mathematics Subject Classification:

37F10, 30D05

This research work of the first author is supported by PhD Faculty fellowship
from University Grants Commission of Nepal

##  1\. Introduction

Throughout this paper, we denote the complex plane by $\mathbb{C}$ and set of
integers greater than zero by $\mathbb{N}$. We assume the function
$f:\mathbb{C}\rightarrow\mathbb{C}$ is transcendental entire function unless
otherwise stated. For any $n\in\mathbb{N},\;\;f^{n}$ always denotes the nth
iterates of $f$. Let $f$ be an entire function. The set of the form

$I(f)=\\{z\in\mathbb{C}:f^{n}(z)\rightarrow\infty\textrm{ as
}n\rightarrow\infty\\}$

is called an escaping set and any point $z\in I(S)$ is called escaping point.
For a transcendental entire function $f$, the escaping set $I(f)$ was first
studied by A. Eremenko [2]. He himself showed that $I(f)\not=\emptyset$; the
boundary of this set is a Julia set $J(f)$ (that is, $J(f)=\partial I(f)$);
$I(f)\cap J(f)\not=\emptyset$; and $\overline{I(f)}$ has no bounded component.
By motivating from this last statement, he posed a question: Is every
component of $I(f)$ unbounded?. This question is considered as an important
open problem of transcendental dynamics and nowadays it is famous as
Eremenko’s conjecture. Note that the complement of Julia set $J(f)$ in complex
plane $\mathbb{C}$ is a Fatou set $F(f)$ and any maximally connected open
subset of a Fatou set is called a Fatou component.

There are two types of points for which the inverse of entire map $f$ is not
well defined, namely critical values and asymptotic values. Recall that the
set $C(f)=\\{z\in\mathbb{C}:f^{\prime}(z)=0\\}$ is the set of critical points
of the transcendental entire function $f$ and the set
$CV(f)=\\{w\in\mathbb{C}:w=f(z)\;\ \text{such that}\;\ f^{\prime}(z)=0\\}$ of
all images of all critical points is called the set of critical values. The
set $AV(f)$ consisting of all $w\in\mathbb{C}$ such that there exists a curve
(asymptotic path) $\Gamma:[0,\infty)\to\mathbb{C}$ so that
$\Gamma(t)\to\infty$ and $f(\Gamma(t))\to w$ as $t\to\infty$ is called the set
of asymptotic values of $f$ and the set $SV(f)=\overline{(CV(f)\cup AV(f))}$
is called the set of singular values of $f$. Note that this set is coincide
with the set of singularities of the inverse function $f^{-1}$ of $f$, and so
this set is also denoted by $\text{Sing}(f^{-1})$. Among the entire functions,
only transcendental entire functions may have asymptotic values: clearly
polynomials cannot have finite asymptotic values. If $SV(f)$ is a bounded set,
then $f$ is said to be of bounded type. The set $\mathscr{B}=\\{f:f\;\
\textrm{is of bounded type}\\}$ is called Eremenko-Lyubich class.

The main concern of this paper is to the study of the completely invariant
escaping set under transcendental semigroup. So we start our formal study from
the notion of transcendental semigroup.

######  Definition 1.1 (Transcendental semigroup and subsemigroup).

Let $A=\\{f_{i}:i\in\mathbb{N}\\}$ be a set of transcendental entire functions
$f_{i}:\mathbb{C}\rightarrow\mathbb{C}$. A transcendental semigroup $S$ is a
semigroup generated by the set $A$ with semigroup operation being the
functional composition. We denote this semigroup by $S=\langle
A\rangle=\langle f_{1},f_{2},f_{3},\cdots,f_{n},\cdots\rangle$. A non-empty
subset $T$ of transcendental semigroup $S$ is a subsemigroup of $S$ if $f\circ
g\in T$ for all $f,\;g\in T$.

A semigroup generated by finitely many functions $f_{i},(i=1,2,3,\ldots,n)$ is
called finitely generated transcendental semigroup. We write $S=\langle
f_{1},f_{2},\ldots,f_{n}\rangle$. If $S$ is generated by only one
transcendental entire function $f$, then $S$ is cyclic transcendental
semigroup. We write $S=\langle f\rangle$. In this case, each $g\in S$ can be
written as $g=f^{n}$, where $f^{n}$ is the nth iterates of $f$ with itself.
Note that in our forthcoming study of transcendental semigroup theory, we say
$S=\langle f\rangle$ is a trivial semigroup.

The transcendental semigroup $S$ is abelian if $f_{i}\circ f_{j}=f_{j}\circ
f_{i}$ for all generators $f_{i}$ and $f_{j}$ of $S$. We say semigroup $S$ is
said to be of bounded type if each generator $f_{i},\;(i\in\mathbb{N})$ of $S$
is taken from the Eremenko-Lyubich class $\mathscr{B}$.

Based on the Fatou-Julia-Eremenko theory of a complex analytic function, the
Fatou set, Julia set and escaping set in the settings of semigroup are defined
as follows.

######  Definition 1.2 (Fatou set, Julia set and escaping set).

Fatou set of the transcendental semigroup $S$ is defined by

$F(S)=\\{z\in\mathbb{C}:S\;\ \textrm{is normal in a neighborhood of}\;\ z\\}$

The Julia set of $S$ is defined by $J(S)=\mathbb{C}-F(S)$ and the escaping set
of $S$ by

$I(S)=\\{z\in\mathbb{C}:\;f^{n}(z)\rightarrow\infty\;\ \textrm{as}\;\
n\rightarrow\infty\;\ \textrm{for all}\;\ f\in S\\}$

We call each point of the set $I(S)$ by escaping point.

Note that above definition of escaping set is slightly different from the
definition of escaping set by Dinesh Kumar and Sanjay Kumar [5, Definition
2.1]. In [9, Theorem 3.2], we proved that our definition of escaping set is
more general than the definition of Dinesh Kumar and Sanjay Kumar.

The following immediate relation hold for any $f\in S$ from the definition 1.2
of escaping set.

######  Theorem 1.1.

Let $S$ be a transcendental semigroup. Then $I(S)\subset I(f)$ for all $f\in
S$ and hence $I(S)\subset\bigcap_{f\in S}I(f)$.

From this theorem 1.1, we can say that the escaping set may be empty. Note
that $I(f)\not=\emptyset$ in classical iteration theory [2]. Dinesh Kumar and
Sanjay Kumar [5, Theorem 2.5] have mentioned the following transcendental
semigroup $S$, where $I(S)$ is an empty set.

######  Theorem 1.2.

The transcendental semigroup $S=\langle f_{1},f_{2}\rangle$ generated by two
functions $f_{1}$ and $f_{2}$ from respectively two parameter families of
functions
$\mathscr{F}=\\{e^{-z+\gamma}+c\;\text{where}\;\gamma,c\in\mathbb{C}\;\text{and}\;Re(\gamma)<0,\;Re(c)\geqslant
1\\}$ and
$\mathscr{F^{\prime}}=\\{e^{z+\mu}+d,\;\text{where}\;\mu,d\in\mathbb{C}\;\text{and}\;Re(\mu)<0,\;Re(d)\leqslant-1\\}$
has empty escaping set $I(S)$.

There are also transcendental semigroups whose escaping sets are non-empty. If
escaping set of the generators of transcendental semigroup $S$ are equal, then
it is not difficult to find escaping set of semigroup as shown in the
following result.

######  Theorem 1.3.

Let $S=\langle f,g\rangle$ be a transcendental semigroup such that
$I(f)=I(g)$. Then $I(S)=I(f)=I(g)$.

###### Proof.

Since $I(f)=I(g)$. It follows $I(f^{n_{i}})=I(g^{n_{j}})$ for all
$n_{i},n_{j}\in\mathbb{N}$. Since any $h\in S$ can be written as
$h=f^{n_{k}}\circ g^{n_{{k-1}}}\circ\ldots\circ g^{n_{1}}$ where
$n_{k},n_{k-1},\ldots n_{1}\in\mathbb{N}$ and some $n_{j}$ are allowed to be
zero as well. These facts concluded that $I(h)=I(f)=I(g)$ for all $h\in S$.
Therefore from the theorem 1.1, $I(S)=I(f)=I(g)$. ∎

There are examples of transcendental entire functions that have same escaping
sets. The following criteria is proved by Dinesh Kumar and Sanjay Kumar [7,
Theorem 2.12].

######  Theorem 1.4.

Let $f$ be a transcendental entire functions of period p and let
$g=f^{k}+p,\;k\in\mathbb{N}$. Then $I(f)=I(g)$.

In [10, Lemma 3.3], we have also proved the following criterion to have same
escaping sets of two transcendental entire functions.

######  Theorem 1.5.

Let $f$ and $g$ be two permutable transcendental entire functions of bounded
type. Then $I(f)=I(g)$.

If semigroup $S$ is generated by two transcendental entire functions as
mentioned in the theorems 1.4 and 1.5, then by theorem 1.3, we have
$I(S)=I(f)=I(g)$.

##  2\. Invariant features of escaping set

We start our study from invariant property which is considered a very basic
and fundamental structure of escaping set $I(S)$ of transcendental semigroup
$S$.

######  Definition 2.1 (Forward, backward and completely invariant set).

For a given semigroup $S$, a set $U\subset\mathbb{C}$ is said to be
$S$-forward invariant if $f(U)\subset U$ for all $f\in S$. It is said to be
$S$-backward invariant if $f^{-1}(U)\subset U$ for all $f\in S$ and it is
called $S$-completely invariant if it is both forward and backward invariant.

Already Dinesh Kumar and Sanjay Kumar [5, Theorem 4.1] and recently we [10,
Theorem 2.3] prove the following result. Note that our proof is based on the
definition 1.2 of escaping set.

######  Theorem 2.1.

The escaping set $I(S)$ of transcendental semigroup $S$ is S-forward
invariant.

It is well-known from [1, 2, 4, 8] that the classical escaping set of
transcendental entire function $f$ is completely invariant.

There are several classes of transcendental semigroups from which we get
backward invariant escaping sets. In [6, Theorem 2.1] Dinesh Kumar and Sanjay
Kumar and in [10, Theorem 2.6] we prove the following result.

######  Theorem 2.2.

The escaping set $I(S)$ of transcendental semigroup $S$ is S-backward
invariant if $S$ is an abelian transcendental semigroup.

With the result of this theorem 2.2, we can conclude that an escaping set
$I(S)$ is S-completely invariant if $S$ is an abelian transcendental
semigroup. For example, the following semigroups

  * •

$\langle z+\gamma\sin z,\;z+\gamma\sin z+2k\pi\rangle$,

  * •

$\langle z+\gamma\sin z,\;-z-\gamma\sin z+2k\pi\rangle$,

  * •

$\langle z+\gamma e^{z},\;z+\gamma e^{z}+2k\pi i\rangle$,

  * •

$\langle z-\sin z,\;z-\sin z+2\pi\rangle$,

are abelian transcendental semigroups, so their escaping sets are S-completely
invariant. Note that it will not better to conclude that escaping set can not
be completely invariant unless the semigroup is abelian.

Eremenko’s [2] result $\partial I(f)=J(f)$ of classical transcendental
dynamics can be generalized to semigroup settings. Dinesh Kumar and Sanjay
Kumar [5, Lemma 4.2 and Theorem 4.3] proved the following result.

######  Theorem 2.3.

Let $S$ be a transcendental entire semigroup. Then

  1. (1)

$int(I(S))\subset F(S)\;\ \text{and}\;\ ext(I(S))\subset F(S)$, where $int$
and $ext$ respectively denote the interior and exterior of $I(S)$.

  2. (2)

$\partial I(S)=J(S)$, where $\partial I(S)$ denotes the boundary of $I(S)$.

Eremenko and Lyubich [3] proved that if transcendental function
$f\in\mathscr{B}$, then $I(f)\subset J(f)$, and $J(f)=\overline{I(f)}$. Dinesh
Kumar and Sanjay Kumar [5, Theorem 4.5] generalized these results to a
finitely generated transcendental semigroup of bounded type as shown below.

######  Theorem 2.4.

For every finitely generated transcendental semigroup $S=\langle f_{1},\\\
f_{2},\ldots,f_{n}\rangle$ in which each generator $f_{i}$ is of bounded type,
then $I(S)\subset J(S)$ and $J(S)=\overline{I(S)}$.

In contrary of the result of the theorem 2.2, the closure of escaping set
$I(S)$, that is, $\overline{I(S)}$ and the Julia set $J(S)$ are S-completely
invariant even though the semigroup $S$ is not abelian. Dinesh Kumar and
Sanjay kumar [6, Theorem 2.4 and Corollary 2.5] proved the following result.

######  Theorem 2.5.

The closure of the escaping set $I(S)$ and Julia set $J(S)$ are S-completely
invariant if a semigroup $S$ is finitely generated and each of its generators
is of bounded type.

##  3\. Completely invariant escaping set

The theorem 2.2 is a condition for completely invariant escaping set of
transcendental semigroup. It is indeed a generalization of completely
invariant property of classical escaping set of single function to more
general settings of semigroup. In this section, we generalize the classical
completely invariant notion of escaping set of single function to the
completely invariant notion of escaping set of transcendental semigroup. That
is, we make completely invariant escaping set of semigroup $S$ in different
manner under each element of $S$ and we show that such type of completely
invariant escaping set is same as escaping set $I(S)$ if and only if $I(S)$ is
S-completely invariant.

######  Definition 3.1.

For a transcendental semigroup $S$, let us define the completely invariant
escaping set of $S$

$\displaystyle K(S)=\bigcap_{i\in\mathbb{N}}\\{E_{i}:E_{i}\;\text{is
completely invariant under each}\;f\in S\;\text{and each}\;E_{i}\;$
$\displaystyle\text{contains points }z\in\mathbb{C}\;\text{such
that}\;f^{n}(z)\to\infty\;\text{as}\;n\to\infty\;\text{for every}\;f\in S\\}$

There are non-trivial transcendental semigroups $S$ for which completely
invariant escaping set $K(S)$ exists. The following example stated in the
theorem 1.3 will be a good source of several other examples.

######  Example 3.1.

Suppose that $S=\langle f,g\rangle$ and $I(f)=I(g)$. Then $K=I(S)$.

Since $I(f)$ is completely invariant under $f$ and $I(g)$ is completely
invariant under $g$. If $I(f)=I(g)$, then $I(h)=I(f)=I(g)=I(S)$ for all $h\in
S$. In this case, $K(S)=I(S)$. A concrete nice example of this case is a semi
group $S=\langle f,g\rangle$ generated by the functions $f(z)=e^{\lambda
z},\;\lambda\in\mathbb{C}\smallsetminus\\{0\\}$ and $g(z)=f^{k}+\frac{2\pi
i}{\lambda},\;k\in\mathbb{N}$. In this, we can find that $I(h)=I(f)=I(g)=I(S)$
for all $h\in S$. Note that the semigroup $S$ that we mentioned here is not
abelian. Another example of same kind is a semigroup $S=\langle f,g\rangle$
generated by the functions $f(z)=\lambda\sin
z,\;\lambda\in\mathbb{C}\smallsetminus\\{0\\}$ and
$g(z)=f^{k}+2\pi,\;k\in\mathbb{N}$. In this case, we also get $K(S)=I(S)$.

From this example 3.1, we can conclude that escaping set $I(S)$ may be
completely invariant even if semigroup $S$ is not abelian. In such a case,
escaping set $I(S)$ is nothing other than the set $K(S)$. There are
tanscendental semigroups $S$ for which the set $I(S)$ as well as the set
$K(S)$ might be empty. For example:

######  Example 3.2.

Suppose that $S=\langle f,g\rangle$, where $f(z)=e^{z}$ and $g(z)=e^{-z}$.
Then both $K(S)$ and $I(S)$ are empty sets. For $z\in I(f)$, then
$g(f^{n}(z))=1/e^{f^{n}(z)}=1/f(f^{n}(z))=1/f^{n+1}(z)\to 0$ as $n\to\infty$.

######  Proposition 3.1.

Let $S$ be a transcendental semigroup. If $K(S)\neq\emptyset$, then it is
S-completely invariant under each $f\in S$ and it contains all points
$z\in\mathbb{C}$ with $f^{n}(z)\to\infty$ as $n\to\infty$ for every $f\in S$.

###### Proof.

By the definition 3.1, the set $K(S)=\bigcap_{i\in\mathbb{N}}E_{i}$, where
$E_{i}$ is completely invariant for each $f\in S$ and it contains points
$z\in\mathbb{C}$ such that $f^{n}(z)\to\infty$ as $n\to\infty$. For all $f\in
S$, we have
$f(K(S))=f(\bigcap_{i\in\mathbb{N}}E_{i})\subseteq\bigcap_{i\in\mathbb{N}}f(E_{i})\subseteq\bigcap_{i\in\mathbb{N}}E_{i}\subset
E_{i}$ for all $i\in\mathbb{N}$. This implies that
$f(K(S))\subset\bigcap_{i\in\mathbb{N}}E_{i}=K(S)$. Backward invariant of
$K(S)$ follows similarly. This proves that $K(S)$ is S-completely invariant.
The next part follows from the definition 3.1. ∎

######  Proposition 3.2.

The set $K(S)\subset I(f)$ for each $f\in S$ and so $K(S)\subset\bigcap_{f\in
S}I(f)$

###### Proof.

Let $z\in K(S)$. Then by the definition 3.1, $z\in E_{i}$ for all $i$ and
$f^{n}(z)\to\infty$ as $n\to\infty$ for every $f\in S$. This proves that $z\in
I(S)$. From the theorem 1.1, we have $I(S)\subset I(f)$ for all $f\in S$.
Thus, we must have $K(S)\subset I(f)$ for all $f\in S$. The last inclusion is
obvious from the assertion $K(S)\subset I(f)$ for all $f\in S$. ∎

The fact of this proposition 3.2 together with fact $I(S)\subset I(f)$ for all
$f\in S$, we can say that $K(S)$ is completely invariant escaping set which is
contained in $I(S)$. The following result shows that $K(S)$ sometime may be
equal to $I(S)$.

######  Proposition 3.3.

Let $S$ be a non-trivial transcendental semigroup. Then $K(S)\\\ =I(S)$ if and
only if $I(S)$ is S-completely invariant.

###### Proof.

Let $K(S)=I(S)$ where $K(S)$ is S-completely invariant by the proposition 3.1.
So, $I(S)$ is S-completely invariant.

Conversely suppose that $I(S)$ is S-completely invariant. Then any $z\in I(S)$
implies that $z\in K(S)$. So, $I(S)\subset K(S)$. On the other hand, $K(S)$ is
completely invariant and consists of points $z\in\mathbb{C}$ such that
$f^{n}(z)\to\infty$ as $n\to\infty$ for every $f\in S$. So, it is nothing
other than set $I(S)$. ∎

A nice example of this proposition is an example 3.1. Next, we construct the
following set which gives an alternative convenient description of the set
$K(S)$. Let $S$ be a transcendental semigroup for which the set
$K(S)\neq\emptyset$. Then by above proposition 3.2, we can write

$K(S)\subset I(h)\;\text{for all}\;h\in S$

Note that as each $I(h)$ is completely invariant, so their intersection
$\bigcap_{h\in S}I(h)$ is also completely invariant. Define

$E_{0}=\bigcap_{h\in S}I(h)$ $E_{1}=\bigcup_{h\in
S}h^{-1}(E_{0})\cup\bigcup_{h\in S}h(E_{0})$
$\ldots\;\;\;\;\;\ldots\;\;\;\;\;\ldots\;\;\;\;\;\ldots\;\;\;\;\;\ldots$
$E_{n+1}=\bigcup_{h\in S}h^{-1}(E_{n})\cup\bigcup_{h\in S}h(E_{n})$

and

(3.1) $E=\bigcap_{n\in\mathbb{N}\cup\\{0\\}}E_{n}$

Here, we have built-up set $E$ from sets $I(h)$ for all $h\in S$ within their
intersection.

######  Proposition 3.4.

The set $E=\bigcap_{n\in\mathbb{N}\cup\\{0\\}}E_{n}$ is non-empty.

###### Proof.

We show that $K(S)\subset E_{n}$ for every $n\in\mathbb{N}\cup\\{0\\}$ by
induction. $K(S)\subset E_{0}$ is obvious by above construction. By the
completely invariant property of $K(S)$ under each $h\in S,\;K(S)$ is subset
of each sets $h^{-1}(E_{0})$ and $h(E_{0})$ for all $\ h\in S$. This shows
$K(S)\subset E_{1}$. Let us suppose $K(S)\subset E_{n}$. Since
$E_{n+1}=h^{-1}(E_{n})\cup h(E_{n})$ for all $h\in S$. By the similar fashion
as above, $K(S)$ is subset of each of the sets $h^{-1}(E_{n})$ and $h(E_{n})$
for all $\ h\in S$. This shows that $K(S)\subset E_{n+1}$ for each
$n\in\mathbb{N}\cup\\{0\\}$. This proves set $E\neq\emptyset$. ∎

The following result will be a convenient description of a completely
invariant escaping set of transcendental semigroup

######  Theorem 3.1.

Let $S$ be a transcendental semigroup. Then $E=K(S)$, where $E$ is a set
defined in 3.1.

First we prove the following lemma.

######  Lemma 3.1.

The closure $\overline{E}$ of any $E\subset\mathbb{C}$ is completely invariant
under a transcendental entire function $f$ if and only if the set $E$ itself
is completely invariant under the same function $f$.

###### Proof.

Let $\overline{E}$ is completely invariant under the given transcendental
entire function $f$. Then $f(\overline{E})\subset\overline{E}$ and
$f^{-1}(\overline{E})\subset\overline{E}$. Let $z\in\overline{E}$, then
$f(z)\in f(\overline{E})$ and so $f(z)\in\overline{E}$. Also,
$z\in\overline{E}\Longrightarrow$ there exists sequence
$(z_{n})_{n\in\mathbb{N}}$ in $E$ such that $z_{n}\to z$ as $n\to\infty$. From
the continuity of the function $f$ we can write $f(z_{n})\to f(z)$ as
$n\to\infty$. As $f(z)\in\overline{E}$, we must have $f(z_{n})\in E$. Note
that $f(z_{n})\in f(E)$ as $z_{n}\in E$. Thus we must have $f(E)\subset E$.

Next, let $z\in\overline{E}$, then $f^{-1}(z)\in
f^{-1}(\overline{E})\subset\overline{E}$. So there exists $f^{-1}(z_{n})\in E$
such that $f^{-1}(z_{n})\to f^{-1}(z)$ as $n\to\infty$. However, it is obvious
that $f^{-1}(z_{n})\in f^{-1}(E)$. Thus we must have $f^{-1}(E)\subset E$.

The converse part of this lemma follows from [bea, Theorem 3.2.3]. ∎

Note that under the assumption of this lemma 3.1, not only the closure of
completely invariant set is invariant but also its complement, interior and
boundary are also completely invariant (see for instance [bea, Theorem
3.2.3]).

###### Proof of the Theorem 3.1.

Since as defined in 3.1, $K(S)$ is completely invariant under each of $h\in S$
and is contained in $I(h)$ for all $h\in S$. Hence, it is contained in $E_{n}$
for all $n\in\mathbb{N}\cup\\{0\\}$. Therefore, $K(S)\subset E$.

On the other hand, set $E$ is contained in $I(h)$ where each of $I(h)$ is
completely invariant. We need to show that $E$ is completely invariant for
each $f\in S$ and for every $z\in E,\;\;f^{n}(z)\to\infty$ as $n\to\infty$ for
every $f\in S$.

Since any $f\in S$ is continuous in $\mathbb{C}$ and $E\subset\mathbb{C}$. So
by the usual topological argument,
$f(\overline{E})\subset\overline{f(E)}\Rightarrow f^{-1}(\overline{f(E)})$ is
closed in $\mathbb{C}$ for every $f\in S\Rightarrow f$ is a continuous closed
map. This shows that $f(\overline{E})$ and $f^{-1}(\overline{E})$ are both
closed sets in $\mathbb{C}$. Since each $f\in S$ is continuous closed map and
$f(\overline{E_{n}})\subset\overline{E_{n}}$ and
$f^{-1}(\overline{E_{n}})\subset\overline{E_{n}}$ for all $n$. Which shows
that $f(\overline{E})\subset\overline{E}$ and
$f^{-1}\overline{(E)}\subset\overline{E}$. By lemma 3.1, it proves that $E$ is
completely invariant under each $f\in S$.

Finally, any $z\in E\Rightarrow z\in E_{n}$ for all $n$. Again $E_{n}$ is a
union of all images and pre-images of $E_{n-1}$ under the each map $f\in S$.
By this way, the point $z$ belongs to the image or pre-image of $E_{0}$ under
each map $f\in S$. Since $E_{0}$ is contained within of all escaping set of
all functions $f\in S$, so this point $z$ goes to infinity under the iteration
of any function in $S$. Hence $E\subset K(S)$. ∎

The following relation holds good in general between the sets $K(S)$ and
$I(S)$ for a non-abelian transcendental semigroup $S$.

######  Proposition 3.5.

Let $S$ be a non-trivial and non-abelian transcendental semigroup. Then
$K(S)\subset I(S)$.

###### Proof.

By the theorem 3.1, $K(S)=E$ and $E\subset I(S)$ by the construction of set
$E$ (that is, $E$ is completely invariant and $I(S)$ is not) in 3.1. Hence the
proposition follows. ∎

We also can construct $I(S)$ by similar fashion as in $K(S)$. Note that
escaping set $I(S)$ is in general only forward invariant. Define

$F_{0}=\bigcap_{h\in S}I(h),$ $F_{1}=\bigcup_{h\in S}(h(F_{0}))$
$\ldots\;\;\;\;\;\;\ldots\;\;\;\;\;\;\ldots$ $F_{n+1}=\bigcup_{h\in
S}(h(F_{n}))$

and

(3.2) $F=\bigcap_{n\in\mathbb{N}}F_{n}$

We can show that $F\neq\emptyset$ by the similar process as in proposition
3.4. Here, we also have built-up set $F$ from sets $I(h)$ for all $h\in S$
within their intersection. The fundamental difference of this set $F$ to that
from the set $E$ of 3.1 is that it is constructed from only forward invariant
property under each $h\in S$. Where as the set $E$ was constructed by
completely invariant property under each $h\in S$. The following theorem
provides an alternative definition of escaping set of transcendental
semigroup.

######  Theorem 3.2.

Let $S$ be a transcendental semigroup. Then $F=I(S)$, where $F$ is defined in
3.2.

###### Proof.

Since $I(S)$ is forward invariant under each $h\in S$ and is contained in each
of $I(h)$, so it is contained in $F$. On the other hand, the set $F$ is
contained in $I(S)$ by above construction. ∎

## References

  * [1] Bergweiler, W.: Iteration of meromorphic functions, Bull. (New Series) Amer. Math. Soc. Vol. 29, Number 2, (1993), 131-188.
  * [2] Eremenko, A.: On the iterations of entire functions, Dynamical System and Ergodic Theory, Banach Center Publication Volume 23, Warsaw, Poland, (1989).
  * [3] Eremenko, A., and Lyubich, M.Y.: Dynamical Properties of Some Classes of Entire Functions,Ann. Inst. Fourier, Grenoble, 42 (1992), 989-1020.
  * [4] Hua, X.H. and Yang, C.C.: Dynamic of transcendental functions, Gordon and Breach Science Publication, (1998).
  * [5] Kumar, D. and Kumar, S.: The dynamics of semigroups of transcendental entire functions-II, arXiv: 1401.0425 v3 (math.DS), May 22, 2014.
  * [6] Kumar, D. and Kumar, S.: Escaping set and Julia set of transcendental semigroups, arXiv:141.2747 v3 [math. DS] October 10, 2014.
  * [7] Kumar, D. and Kumar, S.: On eacping sets of some families of entire functions and dynamics of composite entire functions, arXiv: 1406.2453v4 [math.DS], 4 Oct. 2015.
  * [8] Morosawa, S., Nishimur, Y., Taniguchi, M. and Ueda, T.: Holomorphic dynamics, Cambridge University Press,Cambridge, UK, 2000.
  * [9] Subedi, B. H. and Singh, A.: A short comparison of classical complex dynamics and holomorphic semigroup dynamics, arXiv: 1803.08368v1[math. DS] 22 March 2018.
  * [10] Subedi, B. H. and Singh, A.: Invariant properties of Fatou set, Julia set and escaping set of holomorphic semigroup, arXiv: 1803.09662v1[math. DS] 23 March 2018


[a]Alexander Grigoriev

# Neutrino propagation in moving and polarized matter

Alexander Studenikin  Alexei Ternov

###### Abstract

The rapid development of neutrino astronomy, which is expressed, among other
things, in the emergence of new neutrino mega-projects capable of efficient
registering astrophysical neutrino fluxes requires a detailed knowledge of
neutrino evolution inside neutrino sources (type II supernovae, gamma-ray
bursts). This evolution can be influenced by many factors, each should be
accounted for by a relevant theory. In this work, we develop the theory of
neutrino propagation in moving and/or polarized matter by introducing for the
first time an exact spin integral of motion. This enables us to obtain the
neutrino dispersion under these conditions and resolve it for most important
cases. Our approach opens up the possibility to consistently classify neutrino
states in moving and/or polarized medium and, as a consequence, to give a
systematic description of the related physical phenomena (e.g., neutrino
oscillations, neutrino electromagnetic radiation).

An analytical description of neutrino behavior in external matter and fields
and specifically that of neutrino spin dynamics not only provides the
understanding of the basics of neutrino spin oscillations in astrophysical
media but also enables to discover and give account for another new phenomena.
Examples include the effects of neutrino Spin Light [1], and neutrino beam
self-polarization in matter [1, 2]. All these effects arise due to helicity
dependence of neutrino energy in non-moving matter. In this study we go
further in developing the neutrino theory in a medium and consider matter
motion. In this case the helicity is no longer the exact quantum number and
for consistent treatment one has to find the spin integral of motion. Once
found, it enables easy derivation of the neutrino dispersion relation and
clear physical interpretation of the spin quantum number entered in it.

Our starting point is the modified Dirac equation that describes a neutrino
coherently interacting with particles of the external matter, taking into
account possible effects of matter motion and polarization [1]

$\left\\{i\gamma_{\mu}\partial^{\mu}-\frac{1}{2}\gamma_{\mu}(1+\gamma^{5})f^{\mu}-m\right\\}\Psi(x)=0,$
(1)

where $\gamma^{5}=-i\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}$ and the 4-vector
$f^{\mu}$ is defined once the matter model is chosen, but generally it depends
on matter characteristics – density, velocity and polarization. We will
consider homogenous matter, moving in the given reference frame with an
arbitrary constant speed $\mathbf{v}$ and having the number density $n=\gamma
n_{0}$, where $n_{0}$ is the number density in the matter rest frame
($\gamma=1/\sqrt{1-\mathrm{v}^{2}}$ being the Lorentz gamma factor). We also
take the matter to be composed of electrons (generalization to a more complex
composition is just straightforward), and then within the framework of the
minimally extended Standard Model we have [1]:

$\frac{1}{2}\,f^{\mu}=\gamma\tilde{n}_{0}(1,\,\mathbf{v})=\tilde{n}_{0}v^{\mu},\quad\tilde{n}_{0}=\frac{1}{2\sqrt{2}}G_{F}(1+4\sin^{2}\theta_{W})n_{0}.$
(2)

In order to define the conserved spin quantity one has to find the spin
operator which commutes with the Hamiltonian of motion. From Eq. (1) we find:

$\mathrm{H}=({\bm{\alpha}}{\mathbf{p}})-{\tilde{n}}({\bm{\alpha}}{\mathbf{v}})+\tilde{n}({\bm{\Sigma}}{\mathbf{v}})+\gamma^{5}\tilde{n}+\tilde{n}+\gamma^{0}m,$
(3)

where we have introduced the neutrino momentum $\mathbf{p}$ and used the
notations $\alpha_{i}=\gamma^{0}\gamma^{i}$,
$\Sigma_{i}=\gamma^{0}\gamma^{5}\gamma^{i}$, $\tilde{n}=\gamma\tilde{n}_{0}$.
It should be noted that the same structure of the Hamiltonian also covers
matter polarization, so that our results below can be easily adopted for that
case.

The spin operator is obtained by the following procedure. Let us take the
4-vector spin polarization operator $\mathrm{T}^{\mu}$ (see, for instance,
[3]):

$\mathrm{T}^{\mu}=\gamma^{5}\gamma^{\mu}-\gamma^{5}p^{\mu}/m,$ (4)

and therein do the momentum “extension”
$p^{\mu}\rightarrow\tilde{p}^{\mu}\equiv
p^{\mu}-f^{\mu}=p^{\mu}-\tilde{n}_{0}v^{\mu}$. Denoting the result as
$\widetilde{\mathrm{T}}^{\mu}$ we compose its scalar product with the 4-vector
$v^{\mu}$:
$\widetilde{\mathrm{T}}^{\mu}v_{\mu}\equiv(\widetilde{\mathrm{T}}v)$.
Multiplying finally this quantity by $m$ one can verify that the obtained
result,

$\mathrm{S}=m(\widetilde{\mathrm{T}}v),$ (5)

commutes with the Hamiltonian (3) on solutions of the Dirac equation (i.e.
assuming that the energy operator is realized as follows: $p^{0}\Psi=E\Psi$,
where $E$ is the neutrino energy). Introducing the notation
$\widetilde{\mathrm{H}}=\tilde{p}^{0}=\mathrm{H}-\tilde{n}$, we also present
the found spin operator $\mathrm{S}$ in the expanded form:

$\mathrm{S}=\gamma\left[\gamma^{5}\gamma^{0}m-\gamma^{5}(\widetilde{\mathrm{H}}-\left(\tilde{{\mathbf{p}}}{\mathbf{v}})\right)-m\gamma^{0}({\bm{\Sigma}}{\mathbf{v}})\right].$
(6)

Note that the spin operator (5) is a Lorentz scalar. In our study, it is also
suitable to introduce another scalar operator $P$:

$P=(\tilde{p}^{\mu}v_{\mu})=(\tilde{p}v)=({p}v)-\tilde{n}_{0}.$ (7)

Then, squaring the spin operator $\mathrm{S}$ it is easy to obtain its
eigenvalues and write them through $P$:

$s\sqrt{P^{2}-m^{2}}\equiv s\Lambda,\ \ s=\pm 1.$ (8)

By this means, the operator $\mathrm{S}$ characterizes the longitudinal
particle polarization with respect to the matter velocity 4-vector $v^{\mu}$.
The stationary spin states of the particle are represented by the wave-
function $\Psi_{E,s}$, which is an eigenfunction of operators $\mathrm{H}$ and
$\mathrm{S}$:

$\mathrm{H}\Psi_{E,s}=E\Psi_{E,s},\ \ \mathrm{S}\Psi_{s}=s\Lambda\Psi_{s}.$
(9)

From these equations, it is the standard procedure to derive the neutrino
dispersion in which an explicit dependance on the neutrino spin quantum number
$s$ is embedded:

$p^{2}-m^{2}=2\tilde{n}_{0}(P-s\Lambda),$ (10)

where $p^{2}=E^{2}-{\mathbf{p}}^{2}$. This is another form of the neutrino
dispersion relation in moving matter, previously found in [4] (without spin
dependance) and [5] (with a polarization variable having another physical
meaning).

The obtained dispersion relation (10) enables to easily find neutrino energies
for some basic cases. First of all, on putting ${\mathbf{v}}=0$ and
introducing the “energy sign” $\varepsilon=\pm 1$ we reproduce the well-known
result for neutrino energy in non-moving matter [1]:

$(\sqrt{\widetilde{E}^{2}-m^{2}}+s\tilde{n}_{0})^{2}=p^{2}.$ (11)

By making the matter speed directed along or against the neutrino momentum
($\mathbf{v}{\parallel}\mathbf{p}$) we have:

$E=\varepsilon\sqrt{(\mathrm{p}\mp\mathrm{v}\tilde{n}-s\tilde{n})^{2}+m^{2}}\pm
s\mathrm{v}\tilde{n}+\tilde{n},$ (12)

where the upper signs correspond to the first case and the lower ones to the
second.

In the light of the recent interest in the features of neutrino motion in
transversal matter currents [6], it also makes sense to write out expression
for neutrino energy when $(\mathbf{pv})=0$. In this case, however, Eq. (10) is
the fourth-order equation with respect to energy and we give approximate
values for ultrarelativistic neutrino, $\mathrm{p}\gg\tilde{n}$,
$\mathrm{p}\gg m$, corresponding to $\varepsilon=1$:

$\displaystyle E_{s=+1}$
$\displaystyle=\sqrt{(\mathrm{p}-\tilde{n})^{2}+m^{2}}+\tilde{n},$ (13)
$\displaystyle E_{s=-1}$
$\displaystyle=\sqrt{\mathrm{p}^{2}+4\tilde{n}^{2}\mathrm{v}^{2}+m^{2}}+2\tilde{n}.$
(14)

It is interesting to consider in general terms the impact of Eq. (14) on
neutrino oscillations $\nu_{e}\leftrightarrow\nu_{\mu}$. Associating
$E_{s=-1}$ with left-handed neutrino energy $E_{\nu_{L}}$ and expanding the
root we get

$E_{\nu_{L}}\simeq\mathrm{p}+2\tilde{n}+\frac{2\tilde{n}^{2}\mathrm{v}^{2}}{\mathrm{p}}.$
(15)

The second term leads to the standard effective neutrino oscillation potential
in matter and the third term provides the shift of this value and consequently
the shift of the conventional resonance condition:

$\frac{\Delta}{2\mathrm{p}}\cos
2\theta=\sqrt{2}G_{F}n+8\frac{G_{F}^{2}n^{2}\mathrm{v}^{2}}{\mathrm{p}}\sin^{2}\theta_{W},$
(16)

where the addition is represented by the second term on the right-hand side.
The effect of matter motion becomes substantial (when this addiction is of the
order of the first term), for example, for $\mathrm{v}=0.9$, $\mathrm{p}\sim
10$ keV and extra-dense matter with $n\sim 10^{41}$cm-3.

To conclude, we have established the exact neutrino spin integral of motion
for the general case of neutrino motion and polarization, determined the
corresponding stationary states and obtained the neutrino dispersion relation
with the spin quantum number. Besides the purely theoretical significance,
these results can be important for the theory of astrophysical neutrino
oscillations, for interpreting the data on astrophysical neutrino flux
measurement of large-volume detectors as well as for studying various quantum
processes with neutrinos in initial and final states in dense astrophysical
media.

The work of AS is supported by the Russian Science Foundation under grant
No.22-22-00384.

## References

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