# Representation Edit Distance as a Measure of Novelty

Joshua Alspector
Institute for Defense Analyses
Information Technology and Systems Division
Alexandria, VA
This research was developed with funding from the Defense Advanced Research
Projects Agency (DARPA)
Approved for public release; distribution unlimited
The views, opinions, and/or findings expressed are those of the author and
should not be interpreted
as representing the official views or policies of the Department of Defense or
the U.S. Government.

###### Abstract

> Adaptation to novelty is viewed as learning to change and augment existing
> skills to confront unfamiliar situations. In this paper, we propose that the
> amount of editing of an effective representation (the Representation Edit
> Distance or RED) used in a set of skill programs in an agent’s mental model
> is a measure of difficulty for adaptation to novelty. The RED is an
> intuitive approximation to the change in information content in bit strings
> measured by comparing pre-novelty and post-novelty skill programs. We also
> present some notional examples of how to use RED for predicting difficulty.

##  1 Framework for adaptation to novelty

###  1.1 Introduction

We propose a framework for agents to learn and adapt to novel experiences
based on Algorithmic Information Theory (AIT) (?) and the Minimal Description
Length (MDL) principle (?). We chose AIT because it is based on a small set of
sound principles, because it is widely applicable to all machine learning
methods from traditional AI to deep neural nets, and because instantiations
within the framework are falsifiable by testing. Its use depends only on
information content and generation, their representation structure, and the
probability distributions involved. Agents are viewed as information
processing machines (computers) whose experiences add to their information
content and skills, and, therefore, their complexity. Complexity in the form
of algorithmic entropy is useful for determining difficulty of adaptation to
novel experiences. However, because AIT must be applied to optimal and non-
computable quantities, one must search for methods that can approximate
optimality but are still useful to practically predict adaptation. For this,
we hypothesize that a Representation Edit Distance (RED) using near-optimal
representations can predict approximate difficulty of adaptation.

In our view, novelty is in the eye of the beholder. To an agent, any
sufficiently advanced technology is indistinguishable from magic (?). But if
an agent is familiar with the technology, the novelty, and therefore the
magic, disappears. In competitions, conflicts, and confrontations, a
competitor strives to achieve surprise through novelty to the adversary. The
chessmaster prepares a surprising move, the hacker creates a zero-day exploit,
and the warrior uses a newly developed secret weapon. The agent’s adaptation
to novelty is the subject of this paper. Contributions include 1) a framework
for using any machine learning method that can adapt to novel experiences; 2)
a representation change measure (RED) for determining difficulty of
adaptation; 3) a view of novelty as a mismatch between an agent’s mental model
expectations and its observations; 4) a view of detection, adaptation, and
characterization of novelty using ideas in the framework; and 5) some notional
examples of how to approximate the framework’s representations for prediction
testing.

###  1.2 A general setting for adaptation

We consider a primary agent operating in an environment in which novelty is
introduced. Novelty can be seen as a reconfiguration of previously existing
environmental elements in a new arrangement, or as one or more elements not
previously encountered, or as a previously encountered arrangement of elements
in a novel context. It may be a situation newly encountered by the primary
agent or a novel situation introduced by a secondary agent. The primary agent
possesses a mental model of the environment that may include other agents and
their mental models (Fig. 1). The mental model includes a model of the
environment composed of built-in knowledge and skills (priors in Bayesian
terms) and the ability to acquire new knowledge and skills from experience and
adapt to novel environmental elements and situations that the agent
encounters. Agents have sensors through which they perceive the world, perhaps
incompletely and with distortions, and actuators for interacting with their
environment. The agent may observe the world passively, enabling the ability,
for example, to categorize previously unseen visual object classes. The agent
may also be able to act in the world and affect the environment to better
adapt to novelty, for example, by moving to get a better view. The ability to
adapt to novelty quickly and appropriately is considered a measure of
intelligence by some (?).

Figure 1: An agent, the environment, and its mental model

The agent senses the elements of the environment through its perceptual
system, which is composed of sensors and computational components. The system
does not necessarily render the environment perfectly. This imperfect view is
modified further by the agent’s ability to incorporate the view into its
mental model. The mental model view is its interpretation of the environment,
which includes an interpretive program for prediction encompassing the agent’s
skills in the perceived environmental situation. As in Fig. 1, the
representation may include distorted or missing entities, attributes, or
relations.

We propose a mental model for an agent consisting of a representation portion
and a prediction portion. Following the Minimum Description Length (?) or MDL
principle, the total description length of these two portions, the structure
of the model, plus a third component describing the observation data, the
random noise of the data deviating from the structure of the model, should be
near a minimum. The representation portion may be, for example, a graph of
concepts and relationships and their probability distributions. The complexity
of a chosen representation is unlimited as more concepts and relations are
learned. An example is the representation of the novel coronavirus, which has
become more complex as the world progressed from understanding it as an
infectious disease to developing vaccines that require understanding at an
atomic level including the complexities of protein folding and human
physiology. There are deeper levels of understanding to come. From an
Algorithmic Information Theory (?) or AIT perspective, we view a mental model
of a situation as a hypothesis, $H$, in a two part form. Part 1 is the
representation of the hypothesis, $RH$, such as a function, a neural network,
or a probabilistic graphical model. Part 2 is a prediction program that
analyzes observational data, assesses the situation, and generates
predictions, $PH$. $L(H)$ is the length in bits of the hypothesis description
and is the sum of the descriptions lengths of the representation and the
prediction program. This is the structural part of MDL.

Observation data, $D$, is encoded with the help of the hypothesis, $H$, so
that $L(D|H)$ is the length, in bits, of the data description for use with the
hypothesis model. This is sometimes considered as a noise term. An example
would be a model for the prediction of the location of a planet and directions
for pointing a telescope at it. An instrumented telescope would record the
planet’s observation data as a density of light in a location. The density
might have a gaussian distribution. If the hypothesis is poor, the light
density would not be centered at the predicted location. This would indicate
that the hypothesis needs structural change. If the light density is centered
at the predicted location but the gaussian spread is larger than the expected
parameter encoded in $L(D|H)$, this would indicate that the telescope needs
better focus or an improved lens design because it generates a lot of noise in
the observation.

The prediction portion of the mental model can be seen as a program that reads
the hypothesis representation and operates on it to generate predictions and,
perhaps, select a course of action. The complexity of such a program is
approximately fixed in size. One may enforce constant program size by using
the same generation/prediction program for all representations of a certain
type. This is often the case for knowledge engineering methods. Such a fixed
size is applicable to neural nets as well. For example, a neural net inference
(prediction) program has the same instructions for all neurons and connections
in the network graph. $L(PH)$ is the length in bits of the
generation/prediction program, while $L(RH)$ is the length in bits of the
representation. $L(H)$ is the sum of the two. According to the crude two-part
MDL principle (?), a good model minimizes the sum $L(H)+L(D|H)$. Our
formulation of an MDL-inspired mental model has three parts because of the two
structural portions of the hypothesis (prediction program and representation)
plus the noise term describing the data. We argue that the prediction program
portion can be ignored in the minimization of code lengths because its code
length is close to constant in comparison to the code length of the
representation. Thus, once a machine learning and prediction method is chosen,
a good representation would be one where its description length plus the
description length of observational data is minimized.

##  2 Novelty detection, characterization, and adaptation

###  2.1 Detecting novelty

The prediction program described above operates on a representation that is
learned or built-in. This representation may be adapted if the environmental
data perceived by the agent does not match the expectation generated by the
prediction program. The representation characterizes the perceived state of
the world, and if a novel element is introduced or perceived, the mental model
adapts its characterization in order to correct its representation and,
potentially, improve performance on tasks using the corrected representation.
This ability to adapt the representation can be called characterizing novelty.

Novelty and surprise are factors that arouse interest and motivate exploratory
or avoidance behavior (?). Novelty has not been previously experienced or
encountered, while surprise refers to something unexpected. In our view, the
mechanism for novelty and surprise is the same, stemming from an observation
that does not match an expectation. This mismatch is a novelty or surprise
detection mechanism. Surprise requires introspection to determine if this
entity or situation had been previously encountered. If it has, the mismatch
could result from an unexpected context for the situation leading to surprise
rather than novelty. Finding a live fish in a desert is an example of
surprise, or contextual novelty, but not inherent novelty to the agent.

Prediction errors from novelty and surprise also appear in “predictive coding”
architectures (?). These are layered systems going from a hierarchy of sensory
input levels to levels encoding information in more abstract form. The bottom-
up flow of sensory information to abstract representations is paralleled by
top-down information predictions to the lower levels. This allows higher-level
stages to receive information only through the information mismatch between
their predictions and sensations, so that higher levels receive only
unexpected information. Our view of novelty encountered by an agent depends on
a type of predictive coding within the mental model where the representation
can be considered a description code in the MDL sense.

###  2.2 Characterizing novelty in representation

A variety of representations are depicted in Figure 2. In knowledge
engineering, a knowledge graph of entities and relations is read by the
prediction program and used for inference. In a neural network, nodes and
connections create latent representations during the process of learning.
These may be difficult to disentangle in the manner of a symbolic knowledge
graph, but the simple procedural prediction program produces output inferences
just as the program for a knowledge graph does. The third depiction is for
regression to a set of data points using, for example, the family of
polynomials. Any of these representations plus others such as decision trees,
belief nets, and probabilistic graphical models can be cast into a prediction
program portion and a representation portion to be minimized along with data
for a mental model using MDL techniques. MDL principles (?) can be considered
the practical version of AIT (?). We will motivate our methods using these
ideas.

Figure 2: Various machine learning representations and their prediction
programs

To adapt its mental model to novelty, the agent can utilize priors, $P$, like
prior knowledge and prior mental models along with any new experiences, $E$,
it encounters that seem relevant or valuable to such adaptation. Prior models
and knowledge add an inductive bias to the adaptation due to the structure of
such knowledge and models. For example, if a prior model is in the form of a
fully connected neural net, it will not adapt as well as a convolutional
neural net (CNN) to a novelty in the visual domain. If the model has an
arithmetic and logic unit, as digital computers do, it can be designed to
adapt better to knowledge graph inference or first order logic than a neural
net will. The difficulty of adaptation depends strongly on the prior model and
the task. Adaptation will also depend on the algorithm that optimizes the
model based on the representation, priors, and experience. We will call the
ability to generalize prior models to a novel situation the generalization
difficulty, $GD$ (?), but applied to novelty adaptation rather than a theory
of intelligence.

Regardless of the method used, the change in representation resulting from
adaptation can be considered characterizing novelty. If the representation is
a knowledge graph, an agent’s mental model that adapts to a new entity
discovered by observation must add the entity and any relationships observed
to its knowledge graph for effective adaptation. Since the beginning of the
pandemic caused by the novel coronavirus, the knowledge graph has been
augmented by its genetic code, its transmission modes, its shape and the shape
of variants, the types of infection observed, the effectiveness of various
types of vaccines, etc. If the representation is a neural network,
characterization may consist of latent features embedded in the network layers
and the connection strengths. For regression, characterization consists of
functional forms and parameters. The semantics are clearer in a knowledge
graph where natural language is most likely used, but each representation type
has its own language and semantics.

###  2.3 Approximate information distance

A useful concept is the information distance between two information objects,
say $x$ and $y$, such as computer programs. It is the size of shortest
description that transforms each object into the other (?). Mathematically,
this algorithmic information distance (AID) is the maximum of the conditional
Kolmogorov complexities of the transformation programs up to a logarithmic
additive term. It can be recast to remove conditional terms.

$\begin{split}AID(x,y)=max{[K(x|y),K(y|x)]}\\\
=K(xy)-min[K(x),K(y)]\end{split}$ (1)

It is a universal distance function of shorter length than any other distance
function and also a metric. Here, $xy$ is the concatenation of the strings.
Note that strings $x$ and $y$ may have mutual information to be compressed;
for example, they may be a pre-novelty program to be edited for a post-novelty
program. To adjust for the AID distance being larger for longer strings,
normalize by the maximum of the string lengths to obtain the normalized
information distance

$NID(x,y)=\frac{AID(x,y)}{max[K(x),K(y)]},$ (2)

which is a metric.

One can use compression algorithms like gzip to measure distances between
objects for a practical substitute for the uncomputable Kolmogorov complexity,
$K$, of a string using a compressed but approximate version, $Z$. The
normalized compression distance is

$NCD(x,y)=\frac{Z\left(xy\right)-min\left(Z(x),Z(y)\right)}{max\left(Z(x),Z(y)\right)},$
(3)

which may be useful for bit-string representations.

One can use the statistics of web search engines to measure distances between
semantic ideas, not bit strings. Such concepts come from human knowledge and
require semantic context for their meaning. Using the probability that the
search term $x$ appears in a page-indexed search engine G, one can derive the
normalized web distance, $G(x,y)=NWD(x,y)$, between terms $x$ and $y$ as

$NWD(x,y)=\frac{[G\left(xy\right)]-min[\left(G(x),G(y)\right)]}{max[\left(G(x),G(y)\right)]}.$
(4)

We view the search engine, G, as a compressor using the Web, and $G(x)$ as the
binary length of the compressed version of the set of all pages containing the
search term $x$. This is not a metric because it does not satisfy the triangle
inequality. If concept $x$ (fried) is semantically close to concept $y$
(chicken) and concept $y$ is semantically close to concept $z$ (feather), it
is possible that concept $x$ can be semantically very different from concept
$z$. The $NWD$ is useful where semantic concepts, such as those in knowledge
graphs are the information objects of interest.

###  2.4 Information theoretic measures

#### Difficulty of generalizing a skill program to novelty

The previously mentioned generalization difficulty, $GD$, is a measure of how
much the pre-novelty computational mental model must change to adapt
acceptably to novelty given a set of experiences in a curriculum, $C$, for
learning. It is notionally an edit distance. The concept of adapting
acceptably must be defined for each task, $T$, perhaps by a threshold,
$\theta$, for a performance measure (e.g., 90 percent correct classification)
that the post-novelty computational mental model must reach. Proficiency to
reach the acceptable threshold for each task has a value, $T$, representing
the task’s subjective importance to the agent expressed as a function,
$\omega_{T}(\theta_{T})$.

Algorithmic complexity (AC) measures information content and is also known as
Kolmogorov complexity, denoted by $K$. The AC of a binary string, $K(s)$, is
the length of the shortest program that outputs the string, $s$. $K(s)$ is not
a computable function of $s$, so an approximate measure (?) will have to
suffice for practical use. This allows approximating $K(s)$ with the
compressed version obtained with any lossless compressor $C$ so that $C(s)$ =
$K(s)$ \+ constant. The use of zip-style compression, $Z$, (for strings) and
Google-frequency-style compression, $G$, (for ideas) as distance measures is
motivated by the need for practical measures.

The relative algorithmic complexity ($RAC$), $K(s_{1}|s_{2})$, is the length
of the shortest program that, taking $s_{2}$ as input, produces $s_{1}$. The
RAC, like the AC, can also be estimated by compression (?). To compare the
edit distance required to transform one string to another, one must relate the
edit distance to the size of some program. For strings of random bits, which
have high Shannon entropy, the program size needed, $K$, and the string edit
distance are of the same order. If there are regularities in the bit string,
such as repeating bit sequences, the program size may be much shorter in a
compression code.

Programs can be compared by $RAC$ to determine, for example, how much a pre-
novelty program must be edited or adapted to become a post-novelty program.
For an agent, a skill program represents a snapshot of its task-specific
capabilities including the ability to adapt to novelty within the task.
Consider (?) a task $T$; $Sol^{\theta}_{T}$, the shortest of all possible
solutions of $T$ of threshold $\theta$ (shortest skill program that achieves
at least skill $\theta$ during evaluation, which may be viewed as the agent’s
post-novelty program); and $TrainSol^{opt}_{T,C}$, the shortest optimal
training-time solution given a curriculum (e.g., the pre-novelty program that
has the best adaptability). The (developer-aware) generalization difficulty
($GD$) is defined as the shortest program that, taking as input the initial
system (which may contain built-in priors such as convolutions) plus the
shortest possible program that performs optimally over the situations in
curriculum $C$, produces a skill program that performs at a skill level of at
least $\theta$ during evaluation, normalized by the length of that skill
program. Thus, if the developer builds in the solution to the system at $t=0$,
the $GD$ is low, while, if the developer relies only on training data without
built-in prior bias, the $GD$ is high to reach the solution. $GD$ represents
the edits you would have to make to the shortest training-time solution (what
training should do) to obtain the evaluation-time solution while using the
initial agent’s capabilities without penalty. We will use RED as an
approximation to $GD$.

$GD^{\theta}_{a,T,C}=\frac{K\left(Sol^{\theta}_{T}|TrainSol^{opt}_{T,C,a_{t=0}}\right)}{K\left(Sol^{\theta}_{T}\right)}$
(5)

Note that this quantity is between 0 and 1 by construction. An approximation
to $GD$, like RED, may be used as a measure to judge how difficult a given
task would be. If the environment that the agent confronts is a simulated one,
a novelty generator can test the adaptation ability $A$ of an agent $a$ to
produce a near-optimal skill program. Because such a generator would contain
all testable novelties, a developer should be able to produce a program that
can adapt well to all of them. Relative complexities can be determined based
on the edits needed to turn test agents into the near-optimal agent. We will
use MDL to propose a practical, approximate, intuitive version of $GD$, RED,
which is computable.

A task with high $GD$ is one where the evaluation-time behavior needs to
differ significantly from the simplest possible optimal training-time behavior
in order to achieve sufficient skill. A more generalizable skill program can
handle a wider variety of situations than the ones it was trained on. A
novelty-adaptable agent is one that can detect, characterize, and adapt to
novelty and thus is better able to deal with future uncertainty and novelty.

#### Built-in priors

Priors (labeled $P$) of an agent, $a$, relative to a task, $T$, and a skill
threshold, $\theta$, is the fraction of the AC of the shortest solution of $T$
of skill threshold $\theta$ that is explained by the initial system at the
start of the training phase. This does not consider the training data for an
optimal adaptation program as $GD$ does but only the baseline initial agent at
$t=0$. This is the length (normalized by the $AC$ of the solution) of the
shortest possible program that, taking as input the initial agent skill,
produces the shortest solution of $T$ that performs at a skill level of at
least $\theta$ during evaluation.

$P^{\theta}_{a,T}=\frac{K\left(Sol^{\theta}_{T}\right)-K\left(Sol^{\theta}_{T}\right|a_{t=0})}{K\left(Sol^{\theta}_{T}\right)}$
(6)

If the initial system solves the task, the complexity of the solution
conditioned on the initial system is 0 and the priors are at the maximum value
of 1. If there is no built-in knowledge, the conditional term is high and the
same as the solution complexity, so the priors have value 0. Priors can be
interpreted as the amount of relevant information embedded in the initial
system. Priors may be considered as contained in the performance of a baseline
pre-novelty-adaptation agent that has no training. Components may include a
built-in knowledge base or transfer from pre-training on a similar task or
other components of core knowledge.

#### Experience relevant to novelty adaptation

At each step, the agent is exposed to a situation, responds according to its
current program, and receives feedback from the environment thus gaining
information that may affect the agent’s future responses if the agent is
adaptive enough to use it. The increase in experience at time step t is

$E^{\theta}_{a,T,t}=K\left(Sol^{\theta}_{T}\right|a_{t})-K\left(Sol^{\theta}_{T}\right|a_{t-1},data_{t}).$
(7)

We can define the experience, $E$, accumulated by an intelligent system about
a task during a curriculum as

$E^{\theta}_{a,T,C}=\frac{1}{K\left(Sol^{\theta}_{T}\right)}\sum_{t}E^{\theta}_{a,T,t}.$
(8)

$E$ measures how much easier it is to create the solution program if the agent
makes optimal use of the new information at each step but not necessarily the
actual use. Experience can be considered the amount of accumulated information
that is novel to the agent and relevant to the task. There may be other
information that is not relevant or that is repetitive or redundant to what
the agent already knows. The AC of an optimal adaptive agent should increase
at each step if the information is novel and relevant, but not if the
information is irrelevant. Incremental learners like backpropagation-optimized
neural nets may require several exposures to the information to achieve the
full increase in experience. The potential uncertainty reduction from the data
is that for an optimal learner. Learning curves measure the speed of
adaptation to a task from data and depend strongly on the optimization
procedure. The increase in AC may be a way to measure novel and relevant
information gain if we can approximate $K$ with a computable $C$ from an off-
the-shelf compressor.

#### Adaptability

We define the adaptation ability, $A$, of agent, $a$, rather than the
intelligence (?) of a system, as a measure of its skill-acquisition efficiency
over a scope of tasks, with respect to priors, experience, and generalization
difficulty. For a single task, $T$, adaptability can be expressed as the
difficulty of the task to the agent divided by the total information contained
in the agent’s priors and experience.

$A^{\theta_{T}}_{a,T}=\frac{GD^{\theta_{T}}_{a,T,C}}{P^{\theta_{T}}_{a,T}+E^{\theta_{T}}_{a,T,C}}$
(9)

This adaptability can be viewed as a generalized sample efficiency that
measures how well the agent turns experience and priors into new skills. Put
another way, it measures how well an agent adapts to novelty where $GD$
measures the degree of novelty or difficulty. For a scope of tasks on which
the agent is tested, the adaptability is averaged over the tasks and weighted
by the perceived task value.

$A^{th}_{a,sc}=\underset{T\in sc}{Avg}\left[\omega_{T}(\theta_{T})\sum_{C\in
Cm^{\theta_{T}}_{T}}\left[Pb_{c}\cdot\frac{GD^{\theta_{T}}_{a,T,C}}{P^{\theta_{T}}_{a,T}+E^{\theta_{T}}_{a,T,C}}\right]\right]$
(10)

Sufficient skill threshold, $\theta$, is the subjective threshold of skill
associated with a task, above which a skill program can be said to “solve” the
task $T$. It is a property of a task. The average over tasks has an implied
threshold, $th$. The set of all tasks in scope is $sc$. The probability of a
single curriculum is $Pb$. The agent’s scope is the space of tasks of non-zero
value for which the agent is capable of producing a sufficient solution after
a training phase.

$P+E$ (priors plus experience) represents the total exposure of the system to
information about the problem, including the information it starts with at the
beginning of training. This could be approximated by the information in a set
of experiences. This should also include any start-up information either in a
baseline system or as a knowledge base that may be used as core knowledge.

The sum over a curriculum subspace, weighted by the probability of each
curriculum, represents the expected outcome for the system after training.
This assumes that agents are exposed to multiple curricula or training sets in
a supervised setting for each task. The subjective value placed on achieving
sufficient skill at a task $T$ is $\omega_{T}(\theta_{T})$. The mental model’s
value function captures the relative importance of skill at each task
subjectively. For humanlike intelligence (?), it might be high for tasks
relevant to humans, but low for tasks that are not relevant. It would be low
for nuisance novelty (?), which does not affect task performance.

##  3 Representation Edit Distance (RED)

###  3.1 RED as a measure of difficulty

Prediction error can be the stimulus for learning new concepts and relations.
Error may be used to define novelty as a new situation that an agent failed to
predict accurately. In the real world, any description of the physical
environment is necessarily incomplete and correct prediction is the only test
of the mental model and its complexity is the only one that is relevant. An
exception would be a simulated world where the model of environmental and
agent behavior is fully specified. Note that environment complexity in a
mental model may be reduced by the task the agent must perform because only a
portion of the environmental description is needed for executing the task.

We propose RED as an approximate measure for use in determining difficulty.
Defining this metric for a particular architecture requires some thought and
will depend strongly on the structure of the representation. The size of an
effective representation can be approximated by a type of minimum description
length such that the representation effectively expresses the behavior of the
system it describes in the mental model. For example, if multiple graphical
models, such as random forests, can adequately represent the task and skills
needed to a threshold accuracy, and if such a representation can be reduced
optimally to a few graphs that are accurate enough, the reduced model can be
said to effectively represent the system. We define a near-optimal
representation as the minimum size model that can be used to approximate an
oracle to a threshold accuracy. Our hypothesis is that edit distances on near-
optimal representations can be used for practical information-theoretic
measures of difficulty and adaptability.

###  3.2 Complexity for knowledge graphs

A knowledge graph (KG) (?) is a structured representation of facts, consisting
of entities, relationships, and semantic descriptions. In Figure 2, an
abstract knowledge graph is represented as a set of concept nodes containing
semantic descriptions linked by relational edges. KGs may be hand coded or
learned, and the representation graph may be embedded in a low-dimensional
space. One way the KG may be compressed is to represent each entity in the KG
as a vector of discrete codes and compose embeddings from these codes (?). An
approximate complexity may be derived from such techniques.

KGs are meant to create a semantic model of the environment that may be used
to perform inference. Inferences that do not match observation indicate that
the KG must be changed or edited. This may be due to the introduction of novel
concepts or relations. The edit distance from the previous KG to the new one
is a measure of the difficulty. These distance calculations should ideally be
performed on the most compressed version of the KG, which would be the
Kolmogorov complexity of the graph. For practical use, we posit that a
compressed version, perhaps using NWD, that incorporates semantic concepts is
an acceptable substitute. Such a description of the environment and the task
context should be reduced to its most complete, correct, and consistent but
simplest form using a variety of techniques (?) for use in determining
difficulty.

For an agent, a knowledge and skills graph encompasses concepts, actions,
attributes, relations, and other elements useful in characterizing phenomena.
An agent’s mental model may contain a subset of a larger graph depending on
the agent’s skill and knowledge level. A useful example is the knowledge and
skills needed to adapt to the novel coronavirus. When it first appeared, the
previously held knowledge that it is an infectious disease was sufficient to
promote the standard mitigation measures of quarantine, disinfection of
surfaces, washing hands, and not touching your face. As the virus
characterization revealed aerosol transmission, masks and social distancing
were added as mitigation measures. The level of granularity for
characterization in an ontology of hierarchical concepts had limited success
due to less-than-perfect following of mitigation procedures, and it became
clear that a much finer granularity of knowledge and deeper characterization
would be needed to create vaccines. This involved knowledge at a more atomic
level. Knowledge of the molecular structure from the virus genome and protein
folding physics revealed the structure in sufficient detail for groups around
the world to develop vaccines. Even finer granularity of skills was need for
vaccine technologies, especially for the mRNA vaccines. The level of
characterization granularity reached is the atomic level for vaccines or the
deepest level of hierarchical and relational knowledge sufficient for
executing a task; in this case, developing a vaccine.

Using the refined and reduced semantic knowledge graphs before adaptation and
after adaptation to novelty is a matter of replacing the GD of Eq. 5 with the
normalized difference in description lengths (code-lengths or compression
lengths, C) of the two KGs.

$RED^{\theta}_{a,T,C}=\frac{Cpretr\left(TrSol^{\theta}_{T}\right|a_{t=0})[]to]Cpost\left(Sol^{\theta}_{T}\right)}{Cpost\left(Sol^{\theta}_{T}\right)}$
(11)

RED, like GD, represents the edits ($[to]$ in the equation, which may be
$Cpost-Cpretr$) one would have to make to the shortest training-time solution
(what optimal training should do) to obtain the evaluation-time solution while
using the initial agent’s capabilities without penalty on compressed, near-
optimal versions of the knowledge graphs. This means that $Cpretr$ must
operate near-optimally not only on the pre-novelty data as $Cpre$ does but
also on the training curriculum data. However, it can use the capabilities of
$Cpre$ without being charged for edit distance on the $Cpre$ graph. Such an
optimal use of training data may indicate that $Cpretr$ uses a post-training
graph structure. We will posit that the edit distance can be approximated by
using the structure of the $Cpost$ graph for the $Cpretr$ graph without the
newly learned entities and relations whose edits are charged to RED.

This difference normalized by the post-novelty codelength is the effective RED
in our framework if careful attention is paid to the correct refinement and
reduction of the KGs. To determine difficulty of adaptation, we also need to
estimate the potential increase in complexity of the agent when it experiences
the learning curriculum and adapts optimally. This represents the experience
in Eq. 7. For semantic concepts, this may simply be the addition of new
concepts and relations to the KG. For example, the change in the structure of
a new variant of the coronavirus expressed in atomic form such as a shape
description may be called the approximate effective experience, $Eeff$. An
optimal agent (an entire drug company plus consultants and knowledge bases)
would need just a single exposure to this shape to learn how to make a vaccine
if its prior knowledge included vaccine technology. Of course, this prior
knowledge, Eq. 6, must also be estimated. For KGs, this is simply the
compressed description length, $Cpre$, of the entire pre-novelty knowledge
graph. The compressed description length of the priors, $Pd$, according to Eq.
6, would then be the length of the post-novelty solution graph, $Cpost$ minus
the length, $Cpre$, divided by $Cpost$.

$Pd^{\theta}_{a,T}=\frac{Cpost\left(Sol^{\theta}_{T}\right)-Cpre\left(Sol^{\theta}_{T}\right|a_{t=0})}{Cpost\left(Sol^{\theta}_{T}\right)}$
(12)

Using Eq. 9 and the approximate effective values for priors, experience, and
edit distance, the approximate difficulty of adaptation, $Aeff$, to novelty
for the task is then

$Aeff^{\theta_{T}}_{a,T}=\frac{RED^{\theta_{T}}_{a,T,C}}{Pd^{\theta_{T}}_{a,T}+Eeff^{\theta_{T}}_{a,T,C}}.$
(13)

###  3.3 Complexity in regression

MDL is based on finding structure in individual data sequences (?). Fitting of
data to probability distributions (models) extracts regularities viewed as
representations that express the structural meaning of the data. Variations
around the structure are the random portion of the data in the two-part MDL
formulation. The MDL principle promotes the idea that for a given set of
hypotheses $H$ and data set $D$, one should try to find the hypothesis or
combination of hypotheses in $H$ that compresses $D$ most.

In Fig. 2, a notional polynomial function fits a set of data points. Assume
the family of polynomial functions form the set of hypotheses. The perfect fit
polynomial (of degree one less than the number of data points) would likely
overfit the data and lead to poor generalization to new data from the same
source. A third degree polynomial, the model as shown, seems like a good
enough fit, leaving some random variation to the fitted data but able to
generalize to new data. The particular fitted parameters, say $y=3x^{2}+7x+2$,
of the chosen third-degree model form a point hypothesis in the regression
fit. Two-part MDL chooses a point hypothesis that minimizes the sum of
description lengths of the model, $L(H)$, plus the data encoded with the help
of the model, $L(D|H)$. By using the model of third-degree polynomials to fit
the data, we extract the ’structure’ of the data in a compressed code leaving
only the random errors in the data fit to the hypothesis to encode. If the
random portion is distributed according to a simply described function such as
a normal distribution, the total code-length would be quite short. If,
however, the new data fits a sine function better, the noise portion would be
large. A large amount of noise indicates the model needs adaptation to the
novel data. This is done by choosing a new function or a new family of
functions and a new point hypothesis.

The length of the two-part MDL compressed code is $L(H)+L(D|H)$, and this
quantity is to be minimized in a model selection and regression procedure.
Difficulty of adaptation to novel data can be estimated by using such a
procedure. One may try a family of functions such as polynomials to fit a
training set of data points. For the optimization procedure, one would select
a degree of the polynomial and perform a regression on the data. The sum,
$L(H)+L(D|H)$, would be minimized according to this optimization procedure,
which may be a random search guided by a gradient. This would produce a pre-
novelty model (the function) whose complexity or code-length can be determined
for $Cpre\left(Sol^{\theta}_{T}\right|a_{t=0})$. After novelty is introduced
or discovered in a test set of data, the model selection and regression
procedure is repeated to determine $Cpost\left(Sol^{\theta}_{T}\right)$. The
function family may be different or the degree of the polynomial may be higher
post-novelty. In any case, the priors, $Pd^{\theta}_{a,T}$, can be determined
using Eq. 12. The effective experience, $Eeff^{\theta_{T}}_{a,T,C}$, can be
determined by assessing those data points that are novel and relevant in the
test set by measuring the effect of each point on the model parameters by
estimating the goodness of fit. The $RED^{\theta_{T}}_{a,T,C}$ is the edit
distance from pre-novelty point hypothesis to post-novelty. We will posit, in
analogy with what we did for KGs, that the $Cpretr$ near-optimal function for
training is that of the $Cpost$ function before parameter optimization for
determining RED in Eq. 11. Putting these quantities together in Eq. 13
determines difficulty of adaptation for the agent to the task of regression
for the new data starting from the pre-novelty agent’s mental model.

###  3.4 Complexity in neural nets

In one view of neural network function (?), the layers of a deep net form a
Markov chain of successive internal representations of the input. The
information bottleneck (IB) bound characterizes the optimal representations
that maximally compress the input for a given mutual information (MI) on the
desired output. A trained deep net processes the input through the chain of
hidden layers to the predicted output capturing relevant information in the
process. Each hidden layer can be viewed as being an encoder on the input and
decoder to the output forming a representation characterized by the layer’s MI
to the input and to the output regardless of the features encoded. The MI
measures the number of relevant bits that the input contains about the output
label on average. The optimal learning problem can be seen as the construction
of an optimal encoder of that relevant information via an efficient
representation. The IB tradeoff between the compression of the input and the
prediction of the output leads to an approximate optimal representation. Deep
learning effectively finds efficient representations in the hidden layers that
are approximately optimal, and this plays a large role in accounting for its
success including the surprising lack of overfitting in deep learning.
However, this claim and the connection between compression and generalization
have been disputed (?).

The problem of compressed models in neural nets has been studied in a
supervised setting (?), where inputs to a neural model predict outputs after
training. The predictive model should generalize well to inputs not in the
training set. Using the MDL principle, one could choose, say, a uniform
encoding model with K classes so that probability, $p$, of a class is 1/$K$.
The codelength gain, given true distribution $q$, compared to a uniform code
is limited by the amount of mutual information between input and output. There
are advantages (?) in using a prequential (prediction/sequential) code where
one predicts the $n+1$ label after already having seen $n$ input-output pairs.
First, one trains a default model on a few samples of data, then one trains on
the resulting encodings. Next, one uses this model for the next few data
samples, retrains on all new encodings, and so on. For neural nets, a
trainable deep net encoding data incrementally starting from uniform encoding
works well even though model parameters are never encoded explicitly. For a
convolutional network of depth 8, the authors (?) achieve a description length
with a compression ratio of 50x, with test set accuracy that is better than
that of a variational code (?).

To use these ideas practically requires a compressed coding of a neural
network such as a uniform encoding of the synaptic weight parameters.
Encodings may reduce the precision of the weights from a floating-point value
to a fixed-point value, which, in practice, has been studied extensively,
usually in the context of hardware implementations (?) ; (?). The network
adjacency matrix for the graph and the binary coded connections form the
string to be edited by the optimization method, which may be supervised
learning by back-propagation of errors. This learning can take place at full
precision if required. We are interested in the compressed version for
evaluation of the RED.

The procedure for determining difficulty of adaptation is much the same as in
the previous cases discussed. Use the compressed version of pre-novelty and
post-novelty networks for evaluation of $Cpre$, $Cpretr$, and $Cpost$
complexities. If, for example, weight dropout is used during training,
$Cpretr$ would use the $Cpost$ net structure after training. If no structure
change occurs, then $Cpre$ and $Cpost$ network structures are the same.
Determine the experience in the training curriculum by estimating the increase
in complexity for the agent’s neural network representation, assuming the
agent can make optimal use of the training examples. A way to do this may be
to experiment with batch learning of a representative training sample until
the prediction performance saturates. The change in the compressed network
representation represents the effective (novel and relevant) experience,
$Eeff$. The description length of the priors, $Pd$, is given by Eq. 12. The
RED is given by Eq. 11. Eq. 13 then gives the difficulty of adaptation for the
agent and task.

###  3.5 Intuitive representations may work well

In many domains, there may be a multiplicity of representations producing
similar accuracy results. This multiplicity has been called (?) the Rashomon
effect after the classic movie by Akira Kurosawa where the same facts of a
crime are interpreted in different ways by a multiplicity of characters. A
recent study of this effect (?) noted that a model from a simpler class is
approximately as accurate as the most accurate model within the hypothesis
space. The authors provide a method for determining when a large set of such
equivalent models exist and how likely it is that a simple, interpretable
model exists within the class. This supports our complexity analysis of model
descriptions and helps to justify the notion that a near-optimal
representation, perhaps even a semantic representation, can be edited to
provide a RED measure for determining difficulty.

##  4 Conclusion and future work

We have described a framework based on AIT that has applicability to all
machine learning methods for adapting to novelty. It remains to test these
ideas with approximate instantiations that can provide predictions of
difficulty and adaptability using this framework. Methods are needed for good,
practical approximations to non-computable Kolmogorov measures within the
framework. We propose that RED can be used as an approximate measure to
predict difficulty. Research to provide theoretical justifications for methods
to produce RED and for its use in prediction is needed. Likewise, it is
important to provide experimental tests of predictions of the framework. In
conclusion, both theoretical and experimental work is needed to establish and
verify the framework proposed.

##  5 Acknowledgments

This work was supported by the Science of Artificial Intelligence and Learning
for Open-world Novelty (SAIL-ON) program of the U.S. Defense Advanced Research
Projects Agency. It has benefited greatly from discussions with participants
in the program.

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<EMAIL_ADDRESS><EMAIL_ADDRESS>
# Positron supercritical resonances and spontaneous positron creation in slow
collisions of heavy nuclei

Dmitry A. Telnov (https://orcid.org/0000-0002-2509-2904)  St. Petersburg State
University, St. Peterburg 199034, Russia  N. K. Dulaev
(https://orcid.org/0009-0000-0854-422X)  St. Petersburg State University, St.
Peterburg 199034, Russia  Petersburg Nuclear Physics Institute named by B. P.
Konstantinov of National Research Center ‘‘Kurchatov Institute’’, Gatchina,
Leningrad region 188300, Russia  Y. S. Kozhedub
(https://orcid.org/0000-0003-1273-9008)  St. Petersburg State University, St.
Peterburg 199034, Russia  I. A. Maltsev
(https://orcid.org/0000-0001-8452-639X)  St. Petersburg State University, St.
Peterburg 199034, Russia  R. V. Popov (https://orcid.org/0000-0003-0730-5213)
St. Petersburg State University, St. Peterburg 199034, Russia  Petersburg
Nuclear Physics Institute named by B. P. Konstantinov of National Research
Center ‘‘Kurchatov Institute’’, Gatchina, Leningrad region 188300, Russia  I.
I. Tupitsyn (https://orcid.org/0000-0001-9237-5667)  St. Petersburg State
University, St. Peterburg 199034, Russia  V. M. Shabaev
(https://orcid.org/0000-0002-2769-6891)  St. Petersburg State University, St.
Peterburg 199034, Russia  Petersburg Nuclear Physics Institute named by B. P.
Konstantinov of National Research Center ‘‘Kurchatov Institute’’, Gatchina,
Leningrad region 188300, Russia

###### Abstract

We present a theoretical and computational study of positron supercritical
resonances in systems consisting of two highly-charged bare nuclei. The
resonance positions and widths depending on the internuclear separation are
calculated with the help of the complex-scaling generalized pseudospectral
method in modified prolate spheroidal coordinates. The results are applied to
estimate the probability of spontaneous positron creation in slow U92+–U92+
and Cm96+–Cm96+ collisions.

##  1 Introduction

The standard quantum electrodynamics (QED) theory predicts that spontaneous
electron-positron pair creation by a time-independent electric field can be
observed if the field strength is comparable with some critical value or
exceeds it Fradkin _et al._ (1991). However, to date the spontaneous pair
creation has not been detected in experiments due to extremely high value of
the critical field strength. For the uniform electric field, it is equal to
$1.3\times 10^{18}$ V/m. In the case of the Coulomb field, the supercritical
regime can be achieved in low-energy collisions of two bare nuclei with the
total charge number larger than $173$. In such a collision the originally
neutral vacuum should decay into the charged vacuum and two positrons (see
Refs. Pomeranchuk and Smorodinsky (1945); Gershtein and Zeldovich (1969);
Pieper and Greiner (1969); Zeldovich and Popov (1971); Greiner _et al._
(1985); Voskresensky (2021) and references therein). This phenomenon was
predicted more than $50$ years ago, but so far it was not observed in
experiments. The main obstacle for the experimental observation of this
phenomenon is the fact that the time of the supercritical regime is too short.
As a result, the spontaneous pair creation is strongly masked by the dynamical
(or induced) pair creation, which is due to the time dependence of the field.
This was the main reason why the Frankfurt group, which worked on this topic
for more than $20$ years, concluded that the vacuum decay can be observed only
in collisions with nuclear sticking when the nuclei are bound for some period
of time due to nuclear forces Reinhardt and Greiner (2005). To date, we have
no evidence of the nuclear sticking in collisions of so heavy ions, therefore
this scenario does not seem promising. However, in the recent works by the St.
Petersburg group Maltsev _et al._ (2019); Shabaev _et al._ (2019); Popov _et
al._ (2020, 2023) it was found that the vacuum decay can be observed via
impact-sensitive measurements of pair-production probabilities and
corresponding positron spectra. In these papers it was shown that clear
signatures indicating the transition from the subcritical to the supercritical
regime arise in both pair-creation probabilities and positron spectra if the
collisions are considered along the trajectories, which correspond to a given
minimum internuclear distance.

The calculations by the Frankfurt and St. Petersburg groups showed that the
dynamical and spontaneous pair-creation channels contribute coherently, and,
generally, the total pair-creation probability for a given trajectory does not
exhibit any threshold effects when crossing the border from the subcritical to
the supercritical regime. For this reason, we cannot formally separate these
channels in the total pair-creation probability when considering collisions at
given impact parameters and energies. However, to estimate the spontaneous
contribution to the total probability, one can integrate the resonance width
over the supercritical regime while the nuclei are moving along their
trajectories Popov (1973); Peitz _et al._ (1973). In the present paper, we
perform this calculation for the U92+–U92+ and Cm96+–Cm96+ head-on collisions
at the energies which correspond to the shortest internuclear distance $R_{\rm
min}=17.5$ fm. This distance was chosen as the minimal one in the scenarios
for the experimental observation of the transition to the supercritical regime
in Refs. Popov _et al._ (2020, 2023). Thus, the present work should allow one
to better understand why it is possible to experimentally observe the
transition to the supercritical regime according to the previously proposed
scenarios Maltsev _et al._ (2019); Shabaev _et al._ (2019); Popov _et al._
(2020, 2023).

The paper is organized as follows. Section 2 contains a short account of the
theoretical approach, in section 3 we describe our numerical method, in
section 4 we present and discuss our results, and section 5 contains
concluding remarks. Atomic units are used throughout the paper unless
specified otherwise.

##  2 Theory

###  2.1 Time-independent Dirac equation

To simplify the calculations, we will use Dirac’s hole picture started with
the Dirac equation for positrons Greiner _et al._ (1985); Godunov _et al._
(2017). In this approach, the lower Dirac continuum states as well as discrete
states, detached from the lower continuum to the gap between the lower and
upper continua, are considered occupied by positrons, which are not free and
cannot be observed. A positron becomes free if a transition is made from such
states to the upper continuum. At the same time, a hole in the lower continuum
or in the discrete state is created, which is described as an electron in a
continuum or bound state, respectively. Thus the electron-positron pair
creation can be caused by the energy absorption from an external field or may
occur spontaneously if the parameters of the nuclear system, such as
internuclear separation, make it possible that the energy of some initially
occupied positron state get in the upper continuum.

In our case, as the distance between the two nuclei decreases, the energies of
the discrete positron states in the gap between the lower and upper continua
move towards the onset of the upper continuum. If the total charge of the
system $2Z_{n}$ exceeds $173$, then at least the highest-lying $1s\sigma_{g}$
states eventually cross the boundary of the upper continuum, becoming a
supercritical resonance and opening up a possibility of spontaneous electron-
positron pair creation. The positions and widths of the positron supercritical
resonances are obtained by solving the time-independent Dirac equation for a
one-positron system with two identical nuclei in the center-of-mass frame of
reference:

$H\Psi(\bm{r})=E\Psi(\bm{r}),$ (1)

where $\Psi(\bm{r})$ is a four-component positron wave function and the
Hamiltonian $H$ takes the form:

$H=c(\bm{\alpha}\cdot\bm{p})+c^{2}\beta+U.$ (2)

In Eq. (2), $c$ is the speed of light (equal to the inverse fine-structure
constant in atomic units), $\bm{p}$ is the momentum operator, and
$\bm{\alpha}$ and $\beta$ are the Dirac matrices:

$\bm{\alpha}=\begin{Vmatrix}0_{2}&\bm{\sigma}\\\
\bm{\sigma}&0_{2}\end{Vmatrix},\quad\beta=\begin{Vmatrix}1_{2}&0_{2}\\\
0_{2}&-1_{2}\end{Vmatrix}.$ (3)

In Eq. (3), $0_{2}$ and $1_{2}$ are zero and unit $2\times 2$ matrices:

$0_{2}=\begin{pmatrix}0&0\\\ 0&0\end{pmatrix},\quad
1_{2}=\begin{pmatrix}1&0\\\ 0&1\end{pmatrix},$ (4)

and $\bm{\sigma}$ denotes the vector consisting of the Pauli matrices as
components:

$\sigma_{x}=\begin{pmatrix}0&1\\\ 1&0\end{pmatrix},\
\sigma_{y}=\begin{pmatrix}0&-i\\\ i&0\end{pmatrix},\
\sigma_{z}=\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix}.$ (5)

The interaction with the nuclei $U$ can be described as follows:

$U=U_{n}(|\bm{r}+a\bm{e}_{z}|)+U_{n}(|\bm{r}-a\bm{e}_{z}|).$ (6)

where $U_{n}$ is the potential which depends on the nuclear charge
distribution, $a$ is a half internuclear distance, and $\bm{e}_{z}$ is the
unit vector along the $z$ axis.

When solving Eq. (1), we are interested in complex energy eigenvalues of the
positron resonances:

$E=E_{r}-i\frac{\Gamma}{2},$ (7)

where $E_{r}$ represents the position of the resonance on the energy scale and
$\Gamma$ is the resonance width, a positive-definite number, which has a
meaning of the positron creation probability per unit time. In theory, such
eigenvalues can be obtained by imposing special (outgoing-wave) boundary
conditions on the corresponding wave functions at infinity in the coordinate
space. However, it appears that the absolute value of the resonance wave
function diverges in the real coordinate space as $r\rightarrow\infty$ thus
making the numerical implementation of the eigenvalue problem not
straightforward since the standard routines assume zero boundary conditions at
infinity. Various approaches exist to solve the problem, such as using complex
absorbing potentials Muga _et al._ (2004) and complex scaling of the
coordinates Moiseyev (1998). We make use of the uniform complex scaling, when
the entire $\lambda$ semiaxis is rotated in the complex plane by some
appropriate angle $\Theta$. Upon such rotation, the resonance wave function
vanishes at infinity, so conventional zero boundary conditions can be imposed.

###  2.2 Nuclear charge distribution model

For calculations of the supercritical positron resonances by complex scaling,
the nuclear potential should allow for analytical continuation. We make use of
the Gaussian distribution of the nuclear charge density $\rho_{n}(r)$:

$\rho_{n}(r)=\frac{Z_{n}}{\pi^{3/2}}\left(\frac{s}{R_{n}}\right)^{3}\exp\left[-\left(\frac{sr}{R_{n}}\right)^{2}\right],$
(8)

where $Z_{n}$ is the total charge of a single nucleus, $R_{n}$ is the root-
mean-square nuclear radius, and $s$ is a dimensionless adjustable parameter.
In the calculations, we used $R_{n}=5.8571$ fm for the uranium isotope 238U
and $R_{n}=5.8429$ fm for the curium isotope 244Cm Angeli and Marinova (2013).
In this model, the nuclear potential takes the form:

$U_{n}(r)=\frac{Z_{n}}{r}\mathrm{erf}\left(\frac{sr}{R_{n}}\right),$ (9)

where $\mathrm{erf}(z)$ is the error function. Since the error function is
holomorphic on the whole complex plane, its analytical continuation from the
real axis is straightforward. The parameter $s$ was adjusted to reproduce the
low-lying energy levels of one-electron diatomic quasimolecules at the
internuclear separation $2/Z_{n}$ most closely to those given by the Fermi
nuclear model Parpia and Mohanty (1992), which is considered accurate and is
widely used in the calculations Visscher and Dyall (1997); Shabaev _et al._
(2013); Mironova _et al._ (2015). The best fit is obtained at $s=1.189$. The
ground-state ($1s\sigma_{g}$) and the first excited-state
($2p_{1/2}\sigma_{u}$) energies of U${}_{2}^{183+}$ and Cm${}_{2}^{191+}$
quasimolecules at $R=2/Z_{n}$ are listed in Table 1 for the Fermi model and
the Gaussian model with $s=1.189$. The energies were computed with the CODATA
recommended inverse fine-structure constant value $137.035999084$ Tiesinga _et
al._ (2021).

##  3 Numerical method

The eigenvalue problem (1) is solved with the help of the generalized
pseudospectral (GPS) method in prolate spheroidal coordinates Telnov _et al._
(2018). Prolate spheroidal coordinates Abramowitz and Stegun (1972) are a
natural choice for description of diatomic quantum systems. Here we make use
of slightly modified prolate spheroidal coordinates, more convenient to use
for systems with variable internuclear separation. Namely, we shift the
traditional coordinate $\xi$ and scale it with the parameter $a$, which is a
half internuclear distance:

$\lambda=a(\xi-1).$ (10)

It is assumed that the nuclear charge distributions are centered at the points
$\pm a$ on the Cartesian $z$ axis, and the relations between the Cartesian
coordinates and modified prolate spheroidal coordinates read as

$\begin{split}x&=\sqrt{[(\lambda+a)^{2}-a^{2}][1-\eta^{2}]}\cos\varphi,\\\
y&=\sqrt{[(\lambda+a)^{2}-a^{2}][1-\eta^{2}]}\sin\varphi,\\\
z&=(\lambda+a)\eta.\end{split}$ (11)

At any $a$, the coordinates $\lambda$ and $\eta$ vary within the limits
$[0,\infty]$ and $[-1,1]$, respectively. The angle $\varphi$, as usually,
describes rotation about the $z$ (internuclear) axis and varies within the
range $[0,2\pi]$.

The problem is two-dimensional because the $\varphi$ angle can be eliminated
due to rotational symmetry. In the GPS method, the wave functions and
operators are discretized on a set of collocation points, which are the roots
of orthogonal polynomials or their derivatives. We make use of the Legendre
polynomials and apply the Radau quadrature scheme for the $\lambda$ coordinate
and Gauss quadrature scheme for the $\eta$ coordinate Telnov and Chu (2009).
Since the collocation points lie within the interval $[-1,1]$, a mapping
transformation should be applied to represent the $\lambda$ coordinate:

$\lambda=R_{l}\dfrac{(1+x)^{2}}{1-x}\exp(i\Theta).$ (12)

In Eq. (12), the variable $x$ takes the values in the interval $[-1,1]$ and
$R_{l}$ is a mapping parameter; the latter is used to change the distribution
of the $\lambda$ grid points and thus improve the accuracy of the
calculations. The quadratic dependence of $\lambda$ on $1+x$ ensures a dense
enough distribution of the grid points at small $\lambda$; in particular, the
nuclear potential at very small distances within the nuclear radius can be
well described. Note the complex scaling factor $\exp(i\Theta)$ on the right-
hand side of (12). This factor implements the uniform complex scaling in the
GPS method, so the complex eigenvalues can be obtained by standard computer
routines for diagonalization of non-Hermitian matrices. The complex-scaling
GPS method proves to be reliable in calculations of the resonance states in
atomic and molecular systems Chu and Telnov (2004), and the resulting complex
energy eigenvalues do not depend on $\Theta$ in a wide range of the complex
rotation angle.

The coordinate $\eta$ varies in the same interval $[-1,1]$ where the
collocation points of the Gauss–Legendre quadrature lie. It may appear that no
mapping transformation is required to represent this coordinate. However, it
was pointed out Telnov and Chu (2007) that the wave functions, corresponding
to odd orbital angular momentum projections onto the internuclear axis,
exhibit a square-root singularity as $\eta\rightarrow\pm 1$, which may
significantly reduce the accuracy of numerical differentiation in the GPS
method. To circumvent the problem, a special coordinate mapping was suggested
Telnov and Chu (2009), which can be used for any angular momentum projection
without loss of accuracy. Here we also apply this mapping for the coordinate
$\eta$:

$\eta=\sin\left(\frac{\pi}{2}y\right),$ (13)

where $y$ denotes the collocation points of the Gauss-Legendre quadrature.

We performed calculations of the positron supercritical resonances with the
following numerical parameters: $160$ grid points for the coordinate
$\lambda$, $16$ grid points for the coordinate $\eta$, the mapping parameter
$R_{l}=1/Z_{n}$, the complex rotation angle $\Theta=0.3$.

##  4 Results and discussion

As the internuclear distance decreases, the discrete positron vacuum states in
the gap between the lower and upper continua move up. The highest-lying
$1s\sigma_{g}$ states eventually plunge into the upper continuum, if the total
charge of the two nuclei exceeds $173$. If the total charge of the nuclei is
even larger, the next discrete positron vacuum states may also cross into the
upper continuum. For the collisional systems under consideration with the
shortest internuclear distance $R_{\mathrm{min}}=17.5$ fm, this happens for
the Cm96+–Cm96+ system: the $1s\sigma_{g}$ energy level crosses into the upper
continuum at the internuclear distance $R\approx 52$ fm, while the
$2p_{1/2}\sigma_{u}$ reaches the onset of the upper continuum at $R\approx 20$
fm. For the U92+–U92+ system, only the $1s\sigma_{g}$ energy level reaches the
upper continuum at $R\approx 35$ fm, the $2p_{1/2}\sigma_{u}$ level would do
so at $R\approx 11$ fm, but this distance is unreachable since it is smaller
than $R_{\mathrm{min}}$.

Once a discrete positron vacuum state crosses into the upper continuum, it
becomes an unstable resonance state, which can be characterized by a complex
energy (7). Therefore the vacuum may undergo a spontaneous decay with creation
of a free positron. The instantaneous rate of this process is given by
$\Gamma$, twice the absolute value of the imaginary part of the complex
energy, while the positron energy distribution is centered at $E_{r}$, the
real part of the complex energy, and the width of this distribution is equal
to $\Gamma$.

In Fig. 1, we show the width of the $1s\sigma_{g}$ positron supercritical
resonance as a function of the internuclear distance $R$ for the systems
U92+–U92+ and Cm96+–Cm96+. For U92+–U92+, the width $\Gamma$ is identically
equal to zero at $R>35$ fm, and for Cm96+–Cm96+, it is equal to zero at $R>52$
fm. As one can see in Fig. 1, the resonance width remains quite small unless
the resonance energy level plunges deeply into the upper continuum. That is
why the contribution of the $2p_{1/2}\sigma_{u}$ state to the spontaneous
positron creation in Cm96+–Cm96+ can be neglected: this state becomes a
supercritical resonance on a small range of internuclear distances $17.5<R<20$
fm, does not go far in the upper continuum, and its width is always several
orders of magnitude smaller than that of the $1s\sigma_{g}$ state. Our data
for the supercritical $1s\sigma_{g}$ resonance width in the U92+–U92+ system
are in fair agreement with the recent results of Maltsev et al. Maltsev _et
al._ (2020), who used a one-center approach in spherical coordinates and a
uniformly charged ball for the nuclear charge distribution, and with the
earlier work by Marsman and Horbatsch Marsman and Horbatsch (2011). Unlike
Ref. Maltsev _et al._ (2020), we do not need to search for the optimal complex
rotation angle since our results do not depend on this angle within the range
$0.2$ to $0.5$.

The spontaneous vacuum decay in slow collisions of heavy nuclei follows the
exponential law with the instantaneous rate $\Gamma(t)$ at each time moment
$t$ in the supercritical regime, when $\Gamma(t)$ does not vanish. If the
probability $P_{s}$ of spontaneous positron creation is well below unity, it
can be calculated according to the following equation:

$P_{s}=2\int\limits_{-t_{s}}^{t_{s}}dt\ \Gamma(t),$ (14)

where $-t_{s}$ and $t_{s}$ denote the times of entering and leaving the
supercritical regime while the nuclei move along their trajectories. The
factor $2$ on the right-hand side of Eq. (14) accounts for two possible
projections of the positron angular momentum on the internuclear axis: the
$1s\sigma_{g}$ energy level is degenerate and corresponds to two different
states, with the angular momentum projections $\frac{1}{2}$ and $-\frac{1}{2}$
on the internuclear axis. The dependence of $\Gamma$ on time is determined by
the law of motion of the nuclei. The function $\Gamma(R)$ presented in Fig. 1
is universal but the law of motion $R(t)$ is different for different
trajectories. Below we consider head-on collisions in the U92+–U92+ and
Cm96+–Cm96+ systems. We assume the classical motion of the nuclei, so their
trajectories and laws of motion are the solutions of the well-known Rutherford
scattering problem.

In Figs. 2 and 3, we show the position of the supercritical resonance $E_{r}$
and its width $\Gamma$ as functions of time for head-on collisions with
$R_{\mathrm{min}}=17.5$ fm in systems U92+–U92+ and Cm96+–Cm96+, respectively.
The shortest internuclear distance $R_{\mathrm{min}}$ is reached at time
$t=0$. For the resonance position $E_{r}$, the zero energy corresponds to the
onset of the upper positron continuum. The maximum at $t=0$ in the dependence
$E_{r}(t)$ can serve as an estimate of the largest energy of positrons created
by the spontaneous mechanism in the collisions under consideration. For the
U92+–U92+ system, this energy is approximately $270$ keV; for the Cm96+–Cm96+
system, it is equal to $550$ keV. At larger energies, the energy distribution
of positrons resulting from the spontaneous vacuum decay is expected to
decrease rapidly.

It is instructive to compare spontaneous positron creation probabilities
calculated with the formula (14) and total positron creation probabilities
from solving the time-dependent Dirac equation, which can be found for the
same head-on collisions in the U92+–U92+ and Cm96+–Cm96+ systems in recent
papers Popov _et al._ (2020, 2023). We show such a comparison in Table 2. As
one can see, the contribution of the spontaneous mechanism to the total
positron production is already considerable in the U92+–U92+ system (about
27%) and becomes dominant in the Cm96+–Cm96+ system (about 78%). These results
are fully consistent with the previous findings Maltsev _et al._ (2019);
Shabaev _et al._ (2019); Popov _et al._ (2020, 2023), which indicate that the
observation of spontaneous vacuum decay is possible with the implementation of
some impact-sensitive schemes.

##  5 Conclusion

In this paper, we have reported the results of theoretical and computational
study of positron supercritical resonances in systems of two bare nuclei. The
theoretical approach is based on the time-independent Dirac equation for
positrons, and computations are done with the help of the complex-scaling
generalized pseudospectral method in modified prolate spheroidal coordinates.
The generalized pseudospectral method previously proved to be accurate and
efficient in numerous atomic and molecular calculations. The uniform complex
scaling in modified prolate spheroidal coordinates implemented in this work
appears reliable: the complex energy eigenvalues are insensitive to large
variations of the complex rotation angle, at least in the range $0.2$ to
$0.5$.

Numerical data, obtained in our calculations of the positron supercritical
resonances, are applied to estimate the probabilities of spontaneous positron
creation in slow U92+–U92+ and Cm96+–Cm96+ collisions. Our results show that
contribution of the spontaneous mechanism to the total positron production in
head-on collisions with the shortest internuclear separation
$R_{\mathrm{min}}=17.5$ fm is significant in the U92+–U92+ collision and
becomes dominant in the Cm96+–Cm96+ collision. Of course, for nonzero impact
parameters, with increasing $R_{\mathrm{min}}$, the dynamic mechanism of
positron production becomes more and more important. However, we believe that
experimental observation of the fundamental process of spontaneous vacuum
decay in slow collisions of highly charged atomic nuclei is still possible if
some coincidence scheme is implemented and only positrons resulting from
collisions with small impact parameters (and large scattering angles of the
nuclei) are collected and analyzed. The transition from the subcritical to
supercritical regime can be observed along the scenarios proposed in Refs.
Maltsev _et al._ (2019); Shabaev _et al._ (2019); Popov _et al._ (2020, 2023).

###### Acknowledgements.

This work was supported by the Russian Science Foundation (Grant No.
22-62-00004).

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5mm normal

Figure 1: Width of $1s\sigma_{g}$ positron supercritical resonance vs
internuclear distance in U92+–U92+ and Cm96+–Cm96+ systems.

5mm normal

Figure 2: Position (upper panel) and width (lower panel) of $1s\sigma_{g}$
positron supercritical resonance vs time for U92+–U92+ head-on collision with
shortest internuclear distance $R_{\mathrm{min}}=17.5$ fm.

5mm normal

Figure 3: Position (upper panel) and width (lower panel) of $1s\sigma_{g}$
positron supercritical resonance vs time for Cm96+–Cm96+ head-on collision
with shortest internuclear distance $R_{\mathrm{min}}=17.5$ fm.

0mm flushleft
System Orbital Fermi model Gaussian model U${}_{2}^{183+}$ $1s\sigma_{g}$
$-9956.962$ $-9956.948$ $2p_{1/2}\sigma_{u}$ $-7187.069$ $-7187.046$
Cm${}_{2}^{191+}$ $1s\sigma_{g}$ $-10916.92$ $-10916.93$ $2p_{1/2}\sigma_{u}$
$-8000.568$ $-8000.576$

Table 1: $1s\sigma_{g}$ and $2p_{1/2}\sigma_{u}$ electron energies of
U${}_{2}^{183+}$ and Cm${}_{2}^{191+}$ quasimolecules at $R=2/Z_{n}$ (a.u.).
Zero energy is at the onset of the electron upper continuum.

0mm flushleft
System $P_{s}$ $P_{t}$ $P_{s}/P_{t}$ U92+–U92+ $0.00326$ $0.0120$ $0.27$
Cm96+–Cm96+ $0.0333$ $0.0425$ $0.78$

Table 2: Spontaneous positron creation probability $P_{s}$ due to
supercritical resonance $1s\sigma_{g}$ and total positron creation probability
$P_{t}$ Popov _et al._ (2023) for head-on collisions in U92+–U92+ and
Cm96+–Cm96+ systems with $R_{\mathrm{min}}=17.5$ fm.