Let us try to interpret it. We shall first use the fact that $\|A\|_{\infty}$,
the operator norm of $A$, is the largest possible value of $\|Au\|_{2}$ over
all unit vectors $u$, which in turn is the largest possible value of $\langle
Au,v\rangle$ over all pairs of unit vectors $u$ and $v$. Therefore, we can
find unit vectors $u$ and $v$ such that

$\langle(\mathbb{E}_{x}f(x)\otimes\overline{\rho(x)})u,v\rangle\geq c^{1/2}.$

Now let us rewrite this in coordinate form. Because of the special form of the
$n\times n_{\rho}$ matrix $f(x)\otimes\overline{\rho(x)}$ it is natural to
give it four indices instead of two: we have that
$(f(x)\otimes\overline{\rho(x)})_{ijkl}=f(x)_{ik}\overline{\rho(x)_{jl}}$.
Then indexing $u$ and $v$ in the corresponding way, and writing them as $U$
and $V$ since they have now become matrices, we have that

$\langle(f(x)\otimes\overline{\rho(x)})U,V\rangle=\sum_{i,j,k,l}f(x)_{ik}\overline{\rho(x)_{jl}}U_{kl}\overline{V_{ij}}=\langle
f(x)U\rho(x)^{*},V\rangle,$

where the product in the last expression is just normal matrix multiplication.
Taking expectations, we deduce that

$\mathbb{E}_{x}\langle f(x)U\rho(x)^{*},V\rangle\geq c^{1/2}$

for two $n\times n_{\rho}$ matrices $U$ and $V$ that have Hilbert-Schmidt norm
1. We can write this more symmetrically as

$\mathbb{E}_{x}\langle f(x)U,V\rho(x)\rangle\geq c^{1/2}.$

It is natural to rescale $U$ and $V$ so that they have Hilbert-Schmidt norm
$n_{\rho}^{1/2}$. That is, we can say that there exist an irreducible
representation $\rho$ and matrices $U$ and $V$ with
$\|U\|_{2}^{2}=\|V\|_{2}^{2}=n_{\rho}$ such that

$\mathbb{E}_{x}\langle f(x)U\rho(x)^{*},V\rangle=\mathbb{E}_{x}\langle
f(x)U,V\rho(x)\rangle=\mathbb{E}_{x}\langle f(x),V\rho(x)U^{*}\rangle\geq
c^{1/2}n_{\rho}.$

This seems quite satisfactory, but it falls short of being a true inverse
theorem for the matrix $U^{2}$ norm because the converse does not hold. That
is, if we are given $\rho,U$ and $V$ satisfying the above conditions, we
cannot deduce that $f$ has a large $U^{2}$ norm unless $n_{\rho}$ is
comparable to $n$, which it does not have to be.

Thus, we are in an interesting situation. Earlier, we thought of inverse
theorems as something one settles for when one does not have an inversion
formula. But here we have a clean and easily proved inversion formula that
does not directly yield an inverse theorem.

However, we have not yet exhausted our options. If
$\|f\|_{U^{2}}^{4}=\|\hat{f}\|_{4}^{4}\geq cn$, then we are given that

$\sum_{\rho}n_{\rho}\|\hat{f}(\rho)\|_{4}^{4}\geq cn,$

where $\|\hat{f}(\rho)\|_{4}^{4}$ denotes the sum of the fourth powers of the
singular values of $\hat{f}(\rho)$. Let these singular values be
$\lambda_{\rho,i}$ for $i=1,2,\dots,n_{\rho}$. Then we find that

$\sum_{\rho}n_{\rho}\sum_{i=1}^{n_{\rho}}\lambda_{\rho,i}^{4}\geq cn.$

From Parseval’s inequality and the assumption that each $f(x)$ has operator
norm at most 1 (and hence Hilbert-Schmidt norm at most $n$) we also have that

$\sum_{\rho}n_{\rho}\sum_{i=1}^{n_{\rho}}\lambda_{\rho,i}^{2}\leq n.$

Also, since $\hat{f}(\rho)=\mathbb{E}_{x}f(x)\otimes\overline{\rho(x)}$ is an
average of matrices with operator norm at most 1, every $\lambda_{\rho,i}$ is
at most 1.

Let $\lambda_{1},\dots,\lambda_{m}$ be the singular values $\lambda_{\rho,i}$
arranged in some order, and for each $i$ let $n_{i}$ be the $n_{\rho}$ that
corresponds to $\lambda_{i}$. Then we can rewrite these inequalities as

$\sum_{i}n_{i}\lambda_{i}^{4}\geq cn,$ $\sum_{i}n_{i}\lambda_{i}^{2}\leq n,$

and

$\lambda_{i}\leq 1.$

Note that if $c=1$, then the only way of achieving the above inequalities is
for $\lambda_{i}^{4}$ to equal $\lambda_{i}^{2}$ for every $i$ (assuming that
none of the $n_{i}$ is zero). Thus, each $\lambda_{i}$ is either 0 or 1, and
$\sum\\{n_{i}:\lambda_{i}=1\\}=n$. It is a straightforward exercise to prove
that the more relaxed assumptions above lead to similar but more relaxed
conclusion: we can find a set $A$ and constants $c_{1}>0$ and $C$ that depend
on $c$ only (with a power dependence) such that $c_{1}n\leq\sum_{i\in
A}n_{i}\leq Cn$, and $\lambda_{i}\geq c_{1}$ for every $i\in A$. In short, we
can find a set of large singular values (coming from the various
$\hat{f}(\rho)$ of size roughly comparable to $n$.

With each such singular value $\lambda_{i}$ we can associate $n\times n_{i}$
matrices $U_{i}$ and $V_{i}$ with Hilbert-Schmidt norm $n_{i}$ such that
$\mathbb{E}_{x}\langle f(x)U_{i},V_{i}\rho_{i}\rangle\geq\lambda_{i}n_{i}$,
where $\rho_{i}$ is the representation corresponding to $\lambda_{i}$.
Moreover, if two pairs $(U_{i},V_{i})$ and $(U_{j},V_{j})$ come from the same
$\rho$, then because of the nature of singular value decompositions, we have
that $\langle U_{i},U_{j}\rangle=\langle V_{i},V_{j}\rangle=0$.

It is plausible that we can put together these matrices and irreducible
representations to create a representation-like function that correlates with
$f$, and moreover that gives us an inverse theorem in the sense that the
correlation in its turn implies that $f$ has a large $U^{2}$ norm. Exactly how
the putting together should work is not obvious, but it turns out that it can
be done. It yields the following theorem, due to the author and Omid Hatami
[22]. In the statement, recall that $\|f\|_{\infty}$ means the largest
operator norm of any $f(x)$. Also, we define a _partial unitary_ matrix to be
an $n\times m$ matrix such that the rows are orthonormal if $n\leq m$ and the
columns are orthonormal if $n\geq m$. (In particular, if $n=m$ then the matrix
is unitary.)

######  Theorem 15.1.

Let $G$ be a finite group, let $c>0$ and let $f:G\to$M${}_{n}(\mathbb{C})$ be
a function such that $\|f\|_{\infty}\leq 1$ and $\|f\|_{U^{2}}^{4}\geq cn$.
Then there exists $m$ such that $cn/4\leq m\leq 4n/c$, an $m$-dimensional
representation $\sigma$, and $n\times m$ partial unitary matrices $U$ and $V$,
such that

$\mathbb{E}_{x}\langle f(x)U,V\sigma(x)\rangle\geq c^{2}m/16.$

Note that $\sigma$ will _not_ normally be irreducible. This theorem tells us
that $f$ correlates with the function $V\sigma U^{*}$. The extra strength of
this theorem over what we remarked earlier is that the dimension of $\sigma$
is comparable to that of $f$. It turns out to be simple to deduce a converse
statement – i.e., that if $f$ correlates with a function of the above form,
then $\|f\|_{U^{2}}^{4}\geq c^{\prime}n$ for some suitable $c^{\prime}$.
Therefore, Theorem 15.1 is indeed an inverse theorem for the matrix $U^{2}$
norm.

It is possible to give a more careful argument when $c=1-\epsilon$ for some
small $\epsilon$ that allows us to show that $(1-2\epsilon)n\leq
m\leq(1-4\epsilon)^{-1}n$, and to obtain a lower bound of $(1-16\epsilon)m$ in
the last inequality. From this result it is not too hard to deduce a so-called
_stability theorem_ for near representations. Roughly speaking, it states that
any unitary-valued function that almost obeys the condition to be a
representation is close to a unitary representation.

######  Theorem 15.2.

Let $G$ be a finite group and let $f:G\to U(n)$ be a function such that
$\|f(x)f(y)-f(xy)\|_{HS}\leq\epsilon\sqrt{n}$ for every $x,y\in G$. Then there
exist $m$ with $(1-\epsilon^{2})n\leq m\leq(1-2\epsilon^{2})^{-1}n$, an
$n\times m$ partial unitary matrix $U$, and a unitary representation
$\rho:G\to U(m)$, such that

$\|f(x)-U\rho(x)U^{*}\|\leq 31\epsilon\sqrt{n}$

for every $x\in G$.

When $\epsilon$ is bounded above by $cn^{-1/2}$ for a suitable constant $c$,
the inequality for $m$ forces $m$ to equal $n$. In this regime the result was
known and is due to Grove, Karcher and Ruh [32]. They also proved a stability
result, this time with no restriction on $\epsilon$, with the operator norm
replacing the normalized Hilbert-Schmidt norm [33] (see also [40]).

##  16\. Conclusion

Now that we have seen several different generalizations of Fourier analysis
(though not a complete list), we can draw up a checklist of the properties
that a generalization is likely to need in order to be useful. Ideally we
would have all of the following.

  * •

A Parseval identity

  * •

A convolution identity

  * •

An inversion formula

  * •

A quasirandomness-measuring norm

  * •

An inverse theorem for the quasirandomness-measuring norm

Sometimes we can indeed get all of these, but in situations where we can’t, it
turns out that just having the last two properties is sufficient for some very
interesting applications. In several of these situations, it remains a
fascinating challenge to find new versions of the generalizations with
improved properties.

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# A minimalist account of agency in physics

Ali Barzegar<EMAIL_ADDRESS>Arnold Sommerfeld Center for Theoretical
Physics & Munich Center for Mathematical Philosophy, Ludwig-Maximilians-
Universität München, Germany, EU Emilia Margoni<EMAIL_ADDRESS>Università degli Studi di Firenze/Université de Genève/Università di Pisa
Daniele Oriti<EMAIL_ADDRESS>Arnold Sommerfeld Center for
Theoretical Physics & Munich Center for Mathematical Philosophy, Ludwig-
Maximilians-Universität München, Germany, EU

###### Abstract

We adopt a top-down approach to agency aimed at developing a minimalist,
scalable and naturalized account of it. After providing a general definition,
we explore some possible extensions and refinements, domain of applicability,
as well as a comparison with other recent accounts of agency, and possible
objections to our proposal. With respect to what we classify as strong (such
as Tononi’s) and weak (such as Rovelli’s) characterizations, our notion of
agency situates itself in a middle position – our intent being precisely that
of spelling out the advantages of this median account within a variety of
contexts, such as the interpretation of quantum mechanics, the debate on the
nature of physical laws and bayesianism.

Keywords: _agency, models, representation, naturalism, quantum mechanics,
physical laws_

##  1 Introduction

There is a sense in which agency is everywhere to be found: a river gradually
eroding the rocks that mark its banks, squid ink being released to seed a
predator, a child knocking over a vase while crawling, thus soiling the
carpet. Every time systems interact, they display causal relationships that
result in reciprocal change. However, the notion of agency – which has
prospered throughout the last few decades in a variety of disciplines and
research areas, such as psychology, anthropology, philosophy, and cognitive
neuroscience, as well as physics – is generally tied to a thinner connotation,
whereby an agent gets identified with an entity that is able to act in
specific ways with respect to its surrounding environment – the notion of
agency marking the exertion of this specific function.

While traditionally the ability to act is related to the performance of
intentional faculties – two leading figures of such a characterization of
agency being, within contemporary analytic philosophy, Elizabeth Anscombe
(1957) and Donald Davidson (1963) – more recent debates have centred on
rebutting such a restricted view. The underlying assumption is that, whenever
attention is drawn to entities simpler than human beings, the traditional view
proves too demanding, to such an extent that the notion of agency should be
detached from that of representational and intentional states (see, e.g.,
Silberstein & Chemero, 2011; Hutto & Myin, 2014).

As no agreed-upon definition is available in the literature, speaking about
agency means invoking a variety of kinds and contexts. Some adopted notions
are mental, shared, collective, relational, and artificial agency111For a
thorough discussion about the various characterizations of agency, see
Schlosser (2019)., where in all these declinations the invoked notion can be
more or less conforming to the traditional (intentional) meaning. For
instance, while shared agency refers to the exertion of action by two or more
individuals and collective agency occurs when two or more individuals act as a
group, it is still debatable whether the resulting action should be reduced to
the agency of the single involved individuals, or whether this shared or
collective faculty constitutes something on top of individual agency, thus
opening the question if, within the traditional paradigm, group of individuals
should be accorded intentional or mental states.

All this comes down to the issue of whether there is a way to provide a basic
characterization of agency. Recently, Barandiaran et al. (2009) have worked on
what they dub “minimal agency”, a notion that does not require the possession
of representational or mental states. According to them, agency is defined
whenever a system satisfies three conditions, namely that: a) it establishes
its own individuality; b) it displays an interactional asymmetry with respect
to its surrounding environment; and c) the interaction between the system and
the environment is based on certain normative parameters. On this perspective,
organisms such as bacteria meet the minimal criterion of agency, thus
enlarging the eligible entities with respect to more traditional accounts.

Another recent proposal – which embraces a computer science perspective – is
Biehl, Ikegami & Polani (2016), which advances an “information theoretic
notion of integrated spatiotemporal patterns” as “the basic building block of
an agent definition” (2016, p. 1). Their strategy is the following: 1. define
spatiotemporal patterns for both living, such as bacteria and animals, and
non-living, such as rocks and crystals, persisting systems; 2. establish some
criteria, namely perception, action and goal-directedness, to catalogue those
spatiotemporal patterns; 3. identify, among the total set of persisting
systems, those that meet the agency criterion.

Though intriguing, more flexible and broader as compared to traditional
accounts, both these proposals still refer to notions such as perception,
action, and intentionality mediated through normative parameters. It follows
that the associated accounts of agency can hardly be considered a fully
naturalized, let alone minimalist, characterization of it.

The intent of the present paper is to propose such a minimalist, scalable and
naturalized characterization of agency, based on a minimal notion of
representation, modelling and normativity. From this minimalist basis,
supplementary features can be added which correspond to higher-order levels of
complexity. This account will be based on the exertion of modelling capacities
that a certain system displays with respect to the surrounding environment,
where even the notion of environment will be scrutinized. On this view,
modelling is the hallmark of agency: the more complex the model, the more
complex the agent. By relying on a minimalist characterization of modelling,
our account situates itself between the two extremes of depicting an agent as
every possible interacting system – what we dub a _weak_ account of agency,
such as Rovelli’s (2020) – or of taking only a very narrow set of systems as
agents in virtue of the possession of some very specific features – what we
dub a _strong_ account of agency, such as Tononi’s (2015) and collaborators’
(Albantakis et al. 2020) requirement that an agent should display some minimal
grade of consciousness.

The paper is structured as follows. In the section 2, we provide our
definition of agency and discuss its main features. The 3rd section expands on
its domain of applicability, as well as further extension and refinements. In
the 4th section, a comparison between our account and other recent proposals
is outlined. The 5th section addresses some objections that can be raised
against our proposal. Finally, we draw some concluding remarks and provide
further lines of inquiry.

##  2 A minimalist definition of agency

What is it that we are talking about when making reference to agents? Though
currently available literature shows a proliferation of very different
characterizations, an agent can be generally identified with a system or
entity that interacts – in a way that needs to be further specified – with the
surrounding environment (see, e.g., Ried et al., 2019). Zooming in from this
coarse-grained understanding, we need to spell out more thoroughly which
subset of systems or entities actually count as agents. That is, if an agent
is a system situated within an environment, interacting with and acquiring
information about it, the question becomes what is the hallmark of agency
_vis-à-vis_ generic interactions. Here is our tentative definition:

_An agent is an information-processing system which is characterized by
modelling activity with respect to its surrounding environment, geared towards
successive interaction.
_

Let us emphasize some preliminary aspects of this definition222For this
definition to be relevant for the understanding of scientific epistemology and
more generally scientific activity, it should allow an agent to be able to
model at least some of its own (scientific) activity (E. Curiel, private
communication). In more detail (E. Curiel, 2020), the empirical content of
scientific propositions is grounded in an agent’s capacity to schematize the
observer in the models of experiments, where the notions of observer and
experiment should be conceived broadly. However, our point, as will be
discussed afterwards, is that, moving from this minimal characterization, a
hierarchy of ‘agency level’ can be defined, Curiel’s requirements being
eventually realized from a certain level onward.:

  1. 1.

First of all, note that this definition is a functional one: any system
exhibiting the required properties can be qualified as an agent, our account
being extremely neutral with respect to the nature of the eligible entities.
It thus becomes an empirical question to determine which systems satisfy this
definition of agency, and which do not.

  2. 2.

Second, it is a minimal definition of agency: an agent is an entity that
organizes the information from the environment following the interaction with
it – where this organization is what we identify with the agent’s modelling
capacity, which in turn is based on a minimal notion of representation. Every
other feature of agency can be added to this minimal level as a higher-order
level of complexity. An agent is an information-processing system, and part of
this processing activity is modelling. The latter provides a basis for
successive interactions with the environment, thus implying a minimal notion
of normativity, in the basic sense that the modelling activity informs (it is
used in) successive interactions. On this approach, one can envisage a
spectrum of agency in terms of the way in which the latter reacts to external
inputs – with the two extremes of “simple reflexes to environmental stimuli”
(Briegel, 2012, p. 2), on the one hand, and the “projected simulation […] into
conceivable future situations” (Briegel, 2012, p. 4), on the other.
Importantly, both an extremely complex system not interacting with the
surrounding environment and a system interacting but devoid of the appropriate
internal structure would not meet our criteria for agency.

  3. 3.

Third, our definition of agency is meant to be sufficiently scalable and
malleable. This means that it occupies a middle position between paradigms
defending a strong notion of agency (such as the one which takes the property
of, say, having consciousness to be a necessary condition for speaking about
agents) and those paradigms envisaging a thin notion of agency (such as the
one according to which every interacting system is an agent). Moving from this
minimal, but intermediate standpoint, it is then possible to add features of
agency that may correspond to stronger notions, or to remove those that would
give us back the thinner ones (although in this case, we would argue, we are
not considering proper agency anymore).

After this preliminary analysis, it is necessary to unpack the underlying
assumptions of our proposed definition. In particular, it is worth exploring
the role that modelling and representation play, as well as the physical
requirements necessary for their exertion.

###  2.1 A minimalist account of modelling

Our account associates the capacity of a certain system to model the
environment with agency. A minimal notion of agents as entities capable of
modelling the environment can be articulated further in the following way. The
agent receives data from its interaction with the environment. 333
Schematically, one could interpret these interactions as measurements by the
agent on the environment, thereby acquiring the measurement results as data;
however, while any measurement is an interaction, not every interaction counts
as a measurement, and we are interested in understanding the specific
conditions under which a certain interaction can be taken as providing a
measurement result. From this perspective, for an agent to be able to model
its environment, it should display the capacity to model these data, thereby
providing some structure to them. That is, the agent should be able to build
_data models_ , which corresponds to the minimal level of modelling activity.
Minimality is guaranteed by a categorization scheme of the information
gathered from the surrounding environment – this capacity standing for the
fact that an agent is able to give some structure to such information.444For a
thorough discussion on the relationship between data and models of data see,
e.g., Leonelli (2019).

What, then, is a data model? Formally, a data model can be defined in set-
theoretic 555See, among others, Hegner and Maulucci (1978), Yannakoudakis
(2007). or category-theoretic terms 666See, e.g., Zhao (2019). as a set of
elements or categories with relations holding between them. Based on the
number of elements, relations, and degrees of freedom, one can then define a
measure of complexity for different data models. For instance, one can start
evaluating the complexity of the data model by counting the number of involved
elements. Successively, one can consider binary relations, then binary plus
tertiary relations, and so on, thus adding levels of complexity.

Let us turn to a more concrete analysis of data models. Obviously, they are
models of _data_. However, it should be noted that the notion of data, and
especially _raw data_ , is an utterly ambiguous one. This is related to the
old debate in methodology on the theory-ladenness of observations and
empirical facts. 777As a particularly instructive discussion on the topic, see
Ian Hacking’s (2002; 2009). Furthermore, there is no one-to-one correspondence
between data sets and data models. There are always competing patterns within
data, which implies the adoption of criteria to extract data models. At a
first level of analysis, what you see in a data model is a minimal level of
filtering with respect to raw data or in the patterns among data in which not
all the data are included, as some of them may not be useful and/or reliable.
At a higher level, agents use data models in a future-oriented manner. For
example, the choice of a data model can be articulated in terms of some
subtler function within the agent engaged in the modeling activity such as its
goal-orientedness. From this perspective, data models become goal-oriented
organizations of data, to such an extent that different degrees of goal-
orientedness correspond to different degrees of complexity of the involved
agent. At more refined levels of organization, they use data models in
counterfactual situations, to predict and respond to possible situations
occurring in the future. Within these counterfactual scenarios, and in
contrast to data models of agents at lower levels of complexity, these data
models and successive elaborations are hypothetical instead of actual.

In light of what has been discussed so far, an interesting way to elucidate
the difference between data models of agents characterized by different levels
of complexity is by relying on the relational understanding of data models,
according to which they intertwine and so are defined in relation to other
models the agent constructs and uses (Leonelli, 2016). From this perspective,
there is a feedback loop between different models the agent builds and makes
use of. And these other models available to the agent influence the nature and
complexity of the data models it is able to construct in the first place. As a
consequence of this, by comparing the data models themselves one should be
able to distinguish the corresponding agents. This is due to the fact that the
more developed agent making use of other models, over and above data models,
will construct a more complex data model out of the same data set _vis-à-vis_
an agent who just can build and use them. In other words, a difference at the
level of data models signals a difference at the level of agency. Moreover, by
relying on the relational understanding of models, we can also argue that at a
specific level of complexity the model allows for Bayesian probabilistic
inferences.888This is a topic that will be further investigated in the
following sections.

Following Briegel’s (2012), we can provisionally define a first taxonomy of
agency, roughly corresponding to three levels of complexity by virtue of the
associated modeling activity. At the minimal level, modeling is any kind of
structuring or categorization of the data received from the interaction with
the environment. Then there is a second level of complexity, which is
associated to the existence of feedback loops in the internal information-
processing activity of the agent, thus among its constructed (data) models. At
a higher level, the agent becomes a self-referential system and provides
feedback on its own activity, such as the history of data patterns. When a new
input is received from the environment, the agent reacts to it also by
considering the stored patterns, not just the currently available ones. At
this level, a certain pattern is selected based on the weights the agent
assigns to it, thus resulting in a specific activity. This third level of
complexity can be further expanded if the agent can not only store
information, but also elaborate evidence-based future models. In this case,
the agent can construct counterfactual scenarios and take advantage of them to
simulate or predict its successive interaction with the environment. The more
complex the hypothetical patterns, the more complex the agent.

###  2.2 Physical requirements for modelling activity

So far we have focused our attention on the role played by data models in our
minimalist account of agency, and in providing a possible taxonomy of agents,
based on their ability to process data collected from the environment. Now we
need to spell out what physical requirements enable the production of data
models in the first place. According to our definition, information-gathering
is a necessary condition for a system to count as an agent. However, not every
information-storing system counts as an agent. Over and above gathering and
storing information, an agent organizes and then uses this information for
modelling and then successively interacting with the environment (where,
again, a minimal notion of modelling is here intended as a categorization of
the inputs from the environment). It is then necessary to discriminate between
simply acquiring information about the environment and modelling it. The
latter activity is not measured by the sheer amount of information one
acquires, say by relative information (Rovelli, 2016).

Consequently, there should be a physical basis to sustain this activity of
organization and structure with respect to the acquired information. Since a
minimal form of complexity is required, an agent cannot correspond to an
elementary system, namely a system without parts. Importantly, when we talk of
complexity as a requirement for agency, we do not simply mean that more than
one degree of freedom is involved (though this is a necessary condition). We
are also referring to the peculiar way in which the modelling activity arises
out of the interaction among the involved degrees of freedom. For instance,
Albantakis et al. (2020) argue that one leading aspect of an agent is the fact
that the latter displays a form of irreducibility with respect to its parts,
to such an extent that the integration and relation between its parts
constitute a unitary whole, as “all of its subsets constrain and are being
constrained by other subsets within the system above a background of external
influences” (p. 2). 999For further discussion on this topic see, among others,
Maturana and Varela (1980), Hoel, Albantakis and Tononi (2013) and Marshall et
al. (2017). We concur with this perspective, and argue that (in particular,
higher-level) agents are something over and above the sum of their component
parts, to such an extent that the agent needs something like a central
processing system, based on which subsystems are not independent from one
another; rather, there is some internal interaction between its parts.

To summarize, the type of complexity required to define an agent consists of
two elements: a number of degrees of freedom and some internal, specific
interaction among them. Again, the more structured such internal interaction,
the more complex the associated agent. While for Tononi (2012) this internal
interaction is what gives rise to integrated information systems (IIS) that
are irreducible to their component parts, the _minimal_ modelling activity
here invoked requires a minimal form of organization or interaction within the
system. At higher levels of agency, this discourse translates into the
possible irreducibility of higher-order models into lower-level ones, thus
calling for a hierarchy of models with higher-level ones being irreducible to
the lower-level ones: each and every level of this hierarchy can be regarded
as corresponding to a form of agency, characterized by a specific level of
complexity in terms of internal structure and modelling activity.

To give a concrete example, according to this characterization, an elementary
particle cannot display agency because, by definition, it has no parts.
However, a system of, say, four electrons can manifest a certain structure
which enables it to model the environment.101010Actually, within our account,
as will be discussed later on, even a single electron, provided that it can
categorize the surrounding environment with respect to its two subsets of
degrees of freedom (position and spin), could in principle be taken as an
agent. Then there should be a non-trivial difference between an elementary and
a complex system – a sort of ‘agency threshold’– after which complexity comes
in degrees.

Note that our account is eminently perspectival: an atom would qualify as an
agent insofar as it is regarded as a composition of elementary particles, but
if we take it as an individual, then it would correspond to a singular system,
to which we would not attribute agency. Therefore, our notion of agency is
scalable, whereby there is no need to have a notion of a system which is not
theory-laden. Within this context, perspectivalism boils down to the fact that
the attribution of agency depends on the perspective (such as it is provided
itself by a model or a theory) according to which systems are classified as
elementary or complex. Whatever, according to that perspective, is defined as
elementary cannot produce structured information, in that there is no physical
structure to support it. Still, being qualified as a complex system does not
automatically imply agency: having parts is not sufficient for displaying
structured information (with the notion of structured information giving the
minimal notion of modelling the environment by the agent). As an instructive
example, consider the case of ants. The latter can be regarded as agents, but
so does, in the appropriate context, the anthill, thus leading to a scale-
dependent account of agency. In other words, depending on what you are looking
at, you get a different notion of agent, which is precisely in this sense
perspectival. Importantly, this perspectival account of agency also means that
an agent is always regarded as a sub-system within a larger system: an agent
cannot be completely separate from the environment.

###  2.3 Agency and Model Representation

According to our definition, agency is to a large extent modelling activity.
This requires us to discuss in more detail how models are meant to represent.
Frigg and Nguyen (2020) argues that an account of model-representation needs
to address five topics. First of all, since a model represents a specific part
or aspect of the world, namely its _target system_ , one must justify in
virtue of what a certain model is able to represent it. This implies answering
the _Epistemic Representation Problem_ , namely to explain why a certain model
(or graph, diagram, drawing) represents a certain target system. Second, as
models are not the only epistemic devices, a model-representation should be
analyzed within the larger category of epistemic representation, so much so
that one needs to clarify what is the specificity of (scientific) model
representation _vis-à-vis_ other kinds of epistemic representation – this is
the _Representational Demarcation Problem_. Third, model-representation should
clarify what styles are there and in which way they can be characterized – the
so-called _Problem of Style_. Fourth, it is necessary to define some
_Standards of Accuracy_ , namely when it is possible to state that a certain
model or epistemic device constitutes an accurate representation of a specific
target system under investigation. Finally, a model-representation should
address the _Problem of Ontology_ , namely to clarify what type of object
models are.

Obviously, the debate on the nature of representation, and the way in which
models are connected to representation is extremely vast. Indeed, there are
very different ways models are said to represent.111111For an introductory
discussion on the topics of models and the relationship they bear with
representation see Frigg and Hartmann (2020), Frigg and Nguyen (2021). Other
interesting contributions are Morrison (1999), Isaac (2013), Levy (2015),
Massimi (2018). In the present context, we are interested in detailing the
relationship between modelling activity and agency. To cut a long story short,
here are some of the most accredited characterizations of how models are
assumed to represent. The first is that models represent by way of similarity
between the model and the target system (see, e.g., Cartwright, 1983; Giere,
1988, 2010; Weisberg, 2012). The second draws from a structural relation of
isomorphism between the model and the target system (see, among others,
Bokulich (2011), and King (2016) for a critical discussion). But as similarity
and isomorphism are symmetric properties – whereas representation is founded
on directionality – there is an issue connected to the multiple realizability
as typically associated with these features. Anything can be similar to
anything else in different ways. As different descriptions give different
structures, an underdetermination of the structure from the target system is
at stake. The third view of representation is based on the idea that the model
should allow for inferences to be drawn about the target system (see, e.g.,
Verreault-Julien, 2021). However, for these inferences to be useful there
should be a link between them and the target system. That is, the
representation relation between the model and its target system should allow
for inferences which are about future actions. Otherwise, one is ‘trapped
inside the model’, and cannot ensure that the conclusions inferred from the
model are applicable to the target system. If the agent derives conclusions
about the target system just at the level of propositions, this would imply
the presence of a closed system, and then the only task of the agent becomes
to check the coherence of this set of propositions. One can indeed
conceptualize a model as a set of propositions linked by logical connectors
and then try to check its coherence.

Keeping this in the background, it is important to outline that the only way
to break out of the model is via some empirical connection between it and its
target. Therefore, action stands for the representational link between the
model and the target system. It is only when we take into consideration the
action of the agent building and using the model that we go beyond any kind of
abstract inference. Thus, we have to invoke the agent to be able to say that a
model represents a target system. We believe this is the way to go to bring in
the directionality of model representation. On the one hand, the agent builds
and uses the model to successively interact with the environment. On the other
hand, to the extent that the model helps the agent to successively interact
with the environment, it can be said to provide a directed and/or purposeful
representation of the environment. This is an important point to emphasize:
agency requires models to be defined; they are part of its very definition. On
the other hand, agency feeds back on the notion of models. However, this does
not lead to any circularity in our argument, because this feedback takes place
at the level of the representational function of models, rather than at the
level of their definition: it determines their representational capacities.

##  3 Further refinements and possible applications

To summarize what has been discussed so far, our account is meant to capture
the bare minimum notion of agency, which is in turn based on the bare minimum
of modelling activity, such as a simple categorization of external data. From
this perspective, any system with more than one degree of freedom can engage,
in principle, in modelling activity, thus counting as an agent, provided the
remaining ingredients in our definition (such as its use of the constructed
model to inform successive interactions with the environment) are also
present. For example, an electron with spin and position can categorize its
interactions with the surrounding environment based on these two degrees of
freedom, and would therefore count as an agent. Two electrons likewise count
as an agent. But, as there is something counterintuitive in regarding two
electrons – and even more so a single electron – as agents, perhaps some
explication is required.

First of all, it is important to clarify whether we need to introduce a
further classification on top of the proposed definition of agency, or even
make the same definition more restrictive. In the first case, one should
simply offer a further classification of different levels of agency, based on
different levels of (complexity of) modeling activity. In the second case, we
should raise the minimal threshold for agency, under which this notion cannot
be applied fruitfully, above what we have identified already. We have
explained our minimal notion already excludes (contra some views in the
literature according to which any interacting physical system can be qualified
as an agent) a number of physical systems to be considered as agents . As we
have argued, these minimal ingredients concern the presence of more than one
degree of freedom, internal structure for acquired information, as well as
some correlation between previous information and successive evolution. What
else can be required?

An interesting (and already mentioned) reference is Barandiaran et al’s
(2009), according to which agency should be taken as an “autonomous
organization that adaptively regulates its coupling with its environment and
contributes to sustaining itself as a consequence”(p. 367). They base their
definition on the following necessary conditions:

  1. 1.

A system must define its own _individuality_

  2. 2.

It must be the active source of activity within its environment
(_interactional asymmetry_)

  3. 3.

It must regulate this activity in relation to certain norms (_normativity_)

This definition is very close to ours. With respect to these conditions, we
have seen that our account makes reference to a minimalist notion of
normativity, fixed by the fact that the stored information is then used by the
agent to successively interact with the environment. However, their concept of
individuality is closely tied to the notion of autonomy – where an autonomous
system is one whose components or parts form an irreducible, interrelated
network. By properly tuning their analysis to the purpose of our account, a
three-level distinction comprising all physical systems can be defined. The
hierarchy would then comprise 1. systems 2. autonomous systems 3. agents. In
this way, moving from their definition, our account would correspond to the
third level of the proposed taxonomy. From a certain standpoint, their
definition applies at the second level of the taxonomy, to such an extent that
we can use their definition of agency as a criterion for selecting systems to
which our definition applies. Our account would then correspond to a subclass
of their definition. However, their notion of normativity is stronger than
ours, in this way designating a less minimalist account of agency with respect
to ours.

We now indicate three contexts in which such an account of agency may be
fruitfully applied:

  1. 1.

_Agency in the context of Quantum Mechanics (QM)_ : A possible way to put our
conceptual strategy to the test is to offer a taxonomy of the various
epistemic interpretations based on the underlying characterization of agency,
developing further the analysis in (Barzegar & Oriti, 2022). In this context,
the point is to understand and characterize more precisely what is meant by
the relational nature of quantum states. For instance, if one confines them to
be dependent on a classical measurement context, no agency is involved in
defining such context, and then Bohr’s interpretation is the one that fits
best. If an agent is taken as an observer with a stronger form of agency
(possessing personal experiences), then QBism represents the corresponding
interpretative framework. If we weaken the status of the agent, naturalize it
to identify an ‘observing system’ that can be a suitably complex physical
system, accounting also for further elements of the context of interaction
with the quantum system, and embrace a form of pragmatism, we get closer to
interpretations _à la_ Healey or _à la_ Brukner-Zeilinger. Finally, if we
broaden the notion of agency and ‘observer’ to such an extent of including
every possible interacting physical system, then you get Rovelli’s relational
quantum mechanics. The latter would seem to be excluded by our minimalist
definition of agency, if any agency is actually required for a relational view
of quantum states. Our analysis of agency, and further refinements of the
same, therefore, will correspond to an analysis of the differences between all
these interpretations of quantum mechanics.

  2. 2.

_Agency in the context of the debate about the nature of laws_ : developing a
more detailed account of agency might offer fruitful insights on the nature of
laws. The idea would be to promote an account of laws in which the epistemic
components are defining features of the laws themselves. Starting from the
Better-Best System Account (BBSA) developed by Cohen and Callender (2009)
121212On this account, laws of nature are the axioms or theorems of the system
achieving the best balance between simplicity and strength. However, it seems
that simplicity and strength can only be evaluated based on epistemic features
of the cognitive agents, and their interests. The BBSA explicitly acknowledges
this fact. Laws are constructed by agents to be used by agents. Lawhood should
be epistemically accessible. So, BBSA clearly makes laws dependent on agents
and their epistemic features., the goal would be to deny the (tenability or
usefulness of the) distinction between an objective reality and the content of
a theory or set of theories. On this perspective, laws are produced in a
perspectival manner, to such an extent that models and theories would not be
connoted in terms of truthfulness or correspondence, rather in terms of
similarity. Perception itself would be a kind of modelling, and not a direct
representation of the world (in the sense of isomorphic correspondence). The
agent has thus an irreducible role in constructing reality, including its
‘laws’. An agent can only interact in a specific way with the world, thus
constructing a model of the latter from its own perspective. By taking stock
of Putnam’s analysis on the notion of internal realism (Putnamn, 1976), it is
possible to assert that accessing the world outside a model or a theory is not
possible: every access is from within. But a precise epistemic account of
laws, in this direction, requires a more precise account of agency, with
particular attention to cognitive/epistemic agents as a subclass within our
minimalist account.

  3. 3.

_Agency and Bayesian probabilities_ : There are several issues to be
considered in this context, the general question being what characterizes a
Bayesian agent, i.e. an entity that uses Bayesian probabilities in its
interactions with the environment. The first one is the relationship between
Bayesian probabilities and agency. For instance, a system can be said to use
Bayesian probabilities even in the limited sense of applying a spam filter:
the latter is able to establish, based on relative frequency information, what
corresponds to an incoming email and what is a spam. But this is not enough to
classify the spam filter as a Bayesian agent. A spam filter uses Bayes’
theorem to estimate priors via relative frequencies of the words in a given
email based on its past occurrence in spam and non-spam emails. It calculates,
based on Bayes’ rule, but it has no degree of belief corresponding to the
priors and posteriors. The issue of representing probabilities is different
from the one of using probabilities. How can we translate this use of Bayesian
probabilities into a role for agency? What are the elements in model-building
that correspond to using Bayesian probabilities? The only way one can
ascertain whether a system is using Bayesian calculus is by looking at its
behaviour, but there is a difference between being able to represent
probabilistic behaviour _vs_ using the Bayesian probabilities. What is the
role of future-orientedness? We leave all this to future work.

These three lines of inquiry are deeply entrenched. First of all, one needs to
spell out the relationship between our account of agency and the general
discussion on the laws of nature. In the context of QM, one should explore
whether and under which perspective Quantum Mechanical probabilities can or
should be interpreted as Bayesian probabilities. If that is so, only those
systems capable of using Bayesian probabilities, i.e. Bayesian agents, could
apply Quantum Mechanical laws or play the role of the second system (the
“observer”) with respect to which quantum states are defined in a relational
fashion.

In addition, there is a relation between the complexity of the agent and the
kind of probabilities the agent can support or use. Let us sketch how this
link can be understood, without any pretense of completeness or precision of
details. We have defined agency as modeling activity. At the minimal level,
modelling is any kind of structure or categorization of data received from the
interaction with the environment. A higher level of complexity is found in
terms of the existence of feedback loops in the internal information-
processing activity of the agent. At this level, there can only be
dispositions or propensities defined as objective, or context-dependent
probabilities. At a higher level, the agent becomes a self-referential system
whose successive interactions depend on its own activity. At this level of
complexity, the agent becomes forward-looking, relying on its past history,
where the patterns of past input-output are stored. A certain pattern is
selected based on the weights the agent assigns to them which results in an
output in the form of an action. Here one can talk about objective Bayesian
probabilities which are mainly assigned according to the available evidence.

Furthermore, the agent can imagine or compose patterns which were never
experienced before in its interaction with the environment and can take
advantage of them to simulate or predict its future behavior. The agent
becomes forward-looking and deals with counterfactual scenarios. Here, the
agent moves to the level of subjective Bayesian probabilities because, in
assigning the probabilities to different imagined or expected patterns, it
mainly relies on subjective factors (though past evidence still plays a role).
The more the agent could construct and use hypothetical patterns within its
model, the more it can be viewed as “creative”. Two are the reasons why it is
here possible to define Bayesian probabilities as different from dispositions.
First, the evidence does not fully determine the assigned probabilities.
Second, here the context is not just a generic system, rather it is a complex
and special agent which deserves to be regarded as a subject assigning
probabilities. It is indeed a subject in the sense that it accords
probabilities to stored patterns based on its own, individual features.

##  4 Comparison with other accounts of agency

Let us now compare our account of agency and others available in the
literature:

  1. 1.

_Briegel_ : According to Briegel (2012), an agent is an information processing
system equipped with a memory that allows for projective simulation. In his
words, a memory is “any kind of organ, or physical device, that allows the
agent to store and recall information about past experience” and it grants
“the agent to relate its actions to its past” (Briegel, 2012, p. 2). At the
minimal level of internal information processing, the actions of the system
remain simple reflexes to environmental stimuli. All the behavioral outputs of
the system are just immediate responses to the received data from interaction
with the environment. At a second level, the information stored within the
memory undergoes projective simulation. A memory can be modelized as a
stochastic network of clips, where clips are the basic units of memory
corresponding to very short episodes which are sequences of remembered
percepts and actions. Triggered by a perceptual input, some specific clip in
memory related to the input is excited. This excited clip will then, with a
certain degree of probability, excite a neighboring clip, leading to a
transition within the clip network. This process of generating excited clips
continues until a certain feature is detected and thereby an excited clip
couples out and translates into motor action. At a still higher level of
internal information processing, there are not only transitions between
existing clips, but the latter may themselves be created and varied as part of
the simulation process itself. This composition of new fictitious clips allows
for simulation of plausible future experience and new patterns for future
action. This allows the agent to detach itself from strict dependence on its
surrounding environment by generating a space of possibilities for responding
to environmental stimuli. Briegel’s notion of agency is very close to ours.
More specifically, his notions of memory space and projective simulation
correspond to our concept of modeling activity, thus we would argue that our
definitions agree in making agency defined by “action based on internal
modelling”. However, his definition is not as minimal and scalable as ours,
because his notion of modelling is not as minimal as ours. Moreover, he
invokes higher-order concepts like memory and experience which also distances
his notion of agency from our naturalized account, placing it higher up in our
scale of further classification of kinds of agency within our broader
definition.

  2. 2.

_Hartle_ : The concept of an information gathering and utilizing system (IGUS)
was introduced by Hartle (2005) and has been further developed by Ismael
(2011) and Callender (2017). An IGUS, as its name implies, is any system with
the ability to gather information from its environment and then process this
information to guide its future behavior. An IGUS has a memory register $P$,
in which it stores the information received from the environment. This
information is stored in $n+1$ memory registers with $P_{0}$ being the most
recent data and $P_{n}$ the oldest ones. Each register is a snapshot of a
different instant of time. When the system acquires new information, it is
registered in $P_{0}$ and the data are conveyed to the memory registers, i.e,
$P_{n}$ is erased and replaced by the information in $P_{n-1}$ and so on. In
this way, the IGUS acquires a coarse-grained picture of its surrounding
environment. The IGUS makes use of two kinds of information processing. Via a
process called $U$, it builds a model of the world based on the information
stored in its memory register. Using another process $C$, it takes the newest
information stored in $P_{0}$ together with the model provided by $U$ to make
predictions and act accordingly. The concept of an IGUS comes very close as
well to our notion of an agent, and it is clearly based on the same conceptual
and information-theoretic elements. However, the main difference arises from
the the fact that while an IGUS only uses one model of the world, our
definition of agency invokes a hierarchy of models, thus being more scalable
and minimal. It follows that the concept of an IGUS can be regarded as
identifying a further kind within our broader definition.

  3. 3.

_Moreno and Mossio_ : The notion of agency developed by Moreno & Mossio (2015)
follows the tradition of self-maintenance accounts.131313See, for instance,
Bickhard & Terveen (1995) and as a recent account Jones (2017). On this
perspective, the main focus lies within the context of biological autonomy, to
such an extent that agency is primarily founded on the distinction between a
system (the agent) and the environment. In addition, agents are connoted by
their capacity to exert causal influence and operate via some normative
principles. It is from these assumptions that Moreno & Mossio ground their
notion of agents as those systems which are able to maintain their own
organization (also) through their influence on the surrounding environment,
based on some notion of normativity. According to this account, while bacteria
and higher organisms count as agents, viruses do not, in that the latter lack
the organization which is required to bring about the proper interaction with
the environment. Viruses do not display “agential capacities”, in the sense
that “they exhibit such capacities only insofar as they are integrated into
much more complex systems (typically cells) that are organised, in the
specific sense that they realise a closure of constraints” (Moreno & Mossio,
2015, p. 96). Their account is thus close to Barandiarian et al., but is
definitely distant from ours. Indeed, as our minimalist notion of agency is
explicitly scalable, we do not think it is necessary for a system to count as
an agent that the distinction between the latter and the environment is clear
and fixed once and for all. Rather, we claimed that this distinction can be
context-dependent, thus defined in a perspectival manner. In addition, their
notion of normativity is definitely more burdensome than ours.

  4. 4.

_Rovelli_ : In relational quantum mechanics (RQM) 141414Originally developed
by Rovelli (1996), an introductory discussion is Laudisa & Rovelli (2021)., a
quantum state ($\psi$) provides the information an observing system ($A$)
gathers of an observed system ($S$). As this information is collected via the
interaction between the two systems, the quantum state $\psi$ is relative to
$A$. In this sense RQM is said to provide perspectival, system-dependent
information, thus rebutting the idea of absolute, intrinsic properties of
systems, as well as a notion of absolute and objective evolution through
time.151515Importantly, no hidden variables are posed, meaning that the
quantum state is taken as a complete characterization of the physical system.
Quantum states of a system are only defined relationally, namely with respect
to another system (called an observer or a reference-system) that is in
relation with the former; in fact, this general relational, perspectival and
context-dependent view on quantum states is common to a number of
interpretations of quantum mechanics, collectively labelled ‘epistemic-
pragmatist’or ‘neo-Copenhagen’. We refer to Brukner (2021), Pienaar (2021), as
well as Di Biagio & Rovelli (2021), for a discussion on the role of observers
in RQM, and to Barzegar & Oriti (2022) for a broader account of the general
interpretative framework for QM, common to RQM and other relational or
pragmatist perspectives. Importantly, in RQM the observing or reference-system
can in principle be any other physical system – where the latter, again, is
not defined intrinsically, but always as a set of relations with respect to
other systems. Moreover, as all systems are equivalent, it follows, as
outlined by Van Fraassen (2010), that RQM provides a general expression for
the information that any system can have of any other system. For what matters
in the present discussion, the only way a notion of agency can be associated
to the ‘observing system’ in RQM, not only it would make no reference at all
to consciousness, but it would actually have to be minimized to include every
physical system, no matter how elementary. This notion of agency has been in
fact proposed by Rovelli (2020). Our definition of agency is similarly
naturalized, and similarly based on an information-theoretic perspective, and
in fact it would also suggest (or at least be consistent with) a relational
understanding of quantum states, when applied as a basis for an interpretative
framework for quantum mechanics. However, albeit minimalist, our definition of
agency would not apply to any interacting physical system, and exclude several
systems that would instead counts as agents according to Rovelli (2020). To
the extent in which a notion of agency is considered useful or even needed, as
an attribution of the ‘observing system’ in quantum mechanics, to specify the
(relational, perspectival or context-dependent) nature of quantum states, then
our definition would go in the direction of supporting epistemic-pragmatist
interpretations other than RQM, such as Healey’s or Brukner-Zeilinger, while
not necessarily favouring the more radical direction of QBism.

  5. 5.

_Steward_ : Helen Steward is interested in defining a notion of agency that
can be sufficiently loose to include animals. On her account, some criteria
need to be fulfilled, namely that an agent: i. is able to move a whole, or
some of its sub-components, as something we would call its _body_ (or parts of
its body); ii. displays some form of _subjectivity_ ; iii. exhibits at least
some rudimentary form of _intentionality_ ; iv. controls certain of the
movements of its body (Steward, 2009, p. 226). This account of agency relies
on a notion of intentional states as goal-orientedness, as well as on the idea
that the agent is able to make choices. This account of agency, which is
invoked by Steward to defend a libertarian metaphysics in the context of the
free-will debate, is definitely stronger than our minimalist account and
largely tangential to it.

  6. 6.

_Tononi et al._ : The starting point of Tononi and collaborators’ analysis is
that the reductionist paradigm proves untenable to account for autonomy and
agency, in that the latter implies that there is a distinction between agents
and environment, that agents are associated with macroscopic systems, and that
agents are said to act upon the surrounding environment. Provided that a
minimal notion of agency implies the existence of a system that “dynamically
and informationally interacts with an environment” (Albantakis et al., 2020,
p. 1), the subtler question is whether a subset of a larger system – which is
characterized by informational features – can be qualified as an agent. It is
for this reason that Tononi and collaborators invoke the so-called integrated
information theory (IIT) paradigm: originally proposed as a theory of
consciousness (Tononi, 2015), IIT provides a quantitative framework to specify
the causal structure of discrete dynamical systems. Within this account, there
is a quantity, called $\Phi$, that establishes the extent to which the causal
constraints a certain system exerts on itself is irreducible to its underlying
components. IIT is defined from the intrinsic perspective of the system only
if “this information is maximally irreducible _both_ in the past _and_ in the
future” (Tononi, 2012, p. 301). It is argued that whenever $\Phi>0$, then the
system is said to form a unitary whole, in the sense that “all of its subsets
constrain and are being constrained by other subsets within the system above a
background of external influences” (Albantakis et al., 2020, p. 2).161616For
further discussion on the topic, see, among others, Maturana and Varela,
(1980), Marshall et al., (2017), Albantakis (2018). Keeping this in the
background, information is here defined in terms of alternatives, whereby a
system can be qualified as an agent in case it is able to decide between a set
of alternatives – where this decision cannot be reduced to the causes of the
underlying components. Moreover, Tononi relies on a minimal notion of
consciousness, where minimality is granted by the fact that, for instance, the
advanced notion does not imply self-awareness. Despite the minimality of the
invoked notion of consciousness, we believe agency should be defined
independently of a putative characterization of consciousness. Indeed, we
claim that developing an account of agency in terms of modeling capacity makes
it more naturalistic, scalable and, most importantly, minimal. However, the
approach by Tononi and collaborators, which is based on information
acquisition and processing, as well as the role they assign to the system’s
activity, is clearly consonant to ours.

  7. 7.

_Winning_ : Again, our account associates modeling capacity with agency. In
this sense, it strongly resonates with Winning’s (2020) analysis, according to
which the notion of internal perspectivalism is preliminary to the possibility
of defining an agent. His underlying assumptions are two. The first is a form
of realism with respect to the existence of agents. The second is that agency
is not a natural kind, in the sense that no fixed boundary between what is an
agent and what is not can be ultimately defined. More specifically, he takes
agents as a specific kind of autonomous control systems – where an autonomous
control system is one that displays a degree of influence on the surrounding
environment which is above a certain threshold, a threshold that the system is
able to recognize. To count as autonomous normative systems – thus, as agents
– controllers should display not only a discrete output repertoire, but also a
discrete input repertoire. Put it otherwise, they must be observers. And here
is where Winning’s account approaches ours. For a system displaying
observational properties it must rely on some modeling activity. Indeed, there
are many ways in which models can be related to autonomous control systems. In
his PhD dissertation (Winning, 2019), Winning defines three ways controllers
involve models. The first is the one drawn by Pattee (1995), according to
which every controller mapping inputs for behavioral outputs defines an
implicit model. The second is the adoption of models to retain information
about the environment not presently tracked down, as in the case of forward
models. The third is the use of models to account for hypothetical scenarios.
On this perspective, an agent is a specific type of autonomous controller that
is multidimensional, multivariate, composite, variable, and metamorphic
(Winnig, 2019, pp. 125-126). Despite being close to our account, Winning’s is
still different, and definitely less minimalist than ours. Indeed, he takes
agency as “control by means of preferences” (PhD thesis, p. 127), to such an
extent that being an agent implies the possession of specific features geared
towards preference-based behavior. In some more detail, an agent is a system
that displays a perspective on the surrounding environment, carves it into a
number of states of affairs and offers a perspective on what states of affairs
are at stake and which ones are under its own control. The set of
categorizations of such states of affairs, together with the agent’s beliefs
and preferences constitute its normative standard for how to control the
surrounding environment. In this sense, Winning’s account is closer to
Sterenly’s (2003), despite being more refined than the latter. It could be
made closer also to ours only to the extent in which her notion of preferences
and perspective can be scaled down’to a more minimalist and naturalised level.

##  5 Objections

In this section we consider a number of objections that can be raised against
our account.

First of all, one might be concerned that our account relies too heavily on
the representationalist paradigm, according to which representation plays a
central role in the way agents acquire knowledge or information about the
environment. In other words, one might interpret the notion of modeling as
constructing a copy of the surrounding environment from the inside. Then the
agent uses this internal model to interact with and navigate such an
environment. However, according to the embodied cognition paradigm (see, among
others, Mahon, 2015; Shapiro, 2019), this is not the way agents interact with
and acquire knowledge or information about the world. According to the
embodied paradigm, in contrast to the representationalist paradigm, the priors
in perceptual anticipation are not beliefs and assumptions. Rather, they are
embodied skills and patterns of action-readiness. The agent does not have a
model of the world. Rather, it is a model of the world.

The controversy regarding the representationalist vs embodied paradigms of
cognition is a debate concerning whether one should regard the brain or the
body as the locus of cognitive activities. Here, we are just interested in
portraying a minimalist notion of agency, thus remaining neutral with respect
to such a debate. It can be the body that acts as the modeling element, in the
sense that it displays a form of covariance between its activities and the
environmental stimuli. Alternatively, it can be the brain that acts as the
modeling element, in the sense that there is a covariance between its neuronal
activities and the external stimuli. And importantly, in both cases the
modeling element can, under appropriate conditions, serve the purpose of
constructing hypothetical scenarios geared towards successive interaction with
the environment. Furthermore, we are concerned with a minimal notion of agency
and representation not loaded with cognitive and conscious activities. Of
course, at the higher levels of modeling activity, one can introduce
considerations on cognition or consciousness according to our relational
notion of hierarchy of models. Still, at the minimal level of data models, our
notion of agency has nothing to do with a strong notion of representation, as
used within cognitive science.

A second objection that can be raised, which directly follows from the first
one, is that our notion is too minimal, and in this way is not able to capture
the essential character typically associated to agency. As already discussed
in the introduction, the traditional notion of agency is connected to the idea
of a system endowed with the capacity to act – where the latter is constructed
in terms of intentional states exerted by the agent’s mental faculties. All in
all, it looks as if we have betrayed the traditional apparatus concerning
agency. In this sense, accounts such as Tononi’s and recently Ried et al.’s
(2019) come more equipped to address such traditional requirements. For
instance, Ried et al. (2019)’s definition of agents is “entities that interact
with their environment via explicitly modelled perceptions and actions,
endowed with an internal mechanism for deciding how to respond, and capable of
adapting those responses based on an individual history of interactions and
feedback” (p. 2). Two comments can be added. The first one is that we think
disentangling agency from notions such as intentionality, perception and
mental states is necessary if we aim at naturalizing it, at least to some
extent. The second one is that, as stressed time and again throughout this
paper, our notion of agency is scalable, and can thus be structured in a
hierarchy of levels, the higher ones of which may introduce some of the other
features that are traditionally associated to agency. Finally, our goal was
indeed to achieve a minimalist account of agency that was at the same time not
so minimal as to be trivial or to end up regarding any possible physical
system as an agent. We think this goal was achieved, and we regard our notion
of agency as indeed very minimal (leaving out features like the ones discussed
above) but not too minimal being not obviously satisfied in concrete cases.

A third objection is the opposite of the previous one, and would arise, for
example, in approaches aiming at erasing any demarcation among physical
systems, for example in the context of relational approaches to quantum
physics (and maybe physics in general). The notion of agency that could be
attributed to the observing or reference system in RQM, as discussed, would
have to be weaker than ours, since it has to fit any interacting physical
system. However, within this context, we would argue that the resulting notion
of agency is so much thin, minimalist, and objectified that one may wonder
what is the reason to introduce it in the first place, namely why to
distinguish it from simple interaction. Regardless of the respective merits of
RQM vis-à-vis other interpretations of quantum mechanics, we would maintain
that such extremely weak notion of agency can hardly serve the purpose of
providing fruitful insights in the contexts we outlined – such as the debate
on the nature of physical laws and bayesianism. In fact, we would argue that
it does not even help in the interpretation of quantum mechanics; being so
weak, it does not help to elucidate any aspect of quantum mechanics that RQM
does not elucidate already without it; therefore, it cannot be the basis for
preferring RQM over its alternatives.

##  6 Concluding remarks

In this paper, we have proposed a minimalist, scalable and naturalized account
of agency. To this end, agency has been related to the capacity of a certain
system to display modeling activity – where minimality is guaranteed by the
fact that even a simple classification of the information arising from the
environment can be taken as a kind of model-building (of data models, more
precisely). We have explored some possible refinements of our definition, so
to assure its scalability. We have discussed how our notion compares with
other accounts of agency in the literature, emphasizing the similarities with
those that also are based on information acquisition and processing. Finally,
we have pointed to some contexts – such as the interpretation of quantum
mechanics, the debate on the nature of physical laws and bayesianism in
physics, epistemology and philosophy of science (see Bovens & Hartmann, 2003
and Hartmann & Sprenger, 2019 for general overviews)– in which our account may
provide fruitful insights. We leave the application of our account of agency
to these contexts for future work.

## Acknowledgements

This research has been funded by the FQxI Grant FQXi-RFP-IPW-1908. DO
acknowledges financial support also from DFG research grants OR432/3-1 and
OR432/4-1. EM thanks MCMP and LMU for hospitality during the research work for
this project.

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