# No Substitute for Functionalism - A Reply to ‘Falsification & Consciousness’

Natesh Ganesh
Information Technology Lab, ACMD, NIST Boulder
Dept of Physics, University of Colorado, Boulder
Boulder, Colorado 80305
<EMAIL_ADDRESS>

###### Abstract

In their paper ’Falsification and Consciousness’ [1], Kleiner and Hoel
introduced a formal mathematical model of the process of generating observable
data from experiments and using that data to generate inferences and
predictions onto an experience space. The resulting substitution argument
built on this framework was used to show that any theory of consciousness with
independent inference and prediction data are pre-falsified, if the inference
reports are considered valid. If this argument does indeed pre-falsify many of
the leading theories of consciousness, it would indicate a fundamental problem
affecting the field of consciousness as a whole that would require radical
changes to how consciousness science is performed. In this reply, the author
will identify avenues of expansion for the model proposed in [1] allowing us
to distinguish between different types of variation. Motivated by examples
from neural networks, state machines and Turing machines, we will prove that
substitutions do not exist for a very broad class of Level-1 functionalist
theories, rendering them immune to the aforementioned substitution argument.

_K_ eywords Consciousness $\cdot$ Falsification $\cdot$ Unfolding Argument
$\cdot$ Substitution Argument $\cdot$ IIT $\cdot$ Causal Structure

##  1 Introduction

A formal model of generating data through experiments in consciousness science
and then using this data to make predictions and inferences onto an experience
space was introduced in [1]. This model was then used to propose a clear
definition of falsification, followed by the _‘substitution arguments’_. The
authors also pointed out that the unfolding argument from [2] would be a
special case of their results. It is very interesting work, accessible and
proposes a necessary descriptive mathematical framework that could prove to be
very useful moving forward. The author will assume that the readers are
familiar with the work in [1] and for the sake of clarity, we will try and
borrow the symbols and terminologies from it as much as possible.

The reply is structured as follows - we will start by discussing the main
definitions and theorem(s) from [1] in section 2, identify and correct
specific aspects of the model from the original work. Following this
expansion, we will then use some of the proposed examples for substitutions in
[1] to show why these specific substitutions do not imply pre-falsifications
for a broad class of Level-1 functionalist theories in section 3. In section
4, we will follow it up with a formal definition of this class of
functionalist framework and a proof that no substitutions exist for them. In
section 5, we will briefly explore the difference between Level-1 and Level-2
functionalism. The note will conclude in section 6 summarizing the work and
briefly discussing the implications.

##  2 Understanding the Substitution Argument

We start with one of the central definitions and results from [1] (refer to
Fig.(1) from [1]) that will be the focal point of the discussions here. These
include -

  * (a)

Definition 2.1 \- _falsification_ is defined as _‘there is a falsification at
$o\in O$ if we have $inf(o)\notin pred(o)$’_.

  * (b)

Definition 3.1 \- $o_{r}$_-substitution_ is defined as _‘a
$o_{r}$-substitution if there is a transformation $S:P_{o_{r}}\rightarrow
P_{o_{r}}$ such that at least for one $p\in P_{o_{r}}$ \- $pred\cdot
obs(p)\cap pred\cdot obs(S(p))=\phi$’_.

  * (c)

Definition 3.8 \- Inference and prediction data is defined as independent ‘if
for any $o_{i}$, $o_{i}^{\prime}$ and $o_{r}$, there is a variation
$v:P\rightarrow P$ such that $o_{i}\in obs(p)$, $o_{i}^{\prime}\in obs(v(p))$,
but $o_{r}\in obs(p)$ and $o_{r}\in obs(v(p))$ for some $p\in P$.’

  * (d)

In section (3.4.1), minimally informative is defined as that for every $o\in
O$, there exists an $o^{\prime}\in O$ such that $pred(\bar{o})\cap
pred(\bar{o}^{\prime})=\emptyset$.

  * (e)

The substitution argument is given in Theorem 3.10 \- ‘If inference and
prediction data are independent, either every single inference operation is
wrong or the theory under consideration is already falsified.’

Figure 1: Borrowed from [1] \- ‘This picture illustrates substitutions. Assume
that some data set $o$ with inference content $o_{r}$ is given. A substitution
is a transformation $T$ of physical systems which leaves the inference content
or invariant but which changes the result of the prediction process. Thus
whereas $p$ and $T(p)$ have the same inference content or, the prediction
content of experimental data sets is different. Different in fact to such an
extent that the predictions of consciousness based on these datasets are
incompatible (illustrated by the non-overlapping circles on the right). Here
we have used that by definition of $P_{o_{r}}$, every $p\in P_{o_{r}}$ yields
at least one data set $o^{\prime}$ with the same inference content as $o$ and
have identified as $o$ and $o^{\prime}$ in the drawing.’

One of the major things to note is that while the minimally informative
criterion guarantees that there are at least two sets of observable data with
different predictions, it does not constrain their corresponding inferences in
any manner [3]. Thus frameworks for which we have $pred(\bar{o})\cap
pred(\bar{o}^{\prime})=\emptyset$ could also have $inf(o)\cap
inf(o^{\prime})=\emptyset$ and still be minimally informative. A more robust
and expanded definition of independence can be built based on this observation
by expanding on what definition (3.8) intended to capture - “in most
experiments, the prediction content $o_{i}$ and inference content $o_{r}$
consist of different parts of a dataset. What is more, they are usually
assumed to be independent, in the sense that changes in $o_{i}$ are possible
while keeping $o_{r}$ constant” [1]. While we can have
($o_{i},o_{i}^{\prime}$) pairs that preserve $o_{r}$, it does not necessarily
imply that all ($o_{i},o_{i}^{\prime}$) pairs have to satisfy this constraint
(and it would be erroneous to assume otherwise). Thus we could have a set of
prediction data $\\{o_{i}^{\prime}\\}$ generated by a variation (as defined in
[1]) that maintains the same $o_{r}$, but does not contain
$o_{i}^{\prime\prime}$ (or a set of them) with $inf(o)\cap
inf(o^{\prime\prime})=\emptyset$ that ensures that the minimally informative
criterion ($pred(\bar{o})\cap pred(\bar{o}^{\prime\prime})=\emptyset$ ) is
met. Furthermore varying $o_{i}$ while keeping $o_{r}$ constant does not
necessarily mean we vary the prediction $pred(o_{i})$ i.e. we can have
$o_{i}\neq o_{i}^{\prime}$ but still have $pred(o_{i})=pred(o_{i}^{\prime})$.
This expanded idea of independence is aligned with the underlying motivation,
while not being equivalent to falsification by definition [3].

Utilizing the original definition of variation, we see that for any
($o_{i},o_{i}^{\prime}$) pair that maintain the same $o_{r}$ there are
variations of two types -

  * (i)

Type-1 variations where $o_{i}\neq o_{i}^{\prime}$, but $pred(o)\cap
pred(o^{\prime})\neq\emptyset$.

  * (ii)

Type-2 variations where $o_{i}\neq o_{i}^{\prime}$, but $pred(o)\cap
pred(o^{\prime})=\emptyset$.

It is immediately evident that while both variations allow for prediction and
inference data to be independent, only Type-2 variations imply pre-
falsifications. Thus independence (under this expanded set of variations) does
not necessarily imply falsifications by definition and we could have
frameworks with prediction and inference data independence through only Type-1
variations. Thus Theorem 3.10 is incomplete at best since Definition 3.8 of
_independence_ does not account for Type-1 variations. The correct restatement
of the theorem would then be -

###### Theorem 1

‘If inference and prediction data are independent under a Type-2 variation,
either every single inference operation is wrong or the theory under
consideration is already falsified.’

Since all variations that produce independence do not immediately entail
falsification, the claim that - ’our results show that if the independence of
prediction and inference holds true, as in contemporary cases where report
about experiences is relied upon, it is likely that no current theory of
consciousness is correct’ in [1] only holds if is shown that the variation is
of Type-2 for the theory. The question now becomes one of determining whether
substitutions identified in different existing frameworks are produced by
variations of Type-1 or 2. This will be the focus of the next section with
respect to some of cases suggested in the paper.

The authors of [1] also make the case for why $obs:P\rightarrow O$ and
$pred:O\rightarrow E$ mappings are better characterized as _correspondences_ ,
while $inf$ can be considered to be a function. They argue that while $inf$
can be a correspondence, it can redefined to be a function by redefining the
experience space $E$ to a new space $E^{\prime}\coloneqq\\{inf(o)|o\in O\\}$,
’where every individual element $e^{\prime}\in E^{\prime}$ describes what can
be inferred from one dataset $o\in O$, so that $inf^{\prime}\coloneqq
O\rightarrow E^{\prime}$ is a function.’ While this might be coherent
mathematically, the authors do not explain the effect of redefining $E$ to
$E^{\prime}$ has on the $pred$ correspondence or explore the more general case
of both $pred$ and $inf$ being correspondences (if it does not affect the
actual theorems). If $inf(o)$ is a subset of elements in $E$ as opposed to a
single element, it would entail a change in falsification to $inf(o)\cap
pred(o)=\emptyset$ and complicate both variations and substitutions since we
could now have variations that maintain $o_{r}$ and have $pred(\bar{o})\cap
pred(\bar{o}^{\prime})$ but still not entail falsification since $pred(o)\cap
inf(o)\neq\emptyset$ and $pred(o^{\prime})\cap inf(o^{\prime})\neq\emptyset$.
However this is a minor objection and unrelated to the rest of this reply.
Since this is a first step in formalizing these ideas, it is justifiable for
the authors of [1] to not cover the more general cases in their work yet.
However one hopes that the change in $inf$ from correspondence to function is
explored in detail and if it does not affect the final results as claimed in
the paper, a clear and rigorous derivation of that will be provided in future
work.

##  3 Application of the Substitution Argument to Neural Networks & Turing
Machines

The authors of [1] would have served the readers better by providing a very
clear example of the application of the substitution argument to some of the
proposed cases of data independence stated in the paper. In this section, we
will examine the existence and type of variations in the case of powerful and
widely used models of computation - neural networks, a simple finite state
machines and Turing machines. Let us start with the case of artificial neural
networks discussed in the paper. The authors state - ‘For any ANN, report
(output) is a function of node states. Crucially, this function is non-
injective, i.e., some nodes are not part of the output. For example, in deep
learning, the report is typically taken to consist of the last layer of the
ANN, while the hidden layers are not taken to be part of the output.
Correspondingly, for any given inference data, one can construct a ANN with
arbitrary prediction data by adding nodes, changing connections and changing
those nodes which are not part of the output. Put differently, one can always
substitute a given ANN with another with different internal observables but
identical or near-identical reports. From a mathematical perspective, it is
well-known that both feedforward ANNs and recurrent ANNs can approximate any
given function (Hornik et al. 1989; Schafer and Zimmermann 2007). Since
reports are just some function, it follows that there are viable universal
substitutions.’ It seems like the authors are claiming that the arguments
presented in their discussion imply the existence of _universal substitutions_
for neural networks but is that really the case? Since there is no example
accompanying the claim, we will take up a simple example of a recurrent neural
network and apply the substitution argument as faithfully as possible to the
discussion above. We would like to acknowledge that a bulk of the work in this
section is not very novel and is built on the analysis performed by Hanson and
Walker in [4] and [5].

Consider a general recurrent neural network (RNN) $\mathcal{N}$ with
$\mathcal{I}$ number of input neurons, an arbitrary number $\mathcal{H}$ of
hidden neurons and $\mathcal{T}$ output neurons. Let the input-to-hidden
neuron weights, hidden-to-hidden weights and hidden-to-output weights be given
by $W_{\mathcal{I}}$, $W_{\mathcal{H}}$ and $W_{\mathcal{T}}$. The dynamics of
the hidden states $h_{t}$ and output $o_{t}$ of the recurrent neural network
at time $t$ is given as

$\displaystyle h_{t}$ $\displaystyle=$
$\displaystyle\sigma(W_{\mathcal{H}}h_{t-1}+W_{\mathcal{I}}x_{t})$
$\displaystyle o_{t}$ $\displaystyle=$ $\displaystyle W_{\mathcal{T}}h_{t}$
(1)

where $\sigma$ can be a non-linear activation function like ReLu or sigmoid.
Note that $h_{t}$ is a $\mathcal{H}$-dimensional vector. By the discussion of
such neural networks in [1], we have the hidden states $h_{t}$ as part of the
prediction data and output states $o_{t}$ as the inference data. From existing
literature we know that it is possible to construct a different RNN
$\mathcal{N}^{\prime}$ that can produce the same input-output behavior but
with a different hidden-to-hidden weight matrix. Thus $\mathcal{N}^{\prime}$
would have the same number of input and output neurons $\mathcal{I}$ and
$\mathcal{T}$ respectively as $\mathcal{N}$. However we assume that
$\mathcal{N}^{\prime}$ has a different number of hidden neurons
$\mathcal{H}^{\prime}$. The corresponding weight matrices are given as
$W_{\mathcal{I}}^{\prime}$, $W_{\mathcal{H}}^{\prime}$ and
$W_{\mathcal{T}}^{\prime}$. We will assume that the new hidden state of this
RNN is given as $h_{t}^{\prime}$ (a $\mathcal{H}^{\prime}$-dimensional vector)
while inputs and outputs remain the same at $x_{t}$ and $o_{t}$. The dynamics
are given as

$\displaystyle h_{t}^{\prime}$ $\displaystyle=$
$\displaystyle\sigma(W_{\mathcal{H}}^{\prime}h_{t-1}^{\prime}+W_{\mathcal{I}}^{\prime}x_{t})$
$\displaystyle o_{t}$ $\displaystyle=$ $\displaystyle
W_{\mathcal{T}}^{\prime}h_{t}^{\prime}$ (2)

Thus $\mathcal{N}$ and $\mathcal{N}^{\prime}$ both give the same output
$o_{t}$ (corresponding to inference data) while changing the hidden states
(corresponding to prediction data). $h_{t}\neq h_{t}^{\prime}$ since they are
vectors of different dimensions. Under the definitions used in [1], this would
imply that prediction and inference data are independent and viable
substitutions exist. But whether this particular substitution in the neural
network implies falsification depends on whether or not the variation is of
Type-1 or 2. The answer is that it depends upon the framework or theory of
interest and it’s effect on the $pred$ function.

We will explore this further with a functionalist framework $\mathcal{F}$. We
define a functionalist framework to be one in which - ‘the states are
typically described in terms of functional behaviors ("stop", "walk", "go",
etc.) but what really gives them meaning mathematically is only their
topological relationship with one another. This implies that at this level,
the formal description of the computation is not grounded in any particular
physical representation and could, in fact, be realized by radically different
causal structures. This abstract treatment of computation corresponds to what
Chalmers’ refers to as the “finite-state automaton” (FSA) level of
description, due to the fact it is defined in terms of a global finite-state
automaton. Beneath this level is what Chalmers refers to as the
"combinatorial-state automaton" (CSA) description. The only difference between
the FSA and CSA levels of description is that the latter specifies the
computational states of the former in terms of a specific labeling or encoding
of the subsystems that comprise the global system.’ [5], [6]. We will simply
refer to the FSA and CSA levels of description as Level-1 and Level-2
descriptions respectively in order to be more general and avoid any baggage
with the terms - FSA and CSA.

Under this computational hierarchy, we have $\mathcal{F}$ to be a Level-1
functionalist framework. While specific representations are important to
understand the physical implementation, the $pred$ function for such a
$\mathcal{F}$ would be dependent on the functional states $\\{s\\}$ as defined
in the Level-1 description only and not on the particular of any specific
encoding at Level-2 i.e. Level-1 functionalist frameworks are not
representationalist simply because they employ representation. In the case of
the neural network, the states in the Level-2 description are defined by their
relationship to other states $\\{h_{t-1},x_{t}\\}\rightarrow\\{h_{t},o_{t}\\}$
i.e. their functional structure. Though $h$ and $h^{\prime}$ are vectors of
different dimensions (corresponding to their particular physical
representations in their respective networks), both networks maintain the same
input-output relationship and as a result, we would have for every
$h\in\\{h\\}$ a corresponding $h^{\prime}\in\\{h^{\prime}\\}$ such that
$\\{h_{t-1},x_{t}\\}\rightarrow\\{h_{t},o_{t}\\}$ and
$\\{h_{t-1}^{\prime},x_{t}\\}\rightarrow\\{h_{t}^{\prime},o_{t}\\}$ for all
input-output pairs $(x_{t},o_{t})$. Thus $h_{t}$ and $h_{t}^{\prime}$
correspond to the same functional state $s_{t}$ (say) at Level-1. We can think
of the $pred$ function for $\mathcal{F}$ as composed of two functions - $abs$
which maps the particular physical realization of Level-2 to the higher
Level-1 FSA description, followed by a $pred^{\prime}$ function that maps the
(abstract) functional state $s_{t}$ to the experience space -
$pred()=pred^{\prime}(abs())$. Hence we have the specific encodings of the
hidden states $h_{t}\neq h_{t}^{\prime}$ but would still have
$pred(h_{t})=pred(h_{t}^{\prime})=pred^{\prime}(s_{t})$ (since $pred$ function
only depends upon the Level-1 functional state), while the output remains the
same for both $\mathcal{N}$ and $\mathcal{N}^{\prime}$. Thus the variation
from $\mathcal{N}$ and $\mathcal{N}^{\prime}$ does not imply pre-
falsification. It is trivial to see that if we were drop the time element $t$
from the states, the same arguments can be extended to feedforward neural
networks with varying number of hidden neurons. Since the assumptions made
here are very general, we claim that for a Level-1 functional framework
$\mathcal{F}$, any substitution of ‘a given ANN with another with different
internal observables but identical or near-identical reports’ [1] will always
correspond to a Type-1 variation and never imply pre-falsification of
$\mathcal{F}$ if we take the inferences to be true.

To see the dependence of the above discussion on the framework in question,
let us consider a different framework of consciousness $\mathcal{F}^{\prime}$
in which the result of the prediction function is dependent on the
dimensionality of the hidden-state vector i.e. $pred(h_{t})=g(dim(h_{t}))$. In
this framework we can clearly see that $pred(h_{t})$ need not be equal to
$pred(h_{t}^{\prime})$ since the two vectors are of different dimensions. In
such case, the relationship between $N$ and $N^{\prime}$ corresponds to a
type-2 variation and we would have framework $\mathcal{F}^{\prime}$ to be
falsified by Theorem 1. While the dependence on dimensionality of the hidden
state is a contrived example that is easy to visualize, the same argument
would apply to any framework that distinguished between any $h_{t}$ and
$h_{t}^{\prime}$ (that have the same functional structure as per a Level-1
description) with respect to the $pred$ function as a result of the difference
in their physical encoding/representation i,e, CSA or Level-2 description. We
will see this in greater detail when studying an example comparing systems
that do and do not contain any feedback next.

Diving deeper into the differences between Level-1 (FSA) and Level-2 (CSA)
descriptions - “In digital electronics, as well as models of the human brain,
this encoding is usually given in terms of binary labels that are assigned to
instantiate the functional states of the system. Consequently, transitions
between states in the CSA description fix local dependencies between elements,
as the correct Boolean update must be applied to each “bit” or “neuron” based
on the global state of the system. Furthermore, once a binary representation
is specified it constrains the memory required to instantiate the computation,
as the number of bits that comprise the system is now fixed. The final level
of the hierarchy is the specific choice of logic gates used to implement the
Boolean functions specified at the CSA level. [5]. This is best understood
from the example worked out in [5] where they show that a causal structure
theory of consciousness like Integrated Information Theory (IIT) makes
different predictions for different CSA representations and as a result are
either pre-falsified or are unfalsifiable depending on whether the inference
is made at the FSA/Level-1 or CSA/Level-2 respectively.

Figure 2: Modified from [5] \- We created the mirror image of the Fig.(1) from
[5] by starting from two different physical implementations 1 and 2 of the
tollbooth on the left, having feedback & no feedback respectively. This
picture is closer to the idea of performing experiments and collecting data
from a physical system. Since both implementations maintain the same input-
output relationship while producing different representations of their
internal states, they are variations of each other. The corresponding CSA
encoding representations that constitute the Level-2 description are given in
the middle. We can see that an abstraction function $abs$ will generate the
same higher Level-1 description of the Mod 8 FSA in both cases.

We will borrow Fig.(1) from [5] and modify it to fit the case here to give
Fig.(2). We see that starting with the two physical implementations 1 and 2,
we have two physical systems from which our prediction and inference data is
generated using the $obs$ correspondence. We will have the internal states and
outputs of the implementation correspond to prediction and inference data
respectively. Since both implementations maintain the same output while having
different CSA encoding representations for the internal states, the two
systems can be seen as a variation of each other. This difference in the state
representations correspond to the difference in their physical implementations
- feedback vs no-feedback. We can then generate the higher Level-1 FSA
description that represents the functional structure shared by both Level-2
CSA descriptions using a suitable abstraction functions $abs$. From the
Fig.(2), we have the state $100^{1}$ from CSA representation-1 to have the
same topological relationships as the state $111^{2}$ from CSA
representation-2, and both of these correspond to state $H$ in the Level-1
description. Since the predictions of a Level-1 functionalist framework only
depend on functional states in the Level-1 description, we would have
$pred(100^{1})=pred(111^{2})=pred^{\prime}(H)$. Thus variations like the one
in Fig.(2) that maintain the input-output relationship also maintain the
functional structure and are thus Type-1 variations that do not pre-falsify
the framework. On the other hand, we can see that for frameworks that produce
different predictions based on different CSA representations i.e.
$pred(100^{1})\neq pred(111^{2})$, such a variation would be of Type-2 and
imply pre-falsification if the inferences at the Level-1 description are
considered to be valid.

A similar claim on the existence of substitutions for Turing machine was also
made - ‘Turing machines are extremely different in architecture than ANNs.
Since they are capable of universal computation, they should provide an ideal
candidate for a universal substitution’ [1]. To explore this, we will start
with a mathematical definition of a Turing machine $\mathcal{M}$ from [7] as a
7-tuple $\mathcal{M}=\\{Q,\Lambda,b,\Sigma,\delta,q_{0},F\\}$ where

  * •

$Q$ is a finite, non-empty set of states.

  * •

$\Lambda$ is a finite, non-empty set of tape alphabet symbols.

  * •

$b\in\Lambda$ is the blank symbol.

  * •

$\Sigma\subseteq\Lambda\backslash\\{b\\}$ is the set of input symbols.

  * •

$q_{0}\in Q$ is the initial state.

  * •

$F\subseteq Q$ is the set of final or accepting states.

  * •

$\delta:(Q\backslash F)\times\Lambda\rightarrow Q\times\Lambda\times\\{L,R\\}$
is a partial function called the transition function.

In a Turing machine, the input is encoded into the tape and Turing machine (or
specifically the state registry) starts at an initial starting state. There is
a transition function that maps the current state of the state registry and
the symbol on the tape to a new state of the registry, a change in the symbol
on the tape at the current position and a shift in the read/write head to
either the left or right $(L/R)$ by one position. The complete configuration
of the Turing machine can be seen as the combined state of the state registry
and the symbols on the tape. And we can view the functioning of the Turing
machine as set of transitions between these different complete configurations
$\\{q,x\\}\in Q\times\Sigma$. The machine halts with the state (registry) of
machine in $\\{q_{F}\\}\in F$ and the output on the tape.

A simple example of a general substitution in Turing machines is through the
use of a universal Turing machine $\mathcal{U}$ which can simulate an
arbitrary Turing machine $\mathcal{M}$ on arbitrary input by reading both the
description of the machine to be simulated as well as the input to that
machine from its own tape. Thus a universal Turing machine $\mathcal{U}$ can
also be defined by a similar 7-tuple with an expanded set of input symbols
$\Sigma^{\mathcal{U}}$ (to accept original inputs and description of
$\mathcal{M}$) and states space of the state registry $Q^{\mathcal{U}}$. When
the machine $\mathcal{U}$ halts, the symbols on the tape should correspond to
the same output generated by $\mathcal{M}$ upon halting. The complete
configuration of $\mathcal{U}$ is given by $\\{q^{\prime},x^{\prime}\\}\in
Q^{\mathcal{U}}\times\Sigma^{\mathcal{U}}$. Since $\mathcal{U}$ will produce
the same output on the tape as $\mathcal{M}$ upon halting for the same inputs,
albeit through a different set of registry states from the space
$Q^{\mathcal{U}}$ and transition function $\delta^{\mathcal{U}}$, we can
construct a substitution using the concatenated state
$\\{q_{0}q_{1}q_{2}...q_{F}\\}$ and
$\\{q_{0}^{\mathcal{U}}q_{1}^{\mathcal{U}}q_{2}^{\mathcal{U}}....q_{F}^{\mathcal{U}}\\}$
as prediction data of $\mathcal{M}$ and $\mathcal{U}$ respectively, and the
final output on the tape when the machine halts as the inference data.

Now imagine a Level-1 functionalist theory of consciousness $\mathcal{T}$. We
can see that for any input-output pair $(x,o)$, we can write the corresponding
Turing machine realization as
$(q_{0},x)\rightarrow(\bar{q}=\\{q_{1}q_{2}....q_{F}\\},o)$ and
$(q_{0}^{\mathcal{U}},x\times\mathcal{X}_{\mathcal{M}})\rightarrow(\bar{q}^{\mathcal{U}}=\\{q_{1}^{\mathcal{U}}q_{2}^{\mathcal{U}}....q_{F}^{\mathcal{U}}\\},o)$
in $\mathcal{M}$ and $\mathcal{U}$ respectively, where
$\mathcal{X}_{\mathcal{M}}$ is the description of $\mathcal{M}$ as input to
$\mathcal{U}$. We can rewrite the initial state of $\mathcal{U}$ by combining
$\mathcal{X}_{\mathcal{M}}$ into the initial state $q_{0}^{\mathcal{U}}$ to
produce $q_{0}^{\mathcal{U}*}=(q_{0}^{\mathcal{U}},\mathcal{X}_{\mathcal{M}})$
and thus maintain the same ‘input’ $x$. Plugging this back in for the machine
$\mathcal{U}$, we have
$(q_{0}^{\mathcal{U}*},x)\rightarrow(\bar{q}^{\mathcal{U}}=\\{q_{1}^{\mathcal{U}}\times....q_{F}^{\mathcal{U}}\\},o)$
Even though $\\{q\\}$ and $\\{q^{\mathcal{U}}\\}$ are elements in different
spaces and $\bar{q}$ and $\bar{q}^{\mathcal{U}}$ are of different lengths
($\mathcal{M}$ and $\mathcal{U}$ need not produce the output in the same order
of time-complexity), their topological functional relationship remains the
same across configurations of the two Turing machines. Thus for every
$\bar{q}\in\\{\bar{q}\\}$ with $(q_{0},x)\rightarrow(\bar{q},o)$, there is a
$\bar{q}^{\mathcal{U}}\in\\{\bar{q}^{\mathcal{U}}\\}$ with
$(q_{0}^{\mathcal{U}*},x)\rightarrow(\bar{q}^{\mathcal{U}},o)$ across all
input-output pairs $(x,o)$. This is isomorphic to the example of the hidden
states in the RNN example used earlier in the section. The very same arguments
can be extended here to show that though $\bar{q}\neq\bar{q}^{\mathcal{U}}$,
we would have $pred_{\mathcal{T}}(\bar{q})\cap
pred_{\mathcal{T}}(\bar{q}^{\mathcal{U}})\neq\emptyset$ while output remains
the same. For such a theory of consciousness $\mathcal{T}$, the variation
between Turing machine $\mathcal{M}$ and an universal Turing machine
$\mathcal{U}$ is a Type-1 variation and does not imply pre-falsification. Both
Turing machines here are universal models of computations capable of
simulating any computable function and equivalent to other universal models
like Lambda calculus, cellular automata, recursive functions, etc. It follows
that for a Level-1 functionalist theory $\mathcal{T}$ as defined above, a
variation between any two universal models of computation will be a Type-1
variation with respect to $\mathcal{T}$ and will not imply it’s pre-
falsification. Of course, as in the case of the RNN, we can construct a theory
of consciousness $\mathcal{T}^{\prime}$ that is sensitive to Level-2
descriptions such that the variation from $\mathcal{M}$ to $\mathcal{U}$ is of
Type-2 and $\mathcal{T}^{\prime}$ is pre-falsified by Theorem 1. The examples
discussed in this section strongly hint at the non-existence of Type-2
variations for Level-1 functionalist theories of consciousness. We will prove
that this is indeed the case in the next section.

##  4 No Substitute for Level-1 Functionalism - A Formal Proof

The authors of [5] proposed the following idea - ’For a theory to avoid the
epistemic problems revealed by IIT under the isomorphic transformation we
introduce requires that no transformation or "substitution" exists that
changes the prediction without affecting the inference. This, in turn, implies
that beneath the specified level of inference, a mathematical theory of
consciousness must be invariant with respect to any and all changes that leave
the results from the inference procedure fixed. In other words, if you can
make a change to the physical system that does not affect what will be used to
infer the conscious state of the system, then such a change must not affect
the prediction of the theory either.’ In this section, we will build on the
examples from the previous section and provide a short formal proof (by
contradiction) that no substitutions exist by showing no Type-2 variations
exist for Level-1 functionalist theories (of consciousness). As a result,
these cannot be pre-falsified by the substitution argument of [1]. We have
briefly described Level-1 functional description in the previous section but
will provide a more precise definition here. A Level-1 functionalist framework
$\mathcal{T}_{F}$ is defined in terms of

  * •

Functional states $s^{\mathcal{T}_{F}}\in\mathcal{S}^{\mathcal{T}_{F}}$ which
are defined mathematically in terms of their topological relationship to other
functional states. They are thus independent of any particular encoding or
representation.

  * •

This topological relationship, which we call the functional structure
$\mathcal{G}_{F}$ can be characterized as the map:
$(s_{(t-1)}^{\mathcal{T}_{F}},x_{t})\rightarrow(s_{t}^{\mathcal{T}_{F}},o_{t})$
defined over all $(x_{t},o_{t})$. $x_{t}$ and $o_{t}$ are inputs and outputs
at time $t$. Note that we can also combine $o_{t}$ into the functional state
$s_{t}^{\mathcal{T}_{F}}$ and rewrite the map as
$(s_{(t-1)}^{\mathcal{T}_{F}},x_{t})\rightarrow s_{t}^{\mathcal{T}_{F}}$. If
there are a collection of functional states $\\{s\\}$ and $\\{s^{\prime}\\}$,
and for every $s\in\\{s\\}$, there is (atleast) a
$s^{\prime}\in\\{s^{\prime}\\}$ such that
$(s_{(t-1)},x_{t})\rightarrow(s_{t},o_{t})$ and
$(s^{\prime}_{(t-1)},x_{t})\rightarrow(s^{\prime}_{t},o_{t})$ over all
$(x_{t},o_{t})$ pairs, then $\\{s\\}$ and $\\{s^{\prime}\\}$ share the same
functional structure and the corresponding $s\in\\{s\\}$ and
$s^{\prime}\in\\{s^{\prime}\\}$ are equivalent functional states.

  * •

The result of the prediction correspondence $pred$ only depends upon the set
of functional states $\\{s^{\mathcal{T}_{F}}\\}$. If $s\in\\{s\\}$ and
$s^{\prime}\in\\{s^{\prime}\\}$ are equivalent functional states, then
$pred(s)=pred(s^{\prime})$ (or in a broader sense $pred(s)\cap
pred(s^{\prime})\neq\emptyset$). In general, the $pred$
function/correspondence from [1] defined for prediction data $o\in\mathcal{O}$
can be decomposed into a composition of two functions - an abstraction
function $abs:\mathcal{O}\rightarrow\mathcal{S}^{\mathcal{T}_{F}}$ that maps
prediction data in $\mathcal{O}$ to functional states in
$\mathcal{S}^{\mathcal{T}_{F}}$. And a
$pred^{\mathcal{T}_{F}}:\mathcal{S}^{\mathcal{T}_{F}}\rightarrow\mathcal{E}$
that maps functional states to prediction results in the experience space
$\mathcal{E}$. Thus $pred(o)=pred^{\mathcal{T}_{F}}(abs(o))$.

This abstraction function
$abs:\mathcal{O}\rightarrow\mathcal{S}^{\mathcal{T}_{F}}$ always exists since
it can always be constructed for a Level-1 functionalist framework. In the
prediction data $o_{i}$, let there be a particular physical representation of
states $\\{h^{\mathcal{T}_{F}}\\}\in\mathcal{H}^{\mathcal{T}_{F}}$. If
$\\{h^{\mathcal{T}_{F}}\\}$ is such that for every input-output pair
$(x_{t},o_{t})$, we have
$(h_{(t-1)}^{\mathcal{T}_{F}},x_{t})\rightarrow(h_{t}^{\mathcal{T}_{F}},o_{t})$
and a $s_{(t-1)}^{\mathcal{T}_{F}}\in\\{s^{\mathcal{T}_{F}}\\}$ with
$(s_{(t-1)}^{\mathcal{T}_{F}},x_{t})\rightarrow(s_{t}^{\mathcal{T}_{F}},o_{t})$,
then we define $abs(h_{(t-1)}^{\mathcal{T}_{F}})=s_{(t-1)}^{\mathcal{T}_{F}}$,
thereby constructing it into existence. By the very definition of a Level-1
functionalist theory of consciousness,
$pred^{\mathcal{T}_{F}}:\mathcal{S}^{\mathcal{T}_{F}}\rightarrow\mathcal{E}$
also exists and would be built according to the particular framework. The
output in general would be a non-invertible function of the functional state
i.e. $o_{t}=g(s_{t}^{\mathcal{T}_{F}})$ is a coarse-graining over the
functional states. In the case where $o_{t}$ has been concatenated to the
functional state, the function $g$ simply traces out the non-output parts of
the functional state. Note that since $g$ is assumed to be a non-invertible
function, $g^{-1}$ need not exist and thus
$s_{t}^{\mathcal{T}_{F}}=g^{-1}(o_{t})$ is not necessarily defined. If we view
the output as corresponding to the inference data $o_{r}$, we have that
$\\{s^{\mathcal{T}_{F}}\\}\not\subseteq\\{o\\}$ (i.e $o_{i}\not\subseteq
o_{r}$) and thus we have not defined our Level-1 functionalist framework to be
(pathologically) unfalsifiable as described by the conditions in [1].

We will briefly take a moment here to discuss the Reductio ad absurdum
argument from [1]. The authors argue that for experiments in the natural
sciences - “If there are two quantities of interest whose relation is to be
modeled by a scientific theory, then in all reasonable cases there are two
independent means of collecting information relevant to a test of the theory,
one providing a dataset that is determined by the first quantity, and one
providing a dataset that is determined by the second quantity.” They explain
this with an example of the relationship between temperature $T_{0}$ and it’s
relationship to the energy of microphysical states. They argue that in order
to determine this for any particular model we would make two different
measurements - one that would access the microphysical states and measure
their kinetic energy (say) which would correspond to the prediction data
($o_{m}$) and the other would use a thermometer to obtain a dataset
$o_{T_{0}}$ that replaces the inference dataset. They claim that ‘these
independent means provide independent access to each of the two datasets in
question’ and this ’differs from the case of theories of consciousness
considered here, wherein the physical system determines both datasets.’ We are
not certain as to whether independent is used according to how it has been
defined earlier in the paper. Nonetheless, we do not understand why in this
example, the authors of [1] believe that the temperature measurement of the
system $P$ using the thermometer does not depend upon the physical system $P$?
The measurement obtained from the thermometer is in fact a calibrated value
that provides a coarse-grained macroscopic description of the underlying
microscopic kinetic energies (via a Maxwell-Boltzmann distribution of particle
speeds). So if the authors identify no issues associated with the example they
provided, then it should follow that for the sake of consistency, they will
have no objections to the Level-1 functionalist framework defined above since
the inference data/output is simply a coarse-grained result of the functional
states achieved through an non-invertible function. We can now proceed further
with our framework $\mathcal{T}_{F}$.

Since the functional states
$s^{\mathcal{T}_{F}}\in\mathcal{S}^{\mathcal{T}_{F}}$ are independent of any
specific encoding, we will define the set
$\\{\mathcal{H}_{i}^{\mathcal{T}_{F}}\\}$ as the collection of different
representations/encodings of Level-1 functional states that realize the same
functional structure -
$\\{\mathcal{H}_{i}^{\mathcal{T}_{F}}\\}=\\{\mathcal{H}_{1}^{\mathcal{T}_{F}},\mathcal{H}_{2}^{\mathcal{T}_{F}},...,\mathcal{H}_{N}^{\mathcal{T}_{F}},...\\}$.
Each element
$\mathcal{H}_{j}^{\mathcal{T}_{F}}\in\\{\mathcal{H}_{i}^{\mathcal{T}_{F}}\\}$
is itself a collection of states $\\{h_{i}^{\mathcal{T}_{F}}\\}$ that
corresponds to a very particular physical representation of the individual
functional states that maintain the same functional structure as any other
$\mathcal{H}_{k}^{\mathcal{T}_{F}}\in\\{\mathcal{H}_{i}^{\mathcal{T}_{F}}\\}$.
Under this definition of $\\{\mathcal{H}_{i}^{\mathcal{T}_{F}}\\}$, consider
any pair of encodings
$\mathcal{H},\mathcal{H}^{\prime}\in\\{\mathcal{H}_{i}^{\mathcal{T}_{F}}\\}$
(where we have dropped $\mathcal{T}_{F}$ to keep the equations cleaner). Since
both $\mathcal{H}$ and $\mathcal{H}^{\prime}$ have the same functional
structure $\mathcal{G}_{F}$, by definition we have that for every
$h^{j},h^{j^{\prime}}\in\mathcal{H}$, there is a
$h^{k},h^{k^{\prime}}\in\mathcal{H}^{\prime}$, such that
$(h_{(t-1)}^{j},x^{t})\rightarrow(h_{t}^{j^{\prime}},o^{t})$ and
$(h_{(t-1)}^{k},x^{t})\rightarrow(h_{t}^{k^{\prime}},o^{t})$ for all input-
output pairs $(x^{t},o^{t})$. This means that the different encodings $h^{j}$
and $h^{k}$ correspond to equivalent functional states. We thus have that for
different encodings $\mathcal{H}$ and $\mathcal{H}^{\prime}$ that share the
same functional structure, we can write using the $abs$ and
$pred^{\mathcal{T}_{F}}$ function from before, $s=abs(h^{j})\in\\{s\\}$,
$s^{\prime}=abs(h^{k})\in\\{s^{\prime}\\}$ where
$pred^{\mathcal{T}_{F}}(s)=pred^{\mathcal{T}_{F}}(s^{\prime})$ (or
$pred^{\mathcal{T}_{F}}(s)\cap
pred^{\mathcal{T}_{F}}(s^{\prime})\neq\emptyset$).

We will now prove by contradiction that Type-2 variations do not exist for
Level-1 functionalist frameworks. Let us say that there is a pre-falsification
achieved through the existence of Type-2 variation for the theory
$\mathcal{T}_{F}$. This means that there is a $v:P\rightarrow P$ such that the
inference data $o_{r}$ is kept constant while the prediction data changes from
$o_{i}$ to $o_{i}^{\prime}=obs(v(p))\neq o_{i}$, and the corresponding
predictions do not overlap either i.e. we have $pred(o)\cap
pred(o^{\prime})=\emptyset$ for $v$. From discussions in [1], we take the
prediction $o_{i}$ and inference $o_{r}$ data to correspond to particular
encodings of the functional states and outputs respectively in order to be
able to apply the functionalist model of interest. This means that
$o_{i}\equiv\mathcal{H}$, $obs(v(p))=o_{i}^{\prime}\equiv\mathcal{H}^{\prime}$
and $o_{r}=\\{o\\}$ (outputs). As $o_{i}\neq o_{i}^{\prime}$, we have that the
corresponding encodings $\mathcal{H}$ and $\mathcal{H}^{\prime}$ are different
in some measurable way. Since $v$ is a variation that maintains the inference
data for the same experiments (i.e. maintains all input-output pairs across
the variation), that would mean that for every $h^{j}\in\mathcal{H}$, there is
a $h^{k}\in\mathcal{H}^{\prime}$ such that
$(h_{(t-1)}^{j},x^{t})\rightarrow(h_{t}^{j^{\prime}},o^{t})$ and
$(h_{(t-1)}^{k},x^{t})\rightarrow(h_{t}^{k^{\prime}},o^{t})$ for all input-
output pairs $(x^{t},o^{t})$. This means that both $\mathcal{H}$ and
$\mathcal{H}^{\prime}$ have the same functional structure by definition (and
are elements of $\\{\mathcal{H}_{i}^{\mathcal{T}_{F}}\\})$. This would also
mean that the $h^{j}$ and $h^{k}$ from above correspond to equivalent
functional states (say $s$ and $s^{\prime}$) such that
$s=abs(h^{j})\in\\{s\\}$, $s^{\prime}=abs(h^{k})\in\\{s^{\prime}\\}$ with
$pred^{\mathcal{T}_{F}}(s)=pred^{\mathcal{T}_{F}}(s^{\prime})$ (or
$pred^{\mathcal{T}_{F}}(s)\cap
pred^{\mathcal{T}_{F}}(s^{\prime})\neq\emptyset$).

The $pred$ function for $\mathcal{T}_{F}$ was defined as a composition of two
functions $pred^{\mathcal{T}_{F}}$ and $abs$ functions. Thus for any
(arbitrary) single prediction
$pred(o)=pred(h^{j})=pred^{\mathcal{T}_{F}}(abs(h^{j}))$ and the corresponding
prediction on the data obtained after the variation -
$pred(o^{\prime})=pred(h^{k})=pred^{\mathcal{T}_{F}}(abs(h^{t}_{k}))$. Since
we know that for any $h^{j}\in\mathcal{H}$ and $h^{k}\in\mathcal{H}^{\prime}$
that share the same functional relationships, we have
$pred^{\mathcal{T}_{F}}(abs(h^{j}))=pred^{\mathcal{T}_{F}}(abs(h^{k}))$. This
means that for any single prediction $pred(o)=pred(o^{\prime})$ (or rather
$pred(o)\cap pred(o^{\prime})\neq\emptyset$). Since the choice of $h^{j}$ was
arbitrary, we have $pred(o)\cap pred(o^{\prime})\neq\emptyset$ over the
prediction dataset. But we began with the assumption that $v$ is a Type-2
variation with $pred(o)\cap pred(o^{\prime})=\emptyset$. This is clearly a
contradiction and thus no such Type-2 variation $v$ exists for a Level-1
functionalist theory $\mathcal{T}_{F}$. Thus prediction and inference data are
independent via Type-1 variations and there no substitutions. Hence we have
proved that such functionalist frameworks cannot be pre-falsified by the
substitution argument presented in [1]. While the unfolding argument can be
seen as a special case of the substitution argument, the above proof indicates
that the claim about Level-1 functionalist theories like GWT, HOT and PP
frameworks being unaffected [2] by the unfolding (and now substitution)
argument still hold. While we have shown that no substitutions exist for
Level-1 functionalist theories, there is an underlying question about the
status of Level-2 functionalist frameworks and whether some of the current
leading frameworks fall under that category. This will be the focus of our
next section.

##  5 Level-1 vs Level-2 Functionalism

One might be tempted to characterize frameworks that depend on the CSA
descriptions as Level-2 functionalist theories, since the different states in
each individual representation is defined in terms of it’s relationship to
other states under the same representation. (The author is not sure what the
Level-1 structure corresponds to if functional structure is defined at
Level-2.) However with functional structure defined based on Level-2 state
encoding, we can see how one can look at the different encodings of the same
Level-1 functional state and mistake them to be different Level-1 states - for
eg: both $111^{1}$ and $001^{2}$ in Fig.(2) correspond to same Level-1 state
$E$ in the context of the entire functional structure. However we see that
$111^{1}\rightarrow 011^{1}$ and $001^{2}\rightarrow 101^{2}$ (and
$111^{2}\rightarrow 000^{2}$) which can lead to an erroneous conclusion that
$111^{1}$ and $001^{2}$ correspond to different Level-1 functional states. The
important thing to note is the superscript $1$ and $2$ over the states that
correspond to different Level-2 descriptions and we must view these state
encodings within the context of the entire CSA representation i.e. the overall
functional structure to determine if they correspond to the same Level-1
functional state. If we fail to do that and distinguish between $111^{1}$ and
$001^{2}$, then we are capturing something else about the states - like the
difference in architecture - feedback vs no-feedback that is reflected in the
$\phi$ calculations for IIT [5]. We would then be more accurate in describing
frameworks that differentiate based on these Level-2 descriptions by the
specific factor/constraint that produced the difference in representations in
the 1st place (for eg: it would be more accurate to call the framework that
distinguishes between CSA representations 1 and 2 in Fig.(2) as a feedback
theory of consciousness).

This is the case in [8] where the author discusses a specific type of
functionalism called machine state functionalism where ‘any creature with a
mind can be regarded as a Turing machine (an idealized finite state digital
computer), whose operation can be fully specified by a set of instructions (a
“machine table” or program) each having the form - If the machine is in state
$S_{i}$, and receives input $I_{j}$, it will go into state $S_{k}$ and produce
output $O_{l}$ (for a finite number of states, inputs and outputs)’ [9]. The
machine table as defined above would be a collection of these instructions of
the form $(S_{i},I_{j})\rightarrow(S_{k},O_{l})$ which is similar to how we
characterized functional structure here. However this particular definition of
machine state functionalism is adopted at a Level-2 description in [8], which
leads to Level-2 functionalism (as discussed above). As a result, changes to
the automaton from feedback to feedforward architectures that is reflected in
the Level-2 state encodings are viewed as changes to the functional structure.
Since the Krohn-Rhodes decomposition [10] used to achieve the transformation
between the two state representations assumes the preservation of the
underlying (Level-1) functional structure, there is now confusion as to what
actually constitutes functional structure. We could have different definitions
for Level-1 and Level-2 functional structures, where the Level-1 structure is
preserved across different Level-2 representations (arising from different
physical realizations) and it is not as per the definition of the Level-2
structure. While this would be coherent in principle, results and claims now
trivially become about which definitions have been adopted. In order to avoid
this, it is important that we adopt and practice the use of terms in a
consistent manner by determining which of the two definitions of functional
structure is closer to what we originally intended to capture. This difference
between defining functional structure at Level-1 vs Level-2 descriptions also
corresponds to the unresolved debate between role and realizer functionalism
[9]. Given how functionalism and functional states are conceptualized to be
independent of the realization details plus given the arguments in favor of
role functionalism [9], the author leans more towards a Level-1 definition as
being closer to what functional structure intended to characterize.

Consequently, claims made about Global Workspace theory (GWT), Global Neuronal
Workspace theory (GNWT) along with other functionalist frameworks in [8] based
on Lemma 2.9 do not follow unless one adopts a Level-2 definition of
functional structure (According to Definition 2.8 in [8], a Level-1
functionalist picture as defined in the previous section will not satisfy
Lemma 2.9 for the same reasons it is immune to the substitution argument).
While the description of GNWT from [11] that is used in [8] highlight a very
architecture specific Level-2 type description (which could motivate one to
view GNWT as a Level-2 functionalist framework), the authors of [11] clearly
state in their paper that this particular description is “from a neuronal
architecture standpoint.” Furthermore they start this description by pointing
out that the original GNWT relies on the following main assumptions - “that
conscious access is global information availability: what we subjectively
experience as conscious access is the selection, amplification and global
broadcasting, to many distant areas, of a single piece of information selected
for its salience or relevance to current goals” \- which is more of a Level-1
description that is independent of how selection, amplification and global
broadcasting is realized. The authors in [2] make a similar point with respect
to Global Workspace, Higher-Order Thought (HOT) and Predictive Processing (PP)
theories which they characterize as Level-1 functionalist frameworks - “The
unfolding argument does not apply to these theories because they propose that
systems are conscious in so far as they implement the right kind of
function–independently of the causal structure. Of course, these theories are
usually couched in terms of recurrent or top-down processing, or other
seemingly causal-structure terminology, but they can be formulated in other
kinds of networks too” and provide a feedforward toy model of GWT to further
strengthen their case. We can leave the status of whether these leading
theories of consciousness are Level-1 or Level-2 functionalist frameworks as
open, but reiterate that claims of pre-falsification only apply if their
functional structure is couched in Level-2 descriptions. Since we can always
construct an abstraction function $abs$ to map from the Level-2 to Level-1
description, it is possible to construct a Level-1 version of a Level-2
framework and avoid pre-falsification via substitution. On the other hand, one
could argue that we can always construct a specific Level-2 realization of a
Level-1 theory (by constraining aspects of the architecture, use of neuronal
units, etc), which would then be fallible to the substitution argument. This
would at best only pre-falsify by substitution that specific Level-2 version
and not the underlying Level-1 framework, and at worst bring us back to square
one on whether or not the Level-2 description is a functionalist framework to
begin with. Finally, if we used the definition of functional structure from
[8] and followed the results of [5], it seems like we would have both
functionalist and causal structure being defined on the same Level-2 (CSA)
descriptions which appears to be contradictory to how many view both and would
only lead to further confusion. It would be important to pin this down in a
coherent and consistent manner before they lead to further debates over pre-
falsification.

##  6 Summary & Conclusion

The work presented in [1] represents a good first step in formalizing the
techniques used in the science of consciousness. The corresponding
substitution arguments would have important implications in the field of
consciousness by pointing at fundamental problems in the manner in which
experiments are conducted that would pre-falsify many of the current major
frameworks of consciousness. However we find that claims like - “We come to a
surprising conclusion: a widespread experimental assumption implies that most
contemporary theories of consciousness are already falsified” by the authors
are currently unjustified once you take into account that independence was
defined to trivially imply pre-falsification. In this reply, we introduced a
more complete definition of independence that allowed us to expand variations
from the original paper into Type-1 and Type-2. We then redefined Theorem 3.10
from [1] using Type-2 variations and then explored substitutions in the case
of neural networks, state machines with and without feedback and Turing
machines. We showed that for these particular example cases, the substitutions
of interest were not of Type-2 with respect to a Level-1 functionalist
framework and hence does not entail pre-falsification. We then presented a
formal proof of the non-existence of Type-2 variations for Level-1
functionalist frameworks of consciousness and completed the reply with a
discussion of Level 1 and 2 functionalist theories and where contemporary
theories of consciousness fall into. Currently, we are uncertain as to whether
or not most leading functionalist pictures of consciousness are of Level-1 or
2. What is more certain however is that they have not already been pre-
falsified independent of that determination.

## References

  * [1] Kleiner, Johannes and Hoel, Erik, "Falsification and consciousness." Neuroscience of Consciousness, (2021): niab001.
  * [2] Doerig, Adrien, et al., “The unfolding argument: Why IIT and other causal structure theories cannot explain consciousness.” _Consciousness and Cognition_ , 72 (2019): 49-59.
  * [3] Hanson, Jake R, _Personal Communication_ , 2020.
  * [4] Hanson, Jake R, and Sara I. Walker. “Integrated Information Theory and Isomorphic Feed-Forward Philosophical Zombies,” Entropy 21.11 (2019): 1073.
  * [5] Hanson, Jake R., and Sara I. Walker, “Formalizing Falsification of Causal Structure Theories for Consciousness Across Computational Hierarchies,” arXiv preprint, arXiv:2006.07390 (2020).
  * [6] Chalmers, D. J., “A computational foundation for the study of cognition,” Unpublished, 1993.
  * [7] Hopcroft, John and Ullman, Jeffrey, Introduction to Automata Theory, Languages, and Computation, Addison–Wesley, Reading Mass, 1979.
  * [8] Kleiner, Johannes, “Brain states matter. A reply to the unfolding argument,” _Consciousness and Cognition,_ 85 (2020): 102981.
  * [9] Levin, Janet, “Functionalism,” Stanford Encyclopedia of Philosophy, (2004).
  * [10] Krohn, K, Rhodes, J, “Algebraic theory of machines - I: Prime decomposition theorem for finite semigroups and machines,” Transactions of the American Mathematical Society, 1965, 116, 450–464.
  * [11] Dehaene, Stanislas, Jean-Pierre Changeux, and Lionel Naccache, “The global neuronal workspace model of conscious access: from neuronal architectures to clinical applications,” Characterizing consciousness: From cognition to the clinic?, (2011): 55-84.


# Direct coupling of ferromagnetic moment and ferroelectric polarization in
BiFeO3

Shiro Kawachi,1 Shin Miyahara,2 Toshimitsu Ito,3 Atsushi Miyake,4 Nobuo
Furukawa,5 Jun-ichi Yamaura,1 and Masashi Tokunaga4 1Materials Research Center
for Element Strategy, Tokyo Institute of Technology, Yokohama, Kanagawa
226-8503, Japan.
2Department of Applied Physics, Fukuoka University, Jonan-ku, Fukuoka
814-0180, Japan.
3National Institute of Advanced Industrial Science and Technology (AIST),
Tsukuba, Ibaraki 305-8562, Japan.
4The Institute for Solid State Physics, The University of Tokyo, Kashiwa,
Chiba 277-8581, Japan.
5Department of Physics and Mathematics, Aoyama Gakuin University, Sagamihara,
Kanagawa 229-8558, Japan.

###### Abstract

The spin-driven component of electric polarization in a single crystal of
multiferroic BiFeO3 was experimentally investigated in pulsed high magnetic
fields up to 41 T. Sequential measurements of electric polarization for
various magnetic field directions provide clear evidence of electric
polarization normal to the hexagonal $c$ axis (${\bm{P}}_{\rm t}$) in not only
the cycloidal phase, but also the field-induced canted antiferromagnetic
phase. The direction of ${\bm{P}}_{\rm t}$ is directly coupled with the
ferromagnetic moment in the canted antiferromagnetic phase, and thus
controlled by changing the direction of the applied magnetic field. This
magnetoelectric coupling is reasonably reproduced by the metal–ligand
hybridization model.

††preprint: APS/123-QED

The crossed coupling of magnetism and ferroelectricity in multiferroic
materials has unified two major research areas in condensed matter physics
that were traditionally distinguished. In addition to being of interest from a
point of view of basic science, magnetoelectric coupling is expected to allow
the development of innovative devices, such as electrically controllable
magnetic memory and multi-bit memory Bibes2008 ; Scott2007 . While most
multiferroic materials are active only at low temperatures, BiFeO3, which
exhibits remarkable multiferroic properties at room temperature, has attracted
much interest Catalan2009 .

BiFeO3 crystallizes in the non-centrosymmetric $R3c$ space group within a
trigonal crystal system at temperatures below $\sim 1100$ K Kubel1990 . Atomic
displacement from the perovskite-type structure results in spontaneous giant
electric polarization ${\bm{P}}_{\rm s}$ along the hexagonal $c$ axis, which
corresponds to $\langle 111\rangle_{\rm c}$ in cubic perovskites [the $+Z$
direction in Fig. 1(a)] Catalan2009 ; Shvartsman2007 ; Lebeugle2007 . The
$S=5/2$ spins of ${\rm Fe}^{3+}$ ions form a cycloidal magnetic order with the
$XY$-component of the magnetic propagation vector ${\bm{q}}$ pointing in the
hexagonal direction $\langle 100\rangle_{\rm h}$ below temperatures of $\sim
640$ K Sosnovska1982 . Three-fold rotational symmetry allows the coexistence
of three equivalent magnetic domains Johnson2013 . The direction of $\bm{q}$
can be controlled by flopping ${\bm{P}}_{\rm s}$ with an electric field
Lebeugle2008 ; Lee2008PRB . This flop of ${\bm{P}}_{\rm s}$ is, however,
accompanied by crystal deformation and hence often degrades the crystal, which
is unfavorable for application.

Figure 1: (a) Illustration of the arrangement of Fe ions in BiFeO3. ${\bm{a}}$
and ${\bm{c}}$ represent unit vectors in the hexagonal cell. ${\bm{P}}_{\rm
s}$ indicates spontaneous ferroelectric polarization below $\sim$1100 K. (b)
Schematic drawing of the definition of azimuthal angles for the applied field
and cycloidal ${\bm{q}}$ vector, $\phi_{H}$ and $\phi_{q}$ respectively. (c)
Photograph of the single crystal used for the measurements.

In addition to ${\bm{P}}_{\rm s}$, BiFeO3 has parasitic ferroelectric
polarization originating from the magnetic order. In the cycloidal phase,
BiFeO3 undergoes a magnetic phase transition with the application of a
magnetic field of about 20 T Kadomtseva2004 . Extrapolation of the
magnetization curve in the field-induced phase to a zero magnetic field has a
finite intercept, which suggests the occurrence of the canted
antiferromagnetic (CAFM) order with a net ferromagnetic moment of $\sim 0.04$
µB/Fe on the $XY$ plane Kadomtseva2004 ; Ruette2004 ; Pyatakov2009 ;
Tokunaga2015 ; Kawachi2017 . In general, the cycloidal magnetic order is known
to be accompanied by spin-driven electric polarization Katsura2005 ; Arima2011
; Kaplan2011 ; Pyatakov2015 ; Miyahara2016 . Field-induced suppression of this
phase may affect the electric polarization. Recent experiments using single
crystals revealed spin-driven electric polarization in two directions
longitudinal and transverse to the $Z$ direction Park2011 ; Lee2013 ; Lee2015
; Tokunaga2015 . The transverse electric polarization (${\bm{P}}_{\rm t}$) in
the cycloidal state is explained by the inverse Dzyaloshinskii–Moriya (DM)
model Lee2015 ; Miyahara2016 ; Tokunaga2015 , with which the emergence of
${\bm{P}}_{\rm t}$ normal to the ${\bm{q}}$-${\bm{Z}}$ plane in a zero field
is predicted. Reorientation of magnetic domains is accompanied by the
reorientation of ${\bm{P}}_{\rm t}$. This effect appears as a non-volatile
memory effect that can be controlled by a magnetic field or electric field
even at room temperature Tokunaga2015 ; Kawachi2016 .

In this study, we determined the direction and absolute value of
${\bm{P}}_{\rm t}$ and its field-angle dependence in a single crystal of
BiFeO3 through a series of magnetoelectric measurements by changing the
direction of the applied magnetic field. The results revealed the doubling of
${\bm{P}}_{\rm t}$ at the transition from the cycloidal to the CAFM state.
Semi-quantitative explanation of the results by the metal–ligand hybridization
mechanism clarified the microscopic origin for the spontaneous coupling
between ferromagnetism and ferroelectricity in this material, and demonstrated
that prominent multiferroic coupling could be realized without complicated
spin order.

Single crystals of BiFeO3 were grown adopting the laser-diode heating
floating-zone method Ito2011 . The obtained crystal almost entirely consisted
of a single ferroelectric domain of ${\bm{P}}_{\rm s}$. A crystal used for the
measurements was cut into a rectangular parallelepiped shape of $2.34\times
0.49\times 0.85$ mm3 along the crystal axes as shown in Fig. 1(c). To
determine the crystal orientation with its chirality, we conducted an x-ray
diffraction measurement using a curved imaging plate on a RIGAKU VariMax with
RAPID diffractometer at the wavelength of Mo-K$\alpha$ radiation. The crystal
chirality was identified by the intensities of Bijvoet-pair-related
reflections of $\\{6\bar{3}6\\}_{\rm h}$ and $\\{6\bar{3}\bar{6}\\}_{\rm h}$
on a hexagonal setting owing to the loss of inversion symmetry based on atomic
coordinates given in the literature Kubel1990  (See Supplementary Material).
The determined $a$ and $c$ hexagonal axes are chosen as the $+X$ and $+Z$
directions. By this definition, ${\bm{P}}_{\rm s}$ points in the $+Z$
direction as shown in Fig. 1(a). Magnetic-field-induced changes in electric
polarization were measured by integrating (de)polarization currents
Mitamura2007  induced in pulsed magnetic fields generated using a 55-T magnet
for a duration of 8 ms. These experiments only determined the relative changes
in electric polarization ($\Delta P$) with respect to the initial values for a
zero field. To determine the zero level of electric polarization, we performed
a series of experiments while changing the field angle using a sample rotator
as discussed in the following.

Figure 2:  (a) Magnetic-field-induced changes in electric polarization along
$X$ at $\phi_{H}=45^{\circ}$. Traces in the first and second cycles are
respectively shown by dotted and solid lines. Magnetic-field-induced changes
in electric polarization along (b) $X$, (c) $Y$, and (d) $Z$ directions,
measured at various field angles. Traces are vertically offset so that the
initial point of each curve is the end point of the previous curve. The inset
presents a rectangular parallelepiped sample of BiFeO3 (dark gray area) with
electrodes (light gray area) to visualize the measurement direction.

Typical results are shown in Fig. 2(a). Prior to this experiment, we applied a
magnetic field ($H$) with a magnitude up to 41 T at an azimuthal angle of
$\phi_{H}=91^{\circ}$ ($\sim H\parallel Y$) as defined in Fig. 1(b). We then
measured $\Delta P$ along the $X$ direction ($\Delta P_{X}$) as a function of
the magnetic field at $\phi_{H}=45^{\circ}$. In the first field cycle, $\Delta
P_{X}$ has a hysteretic trace below 10 T [red dotted line in Fig. 2(a)], which
is ascribed to the reorientation of the magnetoelectric domains. Here, the
change in $P_{X}$ from 0 to 20 T becomes small after reorientation of the
magnetoelectric domain. The generalized inverse DM model predicts the
emergence of the largest $P_{X}$ in the reoriented state and absence of it in
the CAFM state. The observed reduction in $\Delta P_{X}$ is therefore a
puzzling phenomenon Tokunaga2015 .

To unambiguously determine the spin-driven components in electric
polarization, we sequentially measured $\Delta P_{X}$ while systematically
changing $\phi_{H}$. Representative data of $\Delta P_{X}$, $\Delta P_{Y}$,
and $\Delta P_{Z}$ are presented in Figs. 2(b)–(d) as functions of $H$ applied
at various $\phi_{H}$. All data presented in the following were recorded at a
temperature of 4.2 K to minimize the effect of leakage current. Throughout
this letter, $P$–$H$ curves obtained in the first and second magnetic field
sweeps for each $\phi_{H}$ are respectively represented by dotted and solid
lines. In Figs. 2(b) and 2(c), all traces are offset on the vertical axis so
that the starting point of the trace coincides with the endpoint of the
previous trace. The curves of $\Delta P_{X}$ in Fig. 2(b) were measured in the
order of $\phi_{H}=91^{\circ}\rightarrow 45^{\circ}\rightarrow
67^{\circ}\rightarrow 45^{\circ}\rightarrow 79^{\circ}\rightarrow
45^{\circ}\rightarrow 117^{\circ}\rightarrow 45^{\circ}\rightarrow
134^{\circ}$. We here determined the zero level of $P_{X}$ by taking the
average of the largest and smallest values of $P_{X}$ for a zero field.
Adopting the present definition of the zero level, we can reasonably reproduce
the observed angular dependence using a theoretical model as will be seen
later.

The data at $\phi_{H}=134^{\circ}$ were obtained after application of a field
at $\phi_{H}=45^{\circ}$. Thereby, an irreversible change in $P_{X}$ below 10
T in the first field sweep is due to the $90^{\circ}$ rotation of the
${\bm{q}}$ vector, which appears as a sign change in $P_{X}$ as shown by the
blue dotted line. A steep change in $P_{X}$ at $\sim 18$ T corresponds to the
transition from the cycloidal phase to the CAFM phase. The largest value of
$P_{X}$ in the cycloidal state is thus determined as $\sim 300$ µC/m2 at a
zero field. Meanwhile, $P_{X}$ in the CAFM phase is almost twice this value.

Moreover, we also carried out for $P_{Y}$ on the same crystal in the order of
$\phi_{H}=177^{\circ}\rightarrow 155^{\circ}\rightarrow 140^{\circ}\rightarrow
126^{\circ}\rightarrow 111^{\circ}\rightarrow 90^{\circ}$. The results are
plotted in Fig. 2(c) adopting a vertical offset as in the previous case. Here,
the zero level of $P_{Y}$ is defined as the average value of $P_{Y}$ for the
zero field after the application of fields along $\phi_{H}=90^{\circ}$ and
$177^{\circ}$. We again observe larger $P_{Y}$ in the CAFM phase for fields
stronger than 18 T.

Meanwhile, $\Delta P_{Z}$ has negligible angular dependence as shown in Fig.
2(d). In this case, we can hardly determine the zero level for $P_{Z}$.
$P_{Z}$ steeply decreases by $\sim 240$ µC/m2; i.e., total $P_{Z}$ slightly
reduces at the cycloid-to-CAFM transition around 18 T independently of the
azimuthal angle of the applied field.

Figure 3:  (a) Angular dependence of the electric polarization along $X$ and
$Y$ directions at µ${}_{0}H=0$ and 10 T in the cycloidal phase shown by solid
circles and crosses, respectively. The solid line indicates the fitting
function of eq. (3) and/or (4) for solid circles at 0 T. Polar plots
presenting the dependencies of (b) $P_{X}$ and (c) $P_{Y}$ on the magnetic
field angle after removal of the magnetic field in the cycloidal phase. Open
symbols are projected points of the experimental data at $\phi_{H}$ (solid
symbols) to $\phi_{H}+\pi$. (d) Schematics of relative directions between
${\bm{P}}_{\rm t}$ and the $XY$ component of the ${\bm{q}}$ vector in the
cycloidal phase.

Figure 3(a) presents $P_{X}$ and $P_{Y}$ as functions of $\phi_{H}$ at 0 T in
the cycloidal phase ($P^{\rm cy}$). Here, the data were extracted in the
reoriented domain state after the first field cycle at each $\phi_{H}$.
According to Bordacs $et$ $al$., the direction of the ${\bm{q}}$ vector for a
zero field may slightly deviate from the direction perpendicular to the
applied field owing to small anisotropy within the $XY$ plane Bordacs2018 ;
Fishman2018 . This deviation, however, plays only a minor role because $P_{X}$
and $P_{Y}$ for a field strength of 10 T, which is higher than the strength of
pinning threshold fields of $\sim 5$ T Bordacs2018 , follow almost the same
functional forms as shown by crosses in Fig. 3(a). Figures 3(b) and 3(c)
exhibit polar plots of the $P_{X}$ and $P_{Y}$ shown in Fig. 3(a). The solid
circles represent the experimental data, while the open ones are projection of
the data at $\phi_{H}$ to $\phi_{H}+\pi$. For some field angles, we confirmed
that the magnetoelectric effect appears symmetric with respect to the sign
change in the applied field as shown in Supplementary Material.

Continuous changes in $P_{X}$ and $P_{Y}$ indicate that $\phi_{q}$ smoothly
follows changes in $\phi_{H}$ in the cycloidal state. As for the slight
difference between the amplitudes of $P_{X}$ and $P_{Y}$, we cannot rule out
possible error in the estimation of the effective surface area of the sample.
$P_{X}$ and $P_{Y}$ respectively trace simple functions of $-P_{\rm
t}\sin{2\phi_{H}}$ and $-P_{\rm t}\cos{2\phi_{H}}$. This means that
${\bm{P}}_{\rm t}$ rotates in the $XY$ plane while the magnitude is almost
unchanged.

Figure 3(d) illustrates the relationship between ${\bm{P}}_{\rm t}$ and
${\bm{q}}$ on the $XY$ plane determined by the experimental data at several
magnetic field angles in the cycloidal phase; ${\bm{q}}\perp{\bm{P}_{\rm t}}$
at $\phi_{H}=(1/2+2n/3)\pi$ and ${\bm{q}}\parallel{\bm{P}_{\rm t}}$ at
$\phi_{H}=2{\pi}n/3$ with integer $n$. ${\bm{P}_{\rm t}}$ rotates in the
opposite direction to the magnetic field.

Figure 4:  (a) Field-angle dependence of the electric polarization along $X$
(orange) and $Y$ (green) directions at µ${}_{0}H=40$ T in the CAFM phase. The
solid line shows the fitting functions of $P_{X}$ and $P_{Y}$ obtained using
eq. (3). Polar plots represent the magnetic-field-angle dependencies of (b)
$P_{X}$ and (c) $P_{Y}$ at 40 T in the CAFM phase. Experimental data and a
theoretical curve are respectively shown by symbols and a solid line. The open
symbols are a projection of experimental data at $\phi_{H}$ (solid symbols) to
$\phi_{H}+\pi$. (d) Schematic relation between the ferromagnetic moment (blue
arrow) and ${\bm{P}}_{\rm t}$ (green arrow) in the CAFM phase.

Figures 4(a)–4(c) show the $\phi_{H}$ dependencies of $P_{X}$ and $P_{Y}$ at
40 T in the CAFM phase. $P_{X}$ and $P_{Y}$ have the same functional formula
as in the cycloidal phase. The relative orientation determined experimentally
between the ferromagnetic component in the CAFM state (${\bm{m}}$) and
${\bm{P}}_{\rm t}$ is schematically shown in Fig. 4(d). ${\bm{m}}$ follows a
curve parallel to ${\bm{H}}$ owing to the Zeeman energy Tokunaga2015 ;
Kawachi2017 , while ${\bm{P}}_{\rm t}$ rotates in the opposite direction to
the rotation of ${\bm{H}}$ by a double angle. This result means that if we
rotate ${\bm{m}}$ at a frequency of $\omega$, we will have rotation of
${\bm{P}}_{\rm t}$ at the frequency of $2\omega$.

Let us discuss the origin for ${\bm{P}}_{\rm t}$ coupled with the magnetic
order. The multiferroic behavior is usually reproduced using spin-dependent
electric polarization:

$\displaystyle{p}^{\alpha}_{i}$ $\displaystyle=$
$\displaystyle\sum_{\beta\gamma}\Lambda^{\alpha}_{\beta\gamma}S_{i}^{\beta}S_{i}^{\gamma}$
(1) $\displaystyle{p}^{\alpha}_{i\,j}$ $\displaystyle=$
$\displaystyle\Pi_{ij}^{\alpha}{\bm{S}}_{i}\cdot{\bm{S}}_{j}+\sum_{\beta}d_{ij}^{\alpha\beta}\left({\bm{S}}_{i}\times{\bm{S}}_{j}\right)^{\beta},$
(2)

where $\alpha,\beta,\gamma=X,Y,Z$ and $\Lambda,\Pi_{ij},d_{ij}$ are coupling
constants Moriya1968 . ${p}_{i}$ is a single-spin term that is induced by
metal–ligand hybridization (ML) and ${p}_{ij}$ is a spin pair term that
includes the electric polarization induced by symmetric exchange striction
(ES) and antisymmetric inverse DM interaction. We consider ${\bm{p}}_{i}$ at
Fe ion sites and ${\bm{p}}_{ij}$ for the nearest neighboring pair of Fe ions.
Locations of Fe ions are represented by
${\bm{r}}_{i}=n_{i}^{x}{\bm{e}}_{x}+n_{i}^{y}{\bm{e}}_{y}+n_{i}^{z}{\bm{e}}_{z}$
for a set of integers $(n^{x}_{i},n^{y}_{i},n^{z}_{i})$ and unit vectors
${\bm{e}}_{x}$, ${\bm{e}}_{y}$, and ${\bm{e}}_{z}$ connecting adjacent Fe
sites as shown in Fig. 1(a). The non-zero terms of ${\bm{p}}_{i}$ in BiFeO3
are originated from a $C_{3}$ point group Matsumoto2017  and any term of
${\bm{p}}^{\alpha}_{ij}$ is allowed to be non-zero from a symmetry argument
Miyahara2016 . We defined the coupling constants of $p^{\alpha}_{ij}$ for one
of the bonds along ${\bm{e}}_{z}$ as eqs. (S2) and (S3) in the Supplementary
Material; (e.g., $\Pi^{X}\equiv\Pi_{ii+z}^{X}$, $d^{XY}\equiv d^{XY}_{ii+z}$),
and the other coupling constants for the inequivalent five bonds are expressed
using the rotational-symmetry operation.

The spin at the $i$-th site can be represented as
${\bm{S}}_{i}=\left(S\cos{\phi_{q}}\sin{{\bm{q}}\cdot{\bm{r}}_{i}},S\sin{\phi_{q}}\sin{{\bm{q}}\cdot{\bm{r}}_{i}},S\cos{{\bm{q}}\cdot{\bm{r}}_{i}}\right)$
and
${\bm{S}}_{i}=\\{S\sin{\left[\eta+(-1)^{n_{i}}\phi_{H}\right]},-(-1)^{n_{i}}S\cos{\left[\eta+(-1)^{n_{i}}\phi_{H}\right]},0\\}$
in cycloidal and CAFM states, respectively, whereas
$\bm{q}=\left(q_{0}\cos{\phi_{q}},q_{0}\sin{\phi_{q}},\sqrt{3}\pi/a_{\rm
c}\right)$ and $q_{0}=2\pi/\lambda$ with the cycloidal period $\lambda=620$ Å
and lattice constant of the pseudocubic cell $a_{\rm c}=3.96$ Å. $\eta$ ($\ll
1$) represents the spin canting angle in the CAFM state. In the cycloidal
state, we ignore the minor spin component normal to the ${\bm{q}}$–${\bm{Z}}$
plane Ramazanoglu2011  for simplicity.

By taking the spatial average in the cycloidal state, we can evaluate
macroscopic polarizations as described in Supplementary Material. In the
cycloidal state, all mechanisms expressed in eqs. (1) and (2) can explain the
emergence of spin-driven electric polarization ${\bm{P}}_{\rm mag}$ in the
form

$\displaystyle{\bm{P}}_{\rm mag}=\left(-P_{\rm t}^{0}\sin{2\phi_{H}},-P_{\rm
t}^{0}\cos{2\phi_{H}},P_{Z}\right).$ (3)

Table 1: Values of $P_{\rm t}^{0}$ in the eq. (3) of IDM, ES, and ML mechanisms in the cycloidal and CAFM phases, respectively. Details of these calculations are described in Supplementary Material.  | IDM | EX | ML
---|---|---|---
Cycloid | $\frac{\sqrt{6}}{4}q_{0}a_{\rm c}S^{2}\left(d^{XY}+d^{YX}\right)$ | $\frac{\left(q_{0}a_{\rm c}\right)^{2}}{2}S^{2}\Pi_{Y}$ | $-\frac{1}{2}S^{2}\Lambda^{X}_{XY}$
CAFM | $0$ | $0$ | $-S^{2}\Lambda^{X}_{XY}$

Here, $P_{\rm t}^{0}$ in each phase is tabulated in Table I. As shown by the
solid lines in Figs. 3(a)–3(c), eq. (3) reasonably reproduces the experimental
results for the cycloidal state when setting $\phi_{H}=\phi_{q}+\pi/2$ and
$P_{\rm t}^{0}\sim 300\,{\rm\mu C/m}^{2}$. In the CAFM phase, $\phi_{H}$
dependence is again expressed by eq. (3). Here, ES and IDM mechanisms suggest
$P_{\rm t}^{0}=0$ contrary to the experimental results. On the other hand, the
ML mechanism predicts doubling of the $P_{\rm t}^{0}$ in the CAFM phase. This
predicted doubling of $P_{\rm t}^{0}$ together with the characteristic angular
dependence in eq. (3) reasonably reproduce the experimental results as shown
in Figs. 4(b) and 4(c). Therefore, spin-induced polarization in the cycloidal
and CAFM phases can mainly be ascribed to the ML mechanism.

In the ML mechanism, spin-driven electric polarization is determined by the
local spin component at each Fe ion. For example, amplitude of $p_{i}^{Y}$ is
proportional to $(S_{i}^{X})^{2}$ when $\phi_{H}=\pi/2$ ($\phi_{q}=0$) as
shown in Supplementary Material. In the cycloidal state, spatial average of
sinusoidally changing $(S_{i}^{X})^{2}$, namely sinusoidally modulating
$p_{i}^{Y}$, results in the reduction of $P_{Y}$ by a factor of 1/2 from that
in the CAFM state.

Finite magnetoelectric effects in the CAFM phase have been argued in some
earlier reports Popov2004 ; Kadomtseva2004 ; Tokunaga2015JMMM ; Kawachi2017 .
Different from the earlier arguments, our results indicate emergence of the
spontaneous electric polarization in the CAFM phase even in the zero-field
limit, which is crucially important to utilize the CAFM states in thin films
or Co-substituted BiFeO3 at zero field Yamamoto2016 ; Sando2013 . Although we
studied the magnetic control of ${\bm{P}}_{\rm t}$ in the CAFM phase at a
temperature of 4.2 K, this phase exists up to a temperature higher than room
temperature Tokunaga2010 . In addition, the $P_{t}^{0}$ remains finite in the
limit of $\eta\rightarrow 0$ meaning simple collinear antiferromagnetic state
can induce spin-driven polarization in this material. In many multiferroic
materials, frustrated spin systems have been widely studied to realize
complicated spin structures, which results in low ordering temperature. The
present result suggests that magnetically controllable electric polarization
of $\sim 600\,{\rm\mu C/m}^{2}$ can be realized even in a simple
antiferromagnet. This finding will pave a way to develop multiferroic
materials useful at room temperature.

In summary, we measured the angular dependence of field-induced changes in the
electric polarization for a single crystal of BiFeO3. The electric
polarization normal to the hexagonal axis was systematically controlled by
changing the azimuthal angle of the applied magnetic field. Results reveal
that this component is strengthened by the field-induced transition from the
cycloidal phase to the canted antiferromagnetic phase. We explained this spin-
driven component of electric polarization using the metal–ligand hybridization
mechanism.

This work was supported by the JSPS Grant No. JP16K05413 and the MEXT Element
Strategy Initiative to Form Core Research Center.

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