Integrated information theory (IIT) 4.0: Formulating the properties of
phenomenal existence in physical terms

Larissa Albantakis1, Leonardo Barbosa1,2, Graham Findlay1,3, Matteo Grasso1,
Andrew M Haun1, William Marshall1,4, William GP Mayner1,3, Alireza
Zaeemzadeh1, Melanie Boly1,5, Bjørn E Juel1,6, Shuntaro Sasai1,7, Keiko
Fujii1, Isaac David1, Jeremiah Hendren1,8, Jonathan P Lang1, Giulio Tononi1*

1 Department of Psychiatry, University of Wisconsin, Madison, WI 53719, USA

2 Fralin Biomedical Research Institute at VTC, Virginia Tech, Roanoke, VA
24016, USA

3 Neuroscience Training Program, University of Wisconsin, Madison, WI 53705,
USA

4 Department of Mathematics and Statistics, Brock University, St. Catharines,
ON L2S 3A1, Canada

5 Department of Neurology, University of Wisconsin, Madison, WI 53719, USA

6 Institute of Basic Medical Sciences, University of Oslo, Oslo, 0372, Norway

7 Araya Inc., Tokyo, 107-0052, Japan

8 Graduate School Language & Literature, Ludwig Maximilian University of
Munich, Munich, 80799, Germany

These authors contributed equally to this work.

*<EMAIL_ADDRESS>
## Abstract

This paper presents Integrated Information Theory (IIT) 4.0. IIT aims to
account for the properties of experience in physical (operational) terms. It
identifies the essential properties of experience (axioms), infers the
necessary and sufficient properties that its substrate must satisfy
(postulates), and expresses them in mathematical terms. In principle, the
postulates can be applied to any system of units in a state to determine
whether it is conscious, to what degree, and in what way. IIT offers a
parsimonious explanation of empirical evidence, makes testable predictions,
and permits inferences and extrapolations. IIT 4.0 incorporates several
developments of the past ten years, including a more accurate translation of
axioms into postulates and mathematical expressions, the introduction of a
unique measure of intrinsic information that is consistent with the
postulates, and an explicit assessment of causal relations. By fully unfolding
a system’s irreducible cause–effect power, the distinctions and relations
specified by a substrate can account for the quality of experience.

## Author summary

IIT aims to account for consciousness and its properties in physical terms.
The theory identifies the essential properties of experience (axioms),
formulates them as physical properties in terms of cause–effect power
(postulates), and provides a mathematical formalism for assessing those
properties. This formalism can be employed to “unfold” a cause–effect
structure from a substrate constituted of units in a state, whose interactions
can be characterized as interventional conditional probabilities. According to
IIT, all the properties of an experience can be accounted for, in physical
terms, by those of a cause–effect structure that satisfies its postulates. The
theory is consistent with neurological data and has led to successful
experimental predictions. As the latest iteration of the theory, IIT 4.0
incorporates several developments pursued over the past ten years.

## Introduction

A scientific theory of consciousness should account for experience, which is
subjective, in objective terms [1]. Being conscious—having an experience—is
understood to mean that “there is something it is like to be” [2]: something
it is like to see a blue sky, hear the ocean roar, dream of a friend’s face,
imagine a melody flow, contemplate a choice, or reflect on the experience one
is having.

IIT aims to account for phenomenal properties—the properties of experience—in
physical terms. IIT’s starting point is experience itself rather than its
behavioral, functional, or neural correlates [1]. Furthermore, in IIT
“physical” is meant in a strictly operational sense—in terms of what can be
observed and manipulated.

The starting point of IIT is the existence of an experience, which is
immediate and irrefutable [3, 4]. From this “zeroth” axiom, IIT sets out to
identify the essential properties of consciousness—those that are immediate
and irrefutably true of every conceivable experience. These are IIT’s five
axioms of phenomenal existence: every experience is for the experiencer
(intrinsicality), specific (information), unitary (integration), definite
(exclusion), and structured (composition).

Unlike phenomenal existence, which is immediate and irrefutable (an axiom),
physical existence is an explanatory construct (a postulate) and it is
assessed operationally from within consciousness: in physical terms, to be is
to have cause–effect power (see Box 2: Principle of being). In other words,
something can be said to exist physically if it can “take and make a
difference”—bear a cause and produce an effect—as judged by a conscious
observer/manipulator.

The next step of IIT is to formulate the essential phenomenal properties (the
axioms) in terms of corresponding physical properties (the postulates). This
formulation is an “inference to a good explanation” and rests on basic
assumptions such as realism, physicalism, and atomism (see Box 1:
Methodological guidelines of IIT). If IIT is correct, the substrate A
substrate should be understood as a set of units that can be observed and
manipulated of consciousness, beyond having cause–effect power (existence),
must satisfy all five essential phenomenal properties in physical terms: its
cause–effect power must be for itself (intrinsicality), specific
(information), unitary (integration), definite (exclusion), and structured
(composition).

On this basis, IIT proposes a fundamental explanatory identity: an experience
is identical to the cause–effect structure unfolded from a maximal substrate
(defined below). Accordingly, all the specific phenomenal properties of any
experience must have a good explanation in terms of the specific physical
properties of the corresponding cause–effect structure, with no additional
ingredients.

IIT formulates the postulates in a mathematical framework that is in principle
applicable to general models of interacting units (but see As mentioned in the
section “Determining maximal unit grains,” a substrate unit must be maximally
irreducible within, which is likely the case for “real” neurons in the brain,
but is not the case for “virtual,” simulated neurons in a computer program). A
mathematical framework is needed (i) to evaluate whether the theory is self-
consistent and compatible with our overall knowledge about the world, (ii) to
make specific predictions regarding the quality and quantity of our
experiences and their substrate within the brain, and (iii) to extrapolate
from our own consciousness to infer the presence (or absence) and nature of
consciousness in beings different from ourselves.

Ultimately, the theory should account for why our consciousness depends on
certain portions of the world and their state, such as certain regions of the
brain and not others, and for why it fades during dreamless sleep, even though
the brain remains active. It should also account for why an experience feels
the way it does—why the sky feels extended, why a melody feels flowing in
time, and so on. Moreover, the theory makes several predictions concerning
both the presence and the quality of experience, some of which have been and
are being tested empirically [7].

While the main tenets of the theory have remained the same, its formal
framework has been progressively refined and extended [8, 9, 10, 11]. Compared
to IIT 1.0 [8, 9], 2.0 [10, 12], and 3.0 [11], IIT 4.0 presents a more
complete, self-consistent formulation and incorporates several recent advances
[13, 14, 15, 16]. Chief among them are a more accurate translation of the
axioms into postulates and mathematical expressions, the introduction of an
Intrinsic Difference (ID) measure [17, 15] that is uniquely consistent with
IIT’s postulates, and the explicit assessment of causal relations [14].

In what follows, after introducing IIT’s axioms and postulates, we provide its
updated mathematical formalism. In the “Results and discussion” section, we
apply the mathematical framework of IIT to representative examples and discuss
some of their implications. The article is meant as a reference for the
theory’s mathematical formalism, a concise demonstration of its internal
consistency, and an illustration of how a substrate’s cause–effect structure
is unfolded computationally. A discussion of the theory’s motivation, its
axioms and postulates, and its assumptions and implications can be found in a
forthcoming book [4] and wiki [18] as well as in several publications [19, 1,
20, 21, 22, 23, 24]. A survey of the explanatory power and experimental
predictions of IIT can be found in [7]. The way IIT’s analysis of cause–effect
power can be applied to actual causation, or “what caused what,” is presented
in [13].

## From phenomenal axioms to physical postulates

### Axioms of phenomenal existence

That experience exists—that “there is something it is like to be”—is immediate
and irrefutable, as everybody can confirm, say, upon awakening from dreamless
sleep. Phenomenal existence is immediate in the sense that my experience is
simply there, directly rather than indirectly: I do not need to infer its
existence from something else. It is irrefutable because the very doubting
that my experience exists is itself an experience that exists—the experience
of doubting [1, 3]. Thus, to claim that my experience does not exist is self-
contradictory or absurd. The existence of experience is IIT’s zeroth axiom.

Existence

Experience exists: there is something.

Traditionally, an axiom is a statement that is assumed to be true, cannot be
inferred from any other statement, and can serve as a starting point for
inferences. The existence of experience is the ultimate axiom—the starting
point for everything, including logic and physics.

On this basis, IIT proceeds by considering whether experience—phenomenal
existence—has some axiomatic or essential properties, properties that are
immediate and irrefutably true of every conceivable experience. Drawing on
introspection and reason, IIT identifies the following five:

Intrinsicality

Experience is intrinsic: it exists for itself.

Information

Experience is specific: it is the way it is.

Integration

Experience is unitary: it is a whole, irreducible to separate experiences.

Exclusion

Experience is definite: it is this whole.

Composition

Experience is structured: it is composed of distinctions and the relations
that bind them together, yielding a phenomenal structure.

To exemplify, if I awaken from dreamless sleep, and experience the white wall
of my room, my bed, and my body, the experience not only exists, immediately
and irrefutably, but 1) it exists for me, 2) it is specific (the wall is a
wall and it is white), 3) it is unitary (the left side is not experienced
separately from the right side, and vice versa), 4) it is definite (it
includes the visual scene in front of me—neither less, say, its left side
only, nor more, say, the wall behind my head), 5) it is structured by
distinctions (the wall, the bed, the body) and relations (the body is on the
bed, the bed in the room).

The axioms are not only immediately given, but they are irrefutably true of
every conceivable experience. For example, once properly understood, the unity
of experience cannot be refuted. Trying to conceive of an experience that were
not unitary leads to conceiving of two separate experiences, each of which is
unitary, which reaffirms the validity of the axiom. Even though each of the
axioms spells out an essential property in its own right, the axioms must be
considered together to properly characterize phenomenal existence.

IIT takes the above set of axioms to be complete: there are no further
properties of experience that are essential. Other properties that might be
considered as candidates for axiomatic status include space (experience
typically takes place in some spatial frame), time (an experience usually
feels like it flows from a past to a future), change (an experience usually
transitions or flows into another), subject–object distinction (an experience
seems to involve both a subject and an object), intentionality (experiences
usually refer to something in the world, or at least to something other than
the subject), a sense of self (many experiences include a reference to one’s
body or even to one’s narrative self), figure–ground segregation (an
experience usually includes some object and some background), situatedness (an
experience is often bound to a time and a place), will (experience offers the
opportunity for action), and affect (experience is often colored by some
mood), among others. However, experiences lacking each of these candidate
properties are conceivable—that is, conceiving of them does not lead to self-
contradiction or absurdity. They are also achievable, as revealed by altered
states of consciousness reached through dreaming, meditative practices, or
drugs.

### Postulates of physical existence

To account for the many regularities of experience (Box 1), it is a good
inference to assume the existence of a world that persists independently of
one’s experience (realism). From within consciousness, we can probe the
physical existence of things outside of our experience operationally—through
observations and manipulations. To be granted physical existence, something
should have the power to “take a difference” (be affected) and “make a
difference” (produce effects) in a reliable way (physicalism). IIT also
assumes operational reductionism: ideally, to establish what exists in
physical terms, one would start from the smallest units that can take and make
a difference, so that nothing is left out (atomism).

By characterizing physical existence operationally as cause–effect power, IIT
can proceed to translate the axioms of phenomenal existence into postulates of
physical existence. This establishes the requirements for the substrate of
consciousness, where “substrate” is meant operationally as a set of units that
can be observed and manipulated.

Existence

The substrate of consciousness must have cause–effect power: its units must
take and make a difference.

Building from this “zeroth” postulate, IIT formulates the five axioms in terms
of postulates of physical existence:

Intrinsicality

The substrate of consciousness must have intrinsic cause–effect power: it must
take and make a difference within itself.

Information

The substrate of consciousness must have specific cause–effect power: it must
select a specific cause–effect state.

This state is the one with maximal intrinsic information
($\operatorname{\mathit{ii}}$), a measure of the difference a system takes or
makes over itself for a given cause state and effect state.

Integration

The substrate of consciousness must have unitary cause–effect power: it must
specify its cause–effect state as a whole set of units, irreducible to
separate subsets of units.

Irreducibility is measured by integrated information ($\varphi$) over the
substrate’s minimum partition.

Exclusion

The substrate of consciousness must have definite cause–effect power: it must
specify its cause–effect state as this set of units.

This is the set of units that is maximally irreducible, as measured by maximum
$\varphi$ ($\varphi^{*}$). This set is called a maximal substrate, also known
as complex [11, 16].

Composition

The substrate of consciousness must have structured cause–effect power:
subsets of its units must specify cause–effect states over subsets of units
(distinctions) that can overlap with one another (relations), yielding a
cause–effect structure or $\varPhi$-structure (“Phi-structure”).

Distinctions and relations, in turn, must also satisfy the postulates of
physical existence: they must have cause–effect power, within the substrate of
consciousness, in a specific, unitary, and definite way (they do not have
components, being components themselves). They thus have an associated
$\varphi$ value. The $\varPhi$-structure unfolded from a complex corresponds
to the quality of consciousness. The sum total of the $\varphi$ values of the
distinctions and relations that compose the $\varPhi$-structure measures its
structured information $\varPhi$ (“big Phi”) and corresponds to the quantity
of consciousness.

According to IIT, the physical properties characterized by the postulates are
necessary and sufficient for an entity to be conscious. They are necessary
because they are needed to account for the properties of experience that are
essential, in the sense that it is inconceivable for an experience to lack any
one of them. They are also sufficient because no additional property of
experience is essential, in the sense that it is conceivable for an experience
to lack that property. Thus, no additional physical property is a necessary
requirement for being a substrate of consciousness.

The postulates of IIT have been and are being applied to account for the
location of the substrate of consciousness in the brain [7] and for its loss
and recovery in physiological and pathological conditions [25, 26].

### The explanatory identity between experiences and $\varPhi$-structures

Having determined the necessary and sufficient conditions for a substrate to
support consciousness, IIT proposes an explanatory identity: every property of
an experience is accounted for in full by the physical properties of the
$\varPhi$-structure unfolded from a maximal substrate (a complex) in its
current state, with no further or “ad hoc” ingredients. That is, there must be
a one-to-one correspondence between the way the experience feels and the way
distinctions and relations are structured. Importantly, the identity is not
meant as a correspondence between the properties of two separate things.
Instead, the identity should be understood in an explanatory sense: the
intrinsic (subjective) feeling of the experience can be explained
extrinsically (objectively, _i.e._ , operationally or physically) in terms of
cause–effect power Strictly speaking, distinctions and relations that can be
singled out phenomenally, such as a spot, a book, and so on, correspond, in
physical terms, to bundles of distinctions and relations (compound
distinctions and relations)—that is, to sub-structures of a
$\varPhi$-structure ($\varPhi$-folds) [14, 1]. This can be understood in
neural terms because attentional mechanisms can only highlight subsets of
units, and thereby all the associated distinctions and relations, rather than
individual distinctions and relations. In other words, introspection is the
starting point for an explanation of experience in physical terms, but it can
only go so far. A full explanation can only be provided through a back-and-
forth between the properties of a substrate, which can be explored in great
detail, and the properties of experience, which can only be characterized
crudely through introspection.

The explanatory identity has been applied to account for how space feels
(spatial extendedness) and which neural substrates may account for it[14].
Ongoing work is applying the identity to provide a basic account of the
feeling of temporal flow [28] and that of objects [29].

### Box 1. Methodological guidelines of IIT

#### Inference to a good explanation

We should generally assume that an explanation is good if it can account for a
broad set of facts (scope), does so in a unified manner (synthesis), can
explain facts precisely (specificity), is internally coherent (self-
consistency), is coherent with our overall understanding of things (system
consistency), is simpler than alternatives (simplicity), and can make testable
predictions (scientific validation). For example, IIT 4.0 aims at expressing
the postulates of intrinsicality, information, integration, and exclusion in a
self-consistent manner when applied to systems, causal distinctions, and
relations (see formulas).

#### Realism

We should assume that something exists (and persists) independently of our own
experience. This is a much better hypothesis than solipsism, which explains
nothing and predicts nothing. Although IIT starts from our own phenomenology,
it aims to account for the many regularities of experience in a way that is
fully consistent with realism.

#### Operational physicalism

To assess what exists independently of our own experience, we should employ an
operational criterion: we should systematically observe and manipulate a
substrate’s units and determine that they can indeed take and make a
difference in a way that is reliable and persisting. Doing so demonstrates a
substrate’s cause–effect power—the signature of physical existence. Ideally,
cause–effect power is fully captured by a substrate’s transition probability
matrix (TPM) (1). This assumption is embedded in IIT’s zeroth postulate.

#### Operational reductionism (“atomism”)

Ideally, we should account for what exists physically in terms of the smallest
units we can observe and manipulate, as captured by unit TPMs. Doing so would
leave nothing unaccounted for. IIT assumes that in principle it should be
possible to account for everything purely in terms of cause–effect
power—cause–effect power “all the way down” to conditional probabilities
between atomic units. Eventually, this would leave neither room nor need to
assume intrinsic properties or laws.

#### Intrinsic perspective

When accounting for experience itself in physical terms, existence should be
evaluated from the intrinsic perspective of an entity—what exists for the
entity itself, not from the perspective of an external observer. This
assumption is embedded in IIT’s postulate of intrinsicality and has several
consequences. One is that, from the intrinsic perspective, the quality and
quantity of existence must be observer-independent and cannot be arbitrary.
For instance, information in IIT must be relative to the specific state the
entity is in, rather than an average of states as assessed by an external
observer. Similarly, it should be evaluated based on the uniform distribution
of possible states, as captured by the entity’s TPM (1), rather than on an
observed probability distribution. By the same token, units outside the entity
should be treated as fixed background conditions that do not contribute
directly to what the system is. The intrinsic perspective also imposes a
tension between expansion and dilution (see below and [17, 15]): from the
intrinsic perspective of a system (or a mechanism within the system), having
more units may increase its informativeness (cause–effect power measured as
deviation from chance), while at the same time diluting its selectivity
(ability to concentrate cause–effect power over a specific state).

## Overview of IIT’s framework

IIT 4.0 aims at providing a formal framework to characterize the cause–effect
structure of a substrate in a given state by expressing IIT’s postulates in
mathematical terms. In line with operational physicalism (Box 1), we
characterize a substrate by the transition probability function of its
constituting units.

On this basis, the IIT formalism first identifies sets of units that fulfill
all required properties of a substrate of consciousness according to the
postulates of physical existence. First, for a candidate system, we determine
a maximal cause–effect state based on the intrinsic information
($\operatorname{\mathit{ii}}$) that the system in its current state specifies
over its possible cause states and effect states. We then determine the
maximal substrate based on the integrated information ($\varphi_{s}$) of the
maximal cause–effect state. To qualify as a substrate of consciousness, a
candidate system must specify a maximum of integrated information
($\varphi^{*}_{s}$) compared to all competing candidate systems with
overlapping units.

The second part of the IIT formalism unfolds the cause–effect structure
specified by a maximal substrate in its current state, its
$\varPhi$-structure. To that end, we determine the distinctions and relations
specified by the substrate’s subsets according to the postulates of physical
existence. Distinctions are cause–effect states specified over subsets of
substrate units (purviews) by subsets of substrate units (mechanisms).
Relations are congruent overlaps among distinctions’ cause and/or effect
states. Distinctions and relations are also characterized by their integrated
information ($\varphi_{d}$, $\varphi_{r}$). The $\varPhi$-structure they
compose corresponds to the quality of the experience specified by the
substrate; the sum of their $\varphi_{d/r}$ values corresponds to its quantity
($\varPhi$).

While IIT must still be considered as work in progress, having undergone
successive refinements, IIT 4.0 is the first formulation of IIT that strives
to characterize $\varPhi$-structures completely and to do so based on measures
that satisfy the postulates uniquely. For a comparison of the updated
framework with IIT 1.0, 2.0, and 3.0, see A.2.

### Substrates, transition probabilities, and cause–effect power

IIT takes physical existence as synonymous with having cause–effect power, the
ability to take and make a difference. Consequently, a substrate $U$ with
state space $\Omega_{U}$ is operationally defined by its potential
interactions, assessed in terms of conditional probabilities (physicalism, Box
1). We denote the complete transition probability function of a substrate $U$
over a system update $u\rightarrow\bar{u}$ as

$\mathcal{T}_{U}\equiv p(\bar{u}\mid u),\quad u,\bar{u}\in\Omega_{U}.$ (1)

A substrate in IIT can be described as a stochastic system
$U=\\{U_{1},U_{2},\ldots,U_{n}\\}$ of $n$ interacting units with state space
$\Omega_{U}=\prod_{i}\Omega_{U_{i}}$ and current state $u\in\Omega_{U}$. We
assume that the system updates in discrete steps, that the state space
$\Omega_{U}$ is finite, and that the individual random variables $U_{i}\in U$
are conditionally independent from each other given the preceding state of
$U$:

$p(\bar{u}\mid u)=\prod_{i=1}^{n}p(\bar{u}_{i}\mid u).$ (2)

Finally, we assume a complete description of the substrate, which means that
we can determine the conditional probabilities in (2) for every system state,
with $p(\bar{u}\mid u)=p(\bar{u}\mid\operatorname{do}(u))$ [13, 30, 31, 32],
where the “do-operator” $\operatorname{do}(u)$ indicates that $u$ is imposed
by intervention. This implies that $U$ must correspond to a causal network
[13], and $\mathcal{T}_{U}$ is a transition probability matrix (TPM) of size
$|\Omega_{U}|$ While the IIT formalism can be applied to hypothetical or
simulated systems (as we do for the example systems in the Results/Discussion
section), for the resulting quantities to capture existence in physical terms
they must be applied to substrate units that can actually be observed and
manipulated in physical terms.

The TPM $\mathcal{T}_{U}$, which forms the starting point of IIT’s analysis,
serves as an overall description of a system’s cause–effect power: what is the
probability that the system will transition into each of its possible states
upon an intervention that initializes it into every possible state (Figure 1)?
(Notably, there is no additional role for intrinsic physical properties or
laws of nature.) In practice, a causal model will be neither complete nor
atomic (capturing the smallest units that can be observed and manipulated),
but will capture the relevant features of what we are trying to explain and
predict As demonstrated in [24], it is possible to extend IIT’s causal
framework to finite quantum systems under unitary evolution, where the
conditional independence assumption (2) applies to non-entangled subsystems.

In the “Results and discussion” section, the IIT formalism will be applied to
extremely simple, simulated networks, rather than causal models of actual
substrates. The cause–effect structures derived from these simple networks
only serve as convenient illustrations of how a hypothetical substrate’s
cause–effect power can be unfolded.

### Implementing the postulates

In what follows, our goal is to evaluate whether a hypothetical substrate
(also called “system”) satisfies all the postulates of IIT. To that end, we
must verify whether the system has cause–effect power that is intrinsic,
specific, integrated, definite, and structured.

#### Existence

According to IIT, existence understood as cause–effect power requires the
capacity to both take and make a difference (see Box 2, Principle of being).
On the basis of a complete description of the system in terms of
interventional conditional probabilities ($\mathcal{T}_{U}$) (1), cause–effect
power can be measured as causal informativeness. Cause informativeness
measures how much a potential cause increases the probability of the current
state, and effect informativeness how much the current state increases the
probability of a potential effect (as compared to chance).

#### Intrinsicality

Building upon the existence postulate, the intrinsicality postulate further
requires that a system exerts cause–effect power within itself. In general,
the systems we want to evaluate are open systems $S\subseteq U$ that are part
of a larger “universe” $U$. From the intrinsic perspective of a system $S$
(see Box 1), the set of the remaining units $W=U\setminus S$ merely act as
background conditions, whose state can be considered as fixed. This is
enforced by causally conditioning the larger TPM ($\mathcal{T}_{U}$) on the
current state $W=w$, which makes $W$ causally inert.

#### Information

The information postulate requires that a system’s cause–effect power be
specific: the system in its current state must select a specific cause–effect
state for its units. Based on the principle of maximal existence (Box 2), this
is the state for which intrinsic information is maximal—the maximal
cause–effect state. Intrinsic information ($\operatorname{\mathit{ii}}$)
measures the difference a system takes or makes over itself for a given cause
and effect state as the product of informativeness and selectivity. As we have
seen (existence), informativeness quantifies the causal power of a system in
its current state as a reduction of uncertainty with respect to chance.
Selectivity measures how much cause–effect power is concentrated over that
specific cause or effect state. Selectivity is reduced by uncertainty in the
cause or effect state with respect to other potential cause and effect states.

From the intrinsic perspective of the system, the product of informativeness
and selectivity leads to a tension between expansion and dilution, whereby a
system comprising more units may show increased deviation from chance but
decreased concentration of cause–effect power over a specific state [17, 15].

#### Integration

By the integration postulate, it is not sufficient for a system to have
cause–effect power within itself and select a specific cause–effect state: it
must also specify its maximal cause–effect state in a way that is irreducible.
This can be assessed by partitioning the set of units that constitute the
system into separate parts. The system integrated information ($\varphi_{s}$)
then quantifies how much the intrinsic information specified by the maximal
state is reduced due to the partition Note that this notion of irreducibility
based on set-partitions differs from typical information-theoretic notions
such as redundancy or synergy [36, 85, 86]. Integrated information is
evaluated over the partition that makes the least difference, the minimum
partition (MIP), in accordance with the principle of minimal existence (see
Box 2).

Integrated information is highly sensitive to the presence of fault
lines—partitions that separate parts of a system that interact weakly or
directionally [16].

#### Exclusion

Many overlapping sets of units may have a positive value of integrated
information ($\varphi_{s}$). However, the exclusion postulate requires that
the substrate of consciousness must be constituted of a definite set of units,
neither less nor more. Moreover, units, updates, and states must have a
definite grain. Operationally, the exclusion postulate is enforced by
selecting the set of units that maximizes integrated information over itself
($\varphi_{s}^{*}$), based again on the principle of maximal existence (see
Box 2). That set of units is called a maximal substrate, or complex. Over a
universal substrate, sets of units for which integrated information is maximal
compared to all competing candidate systems with overlapping units can be
assessed recursively (by identifying the first complex, then the second
complex, and so on).

#### Composition

Once a complex has been identified, composition requires that we characterize
its cause–effect structure by considering all its subsets and fully unfolding
its cause–effect power.

Usually, causal models are conceived in holistic terms, as state transitions
of the system as a whole (1), or in reductionist terms, as a description of
the individual units of the system and their interactions (2) [36]. However,
to account for the structure of experience, considering only the cause–effect
power of the individual units or of the system as a whole would be
insufficient [36, 20]. Instead, by the composition postulate, we have to
evaluate the system’s cause–effect structure by considering the cause–effect
power of its subsets as well as their causal relations.

To contribute to the cause–effect structure of a complex, a system subset must
both take and make a difference (as required by existence) within the system
(as required by intrinsicality). A subset $M\subseteq S$ in state $M=m$ is
called a mechanism if it links a cause and effect state over subsets of units
$Z_{c/e}\subseteq S$, called purviews. A mechanism together with the cause and
effect state it specifies is called a causal distinction. Distinctions are
evaluated based on whether they satisfy all the postulates of IIT (except for
composition). For every mechanism, the cause–effect state is the one having
maximal intrinsic information ($\operatorname{\mathit{ii}}$), and the cause
and effect purviews are those yielding the maximum value of integrated
information ($\varphi_{d}$) within the complex—that is, those that are
maximally irreducible.

Distinctions whose cause or effect states overlap congruently within the
system (over the same subset of units in the same state) are bound together by
causal relations. Relations also have an associated value of integrated
information ($\varphi_{r}$), corresponding to their irreducibility.

The distinctions (and associated relations) that exist for the complex are
only those whose cause–effect state is congruent with the cause–effect state
of the complex as a whole. Together, those distinctions and relations compose
the cause–effect structure of the complex in its current state. The
cause–effect structure specified by a complex is called a $\varPhi$-structure.
The sum of its distinction and relation integrated information amounts to the
structured information ($\varPhi$) of the complex.

In the following, we will provide a formal account of the IIT analysis. The
first part demonstrates how to identify complexes. This requires that we (a)
determine the cause–effect state of a system in its current state, (b)
evaluate the system integrated information ($\varphi_{s}$) over that
cause–effect state, and (c) search iteratively for maxima of integrated
information ($\varphi_{s}^{*}$) within a universe. The second part describes
how the postulates of IIT are applied to unfold the cause–effect structure of
a complex. This requires that we identify the causal distinctions specified by
subsets of units within the complex and the causal relations determined by the
way distinctions overlap, yielding the system’s cause–effect structure and its
structured information $\varPhi$.

## Box 2. Ontological principles of IIT

#### Principle of being

The principle of being states that to be is to have cause–effect power. In
other words, in physical, operational terms, to exist requires being able to
take and make a difference. The principle is closely related to the so-called
Eleatic principle, as found in Plato’s Sophist dialogue [37]: “I say that
everything possessing any kind of power, either to do anything to something
else, or to be affected to the smallest extent by the slightest cause, even on
a single occasion, has real existence: for I claim that entities are nothing
else but power.” A similar principle can be found in the work of the Buddhist
philosopher Dharmakīrti: “Whatever has causal powers, that really exists.”
[38] Note that the Eleatic principle is enunciated as a disjunction (either to
do something… or to be affected…), whereas IIT’s principle of being is
presented as a conjunction (take and make a difference).

#### Principle of maximal existence

The principle of maximal existence states that, with respect to an essential
requirement for existence, what exists is what exists the most. The principle
is offered by IIT as a good explanation for why the system state specified by
the complex and the cause–effect states specified by its mechanisms are what
they are. It also provides a criterion for determining the set of units
constituting a complex—the one with maximally irreducible cause–effect power,
for determining the subsets of units constituting the distinctions and
relations that compose its cause–effect structure, and for determining the
units’ grain. To exemplify, consider a set of candidate complexes overlapping
over the same substrate. By the postulates of integration and exclusion, a
complex must be both unitary and definite. By the maximal existence principle,
the complex should be the one that lays the greatest claim to existence as one
entity, as measured by system integrated information ($\varphi_{s}$). For the
same reason, candidate complexes that overlap over the same substrate but have
a lower value of $\varphi_{s}$ are excluded from existence. In other words, if
having maximal $\varphi_{s}$ is the reason for assigning existence as a
unitary complex to a set of units, it is also the reason to exclude from
existence any overlapping set not having maximal $\varphi_{s}$.

#### Principle of minimal existence

Another key principle of IIT is the principle of minimal existence, which
complements that of maximal existence. The principle states that, with respect
to an essential requirement for existence, nothing exists more than the least
it exists. The principle is offered by IIT as a good explanation for why,
given that a system can only exist as one system if it is irreducible, its
degree of irreducibility should be assessed over the partition across which it
is least irreducible (the minimum partition). Similarly, a distinction within
a system can only exist as one distinction to the extent that it is
irreducible, and its degree of irreducibility should be assessed over the
partition across which it is least irreducible. Moreover, a set of units can
only exist as a system, or as a distinction within the system, if it specifies
both an irreducible cause and an irreducible effect, so its degree of
irreducibility should be the minimum between the irreducibility on the cause
side and on the effect side A principle of IIT not discussed here is the
Principle of becoming, which states that powers become what powers do. That
is, conditional probabilities in the TPM update depending on what happens. The
principle and some of its implications—examined in [4] and forthcoming work.

## Identifying substrates of consciousness through existence, intrinsicality,
information, integration, and exclusion

Our starting point is a substrate $U$ in current state $U=u$ with TPM
$\mathcal{T}_{U}$ (1). We consider any subset $s\subseteq u$ as a possible
complex and refer to a set of units $S\subseteq U$ as a candidate system.
(Note that $s$, in a slight abuse of notation, refers to both a subset of
units in a state, as well as the state itself.)

By the intrinsicality postulate, the units $W=U\setminus S$ are fixed in their
current state $w\in\Omega_{W}$ throughout the analysis of the candidate system
$S$ (causal conditioning). Accordingly, we obtain the TPM $\mathcal{T}_{S}$ of
a candidate system $S$ from its intrinsic transition probability function

$\mathcal{T}_{S}\equiv p(\bar{s}\mid s)=p(\bar{s}\mid s,w),\quad
s,\bar{s}\in\Omega_{S},$ (3)

where $w=u\setminus s$.

The intrinsic information $\operatorname{\mathit{ii}}_{c/e}$ is a measure of
the intrinsic cause/effect power exerted by a system $S$ in its current state
$s$ over itself by selecting a specific cause/effect state $\bar{s}$. The
cause–effect state for which intrinsic information
($\operatorname{\mathit{ii}}_{c}$ and $\operatorname{\mathit{ii}}_{e}$) is
maximal is called the maximal cause–effect state
$s^{\prime}=\\{s^{\prime}_{c},s^{\prime}_{e}\\}$. The integrated information
$\varphi_{s}$ is a measure of the irreducibility of a cause–effect state,
compared to the directional system partition $\theta^{\prime}$ that affects
the maximal cause–effect state the least (minimum partition, or MIP). Systems
for which integrated information is maximal ($\varphi_{s}^{*}$) compared to
any competing candidate system with overlapping units are called maximal
substrates, or complexes.

The IIT 4.0 formalism to measure a system’s integrated information
$\varphi_{s}$ and to identify maximal substrates was first presented in [16].
An example of how to identify complexes in a simple system is given in Fig. 1,
while a comparison with prior accounts (IIT 1.0, IIT 2.0, and IIT 3.0) can be
found in A.2. An outline of the IIT algorithm is included in A.4.

Fig 1: Identifying substrates of consciousness through the postulates of
existence, intrinsicality, information, integration, and exclusion.(A) The
substrate $S=aBC$ in state $(-1,1,1)$ (lowercase letters for units indicated
state “$-1$”, uppercase letters state “$+1$”) is the starting point for
applying the postulates. The substrate updates its state according to the
depicted transition probability matrix (TPM) (each unit follows a logistic
equation (see Results for definition) with k = 4.0 and connection weights as
indicated in the causal model). Existence requires that the substrate must
have cause–effect power, meaning that the TPM among substrate states must
differ from chance. (B) Intrinsicality requires that a candidate substrate,
for example, units $aB$, has cause–effect power over itself. Units outside the
candidate substrate (in this case, unit $C$) are treated as background
conditions by pinning them in their current state. (C) Information requires
that the candidate substrate $aB$ selects a specific cause–effect state
($s^{\prime}$). This is the cause state and effect state for which intrinsic
information (_$\operatorname{\mathit{ii}}$_) is maximal. Dark-colored and gray
bars represent the quantities for informativeness (constrained and
unconstrained), and light colored bars for selectivity. (D) Integration
requires that the substrate specifies its cause–effect state irreducibly (“as
one”). This is established by identifying the minimum partition (MIP) and
measuring the integrated information of the system ($\varphi_{s}$)—the minimum
between cause integrated information ($\varphi_{c}$) and effect integrated
information ($\varphi_{e}$). Here, gray bars represent the partitioned
probability. (E) Exclusion requires that the substrate of consciousness is
definite, including some units and excluding others. This is established by
identifying the candidate substrate with the maximum value of system
integrated information ($\varphi^{*}_{s}$)—the maximal substrate, or complex.
In this case, $aB$ is a complex since its system integrated information
($\varphi_{s}=0.17$) is higher than the one of all other overlapping systems
(for example, subset $a$ with $\varphi_{s}=0.06$ and superset $aBC$ with
$\varphi_{s}=0.13$).

### Existence, intrinsicality, and information: Determining the maximal
cause–effect state of a candidate system

Given a causal model $\mathcal{T}_{S}$ (3), we wish to identify the maximal
cause–effect state specified by a system in its current state over itself and
to quantify the causal power with which it does so. In doing so, we quantify
the cause–effect power of a system from its intrinsic perspective, rather than
from the perspective of an outside observer (see Box 1).

#### System intrinsic information $\operatorname{\mathit{ii}}$

Intrinsic information $\operatorname{\mathit{ii}}(s,\bar{s})$ measures the
causal power of a system $S$ over itself, for its current state $s$, over a
specific cause/effect state $\bar{s}$. Intrinsic information depends on
interventional conditional probabilities and unconstrained probabilities of
cause/effect states and is the product of selectivity and informativeness.

On the effect side, intrinsic effect information
$\operatorname{\mathit{ii}}_{e}$ of the current state $s$ over a possible
effect state $\bar{s}$ is defined as:

$\operatorname{\mathit{ii}}_{e}(s,\bar{s})=p_{e}(\bar{s}\mid
s)\log\left(\frac{p_{e}(\bar{s}\mid s)}{p_{e}(\bar{s})}\right).$ (4)

Above, $p_{e}(\bar{s}\mid s)=p(\bar{s}\mid s)$ (3) is the interventional
conditional probability that the current state $S=s$ produces the effect state
$\bar{s}$, as indicated by $\mathcal{T}_{S}$.

The interventional unconstrained probability $p_{e}(\bar{s})$

$p_{e}(\bar{s})=|\Omega_{S}|^{-1}\sum_{s\in\Omega_{S}}p(\bar{s}\mid s),$ (5)

is defined as the marginal probability of $\bar{s}$, averaged across all
possible current states of $S$ with equal probability (where $|\Omega_{S}|$
denotes the cardinality of the state space $\Omega_{S}$).

On the cause side, intrinsic cause information
$\operatorname{\mathit{ii}}_{c}$ of the current state $s$ over a possible
cause state $\bar{s}$ is defined as:

$\operatorname{\mathit{ii}}_{c}(s,\bar{s})=p_{c}(\bar{s}\mid
s)\log\left(\frac{p_{e}(s\mid\bar{s})}{p_{e}(s)}\right).$ (6)

Above, $p_{c}(\bar{s}\mid s)$ is the interventional conditional probability
that the current state $S=s$ was produced by $\bar{s}$. The latter is derived
from $\mathcal{T}_{S}$ using Bayes’ rule, where we again assign a uniform
prior to the states of $\bar{s}$,

$p_{c}(\bar{s})=|\Omega_{S}|^{-1},~{}\bar{s}\in\Omega_{S},$ (7)

such that

$p_{c}(\bar{s}\mid
s)=\frac{p_{e}(s\mid\bar{s})\cdot|\Omega_{S}|^{-1}}{p_{e}(s)}=\frac{p(s\mid\bar{s})}{\sum\limits_{\hat{s}\in\Omega_{S}}p(s\mid\hat{s})}.$
(8)

#### Informativeness (over chance)

In (4) and (6), the logarithmic term (in base 2 throughout) is called
informativeness. Note that informativeness is expressed in terms of effect
probabilities for both $\operatorname{\mathit{ii}}_{e}$ (4) and
$\operatorname{\mathit{ii}}_{c}$ (6). However,
$\operatorname{\mathit{ii}}_{e}$ (4) evaluates the increase in probability of
the effect state due to the current state, while
$\operatorname{\mathit{ii}}_{c}$ (6) evaluates the increase in probability of
the current state due to the cause state.

In line with the existence postulate, a system $S$ in state $s$ has
cause–effect power (it takes and makes a difference) if it raises the
probability of a possible effect state compared to chance, which is to say
compared to its unconstrained probability,

$\log\left(\frac{p_{e}(\bar{s}\mid s)}{p_{e}(\bar{s})}\right)>0,$ (9)

and if the probability of the current state is raised above chance by a
possible cause state,

$\log\left(\frac{p_{e}(s\mid\bar{s})}{p_{e}(s)}\right)>0.$ (10)

Informativeness is additive over the number of units: if a system specifies a
cause or effect state with probability $p=1$, its causal power increases
additively with the number of units whose states it fully specifies
(expansion), given that the chance probability of all states decreases
exponentially.

#### Selectivity (over states)

From the intrinsic perspective of a system, cause–effect power over a specific
cause or effect state depends not only on the deviation from chance it
produces, but also on how that deviation is concentrated on that state, rather
than being diluted over other states. This is measured by the selectivity term
in front of the logarithmic term in (4) and (6), corresponding to the
conditional probability $p_{c/e}(\bar{s}∣s)$ of that specific cause or effect
state. Selectivity means that if $p<1$, the system’s causal power becomes
subadditive (dilution) (see [17] for details). For example, as shown in [15],
if an unconstrained unit is added to a fully specified unit, intrinsic
information does not just stay the same, but decreases exponentially. From the
intrinsic perspective of the system, the informativeness of a specific
cause/effect state is diluted because it is spread over multiple possible
states, yet the system must select only one state.

Altogether, taking the product of informativeness and selectivity leads to a
tension between expansion and dilution: a larger system will tend to have
higher informativeness than a smaller system because it will deviate more from
chance, but it will also tend to have lower selectivity because it will have a
larger repertoire of states to select from.

Because of the selectivity term, intrinsic information is reduced by
indeterminism and degeneracy. As shown in [16], indeterminism decreases the
probability of the selected effect state because it implies that the same
state can lead to multiple states. In turn, degeneracy decreases the
probability of the selected cause state because it implies that multiple
states can lead to the same state, even in a deterministic system.

The intrinsic information $\operatorname{\mathit{ii}}$ is quantified in units
of _intrinsic bits_ , or _ibits_ , to distinguish it from standard
information-theoretic measures (which are typically additive). Formally, the
_ibit_ corresponds to a point-wise information value (measured in bits)
weighted by a probability.

#### The maximal cause–effect state

Taking the product of informativeness and selectivity on the system’s cause
and effect sides captures the postulates of existence (taking and making a
difference) and intrinsicality (taking and making a difference over itself)
for each possible cause/effect state, as measured by intrinsic information.
However, the information postulate further requires that the system selects a
specific cause/effect state. Which one is determined based on the principle of
maximal existence (Box 1): the cause/effect specified by the system should be
the one that maximizes intrinsic information. On the effect side (and
similarly for the cause side),

$\begin{split}s^{\prime}_{e}(\mathcal{T}_{S},s)=&\operatornamewithlimits{argmax}\limits_{\bar{s}\in\Omega_{S}}\operatorname{\mathit{ii}}_{e}(s,\bar{s})\\\
=&\operatornamewithlimits{argmax}\limits_{\bar{s}\in\Omega_{S}}p_{e}(\bar{s}\mid
s)\log\left(\frac{p_{e}(\bar{s}\mid s)}{p_{e}(\bar{s})}\right).\end{split}$
(11)

The system’s intrinsic effect information is the value of
$\operatorname{\mathit{ii}}_{e}$ (4) for its maximal effect state:

$\operatorname{\mathit{ii}}_{e}(\mathcal{T}_{S},s):=\operatorname{\mathit{ii}}_{e}(s,s^{\prime}_{e})=\max_{\bar{s}\in\Omega_{S}}p_{e}(\bar{s}\mid
s)\log\left(\frac{p_{e}(\bar{s}\mid s)}{p_{e}(\bar{s})}\right).$ (12)

We have made the dependency of $s^{\prime}$ and
$\operatorname{\mathit{ii}}_{e}$ on $\mathcal{T}_{S}$ explicit in (11) and
(12) to highlight that for intrinsic information to properly assess
cause–effect power, all probabilities must be derived from the system’s
interventional transition probability function, while imposing a uniform prior
distribution over all possible system states. If there is no state with
$\operatorname{\mathit{ii}}_{e}(\mathcal{T}_{S},s)>0$, the system $S$ in state
$s$ has no causal power (and likewise for
$\operatorname{\mathit{ii}}_{c}(\mathcal{T}_{S},s)$). Note also that a
system’s intrinsic cause/effect state does not necessarily correspond to the
actual cause/effect state (what actually happened before / will happen after)
in the dynamical evolution of the system, which typically also depends on
extrinsic influences. (For an account of actual causation according to the
causal principles of IIT, see [13].)

Because consciousness is the way it is, the translation of its properties in
physical, operational terms should be unique and based on quantities that
uniquely satisfy the postulates [40, 15]. Intrinsic information (12) is
formally equivalent to a measure of intrinsic difference [17], which uniquely
satisfies three desired properties (causality, specificity, and
intrinsicality) that align with the postulates of IIT, but also have
independent justification.

### Integration: Determining the irreducibility of a candidate system

Having identified the maximal cause–effect state
$s^{\prime}=\\{s^{\prime}_{c},s^{\prime}_{e}\\}$ of a candidate system $S$ in
its current state $s$, the next step is to evaluate whether the system
specifies the cause–effect state of its units in a way that is irreducible, as
required by the integration postulate: a candidate system can only be a
substrate of consciousness if it is one system—that is, if it cannot be
subdivided into subsets of units that exist separately from one another.

#### Directional system partitions

To that end, we define a set of directional system partitions $\Theta(S)$ that
divide $S$ into $k\geq 2$ parts $\\{S^{(i)}\\}^{k}_{i=1}$, such that

$S^{(i)}\neq\varnothing,\ S^{(i)}\cap S^{(j)}=\varnothing,\mathrm{\ and\
}\bigcup_{i=1}^{k}S^{(i)}=S.$ (13)

In words, each part $S^{(i)}$ must contain at least one unit, there must be no
overlap between any two parts $S^{(i)}$ and $S^{(j)}$, and every unit of the
system must appear in exactly one part. For each part $S^{(i)}$, the partition
removes the causal connections of that part with the rest of the system in a
directional manner: either the part’s inputs, outputs, or both are replaced by
independent “noise” (they are “cut” by the partition in the sense that their
causal powers are substituted by chance). Directional partitions are necessary
because, from the intrinsic perspective of a system, a subset of units that
cannot affect the rest of the system, or cannot be affected by it, cannot
truly be a part of the system. In other words, to be a part of a system, a
subset of units must be able to interact with the rest of the system in both
directions (cause and effect).

A partition $\theta\in\Theta(S)$ thus has the form

$\theta=\\{S^{(1)}_{\delta_{1}},S^{(2)}_{\delta_{2}},\ldots,S^{(k)}_{\delta_{k}}\\},$
(14)

where $\delta_{i}\in\\{\leftarrow,\rightarrow,\leftrightarrow\\}$ indicates
whether the inputs ($\leftarrow$), outputs ($\rightarrow$), or both
($\leftrightarrow$) are cut for a given part. For each part $S^{(i)}$, we can
then identify a set of units $X^{(i)}\subseteq S$ whose inputs to $S^{(i)}$
have been cut by the partition, and the complementary set $Y^{(i)}=S\setminus
X^{(i)}$ whose inputs to $S^{(i)}$ are left intact. Specifically,

$X^{(i)}=\begin{cases}S\setminus S^{(i)}&\text{ if
}\delta_{i}\in\\{\leftarrow,\leftrightarrow\\}\\\ &\\\
\bigcup\limits_{\begin{subarray}{c}j\neq i:\\\
\delta_{j}\in\\{\rightarrow,\leftrightarrow\\}\end{subarray}}S^{(j)}&\text{ if
}\delta_{i}\in\\{\rightarrow\\}.\end{cases}$ (15)

Given a partition $\theta\in\Theta(S)$, we define a partitioned transition
probability matrix $\mathcal{T}_{S}^{\theta}$ in which all connections
affected by the partition are “noised.” This is done by combining the
independent contributions of each unit $S_{j}\in S$ in line with the
conditional independence assumption (2), such that

$\mathcal{T}_{S}^{\theta}\equiv p^{\theta}(\bar{s}\mid
s)=\prod_{j=1}^{n}p^{\theta}(\bar{s}_{j}\mid s),~{}\bar{s},s\in\Omega_{S},$
(16)

where the partitioned probability of a unit $S_{j}\in S^{(i)}$ is defined as

$p^{\theta}(\bar{s}_{j}\mid
s)=|\Omega_{X^{(i)}}|^{-1}\sum_{x\in\Omega_{X^{(i)}}}p(\bar{s}_{j}\mid
x,Y^{(i)}=y).$ (17)

This means that all connections to unit $S_{j}$ that are affected by the
partition are causally marginalized (replaced by independent noise).

#### System integrated information $\varphi_{s}$

The integrated effect information $\varphi_{e}$ measures how much the
partition $\theta\in\Theta_{S}$ reduces the probability with which a system
$S=s$ specifies its effect state $s^{\prime}_{e}$ (11),

$\varphi_{e}(\mathcal{T}_{S},s,\theta)=p_{e}(s^{\prime}_{e}\mid
s)\log\left(\frac{p_{e}(s^{\prime}_{e}\mid
s)}{p_{e}^{\theta}(s^{\prime}_{e}\mid s)}\right).$ (18)

Note that $\varphi_{e}$ has the same form as the intrinsic information
$\operatorname{\mathit{ii}}_{e}(s,\bar{s})$ (4), with the partitioned effect
probability taking the place of the unconstrained (marginal) probability.
Likewise, the integrated cause information $\varphi_{c}$ is defined as

$\varphi_{c}(\mathcal{T}_{S},s,\theta)=p_{c}(s^{\prime}_{c}\mid
s)\log\left(\frac{p_{e}(s\mid s^{\prime}_{c})}{p_{e}^{\theta}(s\mid
s^{\prime}_{c})}\right).$ (19)

(By the principle of maximal existence, if two or more cause–effect states are
tied for maximal intrinsic information, the system specifies the one that
maximizes $\varphi_{c/e}$.)

By the zeroth postulate, existence requires cause and effect power, and the
integration postulate requires that its cause–effect power be irreducible. By
the principle of minimal existence (Box 2), then, system integrated
information for a given partition is the minimum of its irreducibility on the
cause and effect sides:

$\varphi_{s}(\mathcal{T}_{S},s,\theta)=\min\\{\varphi_{c}(\mathcal{T}_{S},s,\theta),\varphi_{e}(\mathcal{T}_{S},s,\theta)\\}.$
(20)

Accordingly, the system is reducible if at least one partition
$\theta\in\Theta_{S}$ makes no difference to the cause or effect probability.

Moreover, again by the principle of minimal existence, the integrated
information of a system is given by its irreducibility over its minimum
partition (MIP) $\theta^{\prime}\in\Theta_{S}$, such that

$\varphi_{s}(\mathcal{T}_{S},s):=\varphi_{s}(\mathcal{T}_{S},s,\theta^{\prime}).$
(21)

The MIP is defined as the partition $\theta\in\Theta_{S}$ that minimizes the
system’s integrated information, relative to the maximum possible value it
could take for an arbitrary TPM $\mathcal{T}^{\prime}_{S}$ over the units of
system $S$

$\theta^{\prime}=\operatornamewithlimits{argmin}\limits_{\theta\in\Theta(S)}\frac{\varphi_{s}(\mathcal{T}_{S},s,\theta)}{\displaystyle\max_{\mathcal{T}^{\prime}_{S}}\varphi(\mathcal{T}^{\prime}_{S},s,\theta)}.$
(22)

The normalization ensures that $\varphi_{s}(\mathcal{T}_{S},s)$ is evaluated
fairly over a system’s fault lines by assessing integration relative to its
maximum possible value over a given partition. Using the relative integrated
information quantifies the strength of the interactions between parts in a way
that does not depend on the number of parts and their size. As proven in [16],
the maximal value of $\varphi(\mathcal{T}_{S},s,\theta)$ for a given partition
$\theta$ is the normalization factor
$\displaystyle\max_{\mathcal{T}^{\prime}_{S}}\varphi(\mathcal{T}_{S},s,\theta)=\sum_{i=1}^{k}|S^{(i)}||X^{(i)}|$,
which corresponds to the maximal possible number of “connections” (pairwise
interactions) affected by $\theta$. For example, as shown in [16], the MIP
will correctly identify the fault line dividing a system into two large
subsets of units linked through a few interconnected units (a “bridge”),
rather than defaulting to partitions between individual units and the rest of
the system. Once the minimum partition has been identified, the integrated
information across it is an absolute quantity, quantifying the loss of
intrinsic information due to cutting the minimum partition of the system. (If
two or more partitions $\theta\in\Theta(S)$ minimize Eqn. (22), we select the
partition with the largest unnormalized $\varphi_{s}$ value as
$\theta^{\prime}$, applying the principle of maximal existence.) Defining
$\theta^{\prime}$ as in (22), moreover, ensures that
$\varphi_{s}(\mathcal{T}_{S},s)=0$ if the system is not strongly connected in
graph-theoretic terms.

In summary, the system integrated information
($\varphi_{s}(\mathcal{T}_{S},s)$) quantifies the extent to which system $S$
in state $s$ has cause–effect power over itself as one system (_i.e._ ,
irreducibly). $\varphi_{s}(\mathcal{T}_{S},s)$ is thus a quantifier of
irreducible existence.

### Exclusion: Determining maximal substrates (complexes)

In general, multiple candidate systems with overlapping units may have
positive values of $\varphi_{s}(\mathcal{T}_{S},s)$. By the exclusion
postulate, the substrate of consciousness must be definite; that is, it must
comprise a definite set of units. But which one? Once again, we employ the
principle of maximal existence (Box 2): among candidate systems competing over
the same substrate with respect to an essential requirement for existence, in
this case irreducibility, the one that exists is the one that exists the most.
Accordingly, the maximal substrate, or complex, is the candidate substrate
with the maximum value of system integrated information ($\varphi^{*}_{s}$),
and overlapping substrates with lower $\varphi_{s}$ are excluded from
existence.

#### Determining maximal substrates recursively

Within a universal substrate $U_{0}$ in state $u_{0}$, subsets of units that
specify maxima of irreducible cause–effect power (complexes) can be identified
by an iterative search for the system $S_{k}^{*}\subseteq U_{k}$ with

$\varphi_{s}^{*}(\mathcal{T}_{U_{k}},u_{k})=\max_{S\subseteq
U_{k}}\varphi_{s}(\mathcal{T}_{S},s),$ (23)

such that

$S_{k}^{*}=\operatornamewithlimits{argmax}\limits_{S\subseteq
U_{k}}\varphi_{s}(\mathcal{T}_{S},s),$ (24)

and $U_{k+1}=U_{k}\setminus S_{k}^{*}$ until $U_{k+1}=\emptyset$ or
$U_{k+1}=u_{k}$ (the units in $U_{0}\setminus U_{k+1}$ still serve as
background conditions). If the maximal substrate $S^{*}_{k}$ is not unique,
and all tied systems overlap, the next best system that is unique is chosen
instead; for details see [16], and also [41].

For any complex $S^{*}=s^{*}$, overlapping substrates that specify less
integrated information ($\varphi_{s}<\varphi_{s}(\mathcal{T}_{S^{*}},s^{*})$)
are excluded. Consequently, specifying a maximum of integrated information
$\varphi_{s}^{*}$ compared to all overlapping systems

$S\cap\tilde{S}\neq\varnothing\Rightarrow\varphi_{s}(s)>\varphi_{s}(\tilde{s}),\;\;\forall\tilde{S}\neq
S\subseteq U$ (25)

is a sufficient requirement for a system $S\subseteq U$ to be a complex.

As shown in [16], this recursive search for maximal substrates “condenses” the
universe $U_{0}=u_{0}$ into a disjoint (non-overlapping) and exhaustive set of
complexes—the first complex, second complex, and so on.

#### Determining maximal unit grains

Above, we presented how we can determine the borders of a complex within a
larger system $U$, assuming a particular spatio-temporal grain for the units
$U_{i}\in U$. In principle, however, all possible grains should be considered
[42, 43]. In the brain, for example, the grain of units could be brain
regions, groups of neurons, individual neurons, sub-cellular structures,
molecules, atoms, quarks, or anything finer, down to hypothetical atomic units
of cause–effect power [3, 7]. For any unit grain—neurons, for example—the
grain of updates could be minutes, seconds, milliseconds, micro-seconds, and
so on. And the grain of states could be 2 states (“no spikes / any number of
spikes per neuron over one hundred milliseconds”), 4 states (“no spikes, 1
spike, 2–5 spikes, bursts of $>5$ spikes”), or 256 states (from 0 to 255
spikes in intervals of 1 spike), and so on. However, by the exclusion
postulate, the units that constitute a system $S$ must also be definite, in
the sense of having a definite spatio-temporal grain.

Once again, the grain is defined by the principle of maximal existence: across
the possible micro- and macroscopic levels, the “winning” grain is the one
that ensures maximally irreducible existence ($\varphi_{s}^{*}$) for the
entity to which the units belong [42, 43].

As constituents of a complex upon which its cause–effect power rests, the
units themselves should comply with the postulates of IIT [4]. Otherwise it
would be possible to “make something out of nothing.” Accordingly, units
themselves must also be maximally irreducible, as measured by the unit’s
integrated information ($\varphi_{j}\equiv\varphi_{s}$); otherwise, they would
not be units but “disintegrate” into their constituents. However, in contrast
to systems, units only need to be maximally irreducible within, because they
do not exist as entities in their own right: a unit $J$ qualifies as a
candidate unit of a larger system $S$ if its integrated information
$\varphi(j)$ is higher than that of any of its subsets

$\tilde{J}\subset
J\Rightarrow\varphi_{s}(\mathcal{T}_{J},j)>\varphi_{s}(\mathcal{T}_{\tilde{J}},\tilde{j}),\;\;\forall\tilde{J}\subset
J.$ (26)

Out of all possible sets of such candidate units, the set of (macro) units
that actually exists is the one that maximizes the existence of the complex to
which they belong, rather than their own existence.

## Unfolding the cause–effect structure of a complex through composition

Once a maximal substrate and the associated maximal cause–effect state have
been identified, we must unfold its cause–effect power to reveal its
cause–effect structure of distinctions and relations, in line with the
composition postulate. As components of the cause–effect structure,
distinctions and relations must also satisfy the postulates of IIT (save for
composition).

### Composition and causal distinctions

Causal distinctions capture how the cause–effect power of a substrate is
structured by subsets of units that specify irreducible causes and effects
over subsets of its units. A candidate distinction $d(m)$ consists of  (1) a
mechanism $M\subseteq S$ in state $m\in\Omega_{M}$ inherited from the system
state $s\in\Omega_{S}$;  (2) a maximal cause–effect state
$z^{*}=\\{z^{*}_{c},z^{*}_{e}\\}$ over the cause and effect purviews
($Z_{c},Z_{e}\subseteq S$) linked by the mechanism; and  (3) an associated
value of irreducibility ($\varphi_{d}>0$).  A distinction $d(m)$ is thus
represented by the tuple

$d(m)=\Big{(}m,z^{*},\varphi_{d}\Big{)}.$ (27)

For a given mechanism $m$, our goal is to identify the maximal cause
$Z^{*}_{c}=z^{*}_{c}$ and effect $Z^{*}_{e}=z^{*}_{e}$ of $m$ within the
system, where $Z^{*}_{c},Z^{*}_{e}\subseteq S$,
$z^{*}_{c}\in\Omega_{Z^{*}_{c}}$, $z^{*}_{e}\in\Omega_{Z^{*}_{e}}$.

As above, in line with existence, intrinsicality, and information, we
determine the maximal cause or effect state specified by the mechanism over a
candidate purview within the system based on the value of intrinsic
information $\operatorname{\mathit{ii}}(m,z)$. Next, in line with integration,
we determine the value of integrated information $\varphi_{d}(m,Z,\theta)$
over the minimum partition $\theta^{\prime}$. In line with exclusion, we
determine the maximal cause–effect purviews for that mechanism over all
possible purviews $Z\subseteq S$ based on the associated value of
irreducibility $\varphi_{d}(m,Z,\theta^{\prime})$. Finally, we determine
whether the maximal cause–effect state specified by the mechanism is congruent
with the system’s overall cause–effect state ($z^{*}_{c}\subseteq s^{*}_{c}$,
$z^{*}_{e}\subseteq s^{*}_{e}$), in which case we conclude that it contributes
a distinction to the overall cause–effect structure.

The updated formalism to identify causal distinctions within a system $S$ in
state $s$ was first presented in [15]. Here we provide a summary with minor
adjustments on selecting $z^{*}_{c}$ and $z^{*}_{e}$, the cause integrated
information $\varphi_{c}(m,Z)$, and the requirement that causal distinctions
must be congruent with the system’s maximal cause–effect state (see A.2).

#### Existence, intrinsicality, and information: Determining the cause and
effect state specified by a mechanism over candidate purviews

Like the system as a whole, its subsets must comply with existence,
intrinsicality, and information. As for the system, we begin by quantifying,
in probabilistic terms, the difference a subset of units $M\subseteq S$, in
its current state $m\subseteq s$ takes and makes from and to subsets of units
$Z\subseteq S$ (cause and effect purview). As above, we start by establishing
the interventional conditional probabilities and unconstrained probabilities
from the TPM $\mathcal{T}_{S}$.

When dealing with a mechanism constituted by a subset of system units, it is
important to capture the constraints on a purview state $z$ that are
exclusively due to the mechanism in its state ($m$), removing any potential
contribution from other system units. This is done by causally marginalizing
all variables in $X=S\setminus M$, which corresponds to imposing a uniform
distribution as $p(X_{t})$ [11, 13, 15] (note Units within the candidate
system are causally marginalized to discount their causal contribution if they
are not part of the mechanism or purview under consideration. By contrast,
units outside the candidate system are causally conditioned in their current
state throughout the analysis and act as fixed background conditions). In
addition, product probabilities $\pi(z\mid m)$ are used instead of conditional
probabilities $p(z\mid m)$ to discount correlations from units in
$X=S\setminus M$ with divergent outputs to multiple units in $Z\subseteq S$
[11, 45, 13]. In this way, causal marginalization maintains the conditional
independence between units (2) by applying independent noise to individual
connections. The assumption of conditional independence distinguishes IIT’s
causal analysis from standard information-theoretic analyses of information
flow [31, 13] and corresponds to an assumption that variables are “physical”
units in the sense that they are irreducible within and can be observed and
manipulated independently.

The effect probability of a single unit $Z_{i}\in Z$ conditioned on the
current state $m$ is thus defined as

$\pi_{e}(z_{i}\mid m)=p_{e}(z_{i}\mid
m)=|\Omega_{X}|^{-1}\sum_{x\in\Omega_{X}}p\left(z_{i}\mid m,x\right),\quad
z_{i}\in\Omega_{Z_{i}}$ (28)

and the effect probability over a set $Z$ of $|Z|$ units is defined as the
product of the effect probabilities over individual units

$\pi_{e}(z\mid m)=\prod_{i=1}^{|Z|}\pi_{e}(z_{i}\mid m),\quad z\in\Omega_{Z}.$
(29)

From Eq. (29) we can also define an unconstrained effect probability

$\pi_{e}(z;M)=|\Omega_{M}|^{-1}\sum_{m\in\Omega_{M}}\pi_{e}(z\mid M=m),\quad
z\in\Omega_{Z}.$ (30)

Given the set $Y=S\setminus Z$, the cause probability for a mechanism $m$ with
$|M|$ units is computed using Bayes’ rule over the product distributions

$\pi_{c}(z\mid m)=\frac{\displaystyle\pi_{e}(m\mid
z)\cdot|\Omega_{Z}|^{-1}}{\displaystyle\pi_{e}(m;Z)}=\frac{\displaystyle\prod_{i=1}^{|M|}p_{e}(m_{i}\mid
z)}{\displaystyle\sum_{\hat{z}\in\Omega_{Z}}\prod_{i=1}^{|M|}p_{e}(m_{i}\mid\hat{z})},\quad
z\in\Omega_{Z},$ (31)

where $\displaystyle p_{e}(m_{i}\mid
z)=|\Omega_{Y}|^{-1}\sum_{y\in\Omega_{Y}}p(m_{i}\mid z,y)$ in line with (28).

To correctly quantify intrinsic causal constraints, the unconstrained cause
probability is again set to the uniform distribution
$\pi_{c}(z)=|\Omega_{Z}|^{-1},\;z\in\Omega_{Z}$ (unlike the unconstrained
effect probability, the unconstrained cause distribution does not depend on
the mechanism). Note that the product in (31) is over units of $M$, not of
$Z$. For details see [13, 15]. As above, all probabilities $p$ are obtained
from the TPM $\mathcal{T}_{S}$ (3) and thus correspond to interventional
probabilities throughout.

Having defined cause and effect probabilities, we can now evaluate the
intrinsic information of a mechanism $m$ over a purview state $Z=z$
analogously to the system intrinsic information (4) and (6). The intrinsic
effect information that a mechanism in a state $m$ specifies about a purview
state $z$ is

$\operatorname{\mathit{ii}}_{e}(m,z)=\pi_{e}(z\mid
m)\log\left(\frac{\pi_{e}(z\mid m)}{\pi_{e}(z;M)}\right).$ (32)

The intrinsic cause information that a mechanism in a state $m$ specifies
about a purview state $z$ is

$\operatorname{\mathit{ii}}_{c}(m,z)=\pi_{c}(z\mid
m)\log\left(\frac{\pi_{e}(m\mid z)}{\pi_{e}(m;Z)}\right).$ (33)

As with system intrinsic information, the logarithmic term is the
informativeness, which captures how much causal power is exerted by the
mechanism $m$ on its potential effect $z$ (how much it increases the
probability of that state above chance), or by the potential cause $z$ on the
mechanism $m$. The first term corresponds to the mechanism’s selectivity,
which captures how much the causal power of the mechanism $m$ is concentrated
on a specific state of its purview (as opposed to other states). In the
following we will again focus on the effect side, but an equivalent procedure
applies on the cause side.

Based on the principle of maximal existence, the maximal effect state of $m$
within the purview $Z$ is defined as

$\begin{split}z^{\prime}_{e}(m,Z)=&\operatornamewithlimits{argmax}\limits_{z\in\Omega_{Z}}\operatorname{\mathit{ii}}_{e}(m,z)\\\
=&\operatornamewithlimits{argmax}\limits_{z\in\Omega_{Z}}\left(\pi_{e}(z\mid
m)\log\left(\frac{\pi_{e}(z\mid m)}{\pi_{e}(z;M)}\right)\right),\end{split}$
(34)

which corresponds to the specific effect of $m$ on $Z$. Note that
$z^{\prime}_{e}$ is not always unique (see A.1). The maximal intrinsic
information of mechanism $m$ over a purview $Z$ is then

$\operatorname{\mathit{ii}}_{e}(m,Z):=\operatorname{\mathit{ii}}_{e}(m,z^{\prime}_{e})=\max_{z\in\Omega_{Z}}\operatorname{\mathit{ii}}_{e}(m,z).$
(35)

The intrinsic information of a candidate distinction, like that of the system
as a whole, is sensitive to indeterminism (the same state leading to multiple
states) and degeneracy (multiple states leading to the same state) because
both factors decrease the probability of the selected state. Moreover, the
product of selectivity and informativeness leads to a tension between
expansion and dilution: larger purviews tend to increase informativeness
because conditional probabilities will deviate more from chance, but they also
tend to decrease selectivity because of the larger repertoire of states.

#### Integration: Determining the irreducibility of a candidate distinction

To comply with integration, we must next ask whether the specific effect of
$m$ on $Z$ is irreducible. As for the system, we do so by evaluating the
integrated information $\varphi_{e}(m,Z)$. To that end, we define a set of
“disintegrating” partitions $\Theta(M,Z)$ as

$\Theta(M,Z)=\Bigg{\\{}\\{(M^{(i)},Z^{(i)})\\}_{i=1}^{k}\;:\;k\in\\{2,3,4,\ldots\\},\;M^{(i)}\in\mathbb{P}(M),\;Z^{(i)}\in\mathbb{P}(Z),\\\
\bigcup M^{(i)}=M,\bigcup Z^{(i)}=Z,Z^{(i)}\cap Z^{(j)}=M^{(i)}\cap
M^{(j)}=\emptyset\;\forall\;i\neq j,M^{(i)}=M\Longrightarrow
Z^{(i)}=\emptyset\Bigg{\\}},$ (36)

where $\\{M^{(i)}\\}$ is a partition of $M$ and $\\{Z^{(i)}\\}$ is a partition
of $Z$, but the empty set may also be used as a part ($\mathbb{P}$ denotes the
power set). As introduced in [13, 15], a disintegrating partition
$\theta\in\Theta(M,Z)$ either “cuts” the mechanism into at least two
independent parts if $|M|>1$, or it severs all connections between $M$ and
$Z$, which is always the case if $|M|=1$ (we refer to [13, 15] for details).

Given a partition $\theta\in\Theta(M,Z)$, we can define the partitioned effect
probability

$\pi_{e}^{\theta}(z^{\prime}_{e}\mid
m)=\prod_{i=1}^{k}\pi_{e}(z^{\prime(i)}_{e}\mid m^{(i)}),$ (37)

with $\pi(\varnothing|m^{(i)})=\pi(\varnothing)=1$. In the case of
$m^{(i)}=\varnothing$, $\pi_{e}(z^{\prime(i)}_{e}|\varnothing)$ corresponds to
the fully partitioned effect probability

$\pi_{e}(z\mid\varnothing)=\prod_{i=1}^{|Z|}\pi_{e}(z_{i})=\prod_{i=1}^{|Z|}\sum_{s\in\Omega_{S}}p_{e}(z_{i}\mid
s)|\Omega_{S}|^{-1}.$ (38)

The integrated effect information of mechanism $m$ over a purview $Z\subseteq
S$ with effect state $z^{\prime}_{e}$ for a particular partition
$\theta\in\Theta(M,Z)$ is then defined as

$\varphi_{e}(m,Z,\theta)=\pi_{e}(z^{\prime}_{e}\mid
m)\log\left(\frac{\pi_{e}(z^{\prime}_{e}\mid
m)}{\pi_{e}^{\theta}(z^{\prime}_{e}\mid m)}\right).$ (39)

The effect of $m$ on $z^{\prime}_{e}$ is reducible if at least one partition
$\theta\in\Theta(M,Z)$ makes no difference to the effect probability. In line
with the principle of minimal existence, the total integrated effect
information $\varphi_{e}(m,Z)$ again has to be evaluated over
$\theta^{\prime}$, the minimum partition (MIP)

$\varphi_{e}(m,Z):=\varphi_{e}(m,Z,\theta^{\prime}),$ (40)

which requires a search over all possible partitions $\theta\in\Theta(M,Z)$:

$\theta^{\prime}=\operatornamewithlimits{argmin}\limits_{\theta\in\Theta(M,Z)}\frac{\varphi(m,Z,\theta)}{\displaystyle\max_{\mathcal{T}^{\prime}_{S}}\varphi(m,Z,\theta)}.$
(41)

As in (22), the minimum partition is evaluated against its maximum possible
value across all possible system $\mathcal{T}^{\prime}_{S}$, which again
corresponds to the number of possible pairwise interactions affected by the
partition.

The integrated cause information is defined analogously, as

$\varphi_{c}(m,Z):=\varphi_{c}(m,Z,\theta^{\prime})=\pi_{c}(z^{\prime}_{c}\mid
m)\log\left(\frac{\pi_{e}(m\mid
z^{\prime}_{c})}{\pi_{e}^{\theta^{\prime}}(m\mid z^{\prime}_{c})}\right),$
(42)

where the partitioned probability $\pi^{\theta}_{e}(m\mid z)$ is again a
product distribution over the parts in the partition, as in (37).

Taken together, the intrinsic information (35) determines what cause or effect
state the mechanism $m$ specifies. Its integrated information quantifies to
what extent $m$ specifies its cause or effect in an irreducible manner. Again,
$\varphi(m,Z)$ is a quantifier of irreducible existence.

#### Exclusion: Determining causal distinctions

Finally, to comply with exclusion, a mechanism must select a definite effect
purview, as well as a cause purview, out of a set of candidate purviews.
Resorting again to the principle of maximal existence, the mechanism’s effect
purview and associated effect is the one having the maximum value of
integrated information across all possible purviews $Z\subseteq S$ in state
$z^{\prime}_{e}(m,Z)$ (34)

$z_{e}^{*}(m)=\operatornamewithlimits{argmax}\limits_{\\{z^{\prime}_{e}|Z\subseteq
S\\}}\varphi_{e}(m,Z=z^{\prime}_{e}).$ (43)

The integrated effect information of a mechanism $m$ within $S$ is then

$\varphi_{e}(m):=\varphi_{e}(m,z^{*}_{e})=\max_{\\{z^{\prime}_{e}|Z\subseteq
S\\}}\varphi_{e}(m,Z=z^{\prime}_{e}).$ (44)

The integrated cause information $\varphi_{c}(m)$ and the maximally
irreducible cause $z^{*}_{c}(m)$ are defined in the same way. Based again on
the principle of minimal existence, the irreducibility of the distinction
specified by a mechanism is given by the minimum between its integrated cause
and effect information

$\varphi_{d}(m)=\min\bigl{(}\varphi_{c}(m),\varphi_{e}(m)\bigr{)}.$ (45)

#### Determining the set of causal distinctions that are congruent with the
system cause–effect state

As required by composition, unfolding the full cause–effect structure of the
system $S$ in state $s$ requires assessing the irreducible cause–effect power
of every subset of units within $S$ (Fig. 2). Any $m\subseteq s$ with
$\varphi_{d}>0$ specifies a candidate distinction $d(m)=(m,z^{*},\varphi_{d})$
(27) within the system $S$ in state $s$. However, in order to contribute to
the cause–effect structure of a system, distinctions must also comply with
intrinsicality and information at the system level. This means that the
cause–effect state they specify over subsets of the system
($z^{*}=\\{z^{*}_{c},z^{*}_{e}\\}$) must be congruent with the cause–effect
state specified over itself by the system as a whole $s^{\prime}$.

We thus define the set of all causal distinctions within $S$ in state $s$ as

$D(\mathcal{T}_{S},s)=\\{d(m)~{}:~{}m\subseteq
s,~{}\varphi_{d}(m)>0,~{}z^{*}_{c}(m)\subseteq
s^{\prime}_{c},~{}z^{*}_{e}(m)\subseteq s^{\prime}_{e}\\}.$ (46)

Fig 2: Composition and causal distinctions. Identifying the irreducible causal
distinctions specified by a substrate in a state requires evaluating the
specific causes and effects of every system subset. The candidate substrate is
constituted of two interacting units $S=aB$ (see Fig. 1) and updates its state
according to the depicted transition probability matrix. In addition to the
two first-order mechanisms $a$ and $B$, the second-order mechanism $aB$
specifies its own irreducible cause and effect, as indicated by
$\varphi_{d}>0$.

Altogether, distinctions can be thought of as irreducible “handles” through
which the system can take and make a difference to itself by linking an
intrinsic cause to an intrinsic effect over subsets of itself. As components
within the system, causal distinctions have no inherent structure themselves.
Whatever structure there may be between the units that make up a distinction
is not a property of the distinction but due to the structure of the system,
and thus captured already by its compositional set of distinctions. Similarly,
from an extrinsic perspective, one may uncover additional causes and effects,
both within the system and across its borders, at either macro or micro
grains. However, from the intrinsic perspective of the system causes and
effects that are excluded from its cause–effect structure do not exist [36,
20].

For example, as shown in Fig. 3(A), a system may have a mechanism through
which it specifies, in a maximally irreducible manner, the effect state of a
triplet of units (_e.g._ , $z^{*}_{e}=abc$, a third-order purview; again
lowercase letters for units indicate state “$-1$”, uppercase letters state
“$+1$”). However, if the system lacks a mechanism through which it can specify
the effect state of single units, each taken individually (say, unit $a$, a
first-order effect purview), then, from its intrinsic perspective, that unit
does not exist as a single unit. By the same token, if the system can specify
individually the state of unit $a$, $b$, and $c$, but lacks a way to specify
irreducibly the state of $abc$ together, then, from its intrinsic perspective,
the triplet $abc$ does not exist as a triplet (see Fig. 3(B)). Finally, even
if the system can distinguish the single units $a$, $b$, and $c$, as well as
the triplet $abc$, if it lacks handles to distinguish pairs of units such as
$ab$ and $bc$, it cannot order units in a sequence.

Fig 3: Composition of intrinsic effects. From the intrinsic perspective of the
system, a specific cause or effect is only available to the system if it is
selected by a causal distinction $d\in D(s)$. In (A), only the top-order
effect is specified. From the intrinsic perspective, the system cannot
distinguish the individual units. In (B), only first-order effects are
specified. The system has no “handle” to select all three units together. (C)
If both first- and third-order effects are specified, but no second-order
effects, the system can distinguish individual units and select them together,
but has no way of ordering them sequentially. (D) The system can distinguish
individual units, select them altogether, as well as order them sequentially,
in the sense that it has a handle for $ab$ and $bc$, but not $ac$. The
ordering becomes apparent once the relations among the distinctions are
considered (see below).

### Composition and causal relations

Causal relations capture how the causes and/or effects of a set of
distinctions within a complex overlap with each other. Just as a distinction
specifies which units/states constitute a cause purview and the linked effect
purview, a relation specifies which units/states correspond to which
units/states among the purviews of a set of distinctions. Relations thus
reflect how the cause–effect power of its distinctions is “bound together”
within a complex. The irreducibility due to this binding of cause–effect power
is measured by the relations’ irreducibility ($\varphi_{r}>0$). Relations
between distinctions were first described in [14] (for differences with the
initial presentation see A.2).

A set of distinctions $\bm{d}\subseteq D(s)$ is related if the cause, or
effect, or both the cause and effect of each distinction $d\in\bm{d}$ overlap
congruently over a set of shared units. Below we will denote the cause of a
distinction $d$ as $z^{*}_{c}(d)$ and its effect as $z^{*}_{e}(d)$. For a
given set of distinctions $\bm{d}\subseteq D(s)$, there are potentially many
“relating” sets of causes and/or effects $\bm{z}$ such that

$\bm{z}\;:\;\bm{z}\cap\\{z^{*}_{c}(d),z^{*}_{e}(d)\\}\neq\varnothing\;\;\forall
d\in\bm{d},\;\bigcap_{z\in\bm{z}}z\neq\varnothing,\;|\bm{z}|>1$ (47)

with maximal overlap

$o^{*}(\bm{z})=\bigcap_{z\in\bm{z}}z\neq\varnothing.$ (48)

All possible sets $\bm{z}$ specify unique aspects about a relation $r(\bm{d})$
and constitute the various “faces” of the relation. The maximal overlap
$o^{*}(\bm{z})$ (48) is also called the “face purview.” Note that (47)
includes the case $\bm{z}=\\{z^{*}_{c}(d),z^{*}_{e}(d)\\}$, which indicates a
“self-relation” over the cause and effect of a single distinction $d\in D(s)$.

A relation $r(\bm{d})$ thus consists of a set of distinctions $\bm{d}\subseteq
D(s)$, with an associated set of faces
$\bm{f}(\bm{d})=\\{f(\bm{z})\\}_{\bm{d}}$ and irreducibility $\varphi_{r}>0$,

$r(\bm{d})=\Big{(}\bm{d},\bm{f}(\bm{d}),\varphi_{r}\Big{)}.$ (49)

A relation that binds together $h=|\bm{d}|$ distinctions is a $h$-degree
relation. A relation face $f(\bm{z})\in\bm{f}(\bm{d})$ consists of a set of
causes and effects $\bm{z}$ (as in 47), with associated face purview
$o^{*}(\bm{z})\eqref{eq:overlap}$

$f(\bm{z})=\Big{(}\bm{z},o^{*}(\bm{z})\Big{)}.$ (50)

A relation face over $k=|\bm{z}|$ purviews is a $k$-degree face. Because
$\bm{z}$ may include either the cause, or the effect, or both the cause and
effect of a distinction $d\in\bm{d}$, a relation $r(\bm{d})$ with $|\bm{d}|>1$
may comprise up to $3^{|\bm{d}|}$ faces. If a set of distinctions $\bm{d}\in
D(s)$ does not overlap congruently, it is not related (in that case
$o^{*}(\bm{z})=\varnothing$ for all possible $f(\bm{z})\in\bm{f}(\bm{d})$).

Causal relations inherit existence from the cause–effect power of the
distinctions that compose them. They inherit intrinsicality because the causes
and effects that compose their faces are specified within the substrate.
Moreover, relations are specific because the joint purviews of their faces
must be congruent for all causes and effects $z^{*}\in\bm{z}$. Note that
relation purviews are necessarily congruent with the overall cause and effect
state specified by the system as a whole, because the causes and effects of
the distinctions composing a relation must themselves be congruent.

The irreducibility of a causal relation is measured by “unbinding”
distinctions from their joint purviews, taking into account all faces of the
relation. Distinctions $d\in D(s)$ are already established as maximally
irreducible components, characterized by their value of integrated information
$\varphi_{d}$. To assess the irreducibility of a relation, we thus assume that
the integrated information $\varphi_{d}$ of a distinction is distributed
uniformly across unique cause and effect purview units, such that

$\frac{\varphi_{d}}{|z^{*}_{c}(d)\cup z^{*}_{e}(d)|}$ (51)

is the average irreducible information $\varphi_{d}$ per unique purview unit
for an individual distinction $d\in\bm{d}$ with cause–effect state
$z^{*}(d)=\\{z^{*}_{c}(d),z^{*}_{e}(d)\\}$ (congruent units on the cause and
effect side count as one, while incongruent units are counted separately).

Since distinctions are related by constraining common units (though they may
constrain them in different ways), the effect of “unbinding” a distinction
must be proportional to the number of units jointly constrained in the
relation, _i.e._ the number of unique units over the joint purviews of all
faces in the relation:

$\left|\bigcup_{f\in\bm{f}(\bm{d})}\;o^{*}_{f}\;\right|.$ (52)

This union of the face purviews $o^{*}_{f}$ is also called the “relation
purview” or the “joint purview” of the relation. While any partition of one or
more distinctions from the relation will “unbind” the set of distinctions
$\bm{d}$, by the principle of minimal existence, a relation can only be as
irreducible as the minimal amount of integrated information specified by any
one distinction in the relation. Therefore, the relation integrated
information $\varphi_{r}(\bm{d})$ is defined as

$\varphi_{r}(\bm{d})=\left|\bigcup_{f\in\bm{f}(\bm{d})}\;o^{*}_{f}\;\right|\min_{d\in\bm{d}}\;\frac{\varphi_{d}}{|z^{*}_{c}(d)\cup
z^{*}_{e}(d)|}.$ (53)

Defining $\varphi_{r}$ in this way guarantees that the integrated information
of a relation cannot exceed the integrated information of its weakest
distinction. A maximal relation is thus a relation in which the cause and
effect of each distinction is fully overlapped by other distinctions in the
relation (in that case, $\varphi_{r}=\min_{d\in\bm{d}}\varphi_{d}$). Note also
that a relation satisfies exclusion in that its integrated information is
naturally maximized over the maximally congruent overlap $o^{*}_{f}$ for each
relation face (48) (taking subsets of these overlaps could only reduce the
integrated information of the relation).

Fig 4: Composition and causal relations. Relations between distinctions
specify joint causes and/or effects. The two distinctions $d(a)$ and $d(aB)$
each specify their own cause and effect. In this example, their cause and
effect purviews overlap over the unit $b$ and are congruent, which means that
they all specify $b$ to be in state “-1”. The relation $r(\\{a,ab\\})$ thus
binds the two distinctions together over the same unit. Relation faces are
indicated by the blue lines and surfaces between the distinctions’ causes
and/or effects. Because all four purviews overlap over the same unit, all nine
possible faces exist. Note that the fact that the two distinctions overlap
irreducibly can only be captured by a relation and not by a high-order
distinction.

In summary, just as distinctions link a cause with an effect, relations bind
various combinations of causes and effects that are congruent over the same
units, _i.e._ , constraining those units to be in the same state (Fig. 4). And
just as a distinction captures the irreducibility of an individual
cause–effect linked by a mechanism, a relation captures the irreducibility of
a set of distinctions bound by the joint purviews of their causes and/or
effects.

For a set of distinctions $D$, we define the set of all relations among them
as

${R}(D)=\\{r(\bm{d}):\varphi_{r}(\bm{d})>0\\},\;\forall\bm{d}\subseteq D.$
(54)

In practice, the total number of relations and their
$\sum_{\bm{R(D)}}\varphi_{r}$ can be determined analytically for a given set
of distinctions $D$, which greatly reduces the necessary computations (see
A.3). Together, a set of distinctions $D$ and its associated set of relations
${R}(D)$ compose a cause–effect structure.

Fig 5: Structuring of intrinsic effects by relations. (A) A single
undifferentiated effect has no relations. (B) Likewise, there are no relations
among multiple non-overlapping effects. (C) The set of three first-order
effects and one third-order effect supports three relations, which bind the
effects together. (D) The set of first, second, and third-order effects
supports a large number of relations (10 2-relations (between two effects), 6
3-relations, and 1 4-relation), which bind the effects in a structure that is
ordered sequentially.

### Cause–effect structures and $\varPhi$-structures

A cause–effect structure is defined as the union of the distinctions specified
by a substrate and the relations binding them together:

$C(D)=D\cup{R}(D).$ (55)

The cause–effect structure specified by a maximal substrate—a complex—is also
called a $\varPhi$-structure:

$C(\mathcal{T}_{S^{*}},s^{*})=\Bigg{\\{}\Big{\\{}d(m)=\\{m,z^{*},\varphi_{d}\\}\in
D(\mathcal{T}_{S^{*}},s^{*})\Big{\\}}~{}\bigcup~{}\Big{\\{}r(\bm{d})=\\{\bm{d},\bm{f}(d),\varphi_{r}\\}\in
R\bigl{(}D(\mathcal{T}_{S^{*}},s^{*})\bigr{)}\Big{\\}}\Bigg{\\}}.$ (56)

The sum of the values of integrated information of a substrate’s distinctions
and relations, called $\varPhi$ (“big phi”) corresponds to the structured
information of the $\varPhi$-structure,

$\varPhi(\mathcal{T}_{S^{*}},s^{*})=\sum_{C(\mathcal{T}_{S^{*}},s^{*})}\varphi.$
(57)

In conclusion, a maximal substrate or complex is a set of units $S^{*}=s^{*}$
that satisfies all of IIT’s postulates: it has to have cause–effect power that
is intrinsic, specific, irreducible, definite, and structured. By IIT, a
complex $S^{*}$ does not exist as such, but exists “unfolded” into its
$\varPhi$-structure, with all the causal distinctions and relations that
compose it. In other words, a substrate is what can be observed and
manipulated “operationally” from the extrinsic perspective. From the intrinsic
perspective, what truly exists is a complex with all its causal powers
unfolded—an intrinsic entity that exists for itself, absolutely, rather than
relative to an external observer.

According to the explanatory identity of IIT, an experience is identical to
the $\varPhi$-structure of an intrinsic entity: every property of the
experience should be accounted for by a corresponding property of the
$\varPhi$-structure, with no additional ingredients. If a system $S$ in state
$s$ is a complex, then its $\varPhi$-structure corresponds to the quality of
the experience of $S$ in state $s$, while its $\varPhi$ value corresponds to
its quantity—in other words, the nature and amount of intrinsic content.

## Results and discussion

In this section, we apply the mathematical framework of IIT 4.0 to several
example systems. The goal is to illustrate three critical implications of
IIT’s postulates:

  1. 1.

Consciousness and connectivity: how the way units interact determines whether
a substrate can support a $\varPhi$-structure of high $\varPhi$

  2. 2.

Consciousness and activity: how changes in the activity of a substrate’s units
change $\varPhi$-structures

  3. 3.

Consciousness and functional equivalence: how substrates that are functionally
equivalent may not be equivalent in terms of their $\varPhi$-structures, and
thus in terms of consciousness

The following examples will feature very simple networks constituted of binary
units $U_{i}\in U$ with $\Omega_{U_{i}}=\\{-1,1\\}$ for all $U_{i}$ and a
logistic (sigmoidal) activation function

$p(U_{i,t}=1\mid
u_{t-1})=\frac{1}{1+\exp\left(-k\sum_{j=1}^{n}w_{j,i}u_{j,t-1}\right)},$ (58)

where

$\sum_{j=1}^{n}w_{j,i}=1~{}\forall~{}i.$ (59)

In Eq. (58), the parameter $k$ defines the slope of the logistic function and
allows one to adjust the amount of noise or determinism in the activation
function (higher values signify a steeper slope and thus more determinism).
The units $U_{i}$ can thus be viewed as noisy linear threshold units with
weighted connections among them.

As in Figs. 1 and 2, units denoted by uppercase letters are in state ‘$1$’
(ON, depicted in black), units denoted by lowercase letters are in state
‘$-1$’ (OFF, depicted in white). Cause–effect structures are illustrated as
geometrical shapes projected into 3D space (Fig. 6). Distinctions are depicted
as mechanisms (black labels) tying a cause (red labels) and an effect (green
labels) through a link (orange edges, thickness indicating $\varphi_{d}$).
Relation faces of second- and third-degree relations are depicted as blue
edges or triangular surfaces between the causes and effects of the related
distinctions. While edges always bind pairs of distinctions (a second-degree
relation), triangular surfaces may bind the causes and effects of two or three
distinctions (second- or third-degree relation). Relations of higher degrees
are not depicted.

All examples were computed using the “iit-4.0” feature branch of PyPhi [46].
This branch is currently under development and will be available in the next
official release of the software. An example notebook is available here.

### Consciousness and connectivity

The first set of examples highlights how the organization of connections among
units impacts the ability of a substrate to support a cause–effect structure
with high structured information (high $\varPhi$). Fig. 6 shows five systems,
all in the same state $s=\textit{Abcdef}$ with the same number of units, but
with different connectivity among the units.

Fig 6: Causal analysis of various network architectures. Each panel shows the
network’s causal model and weights on the left. Blue regions indicate
complexes with their respective $\varphi_{s}$ values. In all networks, $k=4$
and the state is $Abcdef$. The $\varPhi$-structure(s) specified by the
network’s complexes are illustrated below or to the right (with only second-
and third-degree relation faces depicted) with a list of their distinctions
for smaller systems and their $\sum\varphi$ values for those systems with many
distinctions and relations. All integrated information values are in ibits.
(A) A degenerate network in which unit $A$ forms a bottleneck with redundant
inputs from and outputs to the remaining units. The first-maximal complex is
$Ab$, which excludes all other subsets with $\varphi_{s}>0$ except for the
individual units $c$, $d$, $e$, and $f$. (B) The modular network condenses
into three complexes along its fault lines (which exclude all subsets and
supersets), each with a maximal $\varphi_{s}$ value, but low $\varPhi$, as the
modules each specify only two or three distinctions and at most five
relations. (C) A directed cycle of six units forms a six-unit complex with
$\varphi_{s}=1.74$ ibits, as no other subset is integrated. However, the
$\varPhi$-structure of the directed cycle is composed of only first-order
distinctions and few relations. (D) A specialized lattice also forms a complex
(which excludes all subsets), but specifies 24 first- and high-order
distinctions, with many relations ($>6*10^{5}$) among them. Its $\varPhi$
value is $6747$ ibits, much larger than the structured information of the
complexes that exist in the other networks. (E) A slightly modified version of
the specialized lattice in which the first-maximal complex is $Abef$. The full
system is not maximally irreducible and is excluded as a complex, despite its
positive $\varphi_{s}$ value (indicated in gray).

#### Degenerate systems, indeterminism, and specificity

Fig. 6A shows a network with medium indeterminism ($k=4$) and high degeneracy,
due to the fact that unit $A$ forms a “bottleneck” with inputs and outputs to
and from the remaining units. The network condenses into one complex of two
units $Ab$ and four complexes corresponding to the individual units $c$, $d$,
$e$, and $f$ (also called “monads”).

The causes and effects of the causal distinctions for the two types of
complexes are shown in the middle, and the corresponding cause–effect
structures are illustrated on the right. In this case, degeneracy (coupled
with indeterminism) undermines the ability of the maximal substrate to grow in
size, which in turn limits the richness of the $\varPhi$-structure that can be
supported. Because of the bottleneck architecture, the current state of
candidate system $Abcdef$ has many possible causes and effects, leading to an
exponential decrease in selectivity (the conditional probabilities of cause
and effect states). This dilutes the value of intrinsic information
($\operatorname{\mathit{ii}}$) for larger subsets of units, which in turn
reduces their value of system integrated information $\varphi_{s}$.
Consequently, the maximal substrates are small, and their $\varPhi$ values are
necessarily low.

This example suggests that to grow and achieve high values of $\varPhi$,
substrates must be constituted of units that are specialized (low degeneracy)
and interact very effectively (low indeterminism). Notably, the organization
of the cerebral cortex, widely considered as the likely substrate of human
consciousness, is characterized by extraordinary specialization of neural
units at all levels [47, 48, 49]. Moreover, if the background conditions are
well controlled, neurons are thought to interact in a highly reproducible,
nearly deterministic manner [50, 51, 52].

#### Modular systems, fault lines, and irreducibility

Fig. 6B shows a network comprising three weakly interconnected modules, each
having two strongly connected units ($k=4$). In this case, the weak inter-
module connections are clear fault lines. Properly normalized, partitions
along these fault lines separating modules yield values of $\varphi_{s}$ that
are much smaller than those yielded by partitions that cut across modules. As
a consequence, the 6-unit system condenses into three complexes (Ab, cd, and
ef), as determined by their maximal $\varphi_{s}$ values. Again, because the
modules are small, their $\varPhi$ values are low. Intriguingly, a brain
region such as the cerebellum, whose anatomical organization is highly
modular, does not contribute to consciousness [53, 54], even though it
contains several times more neurons than the cerebral cortex (and is
indirectly connected to it).

Note that fault lines can be due not just to neuroanatomy but also to
neurophysiological factors. For example, during early slow-wave sleep, the
dense interconnections among neuronal groups in cerebral cortical areas may
break down, becoming causally ineffective due to the bistability of neuronal
excitability. This bistability, brought about by neuromodulatory changes [55],
is associated with the loss of consciousness [56].

#### Directed cycles, structural sparseness, and composition

Fig. 6C shows a directed cycle in which six units are unidirectionally
connected with weight $w=1.0$ and $k=4$. Each unit copies the state of the
unit before it, and its state is copied by the unit after it, with some
indeterminism. The copy cycle constitutes a 6-unit complex with a maximal
$\varphi_{s}=1.74$ ibits. However, despite the “large” substrate, the
$\varPhi$-structure it specifies has low structured information
($\varPhi=7.65$). This is because the system’s $\varPhi$-structure is composed
exclusively of first-order distinctions, and consequently of a small number of
relations.

Highly deterministic directed cycles can easily be extended to constitute
large complexes, being more irreducible than any of their subsets. However,
the lack of cross-connections (“chords” in graph-theoretic terms) greatly
limits the number of components of the $\varPhi$-structures the complexes
specify, and thus their structured information ($\varPhi$). (Note also that
increasing the number of units that constitute the directed cycle would not
change the amount of $\varphi_{s}$ specified by the network as a whole.)

The brain is rich in partially segregated, directed cycles, such as those
originating in cortical areas, sequentially reaching stations in the basal
ganglia and thalamus, and cycling back to cortex [57, 58]. These cycles are
critical for carrying out many cognitive and other functions, but they do not
appear to contribute directly to experience [7].

#### Specialized lattices and $\varPhi$-structures with high structured
information

Fig. 6D shows a network consisting of six heterogeneously connected units—a
“specialized” lattice, again with $k=4$. While many subsystems within the
specialized network have positive values of system integrated information
$\varphi_{s}$ (not shown), the full 6-unit system is the maximal substrate
(excluding all its subsets from being maximal substrates). Out of 63 possible
distinctions, the $\varPhi$-structure comprises 24 distinctions with causes
and effects congruent with the system’s maximal cause–effect state.
Consequently, the full 6-unit system also specifies a much larger number of
causal relations compared to the copy loop system.

Preliminary work (not shown, [59]) indicates that lattices of specialized
units, implementing different input–output functions, but partially
overlapping in their inputs (receptive field) and outputs (projective fields),
are particularly well suited to constituting large substrates that unfold into
extraordinarily rich $\varPhi$-structures. The number of distinctions
specified by an optimally connected, specialized system is bounded above by
$2^{n}-1$, and that of the relations among as many distinctions is bounded by
$2^{(2^{n}-1)}-1$.The structured information composing such structures is
correspondingly large [60].

In the brain, many portions of posterior cortex appear to constitute similarly
organized lattices of specialized units, which makes the posterior cortex and
similarly organized areas a plausible candidate for the substrate of human
consciousness [7, 61, 14, 62]. Note that directed cycles originating and
ending in such lattices typically remain excluded from the first-maximal
complex because minimal partitions across such cycles yield a much lower value
of $\varphi_{s}$ compared to minimal partitions across large lattices [59].

#### Near-maximal substrates, extrinsic entities, and exclusion

Finally, Fig. 6E shows a network of six units, four of which ($Abef$)
constitute a specialized lattice that corresponds to the first complex. Though
integrated, the full set of 6 units happens to be slightly less irreducible
than one of its 4-unit subsets ($\varphi_{s}=0.15$). From the extrinsic
perspective, the 6-unit system undoubtedly behaves as a highly integrated
whole (nearly as much as its 4-unit subset), one that could produce complex
input–output functions due to its rich internal structure. From the intrinsic
perspective of the system, however, only the 4-unit subset satisfies all the
postulates of existence, including maximal irreducibility (accounting for the
definite nature of experience). In this example, the remaining units form a
second complex with low $\varphi_{s}$ and serve as background conditions for
the first complex.

A similar situation may occur in the brain. The brain as a whole is
undoubtedly integrated (not to mention that it is integrated with the body as
a whole), and neural “traffic” is heavy throughout. However, its anatomical
organization may be such that a subset of brain regions, arranged in pyramids
of grids and primarily located in posterior cortex, may achieve a much higher
value of integrated information than any other subset. Those regions would
then constitute the first complex (the “main complex,” [7]), and the remaining
regions might condense into a large number of much smaller complexes.

Taken together, the examples in Fig. 6 demonstrate that the connectivity among
the units of a system has a strong impact on what set of units can constitute
a complex and, thus, on the structured information it can specify. The
examples also demonstrate the role played by the various requirements that
must be satisfied by a substrate of consciousness: existence (causal power),
intrinsicality, specificity, irreducibility, maximal irreducibility
(exclusion), and composition (structure).

### Consciousness and activity: active, inactive, and inactivated units

A substrate exerts cause–effect power by being in its current state. For the
same substrate, changing the state of even one unit may have major
consequences on the distinctions and relations that compose its
$\varPhi$-structure: many may be lost, or gained, and many may change their
value of irreducibility ($\varphi_{d}$ and $\varphi_{r}$).

Fig. 7 shows a network of five binary units that interact through excitatory
and inhibitory connections (weights indicated in the figure). The system is
initially in state $s=ABcdE$ (Fig. 7A) and is a maximal substrate with
$\varphi_{s}=1.1$ ibits and a $\varPhi$-structure composed of 21 distinctions
and their 4760 relations.

Fig 7: Causal analysis of the same system with one of its units set to active,
inactive, or inactivated. In all panels, the same causal model and weights are
shown on the left, but in different states. For all networks $k=4$. The set of
distinctions $D(s)$, their causes and effects, and their $\varphi_{d}$ values
are shown in the middle. The $\varPhi$-structure specified by the network’s
complex is illustrated on the right (again with only second- and third-degree
relation faces depicted). All integrated information values are in ibits. (A)
The system in state $ABcdE$ is a complex with 21 out of 31 distinctions and
$\varPhi=10.79$. (B) The same system in state $ABcde$, where unit $E$ is
inactive (“OFF”) also forms a complex with a similar number of distinctions,
but a somewhat higher $\varPhi$ value due to a higher number of relations
between distinctions. In addition, the system’s $\varPhi$-structure differs
from that in (A), as the system now specifies a different set of compositional
causes and effects. (C) If instead of being inactive, unit $E$ is inactivated
(permanently fixed into the “OFF” state), the inactivated unit cannot
contribute to the complex or $\varPhi$-structure anymore. The complex is now
constituted of four units ($ABcd$), with only 14 distinctions and
significantly reduced structured information ($\varPhi=3.31$).

If we change the state of unit $E$ from ON to OFF (in neural terms, the unit
becomes inactive), the distinctions that the unit contributes to when ON, as
well as the associated relations, may change (Fig. 7B). In the case
illustrated by the Figure, what changes are the purviews and irreducibility of
several distinctions and associated relations, the number of distinctions and
$\varphi_{s}$ change only slightly, while the number of relations is
considerably higher, leading to a larger $\varPhi$ value. In other words, what
a single unit contributes to intrinsic existence is not some small “bit” of
information. Instead, a unit contributes an entire sub-structure, composed of
a very large number of distinctions and relations. The set of distinctions to
which a subset of units contributes as a mechanism, either alone or in
combination with other units, together with their associated relations, is
called a mechanism $\varPhi$-fold. With respect to the neural substrate of
consciousness in the brain, this means that even a change in the state of a
single unit is typically associated with a change in an entire $\varPhi$-fold
within the overall $\varPhi$-structure, with a corresponding change in the
structure of the experience.

In Fig. 7, we see what happens if unit $E$, instead of just turning inactive
(OFF) is inactivated (abolishing its cause–effect power because it no longer
has any counterfactual states). In this case, all the distinctions and
relations to which that unit contributes as a mechanism cease to exist (its
mechanism $\varPhi$-fold collapses). Moreover, all the distinctions and
relations to whose purviews that unit contributes—its purview
$\varPhi$-fold—also collapse or change. The complex also shrinks because it
cannot include that unit. With respect to the neural substrate of
consciousness, this means that while an inactive unit contributes to a
different experience, an inactivated unit ceases to contribute to experience
altogether. The fundamental difference between inactive and inactivated units
leads to the following corollary of IIT: unlike a fully inactivated substrate
which, as would be suspected, cannot support any experience, an inactive
substrate can. If a maximal substrate is in working order and specifies a
large $\varPhi$-structure, it will support a highly structured experience,
such as the experience of empty space [14] or the feeling of “pure presence”
[63].

### Consciousness and functional equivalence: being is not doing

By the intrinsicality postulate, the $\varPhi$-structure of a complex depends
on the causal interactions between system subsets, not on the system’s
interaction with its environment. In general, different physical systems with
different internal causal structure may perform the same input–output
functions.

Fig. 8 shows three simple deterministic systems with binary units (here the
“OFF” state is $0$, and “ON” is $1$) that perform the same input–output
function, treating the internal dynamics of the system as a black box. The
function could be thought of, for example, as an electronic tollbooth
“counting 8 valid coins” (8 times input $I=1$) before opening the gate [64].
Each system receives one binary input ($I$) and has one binary output ($O$).
The output unit switches “ON” on a count of eight positive inputs $I=1$ (when
the global state with label ‘0’ is reached in the cycle), upon which the
system resets (Fig. 8A).

In addition to being functionally equivalent in their outward behavior, the
three systems share the same internal global dynamics, as their internal
states update according to the same global state-transition diagram (Fig. 8B).
Given an input $I=1$, the system updates its state, cycling through all its 8
global states (labeled 0–7) over 8 updates. For an input of $I=0$, the system
remains in its present state. Moreover, all three systems are constituted of
three binary units whose joint states map one-to-one onto the systems’ global
state labels (0–7). However, the mapping is different for different systems
(Fig. 8C, left). This is because the internal binary update sequence depends
on the interactions among the internal units [36, 64], which differ in the
three cases, as can easily be determined through manipulations and
observations.

For consistency in the causal analysis, in all three cases, the global state
“0” that activates the output unit if $I=1$ is selected such that it
corresponds to the binary state “all OFF” ($000$), which is followed by
$1:=100$ and $2:=010$. Also, the $\varPhi$-structure of each system is
unfolded in state $1:=100$ in all three cases.

Despite their functional equivalence and equivalent global dynamics, the
systems differ in how they condense into complexes and in the cause–effect
structures they specify.

Fig 8: Functionally equivalent networks with different $\varPhi$-structures.
(A) The input–output function realized by three different systems (shown in
(C)): a count of eight instances of input $I=1$ leads to output $O=1$. (B) The
global state-transition diagram is also the same for the three systems: if
$I=0$, the systems will remain in their current global state, labeled as 0-7;
if $I=1$, the systems will move one state forward, cycling through their
global states, and activate the output if $S=0$. (C) Three systems constituted
of three binary units but differing in how the units are connected and
interact. As a consequence, the one-to-one mapping between the 3-bit binary
states and the global state labels differ. However, all three systems
initially transition from $000$ to $100$ to $010$. Analyzed in state $100$,
the first system (top) turns out to be a single complex that specifies a
$\varPhi$-structure with six distinctions and many relations, yielding a high
value of $\varPhi$. The second system (middle) is also a complex, with the
same $\varphi_{s}$ value, but it specifies a $\varPhi$-structure with fewer
distinctions and relations, yielding a lower value of $\varPhi$. Finally, the
third system (bottom) is reducible ($\varphi_{s}=0$) and splits into two
smaller complexes (entities) with minimal $\varPhi$-structures and low
$\varPhi$.

As shown in Fig. 8C, the first system forms a 3-unit complex with a relatively
rich $\varPhi$-structure ($\varPhi=21.01$ ibits). While the second system also
forms a 3-unit complex with the same $\varphi_{s}=2$ ibits, it specifies a
completely different set of distinctions and has much lower structured
information ($\varPhi=3.64$ ibits).

Finally, the third system is reducible ($\varphi_{s}=0$ ibits)—in this case,
because there are only feed-forward connections from unit $A$ to units $B$ and
$C$—and it condenses into two complexes with small $\varPhi$-structures.

These examples illustrate a simple scenario of functional equivalence of three
systems characterized by a different architecture. The equivalence is with
respect to a simple input–output function, in this case coin counting, which
they multiply realize. The systems are also equivalent in terms of their
global system dynamics, in the sense that they go through a globally
equivalent sequence of internal states. However, because of their different
substrates, the three systems specify different cause–effect structures.
Therefore, based on the postulates of IIT, they are not phenomenally
equivalent. In other words, they are equivalent in what they do extrinsically,
but not in what they are intrinsically.

This dissociation between phenomenal and functional equivalence has important
implications. As we have seen, a purely feed-forward system necessarily has
$\varphi_{s}=0$. Therefore, it cannot support a cause–effect structure and
cannot be conscious, whereas systems with a recurrent architecture can. On the
other hand, the behavior (input–output function) of any (discrete) recurrent
system can also be implemented by a system with a feed-forward architecture
[65]. This implies that any behavior performed by a conscious system supported
by a recurrent architecture can also be performed by an unconscious system, no
matter how complex the behavior is. More generally, digital computers
implementing programs capable of artificial general intelligence may in
principle be able to emulate any function performed by conscious humans and
yet, because of the way they are physically organized, they would do so
without experiencing anything, or at least anything resembling, in quantity
and quality, what each of us would experience [23].

The examples also show that the overall system dynamics, while often revealing
relevant aspects of a system’s architecture, typically do not and cannot
exhaust the richness of its current cause–effect structure. For example, a
system in a fixed point is dynamically “dead” (and “does” nothing), but it may
be phenomenally quite “alive,” for example, experiencing “pure presence” [63].
Of course, the system’s causal powers can be fully unfolded, and revealed
dynamically, by extensive manipulations and observations of subsets of system
units because they are implicitly captured by the system’s causal model and
ultimately by its transition probability matrix [36].

### Conclusions

IIT attempts to account for the presence and quality of consciousness in
physical terms. It starts from the existence of experience, and proceeds by
characterizing its essential properties—those that are immediate and
irrefutably true of every conceivable experience (axioms). These are then
translated into essential properties of physical existence (postulates), the
necessary and sufficient conditions that a substrate must satisfy to support
an experience—to constitute a complex. “Physical” is understood purely as
cause–effect power, and ”substrate” is meant in purely operational terms—as a
set of units that a conscious observer can observe and manipulate.

The postulates can be assessed based purely on a substrate’s transition
probability matrix, as was illustrated by a few idealized causal models. Thus,
a substrate of consciousness must be able to take and make a difference upon
itself (existence and intrinsicality), it must be able to specify a cause and
an effect state that are highly informative and selective (information), and
it must do so in a way that is both irreducible (integration) and definite
(exclusion). Finally, it must specify its cause and effect in a structured
manner (composition), where the causal powers of its subsets over its subsets
compose a cause–effect structure of distinctions and relations—a
$\varPhi$-structure. Thus, a complex does not exist as such but only
“unfolded” as a $\varPhi$-structure—an intrinsic entity that exists for
itself, absolutely, rather than relative to an external observer.

As shown above, these requirements constrain what substrates can and cannot
support consciousness. Substrates that lack in specificity, due to
indeterminism and/or degeneracy, cannot grow to be large complexes. Substrates
that are weakly integrated, due to architectural or functional fault lines in
their interactions, are less integrated than some of their subsets. Because
they are not maximally irreducible, they do not qualify as complexes. This is
the case even though they may “hang together” well enough from an extrinsic
perspective (having a respectable value of $\varphi_{s}$). Furthermore, even
substrates that are maximally integrated may support $\varPhi$-structures that
are extremely sparse, as in the case of directed cycles. Based on the
postulates of IIT, a universal substrate ultimately “condenses” into a set of
disjoint (non-overlapping) complexes, each constituted of a set of macro or
micro units.

The physical account of consciousness provided by IIT should be understood as
an explanatory identity: every property of an experience should ultimately be
accounted for by a property of the cause–effect structure specified by a
substrate that satisfies its postulates, with no additional ingredients. The
identity is not between two different substances or realms—the phenomenal and
the physical—but between intrinsic (subjective) existence and extrinsic
(objective) existence. Intrinsic existence is immediate and irrefutable, while
extrinsic existence is defined operationally as cause–effect power discovered
through observation and manipulation. The primacy of intrinsic existence (of
experience) in IIT contrasts with standard attempts at accounting for
consciousness as something “generated by” or “emerging from” a substrate
constituted of matter and energy and following physical laws.

The physical correspondent of an experience is not the substrate as such but
the $\varPhi$-structure specified by the substrate in its current state.
Therefore, minor changes in the substrate state can correspond to major
changes in the specified $\varPhi$-structure. For example, if the state of a
single unit changes, an entire $\varPhi$-fold within the $\varPhi$-structure
will change, and if a single inactive unit is inactivated, its associated
$\varPhi$-fold will collapse, even though the current state of the substrate
appears the same (Fig. 7).

Each experience corresponds to a $\varPhi$-structure, not a set of functions.
Said otherwise, consciousness is about being, not doing [1, 4, 66, 36]. This
means that systems with different architectures may be functionally
equivalent—both in terms of global input–output functions and global intrinsic
dynamics—but they will not be phenomenally equivalent. For example, a feed-
forward system can be functionally equivalent to a recurrent system that
constitutes a complex, but feed-forward systems cannot constitute complexes
because they do not satisfy maximal irreducibility. Accordingly, artificial
systems powered by super-intelligent computer programs would experience
nothing (or nearly nothing) because they have the wrong kind of physical
architecture, even though they may be behaviorally indistinguishable from
human beings [23].

Even though the entire framework of IIT is based on just a few axioms and
postulates, it is not possible in practice to exhaustively apply the
postulates to unfold the cause–effect power of realistic systems [40, 67]. It
is not feasible to perform all possible observations and manipulations to
fully characterize a universal TPM, or to perform all calculations on the TPM
that would be necessary to condense it exhaustively into complexes and unfold
their cause–effect power in full. The number of possible systems, of system
partitions, of candidate distinctions—each with their partitions and
relations—is the result of multiple, nested combinatorial explosions.
Moreover, these observations, manipulations, and calculations would need to be
repeated at many different grains, with many rounds of maximizations. For
these reasons, a full analysis of complexes and their cause–effect structure
can only be performed on idealized systems of a few units [46].

On the other hand, we can simplify the computation considerably by using
various assumptions and approximations, as with the “cut one” approximation
described in [46]. Also, while the number of relations vastly exceeds the
number of units and of distinctions (its upper bound for a system of $n$ units
is $2^{(2^{n}-1)}-1$), it can be determined analytically, and so can
$\sum\varphi_{r}$ for a given set of distinctions A.3. Developing tight
approximations, as well as bounded estimates of a system’s integrated
information $\varPhi(s)$, is one of the main areas of ongoing research related
to IIT [60].

Despite the infeasibility of an exhaustive calculation of the relevant
quantities and structures for a realistic system, IIT already provides
considerable explanatory and predictive power in many real-world situations,
making it eminently testable [7, 68, 69]. A fundamental prediction is that
$\varPhi$ should be high in conscious states, such as wakefulness and
dreaming, and low in unconscious states, such as dreamless sleep and
anesthesia. This prediction has already found substantial support in human
studies that have applied a crude approximation of $\varPhi$ to successfully
classify subjects as conscious vs. unconscious [25, 26, 7, 70]. IIT can also
account mechanistically for the loss of consciousness in deep sleep and
anesthesia [56, 7]. Furthermore, it can provide a principled account of why
certain portions of the brain may constitute an ideal substrate of
consciousness and others may not, why the borders of the main complex in the
brain should be where they are, and why the units of the complex should have a
particular grain (the one that yields a maximum of $\varphi_{s}$). A stringent
prediction is that the location of the main complex, as determined by the
overall maximum of $\varphi_{s}$ within the brain, should correspond to its
location as determined through clinical and experimental evidence. Another
prediction that follows from first principles is that constituents of the main
complex can support conscious contents even if they are mostly inactive, but
not if they are inactivated [7, 14]. Yet another prediction is that the
complete inactivation of constituents of the main complex should lead to
absolute agnosia (unawareness that anything is missing).

IIT further predicts that the quality of experience should be accounted for by
the way the $\varPhi$-structure is composed, which in turn depends on the
architecture of the substrate specifying it. This was demonstrated in a recent
paper showing how the fundamental properties of spatial experiences—those that
make space feel “extended”—can be accounted for by those of
$\varPhi$-structures specified by 2D grids of units, such as those found in
much of posterior cortex [14]. This prediction is in line with neurological
evidence of their role in supporting the experience of space [14]. Ongoing
work aims at accounting for the quality of experienced time [28] and that of
experienced objects [29]. A related prediction is that changes in the strength
of connections within the neural substrate of consciousness should be
associated with changes in experience, even if neural activity does not change
[71]. Also, similarities and dissimilarities in the structure of experience
should be accounted for by similarities and dissimilarities among
$\varPhi$-structures and $\varPhi$-folds specified by the neural substrate of
consciousness. Further validation of IIT will depend on a systematic back-and-
forth between phenomenology, theoretical inferences, and neuroscientific
evidence [1].

In addition to empirical work aimed at validating the theory, much remains to