# The Mathematical Structure of Integrated Information Theory

^{1}Ludwig Maximilian University of Munich, Munich, Germany^{2}Cambridge Quantum Computing Limited, Cambridge, United Kingdom

Integrated Information Theory is one of the leading models of consciousness. It aims to describe both the quality and quantity of the conscious experience of a physical system, such as the brain, in a particular state. In this contribution, we propound the mathematical structure of the theory, separating the essentials from auxiliary formal tools. We provide a definition of a generalized IIT which has IIT 3.0 of Tononi et al., as well as the Quantum IIT introduced by Zanardi et al. as special cases. This provides an axiomatic definition of the theory which may serve as the starting point for future formal investigations and as an introduction suitable for researchers with a formal background.

## 1 Introduction

*Integrated Information Theory (IIT)*, developed by Giulio Tononi and collaborators [5, 45–47], has emerged as one of the leading scientific theories of consciousness. At the heart of the latest version of the theory [19, 25, 26, 31, 40] is an algorithm which, based on the level of *integration* of the internal functional relationships of a physical system in a given state, aims to determine both the quality and quantity (‘

While promising in itself [12, 43], the mathematical formulation of the theory is not satisfying to date. The presentation in terms of examples and accompanying explanation veils the essential mathematical structure of the theory and impedes philosophical and scientific analysis. In addition, the current definition of the theory can only be applied to comparably simple classical physical systems [1], which is problematic if the theory is taken to be a fundamental theory of consciousness, and should eventually be reconciled with our present theories of physics.

To resolve these problems, we examine the essentials of the IIT algorithm and formally define a generalized notion of Integrated Information Theory. This notion captures the inherent mathematical structure of IIT and offers a rigorous mathematical definition of the theory which has ‘classical’ IIT 3.0 of Tononi et al. [25, 26, 31] as well as the more recently introduced *Quantum Integrated Information Theory* of Zanardi, Tomka and Venuti [50] as special cases. In addition, this generalization allows us to extend classical IIT, freeing it from a number of simplifying assumptions identified in [3]. Our results are summarised in Figure 1.

**FIGURE 1**. An Integrated Information Theory specifies for every system in a particular state its conscious experience, described formally as an element of an experience space. In our formalization, this is a map **Sys** into a class **Exp** of experience spaces, which, first, sends each system *S* to its space of possible experiences

In the associated article [44] we show more generally how the main notions of IIT, including causation and integration, can be treated, and an IIT defined, starting from any suitable theory of physical systems and processes described in terms of category theory. Restricting to classical or quantum process then yields each of the above as special cases. This treatment makes IIT applicable to a large class of physical systems and helps overcome the current restrictions.

Our definition of IIT may serve as the starting point for further mathematical analysis of IIT, in particular if related to category theory [30, 49]. It also provides a simplification and mathematical clarification of the IIT algorithm which extends the technical analysis of the theory [1, 41, 42] and may contribute to its ongoing critical discussion [2, 4, 8, 23, 27, 28, 33]. The concise presentation of IIT in this article should also help to make IIT more easily accessible for mathematicians, physicists and other researchers with a strongly formal background.

This work is concerned with the most recent version of IIT as proposed in [25, 26, 31, 40] and similar papers quoted below. Thus our constructions recover the specific theory of consciousness referred to as IIT 3.0 or IIT 3.x, which we will call classical IIT in what follows. Earlier proposals by Tononi et al. that also aim to explicate the general idea of an essential connection between consciousness and integrated information constitute alternative theories of consciousness which we do not study here. A yet different approach would be to take the term ‘Integrated Information Theory’ to refer to the general idea of associating conscious experience with some pre-theoretic notion of integrated information, and to explore the different ways that this notion could be defined in formal terms [4, 27, 28, 37].

### Relation to Other Work

This work develops a thorough mathematical perspective of one of the promising contemporary theories of consciousness. As such it is part of a number of recent contributions which seek to explore the role and prospects of mathematical theories of consciousness [11, 15, 18, 30, 49], to help overcome problems of existing models [17, 18, 34] and to eventually develop new proposals [6, 13, 16, 20, 22, 29, 39].

### 1.1 Structure of Article

We begin by introducing the necessary ingredients of a generalised Integrated Information Theory in Sections 2–4, namely physical systems, experience spaces and cause-effect repertoires. Our approach is *axiomatic* in that we state only the precise formal structure which is necessary to apply the IIT algorithm. We neither motivate nor criticize these structures as necessary or suitable to model consciousness. Our goal is simply to recover IIT 3.0. In Section 5, we introduce a simple formal tool which allows us to present the definition of the algorithm of an IIT in a concise form in Sections 6 and 7. Finally, in Section 8, we summarise the full definition of such a theory. The result is the definition of a generalized IIT. We call any application of this definition ‘an IIT’.

Following this we give several examples including IIT 3.0 in Section 9 and Quantum IIT in Section 10. In Section 11 we discuss how our formulation allows one to extend classical IIT in several fundamental ways, before discussing further modifications to our approach and other future work in Section 12. Finally, the appendix includes a detailed explanation of how our generalization of IIT coincides with its usual presentation in the case of classical IIT.

## 2 Systems

The first step in defining an Integrated Information Theory (IIT) is to specify a class *systems* to be studied. Each element *S* come with the following features.

Definition 1. A *system class**S*, called *systems*, come with the following data:

1. A set *states*;

2. for every

3. a set *decompositions*, with a given *trivial decomposition*

4. for each *cut system**cut state*

Moreover, we require that *empty system*, denoted *I*, and that *S*. For the IIT algorithm to work, we need to assume furthermore that the number of subsystems remains the same under cuts or changes of states, i.e. that we have bijections *S* requires specifying a state *s* of *S*. An example class of systems is illustrated in Figure 2. In this article we will assume that each set

**FIGURE 2**. As an example of Definition 1 similar to IIT 3.0, consider simple systems given by sets of nodes (or ‘elements’), with a state assigning each node the state ‘on’ (depicted green) or ‘off’ (red). Each system comes with a time evolution shown by labelling each node with how its state in the next time-step depends on the states of the others. Decompositions of a system *S* correspond to binary partition of the nodes, such as *z* above. The cut system *S* so that the two halves do not interact; in this case all connections between the halves are replaced by sources of noise which send ‘on’ or ‘off’ with equal likelihood, depicted as black dots above. Given a current state *s* of *S*, any subset of the nodes (such as those below the dotted line) determines a subsystem *S* by fixing the nodes in *s*. Note that while in this example any subsystem (subset of *S*) determines a decomposition (partition of *S*) we do not require such a relationship in general.

## 3 Experience

An IIT aims to specify for each system in a particular state its *conscious experience*. As such, it will require a mathematical model of such experiences. Examining classical IIT, we find the following basic features of the final experiential states it describes which are needed for its algorithm.

Firstly, each experience *e* should crucially come with an *intensity*, given by a number *distance d*(*e*,*e*′) between any pair of experiences *e*,*e*′. Finally, the algorithm will require us to be able to *rescale* any given experience *e* to have any given intensity. Mathematically, this is most easily encoded by letting us multiply any experience *e* by any number

Definition 2. An *experience space* is a set *E* with:

1. An *intensity* function

2. A *distance* function

3. A *scalar multiplication*

for all

We remark that this same axiomatisation will apply both to the full space of experiences of a system, as well as to the spaces describing components of the experiences (‘concepts’ and ‘proto-experiences’ defined in later sections). We note that the distance function does not necessarily have to satisfy the axioms of a metric. While this and further natural axioms such as

Example 3. Any metric space

This is the definition used in classical IIT (cf. Section 9 and Appendix A). An important operation on experience spaces is taking their *product*.

Definition 4. For experience spaces *E* and *F*, we define the product to be the space

intensity

## 4 Repertoires

In order to define the experience space and individual experiences of a system *S*, an IIT utilizes basic building blocks called ‘repertoires’, which we will now define. Next to the specification of a system class, this is the essential data necessary for the IIT algorithm to be applied.

Each repertoire describes a way of ‘decomposing’ experiences, in the following sense. Let *D* denote any set with a distinguished element 1, for example the set *S*, where the distinguished element is the trivial decomposition

Definition 5. Let *e* be an element of an experience space *E*. A *decomposition of e over D* is a mapping

Definition 6. A *cause-effect repertoire* at *S* is given by a choice of experience space *proto-experiences*, and for each

and for each of them a decomposition over *S*, as in the following definition.

Definition 7. A *cause-effect structure* is a specification of a cause-effect repertoire for every

The names ‘cause’ and ‘effect’ highlight that the definitions of *s* of the system, when restricted to *M*, constrains the ‘previous’ or ‘next’ state of *P*, respectively.

## 5 Integration

We have now introduced all of the data required to define an IIT; namely, a system class along with a cause-effect structure. From this, we will give an algorithm aiming to specify the conscious experience of a system. Before proceeding to do so, we introduce a conceptual short-cut which allows the algorithm to be stated in a concise form. This captures the core ingredient of an IIT, namely the computation of how integrated an entity is.

Definition 8. Let *E* be an experience space and *e* an element with a decomposition over some set *D*. The *integration level* of *e* relative to this decomposition is

Here, *d* denotes the distance function of *E*, and the minimum is taken over all elements of *D* besides 1. The *integration scaling* of *e* is then the element of *E* defined by

where *normalization* of *e*, defined as

Finally, the *integration scaling* of a pair

where *S* be a system in a state *collection* of decomposable elements, and denote it as

Definition 9. The *core* of the collection ^{1} The *core integration scaling* of the collection is *core integration scaling* of a pair of collections

## 6 Constructions: Mechanism Level

Let *concept space*

For every choice of *M*, they provide elements with decompositions over Sub(S) given by

The *concept* of *M* is then defined as the core integration scaling of this pair of collections,

It is an element of

## 7 Constructions: System Level

The second level of the algorithm specifies the experience of system *S* in state *s*. To this end, all concepts of a system are collected to form its *Q-shape*, defined as

This Is an Element of the Space

where *s* according to our assumptions. We can also define a Q-shape for any cut of *S*. Let

Because of Eq. 4, and since the number of subsystems remains the same when cutting,

which is a decomposition of *S* gives a collection of decompositions defined as

This is the system level-object of relevance and is what specifies the experience of a system according to IIT.

Definition 10. The *experience* of system *S* in the state

The definition implies that *major complex*. It describes which part of system *S* is actually conscious. In most cases there will be a natural embedding *M* of *S*, allowing us to view

## 8 Integrated Information Theories

We can now summarize all that we have said about IITs.

Definition 11. An *Integrated Information Theory* is determined as follows. The *data* of the theory is a system class

into the class *S* to its space of experiences

which determines the experience of the system when in a state *s*, defined in Eq. 14.The *quantity* of the system’s experience is given by

and the quality of the system’s experience is given by the normalized experience *the major complex*, which is a subsystem of *S*.In the next sections we specify the data of several example IITs.

## 9 Classical IIT

In this section we show how IIT 3.0 [25, 26, 31, 48] fits in into the framework developed here. A detailed explanation of how our earlier algorithm fits with the usual presentation of IIT is given in Appendix A. In [44] we give an alternative categorical presentation of the theory.

### 9.1 Systems

We first describe the system class underlying classical IIT. Physical systems *S* are considered to be built up of several components *elements*. Each element *S* is given by specifying a state of each element, so that

We define a metric *d* on *s* occasionally in what follows.

Additionally, each system comes with a probabilistic (discrete) *time evolution operator* or *transition probability matrix*, sending each

Furthermore, the evolution *T* is required to satisfy a property called *conditional independence*, which we define shortly.

The class *I* having only a single element with a single state and trivial time evolution.

### 9.2 Conditioning and Marginalizing

In what follows, we will need to consider two operations on the map *T*. Let *M* be any subset of the elements of a system and *M*, and by *conditioning* [26] of *T* on *p* as the map

where *S*. Next, we define *marginalisation over M* as the map

such that for each

In particular for any map *T* as above we call *marginal* of *T* over *M* and we write *T* may now be defined as the requirement that

where the right-hand side is again a probability distribution over

### 9.3 Subsystems, Decompositions and Cuts

Let a system *S* in a state *S*. For any subset *S*, the corresponding subsystem is also denoted *M* and

where *s* to

The decomposition set *S* consists of all partitions of the set *N* of elements of *S* into two disjoint sets *M* and

For any decomposition *S* but with a new time evolution

where *M*. The corresponding input of the target element is replaced by noise, i.e. the uniform probability distribution over the source element.

### 9.4 Proto-Experiences

For each system *S*, the first Wasserstein metric (or ‘Earth Mover’s Distance’) makes

where

### 9.5 Repertoires

It remains to define the cause-effect repertoires. Fixing a state *s* of *S*, the first step will be to define maps *M* constrains that of *P* in the next or previous time-steps. We begin with the effect repertoire. For a single element purview

where *s* to *M*. While it is natural to use the same definition for arbitrary purviews, IIT 3.0 in fact uses another definition based on consideration of ‘virtual elements’ [25, 26, 48], which also makes calculations more efficient (Supplementary Material S1 of [26]). For general purviews *P*, this definition is

taking the product over all elements *P*. Next, for the cause repertoire, for a single element mechanism

where λ is the unique normalisation scalar making

For General Mechanisms *M*, we Then Define

where the product is over all elements *M* and where

with intensity 1 when viewed as elements of

The distributions *unconstrained cause and effect repertoires* over

Remark 12. It is in fact possible for the right-hand side of Eq. 28 to be equal to 0 for all *M* and *P*, we define

where we have abused notation by equating each subset *S* via the state *s*. This concludes all data necessary to define classical IIT. If the generalized definition of Section 8 is applied to this data, it yields precisely classical IIT 3.0 defined by Tononi et al. In Appendix A, we explain in detail how our definition of IIT, equipped with this data, maps to the usual presentation of the theory.

## 10 Quantum IIT

In this section, we consider Quantum IIT defined in [50]. This is also a special case of the definition in terms of process theories we give in [44].

### 10.1 Systems

Similar to classical IIT, in Quantum IIT systems are conceived as consisting of elements *S* is defined in terms of the element Hilbert spaces as

where

### 10.2 Subsystems, Decompositions and Cuts

Subsystems are again defined to consist of subsets *M* of the elements of the system, with corresponding Hilbert space

where

Decompositions are also defined via partitions *N* into two disjoint subsets *D* and *N*. For any such decomposition, the cut system

where

### 10.3 Proto-Experiences

For any

turns

### 10.4 Repertoires

We finally come to the definition of the cause-effect repertoire. Unlike classical IIT, the definition in [50] does not consider virtual elements. Let a system *S* in state *M* and *P* to

where

each with intensity 1, where

again with intensity 1, where

## 11 Extensions of Classical IIT

The physical systems to which IIT 3.0 may be applied are limited in a number of ways: they must have a discrete time-evolution, satisfy Markovian dynamics and exhibit a discrete set of states [3]. Since many physical systems do not satisfy these requirements, if IIT is to be taken as a fundamental theory about reality, it must be extended to overcome these limitations.

In this section, we show how IIT can be redefined to cope with continuous time, non-Markovian dynamics and non-compact state spaces, by a redefinition of the maps Eqs. 26 and 28 and, in the case of non-compact state spaces, a slightly different choice of Eq. 24, while leaving all of the remaining structure as it is. While we do not think that our particular definitions are satisfying as a general definition of IIT, these results show that the disentanglement of the essential mathematical structure of IIT from auxiliary tools (the particular definition of cause-effect repertoires used to date) can help to overcome fundamental mathematical or conceptual problems.

In Section 11.3, we also explain which solution to the problem of non-canonical metrics is suggested by our formalism.

### 11.1 Discrete Time and Markovian Dynamics

In order to avoid the requirement of a discrete time and Markovian dynamics, instead of working with the time evolution operator Eq. 18, we define the cause- and effect repertoires in reference to a given trajectory of a physical state

Let

In what follows, we utilize the fact that in physics, state spaces are defined such that the dynamical laws of a system allow to determine the trajectory of each state. Thus for every *s*.

The idea behind the following is to define, for every *P* at time *t* is being constrained by imposing the state *s* at time *M*. This gives an alternative definition of the maps (26) and (28), while the rest of classical IIT can be applied as before.

Let now *s* to *P*, which we denote as *v* using the uniform distribution

where *κ* is the unique normalization constant which ensures that

The probability distribution *P* at time *t* is being constrained by imposing the state *s* on *M* at time

So far, our construction can be applied for any time

to replace Eq. 27 and

to replace Eq. 29. The remainder of the definitions of classical IIT can then be applied as before.

### 11.2 Discrete Set of States

The problem with applying the definitions of classical IIT to systems with continuous state spaces (e.g., neuron membrane potentials [3]) is that in certain cases, uniform probability distributions do not exist. E.g., if the state space of a system *S* consists of the positive real numbers *probability* distribution

It is important to note that this problem is less universal than one might think. E.g., if the state space of the system is a closed and bounded subset of

This problem can be resolved for all well-understood physical systems by replacing the uniform probability distribution

In what follows, we explain how the construction of the last section needs to be modified in order to be applied to this case. In all relevant classical physical theories,

As before, the dynamical laws of the physical systems determine for every state

Using the fact that *v*, for every *A* in the σ-algebra of

which is a finite quantity. The

All that remains for this to give a cause-effect repertoire as in the last section, is to make sure that any measure (normalized or not) is an element of

and finally allows one to construct cause-effect repertoires as in the last section.

### 11.3 Non-canonical Metrics

Another criticism of IIT’s mathematical structure mentioned [3] is that the metrics used in IIT’s algorithm are, to a certain extend, chosen arbitrarily. Different choices indeed imply different results of the algorithm, both concerning the quantity and quality of conscious experience, which can be considered problematic.

The resolution of this problem is, however, not so much a technical as a conceptual or philosophical task, for what is needed to resolve this issue is a justification of why a particular metric should be used. Various justifications are conceivable, e.g. identification of desired behavior of the algorithm when applied to simple systems. When considering our mathematical reconstruction of the theory, the following natural justification offers itself.

Implicit in our definition of the theory as a map from systems to experience spaces is the idea that the mathematical structure of experiences spaces (Definition 2) reflects the phenomenological structure of experience. This is so, most crucially, for the distance function *d*, which describes how similar two elements of experience spaces are. Since every element of an experience space corresponds to a conscious experience, it is naturally to demand that the similarly of the two mathematical objects should reflect the similarity of the experiences they describe. Put differently, the distance function *d* of an experience space should in fact mirror (or “model”) the similarity of conscious experiences as experienced by an experiencing subject.

This suggests that the metrics *d* used in the IIT algorithm should, ultimately, be defined in terms of the phenomenological structure of similarity of conscious experiences. For the case of color qualia, this is in fact feasible [18, Example 3.18], [21, 38]. In general, the mathematical structure of experience spaces should be intimately tied to the phenomenology of experience, in our eyes.

## 12 Summary and Outlook

In this article, we have propounded the mathematical structure of Integrated Information Theory. First, we have studied which exact structures the IIT algorithm uses in the mathematical description of physical systems, on the one hand, and in the mathematical description of conscious experience, on the other. Our findings are the basis of definitions of a physical system class

Next, we needed to disentangle the essential mathematics of the theory from auxiliary formal tools used in the contemporary definition. To this end, we have introduced the precise notion of decomposition of elements of an experience space required by the IIT algorithm. The pivotal cause-effect repertoires are examples of decompositions so defined, which allowed us to view any particular choice, e.g. the one of ‘classical’ IIT developed by Tononi et al., or the one of ‘quantum’ IIT recently introduced by Zanardi et al. as data provided to a general IIT algorithm.

The formalization of cause-effect repertoires in terms of decompositions then led us to define the essential ingredients of IIT’s algorithm concisely in terms of integration levels, integration scalings and cores. These definitions describe and unify recurrent mathematical operations in the contemporary presentation, and finally allowed to define IIT completely in terms of a few lines of definition.

Throughout the paper, we have taken great care to make sure our definitions reproduce exactly the contemporary version of IIT 3.0. The result of our work is a mathematically rigorous and general definition of Integrated Information Theory. This definition can be applied to any meaningful notion of systems and cause-effect repertoires, and we have shown that this allows one to overcome most of the mathematical problems of the contemporary definition identified to date in the literature.

We believe that our mathematical reconstruction of the theory can be the basis for refined mathematical and philosophical analysis of IIT. We also hope that this mathematisation may make the theory more amenable to study by mathematicians, physicists, computer scientists and other researchers with a strongly formal background.

### 12.1 Process Theories

Our generalization of IIT is axiomatic in the sense that we have only included those formal structures in the definition which are necessary for the IIT algorithm to be applied. This ensured that our reconstruction is as general as possible, while still true to IIT 3.0. As a result, several notions used in classical IIT, e.g., system decomposition, subsystems or causation, are merely defined abstractly at first, without any reference to the usual interpretation of these concepts in physics.

In the related article [44], we show that these concepts can be meaningfully defined in any suitable *process theory* of physics, formulated in the language of *symmetric monoidal categories*. This approach can describe both classical and Quantum IIT and yields a complete formulation of contemporary IIT in a categorical framework.

### 12.2 Further Development of IIT

IIT is constantly under development, with new and refined definitions being added every few years. We hope that our mathematical analysis of the theory might help to contribute to this development. For example, the working hypothesis that IIT is a fundamental theory, implies that technical problems of the theory need to be resolved. We have shown that our formalization allows one to address the technical problems mentioned in the literature. However, there are others which we have not addressed in this paper.

Most crucially, the IIT algorithm uses a series of maximalization and minimalization operations, unified in the notion of *core* subsystems in our formalization. In general, there is no guarantee that these operations lead to unique results, neither in classical nor Quantum IIT. Using different cores has major impact on the output of the algorithm, including the ^{2}

Furthermore, the contemporary definition of IIT as well as our formalization rely on there being a finite number of subsystems of each system, which might not be the case in reality. Our formalisation may be extendable to the infinite case by assuming that every system has a fixed but potentially infinite indexing set

Finally, concerning more operational questions, it would be desirable to develop the connection to empirical measures such as the Perturbational Complexity Index (PCI) [7, 9] in more detail, as well as to define a controlled approximation of the theory whose calculation is less expensive. Both of these tasks may be achievable by substituting parts of our formalization with simpler mathematical structure.

On the conceptual side of things, it would be desirable to have a more proper understanding of how the mathematical structure of experiences spaces corresponds to the phenomenology of experience, both for the general definition used in our formalization—which comprises the minimal mathematical structure which is required for the IIT algorithm to be applied—and the specific definitions used in classical and Quantum IIT. In particular, it would be desirable to understand how it relates to the important notion of qualia, which is often asserted to have characteristic features such as ineffability, intrinsicality, non-contextuality, transparency or homogeneity [24]. For a first analysis toward this goal, cf [18]. A first proposal to add additional structure to IIT that accounts for relations between elements of consciousness in the case of spatial experiences was recently given in [14].

## Data Availability Statement

The original contributions presented in the study are included in the article/Supplementary Material, further inquiries can be directed to the corresponding author.

## Author Contributions

JK and ST conceived the project together and wrote the article together.

## Conflict of Interest

Author ST was employed by company Cambridge Quantum Computing Limited.

The remaining author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

## Acknowledgments

We would like to thank the organizers and participants of the *Workshop on Information Theory and Consciousness* at the Center for Mathematical Sciences of the University of Cambridge, of the *Modeling Consciousness Workshop* in Dorfgastein and of the *Models of Consciousness Conference* at the Mathematical Institute of the University of Oxford for discussions on this topic. Much of this work was carried out while Sean Tull was under the support of an EPSRC Doctoral Prize at the University of Oxford, from November 2018 to July 2019, and while Johannes Kleiner was under the support of postdoctoral funding at the Institute for Theoretical Physics of the Leibniz University of Hanover. We would like to thank both institutions.

## Footnotes

^{1}If the maximum does not exist, we define the core to be the empty system *I*.

^{2}The problem of ‘unique existence’ has been studied extensively in category theory using *universal properties* and the notion of a *limit*. Rather than requiring that each

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## Appendix A: Comparison with Standard Presentation of IIT 3.0

In Section 9, we have defined the system class and cause-effect repertoires which underlie classical IIT. The goal of this appendix is to explain in detail why applying our definition of the IIT algorithm yields IIT 3.0 defined by Tononi et al. In doing so, we will mainly refer to the terminology used in [25, 26, 31, 48]. We remark that a particularly detailed presentation of the algorithm of the theory, and of how the cause and effect repertoire are calculated, is given in the Supplementary Material S1 of [26].

### A.1 Physical Systems

The systems of classical IIT are given in Section 9.1. They are often represented as graphs whose nodes are the elements

in [25]. For each probability distribution *T* via

### A.2 Cause-Effect Repertoires

In contemporary presentations of the theory ([25], p. 14] or [48]), the effect repertoire is defined as

and

Here, *M* at time *t*. *Z* (designated by *P* in our case), *z* a state of the whole purview. *t*, is to be set to the term in brackets. This is called *perturbing the system* into the state *t*.

In our notation, the right hand side of Eq. 35 is exactly given by the right-hand side of Eq. 25, i.e. *M*. Subsequently, *T* is applied to evolve the system to time

Similarly, the cause repertoire is defined as ([25], p. 14] or [48])

and

where *M*, with the subscript *t* indicating that the state is considered at time *t*. *Z* again denotes a purview, *z* is a state of the purview and *K* denotes a normalization constant and *M*.

Here, the whole right hand side of Eq. 37 gives the probability of finding the purview in state *z* at time *t* − 1 if the system is prepared in state *m*_{i,t} at time *t*. In our terminology this same distribution is given by Eq. 27, where λ is the denominator in Eq. 37. Taking the product of these distributions and re-normalising is then precisely Eq. 28.

As a result, the cause and effect repertoire in the sense of [31] correspond precisely in our notation to *extended* cause and effect repertoires or *expansion to full state space* of the repertoires in [31].

The behavior of the cause- and effect-repertoires when decomposing a system is described, in our formalism, by decompositions (Definition 5). Hence a decomposition *parition* in the classical formalism. For the case of classical IIT, a decomposition is given precisely by a partition of the set of elements of a system, and the cause-effect repertoires belonging to the decomposition are defined in Eq. 30, which corresponds exactly to the definition

in [25], when expanded to the full state space, and equally so for the effect repertoire.

### A.3 Algorithm: Mechanism Level

Next, we explicitly unpack our form of the IIT algorithm to see how it compares in the case of classical IIT with [31]. In our formalism, the integrated information *φ* of a mechanism *M* of system *S* when in state *s* is

defined in Eq. 10. This definition conjoins several steps in the definition of classical IIT. To explain why it corresponds exactly to classical IIT, we disentangle this definition step by step.

First, consider *d* used for comparison is indeed the Earth Mover’s Distance. Since cause-effect repertoires have, by definition, unit intensity, the factor *r* in the definition (1) of the metric does not play a role at this stage. Therefore, the integration level of *integrated cause information*, denoted as

in [48], where *M* in this notation, and *P*. Similarly, the integration level of *integrated effect information*, denoted as

The integration scaling in Eq. 10 simply changes the intensity of an element of

Consider now the collections (9) of decomposition maps. Applying Definition 9, the core of *P* which gives the decomposition *core cause**M*, and similarly the core of *core effect**M*.

Finally, to fully account for Eq. 10, we note that the integration scaling of a pair of decomposition maps rescales both elements to the minimum of the two integration levels. Hence the integration scaling of the pair *integrated information*, denoted as

in [48], where

In summary, the following operations are combined in Eq. 10. The core of *maximally irreducible cause-effect repertoire*, and determines the integration level of each by analysing the behavior with respect to decompositions. Finally, it rescales both to the minimum of the integration levels. Thus it gives exactly what is called *M* in our formalization is given by the tuple

i.e., the pair of maximally irreducible repertoires scaled by *concept*, or sometimes *quale sensu stricto*, in classcial IIT [48], and denoted as

We finally remark that it is also possible in classical IIT that a cause repertoire value

### A.4 Algorithm: System Level

We finally explain how the system level definitions correspond to the usual definition of classical IIT.

The Q-shape *conceptual structure* or *cause-effect structure*. In [31], one does not include a concept for any mechanism *M* with

When comparing *S* by any of its cuts, it is important to note that both are elements of

Using Definition 3 and the fact that each concept’s intensity is

where *S*, and where the right-hand cause and effect repertoires are those of *M*. The factor *extended Earth Mover’s Distance*. If the integrated information

We remark that in Supplementary Material S1 of [26], an additional step is mentioned which is not described in any of the other papers we consider. Namely, if the integrated information of a mechanism is non-zero before cutting but zero after cutting, what is compared is not the distance of the corresponding concepts as in Eq. 40, but in fact the distance of the original concept with a special null concept, defined to be the unconstrained repertoire of the cut system. We have not included this step in our definitions, but it could be included by adding a choice of distinguished point to Example 3 and redefining the metric correspondingly.

In Eq. 14 the above comparison is being conducted for every subsystem of a system *S*. The subsystems of *S* are what is called *candidate systems* in [31], and which describe that ‘part’ of the system that is going to be conscious according to the theory (cf. below). Crucially, candidate systems are subsystems of *S*, whose time evolution is defined in Eq. 22. This definition ensures that the state of the elements of *S* which are not part of the candidate system are fixed in their current state, i.e., constitute *background conditions* as required in the contemporary version of classcial IIT [26].

Eq. 14 then compares the Q-shape of every candidate system to the Q-shape of all of its cuts, using the distance function described above, where the cuts are defined in Eq. 23. The cut system with the smallest distance gives the system-level *minimum information partition* and the *integrated (conceptual) information* of that candidate system, denoted as

The core integration scaling finally picks out that candidate system with the largest integrated information value. This candidate system is the *major complex M* of *S*, the part of *S* which is conscious according to the theory as part of the *exclusion postulate* of IIT. Its Q-shape is the *maximally irreducible conceptual structure (MICS)*, also called *quale sensu lato*. The overall *integrated conceptual information* is, finally, simply the intensity of

### A.5 Constellation in Qualia Space

Expanding our definitions, and denoting the major complex by *M* with state *S* state *s* is

This encodes the Q-shape *quale sensu lato*, which is taken to describe the quality of conscious experience. By construction it also encodes the integrated conceptual information of the major complex, which captures its intensity, since we have *constellation of concepts in qualia space*

Keywords: Integrated Information Theory, experience spaces, mathematical consciousness science, IIT 3.0, IIT 3.x, generalized IIT

Citation: Kleiner J and Tull S (2021) The Mathematical Structure of Integrated Information Theory. *Front. Appl. Math. Stat.* 6:602973. doi: 10.3389/fams.2020.602973

Received: 07 September 2020; Accepted: 23 December 2020;

Published: 04 June 2021.

Edited by:

Heng Liu, Guangxi University for Nationalities, ChinaReviewed by:

Guangming Xue, Guangxi University of Finance and Economics, ChinaShumin Ha, Shaanxi Normal University, China

Copyright © 2021 Kleiner and Tull. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Johannes Kleiner, johannes.kleiner@lmu.de; Sean Tull, sean.tull@cambridgequantum.com