^{1}

^{2}

This article was submitted to Dynamical Systems, a section of the journal Frontiers in Applied Mathematics and Statistics

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.

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.

While promising in itself [

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. [

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

In the associated article [

Our definition of IIT may serve as the starting point for further mathematical analysis of IIT, in particular if related to category theory [

This work is concerned with the most recent version of IIT as proposed in [

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 [

We begin by introducing the necessary ingredients of a generalised Integrated Information Theory in

Following this we give several examples including IIT 3.0 in

The first step in defining an Integrated Information Theory (IIT) is to specify a class

A

A set

for every

a set

for each

Moreover, we require that

Note that taking a subsystem of a system

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

An IIT aims to specify for each system in a particular state its

Firstly, each experience

An

An

A

A

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

The above definition is very general, and in any specific application of IIT, the experiences may come with further mathematical structure. The following example includes the experience spaces used in classical IIT.

Any metric space

This is the definition used in classical IIT (cf.

An important operation on experience spaces is taking their

For experience spaces

In order to define the experience space and individual experiences of a system

Each repertoire describes a way of ‘decomposing’ experiences, in the following sense. Let

Let

In more detail, a repertoire specifies a proto-experience for every pair of subsystems and describes how this experience changes if the subsystems are decomposed. This allows one to assess how integrated the system is with respect to a particular repertoire. Two repertoires are necessary for the IIT algorithm to be applied, together called the cause-effect repertoire.

For subsystems

A

Examples of cause-effect repertoires will be given in

A

The names ‘cause’ and ‘effect’ highlight that the definitions of

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.

Let

Finally, the

We will also need to consider indexed collections of decomposable elements. Let

The ^{1}

Let

For every choice of

The

It is an element of

The second level of the algorithm specifies the experience of system

This Is an Element of the Space

The

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

An

The

In the next sections we specify the data of several example IITs.

In this section we show how IIT 3.0 [

We first describe the system class underlying classical IIT. Physical systems

Additionally, each system comes with a probabilistic (discrete)

The class

In what follows, we will need to consider two operations on the map

In particular for any map

Let a system

The decomposition set

For any decomposition

For each system

It remains to define the cause-effect repertoires. Fixing a state

For General Mechanisms

The distributions

It is in fact possible for the right-hand side of

Finally we must specify the decompositions of these elements over

This concludes all data necessary to define classical IIT. If the generalized definition of

In this section, we consider Quantum IIT defined in [

Similar to classical IIT, in Quantum IIT systems are conceived as consisting of elements

Subsystems are again defined to consist of subsets

Decompositions are also defined via partitions

For any

We finally come to the definition of the cause-effect repertoire. Unlike classical IIT, the definition in [

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 [

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

In

In order to avoid the requirement of a discrete time and Markovian dynamics, instead of working with the time evolution operator

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

The idea behind the following is to define, for every

Let now

The probability distribution

So far, our construction can be applied for any time

The problem with applying the definitions of classical IIT to systems with continuous state spaces (e.g., neuron membrane potentials [

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

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

Another criticism of IIT’s mathematical structure mentioned [

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

This suggests that the metrics

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.

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 [

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 ^{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) [

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 [

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

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

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.

We would like to thank the organizers and participants of the

If the maximum does not exist, we define the core to be the empty system

The problem of ‘unique existence’ has been studied extensively in category theory using

In

The systems of classical IIT are given in

In contemporary presentations of the theory ([

In our notation, the right hand side of

Similarly, the cause repertoire is defined as ([

Here, the whole right hand side of _{
i,t
} at time

As a result, the cause and effect repertoire in the sense of [

The behavior of the cause- and effect-repertoires when decomposing a system is described, in our formalism, by decompositions (Definition 5). Hence a decomposition

Next, we explicitly unpack our form of the IIT algorithm to see how it compares in the case of classical IIT with [

First, consider

The integration scaling in

Consider now the collections (9) of decomposition maps. Applying Definition 9, the core of

Finally, to fully account for

In summary, the following operations are combined in

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

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

The Q-shape

When comparing

We remark that in Supplementary Material S1 of [

In

The core integration scaling finally picks out that candidate system with the largest integrated information value. This candidate system is the

Expanding our definitions, and denoting the major complex by