The 0th postulate states that the constituent units of any substrate of consciousness must have the capacity to take and make a difference (it must have cause–effect power). Our first step, therefore, is to choose the substrate to analyze and model it in a way that can be characterized operationally by cause–effect power. That is, we obtain an explicit substrate model coded as a transition probability matrix (TPM).

When we model a substrate, we have to explicitly define its constituent units. This means defining their constituent grain, update grain, and unit TPMs. For example, taking a brain as a substrate to model, we might consider how a neuron (its constituent grain) takes and makes a difference over 100ms (its update grain) by modeling it with a typical sigmoidal activation function (to obtain its unit TPM). Finally, although IIT usually assumes binary units (i.e., states are ON and OFF), we also have to define what it means to be ON or OFF. For example, we might consider any burst of action potentials within the 100ms update grain as ON, and OFF otherwise. We can combine all the unit TPMs into a single substrate TPM, which uniquely describes the possible state transitions of the substrate. (For more, see FAQ: How do we get a TPM?) 

In theory, the exclusion postulate requires that we try out every possible operational characterization of the substrate. However, this is clearly not feasible in practice, so this first step of defining a substrate is critical: our operational choices here will determine what complexes and Φ-structures we will find. When making our model, therefore, it is essential to carefully consider our best scientific knowledge about substrates and their constituents (brains, neurons, etc.). 

For the explanations on this page, we will work with the substrate ABCDIO, illustrated below—the same system we used throughout the postulates pages. This simple system is not meant to model any specific substrate in the world; it is rather designed to help explain IIT. This means it is as simple as possible yet sophisticated enough to precisely articulate how the postulates of IIT are operationalized. The units have a basic sigmoidal activation function, and they interact through a mix of strong, weak, and intermediate connections—in particular, the four-unit core (ABCD) is tightly interconnected all-to-all, each one sending and receiving exactly one connection of each type (in addition to having self-loops). All units are (at least weakly) connected to all others, and the strength of causal connections is indicated by the thickness of the respective arrows. The units are shown in an indeterminate state on the substrate graph (roman uppercase), and the substrate TPM presents the potential state transitions (italics, lowercase = OFF, uppercase = ON).

With our substrate model in hand, we could assess whether it fulfills the 0th, existence postulate—whether its units “can take and make a difference” (have cause–effect power)—by computing the informativeness of its possible transitions. However, this is typically not done here explicitly; rather, informativeness is assessed in the information step (step 3 below) when computing intrinsic information. 

As described in the postulate, intrinsicality requires that the cause–effect power of the substrate of consciousness be intrinsic—it must take and make a difference within itself. Therefore, when we apply intrinsicality operationally, we start by explicitly selecting a candidate substrate of consciousness (or complex) within the overall substrate we are studying. To do so, we treat all units outside the candidate as background conditions, leaving them causally inert (visualized by “pinning” them).

Mathematically, this is done by causally marginalizing out the background units from the input side of the TPM, conditional on their current state. This operation results in two distinct TPMs for the candidate complex:

• a cause TPM, containing the probabilities of the candidate complex being in all possible current states conditional on each of its possible input states

• an effect TPM, containing the probabilities of the candidate complex transitioning to each of its possible output states conditional on being in any of its possible current states 

The reason we get two distinct TPMs in this step is that the consequences of “pinning” the background units are different whether the current state is considered as an input or an output state for a given transition. For the effect TPM, when the current substrate state is considered as an input state, the background units are marginalized out conditional on a distribution over their states where the current state has a 100% chance (it is certain) and all counterfactual states have 0% chance. In contrast, for the cause TPM, when the current state is considered as an output, the conditional distribution used when marginalizing out background units is not as straightforward. It is found by “backpropagating” the knowledge that the current substrate state occurred (as an output), resulting in a distribution over the possible input states of the background units. This distribution is then used to marginalize out the background units.

Now, with these two TPMs, we are in a position to assess whether the candidate complex “takes and makes a difference within itself”—as the postulate requires. We can do this by computing the informativeness and selectivity factors of transitions on the cause or effect side using the respective TPM. These are the two factors of the intrinsic information formula in the next step. 

As described in the [information postulate], the cause–effect state of the system is determined operationally by measuring intrinsic information (iic and iie) and picking the cause and the effect for which its value is maximal. Intrinsic information captures the specific cause–effect power of a system in its current state and uniquely combines the requirements not only of the information postulate but also of the existence and intrinsicality postulates [1]. Intrinsic information is the product of two factors, informativeness and selectivity, which are calculated through operations on the [cause and effect TPMs], as described below [2].

Informativeness measures the “raw power” of the system in its current state over a specific effect state, or of a specific cause state over the system’s current state. On the effect side, it is given by the logarithm of the ratio between the constrained and unconstrained probabilities ([...] dark green and gray bars). “Constrained” refers to the probability of a specific effect state given that the system is set to its specific current state, while “unconstrained” refers to the average probability of that effect state if the system is initialized in all possible states. Thus, informativeness is positive if and only if the system’s current state raises the probability of a specific effect state above chance, and zero otherwise. The same calculation is done to determine informativeness on the cause side ([...] dark red and gray bars), but here, the power is of the cause state over the current state. Cause informativeness is thus positive if and only if a specific cause state raises the probability of the current system state above chance.

Informativeness can grow with the number of system units due to the increased repertoire of possible states, which reduces their unconstrained probability. More precisely, the log makes informativeness grow by 1 ibit (intrinsic bit) for every additional binary unit whenever the interventional conditional probability p = 1. Informativeness captures the postulate of existence—to exist, something must have cause–effect power, in the sense of taking a difference (being caused) and making a difference (producing an effect). It also fits with the postulate of intrinsicality because cause–effect power is exerted by the system over itself.

Selectivity measures how much the system’s cause–effect power is concentrated over a specific cause or effect state ([...] light red and green bars)—that is, how close the probability of that specific state is to certainty (p = 1). Thus, effect selectivity is maximal (p = 1) if the effect state is determined with certainty and uniquely by the current system state (likewise for cause selectivity) [3]. Selectivity reflects the intrinsicality postulate because, from the intrinsic perspective of the system, any uncertainty (p < 1) about the potential cause–effect states it selects dilutes its specific cause–effect power.

Intrinsic information (ii) is calculated as the product of informativeness and selectivity for every possible cause and effect state. Following the principle of maximal existence [...], the specific cause and effect states selected by the system in its current state are the ones with the highest ii value. 

[...]

For a metaphorical illustration of intrinsic information and the role of informativeness and selectivity, imagine a rider on a chariot shafted to a number of horses [...]. The rider with the reins stands for the system in its current state and the behavior of the horses shafted to the chariot corresponds to the system’s intrinsic effect. In the first case, the chariot is shafted to one horse. The rider has one “horsepower” at his disposal (corresponding to informativeness) and through the reins he can determine the chariot’s course perfectly (corresponding to perfect selectivity). In the second case, the chariot is shafted to two horses. If the second horse is constrained by the reins just as well as the first, the rider has twice as much horsepower (higher informativeness) under his control (and still perfect selectivity). Given the choice, the rider should definitely prefer the two-horse chariot. In the third case, a third horse is shafted to the chariot, but the reins are extremely slack, so the horse is only minimally constrained by the rider’s pull. Now, despite the increase in horsepower, and a small amount of influence over the third horse, the rider’s control over the chariot’s course has diminished: the chariot will hardly ever go where pointed, swerving around according to the whims of the third horse. Clearly, the rider should now prefer the two-horse chariot to stay on course. In other words, what matters from the intrinsic perspective is neither horsepower alone nor control alone, but the product of the two, corresponding to the product between informativeness and selectivity.

The integrated information of a system (φs) quantifies the intrinsic information specified by a candidate substrate over its minimum partition for its maximal cause or effect state. It is determined by calculating φc and φe and taking the minimum between the two. 

The calculation of φc and φe is similar to that of iic and iie, as presented in [the previous step]. The key difference is that the informativeness factor now becomes integrated informativeness, since we aim to assess the degree to which the cause–effect power of the system as a whole cannot be accounted for by the cause–effect power of its subsets. The values of φc and φe are calculated as the product of integrated informativeness and selectivity.

The probabilities required for calculating φc and φe are shown [below]. The same selectivity values from the ii calculations are used (light red and green bars). For integrated informativeness, the “constrained” probabilities from ii carry over as the “unpartitioned” probabilities (dark red and green bars). But now, instead of the “unconstrained” probabilities from ii, we use the “partitioned” probabilities (gray bars)—obtained, in this example, from the minimum partition. Integrated informativeness on the cause or effect side is then calculated as the logarithm of the ratio between the unpartitioned and partitioned probabilities. In other words, integrated informativeness measures the cause or effect power of the system with its internal connections intact compared to the power it would have if its internal connections were severed along the minimal partition. 

The value of φc and φe is obtained by taking the product of integrated informativeness and selectivity for the unidirectional partition that yields the minimal value (the “minimal partition”), following the principle of minimal existence [1]. Following the same principle, we then choose the minimal φ value between the cause and effect sides. Accordingly, the irreducibility (φs) of substrate Abcd is 1.21, corresponding to the φc for the minimal partition. If the φs were zero (or set to zero if its value is negative), then the candidate system would be reducible—that is, it would not have unitary cause–effect power.

Of note, the minimum partition is defined as the partition that minimizes the integrated information relative to the maximum possible integrated information for a given partition [2]. 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. Once the minimum partition has been identified, the integrated information across it is an absolute quantity. 

We can also identify any minor complexes. This amounts to applying the previous four steps to all remaining candidates of the original substrate (in this case, all subsets of units i and o).

Finally, note that these results are only valid for the specific constituent and update grains that we chose at the start. A different choice of grains means using a different substrate TPM. In theory, we would also have to apply the previous steps to all possible constituent and update grains to determine whether another choice yields the main complex (one with an even higher φs value than what we found with Abcd). (For more, see How do we determine the causal grain at which integrated information (φs) is maximal?) 

As described in the postulate, composition requires that the cause–effect power of the substrate of consciousness be structured: subsets of its units must specify cause–effects over subsets of units (distinctions) that can overlap with one another (relations), yielding a cause–effect structure that is the way it is. Hence, now that we have identified our main complex, we unfold its cause–effect power in full by evaluating the cause–effect power of each distinction (measured by φd) and each relation (i.e., the cause–effect power that distinctions exert jointly, measured by φr). Unfolding can be understood in the four steps below, which reapply the postulates (save composition itself), but now at the level of mechanisms rather than the system as a whole.

1. Determine all irreducible distinctions by calculating the intrinsic information (ii) and integrated information (φd) of each candidate.

2. Enforce the congruence of each irreducible distinction with the cause–effect state of the system.

3. Calculate the φr value for each candidate relation, thus measuring the irreducibility of each distinction overlap. 

4. Calculate Φ (“big Phi”) as the sum of all φd and φr values.

In explaining each step, we will focus mainly on the distinctions (and their relations) used elsewhere in the Wiki—namely, first-order distinction c and third-order distinction bcd. 

We start by considering every possible subset of system units as a candidate mechanism. This means that we take the set of system units in their state (A, b, c, and d) and consider all possible subsets (the mechanism powerset), in this case: A, b, c, d, Ab, Ac, Ad, bc, bd, cd, Abc, Abd, Acd, bcd, and Abcd. (For more, see FAQ: Why do we assess the causal power of all orders of mechanisms?)

For each candidate mechanism, we aim to evaluate whether it specifies a distinction—whether it has an intrinsic and irreducible cause and effect. Hence we also assess the whole powerset of units as candidate causes and effects (called purviews). While the states of candidate mechanisms are “inherited” from the system state, the cause and effect purviews will have to be evaluated over every possible state (thus they are depicted in roman uppercase).

First, we compute intrinsic information (ii) for each candidate mechanism over each purview state. This step can be understood as applying the existence, intrinsicality, and information postulates at the mechanism level. Starting on the cause side, take the example of candidate mechanism bcd. We will compute iic for every 1st-order candidate cause purview in every state (e.g., a, A), 2nd-order one (e.g., ac, Ac, aC, AC), and so on for all candidate causes of all orders. For each candidate cause, we find the state with the highest iic. For bcd over AC, for example, the state with highest iic is ac, while for bcd over A, the state with highest iic is a (details in the slides below). 

After finding the state of the candidate cause of our candidate mechanism, we then measure the irreducibility of the mechanism–purview pair. The candidate cause (in its maximal state) with the highest φc is the maximally irreducible cause. This step can be understood as applying the integration and exclusion postulates. In our example, mechanism bcd has a φc of 0.035 over cause purview ac, and of 0.017 over a. The φc value of purview ac is higher than that of a—and of any other candidate cause as well. We therefore conclude that ac is the maximally irreducible cause of bcd, thereby excluding all other candidate causes.

After following these steps for every candidate mechanism on the cause side, we do the same on the effect side (calculating iie and φe). We find, for example, that for candidate mechanism bcd, its maximally irreducible effect is abd, with φe = 0.017. Thus, candidate mechanism bcd is irreducible and specifies a distinction over ac and abd with φd = 0.017 (the minimum between φc and φe for that distinction, according to the principle of minimal existence).

We repeat the process for every candidate mechanism and end up with a set of irreducible distinctions. For our example system, we find eight distinctions out of a possible fifteen given by the mechanism powerset (shown in the slides).

A causal relation obtains whenever distinctions overlap congruently over cause or effect purviews. To assess the irreducibility of a relation, we partition or “unbind” the distinctions that constitute it one at a time, and identify the distinction that contributed least to the overlap (the relation MIP). As an example, consider the two distinctions in the figure: a–c–d and ac–bcd–abd. These distinctions are bound by a 2nd-degree relation (in purple), with relation purview ad, composed of three 2nd-degree faces (blue edges) with face purviews over a and d, and one 3rd-degree face (blue area), with a face purview over a. 

To compute the φr of the relation, we unbind one distinction at a time and see which one makes the least difference. We calculate this by multiplying the average φd per distinct purview unit by the size of the overlap across all faces. 

Here, bcd’s φd = 0.017, its cause is over ac, its effect is over bcd, and together, these purviews contain four unique units (c is repeated). Thus the size of the union is four, and the average φd per distinct purview unit is 0.017/4 = 0.004. Since the relation in question has an overlap over two units (relation purview ad), φr = 0.004*2 = 0.008.

If instead we unbind distinction c, we get a φr of 0.882.

Since the φr from unbinding bcd is less than that from unbinding c, we conclude that the relation MIP is bcd and the φr of the relation is 0.008.