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What Caused What? A Quantitative Account of Actual Causation Using Dynamical Causal Networks

Abstract

Actual causation is concerned with the question: “What caused what?” Consider a transition between two states within a system of interacting elements, such as an artificial neural network, or a biological brain circuit. Which combination of synapses caused the neuron to fire? Which image features caused the classifier to misinterpret the picture? Even detailed knowledge of the system’s causal network, its elements, their states, connectivity, and dynamics does not automatically provide a straightforward answer to the “what caused what?” question. Counterfactual accounts of actual causation, based on graphical models paired with system interventions, have demonstrated initial success in addressing specific problem cases, in line with intuitive causal judgments. Here, we start from a set of basic requirements for causation (realization, composition, information, integration, and exclusion) and develop a rigorous, quantitative account of actual causation, that is generally applicable to discrete dynamical systems. We present a formal framework to evaluate these causal requirements based on system interventions and partitions, which considers all counterfactuals of a state transition. This framework is used to provide a complete causal account of the transition by identifying and quantifying the strength of all actual causes and effects linking the two consecutive system states. Finally, we examine several exemplary cases and paradoxes of causation and show that they can be illuminated by the proposed framework for quantifying actual causation.

1. Introduction

The nature of cause and effect has been much debated in both philosophy and the sciences. To date, there is no single widely-accepted account of causation, and the various sciences focus on different aspects of the issue [1]. In physics, no formal notion of causation seems to even be required for describing the dynamical evolution of a system by a set of mathematical equations. At most, the notion of causation is reduced to the basic requirement that causes must precede and be able to influence their effects—no further constraints are imposed with regard to “what caused what”.

However, a detailed record of “what happened” prior to a particular occurrence rarely provides a satisfactory explanation for why it occurred in causal, mechanistic terms (see Theory 2.2 for a formal definition of the term “occurrence” as a set of random variables in a particular state at a particular time). As an example, take AlphaGo, the deep neural network that repeatedly defeated human champions in the game Go [2]. Understanding why AlphaGo chose a particular move is a non-trivial problem [3], even though all its network parameters and its state evolution can be recorded in detail. Identifying “what caused what” becomes particularly difficult in complex systems with a distributed, recurrent architecture and wide-ranging interactions, as is typical for biological (neural) networks, including the brain [4,5].

Our interest, here, lies in the principled analysis of actual causation in discrete distributed dynamical systems, such as artificial neural networks, computers made of logic gates, or cellular automata, but also simple models of biological brain circuits or gene regulatory networks. In contrast with general (or type) causation, which addresses the question of whether the type of occurrence A generally “brings about” the type of occurrence B, the underlying notion of actual (or token) causation addresses the question of “what caused what”, given a specific occurrence A followed by a specific occurrence B. For example, what part of the particular pattern on the board caused AlphaGo to decide on this particular move? As highlighted by the AlphaGo example, even with detailed knowledge of all circumstances, the prior system state, and the outcome, there often is no straightforward answer to the “what caused what” question. This has also been demonstrated by a long list of controversial examples conceived, analyzed, and debated primarily by philosophers (e.g., [6,7,8,9,10,11,12]).

A number of attempts to operationalize the notion of causation and to give it a formal description have been developed, most notably in computer science, probability theory, statistics [7,13,14,15,16], the law [17], and neuroscience, (e.g., [18]). Graphical methods paired with system interventions [7] have proven to be especially valuable for developing causal explanations. Given a causal network that represents how the state of each variable depends on other system variables by a “structural equation” [7], it is possible to evaluate the effects of interventions imposed from outside the network by setting certain variables to a specific value. This operation has been formalized by Pearl, who introduced the “do-operator”, $\mathrm{d}\mathrm{o}(X=x)$, which signifies that a subset of system variables X has been actively set into state x, rather than being passively observed in this state [7]. As statistical dependence does not imply causal dependence, the conditional probability of occurrence B after observing occurrence A, $p(B\mid A)$, may differ from the probability of occurrence B after enforcing A, $p(B\mid \mathrm{d}\mathrm{o}(A))$. Causal networks are a specific subset of “Bayesian” networks, that explicitly represent causal dependencies consistent with interventional probabilities.

The causal network approach has also been applied to the case of actual causation [7,8,11,19,20,21], where system interventions can be used to evaluate whether (and to what extent) an occurrence was necessary or sufficient for a subsequent occurrence by assessing counterfactuals—alternative occurrences “counter to fact” [7,22,23]—within a given causal model. The objective is to define “what it means for A to be a cause of B in a model M” [12]. Note that counterfactuals, here, strictly refer to the possible states within the system’s state space (other than the actual one) and not to abstract notions, such as other “possible worlds” as in [22] (see also [7], Chapter 7). While promising results have been obtained in specific cases, no single proposal (to date) has characterized actual causation in a universally satisfying manner [10,12]. One concern about existing measures of actual causation is the incremental manner in which they progress; a definition is proposed that satisfies existing examples in the literature, until a new problematic example is discovered, at which point the definition is updated to address the new example [11,24]. While valuable, the problem with such an approach is that one cannot be confident in applying the framework beyond the scope of examples already tested. For example, while these methods are well-explored in simple binary examples, there is less evidence that the methods conform with intuition when we consider the much larger space of non-binary examples. This is especially critical when moving beyond intuitive toy examples to scientific problems where intuition is lacking, such as understanding actual causation in biological or artificial neural networks.

Our goal is to provide a robust framework for assessing actual causation that is based on general causal principles, and can, thus, be expected to naturally extend beyond simple, binary, and deterministic example cases. Below, we present a formal account of actual causation which is generally applicable to discrete Markovian dynamical systems constituted of interacting elements (see Figure 1). The proposed framework is based on five causal principles identified in the context of integrated information theory (IIT)—namely, existence (here: realization), composition, information, integration, and exclusion [25,26]). Originally developed as a theory of consciousness [27,28], IIT provides the tools to characterize potential causation—the causal constraints exerted by a mechanism in a given state.

In particular, our objective is to provide a complete quantitative causal account of “what caused what”, within a transition between consecutive system states. Our approach differs from previous accounts of actual causation in what constitutes a complete causal account: Unlike most accounts of actual causation (e.g., [7,10,12], but see [29]), causal links within a transition are considered from the perspective of both causes and effects. Additionally, we not only evaluate actual causes and effects of individual variables, but also actual causes and effects of high-order occurrences, comprising multiple variables. While some existing accounts of actual causation include the notion of being “part of a cause” [12,21], the possibility of multi-variate causes and effects is rarely addressed, or even outright excluded [11].

Despite the differences in what constitutes a complete causal account, our approach remains compatible with the traditional view of actual causation, which considers only actual causes of individual variables (no high-order causation, and no actual effects). In this context, the main difference between our proposed framework and existing “contingency”-based definitions is that we simultaneously consider all counterfactual states of the transition, rather than a single contingency (e.g., as in [8,11,19,20,21,30,31]). This allows us to express the causal analysis in probabilistic, informational terms [25,32,33,34], which has the additional benefit that our framework naturally extends from deterministic to probabilistic causal networks, and also from binary to multi-valued variables. Finally, it allows us to quantify the strength of all causal links between occurrences and their causes and effects within the transition.

In the following, we will first formally describe the proposed causal framework of actual causation. We, then, demonstrate its utility on a set of examples, which illustrates the benefits of characterizing both causes and effects, the fact that causation can be compositional, and the importance of identifying irreducible causes and effects for obtaining a complete causal account. Finally, we illustrate several prominent paradoxical cases from the actual causation literature, including overdetermination and prevention, as well as a toy model of an image classifier, based on an artificial neural network.

2. Theory

Integrated information theory is concerned with the intrinsic cause-effect power of a physical system (intrinsic existence). The IIT formalism [25,27] starts from a discrete distributed dynamical system in its current state and asks how the system elements, alone and in combination (composition), constrain the potential past and future states of the system (information), and whether they do so above and beyond their parts (integration). The potential causes and effects of a system subset correspond to the set of elements over which the constraints are maximally informative and integrated (exclusion). In the following we aim to translate the IIT account of potential causation into a principled, quantitative framework for actual causation, which allows for the evaluation of all actual causes and effects within a state transition of a dynamical system of interacting elements, such as a biological or artificial neural network (see Figure 1). For maximal generality, we will formulate our account of actual causation in the context of dynamical causal networks [32,34,35].

2.1. Dynamical Causal Networks

Our starting point is a dynamical causal network: A directed acyclic graph (DAG) $G_u=(V,E)$ with edges E that indicate the causal connections among a set of nodes V and a given set of background conditions (state of exogenous variables) $U=u$ (see Figure 1B). The nodes in $G_u$ represent a set of associated random variables (which we also denote by V) with state space $\mathsf{\Omega }={\prod }_i{\mathsf{\Omega }}_{V_i}$ and probability function $p\left(v\right|u),v\in \mathsf{\Omega }$. For any node $V_i\in V$, we can define the parents of $V_i$ in $G_u$ as all nodes with an edge leading into $V_i$, $$pa\left(V_i\right)=\{V_j\mid e_{ji}\in E\}.$$

A causal network $G_u$ is dynamical, in the sense that we can define a partition of its nodes V into $k+1$ temporally ordered “slices”, $V=\{V_0,V_1,\dots ,V_k\}$, starting with an initial slice without parents ($pa\left(V_0\right)=\varnothing$) and such that the parents of each successive slice are fully contained within the previous slice ($pa\left(V_t\right)\subseteq V_{t-1},t=1,\dots ,k$). This definition is similar to the one proposed in [32], but is stricter, requiring that there are no within-slice causal interactions. This restriction prohibits any “instantaneous causation” between variables (see also [7], Section 1.5) and signifies that $G_u$ fulfills the Markov property. Nevertheless, recurrent networks can be represented as dynamical causal models when unfolded in time (see Figure 1B) [20]. The parts of $V=\{V_0,V_1,\dots ,V_k\}$ can thus be interpreted as consecutive time steps of a discrete dynamical system of interacting elements (see Figure 1); a particular state $V=v$, then, corresponds to a system transient over $k+1$ time steps.

In a Bayesian network, the edges of $G_u$ fully capture the dependency structure between nodes V. That is, for a given set of background conditions, each node is conditionally independent of every other node, given its parents in $G_u$, and the probability function can be factored as $$p(v\mid u)=\prod _{i}p(v_i\mid pa\left(v_i\right),u),v\in \mathsf{\Omega }.$$

For a causal network, there is the additional requirement that the edges E capture causal dependencies (rather than just correlations) between nodes. This means that the decomposition of $p(v\mid u)$ holds, even if the parent variables are actively set into their state as opposed to passively observed in that state (“Causal Markov Condition”, [7,15]), $$p(v\mid u)=\prod _{i}p\left(v_i\mid do(pa\left(v_i\right),u)\right),v\in \mathsf{\Omega }.$$

As we assume, here, that U contains all relevant background variables, any statistical dependencies between $V_{t-1}$ and $V_t$ are, in fact, causal dependencies, and cannot be explained by latent external variables (“causal sufficiency”, see [34]). Moreover, because time is explicit in $G_u$ and we assume that there is no instantaneous causation, there is no question of the direction of causal influences—it must be that the earlier variables ($V_{t-1}$) influence the later variables ($V_t$). By definition, $V_{t-1}$ contains all parents of $V_t$ for $t=1,\dots ,k$. In contrast to the variables V within $G_u$, the background variables U are conditioned to a particular state $U=u$ throughout the causal analysis and are, otherwise, not further considered.

Together, these assumptions imply a transition probability function for V, such that the nodes at time t are conditionally independent given the state of the nodes at time $t-1$ (see Figure 1C),

To reiterate, a dynamical causal network $G_u$ describes the causal interactions among a set of nodes (the edges in E describe the causal connections between the nodes in V) conditional on the state of the background variables U, and the transition probability function $p_u(v_t\mid v_{t-1})$ (Equation (1)) fully captures the nature of these causal dependencies. Note that $p_u\left(v_t\right|v_{t-1})$ is generally undefined in the case where $p_u\left(v_{t-1}\right)=0$. However, in the present context, it is defined as $p_u\left(v_t\right|v_{t-1})=p_u\left(v_t\right|do\left(v_{t-1}\right))$ using the $do\left(v_{t-1}\right)$ operation. The interventional probability $p_u\left(v_t\right|do\left(v_{t-1}\right))$ is well-defined for all $v_{t-1}\in \mathsf{\Omega }$ and can typically be inferred from the mechanisms associated with the variables in $V_t$.

In summary, we assume that $G_u$ fully and accurately describes the system of interest for a given set of background conditions. In reality, a causal network reflects assumptions about a system’s elementary mechanisms. Current scientific knowledge must inform which variables to include, what their relevant states are, and how they are related mechanistically [7,36]. Here, we are primarily interested in natural and artificial systems, such as neural networks, for which detailed information about the causal network structure and the mechanisms of individual system elements is often available, or can be obtained through exhaustive experiments. In such systems, counterfactuals can be evaluated by performing experiments or simulations that assess how the system reacts to interventions. The transition probabilities can, in principle, be determined by perturbing the system into all possible states while holding the background variables fixed and observing the resulting transitions. Alternatively, the causal network can be constructed by experimentally identifying the input-output function of each element (i.e., its structural equation [7,34]). Merely observing the system without experimental manipulation is insufficient to identify causal relationships in most situations. Moreover, instantaneous dependencies are frequently observed in (experimentally obtained) time-series data of macroscopic variables, due to unobserved interactions at finer spatio-temporal scales [37]. In this case, a suitable dynamical causal network may still be obtained, simply by discounting such instantaneous dependencies, since these interactions are not due to the macroscopic mechanisms themselves.

Our objective, here, is to formulate a quantitative account of actual causation applicable to any predetermined, dynamical causal network, independent of practical considerations about model selection [12,36]. Confounding issues due to incomplete knowledge, such as estimation biases of probabilities from finite sampling, or latent variables, are, thus, set aside for the present purposes. To what extent and under which conditions the identified actual causes and effects generalize across possible levels of description, or under incomplete knowledge, is an interesting question that we plan to address in future work (see also [38,39]).

2.2. Occurrences and Transitions

In general, actual causation can be evaluated over multiple time steps (e.g., considering indirect causal influences). Here, however, we specifically focus on direct causes and effects without intermediary variables or time steps. For this reason, we only consider causal networks containing nodes from two consecutive time points, $V=\{V_{t-1},V_t\}$, and define a transition, denoted by $v_{t-1}\prec v_t$, as a realization $V=v$ with $v=(v_{t-1},v_t)\in \mathsf{\Omega }$ and $p_u\left(v_t\right|v_{t-1})>0$ (see Figure 1D).

Note that our approach generalizes, in principle, to system transitions across multiple $(k>1)$ time steps, by considering the transition probabilities $p_u(v_t\mid v_{t-k})$, instead of $p_u(v_t\mid v_{t-1})$, in Equation (1). While this practice would correctly identify counterfactual dependencies between $v_{t-k}$ and $v_t$, it ignores the actual states of the intermediate time steps $(v_{t-k+1},\dots ,v_{t-1})$. As a consequence, this approach cannot, at present, address certain issues regarding causal transitivity across multiple paths, incomplete causal processes in probabilistic causal networks [40], or causal dependencies in non-Markovian systems.

Within a dynamical causal network $G_u=(V,E)$ with $V=\{V_{t-1},V_t\}$, our objective is to determine the actual cause or actual effect of occurrences within a transition $v_{t-1}\prec v_t$. Formally, an occurrence is defined to be a sub-state $X_{t-1}=x_{t-1}\subseteq V_{t-1}=v_{t-1}$ or $Y_t=y_t\subseteq V_t=v_t$, corresponding to a subset of elements at a particular time and in a particular state. This corresponds to the general usage of the term “event” in the computer science and probability literature. The term “occurrence” was chosen, instead, to avoid philosophical baggage associated with the term “event”.

2.3. Cause and Effect Repertoires

Before defining the actual cause or actual effect of an occurrence, we first introduce two definitions from IIT which are useful in characterizing the causal powers of occurrences in a causal network: Cause/effect repertoires and partitioned cause/effect repertoires. In IIT, a cause (or effect) repertoire is a conditional probability distribution that describes how an occurrence (set of elements in a state) constrains the potential past (or future) states of other elements in a system [25,26] (see also [27,41] for a general mathematical definition). In the present context of a transition $v_{t-1}\prec v_t$, an effect repertoire specifies how an occurrence $x_{t-1}\subseteq v_{t-1}$ constrains the potential future states of a set of nodes $Y_t\subseteq V_t$. Likewise, a cause repertoire specifies how an occurrence $y_t\subseteq v_t$ constrains the potential past states of a set of nodes $X_{t-1}\subset V_{t-1}$ (see Figure 2).

The effect and cause repertoire can be derived from the system transition probabilities in Equation (1) by conditioning on the state of the occurrence and causally marginalizing the variables outside the occurrence $V_{t-1}\backslash X_{t-1}$ and $V_t\backslash Y_t$ (see Discussion 4.1). Causal marginalization serves to remove any contributions to the repertoire from variables outside the occurrence by averaging over all their possible states. Explicitly, for a single node $Y_{i,t}$, the effect repertoire is:

where $W=V_{t-1}\backslash X_{t-1}$ with state space ${\mathsf{\Omega }}_W$. Note that, for causal marginalization, each possible state $W=w\in {\mathsf{\Omega }}_W$ is given the same weight $|{\mathsf{\Omega }}_W{|}^{-1}$ in the average, which corresponds to imposing a uniform distribution over all $w\in {\mathsf{\Omega }}_W$. This ensures that the repertoire captures the constraints due to the occurrence, and not to whatever external factors might bias the variables in W to one state or another (this is discussed in more detail in Section 4.1).

In graphical terms, causal marginalizing implies that the connections from all $W_i\in W$ to $Y_{i,t}$ are “cut” and independently replaced by an un-biased average across the states of the respective $W_i$, which also removes all dependencies between the variables in W. Causal marginalization, thus, corresponds to the notion of cutting edges proposed in [34]. However, instead of feeding all open ends with the product of the corresponding marginal distributions obtained from the observed joint distribution, as in Equation (7) of [34], here we impose a uniform distribution $p=\frac{1}{|{\mathsf{\Omega }}_W|},\forall w\in {\mathsf{\Omega }}_W$, as we are interested in quantifying mechanistic dependencies, which should not depend on the observed joint distribution.

The complementary cause repertoire of a singleton occurrence $y_{i,t}$, using Bayes’ rule, is:$$\pi (X_{t-1}\mid y_{i,t})=\sum _{w\in {\mathsf{\Omega }}_W}\frac{p_u\left(y_{i,t}\mid \mathrm{d}\mathrm{o}\left(X_{t-1},W=w\right)\right)}{{\sum }_{z\in {\mathsf{\Omega }}_{V_{t-1}}}{p}_u\left(y_{i,t}\mid \mathrm{d}\mathrm{o}\left(V_{t-1}=z\right)\right)}.$$

In the general case of a multi-variate $Y_t$ (or $y_t$), the transition probability function $p_u(Y_t\mid x_{t-1})$ not only contains dependencies of $Y_t$ on $x_{t-1}$, but also correlations between the variables in $Y_t$ due to common inputs from nodes in $W_{t-1}=V_{t-1}\backslash X_{t-1}$, which should not be counted as constraints due to $x_{t-1}$. To discount such correlations, we define the effect repertoire over a set of variables $Y_t$ as the product of the effect repertoires over individual nodes (Equation (2)) (see also [34]):

In the same manner, we define the cause repertoire of a general occurrence $y_t$ over a set of variables $X_{t-1}$ as:

We can also define unconstrained cause and effect repertoires, a special case of cause or effect repertoires where the occurrence that we condition on is the empty set. In this case, the repertoire describes the causal constraints on a set of the nodes due to the structure of the causal network, under maximum uncertainty about the states of variables within the network. With the convention that $\pi (\varnothing )=1$, we can derive these unconstrained repertoires directly from the formulas for the cause and effect repertoires, Equations (3) and (4). The unconstrained cause repertoire simplifies to a uniform distribution, representing the fact that the causal network itself imposes no constraint on the possible states of variables in $V_{t-1}$,

The unconstrained effect repertoire is shaped by the update function of each individual node $Y_{i,t}\in Y_t$ under maximum uncertainty about the state of its parents,

where $W=V_{t-1}\backslash X_{t-1}=V_{t-1}$, since $X_{t-1}=\varnothing$.

In summary, the effect and cause repertoires $\pi (Y_t\mid x_{t-1})$ and $\pi (X_{t-1}\mid y_t)$, respectively, are conditional probability distributions that specify the causal constraints due to an occurrence on the potential past and future states of variables in a causal network $G_u$. The cause and effect repertoires discount constraints that are not specific to the occurrence of interest; possible constraints due to the state of variables outside of the occurrence are causally marginalized from the distribution, and constraints due to common inputs from other nodes are avoided by treating each node in the occurrence independently. Thus, we denote cause and effect repertoires with $\pi$, to highlight that, in general, $\pi (Y_t\mid x_{t-1})\ne p(Y_t\mid x_{t-1})$. However, $\pi (Y_t\mid x_{t-1})$ is equivalent to $p(Y_t\mid x_{t-1})$ (the conditional probability imposing a uniform distribution over the marginalized variables), in the special case that all variables $Y_{i,t}\in Y_t$ are conditionally independent, given $x_{t-1}$ (see also [34], Remark 1). This is the case, for example, if $X_{t-1}$ already includes all inputs (all parents) of $Y_t$, or determines $Y_t$ completely.

An objective of IIT is to evaluate whether the causal constraints of an occurrence on a set of nodes are “integrated”, or “irreducible”; that is, whether the individual variables in the occurrence work together to constrain the past or future states of the set of nodes in a way that is not accounted for by the variables taken independently [25,42]. To this end, the occurrence (together with the set of nodes it constrains) is partitioned into independent parts, by rendering the connection between the parts causally ineffective [25,26,34,42]. The partitioned cause and effect repertoires describe the residual constraints under the partition. Comparing the partitioned cause and effect repertoires to the intact cause and effect repertoires reveals what is lost or changed by the partition.

A partition $\psi$ of the occurrence $x_{t-1}$ (and the nodes it constrains, $Y_t$) into m parts is defined as:

such that ${\left\{x_{j,t-1}\right\}}_{j=1}^m$ is a partition of $x_{t-1}$ and $Y_{j,t}\subseteq Y_t$ with $Y_{j,t}\cap Y_{k,t}=\varnothing ,j\ne k$. Note that this includes the possibility that any $Y_{j,t}=\varnothing$, which may leave a set of nodes $Y_t\backslash {\bigcup }_{j=1}^m{Y}_{j,t}$ completely unconstrained (see Figure 3 for examples and details).

The partitioned effect repertoire of an occurrence $x_{t-1}$ over a set of nodes $Y_t$ under a partition $\psi$ is defined as:

This is the product of the corresponding m effect repertoires, multiplied by the unconstrained effect repertoire (Equation (6)) of the remaining set of nodes $Y_t\backslash {\bigcup }_{j=1}^m{Y}_{j,t}$, as these nodes are no longer constrained by any part of $x_{t-1}$ under the partition.

In the same way, a partition $\psi$ of the occurrence $y_t$ (and the nodes it constrains $X_{t-1}$) into m parts is defined as:

such that ${\left\{y_{i,t}\right\}}_{i=1}^m$ is a partition of $y_t$ and $X_{j,t-1}\subseteq X_{t-1}$ with $X_{j,t-1}\cap X_{k,t-1}=\varnothing ,j\ne k$. The partitioned cause repertoire of an occurrence $y_t$ over a set of nodes $X_{t-1}$ under a partition $\psi$ is defined as:

the product of the corresponding m cause repertoires multiplied by the unconstrained cause repertoire (Equation (6)) of the remaining set of nodes $X_{t-1}\backslash {\bigcup }_{j=1}^m{X}_{j,t-1}$, which are no longer constrained by any part of $y_t$ due to the partition.

2.4. Actual Causes and Actual Effects

The objective of this section is to introduce the notion of a causal account for a transition of interest $v_{t-1}\prec v_t$ in $G_u$ as the set of all causal links between occurrences within the transition. There is a causal link between occurrences $x_{t-1}$ and $y_t$ if $y_t$ is the actual effect of $x_{t-1}$, or if $x_{t-1}$ is the actual cause of $y_t$. Below, we define causal link, actual cause, actual effect, and causal account, following five causal principles: Realization, composition, information, integration, and exclusion.

Realization. A transition $v_{t-1}\prec v_t$ must be consistent with the transition probability function of a dynamical causal network $G_u$, $$p_u\left(v_t\right|v_{t-1})>0.$$

Only occurrences within a transition $v_{t-1}\prec v_t$ may have, or be, an actual cause or actual effect (This requirement corresponds to the first clause (“AC1”) of the Halpern and Pearl account of actual causation [20,21]; that is, for $C=c$ to be an actual cause of $E=e$, both must actually happen in the first place.)

As a first example, we consider the transition $\{(OR,AND{)}_{t-1}=10\}\prec \{(OR,AND{)}_t=10\}$, shown in Figure 1D. This transition is consistent with the conditional transition probabilities of the system, shown in Figure 1C.

Composition. Occurrences and their actual causes and effects can be uni- or multi-variate. For a complete causal account of the transition $v_{t-1}\prec v_t$, all causal links between occurrences $x_{t-1}\subseteq v_{t-1}$ and $y_t\subseteq v_t$ should be considered. For this reason, we evaluate every subset of $x_{t-1}\subseteq v_{t-1}$ as occurrences that may have actual effects and every subset $y_t\subseteq v_t$ as occurrences that may have actual causes (see Figure 4). For a particular occurrence $x_{t-1}$, all subsets $y_t\subseteq v_t$ are considered as candidate effects (Figure 5A). For a particular occurrence $y_t$, all subsets $x_{t-1}\subseteq v_{t-1}$ are considered as candidate causes (see Figure 5B). In what follows, we refer to occurrences consisting of a single variable as “first-order” occurrences and to multi-variate occurrences as “high-order” occurrences, and, likewise, to “first-order” and “high-order” causes and effects.

In the example transition shown in Figure 4, $\{{OR}_{t-1}=1\}$ and $\{{AND}_t=0\}$ are first-order occurrences that could have an actual effect in $v_t$, and $\{(OR,{AND)}_{t-1}=10\}$ is a high-order occurrence that could also have its own actual effect in $v_t$. On the other side, $\{{OR}_t=1\}$, $\{{AND}_t=0\}$ and $\{(OR,{AND)}_t=10\}$ are occurrences (two first-order and one high-order) that could have an actual cause in $v_{t-1}$. To identify the respective actual cause (or effect) of any of these occurrences, we evaluate all possible sets $\{OR=1\}$, $\{AND=0\}$, and $\{\left(OR,AND\right)=10\}$ at time $t-1$ (or t). Note that, in principle, we also consider the empty set, again using the convention that $\pi (\varnothing )=1$ (see “exclusion”, below).

Information. An occurrence must provide information about its actual cause or effect. This means that it should increase the probability of its actual cause or effect compared to its probability if the occurrence is unspecified. To evaluate this, we compare the probability of a candidate effect $y_t$ in the effect repertoire of the occurrence $x_{t-1}$ (Equation (3)) to its corresponding probability in the unconstrained repertoire (Equation (6)). In line with information-theoretical principles, we define the effect information ${\rho }_e$ of the occurrence $x_{t-1}$ about a subsequent occurrence $y_t$ (the candidate effect) as:

In words, the effect information ${\rho }_e$ is the relative increase in probability of an occurrence at t when constrained by an occurrence at $t-1$, compared to when it is unconstrained. A positive effect information ${\rho }_e(x_{t-1},y_t)>0$ means that the occurrence $x_{t-1}$ makes a positive difference in bringing about $y_t$. Similarly, we compare the probability of a candidate cause $x_{t-1}$ in the cause repertoire of the occurrence $y_t$ (Equation (4)) to its corresponding probability in the unconstrained repertoire (Equation (5)). Thus, we define the cause information ${\rho }_c$ of the occurrence $y_t$ about a prior occurrence $x_{t-1}$ (the candidate cause) as:

In words, the cause information ${\rho }_c$ is the relative increase in probability of an occurrence at $t-1$ when constrained by an occurrence at t, compared to when it is unconstrained. Note that the unconstrained repertoire (Equations (5) and (6)) is an average over all possible states of the occurrence. The cause and effect information thus take all possible counterfactual states of the occurrence into account in determining the strength of constraints.

In an information-theoretic context, the formula ${\log }_2\left(p(x\mid y)/p(x)\right)$ is also known as the “pointwise mutual information” (see [43], Chapter 2). While the pointwise mutual information is symmetric, the cause and effect information of an occurrence pair $(x_{t-1},y_t)$ are not always identical, as they are defined based on the product probabilities in Equations (3) and (4). Nevertheless, ${\rho }_e$ and ${\rho }_c$ can be interpreted as the number of bits of information that one occurrence specifies about the other.

In addition to the mutual information, ${\rho }_{e/c}(x_{t-1},y_t)$ is also related to information-theoretic divergences that measure differences in probability distributions, such as the Kullback–Leibler divergence $D_{KL}\left(p\left(x\right|y)\right||p(x))$, which corresponds to an average of ${\log }_2\left(p(x\mid y)/p(x)\right)$ over all states $x\in {\mathsf{\Omega }}_X$, weighted by $p(x\mid y)$. Here, we do not include any such weighting factor, since the transition specifies which states actually occurred. While other definitions of cause and effect information are, in principle, conceivable, ${\rho }_{e/c}(x_{t-1},y_t)$ captures the notion of information in a general sense and in basic terms.

Note that ${\rho }_e>0$ is a necessary, but not sufficient, condition for $y_t$ to be an actual effect of $x_{t-1}$ and ${\rho }_c>0$ is a necessary, but not sufficient, condition for $x_{t-1}$ to be an actual cause of $y_t$. Further, ${\rho }_{c/e}=0$ if and only if conditioning on the occurrence does not change the probability of a potential cause or effect, which is always the case when conditioning on the empty set.

Occurrences $x_{t-1}$ that lower the probability of a subsequent occurrence $y_t$ have been termed “preventative causes” by some [33]. Rather than counting a negative effect information ${\rho }_e(x_{t-1},y_t)<0$ as indicating a possible “preventative effect”, we take the stance that such an occurrence $x_{t-1}$ has no effect on $y_t$, since it actually predicts other occurrences $Y_t=\neg y_t$ that did not happen. By the same logic, a negative cause information ${\rho }_c(x_{t-1},y_t)<0$ means that $x_{t-1}$ is not a cause of $y_t$ within the transition. Nevertheless, the current framework can, in principle, quantify the strength of possible “preventative” causes and effects.

In Figure 5A, the occurrence $\{{OR}_{t-1}=1\}$ raises the probability of $\{{OR}_t=1\}$, and vice versa (Figure 5B), with ${\rho }_e(\{{OR}_{t-1}=1\},\{{OR}_t=1\})={\rho }_c(\{{OR}_t=1\},\{{OR}_{t-1}=1\})=0.415$ bits. By contrast, the occurrence $\{{OR}_{t-1}=1\}$ lowers the probability of occurrence $\{{AND}_t=0\}$ and also of the second-order occurrence $\{(OR,{AND)}_t=10\}$, compared to their unconstrained probabilities. Thus, neither $\{{AND}_t=0\}$ nor $\{(OR,{AND)}_t=10\}$ can be actual effects of $\{{OR}_{t-1}=1\}$. Likewise, the occurrence $\{{OR}_t=1\}$ lowers the probability of $\{{AND}_{t-1}=0\}$, which can, thus, not be its actual cause.

Integration. A high-order occurrence must specify more information about its actual cause or effect than its parts when they are considered independently. This means that the high-order occurrence must increase the probability of its actual cause or effect beyond the value specified by its parts.

As outlined in Section 2.3, a partitioned cause or effect repertoire specifies the residual constraints of an occurrence after applying a partition $\psi$. We quantify the amount of information specified by the parts of an occurrence based on partitioned cause/effect repertoires (Equations (8) and (10)). We define the effect information under a partition $\psi$ as

and the cause information under a partition $\psi$ as

The information a high-order occurrence specifies about its actual cause or effect is integrated to the extent that it exceeds the information specified under any partition $\psi$. Out of all permissible partitions $\mathsf{\Psi }(x_{t-1},Y_t)$ (Equation (7)), or $\mathsf{\Psi }(X_{t-1},y_t)$ (Equation (9)), the partition that reduces the effect or cause information the least is denoted the “minimum information partition” (MIP) [25,26], respectively:$$\mathrm{M}\mathrm{I}\mathrm{P}={\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{m}\mathrm{i}\mathrm{n}}_{\psi \in \mathsf{\Psi }(x_{t-1},Y_t)}\left({\rho }_e(x_{t-1},y_t)-{\rho }_e{(x_{t-1},y_t)}_{\psi }\right)$$ or $$\mathrm{M}\mathrm{I}\mathrm{P}={\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{m}\mathrm{i}\mathrm{n}}_{\psi \in \mathsf{\Psi }(X_{t-1},y_t)}\left({\rho }_c(x_{t-1},y_t)-{\rho }_c{(x_{t-1},y_t)}_{\psi }\right).$$

We can, then, define the integrated effect information ${\alpha }_e$ as the difference between the effect information and the information under the MIP:

and the integrated cause information ${\alpha }_c$ as:

For first-order occurrences $x_{i,t-1}$ or $y_{i,t-1}$, there is only one way to partition the occurrence ($\psi =\left\{(x_{i,t-1},\varnothing )\right\}$ or $\psi =\left\{(y_{i,t},\varnothing )\right\}$), which is necessarily the MIP, leading to ${\alpha }_e(x_{i,t-1},y_t)={\rho }_e(x_{i,t-1},y_t)$ or ${\alpha }_c(x_{t-1},y_{i,t})={\rho }_c(x_{t-1},y_{i,t})$, respectively.

A positive integrated effect information (${\alpha }_e(x_{t-1},y_t)>0$) signifies that the occurrence $x_{t-1}$ has an irreducible effect on $y_t$, which is necessary, but not sufficient, for $y_t$ to be an actual effect of $x_{t-1}$. Likewise, a positive integrated cause information (${\alpha }_c(x_{t-1},y_t)>0$) means that $y_t$ has an irreducible cause in $x_{t-1}$, which is a necessary, but not sufficient, condition for $x_{t-1}$ to be an actual cause of $y_t$.

In our example transition, the occurrence $\{(OR,{AND)}_{t-1}=10\}$ (Figure 5C) is reducible. This is because $\{{OR}_{t-1}=1\}$ is sufficient to determine that $\{{OR}_t=1\}$ with probability 1 and $\{{AND}_{t-1}=0\}$ is sufficient to determine that $\{AND=0\}$ with probability 1. Thus, there is nothing to be gained by considering the two nodes together as a second-order occurrence. By contrast, the occurrence $\{(OR,{AND)}_t=10\}$ determines the particular past state $\{(OR,{AND)}_{t-1}=10\}$ with higher probability than the two first-order occurrences $\{{OR}_t=1\}$ and $\{{AND}_t=0\}$, taken separately (Figure 5D, right). Thus, the second-order occurrence $\{(OR,{AND)}_t=10\}$ is irreducible over the candidate cause $\{(OR,{AND)}_{t-1}=10\}$ with ${\alpha }_c(\{(OR,{AND)}_{t-1}=10\},\{(OR,{AND)}_t=10\})=0.17$ bits (see Discussion 4.4).

Exclusion: An occurrence should have at most one actual cause and one actual effect (which, however, can be multi-variate; that is, a high-order occurrence). In other words, only one occurrence $y_t\subseteq v_t$ can be the actual effect of an occurrence $x_{t-1}$, and only one occurrence $x_{t-1}\subseteq v_{t-1}$ can be the actual cause of an occurrence $y_t$.

It is possible that there are multiple occurrences $y_t\subseteq v_t$ over which $x_{t-1}$ is irreducible (${\alpha }_e(x_{t-1},y_t)>0$), as well as multiple occurrences $x_{t-1}\subseteq v_{t-1}$ over which $y_t$ is irreducible (${\alpha }_c(x_{t-1},y_t)>0$). The integrated effect or cause information of an occurrence quantifies the strength of its causal constraint on a candidate effect or cause. When there are multiple candidate causes or effects for which ${\alpha }_{c/e}(x_{t-1},y_t)>0$, we select the strongest of those constraints as its actual cause or effect (that is, the one that maximizes $\alpha$). Note that adding unconstrained variables to a candidate cause (or effect) does not change the value of $\alpha$, as the occurrence still specifies the same irreducible constraints about the state of the extended candidate cause (or effect). For this reason, we include a “minimality” condition, such that no subset of an actual cause or effect should have the same integrated cause or effect information. This minimality condition between overlapping candidate causes or effects is related to the third clause (“AC3”) in the various Halpern–Pearl (HP) accounts of actual causation [20,21], which states that no subset of an actual cause should also satisfy the conditions for being an actual cause. Under uncertainty about the causal model, or other practical considerations, the minimality condition could, in principle, be replaced by a more elaborate criterion, similar to, for example, the Akaike information criterion (AIC) that weighs increases in causal strength, as measured here, against the number of variables included in the candidate cause or effect.

We define the irreducibility of an occurrence as its maximum integrated effect (or cause) information over all candidate effects (or causes), $${\alpha }_e^{max}\left(x_{t-1}\right)=\underset{y_t\subseteq v_t}{max}{\alpha }_e(x_{t-1},y_t),$$ and $${\alpha }_c^{max}\left(y_t\right)=\underset{x_{t-1}\subseteq v_{t-1}}{max}{\alpha }_c(x_{t-1},y_t).$$

Considering the empty set as a possible cause or effect guarantees that the minimal value that ${\alpha }^{max}$ can take is 0. Accordingly, if ${\alpha }^{max}=0$, then the occurrence is said to be reducible, and it has is no actual cause or effect.

For the example in Figure 2A, $\{{OR}_t=1\}$ has two candidate causes with ${\alpha }_c^{max}\left(\{{OR}_t=1\}\right)=0.415$ bits, the first-order occurrence $\{{OR}_{t-1}=1\}$ and the second-order occurrence $\{(OR,{AND)}_{t-1}=10\}$. In this case, $\{{OR}_{t-1}=1\}$ is the actual cause of $\{{OR}_t=1\}$, by the minimality condition across overlapping candidate causes.

The exclusion principle avoids causal over-determination, which arises from counting multiple causes or effects for a single occurrence. Note, however, that symmetries in $G_u$ can give rise to genuine indeterminism about the actual cause or effect (see Results 3). This is the case if multiple candidate causes (or effects) are maximally irreducible and they are not simple sub- or super-sets of each other. Upholding the causal exclusion principle, such degenerate cases are resolved by stipulating that the one actual cause remains undetermined between all minimal candidate causes (or effects).

To summarize, we formally translate the five causal principles of IIT into the following requirements for actual causation:

Having established the above causal principles, we now formally define the actual cause and the actual effect of an occurrence within a transition $v_{t-1}\prec v_t$ of the dynamical causal network $G_u$:

Within a transition $v_{t-1}\prec v_t$ of a dynamical causal network $G_u$, the actual cause of an occurrence $y_t\subseteq v_t$ is an occurrence $x_{t-1}\subseteq v_{t-1}$ which satisfies the following conditions:

1. The integrated cause information of $y_t$ over $x_{t-1}$ is maximal$${\alpha }_c(x_{t-1},y_t)={\alpha }^{max}\left(y_t\right);\mathit{a}\mathit{n}\mathit{d}$$

2. No subset of $x_{t-1}$ satisfies condition (1)$${\alpha }_c(x_{t-1}^{\prime },y_t)={\alpha }^{max}\left(y_t\right)\Rightarrow x_{t-1}^{\prime }\not\subset x_{t-1}.$$

Define the set of all occurrences that satisfy the above conditions as $x^{*}\left(y_t\right)$. As an occurrence can have, at most, one actual cause, there are three potential outcomes:

1. If $x^{*}\left(y_t\right)=\left\{x_{t-1}\right\}$, then $x_{t-1}$ is the actual cause of $y_t$;

2. if $|x^{*}\left(y_t\right)|>1$ then the actual cause of $y_t$ is indeterminate; and

3. if $x^{*}\left(y_t\right)=\{\varnothing \}$, then $y_t$ has no actual cause.

Within a transition $v_{t-1}\prec v_t$ of a dynamical causal network $G_u$, the actual effect of an occurrence $x_{t-1}\subseteq v_{t-1}$ is an occurrence $y_t\subseteq v_t$ which satisfies the following conditions:

1. The integrated effect information of $x_{t-1}$ over $y_t$ is maximal $${\alpha }_e(x_{t-1},y_t)={\alpha }^{max}\left(x_{t-1}\right);\mathit{a}\mathit{n}\mathit{d}$$

2. No subset of $y_t$ satisfies condition (1) $${\alpha }_e(x_{t-1},y_t^{\prime })={\alpha }^{max}\left(x_{t-1}\right)\Rightarrow y_t^{\prime }\not\subset y_t.$$

Define the set of all occurrences that satisfy the above conditions as $y^{*}\left(x_{t-1}\right)$. As an occurrence can have, at most, one actual effect, there are three potential outcomes:

1. If $y^{*}\left(x_{t-1}\right)=\left\{y_t\right\}$, then $y_t$ is the actual effect of $x_{t-1}$;

2. if $|y^{*}\left(x_{t-1}\right)|>1$ then the actual effect of $x_{t-1}$ is indeterminate; and

3. if $y^{*}\left(x_{t-1}\right)=\{\varnothing \}$, then $x_{t-1}$ has no actual effect.

Based on Definitions 1 and 2:

Within a transition $v_{t-1}\prec v_t$ of a dynamical causal network $G_u$, a causal link is an occurrence $x_{t-1}\subseteq v_{t-1}$ with ${\alpha }_e^{max}\left(x_{t-1}\right)>0$ and actual effect $y^{*}\left(x_{t-1}\right)$,$$x_{t-1}\to y^{*}\left(x_{t-1}\right),$$or an occurrence $y_t\subseteq v_t$ with ${\alpha }_c^{max}\left(y_t\right)>0$ and actual cause $x^{*}\left(y_t\right)$,$$x^{*}\left(y_t\right)\leftarrow y_t.$$

An integrated occurrence defines a single causal link, regardless of whether the actual cause (or effect) is unique or indeterminate. When the actual cause (or effect) is unique, we sometimes refer to the actual cause (or effect) explicitly in the causal link, $x_{t-1}\leftarrow y_t$ (or $x_{t-1}\to y_t$). The strength of a causal link is determined by its ${\alpha }_e^{max}$ or ${\alpha }_c^{max}$ value. Reducible occurrences (${\alpha }^{max}=0$) cannot form a causal link.

For a transition $v_{t-1}\prec v_t$ of a dynamical causal network $G_u$, the causal account, $𝒞(v_{t-1}\prec v_t)$, is the set of all causal links $x_{t-1}\to y^{*}\left(x_{t-1}\right)$ and $x^{*}\left(y_t\right)\leftarrow y_t$ within the transition.

Under this definition, all actual causes and actual effects contribute to the causal account $𝒞(v_{t-1}\prec v_t)$. Notably, the fact that there is a causal link $x_{t-1}\to y_t$ does not necessarily imply that the reverse causal link $x_{t-1}\leftarrow y_t$ is also present, and vice versa. In other words, just because $y_t$ is the actual effect of $x_{t-1}$, the occurrence $x_{t-1}$ does not have to be the actual cause of $y_t$. It is, therefore, not redundant to include both directions in $𝒞(v_{t-1}\prec v_t)$, as illustrated by the examples of over-determination and prevention in the Results section (see, also, Discussion 4.2).

Figure 6 shows the entire causal account of our example transition. Intuitively, in this simple example, $\{{OR}_{t-1}=1\}$ has the actual effect $\{{OR}_t=1\}$ and is also the actual cause of $\{{OR}_t=1\}$, and the same for $\{{AND}_{t-1}=0\}$ and $\{AND=0\}$. Nevertheless, there is also a causal link between the second-order occurrence $\{(OR,{AND)}_t=10\}$ and its actual cause $\{(OR,{AND)}_{t-1}=10\}$, which is irreducible to its parts, as shown in Figure 5D (right). However, there is no complementary link from $\{(OR,{AND)}_t=10\}$ to $\{(OR,{AND)}_{t-1}=10\}$, as it is reducible (Figure 5C, right). The causal account, shown in Figure 6, provides a complete causal explanation for “what happened” and “what caused what” in the transition $\{(OR,{AND)}_{t-1}=10\}\prec \{(OR,{AND)}_t=10\}$.

Similar to the notion of system-level integration in IIT [25,26], the principle of integration can also be applied to the causal account as a whole, not only to individual causal links (see Appendix A). In this way, it is possible to evaluate to what extent the transition $v_{t-1}\prec v_t$ is irreducible to its parts, which is quantified by $𝒜(v_{t-1}\prec v_t)$.

In summary, the measures defined in this section provide the means to exhaustively assess “what caused what” in a transition $v_{t-1}\prec v_t$, and to evaluate the strength of specific causal links of interest under a particular set of background conditions, $U=u$.

Software to analyze transitions in dynamical causal networks with binary variables is freely available within the “PyPhi” toolbox for integrated information theory [44] at https://github.com/wmayner/pyphi, including documentation at https://pyphi.readthedocs.io/en/stable/examples/actual_causation.html.

3. Results

In the following, we will present a series of examples to illustrate the quantities and objects defined in the theory section and address several dilemmas taken from the literature on actual causation. While indeterminism may play a fundamental role in physical causal models, the existing literature on actual causation largely focuses on deterministic problem cases. For ease of comparison, most causal networks analyzed in the following are, thus, deterministic, corresponding to prominent test cases of counterfactual accounts of actual causation (e.g., [8,11,19,20,21,45]).

3.1. Same Transition, Different Mechanism: Disjunction, Conjunction, Bi-Conditional, and Prevention

Figure 7 shows four causal networks of different types of logic gates with two inputs each, all transitioning from the input state $v_{t-1}=\{AB=11\}$ to the output state $v_t=\{C=1\}$, $\{D=1\}$, $\{E=1\}$, or $\{F=1\}$. From a dynamical point of view, without taking the causal structure of the mechanisms into account, the same occurrences happen in all four situations. However, analyzing the causal accounts of these transitions reveals differences in the number, type, and strength of causal links between occurrences and their actual causes or effects.

Disjunction: The first example (Figure 7A, OR-gate), is a case of symmetric over-determination ([7], Chapter 10): each input to C would have been sufficient for $\{C=1\}$, yet both $\{A=1\}$ and $\{B=1\}$ occurred at $t-1$. In this case, each of the inputs to C has an actual effect, $\{A=1\}\to \{C=1\}$ and $\{B=1\}\to \{C=1\}$, as they raise the probability of $\{C=1\}$ when compared to its unconstrained probability. The high-order occurrence $\{AB=11\}$, however, is reducible (with ${\alpha }_e=0$). While both $\{A=1\}$ and $\{B=1\}$ have actual effects, by the causal exclusion principle, the occurrence $\{C=1\}$ can only have one actual cause. As both $\{A=1\}\leftarrow \{C=1\}$ and $\{B=1\}\leftarrow \{C=1\}$ have ${\alpha }_c={\alpha }_c^{max}=0.415$ bits, the actual cause of $\{C=1\}$ is either $\{A=1\}$ or $\{B=1\}$, by Definition 1; which of the two inputs it is remains undetermined, since they are perfectly symmetric in this example. Note that $\{AB=11\}\leftarrow \{C=1\}$ also has ${\alpha }_c=0.415$ bits, but $\{AB=11\}$ is excluded from being a cause by the minimality condition.

Conjunction: In the second example (Figure 7B, AND-gate), both $\{A=1\}$ and $\{B=1\}$ are necessary for $\{D=1\}$. In this case, each input alone has an actual effect, $\{A=1\}\to \{C=1\}$ and $\{B=1\}\to \{C=1\}$ (with higher strength than in the disjunctive case); here, also, the second-order occurrence of both inputs together has an actual effect, $\{AB=11\}\to \{D=1\}.$ Thus, there is a composition of actual effects. Again, the occurrence $\{D=1\}$ can only have one actual cause; here, it is the second-order cause $\{AB=11\}$, the only occurrence that satisfies the conditions in Definition 1 with ${\alpha }_c={\alpha }_c^{max}=2.0$.

The two examples in Figure 7A,B are often referred to as the disjunctive and conjunctive versions of the “forest-fire” example [12,20,21], where lightning and/or a match being dropped result in a forest fire. In the case that lightning strikes and the match is dropped, $\{A=1\}$ and $\{B=1\}$ are typically considered two separate (first-order) causes in both the disjunctive and conjunctive version (e.g., [20]). This result is not a valid solution within our proposed account of actual causation, as it violates the causal exclusion principle. We explicitly evaluate the high-order occurrence $\{AB=11\}$ as a candidate cause, in addition to $\{A=1\}$ and $\{B=1\}$. In line with the distinct logic structure of the two examples, we identify the high-order occurrence $\{AB=11\}$ as the actual cause of $\{D=1\}$ in the conjunctive case, while we identify either $\{A=1\}$ or $\{B=1\}$ as the actual cause of $\{C=1\}$ in the disjunctive case, but not both. By separating actual causes from actual effects, acknowledging causal composition, and respecting the causal exclusion principle, our proposed causal analysis can illuminate and distinguish all situations displayed in Figure 7.

Bi-conditional: The significance of high-order occurrences is further emphasized by the third example (Figure 7C), where E is a “logical bi-conditional” (an XNOR) of its two inputs. In this case, the individual occurrences $\{A=1\}$ and $\{B=1\}$ by themselves make no difference in bringing about $\{E=1\}$; their effect information is zero. For this reason, they cannot have actual effects and cannot be actual causes. Only the second-order occurrence $\{AB=11\}$ specifies $\{E=1\}$, which is its actual effect $\{AB=11\}\to \{E=1\}$. Likewise, $\{E=1\}$ only specifies the second-order occurrence $\{AB=11\}$, which is its actual cause $\{AB=11\}\leftarrow \{E=1\}$, but not its parts taken separately. Note that the causal strength in this example is lower than in the case of the AND-gate, since, everything else being equal, $\{D=1\}$ is, mechanistically, a less-likely output than $\{E=1\}$.

Prevention: In the final example, Figure 7D, all input states but $\{AB=10\}$ lead to $\{F=1\}$. Here, $\{B=1\}\to \{F=1\}$ and $\{B=1\}\leftarrow \{F=1\}$, whereas $\{A=1\}$ does not have an actual effect and is not an actual cause. For this reason, the transition $v_{t-1}\prec v_t$ is reducible ($𝒜(v_{t-1}\prec v_t)=0$, see Appendix A), since A could be partitioned away without loss. This example can be seen as a case of prevention: $\{B=1\}$ causes $\{F=1\}$, which prevents any effect of $\{A=1\}$. In a popular narrative accompanying this example, $\{A=1\}$ is an assassin putting poison in the King’s tea, while a bodyguard administers an antidote $\{B=1\}$, and the King survives $\{F=1\}$ [12]. The bodyguard thus “prevents” the King’s death (However, the causal model is also equivalent to an OR-gate, as can be seen by switching the state labels of A from ‘0’ to ‘1’ and vice versa. The discussed transition would correspond to the case of one input to the OR-gate being ‘1’ and the other ‘0’. As the OR-gate switches on (‘1’) in this case, the ‘0’ input has no effect and is not a cause). Note that the causal account is state-dependent: For a different transition, A may have an actual effect or contribute to an actual cause; if the bodyguard does not administer the antidote ($\{B=0\})$, whether the King survives depends on the assassin (the state of A).

Taken together, the above examples demonstrate that the causal account and the causal strength of individual causal links within the account capture differences in sufficiency and necessity of the various occurrences in their respective transitions. Including both actual causes and effects, moreover, contributes to a mechanistic understanding of the transition, since not all occurrences at $t-1$ with actual effects end up being actual causes of occurrences at t.

3.2. Linear Threshold Units

A generalization of simple, linear logic gates, such as OR- and AND-gates, are binary linear threshold units (LTUs). Given n equivalent inputs $V_{t-1}=\{V_{1,t-1},V_{2,t-1},\dots ,V_{n,t-1}\}$ to a single LTU $V_t$, $V_t$ will turn on (‘1’) if the number of inputs in state ‘1’ exceeds a given threshold k,

LTUs are of great interest, for example, in the field of neural networks, since they comprise one of the simplest model mechanisms for neurons; capturing the notion that a neuron fires if it received sufficient synaptic inputs. One example is a Majority-gate, which outputs ‘1’ if and only if more than half of its inputs are ‘1’.

Figure 8A displays the causal account of a Majority-gate M with four inputs for the transition $v_{t-1}=\{ABCD=1110\}\to v_t=\{M=1\}$. All of the inputs in state ‘1’, as well as their high-order occurrences, have actual effects on $\{M=1\}$. Occurrence $\{D=0\}$, however, does not work towards bringing about $\{M=1\}$: It reduces the probability for $\{M=1\}$ and, thus, does not contribute to any actual effects or the actual cause. As with the AND-gate in the previous section, there is a composition of actual effects in the causal account. Yet, there is only one actual cause, $\{ABC=111\}\leftarrow \{M=1\}$. In this case, it happens to be that the third-order occurrence $\{ABC=111\}$ is minimally sufficient for $\{M=1\}$—no smaller set of inputs would suffice. Note, however, that the actual cause is not determined based on sufficiency, but because $\{ABC=111\}$ is the set of nodes maximally constrained by the occurrence $\{M=1\}$. Nevertheless, causal analysis, as illustrated here, will always identify a minimally sufficient set of inputs as the actual cause of an LTU $v_t=1$, for any number of inputs n and any threshold k. Furthermore, any occurrence of input variables $x_{t-1}\subseteq v_{t-1}$ with at most k nodes, all in state ‘1’, will be irreducible, with the LTU $v_t=1$ as their actual effect.

Consider a dynamical causal network $G_u$, such that $V_t=\left\{Y_t\right\}$ is a linear threshold unit with n inputs and threshold $k\le n$, and $V_{t-1}$ is the set of n inputs to $Y_t$. For a transition $v_{t-1}\prec v_t$, with $y_t=1$ and $\sum v_{t-1}\ge k$, the following holds: