SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Consolidation is a powerful tool for transforming SCMs, while preserving the causal aspect of interventions. In addition, consolidating SCMs can lead to more general models. For example, recall the tool wear example of Sec. 4.2. While the initial causal graph operates on discrete time steps our consolidated function provides a continuous relaxation of the causal process. We can evaluate it at any point in time  $t \in \mathbb{R}$  and are no longer dependent on the day-to-day basis which was modeled by the initial graph. Additionally, the consolidated equation of our motivating example of rows of dominoes (compare Appendix D.1), yields a generalized formula that is independent of the actual number of dominoes that compose the row, by making use of first-order quantifiers. In our discussion of Sec. 4 we saw that we can further benefit from consolidation in all cases where the initial SCM does not already represent the smallest possible causal model. Lastly, these simplifications align with our goal of making SCMs more interpretable. Consider that the initial domino SCM only provides a 'local' view on the system, by only providing equations for every individual stone, "If stone A falls it pushes over stone B, except in the case of an intervention. If stone B falls, ..." and so on. The equation of the consolidated SCM can be directly translated into a single natural language sentence, e.g. "The last domino will fall, if the first domino is pushed over, except in the case of holding onto or pushing over a stone along the way", capturing the causal mechanisms of the system much more intuitively. Our last example of Sec. 4.3 strikingly revealed the sub-optimal, greedy agent behaviour in a game setting. While we illustrated examples that are well suited for consolidation, we are positively inclined to expect consolidation to be helpful towards a broad range of applications.
Limitations and Broader Impact. Throughout the paper we considered exact consolidations, in that Def. 2 requires strict equality between  $\rho_{\mathbf{E}}(\mathbf{U},\mathbf{I})$  and  $\mathrm{P_E^I}$ . This assumption might be met in logic and idealized scenarios, but may hinder consolidation of SCM in other applications due to noise inherent to the system. The definition might be relaxed by allowing for small deviations of  $\rho_{\mathbf{E}}$  from the distribution of the unconsolidated SCM. Thus, relaxing the strict equality  $\mathrm{P}_{X_{\mathrm{E}}^{\mathrm{I}}} = \mathrm{P}_{\mathrm{E}}^{\mathrm{I}}$  with  $|\mathrm{P}_{X_{\mathrm{E}}^{\mathrm{I}}} - \mathrm{P}_{\mathrm{E}}^{\mathrm{I}}| < \epsilon$  for some small  $\epsilon >0$ , provides a relaxed consolidation constraint for noisy systems. SCM constitute a well suited framework for representing causal knowledge in the form of graphical models. The ability to trace effects through structural equations that yield explanations about the role of variables within causal models is required to make results accessible to non-experts. Actual causation, and only recently, causal abstractions and constrained causal models have come to attention in the field of causality [Halpern, 2016, Zennaro et al., 2023, Beckers and Halpern, 2019, Blom et al., 2020] and might be beneficial for future works on consolidation. Apart from computational advantages, consolidation of SCMs presents itself as a method that enables researchers to break down complex structures and present aspects of causal systems in a broadly accessible manner. Without such tools, SCMs run the danger of being only useful to specialized experts.
The authors acknowledge the support of the German Science Foundation (DFG) project "Causality, Argumentation, and Machine Learning" (CAML2, KE 1686/3-2) of the SPP 1999 "Robust Argumentation Machines" (RATIO). The work was supported by the Hessian Ministry of Higher Education Research, Science and the Arts (HMWK) via the DEPTH group CAUSE of the Hessian Center for AI (hessian.ai). This work was partly funded by the ICT-48 Network of AI Research Excellence Center "TAILOR" (EU Horizon 2020, GA No 952215) and by the Federal Ministry of Education and Research (BMBF; project "PlexPlain", FKZ 01IS19081). It benefited from the Hessian research priority programme LOEWE within the project WhiteBox, the HMWK cluster project "The Third Wave of AI." and the Collaboration Lab "AI in Construction" (AICO) of the TU Darmstadt and HOCHTIEF. We thank the anonymous reviewers for their valuable feedback and constructive criticism, which significantly improved this paper. We sincerely appreciate their time and expertise in evaluating our research. We acknowledge the use of ChatGPT for restructuring some sentences while writing the paper.
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"Do Not Marginalize Mechanisms, Rather Consolidate!"
A Evaluation of Partitioned SCM
A partitioned SCM  $\mathcal{M}_{\mathcal{A}}$  consists of several sub SCM  $\mathcal{M}_{\mathbf{A}}$ , that, in sum, cover all variables and structural equations of an initial SCM  $\mathcal{M}$ . Thus, evaluation of a partitioned SCM yields the same set of values  $\mathbf{v} \in \mathbf{V}$  as the original  $\mathcal{M}$ . Similar to the evaluation of structural equation in the initial  $\mathcal{M}$ , sub SCM need to be evaluated in a specific order to guarantee all  $\mathbf{u} \in \mathcal{M}_{\mathbf{U}}^{\prime}$  exist. As such, sub SCM can be considered multivariate variables that establish another high-level DAG. The evaluation order is determined via the relation  $\mathrm{R_X}$  as defined in Sec. 3.1 and depends on the graph partition  $\mathcal{A}$  and the order of  $\mathbf{X}$  imposed by the initial SCM.
Algorithm 2 Evaluation of partitioned SCM
1: procedure PARTITIONEDSCMEVAL( $\mathcal{M}_{\mathcal{A}}$ ,  $\mathbf{u}$ ,  $\mathbf{I}$ )  
2:  $\mathbf{x} \gets \mathbf{u}$  ▷  $\mathbf{x}$  will gradually collect all values  $\mathbf{x} \in \mathbf{X}$  of  $\mathcal{M}$   
3: for  $\mathbf{A}$  in sort( $\mathcal{A}$ ,  $\mathbf{R}_{\mathbf{X}}$ ) do ▷ Sort clusters by strict partial order imposed by  $\mathcal{M}$   
4:  $\mathcal{M}_{\mathbf{A}}^{\prime} \gets \mathcal{M}_{\mathbf{A}^{\prime}}^{\prime} \in \mathcal{M}_{\mathcal{A}}$  where  $\mathbf{A}^{\prime} = \mathbf{A}$   
5:  $\mathbf{u}^{\prime} \gets \{x_{i} \in \mathbf{x} \mid \mathbf{X}_{i} \in \mathcal{M}_{\mathbf{U}}^{\prime}\}$   
6:  $\mathbf{I}^{\prime} \gets \psi_{\mathbf{A}}(\mathbf{I})$   
7:  $\mathbf{v} = \mathcal{M}_{\mathbf{A}}^{\prime}(\mathbf{u}^{\prime})$   
8:  $\mathbf{x} = \mathbf{x} \cup \mathbf{v}$   
9: end for  
10:  $\mathbf{v} = \{x_{i} \in \mathbf{x} \mid \mathbf{X}_{i} \in \mathcal{M}_{\mathbf{V}}^{\prime}\}$  ▷ Filter all  $\mathbf{u} \in \mathbf{U}$  to get  $\mathbf{v} \in \mathbf{V}$   
11: return  $\mathbf{v}$   
12: end procedure
Algorithm 2 shows the evaluation of partitioned SCM, where  $\mathcal{M}_{\mathcal{A}}$  is the partitioned SCM we want to evaluate,  $\mathbf{u}$  are the values of exogenous variables to the initial model  $\mathcal{M}$  and  $\mathbf{I}$  is the set of applied interventions. The outcomes of sub SCM that are not related via  $\mathrm{R_X}$  are invariant to the evaluation order among each other. Even though  $\mathrm{R_X}$  defines the ordering of sub SCM only up to some partial order,  $\operatorname{sort}(\mathcal{A}, \mathrm{R_X})$  can pick any total ordering that is valid with  $\mathrm{R_X}$ .
Proof 1 (Consistency of Partitioned SCM Evaluation) Evaluations of  $\mathcal{M}_{\mathbf{A}}^{\prime}$  every, in step 7, compute all variables  $\mathbf{V}_i \in \mathbf{A}$  by evaluating  $f_i$  of the original SCM, yielding the same values as the evaluation of  $\mathbf{A}$  in  $\mathcal{M}$ . Therefore  $\mathrm{P}_{\mathcal{M}_{\mathbf{A}}^{\prime}} = \mathrm{P}_{\mathcal{M}_{\mathbf{A}}}$ . By Def. 4 every variable  $V \in \mathbf{V}$  is contained within some sub SCM  $\mathcal{M}_{\mathbf{A}}^{\prime}$ . The evaluation of PartitionedSCMEval is complete, in the sense that all  $\mathbf{V} = \bigcup \mathcal{A} = \bigcup_{\mathbf{A} \in \mathcal{A}} \mathbf{A}$  are evaluated, as the evaluation of all  $\mathcal{M}_{\mathbf{A}}^{\prime} \in \mathcal{M}_{\mathcal{A}}$  is guaranteed by iterating over all  $\mathbf{A}$  in step 2. Finally  $\mathrm{P}_{\mathcal{M}_{\mathbf{A}}^{\prime}} = \bigcup_{\mathbf{A} \in \mathcal{A}} \mathrm{P}_{\mathcal{M}_{\mathbf{A}}^{\prime}} = \bigcup_{\mathbf{A} \in \mathcal{A}} \mathrm{P}_{\mathcal{M}_{\mathbf{A}}} = \mathrm{P}_{\mathcal{M}_{\mathbf{V}}}$ .
B Complexity reduction in function composition
Reduction of encoding length might vary depending on the type and structure of the equations under consideration. No compression of structural equation is gained when the system of consolidated equations is already minimal. Compression of equation to an identity function is showcased in the following.
B.1 Compression of chained inverses
Reduction to constant complexity for the unintervened system is reached in the case of  $f_{B} = f_{A}^{-1}$ . Consider the equation chain of  $X \to A \to B$  with  $A$  getting marginalized. Immediately  $f_{B}^{\prime} \coloneqq f_{B} \circ f_{A} = f_{A}^{-1} \circ f_{A} = \mathrm{Id}$  follows. Therefore,  $B \coloneqq X$ , which is a single assignment of the value(s) of  $X$  into  $B$ . Remaining complexity within the consolidated function is then only due to conditional branching in cases of  $do(A = a)$ ,  $do(B = b) \in \mathbf{I}$ .
B.2 Matrix composition is not sufficient for compressing equations
The operation of matrix multiplication, as a way of expressing composition of linear functions, stays within the class of matrices. Matrix multiplication, therefore, serves as a possible candidate to be considered when consolidating equations and reducing the encoding length of a linear structural systems. When written down an 'high-level' view, matrices can be expressed in terms of single variables  $A, B \in \mathbb{R}^{M \times N}$  and matrix multiplication  $\times : \mathbb{R}^{M \times N} \times \mathbb{R}^{N \times O} \to \mathbb{R}^{M \times O}$ . Assuming equations  $f_{Y} \coloneqq A \times X$  and  $f_{Z} \coloneqq B \times X$ , we can reduce the length of the composed equation  $f_{Z}' \coloneqq A \times B \times X$  by multiplying the matrices  $A$  and  $B$  together,  $f_{i} = C \times X$  with  $C = A \times B$ . While we effectively reduced the number of high-level symbols written in the equation, we are hiding computational complexity in the structure of the matrix  $C$ . The following simple counterexample demonstrates a situation where the size, as well as, the number of non-zero entries even increases:
$$
\begin{array}{c c} C & A \\ \left[ \begin{array}{c c c} 0 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 1 & 1 \end{array} \right] & = \left[ \begin{array}{c c} 0 & 1 \\ 0 & 1 \\ 0 & 1 \end{array} \right] \end{array} \times \left[ \begin{array}{c c c} 0 & 0 & 0 \\ 0 & 1 & 1 \end{array} \right]
$$
Thus, proving that pure matrix multiplication, is not suitable to keep, or even minimize, the size of composed function representations.
B.3 Compression over Finite Discrete Domains
Consolidation may reduce the number of variables within a graph, but burdens the remaining equations with the complexity of the consolidated variables. Without the need to explicitly compute values of consolidated variables, we might leverage cancellation effects to simplify equations, as outlined in the main paper. In terms of compression, no guarantees can be given in the general case. However, we will now show, that the often considered case of chained maps between finite discrete domains simplifies or at least preserves complexity.
The cardinality of the image of a deterministic function  $f: \mathcal{X} \to \mathcal{Y}$  between two finite discrete sets  $\mathcal{X}, \mathcal{Y}$  is bounded by the cardinality of its domain:  $|\operatorname{Img}(f)| \leq |\operatorname{Dom}(f)| \leq |\mathcal{X}|$ , where  $\operatorname{Img}(f)$  is the image and  $\operatorname{Dom}(f)$  the domain of  $f$ . In particular, the strict inequality  $|\operatorname{Img}(f)| < |\operatorname{Dom}(f)|$  holds for all non-injective maps. Function composition may further reduce the 'effective' domain  $\operatorname{Dom}_{\text{effective}}(f)$  of a function, by only considering values of the image of the previous map as inputs to the next function. In contrast considering to all possible values of  $\mathcal{X}$  in the case of the non-composed map, the image of the previous function may only be a subset of  $\mathcal{X}$ . Therefore,  $f_2 \circ f_1 \Rightarrow |\operatorname{Img}_{\text{effective}}(f_2)| \leq |\operatorname{Dom}_{\text{effective}}(f_2)| = |\operatorname{Img}(f_1)| \leq |\operatorname{Dom}(f_1)|$ . In particular, the effective image of a composition chain  $f_n \circ \dots \circ f_1$  is bounded by the function with the smallest image:  $|\operatorname{Img}_{\text{effective}}(f_n \circ \dots \circ f_1)| \leq \min |\operatorname{Img}(f_i)|$ . Thus, equation chains over finite discrete domains strictly preserve or reduce the effective size of the image, allowing for a possibly simpler combined representation in comparison to representing the functions individually.
C Reparameterization of non-deterministic structural equations.
Consolidation of structural equations might lead to duplication of non-deterministic terms within consolidated systems. For example when consolidating fork structures (compare to Sec. 4.1). Without further precautions, different values might be sampled from the duplicated non-deterministic equations. An example where consolidating a variable  $B$  with a non-deterministic equation  $f_{B}$  (indicated by a squiggly line) leads to inconsistent behaviour is shown in 6. In  $\mathcal{M}_1$ ,  $C$  and  $D$  both copy on the value of  $B$ . Therefore,  $c = d$  yields always.  $\mathcal{M}_{1'}$  shows a graph where  $B$  is consolidated from  $\mathcal{M}_1$ . As a result the non-deterministic equation  $f_{B}$  is duplicated into the equations of  $C$  and  $D$ , such that  $f_{C} := \operatorname{Bern}(A)$  and  $f_{D} := \operatorname{Bern}(A)$ . Within the consolidated model  $\mathcal{M}_{1'}$  different values might be sampled from the different noise terms  $\operatorname{Bern}(A)$  in  $f_{C}$  and  $f_{D}$ . Consequently  $c \neq d$  might occur in  $\mathcal{M}_{1'}$ . To obtain consistent behaviour with the initial  $\mathcal{M}_1$ , we need to ensure agreement about the value of  $\operatorname{Bern}(A)$  across all instances of the duplicated equation. To do so, we reparameterize  $\mathcal{M}_1$  and explicitly store a fixed value, sampled from  $\operatorname{Bern}(A)$ , into a new exogenous variable  $R$ . The equation  $f_{B}$  is then reparameterized into a deterministic structural equation taking the variable  $R$  as an additional argument, resulting in  $\mathcal{M}_2$ . When consolidating  $B$  within  $\mathcal{M}_2$ , all instances of  $f_{B}$  now yield the same value, as the noise term is fixed via  $R$  and finally  $\mathrm{P}_{\mathcal{M}_2'} = \mathrm{P}_{\mathcal{M}_1}$ .
(1) Non-Deterministic Model. Yields  $C = D$
(2) Inconsistent Behaviour:  $\diamondsuit C\neq D$
Figure 6: Reparameterization of non-deterministic models. The SCM  $\mathcal{M}_1$  contains a non-deterministic equation  $B := \operatorname{Bern}(A)$  (marked with a squiggly line). With  $C := B$  and  $D := B$ ,  $\mathcal{M}_1$  always yields  $C = D$ . Simply consolidating (or marginalizing)  $B$  creates a model  $\mathcal{M}_{1'}$  with  $C := \operatorname{Bern}(A)$  and  $D := \operatorname{Bern}(A)$ , such that possibly  $C \neq D$ . Reparameterizing  $f_B$  by introducing an exogenous random variable  $R := \mathcal{U}(0,1)$  and  $B := A < R$ , yields the SCM  $\mathcal{M}_2$  with only deterministic equations. Consolidating (or marginalizing)  $B$  in  $\mathcal{M}_2$  leads to  $\mathcal{M}_{2'}$  where  $C := A < R$  and  $D := A < R$ , thus always  $C = D$ .
(3) Reparameterization to Deterministic Eqs.
(4) Consistent Model Behaviour:  $C = D$
D Consolidation Examples
In this section we show further detailed applications of consolidation. Section D.1 presents the worked out consolidation of the dominoes motivating example of the paper, with regard to generalizing abilities of consolidates models. Section D.2 considers consolidation of the classical firing squad example. In contrast to the other examples, we focus on consolidating graphs with multiple edges in the causal graph. Lastly we provide the causal graph and structural equations of the game agent policy discussed in the main paper, in Section D.4.
D.1 Motivating Example: Dominoes
While we applied consolidation to a particular SCMs in the main paper, we will discuss the motivating example with focus on obtaining representations that cover generalize over populations of SCM. We demonstrate this on the particular example of a rows of dominoes, as a simple SCM with highly homogenous structure. Regardless of whether the SCM is obtained by using methods for direct identification of causal graphs from image data, as presented by Brehmer et al. [2022], or abstracting physical simulation using  $\tau$ -abstractions [Beckers and Halpern, 2019]; we assume to be provided with a binary representation of the domino stones. The state of every domino  $S_{i}$  indicates whether it is standing up or getting pushed over. In this case, the structural equations for all dominoes are the same:  $f_{i} := S_{i-1}$ . As a result tipping over the first stone in a row will lead to all stones falling. Also, we are only interested in the final outcome of the chain. That is, whether the last stone will fall or not ( $\mathbf{E} = \{S_{n}\}$ ). Again, we use consolidation to collapse the structural equations in the unintervened case:  $S_{n} := f_{n} \circ \dots \circ f_{1} := S_{1}$ . We consider a single active allowed intervention of holding up any of the dominoes or tipping it over,  $\mathcal{I} = \{do(S_{i} = 0), do(S_{i} = 1)\}$ . Upon evaluation, the unconsolidated model needs to check for every domino if it is being intervened or not, requiring  $n$  conditional branches. Using the fact that perfect interventions 'overwrite' the variable state for the following dominoes, we introduce a first order quantifier that handles all intervention in a unified way. Finally, by combining the formulas of the intervened and unintervened case, we find the following simple equation:
$$
S _ {n} := \left\{ \begin{array}{l l} x _ {i} & \text {i f} \exists d o (S _ {i} = x _ {i}) \in \mathbf {I} \\ S _ {1} & \text {e l s e} \end{array} \right.
$$
The resulting equation no longer has a notion of the actual number of dominoes and, in fact, it is invariant to it. We realise that introducing the first-order for-all  $\forall$  and exists  $\exists$  quantifiers allows for a unified representation of arbitrary chains of dominoes. Similar observations are discussed in Peters and Halpern [2021] and Halpern and Peters [2022] which introduce generalized SEM (GSEM). As intermediate the equations are no longer computed explicitly, the structural equations of consolidated models for different row lengths only differ in the set of allowed interventions  $\mathcal{I}$ . That is, for a row of three domino stones  $\mathcal{I} = \{do(V_1 = v_1), do(V_2 = v_1), do(V_3 = v_1)\}$ , while for four stones the additional  $do(V_4 = v_1)$  is defined. As set out in the introduction of this paper, we consider