A Counterfactual Semantics for Hybrid Dynamical Systems
Authors: Andy Zane
Dmitry Batenkov
Rafal Urbaniak
Jeremy D Zucker
Sam Witty
Venue: NeurIPS 2025 poster ()
Keywords: counterfactual
causal
hybrid dynamical systems
differential equations
intervention
twin world
Primary area: probabilistic_methods
TL;DR: We provide a counterfactual semantics for hybrid dynamical systems and prove that intervention preserves sufficient conditions for solution existence, uniqueness, and measurability.

Abstract: Models of hybrid dynamical systems are widely used to answer questions about the causes and effects of dynamic events in time. Unfortunately, existing causal reasoning formalisms lack support for queries involving the dynamically triggered, discontinuous interventions that characterize hybrid dynamical systems. This mismatch can lead to ad-hoc and error-prone causal analysis workflows in practice. To bridge the gap between the needs of hybrid systems users and current causal inference capabilities, we develop a rigorous counterfactual semantics by formalizing interventions as transformations to the constraints of hybrid systems. Unlike interventions in a typical structural causal model, however, interventions in hybrid systems can easily render the model ill-posed. Thus, we identify mild conditions under which our interventions maintain solution existence, uniqueness, and measurability by making explicit connections to established hybrid systems theory. To illustrate the utility of our framework, we formalize a number of canonical causal estimands and explore a case study on the probabilities of causation with applications to fishery management. Our work simultaneously expands the modeling possibilities available to causal inference practitioners and begins to unlock decades of causality research for users of hybrid systems.

A Counterfactual Semantics for Hybrid Dynamical Systems
Andy Zane 1.2 andy@basis ai
Dmitry Batenkov' dima@basis ai
Rafal Urbaniak' rafal@basha ai
Jeremy Zucker srremmyzzucker@panl...
Sam Witty sam@sorbus ai
Basis Research Institute, New York, NY University of Massachusetts Amherst, Amherst, MA Pacifii Northwest National Laboratory, Richland, WA Sorbus Al
Abstract
Models of hybrid dynamical systems are widely used to answer questions about the causes and effects of dynamic events time. Unfortunately, existing causal reasoning formalisms lack support for queries involving the dynamically triggered, discontinuous interventions that characterize hybrid dynamical systems. This mismatch can lead to ad-hoc and error- causal analysis workflows in prac- tice. To bridge the gap between the needs of hybrid systems users and current causal inference capabilities, we develop a rigorous counterfactual semantics by formalizing interventions as transformations the constraints of hybrid systems. Unlike interventions in typical structural causal model, however, interventions in hybrid systems can easily render the model ill ill-posed. Thus, we identify mild conditions under which our interventions maintain solution existence, uniqueness, and measurability by making explicit connections established hybrid systems theory. To illustrate the utility of our framework, we formalize number of canon- ical causal estimands and explore a case study on the probabilities of causation with applications fishery management. Our work simultaneously expands the modeling possibilitics available to causal inference practitioners and begins to unlock decades of causality research for users of hybrid systems.
Introduction
Models of continuous-time dynamical systems are powerful tools for describing real-world mecha- nisms. From contrastive queries about system behavior under different control policies (Kirk) 2004). to sensitivity analyses designed to aid in understanding which parameters drive system variation (Cacuci 2203). scientists, policy makers, and engineers often use such models answer their "what-if" and causal questions. Unfortunately, causal reasoning with continuous systems can be ad-hoc, manual, and error-prone in daily practice.
In parallel, researchers in causal inference have built rigorous tools for answering an expansive taxonomy of causal queries. For example, causal questions about effect estimation (Pearl 2009 Rubin 1974 Imbens & Rubin 2013), counterfactual reasoning (Pearl) 2009 Ch. 7), mediation
Research conducted at Basis.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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analysis (Pearl 2001). responsibility, blame (Chockler & Halpern 2004).
attribution, and explanation (Halpern & Pearl 2005 bj Beckers 2022) can all be succinctly expressed as estimands constructed
from parallel worlds (Balke & Pearl 1994 Avin et al. 2005 Shpitser & Pearl 2008) or potential
outcomes (Rubin 1974). The toolkit also affords a formal means of determining when those estimands
can be reduced to computationally tractable, probabilistic estimation problems (Pearl) 1995 [hhptser
& Pearl [0066] Hernán & Robins 2023). These insights have made it possible to build general-purposs purpose
technology for causal reasoning. such as the causal probabilistic programming language ChiRho
(Bingham et al. 2021 Witty 2023 Basis-Research 2025
Despite significant progress over the last decade (Mooij et al. 2013 Hansen & Sokol 2014 00140 B Blom
al. 2019 Forré& & Mooij 2020, Peters al. 20200 Blom al. [021] Bongers [022]] Blom &
Mooij 2023] [Boeken & Mooij [024 Peters & Halpern 2025). however, gaps remain in the technical
capacity modern causal reasoning machinery to operate on the full breadth of interventions that can be encoded in continuous-time dynamical systems. In particular, a counterfactual semantics for dynamically triggered, instantaneous intervention has not yet been established. With such an intervention semantics in hand, causal reasoning can be more fully mechanized for causal questions about dynamic temporal events, dramatically expanding the rigor and variety of queries available to users of continuous-time dynamical systems.
Such interventions underpin many closed-loop control problems: for example, HVAC systems activate when temperature thresholds are reached; lockdown and masking measures can be implemented according levels of V-2 in wastewater (Kappus-Krn) Kron al.] (024);; commercial fishing
pressure can be reduced once annual harvest limits are reached (Anon] 2007b Warlick et al. 2018)) ;
central banks adjust interest rates depending on economic indicators like inflation and unemployment
reservoir managers release water depending on storage thresholds and agricultural needs (Ray) 2003); ;
and power grids activate "peaker plants" (or stored energy) when demand exceeds certain thresholds (Zhuk et al. 2016). Despite limited attention from the causality community, these systems have garnered significant interest from control theorists in the form continuous-time, hybrid dynamical systems (Schaft et al. 20000 Goebel et al. 2012 Sanfelice 202T) that encode both continuous and instantaneous dynamics in a set of differential and difference constraining equations.
To construct counterfactual semantics for state-dependent, instantaneous intervention, we formalize a class of transformations on hybrid system constraints that induce the desired counterfactual behavior. An intervention creates twin, parallel world with transformed constraints, but in way that ensures both the twin and original worlds share randomly sampled values for initial conditions and parameters. This induces a familiar joint distribution over counterfactual outcomes (Rubin) 1974, Balke & Pearl 1994, Shpitser Pearl 22066] 2008) that can, in turn, be used as input to established causal estimands, such an expected treatment effect or the probabilities of necessary and sufficient causation.
Our contributions are:
1. A formal, counterfactual semantics for dynamically triggered, instantancous interventions in continuous-timg dynamical systems.
2. Under minimal requirements interventional specifications, proof that sufficient condi- tions for solution existence, uniqueness, and finite-time time measurability are preserved in the intervened system. Our framework also explicitly connects to established well-posedness, conditions on hybrid dynamical systems.
3. A case study on the probabilities of necessary and sufficient causation applied to fishery management, demonstrating extensibility to non-trivial causal estimands rarely applied to dynamical systems.
2 Related Work
In Causality. Many researchers have contributed to the systematization of causality for dynamical systems. Hansen & Sokol (0144), for example, show that dynamical systems can be unrolled into directed, structural causal models (SCMs) In the context of ordinary differential equations (ODEs),
"Structural causal models (SCMs) are systems of deterministic functions on endogenous variables that additionally incorporate exogenous random noise. SCMs come equipped with a widely studied interventional semantics. We refer the reader highly influential work of Pearl (0099) for further background.
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if f is the right-hand side of the continuoustimme differential equation x' = f(z). we can write structural equations 2, = It-At. + f(z;-Ari u) where > 0, Ie E RR is the value of the state variable x: at time I, u E R is fixed realization of exogenous noise, and xo is fixed. Taking At - O, we can recover the system's ' dynamies arbitrarily well. This limit results SCMs with infinitely many variables-
modality that has been recently studied as "Generalized Structural Eguation Models" (GSEMs) (Peters & Halpern 2021 Halpern & Peters, 2022, Peters & Halpern 2025). With At> > 0, this becomes the familiar discrete time approximation. which has been widely rescarched in
causal inference (Spirtes) 2013 Pearl 2009; Murphy 2002; Wang et al. 2018 Assaad et al. 2022
Runge et al. 2023 Zan et al. 2024).
This forward- -uler representation, however, is not the preferred tool of hybrid systems theorists, making it ill-suited for identifying conditions under which intervention preserves established well- posedness conditions. Additionally, under the forward-Fule representation, interventional transfor- mations that induce state-dependent jumps require "soft" intervention (Correa & Bareinboim, (020) on all endogenous nodes that might jump. Indeed, the state-dependent jump conditions must be "checked" at all points in time. We discuss this more precisely in appendix J
Somewhat sidestepping the temporal representation issue, most causal research on continuous-time dynamical systems has employed foundational ideas in cyclic graphical models (Iwasaki & Simon
1994, Spirtes 2013 Lacerda et al., 2008 Hyttinen al., 2012) to develop causal abstractions of
system's s equilibrim behavior (Dash, [2003] Mooij et al. 2013 HHansen & Sokol [0144] Blom et al.
2019 Forré& & Mooij 2020 Bongers| 2022 Blom & Mooij (023). Equilibrium focusee frameworks,
however, can fail to expose complex causal relationships in transient dynamics (Peters et al. 2020).
Extensions such as the "time-splitting" operation (Boeken & Mooij] 2024), or the application of GSEMs to hybrid automata by |Peters& & Halpern (0025), enhance the expressiveness of graphical approaches by supporting static-time discontinuities. In contrast, our work targets dynamically triggered interventions, which cannot be straightforwardly analyzed using methods like time-splitting. Indeed, the order and, therefore, the induced time-split graph structure of dynamically triggered interventions depends on state evolution, and therefore on exogenous noise. Our approach avoids these issues by directly defining counterfactual interventions on hybrid system constraints. While the non- graphical framing means that standard graphical identifiability criteria are not immediately available, building our semantics on established hybrid systems theory opens pathways to leveraging longstanding methods and conditions for system identification of dynamical systems (Walter & Pronzato| 1997 [ung 2012, Raue et al.] 2009 Stuart 2010), such as the "persistence of excitation" ". which has been studied directly in the context of hybrid systems (Johnson 2023 Saoud et al., 2024).
Our approach follows the spirit of recent developments in constraint-based causal modeling. For example, [eekers et al.] (2023) extend SCMs in order to handle logical constraints (such as unit conversions), while Blom al. (2019) interpret equilibrium equations of dynamical systems, along with their corresponding algebraic invariants, as collection of constraints. In both cases, a model characterized by a collection of constraints, and interventions are defined as transformations of those constraints (e.g., by changing, disabling, or enabling them). At high level, we take a similar approach. Hybrid systems, however, are characterized by unique class of constraints requiring special considerations around Zeno behavior, set-valued theory, non-uniqueness even in "well-posed" cases, set-valued stable points, etc. In short, analyzing the post-intervention properties of hybrid systems is made easier via direct use of existing hybrid systems frameworks, rather than existing causal frameworks. Naturally, each school of thought is best suited to different tasks, and we look forward to future work that deftly exercises the comparative advantages of each.
In Control Theory. Control theory and causality share overlapping goals, yet historically oper- ate separately. This paper integrates causal reasoning directly into established. hybrid dynamical systems frameworks (Goebel et al. 20122 Sanfelice, 2021). In particular, our formalization of dy- namically triggered intervention as constraint transtormation mirrors controller-plant compositions from hybrid control theory, which are also shown to preserve established conditions for system well- well-posedness: (Sanfelice [202T). Hybrid system theory presents challenges, however, due to po- tential non- -on-uniquesess of solutions under general conditions (Goebel et al.; (012)), complicating counterfactual reasoning. To address this practically, we follow common simulation practices (e.g.. preferring flowing solutions when multiple are possible) and explicitly formalize these assumptions (Sanfelice et al. 2023a). Our contributions thus link causal semanties to established hybrid systems theory and practice, enabling rigorous and computationally feasible causal analysis.
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Original
Static Time
State Stele-Dependent Time
(a)
(b)
(c)
Figure 1: Three parallel worlds constructed by starting with dose-decay model (fig. la and then transforming that model to reflect dosage at fixed, static time (fig. [b], and dosage when the concentration hits threshold (fig. Ic and example []. This paper develops the first explicitly counterfactual semantics for the dynamically triggered, state-dependent case (fig. [c]. Three sample trajectories are shown for each world, with initial condition and desey rate held fixed across worlds for each sample trajectory. Notice that the state sate-dependent interventions occur at different times for different trajectories induced by different initial conditions and/or parameters.
3 Parameterized Hybrid Systems
As first approximation, the present work focuses continuous, ordinary differential equations models with random initial conditions and parameters. Many intuitive interventions, however, can be conveniently defined as instantaneous (discontinuous) changes to the dynamically evolving state. Thus, we focus on hybrid systems that afford both continuous "flow" and event-based "jumps" in state. Jumps can arise system. With state space S I" and following the framework laid out by Goebel et al. (2012), many hybrid systems can be characterized as comprising four elements: flow set C S differential inclusion S = IR:; a jump set c S: and a set-valued jump map G S = S. In general, the system evolves according to its differential inclusion F when its state is in the flow set C and according to the jump map G when the jump set D. Readers who are unfamiliar with inclusions and set-valued maps should refer appendix | Hybrid systems often alternate between continuous flow and discontinuous jumps, though consecutive jumps remain well-defined. Many hybrid systems, then, be characterized with the tple (C, F, D, G). We ground this out in the following example.
product of interventions and/or discontinuous dynamics in the unintervened
Example (Dosage Model), Consider modeling the exponential decay of drug concentration rate B, where medical providers intervene to administer additional dosage when x reaches a threshold seek state evolutions obeying
y. To model these dynamics,
x C R3D x D = (2EERDEE
be F (x) 1-3x
x E G(x)= = [ +1
where å denotes the time derivative of the state, and x the state immediately following a jump. The solution map of a hybrid system typically takes as "input" an initial condition & E S, but can also be parameterized to additionally incorporate a vector
example
1
[B, 7
Definition (Parameterized Hybrid System). Let S C R. O
R m A parametrized hybrid system P is tuple P = (H, S, θ) where for each 0 E θ, H(()) = (C(O), Fe. D(O), Ge) is standard hybrid system (Goebel al. 2012 Def. 2.2), i.e.
C:: θ = S is set-valued mapping returning the flow set,
Fg (æ) F (ax 0) Vz S and V0 θ, where F : x θ R is differential inclusion, with C(0) = dom Fe S S for all 0 O.
DD 1 S is a set-valued mapping returning the jump set, and
Go (x) G (xx 0) Væ E S and VO E O where G : S x θ = S is an ordered (i.e. returning ordered collection of sets to keep track of interventions, cf. definition 6 set-valued jump map, with D(0) = dom Gg S S for all 0 e.
Without explicit parametrization, we write H (C, F,D.G) G), and also often expand H in P, writing cquivalently P (H, +) (C.F

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Canonically, solutions to hybrid systems are functions of both continuous time t E R30 and discrete event indices j E N. Following (Goebel et al. 2012 Sects. 2.2-2.3), we define, for each possible parameterization O Q and initial condition
S, a "solution" to H((0) to be "hybrid arc".
which is formally a set-valued map ( E. 0) Rg x N = TR. We review Goebel et a [ (2012)
rigorous characterization of hybrid ares as solutions to hybrid systems in appendix AA.3 For ease of exposition in the main body of this paper, however, we use the concept of sime-parameterizer
solution map yp, which we describe informally, below, in definition 2 An expanded, formal treatment of time-parameterizer solution maps can be found in appendix A.4
Definition 2 ((nnormall Time-Parameterizer Solution Map). Let E 0) [0,,tt R be called the time-parane teme-parmmttrrieed solution map of P = (H, S, ②), where (+ = ming sup, dom o(; E,; 0) and where the hybrid are ct:: E, 0) uniquely satisfies (0) from initial state &. VE, 0 S O.
The reader will note that (+) = R. In this paper, we focus strictly on finite time horizons, leaving the analysis of hybrid equilibria to future work- indeed, only the simplest hybrid systems equilibrate to a point, so equilibrium states are most productively defined as belonging to a set. Analyzing the causally relevant behavior of such sets requires machinery beyond our current scope, but our direct connection to established hybrid systems theory, in conjunction with the rich history of causal research on equilibrium models, provides a firm foundation to explore this in the future. Additionally. because hybrid arcs can dynamically evolve in event indices, Zeno and non- non-flowing solutions are possible, which can make t 0 (if it only jumps) or arbitrarily small (if allowable initial conditions are close to Zeno accumulation points). We do not provide universal criteria in this paper under which ++ is arbitrarily large.
While we take the hybrid system P to accurately describe causally relevant mechanisms in the world, we impose assumptions on P indirectly. In particular, we assume that some auxiliary "upstream" system Pf can be "lowered" to produce P = lo,e and that the upstream Py satisfies standard hybrid well-posedness conditions from the literature (the so-called hybrid basic conditions, detailed in ossumptionn4Jof the appendix. and folded into assumption [eelow). While these conditions support our theoretical results and facilitate future extensions (e.g.. to stability analyses). they inherently admit solution non-uniqueness, particularly at state-space boundaries where solutions could either jump or flow. Non- Non-uniqueness, however, complicates both measurability arguments and downstream causal analysis. In this work, then, we formalize a practical approach that is standard in simulating hybrid systems by specifying that the solutions should be "flow preferring" -iff solution could both flow and jump, we choose the solution that flows (Sanfelice et al. 2023a: Sanfelice & Teel 2010 F Note also that a flow-preferring specification is consistent with computational implementations that trigger jumps when the jump-set boundary is crossed
key component of the hybrid basic conditions is the outer semi-continuity of the jump set G in the upstream system. Maintaining this property through intervention requires some bookkeeping on the boundaries between interventional jump sets, but must be handled such that "lowering" favors more recently applied model transformations. We achieve this bookkeeping through the use of an ordered set-valued map G set-valued map in the appendix (c.f. definition
+ U Gx(z),, where last(G) Gk. We fully formalize the ordered
Definition 3. Let P- (C, D,G, θ) = (H, O). We set
preferflow(D, (D,C,F) 0 - D(0), IE E S there is a flowing solution to H()) from E lower(P) = FC, S, 0,));
D' = preferflow(D,C.F) G' = las
The existence of a flowing solution from & is meant in the sense established in the hybrid systems literature. See appendix A.3 (definitions [3 and 14] for details. We can now state our collected assumptions on a hybrid system, and prove the sufficiency of those assumptions for the existence, uniqueness, and measurability of the system's solution. See appendix A.6 (assumptions 3 to 5 for
details, and appendix F for proof of lemma |
Assumption 1. The parameterized hybrid system can be written as P lower (Pt), where P, (C, D, G, S, θ) = (H, S, +), and the following hold for all & E S, 0 E O:
*Other approaches include preferring solutions that jump, or by resolving ambiguities randomly Teel & Hespanhal 2015).
We should say. of a thick jump set, similar to what we have described in definition 17 Thickening jump sets is also common in practical computational environments (Sanfelice et al. 20236
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1. there exists a unique, nontrivial solution to the differential inclusion F (i.e. the continuous part of H (0) that Borel measurable in &. at any fixed e (0,
2. C' outer somi-continuous (osc) and C(0) closed; osc, locally bounded, and F(x, 0) is convex Vzz S;
3. ( is closed and G(D) is Borel; G' is osc, locally bounded; last(G) is single-valued and Borel measurable &, 0 at any fixed & E [0, z0).
Lemma 1. Let satisfy assumption Then has unique time-parameterized solution c(=; 0) [0,t+) R that is morel--masurable in initial conditions and parameters O at any fixed
te te,
Instantaneous Interventions as Constraint Transformations
We now formally define a general class of instantaneous interventions. We show that under certain natural assumptions, the class of systems meeting assumption i "closed" under intervention - i.e. intervened systems will meet assumption Tif the original system does.
An instantancous intervention can be implemented via modifications to the jump map and the jump set functions, respectively G and D in definition To support parameterized interventions and/or stateful jumps, one can simply augment the state space and the parameter space O essentially preserving all properties of interest (appendix [
Definition (Instantaneous Intervention). Consider set-valued mappings D : = S and G Now, let
S θ 3 S and parameterized hybrid system P =
(²()) = C(O)iitt D())
(1)
D'(0) = preferflow (D,C,,F) (0) u D(0)
(2)
G' (x. 0) = Ø D(O)D(O))
G (xx
(3)
a D(0)
(æ,
then
p'= = instint (,D,,))(CC,E.D',C
In words, 5 defines when (or where in the state space) the intervention will occur, while G defines the state transition induced by the intervention. We make two important set-subtractions in this definition to preserve some useful and simplifying properties. First, we define G' in eq. (3) as preferring the new (i.e., the interventional) jump map G wherever the original and new jump sets overlap. Second, the new flow set in eq. (1) has the interior of the new jump set removed. This preserves non-overlap between flow and jump sets, except possibly on the boundary, which discuss below.
D(( is defined (eq. 2 as the union over jump sets, except that a flow now-preferring subtraction (definition 3) is made first on D(Ø). Because (0) has the interior of D(0) already removed, this subtraction operates only on the boundary of D(0). In other words, D'(0) will always contain the interior of D(0), and will have the parts of its boundary removed where Fg flows tangentially to or away from the interventional jump set.
4.1 Intervention Preserves Existence, Uniqueness, and Measurability
Interventional transformations should preserve key model properties. For causal models with explicit forward simulations, this largely trivial. Hybrid systems, however, only implicitly characterize forward simulations (i.e., solutions) by specifying set of constraints. Transformations to these constraints can easily fail to maintain key properties in the intervened world, such as whether a
"This assumption is insufficient, its own, to guarantee unique, measurable solution for the full hybrid system. Indeed, this insufficiency constitutes key challenge addressed by hybrid systems researchers.
"This covers interventions with random parameters, in addition to those that require some "memory" of past system events. For example, if jump event should only occur & e N times, that event could increment counter i, and include only < l in its interventional jump set.

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P.
à exists;
exists;
unique flow;
instint, (P,,
unique flow; P basic
Pt basic
lower (Pt)
lower (P:)
P
P
exists, unique, measurable
instint
P,,
P'
exiists, unique, measurable
Figure 2: Depiction of our "lowering" proof strategy for theorem 1 a simple inductive generalization from lemma which asserts that single interventional trans- formation preserves key system properties, and is what we visualize here. Solid arrows indicate constraint transformations, while dotted arrows indicate that properties of one system imply properties of another. Assume that the parameterized hybrid system P accuuately describes domain of interest, and that it can constructed by applying the lower transformation (definition 3 to a system P that fulfills assumption TT. Applying lower to such system preserves existence and induces uniqueness and mcasurability (lemma T]. To simulate the effects of an intervention, we transform P into the model P' that describes the intervened world. P' can also be constructed, however, by applying slightly modified intervention (instint,, definition 20) to Py and then "lowering". Intervention on Pr maintains key properties in P,, which can, before, be lowered system P that must have unique, measurable solution (lemma)
Proving theorem 1 requires
solution is unique and measurable, or exists at all. The key theoretical contribution of this work, then, identify assumptions sufficient ensure our interventional semantics preserves these properties through model transformation. Formal proof of theorem iis provided in appendix D but we also include fig. 2 as visual aid and proof sketch.
Assumption 2 (Assumptions Interventional Specifications). Consider mappings D e = S and G: θ 3 S and parameterized hybrid system P (C, S,
θ). For all 0 θ, assume
(II) D(0) is closed and well-bebaved relative to P (assumption 6 Additionally, the interior graph Gint D) is oper and the graph G (D) is Borel;
(12) Go S 3 S is outer semi-continuous and locally bounded relative to D(0), and D(0) dom Gg. Additionally, G single-valued and Borel-measurable.
Theorem (Compositions of Instantaneous Interventions Preserve Key Properties). Consider parameterized hybrid system P that meets assumption [ and any finite sequence of K set- valued mappings (Dx) and (Gx). where each D3 and G fulfill assumption 2 relative to P. Let instint; = instint(. DR, Gk) (definition 4) and
o... p' = (instint, .... iistint......nnssintt (P):
(4)
pl then meets assumption 1 and by 7 has a unique cann-aaammtteiiee solution E, 0)
[0,tt) - R", moasallmasurable in initial conditions and parameters O at any fixed t [0, t+).
Causal Estimands as Functionals of Twin Distributions
In this section, exercise framework define three basic causal estimands. Importantly, many of the more complex causal analyses build on these basic inference capabilities. Most targets causal inference take the form of (conditional) expectations, and must now
measurability
results to define those expectations with respect to random solution maps 10) First, we will generalize
$ An assumption that interventional jump set well-behaved reduces, essentially, to asserting that flowing solution cannot oscillate across the boundary of π) infinitely often. This is satisfied by many systems of interest under reasonable regularity assumptions for instance, if the flow map is analytic and aD is Lipschitz. varies.
"This ensures its interior does not suddenly appear/disappear
IIIn this paper, we do not address the random dynamics that characterize stochastic differential constraints
Oksendal 2003 Cassandras & Lygeros 2010 Hansen & Soko] 2014 Boeken & Mooij 2024). or independent, per-jump randomness (Teel 2013 Teel et al. 2014 Teel & Hespanha 2015).
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the parameterized hybrid system to include random initial conditions and parameters. Then, we will establish some notation and define the expected treatment effect, data-conditional treatment effect, and the basic counterfactual query using our machinery.
Definition 5 (Hybrid System with Random Inputs). A parameterized hybrid system with random inputs is characterized by the tuples
R (P.L.0); P (C,F,
We take the probability space (f2, F, as implied by R, where & : - S and θ : S - θ are measurable with respect to F. and the Borel o-algebras on S = TR" and e = Rmm.
When clear from context, for some E Ω, we often write & and O in place of & (w) and 0 (w), respectively. We distinguish random variables & and 0 from possible values & e and 0 θ by the upright font. From here, we directly consider evaluations of the solution as a measurable random variable. A direct consequence of lemma T applied to system that fulfills assumption is the following.
which states that lower induces measurability when
Corollary 1 (Random Time-Parameterivee Solution is Measurable). Consider parameterized hybrid system with random imputs R (P, E. 0),, where P satisfies assumption [ Then, by lemmal 7 P has unique, time-paree ame--prramttrrized solution map ip, and the composition w <p(;; E(w), ((w)) defines an F-measurable random variable at any fixed E {0,,tt).
Having established conditions under which intervention preserves measurability, constructing estimands from the parallel worlds created through intervention. In estimands, we use symbolic subscripts to delineate parallel worlds. Consider original system Rs = (Po. & θ). We might then apply intervention characterized by D, and G2 to yield P. instint Da, G.). By convention, we use R. = (Pa, &, 0) in reference to the full specification for the intervened world, and sp &, 0) for its random, tine-parameterized solution (corollary []. We often write pf in place of (t; E 0) for brevity. Lastly, supposing we wish to focus on a particular element of the state vector time t, we sometimes define a random function that appropriately indexes into the solution vector. For example, we might have that (t; &, 0) = (!o &, 0) always. where represents the solution map for the (i) 'tt state element. We similarly sometimes use h ht & 0). We can exercise this notation with the following examples.
begin
Example 2 (Expected Treatment Effect). Consider Ra (Po, E. 0) and interventional jump set 1> and map G. Assume these components fulfill assumptions and Let P1 = instint(Po, D, C) and iPu and iP1 be the time-paramen pane-parameterized solution maps of the original and intervened worlds. Let yo and y1 be the solution maps for the (i) th element of the state vector. The expected treatment effect some time
[0, min [t (T] [0, {+)
can be written equivalently
10 E Tv - 361 = E 3/ (T; E, 9) (T; E. θ)e = E 9
(T: (r:
Example 3 DData-Conditiona Treatment Effect). Building immediately off example 2 we can specify contttiooiii treatment effect that takes factual observations into account] Let wp be the solution map for some element of the state vector. For some finite set of observation times [tx) = [0, to) ), the data-conditional treatment effecct can then be written as
E [vT vō
vol: Vo N (w)):0): wo
[uu]*] 1==1"
Example 4 (Counterfactual Outcome). Also building off example 2 consider factual outcome event that (T( E. 0) T6 R. The counterfactual outcome, then, can be derived by conditioning on that factual event.
Elyi vo m6l = [[yn(T; E, 0) (o
"While identification results for specific causal estimands are beyond the scope of this paper, system identification has already been studied for hybrid systems under the condition of "persistence of excitation" Johnson 2023 Saoud et al. 2024). Under such conditions, a posterior density p(ç. 0 vo D), for example, where 2> is realization of Up. sufficiently well-behaved to estimate targets defined in this paper. Withoot loss of generality, we write that the data are subject to Gaussian observation noise. Many practical settings call for observation noise, but we also note that the deterministic relationship between inputs (E, 0) and state trajectories means that inference behaves poorly without observation noise.
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query
outcome
probability
nec
Y 1/b
(Y 0 X = 1,Y - =) Pr
D &
7
by by
suf.
Y= 1bb y2
(Y = 0,Y 0) Pr
nec. and suf.
Y == 1/b
(r 1, = 0) = Pr (b,,
Table I: Identities for the probabilities of causation in the fishery management example. Under TAC
quota q. the biomass of the fished species at time is given by by. The outcome is achieved when that biomass meets or exceeds y. We rely on the standard exogeneity conditions Y, IL X and Yz IL X 1 Fnd the fact that, conditioned on X 1 (X2), Y reduces to the outcome only in the world with allowable catch set 'i (42).
Many of the more complex causal inference tasks - such as mediation analysis, the estimation of population-level conditional average treatment effects, or even actual cause assessments are constructed from the counterfactual building blocks we propose here. Indeed, once counterfactual semantics is established, and twin-world or potential-outoomes syntax (e.g.. differentiating yo from vi) is enumerated, many estimands are straightforward and familiar to develop. In the next section, we explore just such a class of estimand: the probabilities of causation.
Case Study: Necessary and Sufficient Causation
To illustrate more sophisticated application of our interventional semantics, we map the standard definitions for the probabilities of necessary and sufficient causation (originally formalized by Pearl (1999)) onto dynamically triggered, discontinuous interventions in hybrid systems. In particular, we work the fishery management domain where regulators employ Total Allowable Catch (TAC) policies to dynamically end the commercial fishing season after caught biomass reaches certain quotas. interested, the reader may wish to review appendix G.I] in which we provide motivating historical context for this domain. Additionally, we review Pearl s original formulation of the probabilities of causation (PoC) in appendix G.2 Throughout appendix G. we provide full simulation analyses the case study. Code is available here and relies the dynamical systems package from the causal probabilistic programming language ChiRho BBasis-Rescaree 2025
14
We focus on hypothetical fishery involving three trophic levels apex predators, intermediate predators (the fished species), and forage fish - with dynamies captured by the differential equations presented by Zhou & Smith (2017). Throughout single season, fishing pressure is modeled at a constant rate applied the intermediate predator, plus some bycatch on the apex trophic level. Regulators intervene by ending the fishing season (setting the catch rate to zero) when the integrated catch reaches a predefined TAC quota. The goal of these policies is to ensure that the biomass of the target fishery species recovers to sustainable level y by the beginning of the next season.
In this context, stakeholders may debate the necessity and/or sufficiency of certain regulatory policies in maintaining joint ecological and economic goals for the fishery. The probabilities of causation are formal tools supporting the assessment of causal attribution between causes and their (supposed) effects. |Parr| (1999) first formalized the PoC for binary treatments and outcomes here, however, both the TAC quota and the biomass are scalar valued. We therefore follow Kawakami et al.]s (2024) generalization of the PoCs to support contrastive queries between scalar-valued treatments and their thresholded outcomes (see their Def. 3.1). Consider two TAC quotas qI and 92, and the following we provide the formalized estimands written in our notation.
natural language queries. In table] I
necessity: in worlds where the ond-of-year biomass levels exceed the target level (success) under quota 91. what is the probability of failure had regulators used quota q2 instead?
sufficiency: in worlds where the end-of-year biomass levels remain below the target level (failure) under 91, what is the probability of success had regulators used Φ2 instead?
necessity and sufficiency: what is the probability that both (1) d1 results in success and (2) Q2 results in failure?
https://githhb.. com wwt..//aasseeeeeaaeccaaaaa/e//ee chirho
'If some parameter or initial condition were influenced by confounder that also influenced X, this would not be the case, and conditioning on X would be required in the identities listed in table I

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For readers less familiar with the applications of the PoC to decision and policy making, we provide an expanded narrative scaffolding for this example in appendix G.4] In appendix [G.] we provide an additional example designed to highlight how certain natural language ambiguities in causal attribution queries - particularly those involving multi-faceted, real world events and policies can be formally clarified.
The PoC queries above rely on the construction of twin, contrastive worlds worlds-one one with TAC quota QI, and the other with Q2. To model these worlds, we start with system P characterizing year-round fishing pressure (i.e., no regulatory intervention), and then transform its constraints add dynamic, season-ending intervention. Notationally, let h2 represent the harvest rate, and b, the biomass, at trophic level i, and let 2 be the total catch (integral of & h2bb) the intermediate trophic level. The system state can be conceptualized as [2, h1:3, b1:3] = E S R.
The regulatory, season-ending intervention can be modeled by dynamically setting harvest rates to zero when the catch exceeds a threshold qi (with i E {1,2 By using our interventional semantics, we can construct parallel worlds with the same random initial conditions and parameters. See appendix G.3 for a generalization of this model to multi-season time scales.
D. {2 Rz0 3 q:} R
Gg. 0) = [[2, 0,0,0,
(5)
R, = (); R.y = (P E, R₂² = (P & 0); Pg. = instint(P, Da. G.))
(6)
7 Limitations and Future Work
Most research developing causal inference tools starts by casting problem in the format of structural causal models (SCMs) (Pearl) 2009). Our work differs in that we construct our counterfactual semantics directly in the parlance of hybrid systems. These two tacks are compatible, however. For example, with our measurability results in hand, the time- same--arammterizee solution map oo can be treated as structural equation with initial conditions and parameters @ viewed parent variables in larger SCM. Our interventional semantics, then, exposes the causal dynamics of the hybrid system for manipulation. When &e is interpreted structural equation, our semantics could be viewed as characterizing a family of "soft-interventions" (Correa & Bareinboim] 2020) on the solution map. Importantly, the adjoint method (Chen et al.) 2018) can be used in tandem with auto-differentiation machinery to learn "event function" (i.e. jump set) parameters T6 thereby supporting end-to-end differentiation of composite SCM and hybrid system models. Relatedly. equivalent forward Bawwerd-uuler representations may prove useful in actual cause analysis of hybrid systems (Halpern & Peters 2022).
This leaves few limitations to review. First, we do not present non-parametric, estimand-specifee identification results - indeed, there may exist sufficient conditions for estimand identification that weaker than those established for full system identification. Second, discussed following I.
definition we focus only on finite time regimes, leaving analysis of hybrid equilibria to future work. Furthermore, we do not provide conditions under which intervention preserves non-Zeno behavior.
8 Conclusion
This paper has strengthened the connection between the modeling capabilities offered by hybrid systems theory and the causal reasoning capabilities developed by the causal inference research community.
We characterize and demonstrate counterfactual semantics for class of dynamically triggered, instantaneous interventions that underpin many closed-loop control problems. Bypassing an explicit re-casting of hybrid systems as structural causal models, we use hybrid systems the primary modeling substrate. This allows clear connections to the extensive body of work on hybrid systems theory, in which we can derive and characterize mild conditions under which solution existence. uniqueness and measurability are preserved in the intervened system.
Finally, we illustrate the flexibility and power of the resulting framework by first formalizing common causal estimands for hybrid systems, and then by developing a case study using the three probabilities of causation in the context of fishery management.
1eSee https: //gi thub. httpt//wwwoc.ccccooooo/////////a ball. for an example.
10

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Acknowledgments and Disclosure of Funding
AZ, DB, RU, JZ. and SW were supported on DARPA Automating Scientific Knowledge Extraction and Modeling (ASKEM) program Grant HR001 12220036. We thank Anirban Chaudhuri, Sabina Altus, Joseph Cottam, and Neeraj Kumar for their insights and contributions throughout the ASKEM program; our colleagues at Basis for helpful comments and discussions; Eli Bingham for guiding our thinking and software design choices around these ideas; Paul Wintz for answering many questions about hybrid systems and pointing us to relevant literature; and David Jensen for helpful comments and discussion. Pacific Northwest National Laboratory (PNNL) a multiprogram national laboratory operated by Battelle for the DOE under Contract DEAC05-76RLO 1830. The views, opinions and/or findings expressed are those of the author and should not be interpreted as representing the official views or policies of the Department of Defense the U.S. Government, Distribution Statement "A" (Approved for Public Release, Distribution Unlimited).
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A Supplementary Definitions and Standard Assumptions
A.1 Differential Inclusions and Set- Valued Maps
We follow Goebel et al. (2012) in generalizing to hybrid systems with inclusion constraints. A
differential inclusion F SS= = R", for example, specifies the constraint that the time derivative at of the state must be included in the set F ((x) S RR.. Note that the equality constraint 2 = (œ) for some f : _º is a special case of the broader notion of differential inclusion. To clarify,
the stacked double arrows in, for example, S = IR" indicate a set-valued mapping from S to a subset of R Goebel et al. (2012) define the domain of set-valued mapping V X = 3 as
dom V = { E X: x # Ø The graph of V is then
G(V) {(,, E X' x y a dom V, y E V (x))
A.2 Ordered Set- VeluVelued Maps
Ordered set-valued maps are special cases of set-valued maps, which we use in this paper to keep track of interventions.
Definition 6 (Ordered Set- Valued Map). Let G = (Gi,..., .k) be a finite sequence of set-valued maps. We call G an ordered sel-valued map, which means is equipped with the following operation:
K
G = x
U Gk(x); R-1
last(G) = GR-
Therefore, dom G - UK= 1 dom Gk. Given two sequences G = (G1..... Gk) and H (H....., HL), we denote G L H (GI....,, Gx. H1,.... Hz). By slight abuse of notation, we sometimes identify a map G with the corresponding one-element sequence (G), and also use G in place of Gt when the context requires a "vanilla" set-valued map.
A.3 Solution Concept
The following definitions and propositions are given almost exactly as stated by Goebel et al. (2012), except that we adapt them slightly for explicitly parameterized hybrid systems (definition
The nature of hybrid systems implies that their solutions should be functions of both continuous time t e Rza and discrete time j N. Let t, denote the time of the j-th discrete event, with t, & t,+1 for all j E N and to = O. Following (Goebel et al., 2012, Sects. 2.2-2.3), we define, for each possible parameterization O E θ and initial condition E S, "solution" to H())) to be a "hybrid are", which is formally a set-valued map & 0) IR x N = IR.. We can formalize this time-event space (of which dom is example) as follows:
Definition 7 (Hybrid Time Domain from Goebel et al. (0122) (Def. 2.3)). E = R x ii a compact
hybrid time domain if it is finite union of sequence of closed intervals E = U ([t, f++1] 1 (i)). where 0- to & 4 & ... LJ, and E is hybrid time domain if for each (T,J) E E, the set E o (I0, T] x (0.1,...,JJ) is compact hybrid time domain.
J})
Generally, dom ø is unknown until after a particular solution ø is found, it depends the exact sequence of state-dependent jump times; therefore, it is natural to consider ş as a-priori set-valued. Definition 8 (Solution Concept adapted from Goebel et al. 2012) (Def 2.6).. Consider parameterized For e E θ, any solution E( to H(() must satisfy @ 0; &, 0) & C(O) u D(O) and the constraints implied by H((0), i.c.:
hybrid system P = (H. θ), with H. =
1. for all j N such that _ (t : (t,j) dom 3) has nonempty interior, we have (tt,j) C(O), Vt int 15, and o(t, Fo(o(t,))) for almost all e 13; [continuous flow regime]
2. for all (1.j) E dom @ s.t. (t.j 1) E dom ø, we have ((t,j:E.0) E D(0) and @(t.j + g 0) E Golo(t,j: & 0)) [discrete jump regime].
convenient to work with solutions that cannot be extended, as formalized by the following concept.
17

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Definition 9 (Maximal Solutions adapted from Goebel et al. (2012) (Def 2.7)). A solution E. E to H((@) (as in definition [) is maximal if there does not exist another solution E'(:;E, 0) to H(() such that dom <(:; &. 0) is proper subset of dom ( 0) and (ll.j: 0) (1, e, 0) for all (1, E dom (f:;E, f)-
Unless specified otherwise, we always consider maximal solutions in this paper. With the solution concept established, we can now state conditions for the existence and uniqueness of solutions. We again borrow from Goebel al.[ (2012), and adapt accordingly to support parameterized systems (definition I).
Proposition (Basic Existence adapted from Goebel et al.
(012) (Proposition 2.10)). Consider and a standard hybrid system H(O)- = (C(0), Fo, D(O), Go) for some O E θ. Let C(O) D(O). If E D(Ø) or
parameterized hybrid system P = (H, θ)
(VC) there exists > O and an absolutely continuous function :: |0, c] - R² such that z(0) &.
E() e Fo(z(t)) for almost all E [0, c] and z(t) E C(0) for all (0,c).
then there exists non-trivial solution E, H with (0, 0)
E - (f(VC) holds for every
& E C(0) D()), then there exists nontrivial solution to H(0)) from every point of C()) u D(0). If the foregoing further holds for H()) at every O E O we say P fulfills the conditions for basic existence.
Proposition 2 (Basic Uniqueness adapted from Goebel et al. 2012) (Proposition 2.11)). Consider parameterized hybrid system P (H, S θ) [C, D.F,D,CG. G, and a standard hybrid system H(() = (C(0), Fe. D(0), Go) for some 0 O. For every & E C(0) D(0) there exists a unique maximal solution
0) with @00 0; 0) = E provided that the following conditions hold.
(a) For every & E C(O) T 0, if two absolutely continuous 21, [0, T] - S are such that (t) E (z) for almost all t E [O, 2+())= E C(0) for all t E (0,T), and (0) = &, i 1, 2, then (t) = (2) for all t E [0,T]:
(b) for every & C(O) n D(O), there does not exist e > 0 and an absolutely continuous f z: [0, S such that z(0) &. =(t) Fo(z(t)) for almost all t E [0, e] and z(t) E C(0) for all E e0,
(c) for every & E D(O), Go(E) consists of one point.
If the foregoing further holds at every 0 O we say that P fulfills the conditions for basic uniqueness.
A.4 Finite-Time Mcasurability of Solution Initial Conditions and Parameters
Measurability is key coherently defining causal estimands (conditional) expectations. In particular, we use the measurability of time- parameterized "solution map" jointly in the initial state and parameters. By "solution map", we refer either functions & or & that, when provided some & E S and O E θ, yield hybrid are (t.j) - Et and time-parameterized function t -tt 0) respectively.
As stated following definition [1 in this paper, we focus strictly on finite time horizons. Definition TO below, makes this finite-time limitation precise, and then employs that definition to formalize the time-parameterized solution map and its measurability.
Definition 10 (t* Uniquely Evaluable). Consider parameterized hybrid system P (C, D. S.S,)) Q) that fulfills conditions for basic existence and uniqueness (propositions and [
Define 1+ ming,o sup, dom E, meaning that for every & S, 0 O yielding unique
solution J + & (I 0), VL E [0, there exists j E N such that (t, E dom d (. 0).. We
then say that P is it uniquely evaluable 18
A non- con-trivial solution is one with more than a single point in its domain Goebel et al. 2012 2.5) 18 Unless the space of initial conditions and parameters are limited to exclude reachable states arbitrarily close to Zeno points, Zeno systems will always have arbitrarily small x*_ Additionally, this definition implicitly excludes evaluation times at the end (in continuous time) of eventually discrete solutions. If such solutions are not complete, one might wish to include the final time in the evaluable interval.
18

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Definition 11 (Time Parameterized Solution Map). Consider t uniquely evaluable parameterized hybrid system (C, D. and its solution map ø. Define for all E [0, (0,t*) & S. O E θ its time-parameterized solution
o ( Φ Jt (E, (E.
(9)
where ji (E.0) is the index of the last discrete jump at time t
Jt (E, max (j (1, dom @ (5
(10)
Definition 12 (L* Measurable). Consider 1t uniquely evaluable parameterized hybrid system P and its pamempprrmeterizee solution map yo If, for every fixed / e (0, E, 0 - yp E 0) is Borel-measurable function, we say that has ( measurable time-parameterized solution map sp
A.5 Flow Preferring Subtraction and Lowering
Definition 13 (Flow-Preferring Subtraction). Consider parameterized hybrid system P FC, F, D, Q) that meets the hybrid basic conditions (assumption 4). We borrow the following viability condition from proposition oon point & ε S, for some 0 E O
(VC) there exists >> 0 and an absolutely continuous function 2 c - R" such that z(0) &. a(t) E F(z(t), 0) for almost all E [0, e] and z(t) E C(0) for all t (0,
We can then transform D to be flow preferring by writing
preferflow C, F) = 0 - D(O) LE E (VC) holds for 0 from E
(11)
Recall the definition of ordered set-values maps (definition affording the last(G) operation on G, the jump map.
Definition 14 (Lowering). Consider parameterized hybrid system P = (,, that meets the hybrid basic conditions (assumption 4.. We write that
D' preferflow(D,C.F)
(12)
G' last(G),
(13)
lower(P) D',,F,,,,GG..
(14)
A.6 Collected Assumptions the Hybrid System
Assumption 3 (Unique, Complete, and Borel Solution Exists for Differential Inclusion for all S). Consider parameterized hybrid system P (C,F, Assume that
(F1) for every S. θ, T that (t()) E (t(z for almost all t E [0. T1, (f) e S for all t (0,77, and 2,( = g. 1, 2, then 21 (l) 22) for / E [0, T]:
if two absolutely continuous 21, 22 :00,7] + S are such
(F2) for all E E S and 0 E θ, such 21 exists for every T'E E (0, o0);
(F3) with z(t; 0) (1) for all & e S, 0 θ, and / e (0, 0. =(1:E,0) is Borel-measurable function for every E (0)
Importantly, note that assumption Jonly relates to the differential inclusion, and does not preclude P from jumping, or from pathologies associated with jumps. Additionally, observe that 2(t; 0) = ø(t, O; &, ) that is, statements z trivially apply the solution mapping up to and including the time of the first jump.
Assumption 4 (Hybrid Basic Conditions adapted from Goebel et al. (2012) (Assump. 6.5)), Consider parameterized hybrid system P = (C, F, D,G.S, S, θ), and assume for all E θ that the following hold.
(A1) C(A) and D(0) are closed subsets of S:
(A2) Fe S 3 IR" is outer semi-continuous and locally bounded relative to C(O), C(0) c dom Fe. and F(x, 0) is convex for every C(O);
(A3) Gg S = is outer semi-continuous and locally bounded relative to D(0), and D(0) c
dom Go.
19

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In particular, (A1) implies that D(Ø) and C(Ø) must overlap on any shared boundary - solutions that start at or graze this boundary can, non non-uniquely, either jump or flow. Additionally, the outer semi-continuity of Go (A3) requires that, at the boundaries of the pieces in piecewise Go. Go must return values from multiple pieces. Solutions hitting those boundaries can jump to multiple states. Assumption (Collected Assumptions on the Original System). The parameterized hybrid system P can be constructed as P lower (Pt), where Pt (C, S, such that:
(PI) Pt satisfies assumption 4
(P2) P1 fulfills the conditions for basic existence (proposition g
(P3) Pt has unique solution to its differential inclusion F from everywhere in S and @ (assump- tion 3
(P4) C(O) is outer semi-continoous at every 0 E O;
(P5) the graph ((D) of the jump set mapping D is Borel;
(P6) last(G) is ainle-valuedd on its domain, with last(G)(z, 0) {g(x,, 0)}, and g Borel- measurable for all æ, 0 E dom last(G).
A.7 Well-Behaved Jump Set
Definition 15 (Well-Behaved Set). Consider θ
S I2m S R", arbitrary set-valued mapping θ R Suppose that for every e θ and &
A e = S, and differential inclusion F' where
(VCs) there exists 0 and an absolutely continuous function c - R" such that =(0) E. a(t E F(=(t), 0) for almost all E [o, c] and z(t) e S for all E (0, c].
there also exists some e' E (0, c] such that z(00, e'I) s int A(0) or x(00 e'I, s Sint A(O)
(15)
In such a case, we say that is well-behaved relative to S for è and F. For parameterized hybrid system p= (C,
F, D, S, θ), we sometimes say that A is well-behaved relative P.
Assumption 6 -eehveBe Interventional Subset). Consider set-valued mapping D O = S and parameterized hybrid system P. Assume D is well- relative to P (definition 15).
Observation 1 (Flow into Subdivisions of C by D). Consider set-valued mapping D D: θ = S that meets assumption relative to some parameterized hybrid system P F, the case that, for every O E θ and & S where
it then
(VC)
there exists >> O and an absolutely continuous function 2: [ R" such that =(0) E. ²(1) E F(=(t), 0) for almost all E [0,e] c] and z(1) E C(0) for all E (0,c).
there also exists some e' E (0, <] such that
z((0, e'J) C int D(0) or 2((0, E'I) s C(O) int D(O).
(16)
Proof. Suppose the proposed antecedent and note that a trajectory =((0, e]) s C(0) S S fulfills the antecedent of the assumed well-behaved property of D relative to S (assumption implies that there exists e' E eO such that either 2((0, e'J) S int D(0) or 2((0, e'I) s in D(O). 2((0, e'I) S int D(0) is precisely the first case of our desired consequent. Thus, we need only show that 2( S int D(O) and =((0,e) e]) S C(0) imply z((0, é'I) S C(O))intt D(0). We have =((0,e')) S e)) C(O), and can thus take the intersection see this implies the second case of the desired consequent:
This
=((0, eT) = C(()) n (Siitt D(0)) = C(O) int D(0).
Remark 1 (Universality of Assumption 6) If and only if assumption 6 holds relative to some parameterized hybrid system P, then it also holds relative to instint(P), instint;(P) (P), and lower(P).
Proof. Assumption 6 holding relative to P = (C, F, G,S,S,)) pertains only to F,S, O, which are unaffected by instint, instint,, and lower.
D
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B Space Augmentation
is often useful parameterize interventions, and fully expressive interventional semantics benefits from stateful jump maps/sets. Thus, it will be useful to establish primitive transformation that
simply augments the parameter and state spaces, without changing the component functions of the system. Subsequent transformations can then operate on this augmented system. Note that, in eq. [22 we write the transformed jump map in its expanded form as an ordered sel-velued map (definition
Definition 16. (Space Augmentation) Consider S R", and Q S R For any parameterized
hybrid system P F.F.D, with G (G1, GL), let, for all æ S, x S, 0 O
E θ,
x' = [x. a] 0' 0,0
(17)
C² (0') C(0) x
(18)
F' (z', F x)x,0)xx0
(19)
D' (0')
(20)
(z, 0) Ge (x, (i:
(21)
G' (x',0') (Gi (x, ...... G (z,00)
(22)
S' 8 5, 0
(23)
then
spaug ( (,,,
(24)
Observation 2 (Compositions of Space Augmentation Preserves Key Properties). Consider parame- terized hybrid system that meets assumption and any finite sequence (S,, Q₂) of length K such that Sy = TR and θ₂ = R.m. Let spaug, spaug(,, Sx. θx) (definition 16 and
p' (spaug,
spaug,
spaug (P).
(25)
P' then meets assumption and has a unique solution (propositions (definition) 12 time-parameterized solution map y (definition
and
2 with a 1+ measurable
Proof. The proposition follows from induction, K < 00, and the fact that the space augmentation operation fulfills the same pattern described in fig. 2 for instint. That is, spaug commutes with lower, and it preserves (P1-6) (assumption 5) on an upstream system P = spaug(P,... ........ For commutativity, recall that lowering makes a flow-preferring subtraction from the jump set (defini- tion 13), and chooses the last map the ordered jump map. A flow nlw-preferriig subtraction D'(0) D(0) S is dictated entirely by the behavior of F(0) on C(0) - i.e. preferflow D' C"" F*) preferflow C, F) S, which implies commutativity on D'. Commutativity of the jump map more straightforward, as, by construction, last(G") = Gz(Zz 0) x {Ø} = last (G) 0) x {z). Assumptions (P1-6) (collected in assumption straightforwardly follow after noting, as we have used in the proof of observation 33, that since every topological space is both open and closed in itself (i.e., clopen), any product with such a space as factor inherits the open (or closed) property from the other factor relative to the product topology. From here, along similar lines argued in the proof of observation 3 properties like graph closure, outer semi-continuity, Borelness, etc, are preserved obviously by construction.
C Static-Time and Do Interventions as Special Cases
As a special case of instint, we can also define an intervention that occurs at a fixed, predefined time
19 Note that, while this definition lets us analyze static time interventions in this theoretical framework, we do find computational implementations in line with the time- splittiin operation (Bocken & Mooij 2024) more practical.
21

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Definition 17 (Static-Timg Intervention). Consider parameterized hybrid system P defined as S. O). Let time be tracked in the first dimension of the state space.
the tuple (C, F, D, G.R2, R3 and, in the second dimension, variable recording whether the intervention has occurred, such that (1, k, a:) E RR x Assume k 0 at 0 by convention and that F is such that dk/dt
always. Let D = [A, A e] x [0, _J]]x S for all 0 E O, a fixed À > O, and any >
20
For some
S x + = S and all (t, kt n 0) TR x S x O. We then define
( x), 0) {t, 13 G (xx 0)
(26)
statint C) instint (D, (P)
(27)
The definition above, it should be noted, is a special case of a more general "repeated" static-time intervention rstatint (definition T9p, which is shown to satisfy the same existence, uniqueness, and measurability theory that we establish below for instint.
Driving one level more granular, we arrive at a transformation representing something akin to the canonical "do" intervention - again as special case instint. This notion has been defined for dynamical systems both via time-spliting mechanism (Boeken & Mooij] 2024) and by casting continuous time system as its infinitely precise Euler approximation interpreted as an SCM (Hansen & Sokol 2014).
Definition 18 (Do-Intervention). Building directly off definition 17 if Ō (2x, 0) {(v} for some fixed u E S and all æ, 0 E S x O, then for some fixed À 3 we write
do (P.x(M) aAA) =v) = statint (P.A.G)
(28)
Alternatively, one might wish to fix an index i e (1....,,) n) and a value a e R!. With G(xx, 0) = [x(1)-)), v, ((itin)] Væ S and v0 E θ, we write instead do (P, xx =).
(i)
These interventional classes form a sort of hierarchy. The jump map of a static-time intervention can be considered the "pre-treatment" model for a do intervention, and the trigger mechanism encoded in the jump set can be considered a pre-treatment model for when a static intervention occurs. At the highest level, a state-dependent intervention especially those that can be triggered many times can be thought of as a soft intervention on system dynamics. By couching these interventions directly in the language of established hybrid systems theory, we can more easily borrow theoretical results from that vast body of literature.
C.1 Repeated Static-Time Intervention
Definition 19 (Repeated Static-Tinn Tami Intervention). Consider parameterized hybrid system P defined as the tuple (C, F, D, G, TR x S,O). Without loss of generality with respect to positioning in the state vector, let time be tracked in the first dimension of the state space, and, in the second dimension, a variable recording whether a specified static intervention has recently occurred, such that (t, k, æ) R2 x S. Assume
0 at 0 by convention and that F' is such that dkjdt 0 always. Also, assume that, for some countable set of unique intervention times A C B there exists such that 0 < inf {|A1
>2l }1 Aa E A2, A * 22 o (.1). For all (t, k, x,, 0) IR3 S x θ
The construction ensures the jump set is "thick". With measure-zero jump set, there can exist solution that reaches the jump set and immediately flows through it, never jumping. A flowing solution is viable from any closed boundary of a jump set where a vector field implied by Fe points into the flow set. Incidentally, (uniformly) "thick" static-time jump sets also ensure that jumps cannot occur infinitely often. We do not include an argument here, as intervention's 's behavior is identical regardless of the choice of < > 0. The counter, k. is required to avoid repeated jumps from the thick jump following À time.
22

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and some G: θ Setet
Di(0)= (UIAA+ x [0..1] x
(29)
AEA
D2
U
[1,1.1] 8
(30)
AEA
Gi (t, x) 0) (tt,k + 1)} x (x.
(31)
G2 k, x)) 0) {(t, - 1)) x (x,
(32)
With instint;() instint( Dj, G.) for i {1, we can define
p' rstatint (P, G) (instinte instinti) (P)
(33)
Observation 3 (rstatint Preserves Collected Assumptions). Continuing from definition 19 if G
meets assumption then Pl, Dú, and G. meet assumption and theorem 1 would thus apply to rs istatint.
2 comprises cub-conditions (II) and (I2). (II) first needs that (0) is closed vo e O, which follows here from (.) being product of closed sets with the topological space S. Since every topological space is both open and closed in itself (i.e., clopen), any product with such space as factor inherits the open (or closed) property from the other factor relative to the product topology. Note that the intervals in the unions over intervals constructed from À E are guaranteed to be disjoint and uniformly separated by selecting to be positive and smaller than the closest two intervention times, which means the corresponding countable union must be closed. Similarly, by uniform separation, we have the (el)-reueree well-behavedness (definition 15) (+) relative to P. (II) also requires that the graph Gint Đ) is open. Note that these jump sets are constant in θ, and therefore their interior graph is the product @ A, where A R" is open. θ is topological space, so the product with A inherits the openness of A - thus Gint D) is open. By a similar argument, D, is closed, which implies that G(D:) is closed, thereby ensuring ((D)) is Borel as required by (II). (I2) asserts straightforward requirements on G, none of which are affected by taking cartesian product with the value,, continuous (and therefore both inner and outer semi-continuous) set valued mappings f, k - {[t, k + 1)}. Theorem | then, applies here because rstatint is composition of instint operations with specifications that meet assumption 2
Proof. Assumption
D Proof of Theorem
The following proof refers to assumption 5 which is an expanded version of assumption I that is referenced by theorem iin the main text.
Proof. By induction and K < cc, we have via lemma 2 that P' will meet assumption 5 Note that, by remark [ if assumption Sasuupptionhhhllle holdd relative to P, it will hold relative to any intermediate system in the chain of Transformations from P to P Then, by lemma existence, uniqueness, and measurability follow from P' fulfilling assumption
E Proof that Instantaneous Intervention Preserves Key Properties
Lemma 2 (Instantaneous Intervention Preserves Key Properties). Consider a parameterized hybrid
system P that meets assumption
5 Now, consider set-valued mappings D = S and G S that fulfill assumption relative to P. The intervened system P' instint(()) (definition 4 will then also meet assumption 5 and therefore will have unique and 1t measurable solution for each 0 E A, C(Ø) D(0) according to lemma 5
23

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Proof. The proof closely follows fig. 2 In assumption 5 we have that P can be constructed by "lowering" (definition [4] from a system Pf that fulfillss certain conditions. In lemma B we prove that assumptions] 2 and imply that the system P is equivalent to system reached by performing a slightly modified intervention on Pt (definition 20),. and then applying lower. The intervention on P1 is proven in lemma
E to preserve properties on the higher system sufficient to say that the
lowered system p' meets assumption Intermediate statements and proofs for lemmas 3 and
and definition 20 can be found in appendix
E.1 Intermediate Results for Lemma
Lemma 2 argues that an intervened system P' instint(P) can also be constructed by applying a slightly different interventional transformation to a different system Pt, and then "lowering" (definition 14). Additionally, if P meets assumption ( by way of Pt, then P' must also meet 5 This can be established by showing that the intervention on P1 preserves properties that allow be properly lowered. First, we will define this alternative intervention, then prove commutativity between intervention and lowering, and finally prove that the alternative intervention preserves the properties listed in assumption 5
assumption
In the following definition, we use the fact that G is an ordered set-valued map (definition 6). which supports appending G to the sequence of maps that compose G.
Definition 20 (Instantaneous Intervention for Higher System). Consider set-valued mappings π @ = S and S θ = S and parameterized hybrid system Pt (C, G, Now, let
(0) C(0) int D(0)
D'(0) D(0) D(0)
x E D(0) ((0)
last last(G) (ax, 0) G (z, 0)
GD (c.0) 0)
æ D(0)
E D(0)
G G 0)
GD(
I D(O)
G (zx,0)
G' = Gn u Gp
then
P = instint,
(34)
Since G' is an ordered set-valued map (definition 6) we can derive the following identity, which helps establish some useful intuitions.
G'(x, 0) = Gp(z, 0) U Gp(,, 0)
x E D(O))
G 0) G (x. u (x, 0) G (xx 0)
(35)
E D(0) D(0)
E D(O))((O)
Below, we additionally use the fact that last(G') GG-
Lemma 3 (Commutativity of instint and lower). Consider parameterized hybrid system P
(C, F, D, G, S, θ) that meets assumption 5 and set-valued mappings D θ = and G S x θ 3 S that meet assumption 2 relative to Pt. The following equality then holds, with instint acting
in definition and instint, as in definition 20
instint (lower D,)) lower (instint, (P,,D.C)).
Proof. First, we adopt the subscript convention, where we use as a symbol (not a variable) mapping to the intervention operation, and as symbol mapping to the lowering operation. The subscript il,
24

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for example, indicates system that has been intervened upon and then lowered. With this convention, we have
lower (Pt) = P1 (C,, Fi, Dj. Sr 01) instint, (Pr,D,C) = =P = -C,, Dj S, + instint (lower (P.), = =P = -Cli, Fls, Dis, Gli, St. 11))
lower (instint, (P.. - Pa (Ca, Fu, Da, Ga, Sa, Oat)
We now want to show that every element of the tuple Pi; equals to the corresponding element in the tuple Pu. We begin with tuple elements that unaffected by both lower and instint. These include the parameter space, the state space, and the flow map, meaning we trivially have that
(Fis, St. ① = (Fa, Sa, Qa) = (Fi, St. θ1) = (F, S, O,) S)
(36)
For the flow set, note that lower leaves it unmodified and that both the higher and lower overloads of instint list the exact same transformation on the flow set. Thus C₁. C. C, 0 C(0) int D(O).
We now show equivalence in the jump map largely straightforward effort despite its verbosity. Consider the system instint,(pt) (C,, F, D, G,, O). We have that G, is an ordered set- valued map that, when lowered, yields its last component:
D(O)D(() D(0)
last(G) (æc 0) G (xx 0)
Ga (x, 0) last(G.)(z.0)
(37)
Now we consider the path wherein lowering occurs first. We have that Gi (x. 0) last(G) (ar, 0). By plugging Gi into the definition of instint for a lowered system (eq. 33 equivalence between GR and G2 becomes clear.
Finally, we show equivalence in the jump set. In what follows, let C* C(O)iii D()) for all O e θ and let D J D refer to - D(0) D(0) we drop explicit dependence generally on 0 for brevity. Also, let the set VA {E S (VC) holds for relative to flow set A(0) and Fo). We can write the "intervention first" path as
D. Dun D Du preferflow (Di,
preferflow (DUD,C'
[DDUD] Vo Di f LDV
[
preferflow (D,C'F) preferflow (D, F).
Following the "lower first" path and looking instint as applied lowered systems (definition we have
D. preferflow (D,C,F)
D3 preferflow (D,C,F) DD
preferflow (D.C,F) preferflow (D,
Now, note that because ("( S C()), have Vor s Ve, and therefore:
Da preferflow (D,Cr,
preferflow (D,C,,F)
Divo DVVe
[iVo [DVVc] = Dn.
25

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Additionally, by assumption and observation we have that Ve S V D Vor S Du Ves, where the second subset relation follows from the closure of D- nothing can "flow into" the interior of D without being in the closure that interior. This leads to
Da: preferflow (D,C',F F)
preferflow
DiVo
DVVel
DiVo
D (D
DiVo (DVo nD
(DIVO) u (D(Vo.)
[DDVor) ucD]
(DIVo) (W ( (IVorcc)
(DIVa) u (D(V 1
2 (DVVo) u (D\Vor) VV
= DVee u =
With both Dú: 2 Da and D 2 Du. must be that Dj; Da-
This concludes the proof of equivalence between every element of Pis and Pü, meaning Pa Pa-
Lemma 4 (Intervention on Higher System Preserves Key Properties). Consider parameterized
hybrid system P = lower that meets assumption
5 and set-valued mappings D θ = S and S θ = S that meet assumption 2 relative P. Now, consider the following systems:
p instint (eatiin,,P,,,D.... D.,
(38)
p' lower (Pt)
(39)
Then P' satisfies assumption 5
Proof. We break the proof into six parts, one for each of the preserved assumptions listed in assump- tion Recall the explicit form of Pi given by eq. (34).
Basic Hybrid Conditions (P1). To show that instint, (definition 20 preserves assumption 4
we proceed through the three sub-conditions (A1), (A2), and (A3).
For (AI), we must demonstrate closure of the intervened jump and flow sets. For the flow set, note that by definition 20 ()) C(0) int D(0). and that by (A1) holding for Pt, C(Ø) is closed. ("()) is thus the result of subtracting an open set from closed set, and therefore closed. We have similarly required that ((0) closed. Definition 20 specifies that D' D(0) J ((0), which the union of closed sets and therefore closed.
For (A2), note that since the flow map Pg unchanged, it trivially remains outer semi-continuous. We then have that (²( S C(0), from which we can conclude that local boundedness relative C(0) implies those properties relative to C²( Additionally, we have that C" (0) c C(0) dom Fg.
C(Ø) and convexity of F(x, (( 0) for every
Finally, for (A3), require the outer remiicontinuing of G²g. its local boundedness relative to D)) and that D'(0) c dom G²- The following arguments closely mimick the developments in Definition 2.11 and Lemma 2.21 from Trom[Sanfelice (2021) they show that the composition of hybrid "plant" and hybrid "controller" into a closed loop hybrid system will meet the basic conditions if the plant and controller meet those conditions. In what follows, work with the identity of G" derived in eq. (35).
Outer semi-continunity of G" means that for every convergent sequence (æx)) D'( to and every convergent sequence (x:² S to t where 2, e G' (xx 0) for each i, we have that t e G' G'(x,, 0).
26

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Note that this is equivalent to graph closure. Now, by closure of D(0), ()) G(Go), and ((G)). the only potentially problematic limiting points of sequences lying in D(O) D()), or D(0): D must lie on the intersection (() D()). The intersecting piece, however, returns G(æ, 0) (æx, 0), which will necessarily contain those limiting points.
Local boundedness of Gg relative to (O),, then, follows from the local boundedness of Ge relative to D()) and of Go relative to (()), and the fact that Go and Go are queried by G only from the sets on which they are locally bounded.
Finally, we need that D') = dom Gig. Recall the piecewise construction of G. in definition 20 and that (()) dom Ge. D(O) dom Gg. We can then write the following, where we again drop dependence on 0 for brevity and write G' u G in place of G(æ, 0) G(z,
[(DVD) o dom c] u [(Dn)) o dom G c [ (B!D) dom c]
dom G'
c DU (DD dom G G| u - DuD D'
Thus, we have that (AI), (A2) and (A3) are all preserved in P7 = instinty (Pt), meaning it meets assumption
Basic Existence (P2). To show that conditions for proposition are preserved, we recall from the proof of lemma 6 that it is sufficient to show that (VC) met (with respect PP) for all &E D'()) for any O E e.
Ignoring whether the flow appropriately remains in the transformed flow set ("))), we know by assumption 3 that there must be some c > 0 amount of time from which some continuous function can flow from every & e S while respecting the differential inclusion. To confirm that (VC) holds at & for P/, we can check whether some c' E (0, c| exists where 2tt) E C"(@) for all t e (0, él1. First, we can decompose the region where (VC) must hold into a union over two cases.
C'O)) int (*(O)) [aC((O))((O))
(40)
E int ('(O) s int (O),), there must be some such e' by the openness of int ("()) in (O)
We can then decompose the boundary region (C) follows, where we've dropped the dependence on for brevity.
aC% =à =olco Cint nc [uu] [DuuD
(oC o Cint D) u (C n Mint D) n CD
(EC n Cint D) u (C
kCDnCD
JaCo Cint D n CD CD u (CneDnCDocD)
[oor (DuD) [cn (aD.D) ct
-eo, (D D) = ECAD.
The first equality in the final line follows from the assumed closure of D(0) implying that aD(o) ((0) = Ø.
By analogous decomposition eq. 440 we have that & E (C(O))D(O) must meet (VC) with respect to P,. By assumption 6 and observation [ we then know that the solution must "flow into" either int Đ or into C(0) int (()) Because D is closed, flow into its interior requires that E D(0) = (())), which we need not consider. This leaves only flows into C(0) int D(0), which by construction (definition 20 is equivalent to C²()), and therefore satisfies (VC) with respect to P with e' as described in observation existence outlined in proposition
Thus, instint, (definition 20) preserves the conditions for
27

---

Unique Flowing Solution Everywhere (P3). Assumption 3 is preserved trivially, since instint (definition 20) does alter F, O or S, which are the only system elements involved assumption
In the remaining results, we use the following observation, leaving its verification from definitions to the reader.
Observation 4. Let A, B.B: θ = S be two set-valued maps. Then the graph and set operations commute:
((ABB) ((A))(()) G(AnB) - O
-)
Outer Semi-Continuite of the Flow Set (P4). We need to show the outer semi-continuity of
C" at every 0 θ. Recall from definition 20 that (*) int D(0). By observation 4 g(C") ((C(O)) ((int D()).. By assumption
have the outer semi-continuity of C, which directly implies the closure of its graph. By assumption 2 have that (int D(0)) open. Thus,
((C") is closed, and therefore by (Goebel et al. 2012 Lemma 5.10) C" is outer comiccontinuous
Borel Jump Set Graph (P5). Recall from definition 20 that D'(0)= D(0) o D(O). The Borel a-algebra is closed under unions, and thus by observation the graph G(D') must also be Borel.
Borel Measurable, Single- Valued Jump Map (P6). We want to show both that last 0) = {g' } for some g' and all 0 dom last (G') - i.. that last(G") is single valued and that g' is a measurble function of initial conditions and parameters on the domain of the intervened jump map. By definition 20
we have that
x D(O) D())
last(G) (x, 0) G (zx 0)
last(G") (x. 0)
(41)
Now, we have by (I2) (assumption 2 that G single-valued, and by (P6) (assumption 5) that G is single-valued. Thus, there must be some g' such that last(G') 0) [g' 0)} on the domain of last(G").
We now want to show that g' is equivalently write the following, where use the lower-case and g in reference to the functions that yield the singletons arising from evaluations of & last(G).
for every æ, E dom Last(G"). Note that we can
g' [z, 0) = g(x 0) I x D(O).D(O)) Z(æ, 0) T D(O)).
(42)
Note now that the indicator functions involving the jump sets can be written as piecewise functions over a partition defined by the graph of the jump sets. We have assumed that the graphs of D and D Borel. Further, by observation 4 both indicator functions can be written piecewise over Borel partitions, meaning they must be Borel measurable. Again, by (12) (assumption) 2 we have that and f are Borel-measurable. Therefore, o' must also be Borel-measurable.
Having shown the preservation of each sub-condition listed assumption this concludes the proof. Indeed, if P, meets those uub-conditions, then P/ will well. In other proofs, this result can be trivially applied to conclude that lower(P() fulfills assumption 5
F
Proof that Lowering Induces Existence, Uniqueness, and Measurability
Lemma follows immediately from the following, more precise statement.
Lemma 5 (Existence, Uniqueness, and Measurability of P). Consider parameterized hybrid system P that meets assumption uniqueness (proposition and tt measurability (definition 12
S P. then, fulfills the conditions for basic existence (proposition) 7]. basic
Proof. This result follows directly from combining lemma and corollary both stated in the following sections.
D
28

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F11 Lowering Preserves Existence and Induces Uniqueness
Lemma (Lowering Preserves Existence and Induces Uniqueness). Consider parameterized hybrid
system Pt that can be lowered (definition 14 construct system P meeting assumption 3 P then fulfills conditions for basic existence and uniqueness (propositions) 2).
We split this proof into two components, one for the preservation of existence, and another for the induction of uniqueness.
Lowering Preserves Existence
Proof. Recall from definition 14 (lowering) that P' = (C, D, D', G', with
D' - (( S (VC) holds for E
(43)
Now, pick some E θ and note that C(θ) = C(0). which follows from the basic condi- tions P asserting that C(0) is closed. By P fulfilling proposition [ have that every ((0))(()) must meet (VC). It will be sufficient, analogously, to show that every & C(O) D' also meets (VC), which precisely the same condition because lower affects neither C' nor F. Note that
E [C(() u D(O)
C()))D'(0)
((0)) (5 (VC) holds for E}]
C(0) [[D(0) (g S (VC) holds for E)]
=C(0) o [CD(() u { S (VC) holds for E]]
= (
{ E S: (VC) holds for {]]
- {[C(O))DO))
(E S: (VC) holds for E
This yields two cases under which we must check that (VC) holds. For the first case, we know already that (VC) holds for all & E [C(0)) D 0)). For the second, we have by construction that (VC) holds. By the subset relation, these cases subsume the desired set.
We thus have our sufficient condition, that (VC) met for any & e D(O)) As we have placed no constraints on 0, this holds for the entirety of b.
Lowering Induces Uniqueness
Proof. Uniqueness for P' involves three conditions on the hybrid system as stated in proposition 2 which we will review in reverse order of complexity. Before proceeding, recall from definition [4 (lowering) that P = F,, D', G., S, O), where for all 0 E and æ E S
D'(0) (E S (VC) holds for {
(44)
G' 0) last(G)(z,
(45)
Recall, also, the convention that (z(xx) Gv 0) for all 0 S x O. Now, pick some 0 O.
Condition (a) requires uniqueness solutions to the differential inclusion on flow set. This condition is precisely what we have presupposed in assumption 3 except that we make the stronger claim that uniqueness holds for any flow in S C(0).
Condition (c) requires that Gy is single-valued on the jump set. We have assumed in (P6) that last(G) is more than single-valued on dom G. which implies that it is single valued on dom Go for every fixed e O. Additionally, by construction that
and the basic conditions on P. we have
D' = D(0) = dom Go = dom G
(46)
Thus, G¹ is exactly single valued on D'.
29

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Condition (b) requires that the solution cannot flow from the overlap of the jump and flow sets. Precisely, for every & E C(0) o O)), (VC) used in definition 13 does not hold. By assumption on P. have that C(0) C(O), and recalling definition 13 is sufficient to show that (VC) does not hold for any & in the following set:
((0) D') =( o ( (VC) holds for E] =0(0) n D(O) 6 (VC) holds for { =C(0) n D(0) ( -(VC) holds for {
{ S: -(VC) holds for E)
This concludes the proof.
F.2 Lowering Induces Measurability
We first state sufficient conditions for measurability, and then prove that sufficiency. Ultimately, this yields corollary stating that lowering induces measurability, We make use of the intermediate results and definitions established appendix F3
Assumption 7 (Collected Conditions for Measurability). Consider parameterized hybrid system (C, F. G, O). Assume that P
(MI) is 1 uniquely evaluable (definition 10
(M2) has unique solution to its differential inclusion everywhere (assumption 3);
(M3) has an outer semi-continuous and closed flow set C(0) every 0 O;
(M4) G is single-valued on dom G, with G(x, 0) {g(x; xx 0)), and g Borel-measurabe for all
x, 0 dom G.
Theorem 2 (Measurability of Solution). Consider parameterized hybrid system P
(C, F. D, S.S.S (,,SS.) θ) and its time-parameterized solution map yp (definition 77'. If meets assump- tion then yp is t+ measurable (definition 12
Proof. Under assumption 7 finite jump times and values are Borel measurable in E, 0 (lemma 7).
Additionally, under assumption 3 the solution is mossl-measurable in up the first jump (F2-3). We are thus able to write the tine-parameterized solution as follows, where to 0) = 0 always. For all & [0, (. & S, and 0 O:
8 Σ T()-1((5.0) a EL< Φ (1-0--(,) (,,,,)))))))),,, (47)
El::E,
which comprises a countable sum over Borel-measurable functions of &, 0, and is therefore it- self Borel measurable. Note that, while we have only shown Boael-measurability for Ey 0) o(t,-1(5, f)
J: E, 0) when (E < tt, the joint requirement that (y-1 (E, 0) & t < avoids those unmeasurable cases.
Corollary (Lowering Induces Measurability). Consider parameterized hybrid system Pt that can be lowered (definition 14) to construct a system P meeting assumption 5 P then fulfills conditions for the t+ measurability of its time-parameterized solution map yp (definition 11
Proof. Assumption 5 when combined with the fact that "lowering" (definition 14) induces unique- ness and preserves existence (lemma 6). subsumes or implies conditions sufficient for the result (assumption and theorem 2 In particular, (M2) maps to (P3), (M3) maps to (A1) and (P4), and (M4) maps to (P6). For (M1), note that 4 measurability requires only 1* 0 addition to existence and uniqueness, which come from lemma
Proof of lemma 5 This result follows directly from combining lemma 6 appendix F. and corol- lary 2 above.
30

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F33 Measurability of Jump Times and Values
Definition 21 (Flowable Region), Consider parameterized hybrid system P (C,
For all 0 E θ let Cp (0) denote the set of states from which there exist flowing solution (respecting Fo) that remains in C (0) after its start. Precisely, this means that there exist 6 > and an absolutely continuous function
S such that z (0) CF (0) and & (1) E (= for almost all t E [0, cI and z (1) C' (0) for all i E (0,
[0,
Observation 5. Consider parameterized hybrid system p D, S, ②) that has unique solution to its differential inclusion everywhere (assumption 3 and where C(0) is closed for every 0 E O In this case, the closure of the flowable region is (+( C(0).
Proof. From assumption 3] for every E θ, we have that an absolutely continuous function & exists from every =0)
S Cr(0) that satisfies Fe. Every interior point E int C(0), then, must be in Cr(O), as some flow must be possible while remaining in C(O). With the closure of the flow set, we thus have C(0) int C()) _ Cf))). Now, for points & E inint C(0), note that flow into C(0) is only possible from 2C( S C(0). This ensures that Cr(0) cannot contain points outside of (()), further implying that (0) S C(0) = C(O). Thus, by a two-sided inclusion, we have
(0( C(O) for every 0 E O
Lemma 7 (Measurability of Jump Times and Values). Consider parameterized hybrid system P (C, F, D, G, Q) and its solution map &. If P meets assumption 7 then the time of the
j > O'th jump,
t, (E. 0) sup(t (t.j-1) 1) dom o (..:E,0)) R30 U (00)
(48)
Borel measurable function of &. 9.
Additionally, the solution values at these jump times
Ej+i 0) ott (E.0) ;)
(49)
are also Borel measurable functions of f, 0 (E, < i+.
Proof. Let t ().0)- = =supft (t, e dom o E 0)} be the first jump time. Note that if the set [(S, 0) (E 0) > a) is Borel for all E R. then & must be Borel measurable. Indeed, we can write that set as a countable intersection of Borel sets, which implies Borelness. Below, we use the closure of the "flowable region" Cr()) (definition 21 to rewrite the pre-image on S x θ of the first its closure in order include the time
jump occurring at after time a. In particular, which the jump occurs (by including states that flow can reach but not flow from). Note also that, by observation 5
we have that (0) C(0) under conditions already provided in assumption We
have
(): :1 (E.0) > a} = (S. 0) & (T, (T,00; 0) E CF (0) = C(O) Vr E [0,
((E. 0): (0, 0 (T. 0: ( (C))
Assumption requires that C(0) is outer semi-continuous all 0 θ. This holds if and only if its graph 9 (C') is closed (Sanfelice, 2021 pg. 49). Closed sets are Borel, so g (C) must be Borel.
The Borelness of {(E, 0): (0, 6 (r. 0: &. 0)) g (C)} follows from the Borelness of G (C) and ø being continuous c, 0 on [0, (]. and therefore Borel measurable.
The ability to write the set as a countable intersection over rationals is justified by the standard argument. For any fixed 7 E [O. choose a sequence (7,) c Q [0, al such that 7n - T. This is always possible due to the density of Q n [0, a] in [0, If for each n N we have
(0, TT 0; &, 0)) e G(C),
(50)
then, because & is continuous in time and ((C)) is closed, the limit is also included
0 lim (TT,, &, 0)) (0, (0, 0) e G(C). 10000
(51)
31

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Countable intersections of Borel sets are Borel, and thus ti must be Borel measurable.
We must now expand from the measurability of the first jump to the measurability of all jumps. We can write the second jump time as follows, with being a function that, when evaluated, returns the single value of G (M4).
E2 0) ( (t1 (E. (.00)00:000) 0).,0))00)
(52)
12 (E. 0) L1 (E. 0) t (E2(5, 0).
(53)
We have that g is hybrid system (definition [].. D = dom Gg. and therefore know that g will only be evaluated where it is assumed to be measurable (M4). Additionally. we have that c is
the domain of G (M4) and that, by definition of a parameterized
for
j = 0 up to and including t (E, 0) (F2-3). Thus, &2 is the composition of Borel- measurbee functions and is therefore itself Borel-measurable. 12, subsequently, is the sum of meael- function and the composition of Borel-measurable functions. The measurability of t for j > 0 then follows
from its recursive form. We use h h!')(r) to represent the n n-fold composition of h with itself. By
standard inductive arguments we have
ho(x) 9 (0 (t1 (z, 0), 0, 0), 0)
(54)
f (5,0) 0) = (E); (1 0)
(55)
ty+n (E. = Σ (E. 0)
(56)
As l5+1 comprises only sums of compositions of Borel-measurable functions, it must also be Borel-
measurable. Additionally, note that (,+1(E, 0) o(l, (E, Ej: 0) can also be written as the composition of Borel-measurable functions, thereby proving the measurability of jump values. D
G Probabilities of Causation and Fishery Management
G.1 Historical Context for the Fishery Management Problem
Notions of causal necessity and sufficiency are often productively employed in policy discourse, especially where competing interests require sammanunderstandabbl justifications as whether particular policy is sufficient and/or necessary to achieve desired outcomes. Recall the control theoretic settings involving state-dependent, instantancous interventions that we have enumerated in the introduction: health-related lockdown measures, interest rate adjustments, and many engineering problems involve cost benefit tradeoffs, where policies are designed to be sufficient for the benefits, but only as costly as necessary. In modern resource management, for example, tragedies of the commons frequently demand a challenging balance between ecological objectives and short and long-term economic outcomes. Additionally, such cases often involve models that our interventional semantics is designed to operate on.
Fishery management offers particularly rich set of problems where the probabilities of causation can help streamline policy discourse. Over the last few decades, numerous fishery management crises have followed similar arc: first, growing markets and new technologies result in overfishing to unsustainable biomass levels; then, regulators impose strict catch quotas, gear restrictions, data collection requirements, area closures, and other measures designed to allow stocks to rebuild; after rebuilding stocks, fishing resumes, ideally at more sustainable levels. In 2000) for example, the NMFS and NOA 2T announced emergency regulatory measures in response to the failure of the Pacific coast groundfish fishery (Anon) 2000). This was followed by an economically tumultuous rebuilding period of around 10 years (Warlick et al. (2018), after which fishing restrictions changed and loosened (Anon 2010a). Similarly, the 1990s saw significant declines in the Atlantic swordfish fishery (Neilson et al. (2013). In 2000, the ICCAT established an ultimately successful 10-year plan to rebuild the stock (Neilson et al., 2013 Anon 2010b).
Naturally, these measures were not without significant economic consequences and backlash, both short and long term (Anon 2000) Cramer et al. 2018 Anon 2007b). Indeed, the United States,
Thaat is, National Marine Fisheries Service and National Oceanic and Atmospheric Administration. Thaat is International Commission for the Conservation of Atlantic Tunas.
32

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the Magnuson- Magnuson-Stevens Act (MSA) mandates the multi-objective of avoiding unnecessary economic sacrifice while pursuing long-term economic and ecological outcomes (Anon 1976 1996 2007a
2018). Myriad ecological and bio-economic dynamical systems approaches were developed during and after these crises to better balance competing objectives ([.ee et al. 2000) Ortiz et al. 2010
Restrepol et al. 2011 Taylor et al. 2022). On some occasions, post-mortems were employed to, for example, determine the degree to which rebuilding success was due to management actions or to natural factors such as species biology Neilson et al. (2013). In essence, the goal of such efforts, as stated in the MSA, is to identify and implement suffcienn rebuilding measures that would induce no more economic hardship than necessary.
G.2 Formal Probabilities of Causation
The formal definitions of the probabilities of causation were originally provided by Pearl 1999). These queries are traditionally defined for binary treatment X and outcome Y variables we enumerate those binary definitions here, and then develop some intuition. our fishery management example (section 6). we expand to the non-binary setting in keeping with definitions provided by Kawakami et al. (2024) for scalar treatment and outcome variables.
Definition 22 (Probabilities of Causation). Let X, Y be binary variables within structural causal model M, and let æt x', y- y', denote the propositions X 1, X 0, Y 1, 0 respectively. Denote by Yr and Yu the counterfactual outcomes obtained by performing the do-interventions do(X = 1) and do(X- = 0) 23 The probabilities of causation Pearl 1999), then, are defined as
follows:
(N(x,y) = P(Y, = 0 x, 2,,),
(57)
PS(x,y) = P(Y, ,', PNS(,, =- P(Y, 1, 0).
(58)
(59)
(N y) quantifies the probability that x was necessary to produce outcome y; PS(x, y) quantifies the probability that x alone would suffice to produce y; and PNS(x, v) jointly quantifies the event that a is both necessary and sufficient for outcome y.
To compute the probability of necessity, we consider only (condition on) worlds where the events a and y occurred, and then evaluate the probability of Y being false if we intervene to make X false. Similarly, the capacity to produce an outcome - the probability of sufficiency - is computed by conditioning on X and Y being false, and evaluating the probability of Y being true we now intervene to make X true. A notion balancing the dimensions of necessity and sufficiency is the probability of necessity and sufficiency, which not function of the separate probabilities. To evaluate PNS, we do not condition either way 24 but rather evaluate the probability that both intervening to make true results in Yz and intervening to make X false results in Y, 1.
G.3 Multi-Year Year Horizon
In our analysis of the fishery management problem, we analyze only the year-long time scale, but we can define multi-season model with an arbitrarily long time horizon. Note that, here, we will need to prepend time to the state vector, which becomes [t, z, h1.3, b1:3] = x E S = R$. See table for full labeling of model parameters and states.
First, model the season's 's starting condition via a jump set that triggers at the beginning of each year. Jumps at the scason's start obey map that (1) resets the integrated catch to zero and (2) sets fishing harvest rates to their noisy, non-null values. Let 0,,, ~ ~N(,7,.07) and On, ~ (V07, .007) be elements of 0, season-start times be Zzo (i.e. the beginning of each year), and rstatint (definition 19 be a generalization of statint (definition [7) that applies the jump map at countably many times
Let Po
F, S, +) describe the fishery in its unfished, natural state, where
The canonical do intervention do(Xx, a.)) fixes the structural equation for X; to a constant, ie.. 8; = 21. AAny conditioning here would bias the outcome. Suppose X and Y are causally disconnected. If you condition æ, y. you make Y, true by default, and you condition x', v/ you make Y, false, effectively making one of the components satisfied for free, which is undesirable.
"Definition 19 also requires binary auxiliary variable to be part of the stae-space - without loss of generality. however, omit that here for smoother exposition. Additionally, this construction assumes the same
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Trajectories of harvesting policies apex pressure)
27.
Intermediate pressure)
intermediate
(queta
127-
19 32
10
Catch under three harvesting policies pressure
pressure
custa-50
60
10
6 1011
13311
maatth
mardh
Figure 3: Examples of the biomass trajectories of apex and intermediate predators, as simulated from the model proposed by Zhou & Smith (2017). The panels comprise simulations with, (left) no fishing pressure, (center) fishing pressure kept up throughout the year, (right) fishing regulators ending the season when reported catch meets the total allowable catch (TAC) quota of 50 biomass units.
name
notation in season
after season
time
total catch
fishing pressure forage
h1
0
0
fishing pressure intermediate fishing pressure apex
h2
h2 Normal(.7,.00
0
f13
h3 Normal(.07,.00m
0
biomass forage
b1
biomass intermediate
by
biomass apex
by
desired outcome lower threshold
Y
130
130
TAC quota
qi
Table 2: Parameters and notation for the fishery example.
C R 20. and ssimlyy indicates the irrelevance of the jump map in the natural state of the
=
fishery.
G. (x. 0) ([t, Oxgr d1:a]}
(60)
P. rstatint (Po, 2,0
(61)
The season's end can be described by setting the harvest rates to zero when the catch exceeds threshold (with i E 21). From these, we can construct parallel worlds with the same random initial conditions and parameters.
fishing pressure year over year. This can be generalized to independently sampled pressures at each year, though some additional theoretical machinery would be required for an infinite time horizon. See Teel & Hespanhal (015)) for one possibility.
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Da. (0) Rş0 f Rs0 2 ai) R
(62)
Gg. (z. 0) {[t, 2,0,0,,, b1:31}
(63)
Pg. instint (ps: Du Gg.)
(64)
R, (Ps, E, 0); R. (P E, 0); Ruy = (P &, 0)
(65)
G.4 Narrative Fishery Management Example
In the main body of the paper, we emphasized the construction of the probabilities of causation, rather than their application. Still, some readers may appreciate a more narrative structure around these concepts. We provide that here.
Example 5 (Probabilities of Causation for Total Allowable Catch (TAC) Quotas). Now, suppose new commercial fishery is being opened up and that, in the first year, fishery managers allow commercial fishing year-round. R2 models this world (or equivalently, Ry when the TAC quota q1 is sufficiently large so to have zero probability of being reached). This results in a failure to preserve the intermediate level biomass above the desired level y. Suppose y = 130 units. Facing ecological scrutiny, fishery managers ask: given that we allowed year-round fishing and failed achieve our outcome, what TAC quota would have a high probability of being sufficient for success? This is a probability of sufficiency query. They introduce a strict TAC quota of 30 units with a relatively high probability of sufficiency (fig. 4. In the next season, they succeed in meeting biomass targets. Subsequently, however, economic interests and local representatives insist that such a low, strict TAC was not necessary to achieve this outcome. They point out that, in comparison to a more lenient TAC of 50 units, there is a low probability that the strict TAC of 30 was necessary (fig.5). Fishery managers, in turn, worry that the probability of success with a TAC of 50 might be too low. Before the start of the next season, stakeholders resolve the disagreement by identifying a TAC that yields a high probability of sufficiency and necessity when contrasted with year-round fishing (R,),, all while avoiding stricter catch limitations that are not justified by gains in the probability of necessity and sufficiency (fig. 6
The simulated results presented here ran on a consumer grade laptop in the order one hour.
G.5 Event-Time Attribution
In the main body of the paper (and in example we analyzed the probabilities of causation as they relate to contrastive policies. In other words, we asked causal attribution questions at the policy level. Queries about the probabilities of causation, however, such as "was x necessary to achieve y", are ambiguous when a real world event x: is multi-faceted and potential alternative actions are plentiful. In example 5 we mapped the events x and x' onto particular TAC quotas. In this next example, however, we will define our event of interest involving the time at which an intervention occurs. We can make this precise by constructing twin worlds using the tools provided in this paper pparticularly by additionally employing the static intervention statint (definition 17)
Example 6 (Probability of Necessity of State- Deppedeenndntt Intervention Timing). Consider worlds where the season ends before some time X, and biomass goal 7 is achieved later time (for instance, at the end of the year). Fishery managers wonder whether they might fail to meet their biomass goals if, contrary to fact in those cases, the season had ended or after time X. The relevant question is: was ending the season before time À necessary for achieving the biomass goal? We can answer this question by asking a probability of necessity query. Unlike in example however, the causal attribution question relates to the time at which the intervention occurs. Consider the following binary predicates, where T (vpg.) extracts the time at which the season ends due to the TAC quota 26
q
X I[T()))) << X]:
Y= I 7
(66)
26 See appendix ! for information and general notation for extracting event times from solutions to hybrid systems. And note that, in lemma 7 we prove that jump times are measa-measurable in and 6.
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Probability of sufficiency: 0.87
Probabilities of sufficiency
1.05
Intermediate biomass faciual
175
factual qunts
75.0
125
0.45
100
0.30
25.0
0.15
0.00
100
250 Intermediate biomass, counterfactual
125
150
175
200
225
275
0.0
25.0
50.0
75.0
100.0
counterractual quota
Figure 4: Step the example narrative (example 5 Within the first year commercial fishing, the fishery has quota (here it is enough to set it to q 120, which is never met), and falls below sustainable biomass. Conditioning on this failure, the regulators seek an intervention with a high probability of changing this outcome next time along the counterfactual dimension. They implement a strict TAC of 30, evaluating the probability of sufficiency (the probability of achieving sustainable biomass above the desired threshold of 130) to be 0.87.
Probability of necessity: 0.18
Probabilities of necessity
Intermed lete biomess, feciual
0.7
100.0
quota quota
200
75.0
0.5
factual
on
150
50.0
0.3
0.2
25.0
out
out
100
75
100
125
150
175
200
225
250
0.0
0.0
25.0
50.0
75.0
100.0
0.0
Intermediate counterfactual
counterfactual
The season ran with a TAC 30 units, and the intermediate biomass target reference limit 130) was met. Conditioning interested in increasing the fishing quota ask whether such a low TAC was necessary. They seek an alternative quota along the counterfactual dimension that, when contrasted with the factual TAC of 30, reveals the factual TAC as probably unnecessary. As a counterexample, they choose a TAC of 50, which yields relatively low probability of necessity (.18).
Figure 5: Step the example narrative (example 5
this, parties
Probability of necessity and sufficiency: 0.61
Probabilities of necessity and sufficiency
Intermediale tiamess, baveline
0.7
180
0.6
160
status qio
75.0
0.5
0.4
120
50.0
0.3
100
0.2
60
fium
0.0
0.0
100
175 Intermediate
200 225
0.0
25.0
50.0
75.0
100.0
proposed quota
Figure 6: Step the example narrative (example This time, before the season starts and prior seeing what the outcome will be, both sides aim to find quota with large probability of both necessity and sufficiency. They contrast proposed TAC quotas with baseline, status quo TAC of 120 units (never met). They notice that the probability surface flattens out above -60, meaning further improvement in the probability of necessity and sufficiency would require excessive limitations in quota. Ultimately, they agree on quota that results in value above 0.6, i.e., TAC quota of 0.35.
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Conditioned counterfactual biomass trajectori es for three time thresholds
end
Jun-29
209
and date Aug22
182
biomme3
360-
242
166 14b
167 1aa
14n 76g
App D0c
month
Figure 7: Samples from the Bayesian dynamics based on the fishery model presented by Zhou & Smith (2017), but with the season ending at different times. We show three end dates and their effect on the biomass at the intermediate trophic level.
Recall that the probability of necessity is P(Yao(X-0) 0 x 1) (with shorthand P(u', x. v). see table To coherently characterize this in our example, we must define what means to perform the intervention do(X = 0). Given our definition of the predicate X above, do(X = 0) suggests intervention that results in world where the season ends at after X, with all else (such as the noise or the resulting fishing pressure of O) remaining equal. Importantly, there are many such worlds, which means the probability of necessity must adopt existential flavor: "under exogenous noise where = and Y 1, what the probability that all worlds consistent with intervention do(XX= 0) fail to meet the outcome? To precisely define this set of interventional worlds, we build off notation from example55 [3] and consider a twin world under a static intervention occurring some time X X, but with the same interventional jump map utilized in the world Pa
PPx x }} where Px = statint (Ps,X,
(67)
As described following definition 18 the trigger dynamics of the detendent intervention can also be considered a sort of "treatment mechanism" determining the time which static intervention occurs. By constructing a world where direct control over the intervention timing is possible, we able disentangle these mechanisms. Importantly, note that while Px is constructed via transformation on P., it is equivalent to single-season world constructed from an intervention on Pg, that directly controls the season-ending time independently of causally upstream events in the system's 's simulation.
(o(x-0) =0 = I[b 7 AX > A
(68)
Note that by, monotonically decreases as X' - 30, then we can equivalently write the event y as L[b²> <.] Indeed, under our model and distributions on & and 0, this is the case, and so we can finally precisely express the probability that ending the season before time À causally necessary to achieve the biomass outcome:
Ply x. = [b <7] [T())) A], > >])
(69)
Unlike in example 5 conditioning on the factual interventional event is required, because knowing that the season ended before À carries information about the model parameters: the carlier the TAC quota is met (at times prior to A), the faster the catch rate. Faster catch rates stem from some combination of higher fishing pressure (f1:)), higher initial biomass (b1.3 at = 0), higher growth
This can equivalently be stated: ...ooobabilityy that there does not exist world under do(X = 1) where the outcome is met." See recent work by [i & Pearl (2024) and Kawakami et al, (2024) for more information on how non-binan binayy variables lead to these kinds of existential statements. This also relates to abstract interventions. which has been studied by Beckers & Halpem 2019). Beckers et al. (2019).
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Probabiitt necessiry
Impect of intervention times on probebility of necessity and on blomass
1.0
conditoonal countatlacccel
uncend tooed courterrsctuul
Blomesss at
130
pay $40.00
340000
parter
$4.05
Juay 12 Oug
DMV
Figure 8: For each X; from a grid of intervention times we (1) condition on the season ending before X and on the successful outcome, and (2) we intervene so that the end of the season occurs at X, The top panel shows the probability that ending the season before X; was necessary to achieve biomass targets, while the bottom panel shows the counterfactual biomass distribution under interventions ending the season at various Ag. the violin plot, we differentiate between counterfactual uncertainty for all worlds (orange and gray), and counterfactual uncertainty after selecting only worlds where the TAC quota was reached before X, and biomass goals were met at т. In other words, the gray violins show biomass probabilities if regulators had ended the season at X in cases where they ended before Dj and met biomass goals. The probability of necessity, then, the proportion of the gray distribution falling below the target level of y-
rates, etc., all of which influence whether the biomass target will be achieved under alternative season-ending times, even after conditioning on success in the factual world 28
Returning to the example, consider range over fixed threshold X. approximated by a finite sequence (A,). For each A we (1) condition on the season ending before X and on the intermediate biomass at T being above y, and (2) intervene that the end of the season occurs at X; (and not earlier). The relevant probability of necessity query is whether intervening before X; was necessary for the success. That is, for each X inspect the posterior predictive distribution of the intermediate biomass at under the intervention, and inspect the probability that this outcome is below yy The results of an estimation are available in fig. g
The simulated results presented here ran on a consumer grade laptop the order one hour.
H Folling-Tannee Fishery Model
The fishery management model presented by Zhou & Smith (2017) describes the population dynamics for given trophic level according the Dolling-Tannee model:
dB dt = rB 1 -
B K
(70)
where B is the biomass of the species, r is the intrinsic growth rate, K is the carrying capacity, M the mortality rate due to predation, and F' is the fishing mortality rate. Elsewhere in the paper, we have avoided using Zhou & Smith's notation, so-as to avoid overloads with the hybrid system literature. In our paper, we use instead of F, and the lowercase b for biomass, with subscript indicating trophic level. In this appendix section, however, we will use Zhou & Smith ss notation.
"IIn other words, we do not necessarily have exogeneity here: Mx Y. This further aligns with understanding the state-dependent trigger dynamics a sort of "treatment mechanism." Exogeneity, here, is violated because & and 0 can be considered "parents" of both X and Y. In violating exogeneity, por-maraettric identification results for this example may be out of reach (Li Pearl (2024] Kawakami al. (024)). As this paper does not address non- parameerrisss however. a parametric identification of initial conditions and parameters would be sufficient for the identifiability of PNS/PN/PS. Parametric identification in hybrid systems has been studied by Johnson (2023).
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The mortality rate due to predation is modeled as:
ppred
M = D B
(71)
where p is the maximum predation rate, Bpred is the biomass the predator, and D is the biomass at which predation reaches half its maximum.
The carrying capacity for predator species is given by:
K
Bprey
(72)
where e the efficiency of converting prey biomass into predator biomass.
The bottom trophic level dynamics follows:
d.Brrrage dt
Brorage
(1-
(1-
Biorage
Frorage Biorage
(73)
K₁
where M12 the mortality rate due to predation from intermediate predators. Species in the intermediate level act as both predator and prey:
dBsternndiate = inttermediatee
(1
Bintermodiate
M2g Bittrnediatee
F Fintermediate B Inttermediate- (74)
dt
The top trophic level follows:
dBapp dt
rg Bupex
(1-
Bapex
Mg Bupex
Fopex Bupex.
(75)
C23 1223Bmnene Bintermediate
The catch rate for the intermediate trophic level is given by the following note that, in the main body of our paper, we use for the integrated catch, meaning Catch below corresponds to é.
Catchinermedinte mnterrmedite BinemnedddieeBnnnneee Bintenneditee
(76)
Fishing efforts for each trophic level are assumed to remain constant over time unless intervened on.
dFF
0, ie forage, intermediate, apex.
(77)
dt
I Practical Utilities for Tracking Intervention Times and Values
In many counterfactual estimands, must translate event's characteristies from one world to another. To do so, we require the ability to extract certain event properties from a hybrid system's 's solution. By recording event specifications in auxiliary state variables, these be straightforwardly read off of solution evaluations at any particular time. First, consider an original system P with time recorded faithfully in the first dimension, and compatible interventional jump set D and jump map G. Assume the intervention preserves the faithful recording time and that single valued everywhere.
To start, we augment state space with an intervention jump counter j. an intervention time lx, and an intervention value vı. Our goal is to record the time and jump value corresponding to the k th
occurrence of the intervention. Let S S R2 IR and θ Ø, and augment the state space accordingly.
p' spaug ())
(78)
Now, augment the original interventional specification appropriately track event details these auxiliary state variables. For all admissible inputs, and fixed integer kt, let
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D'(0) D(0) x x G² (x, J.t, uk 8 (z,,,
(79)
1 + 1, 1,4,, m..
k=j+1 k k
(80)
The intervention time and value can then be read directly off of solution satisfying the constraints of instint (P', D' (.). To track whether an event occurred k times, we can initially set, for example, la 1. A positive 4 would indicate that the event had occurred. This can be switched to co instead for more natural time inequalities if boundedness of the states is not concern.
L11 Notation for Extracting Times and Values
We now describe some general notation for extracting values of ty and iy2 from solution. Consider a hybrid o; satisfying system arising from the i'th instint transformation of that system, where the interventional jump set and map had been augmented to record its k d'th jump. Given the solution's
time parameterization t - 4p (t; E. 0), use TT(A) (Pmax(:; &, 0)) for the time at which the k'th
jump occurred, and (=(() (fmm>i('; &, 0)) to extract the state's value immediately following the jump. When clear from context, the function V extracts only element of the state. The caveat that > simply specifies that properties of the k'th jump due to transformation can be read off of a solution any further transformed system. As shorthand, we sometimes write (mai = fmaii & 0), taking the random inputs as implicit. Additionally, in settings involving interventions that occur only once in the relevant time window, where the order of interventional transformation is clear and denoted using a symbolic subscript like we use, for example, T2(p,) to extract the event's time.
J State-Dependent Intervention in the Forward Euler Representation
Consider forward-Eules approximation of a system ODEs, where, for simplicity, we will assume that At > 0. If f is the right-hand side of the continuous-time differential equation x' f(z), we can write structural equations 24 = Tt-At At + f(z AV, where / > 0, 2e is the value of the state variable at time t, IR is fixed realization of exogenous noise (representing unknown parameters, for example), and zo is fixed to some constant initial condition. Suppose now that we wish to intervene such that the system jumps according to function g to some threshold т. To implement this, we must modify the structural equation for o; for all t under question. That is, replace the original structural equation with the following piecewise construction.
+ when æ falls
Let 7. Zt-Ar At f(I,-At, u) At denote the value that would be obtained under the original (non- intervened) Euler update, and let D = {ar IR x & 7} denote domain which the jump is triggered. Then the intervened structural equation can be approximated as follows:
D
g Ft.
Ft
(81)
Tt $ D
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NeurIPS Paper Checklist
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Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
Answer: [Yes]
Our claims are described in the abstract and clearly enumerated the end the introduction. The first manifests the definition an instantaneous intervention (definition 4) the second in theorem T and its proof (appendix D and the third in section 6
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43


Reviews:
[{"review_id": "YrAnq2LWJW", "reviewer": "Reviewer_cV92", "summary": "This paper presents a new intervention semantics for hybrid dynamical systems where the dynamics is continuous but the changes to the dynamics can be discrete. The main contribution is developing a formal intervention semantics in a hybrid dynamical system, as the changes in the constraints due to the intervention. Many of the mathematical contents addressing the formalism are provided in the Appendix. The causal effects and the counterfactuals are also defined in the examples in Section 5. Application to counterfactual inference is presented in a case study and the Appendix.", "questions": "In the introduction, the HVAC system is mentioned as an example of a hybrid dynamical system.\nCould you provide a brief hypothetical scenario and example to illustrate how the proposed ideas and theory align?\nAlternatively, it could involve continuing the dosage example from Example 1.\nI am unable to follow all the new notations in the paper and the meanings of the mathematical definitions.\n\nTheorem 1 looks interesting. Can you explain it in practical terms?\n\nFor the counterfactual outcome in Example 4, what are the assumptions around unobserved confounders in the hybrid dynamical systems?", "limitations": "YES", "rating": 5, "confidence": 1, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 4, "originality": 4, "strengths_and_weaknesses": "**Strength**\nThe main strength of this paper is in its theoretical contribution to developing a new intervention semantics for hybrid dynamical systems. \n\n**Weakness**\nDue to the emphasis on the theory, I think this paper is too abstract and not suitable for a general AI/ML audience.", "was_revised": true, "final_justification": "I will keep the current score.", "rebuttal": "Thanks to the reviewer for their questions and feedback. Below, we address the reviewer's questions and concerns in the order they appear in the review.\n\n## On Our Theoretical Emphasis  \n\n*The reviewer commented: \"Due to the emphasis on the theory, I think this paper is too abstract and not suitable for a general AI/ML audience.\"*\n  \nWe believe an important practical contribution of our work is to create a conceptual bridge between previously disconnected research communities, which benefits both. For the causal inference community, our work extends the reach of causal inference to widely used hybrid dynamical systems. For the hybrid dynamical systems community, our work brings a taxonomy of causal questions, and a mechanized procedure for answering those questions from models and data. \n\nBeyond that, our theoretical results serve immediate practical ends. For example, without ensuring the existence and measurability of intervened outcomes, one can neither write coherent causal estimands nor reliably implement them in software (see below for discussion of theorem 1, which may help clarify this). We also note that our results apply to any parametric hybrid system fitting our definition, which includes those that characterize the flow map with neural ODEs (Chen et al., 2018).\n\n## On HVAC and Dosage  \n\n*The reviewer commented: \"In the introduction, the HVAC system is mentioned as an example of a hybrid dynamical system. Could you provide a brief hypothetical scenario and example to illustrate how the proposed ideas and theory align? Alternatively, it could involve continuing the dosage example from Example 1. I am unable to follow all the new notations in the paper and the meanings of the mathematical definitions.\"*\n  \nBoth the HVAC and dosage cases involve a thresholding behavior. When the state reaches a particular threshold (temperature/drug concentration, respectively), the state is instantaneously modified (the cooler turns on/a dosage is administered). Such thresholding behavior can be encoded as a transformation of the constraints that characterize a hybrid system. If the transformation is used to construct parallel worlds — some with the thresholding behavior, and others without — then outcomes in those worlds can be compared using standard causal inference machinery. In other words, we can coherently reason about the effects of an HVAC controller even though *when* the controller activates may depend on exogenous randomness. We will ensure these intuitions are communicated in a way that more clearly connects the dosage example to those enumerated in the introduction (including the HVAC example).\n\n## On Theorem 1\n\n*The reviewer commented: \"Theorem 1 looks interesting. Can you explain it in practical terms?\"*\n  \nTheorem 1 states that a finite composition of interventions (defined here as a particular class of transformation to the constraints that define a hybrid system) will maintain conditions sufficient for the existence, uniqueness, and measurability of the intervened system's solution. This is a key requirement when comparing outcomes across original and intervened worlds. If the intervened world's outcome does not exist, for example, that comparison is not possible. Similarly, if the outcomes are not measurable, we cannot coherently define distributions on those outcomes.\n\n## On Example 4 and Unobserved Confounding\n\n*The reviewer commented: \"For the counterfactual outcome in Example 4, what are the assumptions around unobserved confounders in the hybrid dynamical systems?\"*\n  \nBecause a hybrid system's solution is a deterministic function of initial conditions $\\xi$ and system parameters $\\theta$ (note that a structural equation is also deterministic with respect to its inputs), the conditional $\\xi, \\theta \\mid y_0^\\tau = \\bar{y}_0^\\tau$ will often result in a degenerate distribution — for example, a delta in the identified case, or perhaps a Gaussian with a singular covariance matrix in a non-identified case. Importantly, even in the non-identified case, the expectation remains well defined due to Theorem 1. Counterfactual conditionals behave similarly in standard acyclic SCMs. Because we do not address identification in this paper, we omit discussion of unmeasured confounding (or other sources of non-identification) for these examples, instead emphasizing the coherence of the estimands as a consequence of Theorem 1. In practice, of course, you would often need to condition on additional data or have sufficient domain knowledge in order to properly constrain the problem.\n\nRegarding unmeasured confounding specifically — while that is a common source of non-identification in acyclic structural causal models, a more frequent concern in hybrid systems relates to the \"persistence of excitation\" (Johnson, 2023). In short, you often need to observe *transient* dynamics (as opposed to equilibrated dynamics) in order to identify the effects of interventions. Of course, latent state elements also cause problems for identification here, though the deterministic coupling of the state elements in a hybrid system requires careful handling.\n\nFor further discussion regarding identification, see the heading \"Counterfactuals and Identifiability Assumptions\" in our response to reviewer 2WL1.\n\n## References\n\nChen, R. T. Q., Rubanova, Y., Bettencourt, J., and Duvenaud, D. K. (2018). Neural ordinary differential equations. Advances in Neural Information Processing Systems, 31. https://proceedings.neurips.cc/paper_files/paper/2018/file/69386f6bb1dfed68692a24c8686939b9-Paper.pdf\n\nJohnson, R. (2023). Parameter Estimation for Hybrid Dynamical Systems. University of California Santa Cruz."}, {"review_id": "FOjxs1hvc1", "reviewer": "Reviewer_27qb", "summary": "This work develops a counterfactual semantics for hybrid dynamical systems, which are models that combine continuous dynamics (e.g., physical processes) with instantaneous, state-triggered events (e.g., control policies, thresholds). While these systems are widely used in engineering and applied sciences, existing causal inference tools do not adequately support reasoning about the discontinuous, dynamically triggered interventions these models involve.\nTo address this gap, the authors formalize interventions as transformations to system constraints, rather than structural equations, and establish conditions under which these interventions preserve key system properties (solution existence, uniqueness, and measurability). This approach enables the use of standard causal estimands, like treatment effects or probabilities of causation, in a hybrid systems context.\nA case study on fishery management illustrates the practical use of the framework.", "questions": "* It is not clear to me why hybrid system constraints sacrifices immediate graphical identifiability criteria. Is it possible to express the same problems via (parametric) SCMs with different states?\n\n* While I understand that this work introduces a fundamentally different approach, it would be helpful to explicitly clarify why classical frameworkd, particularly those based on SCMs, are inadequate in this context. How does the proposed framework go beyond SCMs and their dynamic extensions? For instance, what is the added value compared to existing approaches such as those by Mooij and collaborators (e.g., Blom & Mooij, 2023; Boeken & Mooij, 2024; Rubenstein et al., 2018) (I understand that these focused on equilibrium, but technically why they cannot be easily extended to the context of this oaoer)? What about threshold-based dynamic SCMs (Zan et al, 2024), or discrete-time dynamic SCMs (Wang et al., 2016; Assaad, 2025)?\nIn other words, why does modeling continuous-time systems with state-triggered interventions necessitate a new framework? Although this is briefly discussed in the related work section, I believe the paper would benefit from more concrete examples or illustrations that clearly highlight where existing methods fall short and how the proposed approach overcomes those limitations. This would greatly help readers appreciate the significance and practical impact of the contribution.\n\n\n* In your framework, you adopt a flow-preferring resolution to handle solution non-uniqueness at the boundary of the flow and jump sets (Definition 3). Given that other strategies like jump-preferring are also used (Teel & Hespanha, 2015), could you elaborate on: 1) the implications of choosing a flow-preferring convention on the causal interpretation of interventions, and 2) whether your theoretical results (especially measurability and uniqueness of solutions under intervention) are robust to alternative resolution strategies?\n\n\n* I think (but not sure) that the citation in line 59 should be replaced by Shpitser and Pearl, 2008. In their 2006 paper they focused on interventional distributions whereas in their 2008 papers they extended the results to counterfactuals.\n\n* The authors also point out the limitation that they focuss on a finite time regimes. I do not see this as a real limitation, as I think in most systems the number of regimes are finite. Do the authors agree or am I missing something? DO think most systems have finite or infinite regimes?\nNevertheless, extending this work to infinite time regimes is an interesting future work ....\n\n\nReferences:\n\n* T. Blom and J. Mooi. Causality and independence in perfectly adapted dynamical systems. 2023\n\n* P. Boeken, J. Mooij. Dynamic Structural Causal Models, 2024\n\n* P. Rubenstein, S. Bongers, B. Schoelkopf, J. Mooij, From Deterministic ODEs to Dynamic Structural Causal Models, 2018\n\n* L. Zan, C. Assaad, E. Devijver, E. Gaussier, A. Aït-Bachir. On the Fly Detection of Root Causes from Observed Data with Application to IT Systems, 2024\n\n* Z. Wang, Y. Liang, D. Zhu and T. Li. The Relationship of Discrete DCM and Directed\nInformation in fMRI based Causality Analysis, 2016\n\n* C. Assaad. Towards identifiability of micro total effects in summary causal graphs with latent confounding: extension of the front-door criterion, 2025", "limitations": "Yes", "rating": 5, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 4, "strengths_and_weaknesses": "Strengths:\n\nThe work bridges causal inference and hybrid systems theory, enabling more reliable and expressive causal reasoning in settings involving closed-loop control and real-time decision-making. \n\nWeaknesses:\n\nIn my opinion this paper does not motivate enough the need of this work, it would be interesting to explain more why other approches fail in this context. In addition, as it is acknowledged by the authors, graphical identifiability criteria are not provided and unlike classical graphical criteria, the criteria given int his paper are parametric.\nMoreover, I think, for most of the audience of the conference, it would be very complicated to understand the paper without going through the appendix.", "was_revised": true, "final_justification": "The authors addressed all my concerns. I was convinced and I’m now even more convinced of the relevance and value of this work.", "rebuttal": "We thank the reviewer for the positive feedback and for pushing us to improve the motivation — their critiques have helped us to further clarify the ways in which this contribution relates to existing work. We'll address the motivation questions first.\n\n## Motivation\n  \n*The reviewer commented: \"While I understand that this work introduces a fundamentally different approach, it would be helpful to explicitly clarify why classical frameworks, particularly those based on SCMs, are inadequate in this context...what is the added value compared to existing approaches such as those by Mooij and collaborators...? What about threshold-based dynamic SCMs (Zan et al, 2024), or discrete-time dynamic SCMs (Wang et al., 2016; Assaad, 2025)? In other words, why does modeling continuous-time systems with state-triggered interventions necessitate a new framework? ...\"*\n  \nThis work builds a bridge between two very mature fields, and making such a connection necessitates a natural choice: should we extend existing causal frameworks or hybrid systems frameworks? Primarily motivated by the specific challenges we hoped to address, we chose the latter for reasons we discuss below. That said, there certainly are useful connections to be made to existing causal frameworks.\n\nOverall, we thank the reviewer for pressing for this context, and agree that it will be quite helpful to our readers. We will expand on these comparisons in our related works section.\n\n**Extending (Discrete Time) SCMs?**\n  \n*The reviewer commented: \"...[why do] hybrid system constraints [sacrifice] immediate graphical identifiability criteria. Is it possible to express the same problems via (parametric) SCMs with different states?\"*\n  \nNotwithstanding the challenges associated with discretizing continuous time systems (e.g., from stiffness), discrete time structural causal models (such as those employed by Zan et al. (2024), Wang et al. (2016), and Assaad (2025)) do afford an instructive mapping to the continuous time domains. As Hansen and Sokol (2014) point out, the discrete time setting can be viewed as an Euler approximation to a continuous state trajectory. Under this interpretation (assume we take it to the limit with infinitesimally small $\\Delta t$), we can write an infinite sequence of structural equations, where $f$ is the right-hand side of the continuous time differential equation $x' = f(x)$:\n\n$$x_t = x_{t-\\Delta t} + f\\left(x_{t-\\Delta t},\\theta\\right) \\Delta t.$$\n\nA static-time intervention with a state-dependent jump value (though not a state-dependent jump *time*), then, can be encoded as a soft intervention (Correa and Barenboim, 2020) that swaps the above structural equation with one governed by a difference (as opposed to differential) equation:\n\n$$x_t = g\\left(x_{t-\\Delta t},\\theta\\right).$$\n\nOne can even encode a *state-dependent* intervention, but it requires a soft-intervention at every point in time. Consider:\n\n$$\nx_t = \n\\begin{cases}\ng(x_{t - \\Delta t}, \\theta) & \\text{if } x_{t - \\Delta t} \\in D(\\theta) \\\\\\\\\nx_{t - \\Delta t} + f(x_{t - \\Delta t}, \\theta)\\ \\Delta t & \\text{if } x_{t - \\Delta t} \\in C(\\theta)\n\\end{cases}\n$$\n\nWe'll highlight two downsides here. First, representing a state-dependent intervention as a soft intervention affecting all endogenous nodes is challenging to work with under graphical theory. Secondly, the Euler representation is not the typical choice of hybrid systems theorists. This makes post-intervention solution existence, uniqueness, stability, etc. challenging to prove, without much to gain from the existing causal inference literature.\n\n**Extending Constraint-Based Causal Models?**\n  \nRecent developments in constraint-based causal modeling offer another potential starting point.\n\nFor example, Beckers et al. (2023) extend SCMs in order to handle logical constraints (such as unit conversions), while Blom et al. (2019) interpret equilibrium equations of dynamical systems, along with their corresponding algebraic invariants, as a collection of constraints. In both cases, a model is characterized by a collection of constraints, and interventions are defined as transformations of those constraints (e.g., by changing, disabling, or enabling them). At a high level, our work operates very similarly.\n\nHybrid systems, however, are characterized by a unique class of constraints requiring special considerations around Zeno behavior, set-valued theory, non-uniqueness even in \"well-posed\" cases, set-valued stable points, etc. In short, analyzing the complexities of hybrid systems is made easier via direct use of existing hybrid systems frameworks, rather than existing causal frameworks.\n\nThat said, we ultimately consider our work to be complementary to that of Mooij and colleagues (and other constraint based causal machinery), and we look forward to future integrations of that developing framework with work on theory, simulation, and inference involving hybrid systems.\n\n## Flow Preference and Lowering\n\nTwo reviewers made very similar comments, so we have combined our response.\n\n*Reviewer 27qb commented: \"...could you elaborate on: 1) the implications of choosing a flow-preferring convention on the causal interpretation of interventions, and 2) whether your theoretical results (especially measurability and uniqueness of solutions under intervention) are robust to alternative resolution strategies?\"*\n\n*Reviewer 2WL1 commented: \"By “lowering” an “upstream” system into a lower one, does this only mean that the system is adjusted to prefer flowing solutions to solutions with many jumps, or are there other differences as well? Could you clarify how the “preferflow” condition selects flowing solutions over solutions with many jumps...?\"*\n\nOur choice of a flow-preferring lowering operation comes largely from its consistency with solver implementations that rely on \"crossing events\" along the jump set boundary. In such implementations (and in theory), the solution is allowed to flow *along* the boundary as long as it doesn't cross into it. This implies that \"grazing\" the boundary will not cause a jump. This boundary behavior sums up the implications on interventions. It is also important to note that, in many practical settings, numerical inaccuracies of solutions along set boundaries can dominate the theoretical nuances of whether those act as open or closed.\n\nA jump-preferring approach can be more challenging to compute with (detecting jump-inducing boundary grazes, for example, is a challenge), but does still induce uniqueness (Sanfelice et al., 2023). We conjecture that measurability also holds under a jump-preferring lowering operation, but a formal proof remains future work. Unfortunately, since we rely on a closed flow set in our measurability proof (line 1033), confirming measurability under a jump preferring lowering operation doesn't immediately follow.\n\n## Finite Time Regimes\n\n*The reviewer commented: \"...I do not see [the focus on finite time regimes] as a real limitation, as I think in most systems the number of regimes are finite...extending this work to infinite time regimes is an interesting future work.\"*\n  \nIn the context of dynamical systems, \"finite time\" essentially refers to \"transient\" as opposed to \"equilibrium\" regimes. We agree with the reviewer that this is more a matter of paper scope than a limitation. Indeed, asymptotic stability results already exist that rely primarily on the \"hybrid basic conditions\" we borrow from Goebel et al. (2012) (see our Assumption 4 in the appendix, and Ch. 3 by Goebel et al.). One challenge here is that hybrid systems often equilibrate to periodic behavior/sets rather than to points. In fact, \"non-constant asymptotic dynamics\" have been considered already for non-hybrid systems by Rubenstein et al. (2018). In sum, proper handling of infinite time/equilibrium regimes for hybrid system counterfactuals is well within striking distance of future work.\n\n## Citation Issue\n*The reviewer commented: \"...the citation in line 59 should be replaced by Shpitser and Pearl, 2008...\"*\n\nYes, this is correct. The 2008 paper also further develops the twin network setting, which aligns with understanding intervention as transformation.\n\n## References\n\nL. Zan, C. Assaad, E. Devijver, E. Gaussier, A. Aït-Bachir. On the Fly Detection of Root Causes from Observed Data with Application to IT Systems, 2024\n\nZ. Wang, Y. Liang, D. Zhu and T. Li. The Relationship of Discrete DCM and Directed Information in fMRI based Causality Analysis, 2016\n\nC. Assaad. Towards identifiability of micro total effects in summary causal graphs with latent confounding: extension of the front-door criterion, 2025\n\nHansen, N. and Sokol, A. (2014). Causal interpretation of stochastic differential equations. Electronic Journal of Probability, 19(none). https://doi.org/10.1214/EJP.v19-2891\n\nCorrea, J. and Bareinboim, E. (2020). General transportability of soft interventions: Completeness results. Advances in neural information processing systems (R-68; Vol. 33, pp. 10902–10912).\n\nBeckers, S., Halpern, J. Y., and Hitchcock, C. (2023). Causal Models with Constraints. https://proceedings.mlr.press/v213/beckers23a/beckers23a.pdf\n\nBlom, T., Bongers, S., and Mooij, J. M. (2019). Beyond Structural Causal Models: Causal Constraints Models (arXiv:1805.06539).\n\nSanfelice, R., Wintz, P., Copp, D., and Nanez, P. (2023). Behavior in the Intersection of C and D. https://hyeq.github.io/simulink/intersection-of-C-and-D\n\nGoebel, R., Sanfelice, R. G., and Teel, A. R. (2012). Hybrid dynamical systems: Modeling, stability, and robustness. Princeton university press.\n\nRubenstein, P. K., Bongers, S., Schoelkopf, B., and Mooij, J. M. (2018). From Deterministic ODEs to Dynamic Structural Causal Models (arXiv:1608.08028)."}, {"review_id": "NMIc5dLNdj", "reviewer": "Reviewer_2WL1", "summary": "This paper introduces a new formal semantics for counterfactual inference in hybrid dynamical systems, framing interventions as transformations of the system’s constraints. Under minimal requirements over the interventions, it is proven that these interventions preserve existence, uniqueness, and measurability of solutions. This framework is then applied to a fishery management case study, demonstrating its ability to answer counterfactual questions in hybrid dynamical systems.", "questions": "- By “lowering” an “upstream” system into a lower one, does this only mean that the system is adjusted to prefer flowing solutions to solutions with many jumps, or are there other differences as well? Could you clarify how the “preferflow” condition selects flowing solutions over solutions with many jumps, for example, in the case where you might have two solutions like “flow-jump-flow\" vs. “flow-flow-jump\"? \n\n- To make counterfactuals identifiable, does this framework assume that the outcome and/or functions are monotonic? Since counterfactuals are generally not identifiable without assumptions, but the counterfactual probabilities in the Appendix were point-identified rather than bounded, it seemed to imply that some assumptions were being made?", "limitations": "addressed in weaknesses", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "## Strengths\n\n- The paper presents a strong theoretic contribution by introducing a novel framework for counterfactual inference in hybrid dynamical systems, which is particularly valuable for comparing and verifying policies within these systems.\n\n- The paper is very well-written and easy to follow. I found the case study especially helpful for understanding how the framework can be applied to real-world models of dynamical systems to estimate probabilities of causation.\n\n- The proofs appear to be comprehensive and detailed, although I have not been able to go through them thoroughly.\n\n## Weaknesses\n\n- The paper currently does not contain a link to the anonymised codebase. However, the authors have stated in the NeurIPS checklist that they will provide this in the camera-ready version.\n\n- It may be beneficial to include some background on causal inference (e.g., definitions of SCMs, interventions, and counterfactuals) for readers less familiar with this work.\n\n- Some of the assumptions made which enable counterfactual identifiability (e.g., monotonicity over outcomes/functions) could be stated explicitly to improve clarity.", "was_revised": true, "final_justification": "The authors have provided a detailed response that addressed my questions, and I will maintain my positive score.", "rebuttal": "We thank the reviewer for their kind and critical comments, and are especially pleased that they found the paper easy to follow.\n\n## On Background\n\n*The reviewer commented: \"It may be beneficial to include some background on causal inference (e.g., definitions of SCMs, interventions, and counterfactuals) for readers less familiar with this work.\"*\n\nMotivated by the reviewer's feedback (and that of 27qb), we will be adding some additional context and background connecting our work to common intuitions relating to SCMs.\n\n## On the Lowering Operation\n\nTwo reviewers made very similar comments, so we have combined our response.\n\n*Reviewer 27qb commented: \"...could you elaborate on: 1) the implications of choosing a flow-preferring convention on the causal interpretation of interventions, and 2) whether your theoretical results (especially measurability and uniqueness of solutions under intervention) are robust to alternative resolution strategies?\"*\n\n*Reviewer 2WL1 commented: \"By “lowering” an “upstream” system into a lower one, does this only mean that the system is adjusted to prefer flowing solutions to solutions with many jumps, or are there other differences as well? Could you clarify how the “preferflow” condition selects flowing solutions over solutions with many jumps...?\"*\n\nOur choice of a flow-preferring lowering operation comes largely from its consistency with solver implementations that rely on \"crossing events\" along the jump set boundary. In such implementations (and in theory), the solution is allowed to flow *along* the boundary as long as it doesn't cross into it. This implies that \"grazing\" the boundary will not cause a jump. This boundary behavior sums up the implications on interventions. It is also important to note that, in many practical settings, numerical inaccuracies of solutions along set boundaries can dominate the theoretical nuances of whether those act as open or closed.\n\nA jump-preferring approach can be more challenging to compute with (detecting jump-inducing boundary grazes, for example, is a challenge), but does still induce uniqueness (Sanfelice et al., 2023). We conjecture that measurability also holds under a jump-preferring lowering operation, but a formal proof remains future work. Unfortunately, since we rely on a closed flow set in our measurability proof (line 1033), confirming measurability under a jump preferring lowering operation doesn't immediately follow.\n\n## Counterfactuals and Identifiability Assumptions\n\n*The reviewer commented: \"Some of the assumptions made which enable counterfactual identifiability (e.g., monotonicity over outcomes/functions) could be stated explicitly to improve clarity.\"*\n\n*The reviewer commented: \"To make counterfactuals identifiable, does this framework assume that the outcome and/or functions are monotonic? Since counterfactuals are generally not identifiable without assumptions, but the counterfactual probabilities in the Appendix were point-identified rather than bounded, it seemed to imply that some assumptions were being made?\"*\n\nRegarding the counterfactual probabilities in the appendix (and indeed any of the estimands we present here, including the PNS quantities), note that any measurable, deterministic function can act as a push-forward of random inputs. The resulting random outputs can then be compared and (conditional) expectations can be taken thereof. Interestingly, as long as $\\xi$ and $\\theta$ start with a well-defined distribution (e.g., a minimally informed Bayesian prior), causal estimands remain well defined even in non-identified regimes — they may not be particularly informative, however, if they require averaging over irreducible uncertainty. With infinite data in the Bayesian setting, for example, posteriors will reduce to degenerate distributions that amount to \"slices\" of the prior distribution. Witty et al. (2022) provide further discussion and intuition. \n\nIn our case study on the probabilities of causation, we effectively push a prior over $\\xi$ and $\\theta$ through the measurable solution map, and then through conditional expectations. Importantly, we do *not* first try to recover non-parametric causal relationships — after only observing, for example, a single, factual sequence of events — and then evaluate PNS quantities given our inferences about those relationships. That task does, indeed, require a monotonic relationship between treatment and outcome if point estimates of PNS are desired (Kawakami et al., 2024). Alternatively, as mentioned, one can also average over irreducible uncertainty. Put differently, if you are working directly with the full joint distribution over potential outcomes (like we are here), then you can estimate the PNS quantities directly — non-identification in this setting simply means that some of your uncertainty in that joint distribution cannot be resolved with more data.\n\nWe should also clarify a potential point of confusion: while monotonicity is not required to coherently define and evaluate PNS quantities in our setting, it is computationally advantageous when working with the existential predicates present when computing PNS quantities on continuous-valued variables. On line 1239 (in the appendix), for example, we employ monotonicity in the relationship between biomass and season length to redefine an otherwise existential predicate as a single inequality. Without monotonicity, we would instead need to approximate the existential predicate using a sampling procedure and some continuity assumptions.\n\nWhile identification concerns are outside the scope of this paper, future work is well poised to address the *statistical* question of whether certain estimands (e.g., differences in outcomes between an original and intervened world) can be asymptotically identified given data only from the original world. Naturally, without a coherent counterfactual semantics and proof that intervened outcomes exist and are measurable, this statistical task would not be possible. A promising place to start is with Johnson's (2023) work on the \"persistence of excitation\", which studies conditions under which hybrid system parameters are identifiable from data.\n\n## References\n\nWitty, S., Jensen, D., and Mansinghka, V. (2022). SBI: A Simulation-Based Test of Identifiability for Bayesian Causal Inference (arXiv:2102.11761). arXiv. http://arxiv.org/abs/2102.11761\n\nKawakami, Y., Kuroki, M., and Tian, J. (2024). Probabilities of causation for continuous and vector variables. In N. Kiyavash and J. M. Mooij (Eds.), Proceedings of the fortieth conference on uncertainty in artificial intelligence (Vol. 244, pp. 1901–1921). PMLR. https://proceedings.mlr.press/v244/kawakami24a.html\n\nJohnson, R. (2023). Parameter Estimation for Hybrid Dynamical Systems. University of California Santa Cruz."}, {"review_id": "dMJamiDBiB", "reviewer": "Reviewer_o1zp", "summary": "This paper proposes to endow the theory of hybrid systems with the notion of causality, by defining the notion of interventions in this theory. The main result is that this intervention preserves the unique existence of a measurable solution if the original system also had this property.", "questions": "In Definition 4, Equation (3), should $D(\\theta)$ be $D'(\\theta)$?\n\nIn Example 3, what is the reason you condition on a normal distribution with mean at the observations and arbitrary variance $\\sigma^2$, rather than just at the observations?", "limitations": "Yes", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 2, "originality": 4, "strengths_and_weaknesses": "**Strengths**\n\nI am a researcher in causality but I am not familiar at all with the theory of hybrid systems, but I found it a refreshing attempt to model causality for dynamical systems in this theory, without the need to rely on SCMs. I am aware of a few works that attempt to extend the theory of SCMs to dynamical systems (or stochastic processes in general), but I have always been somewhat of a sceptic of their success (and indeed, I'm a sceptic of whether these attempts *can* even be successful). \n\nI'm not convinced from this paper that hybrid systems will be *the* theory in which causality should be mathematised for dynamical systems, and in particular, I think a theory of causality that can naturally incorporate a time element should subsume this theory. This theory, from my understanding, is explicitly designed for dynamical systems. However, I still find it an interesting attempt, and give a high evaluation for its originality. \n\nI didn't go through the long proof of Theorem 1 in the appendix, but the result is believable and I trust the maths will be sound. The statement and the purpose of Theorem 1 is clear in the context of the theory the authors are trying to build. \n\nThe paper is in general a well-written paper; it was a pleasure to read. I am grateful to the authors for putting effort into the writing. \n\n**Weaknesses**\n\nDespite the fact that this is in general a well-written paper, there are some places where extensive references to the appendix makes the structure hard to follow. The reference to Appendix A.1 on line 112 is fine, as this is just an easy preliminary definition, but I think writing out Definition 6 in the main body rather than just referring to it (e.g. on line 129) would make it easier to follow the mathematics. The same goes for references to Appendices A.3 and A.4 on lines 137 and 138. What is mathematically problematic is using the notation $\\mathcal{G}$ on line 178, since this was only introduced in Appendix A.1 in Equation (7) and not defined in the main body. Of course, I understand the space limitations, and the authors are in the best position to decide how to use the space allocated in the main body, but their decision does make it slightly more difficult to follow the maths for those who want to. \n\nAlso, for me, Assumption 1 and the subsequent result in Lemma 1 was slightly disappointing. I think I was expecting a stronger result that guarantees unique existence of a solution, rather than requiring the system to be \"lowered\" from another system that is assumed to have a unique solution. \n\n**Minor Comments**\n\nIn Theorem 1, instint needs to be non-italic, to be consistent with the rest of the paper. \n\nIn Definition 5, the displayed equation needs a full stop.\n\nIn the displayed equation of Example 3, the last semi-colon needs to be replaced by a full-stop. Also, in this displayed equation, in the subscript of the expectation, you use $p$, a notation that hasn't been introduced. Of course it is obvious that this is the distribution of $\\xi$ and $\\theta$, but still you should introduce notations. Personally, I do not like using random variables as arguments of distributions. Or are you assuming a density here?\n\nThe displayed equation in Example 4 also needs a full stop.", "was_revised": true, "final_justification": "I see no reason to change my positive evaluation of the paper after the authors' rebuttals.", "rebuttal": "We thank the reviewer for their interest and engagement, and appreciate that they found the paper a pleasant read! Indeed, regardless of whether hybrid systems should be *the* substrate for dynamical causal theory (or whether there can be a single theory), we certainly hope to have demonstrated some value in bringing the rich literature on hybrid systems to bear for causal aims. This paper represents an initial test of a meta-hypothesis that we may be able to make rapid progress by equipping established modeling paradigms with causal semantics based on an understanding of intervention as model transformation.\n\n## Appendix References\n\n*The reviewer commented: \"...there are some places where extensive references to the appendix makes the structure hard to follow...\"* \n \nWe sincerely thank the reviewer for identifying some of these flow issues — these can be difficult for authors familiar with the work to pinpoint. Especially with an additional camera-ready page, we expect to be able to resolve a number of the problems you highlighted.\n\n## Assumption 1/Lemma 1\n\n*The reviewer commented: \"...Assumption 1 and the subsequent result in Lemma 1 was slightly disappointing. I think I was expecting a stronger result that guarantees unique existence of a solution, rather than requiring the system to be \"lowered\" from another system that is assumed to have a unique solution.\"*\n\nAssumption 1 supposes only that a unique solution exists for the continuous part of the original system. For hybrid systems especially, this is not sufficient for the uniqueness or even existence of non-trivial solutions. The primary logical step taken by Assumption 1/Lemma 1 is to move from the \"hybrid basic conditions\" (as proposed by Goebel et al. (2012)) to a unique, measurable solution. Importantly, those basic conditions imply neither solution uniqueness nor measurability. We should also point out that the only additions to Assumption 1 over and above the \"basic conditions\" from Goebel et al. (2012) (see our Assumption 4 for a restatement of their basic conditions) are the unique solution of the differential inclusion, the outer-semi continuity of $C$, and that $\\mathcal{G}(D)$ is Borel. Ultimately, these additional assumptions on the original system are quite lenient, and are added primarily for their role in the measurability result.\n\n## On Minor Comments\n\n*The reviewer commented: \"In the displayed equation of Example 3, the last semi-colon needs to be replaced by a full-stop. Also, in this displayed equation, in the subscript of the expectation, you use $p$, a notation that hasn't been introduced. Of course it is obvious that this is the distribution of and $\\xi$ and $\\theta$, but still you should introduce notations. Personally, I do not like using random variables as arguments of distributions. Or are you assuming a density here?\"*\n\nOn further review, since we do not use this notation elsewhere in the paper, we will just remove that subscript $p$ and condition on $\\boldsymbol{v}_0$ in the expectation.\n\nWe will also implement the other suggested corrections described here regarding equation stops, etc.\n\n## In Definition 4, Equation (3), should $D$ be $D'$?\n  \nUsing $D$ here simplifies the theory a bit. The required relationship (from Goebel et al.'s basic conditions) is simply that $D'(\\theta) \\subset \\textrm{dom } G'(\\cdot, \\theta)$, which is satisfied here without requiring that the more complex expression for $D'$ be used in the definition of $G'$. Essentially, the domain of $G'$ might cover regions outside of the jump set, but $G'$ will not be \"queried\" there.\n\n## On the Noise Model of Example 3\n\n*The reviewer commented: \"In Example 3, what is the reason you condition on a normal distribution with mean at the observations and arbitrary variance $\\sigma^2$, rather than just at the observations?\"*\n\nBecause the state $\\boldsymbol{w}_0$ is a deterministic function of initial conditions and parameters, conditioning on it directly leads to collapsed posteriors on those unknowns. In theory, this is fine, but numerically, this is challenging to work with. Many practical models, therefore, include some kind of independent observation noise (ideally, of course, one that maps to a real measurement phenomenon). Our inclusion of the noise model is simply in an effort to better align with practical usage.\n\nThis can be resolved other ways, of course — the most robust being to use a hybrid system with stochastic dynamics (i.e. Hybrid SDEs), which we are excited to explore in future work.\n\n## References\n\nGoebel, R., Sanfelice, R. G., and Teel, A. R. (2012). Hybrid dynamical systems: Modeling, stability, and robustness. Princeton university press."}]

Author rebuttal:

Meta-review: As emphasized by reviewers, this paper make an interesting attempt at connecting hybrid dynamical systems to causality by introducing a well grounded counterfactual semantics. Despite the formalism required by this kind of work, reviewers have appreciated the clarity of the presentation and the potential it has to bridge the field of causality and dynamical systems.
Decision: Accept (poster)