{"id":"76aa3b9c-ff0e-436b-86bd-b1560ba3536a","arxiv_id":"2501.12914","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Counterfactuals are recast as terminal states of a minimum-energy optimal control problem, computed via occupation measures and moment-SOS relaxations.","lead":"The paper redefines counterfactual explanations as solutions to an optimal control problem: from an initial unsafe state, find the minimum control effort needed to reach a safe set. It solves this problem with occupation measures and the moment-SOS hierarchy, then demonstrates the idea on a glucose-insulin regulation model.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The extraction of a counterfactual as y_{1,τ} is unjustified: mass(μτ)=1 does not imply μτ is a Dirac delta, and finite-order SDP solutions give no atomicity certificate, so the output may be a non-reachable barycenter.","rationale":"The reader's weakest assumption identifies the same load-bearing issue: the final measure need not be atomic, so extracting a counterfactual from the first moment is not justified. The paper's Proposition 1 is not merely missing a technical detail; it asserts a false implication (mass one implies Dirac), and the same error is repeated in the uncertain-systems section. This directly threatens the central claim that a counterfactual can be computed from the moment-SOS hierarchy, because the output of Algorithm 1 and Algorithm 3 may be a convex combination of candidate endpoints rather than a point satisfying the definition of CS-counterfactual. The proposed double-integrator test isolates the issue cleanly: it has two symmetric optimal endpoints with equal cost, so the measure LP admits non-atomic optimal terminal measures, and the first moment can be the origin, which is neither in the terminal set nor reachable as an endpoint of an optimal trajectory. If the SDP returns a rank-one terminal moment matrix in that example, the specific failure does not materialize there, but the paper still needs an explicit atomicity certificate; otherwise the extraction step remains unsupported. This does not change the reader's conditional verdict: the framework may be salvageable via flat-extension conditions or by relying on Algorithm 2's trajectory simulation, but the theoretical justification as written is incorrect.","tokens_in":13678,"tokens_out":6664,"duration_ms":82180,"concrete_test":"Run the proposed SDP (8) on the double integrator x1dot = x2, x2dot = u, with X0 = {(0,0)}, X_T = {x : x1^2 - 1 = 0, x2 = 0}, cost ∫u^2, and a fixed normalized horizon T = 1 after time scaling as in §3.5. At relaxation orders d = 4, 6, 8, compute rank(M_d(yτ)) and y_{1,τ}. If rank > 1 and y_{1,τ} = (0,0) (or any point outside X_T), the first-moment extraction is producing a barycenter, not a reachable counterfactual, and the concern lands. If rank = 1 and y_{1,τ} is one of (±1,0), the extraction happens to work on this example, but the paper should still add a flat-extension or rank-one certificate as a mandatory stopping rule before outputting a counterfactual.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central computational claim is that a counterfactual can be read off as x_cf = y_{1,τ}, the first moment of the terminal measure from the moment-SOS relaxation (8). This is legitimate only if μτ is a Dirac delta supported on a reachable endpoint. Proposition 1 tries to establish this from mass(μτ)=1, but mass-one measures need not be atomic: a uniform measure on an interval has mass 1 and first moment equal to its center, which need not be a trajectory endpoint. The same non sequitur reappears in §4.2, where the authors infer atomicity of μ_{t*} from ⟨1, μ0⟩ = ⟨1, μ_{t*}⟩. At finite relaxation order d, the truncated moment vector yτ need not even have a representing measure; without a flat-extension or rank-one certificate on M_d(yτ), y_{1,τ} is at best a barycenter of a set of candidate endpoints. The remark in §3.4 also assumes only finitely many optimal solutions and ignores that the measure LP is convex, so any convex combination of optimal terminal measures is again optimal; the set of optimal endpoints can be a continuum, and the averaged endpoint can lie outside X_T or be dynamically unreachable.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a control-theoretic definition of counterfactuals: given a factual initial condition in an unsafe set, the counterfactual is the terminal state of a minimum-effort optimal control problem that steers the system to a safe set with free terminal time. The problem is formulated as an infinite-dimensional linear program in occupation measures and solved through the moment-SOS hierarchy. Three extraction algorithms are proposed: reading the counterfactual as the first-order moment of the terminal measure (Algorithms 1 and 3), and simulating a closed-loop trajectory from the approximate dual value function (Algorithm 2). The approach is extended to systems with uncertain parameters by augmenting the state with the uncertain parameters, and numerical experiments are reported on a Bergman glucose-insulin model.","tokens_in":1679,"tokens_out":2361,"duration_ms":101680,"significance":"The idea of grounding counterfactuals in dynamical reachability and minimum effort, rather than in classifier geometry alone, is timely and potentially valuable for safety-critical and biomedical applications. The paper correctly identifies occupation measures and the moment-SOS hierarchy as natural tools for this problem, and the robust extension in Section 4 is a useful addition. However, the central computational claim, that a counterfactual can be read off as the first moment of the terminal measure from a finite relaxation, is not justified and is false in general. Because the numerical results in Sections 3.5 and 4.5 rely on this extraction, the contribution as written is not yet established. If the extraction step is repaired with atomicity certificates or a certified trajectory-recovery procedure, the framework could become a useful addition to the counterfactual-explanation literature.","major_comments":[{"comment":"The proposition infers from mass(mu_tau)=1 that mu_tau is a Dirac measure. This inference is invalid: a non-atomic measure can have total mass 1, and a finite-order truncated moment vector need not even have a representing measure. Consequently, xcf = y_1,tau may be a barycenter of several candidate endpoints and need not be a reachable terminal state of any feasible trajectory. The extraction step must be justified by a flat-extension or rank-one certificate on M_d(y_tau), or replaced by a recovery method that reconstructs an atomic terminal measure; the numerical sections should report such certificates.","section":"§3.4, Proposition 1 and Algorithm 1"},{"comment":"The claim that constraint (22c) together with Liouville's equation implies that the final measure mu_t* is atomic is incorrect. Choosing the test function v=1 in (20b) only gives conservation of mass, <1,mu_0>=<1,mu_t*>, which says nothing about atomicity. The subsequent statement that if (22) admits s optimal solutions then mu_t* is s-atomic is also false: the feasible set of (22) is convex, so any convex combination of optimal terminal measures is again optimal, and the set of optimal endpoints can be a continuum. These errors invalidate Proposition 3 and Algorithm 3 as stated.","section":"§4.2, equations (20b)-(22c)"},{"comment":"The dual-based trajectory recovery is not certified. For a finite relaxation order d, the dual solution v_d is only a subsolution of the value function, and the control u*_d(x) obtained from it is not guaranteed to steer the closed-loop system to X_T at time tau=mass(mu). No reachability or optimality proof is provided for Algorithm 2, so this alternative extraction path also does not currently support the paper's claims.","section":"§3.4, Proposition 2 and Algorithm 2"},{"comment":"The formal results are either near-tautological or not established. Theorem 1 only asserts boundedness of trajectories, which follows from compactness of X when the setup is well posed. Theorem 2's proof assumes V(x)=0 on X_T and V(x)>0 elsewhere, which is not established by (6a)-(6d), and the inequality L_f V <= -phi with phi=u^2 does not imply asymptotic stability of X_T because u^2 is a control-dependent quantity rather than a positive definite function of the state alone. Theorem 3 is a boundedness statement on a compact set and provides no convergence or reachability guarantee. These results should be corrected, or their statements and proofs removed, so that the advertised guarantees match what is actually shown.","section":"§3.4, Theorems 1-3"}],"minor_comments":[{"comment":"The sentence 'For alpha=1 we get the mass of the measure' is incorrect in multi-index notation: the mass is y_0=<1,mu>, not the first-order moment y_1.","section":"§2.1"},{"comment":"The line defining the adjoint operator says 'H*=H', which is not generally true; the correct identity is <Hv,mu>=<v,H*mu>.","section":"§2.1"},{"comment":"The constraint 'p(x)-sum_i a_i(x)v_i >= 0' is written with 'for all i=1,...,I_max', but the positivity condition applies to the single polynomial in the sum, not to each index i separately.","section":"§2.3, equation (2)"},{"comment":"The expression 'xcf = spt(mu_tau=delta_x_tau)=sum_i beta_i*x_tau_i' is not well-formed: spt of a measure is a set of atoms, not a weighted sum. The intended meaning should be written with the measure itself, e.g. mu_tau=sum_i beta_i*delta_x_tau_i.","section":"§3.4, Remark after Proposition 1"},{"comment":"Equation (13) is labeled with two numbers, (13) and (14), for a single displayed polynomial; the duplicate label should be removed.","section":"§3.5"},{"comment":"The text says 'parameters p3 and p3 are related to the dynamics'; the second occurrence should presumably be p2.","section":"§4.5"},{"comment":"The observation that counterfactuals lie near the boundary of the safe set is a direct consequence of the free-terminal-time, no-terminal-cost minimum-effort formulation, not an independent empirical discovery; the paper should present it as such.","section":"§4.5"}],"recommendation":"major_revision","confidential_remarks":"The central extraction step is mathematically unsound as written, and the same flaw propagates to the robust extension. I would be willing to reconsider after the authors add a certified atomicity/recovery procedure, report flat-extension or rank-one certificates in the numerics, and either repair or remove Theorems 1-3. The paper also appears to be a merger of two submitted conference papers [1,2]; the relationship to those works should be clarified in the final version."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The idea here is genuinely useful: define a counterfactual as the terminal state of a minimum-effort optimal control problem that steers a factual from an unsafe set to a safe set, then compute it with occupation measures and moment-SOS. That gives a physics-aware notion of counterfactual, which matters for safety-critical applications. The Bergman glucose-insulin demo is a reasonable proof of concept, and the authors are honest that most of the theoretical machinery comes from earlier occupation-measure work. Proposition 2's trajectory-recovery approach is a sensible alternative to the moment-matrix extraction.\n\nThe soft spots are in the extraction step, and the stress-test note lands. Proposition 1 claims mass(mu_tau)=1 implies the terminal measure is a Dirac delta; that is simply false. A mass-one measure can be uniform on a continuum, and at finite relaxation order you have no certificate that the truncated moment vector represents an atomic measure. So y_{1,tau} is at best a barycenter of candidate endpoints, and there is no guarantee it is a reachable state or even lies in the safe set. The remark about r optimal solutions is also incorrect: the measure LP is convex, so the set of optimal terminal measures is convex, and an optimal measure can be a convex combination of atomic optimizers. The same non sequitur appears in Section 4.2, where atomicity is inferred from mass conservation. Theorems 1 and 3 are near-trivial consequences of compactness, and Theorem 2's Lyapunov argument skips the key step: V(x)=0 on X_T and V(x)>0 elsewhere is not established by the dual program.\n\nThe boundary clustering of counterfactuals in the examples is a direct consequence of the definition: a minimum-effort transfer stops as soon as the safe set is reached. That is not a flaw, but it should be framed as a design property, not an empirical discovery. The numerical results are qualitative and would benefit from reproducible artifacts.\n\nWho gets value from this? Researchers looking for a control-theoretic angle on counterfactual explanations, especially in life-science applications. It is a good position paper and a promising starting point, but the current computational guarantees do not justify clinical or safety-critical use. I would send it to peer review because the direction deserves scrutiny, but I would expect significant revision, especially around the extraction step and the stability claims.","headline":"A sensible reframing of counterfactuals as minimum-effort optimal control, but the moment-based extraction step relies on an unjustified atomicity assumption that can produce non-reachable barycenters.","tokens_in":14469,"tokens_out":2094,"would_cite":false,"duration_ms":23861,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["49K15","90C22"],"pacs":[],"model":"deepseek-v4-flash","headline":"Counterfactual explanations recast as minimum-effort endpoints of a control system.","keywords":["counterfactual explanations","optimal control","occupation measures","moment-SOS hierarchy","minimum-effort control","glucose-insulin regulation","uncertain systems","reachability"],"falsifier":"For a two-state linear system with two equally cheap safe endpoints, solving the moment relaxation and checking whether the first moment of the terminal measure is a reachable endpoint under any admissible control would settle the extraction claim: if it is not, Algorithm 1's output is a barycenter rather than a counterfactual.","tokens_in":13451,"feed_emoji":"🎛️","tokens_out":7331,"duration_ms":69567,"temperature":0.7,"pith_summary":"Counterfactual explanations in machine learning usually perturb feature values without checking whether the resulting state is physically reachable. This paper proposes a control-theoretic alternative: define the factual as an initial state in an unsafe set and the counterfactual as the terminal state reached by the minimum-effort control that steers the system into a safe set. It solves this optimal control problem through occupation measures and the moment-SOS hierarchy, so the answer respects the system's dynamics, and it extends the construction to systems with uncertain parameters. The approach is demonstrated on a standard glucose-insulin regulation model, where the resulting counterfactuals lie near the safe-set boundary for known dynamics and shift deeper into the safe set when parameter uncertainty is taken into account.","feed_headline":"Minimum-effort control makes counterfactuals reachable","feed_subtitle":"Solving for the least control that steers an unsafe state into the safe set yields dynamics-aware what-if endpoints.","key_machinery":"The engine is the occupation-measure formulation of optimal control. The trajectory is represented by three measures, the initial measure $\\mu_0$, the occupation measure $\\mu$, and the terminal measure $\\mu_\\tau$, linked by Liouville's equation $\\delta_T \\otimes \\mu_T = \\delta_0 \\otimes \\mu_0 + L^*\\mu$. This turns the nonlinear reachability-and-effort problem into an infinite-dimensional linear program, which the moment-SOS hierarchy approximates by a sequence of semidefinite programs in truncated moment space. The terminal measure's first-order moment $y_{1,\\tau}$, or in the dual a polynomial subsolution $v$ of the Hamilton-Jacobi-Bellman equation, is what the extraction algorithms use to read off the counterfactual.","core_discovery":"The paper's central claim is that a counterfactual can be defined without reference to any classifier, purely from the system dynamics: given a factual $x_f \\in X_0$, the CS-counterfactual is $x_{cf} = x(\\tau \\mid x_f, u^*)$, the endpoint of the trajectory generated by the control $u^*$ that minimizes $\\int_0^\\tau u^2\\,dt$ subject to the dynamics and $x(\\tau) \\in X_T$. This minimum-effort steering problem is lifted to a linear program over occupation measures, where Liouville's equation encodes the dynamics as a linear constraint. Solving a finite-dimensional moment-SOS relaxation yields a lower bound on the true cost; under the atomicity condition, the first-order moment of the terminal measure gives the counterfactual, and the dual polynomial provides a control law that can be simulated to recover the full trajectory. For uncertain parameters, the same construction is applied to an extended system in which unknown parameters become zero-dynamics states, producing robust counterfactuals that respect the whole range of parameter values.","pith_inferences":["Editorial inference: the same LP-in-measures machinery could certify infeasibility, meaning that if no measure satisfies the constraints, no control of the allowed form can reach the safe set, which amounts to a formal certificate that a counterfactual does not exist within budget.","Editorial inference: the dense clusters of counterfactuals seen in the experiments are likely shaped by the system's controllability and the value function's level sets; one could test this by comparing cluster locations against level sets of $v$ from the dual.","Editorial inference: when the terminal measure is not atomic, a more honest output would be a set or distribution of safe endpoints; checking the rank of the moment matrix $M_d(y_\\tau)$ could flag this before extracting a single point.","Editorial inference: the framework could be extended to stochastic dynamics by replacing Liouville's equation with the Fokker-Planck equation, giving counterfactuals that are reachable with high probability rather than deterministically."],"forward_implications":["Counterfactuals produced this way are actual trajectory endpoints, so they are reachable under the system's dynamics rather than arbitrary feature perturbations.","The method applies to general polynomial, control-affine nonlinear systems and does not require a trained classifier, only a safe set and a dynamic model.","For uncertain parameter values, the extended-state formulation yields counterfactuals that remain safe for every parameter in the specified set, at the price of lying farther from the boundary.","The dual solution gives a feedback law, so one can simulate the full factual-to-counterfactual trajectory, not just the terminal point.","The measure relaxation also computes a lower bound on the minimum control effort, which could serve as a quantitative measure of how hard it is to transition from an unsafe to a safe state."],"supporting_citations":[{"why":"Defines the AI counterfactual notion that the paper redefines in control terms.","marker":"[3]"},{"why":"Introduces the occupation-measure linear program formulation for nonlinear optimal control.","marker":"[11]"},{"why":"Provides the moment/LMI relaxation and convergence results that justify the equations used.","marker":"[12]"},{"why":"Serves as the reference for the moment-SOS hierarchy as a computational framework.","marker":"[13]"},{"why":"Supplies the infinite-dimensional LP duality and moment representability background.","marker":"[14]"},{"why":"Supplies the glucose-insulin regulation dynamics used in the numerical case study.","marker":"[20]"},{"why":"Supports the extended-state representation of uncertain parameters in measure-based robust verification.","marker":"[22]"}],"fun_headline_variants":["Optimal control yields physics-grounded counterfactuals","Minimum-effort trajectories define dynamics-aware counterfactuals","Counterfactuals as optimal steering without classifiers","Control-theoretic counterfactuals from least-effort steering"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The extraction step assumes that the optimal terminal measure is concentrated on a single point, so its first-order moment is itself a reachable counterfactual state; finite moment relaxations do not guarantee this atomicity.","fun_headline_variants_meta":{"raw":{"variants":["Optimal control yields physics-grounded counterfactuals","Minimum-effort trajectories define dynamics-aware counterfactuals","Counterfactuals as optimal steering without classifiers","Control-theoretic counterfactuals from least-effort steering"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000274,"raw_usage":{"total_tokens":1640,"prompt_tokens":947,"completion_tokens":693,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":563,"completion_tokens_details":{"reasoning_tokens":631}},"tokens_in":563,"tokens_out":693,"duration_ms":8097,"temperature":1.0,"reasoning_tokens":631,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T16:37:27.275949+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a two-state linear system with two equally cheap safe endpoints, solving the moment relaxation and checking whether the first moment of the terminal measure is a reachable endpoint under any admissible control would settle the extraction claim: if it is not, Algorithm 1's output is a barycenter rather than a counterfactual.","supporting_citations":[{"cited_title":"Counterfactual explanations and how to fin d them: literature re- view and benchmarking,","cited_arxiv_id":null,"evidence_quote":"Defines the AI counterfactual notion that the paper redefines in control terms."},{"cited_title":"Nonlinea r optimal control syn- thesis via occupation measures,","cited_arxiv_id":null,"evidence_quote":"Introduces the occupation-measure linear program formulation for nonlinear optimal control."},{"cited_title":"N onlinear optimal control via occupation measures and lmi-relaxations,","cited_arxiv_id":null,"evidence_quote":"Provides the moment/LMI relaxation and convergence results that justify the equations used."},{"cited_title":"Henrion, M","cited_arxiv_id":null,"evidence_quote":"Serves as the reference for the moment-SOS hierarchy as a computational framework."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the infinite-dimensional LP duality and moment representability background."},{"cited_title":"Q uantitative estima- tion of insulin sensitivity.,","cited_arxiv_id":null,"evidence_quote":"Supplies the glucose-insulin regulation dynamics used in the numerical case study."},{"cited_title":"Measu res and lmi for space launcher robust control validation,","cited_arxiv_id":null,"evidence_quote":"Supports the extended-state representation of uncertain parameters in measure-based robust verification."}],"review_version":1}