{"id":"e8823de5-9fc4-4fb9-b680-47ad1ddc5e60","arxiv_id":"2502.04171","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Cyclic functional causal models over finite variables get a unique probability rule and a sound and complete graph-separation property (p-separation) that reduces to d-separation in acyclic graphs.","lead":"The paper gives a way to assign probabilities and to read off conditional independences from the graph for cyclic causal models with finite-valued variables, even when the model has multiple or no solutions. It introduces a new graph-separation rule, p-separation, and proves it is sound and complete for all consistent such models.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Completeness proof of Theorem 20 rests on an informal general-case reduction; the XOR/post-selection construction also does not obviously satisfy the paper's teleportation-protocol condition as written.","rationale":"The reader correctly identified the informal completeness argument in Appendix B as the weakest assumption, and I agree that the finite-cardinality restriction is a clearly stated boundary rather than a flaw. My stress-test sharpens the concern: the gap is not merely 'the general case is argued through examples,' but also that the specific XOR construction with post-selection on 0 is not obviously a valid instance of the paper's own teleportation-protocol formalism, which post-selects on the function value 1 and requires Lemma 5. This is a proof-level gap, not a demonstrated counterexample; the construction may be repairable by relabeling the success event and formalizing the collider reduction, but until that is done the completeness half of Theorem 20 is not fully established. Soundness, the probability rule's independence claims, and the acyclic reduction appear internally consistent. The novelty and scope of the results justify a conditional accept rather than a rejection, pending a complete derivation of the general completeness case.","tokens_in":37993,"tokens_out":20098,"duration_ms":213863,"concrete_test":"Take the graph built from equation (113) by inserting, on the path between T and T', a collider A with a descendant D in V3 where D also has a parent E outside the path. Construct the model exactly as in Appendix B, but with f_T = δ_{x_v, x_R} and post-selection on T=1 to match Definition 9. Compute the induced cyclic distribution using equation (33) and check whether (X1⊥⊥X2|X3)_Pr fails for some values with positive probability. If the dependence disappears, the informal general-case reduction is false; if it survives, the proof can likely be repaired by formalizing this relabeling.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim is Theorem 20, and its completeness half is the load-bearing step. The proof in Appendix B treats the case where d- and p-connections coincide cleanly, but the nontrivial case relies on the 'Proof for the general case' (Appendix B, after equation (116)). There, the authors assert that arbitrary d-connection patterns can be realized by XOR functions with post-selection on 0, supporting this with two examples and the statement that the argument 'generalizes in a straightforward manner.' This is not a complete derivation, and it matters because the construction must produce a functional model that falls inside the paper's own framework. In particular, Definition 31 assigns to every post-selection vertex T the function x_T = XOR of its parents, and the proof then post-selects on T=0. But Definition 3 and Definition 9 require the post-selection vertex function f to satisfy Lemma 5, sum_b δ_{f(a,b,c),1} P_B(b) P_C(c) = ptp δ_{a,c}, with success event 1. An XOR function satisfies this only after relabeling success as 0 and replacing f by the equality indicator δ_{a,c}; the proof never makes that substitution explicitly. Additionally, the reduction to a single conditioned collider C in V3 assumes that every d-connecting path between two post-selection vertices can be compressed to a simple collider structure after setting V3 to 0. If a d-connecting pattern requires a collider whose only unblocking descendant in V3 has parents outside the path, the claimed equality wiring can fail, and the induced cyclic model may not realize the required conditional dependence. Since the completeness of p-separation is the paper's headline contribution, this gap is load-bearing; it may be repairable, but as written the theorem is not fully proven.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a probability rule for finite-cardinality functional causal models (fCMs) on arbitrary directed graphs, including cyclic and non-uniquely solvable models, by mapping a cyclic model to a family of acyclic 'classical teleportation graphs' with post-selection. It defines a new graph-separation property, p-separation, and claims in Theorem 20 that p-separation is sound and complete for all consistent finite-cardinality fCMs, recovering classical d-separation on DAGs. The paper also introduces 'average unique solvability' and proves, in Corollary 29, that this property characterizes when the probability rule yields a Markov factorization.","tokens_in":38345,"tokens_out":10536,"duration_ms":111610,"significance":"If Theorem 20 is fully established, this would be the first sound and complete graph-separation property for non-uniquely solvable cyclic functional causal models over finite variables, a genuine open problem in the area. The probability rule (Eq. (33)) is explicit, simple, and recovers the standard rule for uniquely solvable models, and the solvability results are clean and well-motivated. The connection to classical post-selected teleportation is conceptually interesting and links the work to quantum causal models. The main limitation is the finite-cardinality restriction, which the authors acknowledge, and the present incompleteness of the general-case completeness proof for Theorem 20, which is load-bearing for the universality claim.","major_comments":[{"comment":"The completeness half of Theorem 20 is not proven for the general case. After reducing the problem to the three d-connections of Eq. (109), the proof treats three special path shapes (Eqs. (111)-(113)) and then asserts that the argument 'generalizes in a straightforward manner.' This is load-bearing because Theorem 20 claims universality over all directed graphs and all p-connection patterns. A complete proof must either provide an inductive or case-based argument showing that arbitrary unblocked paths among v1, v2 and the post-selection vertices reduce to the structure of Eq. (114), or explicitly construct the required dependence for the general path geometry. The current text leaves the general case at the level of plausibility rather than proof.","section":"Appendix B, 'Proof for the general case' (after Eq. (116))"},{"comment":"The constructed model assigns x_T = XOR of parents to post-selection vertices and then post-selects on t_T = 0, but Definition 9 requires the post-selection vertex to implement a classical post-selected teleportation protocol from Definition 3, whose success event is t_T = 1 and whose function f satisfies Lemma 5. The proof must explicitly define f'_T = 1 - (x_v xor x_R), or otherwise relabel the success event, and verify that the resulting model on Gtp belongs to the family of classical teleportation functional models used to define Pr_G in Definition 12. As written, the construction does not obviously fall inside the framework whose probability rule is being invoked.","section":"Appendix B, Definition 31 and the following paragraph"},{"comment":"The step from conditional dependence in the acyclic model on Gtp to conditional dependence in Pr_G is asserted rather than derived. Pr_G is obtained by conditioning on the post-selection event and marginalizing over pre-selection and other auxiliary variables, and the proof does not explicitly show that the equality x_T xor x_v1 = x_T' xor x_v2, established under x_C = 0, survives this marginalization for arbitrary choices of the remaining vertices in Gtp. This is likely fixable, but it is part of the same completeness gap and should be made explicit.","section":"Appendix B, paragraph around Eq. (115)"}],"minor_comments":[{"comment":"The statement refers to 'the probability rule in definition 2' but it should refer to Definition 12, since the proposition gives an alternative expression for the cyclic-model probability rule.","section":"Proposition 13 (statement)"},{"comment":"There is a typo: 'Finally, we we describe the relations' should read 'Finally, we describe the relations.'","section":"Section 4.5, first paragraph"},{"comment":"The entries [LMGP+11a] and [LMGP+11b] appear to be identical duplicate references to the same paper; this should be cleaned up.","section":"References"},{"comment":"Notation such as 'T = 0' is used ambiguously: it should be clear whether one conditions on the event {T = 0} or on the random variable T as a conditioning variable, especially because Definition 17 uses conditioning on random variables.","section":"Appendix B, Eq. (115) and surrounding text"},{"comment":"The parenthetical 'if the equality holds in this case, the general result follows from corollary 14' creates an apparent forward-reference to a corollary that is proved later; the special-case calculation appears to establish the general implementation case directly for shared split vertices, so please clarify or reorder the argument to avoid an apparent circularity.","section":"Appendix A, proof of Proposition 11"}],"recommendation":"major_revision","confidential_remarks":"The paper is likely to be an important contribution if the completeness proof of Theorem 20 is completed. The current gap is concentrated in one part of Appendix B, but it is the central theorem's load-bearing direction, so I could not recommend acceptance in the present form. I would be willing to review a revised version."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First, the punchline: this paper has a real new result, but the completeness proof of the main theorem (Theorem 20) is not fully written out. I'd send it to review and ask for a proper general-case proof.\n\nWhat's new and good: The framework itself is a genuine advance. The probability rule via classical post-selected teleportation is well-defined, independent of the choice of teleportation graph and protocol (Propositions 11 and 13), and reduces to the known rule for uniquely solvable models. The definition of p-separation is natural, and the soundness proof is clean. The averagely uniquely solvable class and its characterization as the largest class where Markov factorization holds (Corollary 29) is a nice contribution. The paper also does a good job of flagging its own boundary: finite-cardinality only, post-selection event must have positive probability, and no extension to continuous variables.\n\nWhere the soft spots are: The completeness half of Theorem 20 is the load-bearing step, and the proof in Appendix B is informal exactly there. After treating the case where d- and p-connections coincide, the general case (where p-connection comes from conditioning on post-selection vertices) is argued through two examples and the phrase “generalizes in a straightforward manner.” That is not good enough for the central claim. Two specific issues stand out. First, the construction in Definition 31 uses XOR functions and post-selects on 0, but the teleportation protocol in Definition 3 is stated with success event 1; the needed relabeling/equality-indicator substitution is never explicitly made. That one is minor and easy to fix. Second, and more serious, the reduction to a simple collider structure assumes that any d-connecting pattern between post-selection vertices can be compressed to a common-cause form after setting the conditioning variables to 0. If the unblocking collider has a descendant in V3 with parents outside the pathway, the equality wiring can fail. The linear constraints may still create dependence, but that is not demonstrated. This is a real gap, not a stylistic nitpick.\n\nMy overall take: the architecture of the paper is sound, the probability rule and p-separation are likely correct, and the result deserves a serious referee. But as written, the completeness proof is not complete. I would recommend sending it to peer review with a request to either provide a full derivation for the general case or restrict the completeness claim to what is proven. The paper is for researchers in causal modeling and causal discovery, and for people working on graphical criteria for cyclic models. It is not a desk reject.","headline":"A serious and likely-correct contribution to cyclic causal modeling, but the completeness proof of the main p-separation theorem is informal in the general case and needs a full derivation before the result is fully proven.","tokens_in":38886,"tokens_out":4634,"would_cite":true,"duration_ms":44019,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C90","62H05","68T37"],"pacs":[],"model":"deepseek-v4-flash","headline":"Cyclic causal models get a sound and complete graph-separation rule, p-separation, for finite-valued variables.","keywords":["functional causal models","cyclic graphs","p-separation","d-separation","post-selected teleportation","Markov factorization","average unique solvability","causal loops"],"falsifier":"Exhibit any consistent finite-cardinality cyclic fCM and a triple $V_1,V_2,V_3$ where p-separation holds but the distribution given by the paper's Eq. (33) violates the promised conditional independence, or where p-connection holds but every fCM on $G$ satisfies that independence. A concrete place to look is a continuous-variable model on the graph of Eq. (66) that realizes the sigma-separation correlation; if such a model can be approximated by finite-cardinality models while preserving the dependence, the soundness claim would be at risk.","tokens_in":37814,"feed_emoji":"🔄","tokens_out":6024,"duration_ms":56792,"temperature":0.7,"pith_summary":"This paper extends functional causal models (fCMs) from acyclic to cyclic graphs with finite-valued variables, including models that are not uniquely solvable and therefore previously had no well-defined probability distribution. It introduces a post-selection-based probability rule that assigns a unique distribution to every consistent cyclic fCM and reduces to the standard rule when solutions are unique. On top of that it defines a graph-separation property, p-separation, and proves it sound and complete: a p-separation between vertex sets holds exactly when every consistent model on the graph must satisfy the corresponding conditional independence. In acyclic graphs p-separation collapses to the classical d-separation theorem. If correct, this gives the first graph-theoretic criterion that fully captures conditional independences for non-uniquely solvable cyclic causal models over finite variables.","feed_headline":"p-separation is the sound, complete rule for cyclic causal models","feed_subtitle":"For finite-valued variables it decides every conditional independence, reducing to d-separation on acyclic graphs.","key_machinery":"The central object is the classical post-selected teleportation protocol. A teleportation protocol (Definition 3) is a function $f(a,b,c)$ and priors $P_B,P_C$ such that post-selecting on $f=1$ copies any distribution $p_A$ onto $C$; the canonical 'uniform prior' protocol uses $f(a,c)=\\delta_{a,c}$ and $P_C$ uniform, with success probability $1/|X_A|$. Definition 7 turns any cyclic graph $G$ into a family of acyclic teleportation graphs $G_{tp}(G)$ by replacing selected vertices $v$ with a pre-selection copy $R_v$ and a post-selection vertex $T_v$, and Definition 9 builds acyclic fCMs on these graphs. Post-selecting $T_v=1$ and conditioning yields the probability rule $\\Pr(x)_G = \\frac{\\sum_u \\prod_v p_v(u_v)\\delta_{x_v,f_v(x_{Pa(v)},u_v)}}{\\sum_y \\sum_u \\prod_v p_v(u_v)\\delta_{y_v,f_v(y_{Pa(v)},u_v)}}$, which is independent of the choice of $G_{tp}$ and of the teleportation implementation. p-separation is then defined as: there exists $G_{tp}\\in G_{tp}(G)$ with $(V_1 \\perp_d V_2 | V_3 \\cup V_{post})_{G_{tp}}$. This machinery transfers d-separation arguments from acyclic to cyclic models by making the hidden collider effect of cycles explicit.","core_discovery":"The paper's central theorem (Theorem 20) states: for any directed graph $G$ and three disjoint non-empty vertex sets $V_1,V_2,V_3$, $(V_1 \\perp_p V_2 | V_3)_G$ holds if and only if every consistent finite-cardinality functional causal model on $G$ satisfies $(X_1 \\perp\\!\\!\\perp X_2 | X_3)_P$. p-separation is defined by passing to an acyclic teleportation graph $G_{tp}$ obtained by splitting vertices into pre- and post-selection copies, and asking for d-separation of $V_1$ from $V_2$ given $V_3$ together with all post-selection vertices. The same construction yields the probability rule, which counts all solutions of the functional equations and renormalizes, so inconsistent models (zero solutions on average) are exactly those with zero post-selection success probability. The paper also identifies averagely uniquely solvable fCMs --- those with average number of solutions equal to one --- as exactly the class whose probabilities admit the Markov factorization.","pith_inferences":["Editorial: The probability rule effectively weights each consistent solution equally; if this is accepted as a resolution principle, non-uniquely solvable cyclic models become predictable without extra modeling assumptions.","Editorial: The same post-selection construction suggests finite cardinality is not a technical convenience: for continuous variables the post-selection event has probability zero, so p-separation may have no direct continuous analogue.","Editorial: Because d- and p-separation agree on the graph of Eq. (66) while sigma-separation does not, the paper implies that every finite-cardinality model on that graph is fine-tuned relative to sigma-separation; a full taxonomy of separation properties on cyclic graphs is a natural next step."],"forward_implications":["Every consistent finite-cardinality cyclic fCM gets a unique probability distribution, resolving the ambiguity in non-uniquely solvable models.","p-separation gives sound and complete conditional-independence semantics for all such models, and reduces to d-separation for acyclic graphs.","Averagely uniquely solvable models are exactly the Markovian ones; uniquely solvable models are a strict subset of them.","The post-selection success probability is proportional to the average number of solutions, connecting consistency to solution counting.","Because p-separation is sound and complete, it could support causal discovery and causal-compatibility tests for cyclic finite-valued causal structures."],"supporting_citations":[{"why":"Proves the d-separation theorem for acyclic graphs that p-separation generalizes.","marker":"[VP90]"},{"why":"Supplies the standard acyclic probability rule and d-separation theory that the cyclic construction extends.","marker":"[Pea09]"},{"why":"Gives a uniquely solvable cyclic fCM where d-separation soundness fails, the motivating counterexample.","marker":"[Nea00]"},{"why":"Introduces sigma-separation for modular structural equation models, the main alternative graph-separation property contrasted throughout.","marker":"[FM17]"},{"why":"Quantum teleportation protocol whose classical post-selected analogue underlies the probability rule.","marker":"[BBC+93]"},{"why":"Companion paper establishing the quantum analogue and the equivalence of the two p-separation definitions.","marker":"[FGV25]"},{"why":"Provides a non-uniquely solvable cyclic model whose probability rule the new framework recovers and generalizes.","marker":"[VC22b]"},{"why":"Foundational treatment of cyclic structural causal models and unique solvability that defines the problem setting.","marker":"[BFPM21]"}],"fun_headline_variants":["Cyclic causal models finally get a complete separation rule","p-separation: the d-separation for cyclic causal graphs","Complete conditional independence for all finite cyclic causal models","One separation rule to rule cyclic causal independences","p-separation solves cyclic causal independence completely"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole construction assumes variables with finitely many values, because the probability rule post-selects on exact variable values; and the completeness proof for the general d-connection case is argued through examples rather than a full derivation.","fun_headline_variants_meta":{"raw":{"variants":["Cyclic causal models finally get a complete separation rule","p-separation: the d-separation for cyclic causal graphs","Complete conditional independence for all finite cyclic causal models","One separation rule to rule cyclic causal independences","p-separation solves cyclic causal independence completely"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000605,"raw_usage":{"total_tokens":2882,"prompt_tokens":1068,"completion_tokens":1814,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":684,"completion_tokens_details":{"reasoning_tokens":1740}},"tokens_in":684,"tokens_out":1814,"duration_ms":13011,"temperature":1.0,"reasoning_tokens":1740,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T23:16:34.839762+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Exhibit any consistent finite-cardinality cyclic fCM and a triple $V_1,V_2,V_3$ where p-separation holds but the distribution given by the paper's Eq. (33) violates the promised conditional independence, or where p-connection holds but every fCM on $G$ satisfies that independence. A concrete place to look is a continuous-variable model on the graph of Eq. (66) that realizes the sigma-separation correlation; if such a model can be approximated by finite-cardinality models while preserving the dependence, the soundness claim would be at risk.","supporting_citations":[],"review_version":1}