Optimal Correction Sets for Provable Redress of Contestable Causal Discovery

Fabrizio Russo; Giuseppe Mazzotta; Carmine Dodaro; Francesco Ricca; Francesca Toni

cs.AIOct 23, 2026

Optimal Correction Sets for Provable Redress of Contestable Causal Discovery

Authors: Fabrizio Russo, Giuseppe Mazzotta, Carmine Dodaro, Francesco Ricca, Francesca Toni

Organizations: Imperial College London, London, United Kingdom · University of Calabria, Rende, Italy

Abstract

Constraint-based causal discovery relies on conditional-independence (CI) tests that may conflict in finite samples, leaving no causal graph that satisfies them all. Causal Assumption-based Argumentation (ABA) makes these tests explicit as defeasible commitments, and offers correspondence guarantees with a subset of them, but its existing repair procedure releases facts heuristically and offers no formal response to a user's contestation of a release. We argue that contestable causal discovery requires an inspectable repair object, a justification for it, and a predictable revision procedure for such contestations. We therefore formulate CI-fact repair as a minimum-cost correction problem and implement it in Answer Set Programming (ASP). Building on Causal ABA, we provide a compact and efficient ASP encoding of causal consistency, with soundness and completeness guarantees, and a two-way projection correspondence between optimal stable models and minimum-cost correction sets. These results establish fidelity to retained facts, subset-minimality and cost optimality of the repair, conflict-based justifications for released facts, and formally guaranteed redress under user contestations. An empirical evaluation supports the formal account: OptABA-PC retains more correct facts, reduces ambiguity in the compatible outputs, and improves graph reconstruction over Causal ABA on common benchmarks.

Figures & tables

Dataset	$\Delta\#\mathrm{CPDAG}\downarrow$	$\Delta\mathrm{F1}\uparrow$	$\Delta\mathrm{SHD}\downarrow$
Cancer (5)	$-14.1\ (-93.4\%)$	$+0.054\ (+10.3\%)$	$-0.974\ (-34.6\%)$
Survey (6)	$-1277.2\ (-99.9\%)$	$+0.159\ (+36.3\%)$	$-2.790\ (-39.1\%)$
Asia (8)	$-10821.5\ (-100.0\%)$	$+0.313\ (+139.7\%)$	$-10.507\ (-73.1\%)$
ER (5)	$-5.2\ (-63.6\%)$	$+0.027\ (+4.9\%)$	$-0.792\ (-27.5\%)$
ER (8)	$-3835.3\ (-99.9\%)$	$+0.150\ (+31.3\%)$	$-4.661\ (-56.9\%)$
SF (5)	$-3.3\ (-3.7\%)$	$+0.067\ (+10.1\%)$	$-0.741\ (-26.8\%)$

Table 1: Paired OptABA-PC–ABA-PC effects on seven bnlearn and synthetic datasets (nodes in brackets). Entries are mean within-seed differences

\Delta

; parentheses give relative change from ABA-PC, except runtime (

t_{\mathrm{Opt}}/t_{\mathrm{ABA}}

). Arrows indicate preferred directions.

\Delta n_{\mathrm{acc}}^{T}

counts additional true CI facts retained; #CPDAG counts saved MECs satisfying every retained fact. Counts and graph means describe saved, budget-limited sets. Tables 9 and 10 in the supplement give absolute mean

\pm

standard deviation and significance for all datasets; Earthquake is omitted because outcomes do not differ.

Appendix figures & tables17 assets

Supplementary material from the paper’s appendix.

Appendix

Contested fact $f$	$w_{f}$	$c_{\rm keep}$	$\mu(f)$	One private MUS $U_{f}$
indep(e,o,{i})	1000	2070	30	$\{f,\texttt{dep(e,o,\lx@text@lbrace\lx@text@rbrace)},$ $\quad\texttt{indep(e,i,\lx@text@lbrace o\lx@text@rbrace)}\}$
indep(r,i,{e})	530	2540	500	$\{\texttt{dep(r,i,\lx@text@lbrace\lx@text@rbrace)},f,$ $\quad\texttt{indep(e,r,\lx@text@lbrace\lx@text@rbrace)}\}$
indep(o,i,{})	510	2910	870	$\{\texttt{indep(r,i,\lx@text@lbrace o\lx@text@rbrace)},$ $\quad\texttt{dep(r,i,\lx@text@lbrace\lx@text@rbrace)},f\}$

Appendix

Table 2: Contestation certificates and exact integer margins for the three facts released in the 14-fact four-node trace. The weights are the main-paper strengths multiplied by

1000

;

c_{\rm keep}

is the minimum repair cost after making the contested fact mandatory, and

\mu(f)=c_{\rm keep}(f)-2040

Method	Role and implementation	Final settings
MPC	Majority-PC ( Colombo and Maathuis, 2014 ) , an order-independent PC variant, learns a skeleton by CI testing, orients colliders by majority evidence across separating sets, and applies Meek propagation to return one CPDAG.	Stable skeleton search; $G^{2}$ test; $\alpha=0.01$ ; majority collider rule (implementation uc_rule=5 ); collider priority $2$ ; no conditioning-depth limit.
ABA-PC	Causal ABA-PC ( Russo et al., 2024 ) receives MPC’s CI trace and releases facts from lowest to highest strength until its Bayes-ball Causal ABA program is coherent.	Same MPC trace; no conditioning-set-size multiplier; strength $s(p)=1-p/(2\alpha)$ for $p<\alpha$ and $s(p)=(\alpha-p/2-1/2)/(\alpha-1)$ otherwise; integer weight $\max\{1,\operatorname{round}(9{,}999{,}999s)\}$ ; incremental Bayes-ball encoding.
OptABA-PC	Our method uses the same facts, weights, and candidate-DAG encoding as ABA-PC, replacing sequential release with one minimum-cost correction-set optimisation.	MUS-reified weak constraints; clingo branch-and-bound ( bb ); optN enumeration of optimal stable models; minimum total released weight; four solver threads.
FGS	Fast Greedy Search ( Ramsey et al., 2017 ) is the Tetrad/ py-causal fast implementation of GES ( Chickering, 2002b ) ; it greedily inserts and removes edges under a score and returns one CPDAG.	BDeu score ( bdeu-score ); discrete-data interface; sample prior $15$ ; structure prior $1$ ; unrestricted maximum degree ( -1 ); faithfulness assumed; symmetric first step disabled.
ASPCR-DAG	The ASP method of Hyttinen et al. (2014) , restricted here to causally sufficient DAGs, assigns a cost to every graph–test disagreement and returns one optimal DAG.	Exhaustive conditioning schedule $n-2$ ; Bayesian CI test; log weights; prior independence $0.4$ ; Bayesian $\alpha=20$ ; acyclic sufficient encoding; clingo crafty configuration; comparison time limit $300$ s.

Appendix

Table 3: Baseline descriptions and final method settings. Shared data, seed, CI, and computational settings are given in the surrounding protocol.

Dataset	Completed		Paired $n$	End-to-end time (s)		BB/original
	Active paths	Bayes-ball		Active paths	Bayes-ball
Cancer (5)	50/50	50/50	50	$0.118\pm 0.027$	$\mathbf{0.102\pm 0.014}^{***}$	$0.86\times$
Earthquake (5)	50/50	50/50	50	$\mathbf{0.068\pm 0.013}^{***}$	$0.099\pm 0.013$	$1.45\times$
Survey (6)	50/50	50/50	50	$0.385\pm 0.119$	$\mathbf{0.148\pm 0.058}^{***}$	$0.38\times$
Asia (8)	50/50	50/50	50	$51.66\pm 50.27$	$\mathbf{2.74\pm 4.48}^{***}$	$0.05\times$
ER (5)	50/50	50/50	50	$0.078\pm 0.029$	$0.084\pm 0.007$	$1.08\times$

Appendix

Table 4: Matched encoding-efficiency ablation over seeds 2026–2075, with

N=5000

G^{2}

tests at

\alpha=0.01

, identical test traces and weights, and the same sequential ABA-PC repair policy. Only the causal-consistency encoding changes. Times are mean

\pm

standard deviation for completed seed pairs and include program construction and solving; “BB/original” is the ratio of those means, so values below one favour Bayes-ball. Completion is over all 50 runs. Bold marks the significantly faster implementation under paired two-sided Wilcoxon tests with BH adjustment across datasets ( ∗ , ∗∗ , and ∗∗∗ for adjusted

p<0.05

0.01

, and

0.001

). Every completed pair returned identical retained-fact counts, weights, and model counts.

Method	Run status	Feasible traces	Full-trace matches	Explicit repairs	Selected repair enforced
OptABA-PC	400/400 saved: 368 completed, 32 timeout incumbents	88/400	88/400	312/312	400/400
ABA-PC	400/400 saved and completed	88/400	88/400	312/312	400/400
ASPCR-DAG	250/250 saved; all optima proved	32/250	32/250	–	–
MPC	400/400 saved and completed	88/400	80/400	–	–
FGS	400/400 saved and completed	–	–	–	–

Appendix

Table 5: Evidence correspondence, run completion, and repair coverage. A trace is feasible if some DAG satisfies every recorded CI outcome; a full-trace output is the method’s returned graph or class satisfying all of them. A timeout incumbent is a valid saved repair whose optimality was not proved before the limit. An explicit repair names excluded facts, and selected-repair enforcement verifies every retained fact against the output. ASPCR-DAG uses its primary matched run set; FGS has no CI trace. A dash denotes an undefined object.

Method	Scope	Baseline decision evidence	Forced-fact evidence	Certificate supplied
OptABA-PC	$G^{2}$ , 250	Minimum repair on 111/111 inconsistent traces; lower cost than ABA-PC on 108 , tied on 3.	742/742 proved; zero gaps 0/742; median $\mu/w_{f}=0.455$ [0.182,0.912]. 11.6% are $\leq 0.1$ ; 22.5% are $>1$ .	Exact user-contestation threshold.
ABA-PC	$G^{2}$ , 250	Selected repair enforced on 250/250; minimum cost on only 3/111 inconsistent traces (108 higher).	–	Heuristic repair; no exact response threshold.
MPC	$G^{2}$ , 250	Compatible on 126/139 feasible traces; 13 avoidable gaps; no selected repair.	–	No repair or contestation object.
ASPCR-DAG	native, 250	Full trace on 23/250; graph optimum proved on 250/250.	4330/4330 proved; zero gaps 0/4330; median $\mathrm{gap}/w_{f}=2.828$ [0.890,8.647].	Post-hoc graph-objective sensitivity.

Appendix

Table 6: Decision and forced-fact evidence. OptABA-PC, ABA-PC, and MPC use the exact 250-trace

G^{2}

audit; ASPCR-DAG uses a separate 250-run native Bayesian audit. A forced OptABA-PC solve is a user contestation and

\mu(f)/w_{f}

is its exact threshold. ASPCR-DAG’s forced solve is a post-hoc sensitivity test of a soft graph-disagreement objective, so its

\mathrm{gap}/w_{f}

is not numerically comparable. ABA-PC and MPC define neither category optima nor a forced-fact certificate. Brackets give

[Q_{1},Q_{3}]

; a dash means that the object is undefined. Bold marks OptABA-PC’s exact certified result.

Dataset	Feasible / inputs	Lower / inconsistent	Same cost	Median saving	Full trace / inputs	Gap / feasible
	Input status	Saved OptABA-PC repair vs. ABA-PC			MPC output
Cancer (5)	0/50	50/50	0	40.2%	0/50	–
Earthquake (5)	47/50	1/3	2	61.8%	47/50	0/47
Survey (6)	0/50	50/50	0	57.1%	0/50	–
Asia (8)	0/50	50/50	0	73.0%	0/50	–
ER (5)	16/50	26/34	8	48.4%	11/50	5/16

Appendix

Table 7: Repair quality and MPC enforcement on the same 400 primary

G^{2}

traces. Feasible inputs admit a DAG satisfying the entire trace; all other inputs require a repair. Lower/inconsistent counts saved OptABA-PC repairs with lower release cost than ABA-PC among the inconsistent inputs, and same cost is the complementary count; ABA-PC is never cheaper. Median saving is conditional on a lower saved cost. Thirty-two OptABA-PC rows are valid timeout incumbents; Table 6 separately gives the 250-instance exact audit with proved optima. MPC full trace counts returned classes satisfying every input test, and gap counts feasible traces that MPC nevertheless fails to satisfy.

	OptABA-PC contestations			ASPCR-DAG sensitivity probes
Dataset	Proved	Median $\mu/w_{f}$ $[Q_{1},Q_{3}]$	Median $\mu/c^{*}$	Proved	Median $\mu_{\rm ASP}/w_{f}$ $[Q_{1},Q_{3}]$
Cancer (5)	207/207	0.297 [0.158,0.609]	0.072	959/959	4.041 [1.615,7.946]
Earthquake (5)	4/4	4.170 [1.628,6.332]	3.544	80/80	3.625 [0.990,6.349]
Survey (6)	507/507	0.517 [0.200,1.026]	0.051	2094/2094	2.376 [0.537,14.074]
ER (5)	0/0	–	–	503/503	2.163 [0.773,5.555]
SF (5)	24/24	0.734 [0.308,1.344]	0.182	694/694	2.437 [0.958,6.108]

Appendix

Table 8: Dataset-level forced-fact evidence over 250 validated baselines per method (50 per dataset). Each target is a fact rejected by the baseline optimum; all re-solves prove optimality and none has zero gap. OptABA-PC’s

\mu(f)/w_{f}

is an exact user-contestation threshold;

c^{*}

denotes the complete baseline optimum repair cost. ASPCR-DAG’s

\mu_{\rm ASP}(f)/w_{f}

is instead a post-hoc sensitivity gap for its soft graph-disagreement objective. These are not paired effect sizes: the methods use different traces, weights, and objectives, and their synthetic audits use different graph densities.

Dataset	Method	Correct retained $\uparrow$	Fact F1 $\uparrow$	Time (s) $\downarrow$
Cancer (5)	ABA-PC	$24.0\pm 8.7$	$0.904\pm 0.046$	$1.51\pm 0.48$
	OptABA-PC	$\mathbf{25.6\pm 9.8}^{***}$	$0.900\pm 0.064$	$\mathbf{1.28\pm 0.38}^{***}$
Earthquake (5)	ABA-PC	$54.1\pm 0.8$	$0.999\pm 0.007$	$1.83\pm 0.83$
	OptABA-PC	$54.2\pm 0.8$	$0.999\pm 0.003$	$\mathbf{1.68\pm 0.81}^{***}$
Survey (6)	ABA-PC	$24.8\pm 9.6$	$\mathbf{0.789\pm 0.075}^{**}$	$15.7\pm 30.2$
	OptABA-PC	$\mathbf{29.7\pm 12.3}^{***}$	$0.742\pm 0.107$	$2.94\pm 1.75$

Appendix

Table 9: Absolute retained-fact quality and runtime for the primary ABA-PC–OptABA-PC comparison (mean

\pm

standard deviation over 50 matched seeds,

N=5000

\alpha=0.01

). Both methods repair the same recorded CI tests. All eight datasets are included, with node counts in brackets. Correct retained counts facts agreeing with

d

-separation in the generating DAG; Fact F1 scores retained-fact classification. Time includes program construction, solving, and compatible-set evaluation. Arrows indicate preferred directions. Only a significant paired ABA-PC–OptABA-PC winner is bold and starred: one, two, and three stars denote BH-adjusted

p<0.05

0.01

, and

0.001

from two-sided Wilcoxon tests. CPDAG evaluation is in Table 10 .

Dataset	Method	#CPDAGs $\downarrow$	CPDAG F1 $\uparrow$	CPDAG SHD $\downarrow$
Cancer (5)	ABA-PC	$15.1\pm 28.8$	$0.522\pm 0.214$	$2.81\pm 1.18$
	OptABA-PC	$\mathbf{1.0\pm 0.0}^{***}$	$\mathbf{0.576\pm 0.259}^{*}$	$\mathbf{1.84\pm 1.11}^{***}$
Earthquake (5)	ABA-PC	$1.0\pm 0.3$	$0.991\pm 0.033$	$0.09\pm 0.30$
	OptABA-PC	$1.0\pm 0.0$	$0.991\pm 0.030$	$0.08\pm 0.27$
Survey (6)	ABA-PC	$1278.5\pm 2807.2$	$0.438\pm 0.088$	$7.14\pm 1.39$
	OptABA-PC	$\mathbf{1.2\pm 0.7}^{***}$	$\mathbf{0.597\pm 0.127}^{***}$	$\mathbf{4.35\pm 1.09}^{***}$

Appendix

Table 10: Absolute CPDAG evaluation for the primary ABA-PC–OptABA-PC comparison on the same 50 matched seeds per dataset as Table 9 . Counts, F1, and SHD are mean

\pm

standard deviation. Node counts are in brackets. #CPDAGs counts saved equivalence classes satisfying every retained post-repair fact; F1 and SHD average reconstruction quality over these budget-limited sets. Arrows indicate preferred directions. Only a significant paired ABA-PC–OptABA-PC winner is bold and starred: one, two, and three stars denote BH-adjusted

p<0.05

0.01

, and

0.001

from two-sided Wilcoxon tests.

Dataset	Method	CPDAG SHD $\downarrow$	CPDAG F1 $\uparrow$	DAG SHD $\downarrow$	DAG F1 $\uparrow$	Compat. CPDAGs $\downarrow$
Cancer (5)	ABA-PC	$2.81\pm 1.18$	$0.522\pm 0.214$	$2.98\pm 1.20$	$0.483\pm 0.223$	$15.1\pm 28.8$
	OptABA-PC	$\mathbf{1.84\pm 1.11}^{***}$	$\mathbf{0.576\pm 0.259}^{*}$	$\mathbf{1.89\pm 1.14}^{***}$	$\mathbf{0.563\pm 0.262}^{*}$	$\mathbf{1.0\pm 0.0}^{***}$
	MPC	$1.58\pm 0.97$	$\mathbf{0.661\pm 0.202}^{\dagger}$	$1.70\pm 1.01$	$0.623\pm 0.222$	$0.0\pm 0.0$
	FGS	$\mathbf{1.12\pm 0.96}^{\dagger\dagger}$	$\mathbf{0.742\pm 0.212}^{\dagger\dagger}$	$\mathbf{1.11\pm 0.94}^{\dagger\dagger\dagger}$	$\mathbf{0.739\pm 0.215}^{\dagger\dagger}$	–
	ASPCR-DAG	$2.24\pm 0.56^{\dagger}$	$0.490\pm 0.123^{\dagger}$	$2.34\pm 0.89^{\dagger}$	$0.462\pm 0.225^{\dagger}$	$0.0\pm 0.0$
Earthquake (5)	ABA-PC	$0.09\pm 0.30$	$0.991\pm 0.033$	$0.10\pm 0.34$	$0.989\pm 0.039$	$1.0\pm 0.3$

Appendix

Table 11: Graph reconstruction and CI-compatible equivalence classes on the matched primary experiments (mean

\pm

standard deviation over seeds). Numbers in brackets after dataset names give the number of nodes. ABA-PC and OptABA-PC CPDAG values average over every equivalence class admitted by the selected repair; MPC and FGS return one CPDAG when their output is structurally valid, and ASPCR-DAG returns one DAG whose equivalence class supplies its CPDAG score. DAG columns average every compatible DAG for ABA-PC/OptABA-PC and every consistent extension of a valid MPC/FGS CPDAG; ASPCR-DAG is scored on its returned DAG. The final column counts CPDAGs satisfying the enforced evidence: retained post-repair facts for ABA-PC/OptABA-PC and every recorded test for MPC/ASPCR-DAG; it is undefined for score-based FGS (‘–’). Stars identify the significant ABA-PC–OptABA-PC winner. Daggers on an external method indicate a paired difference from OptABA-PC; bold daggers favour that external method, whereas unbold daggers favour OptABA-PC. One, two, and three symbols denote BH-adjusted

p<0.05

0.01

, and

0.001

. Subscripts occur only for MPC DAG-extension metrics when its endpoint output is invalid:

n=38

on Survey and

n=49

on ER(8).

Dataset	Method	Correct available	Correct kept	False kept
Cancer (5)	ABA-PC	$27.2\pm 10.1$	$24.0\pm 8.7$	$\mathbf{1.6\pm 1.5}^{***}$
	OptABA-PC	$27.2\pm 10.1$	$\mathbf{25.6\pm 9.8}^{***}$	$3.3\pm 2.2$
	ASPCR-DAG	$53.7\pm 3.4$	$48.0\pm 7.2$	$12.9\pm 8.5$
Earthquake (5)	ABA-PC	$54.2\pm 0.9$	$54.1\pm 0.8$	$0.1\pm 0.3$
	OptABA-PC	$54.2\pm 0.9$	$54.2\pm 0.8$	$0.1\pm 0.3$
	ASPCR-DAG	$75.6\pm 2.9$	$74.7\pm 3.8$	$3.7\pm 3.0$

Appendix

Table 12: CI-fact counts on the matched primary experiments (mean

\pm

standard deviation over 50 seeds per reported row). Node counts are in brackets. Correct available counts recorded tests agreeing with

d

-separation in the generating DAG; correct/false kept split retained facts. ABA-PC and OptABA-PC repair the identical test set. Only a significant paired ABA-PC–OptABA-PC winner is bold and starred: one, two, and three stars denote BH-adjusted

p<0.05

0.01

, and

0.001

from two-sided Wilcoxon tests. For ASPCR-DAG, kept means its returned DAG satisfies a native soft test relation: descriptive graph–test agreement, not an explicit retained CI set. MPC and FGS expose no comparable fact accounting. Precision, recall, and F1 are in Table 13 .

Dataset	Method	Precision	Recall	Fact F1
Cancer (5)	ABA-PC	$\mathbf{0.927\pm 0.069}^{***}$	$0.887\pm 0.058$	$0.904\pm 0.046$
	OptABA-PC	$0.866\pm 0.091$	$\mathbf{0.941\pm 0.045}^{***}$	$0.900\pm 0.064$
	ASPCR-DAG	$0.792\pm 0.132$	$0.892\pm 0.114$	$0.837\pm 0.120$
Earthquake (5)	ABA-PC	$0.999\pm 0.005$	$0.999\pm 0.010$	$0.999\pm 0.007$
	OptABA-PC	$0.999\pm 0.005$	$1.000\pm 0.002$	$0.999\pm 0.003$
	ASPCR-DAG	$0.952\pm 0.039$	$0.987\pm 0.018$	$0.969\pm 0.026$

Appendix

Table 13: CI-fact precision, recall, and F1 on the matched primary experiments (mean

\pm

standard deviation over 50 seeds per reported row; higher is better). Node counts are in brackets. Retained correct tests are true positives, retained false tests are false positives, and released correct tests are false negatives; counts are in Table 12 . For ASPCR-DAG, retention denotes satisfaction of native soft tests by its returned DAG, not an explicit retained CI set. MPC and FGS expose no comparable fact accounting. Only a significant paired ABA-PC–OptABA-PC winner is bold and starred: one, two, and three stars denote BH-adjusted

p<0.05

0.01

, and

0.001

from two-sided Wilcoxon tests.

Dataset	Method	SHD avg	SHD best	SHD worst
Cancer (5)	ABA-PC	$2.81\pm 1.18$	$1.58\pm 1.18$	$3.84\pm 1.80$
	OptABA-PC	$\mathbf{1.84\pm 1.11}^{***}$	$1.84\pm 1.11$	$\mathbf{1.84\pm 1.11}^{***}$
	MPC	$1.58\pm 0.97$	$1.58\pm 0.97$	$1.58\pm 0.97$
	FGS	$\mathbf{1.12\pm 0.96}^{\dagger\dagger}$	$\mathbf{1.12\pm 0.96}^{\dagger\dagger}$	$\mathbf{1.12\pm 0.96}^{\dagger\dagger}$
	ASPCR-DAG	$2.24\pm 0.56^{\dagger}$	$2.24\pm 0.56^{\dagger}$	$2.24\pm 0.56^{\dagger}$
Earthquake (5)	ABA-PC	$0.09\pm 0.30$	$0.08\pm 0.27$	$0.10\pm 0.36$

Appendix

Table 14: Average, best, and worst CPDAG SHD within each run of the matched primary experiments (mean

\pm

standard deviation over seeds; lower is better). ABA-PC and OptABA-PC are scored over the saved classes satisfying their selected facts. MPC and FGS return one class when structurally valid; ASPCR-DAG’s DAG supplies one class, so their average, best, and worst values coincide within a run. Stars mark the significant ABA-PC–OptABA-PC winner. Daggers compare an external method with OptABA-PC; bold daggers favour that method, unbold daggers favour OptABA-PC. One, two, and three symbols denote BH-adjusted

p<0.05

0.01

, and

0.001

. Compatibility counts are in Table 18 .

Dataset	Method	F1 avg	F1 best	F1 worst
Cancer (5)	ABA-PC	$0.522\pm 0.214$	$0.646\pm 0.262$	$0.440\pm 0.222$
	OptABA-PC	$\mathbf{0.576\pm 0.259}^{*}$	$0.576\pm 0.259$	$\mathbf{0.576\pm 0.259}^{**}$
	MPC	$\mathbf{0.661\pm 0.202}^{\dagger}$	$\mathbf{0.661\pm 0.202}^{\dagger}$	$\mathbf{0.661\pm 0.202}^{\dagger}$
	FGS	$\mathbf{0.742\pm 0.212}^{\dagger\dagger}$	$\mathbf{0.742\pm 0.212}^{\dagger\dagger}$	$\mathbf{0.742\pm 0.212}^{\dagger\dagger}$
	ASPCR-DAG	$0.490\pm 0.123^{\dagger}$	$0.490\pm 0.123^{\dagger}$	$0.490\pm 0.123^{\dagger}$
Earthquake (5)	ABA-PC	$0.991\pm 0.033$	$0.991\pm 0.030$	$0.989\pm 0.038$

Appendix

Table 15: Average, best, and worst CPDAG F1 within each run of the matched primary experiments (mean

\pm

standard deviation over seeds; higher is better). ABA-PC and OptABA-PC are scored over the saved classes satisfying their selected facts. MPC and FGS return one class when structurally valid; ASPCR-DAG’s DAG supplies one class, so their average, best, and worst values coincide within a run. Stars mark the significant ABA-PC–OptABA-PC winner. Daggers compare an external method with OptABA-PC; bold daggers favour that method, unbold daggers favour OptABA-PC. One, two, and three symbols denote BH-adjusted

p<0.05

0.01

, and

0.001

. Compatibility counts are in Table 18 .

Dataset	Method	SHD avg	SHD best	SHD worst
Cancer (5)	ABA-PC	$2.98\pm 1.20$	$1.44\pm 1.18$	$4.18\pm 1.96$
	OptABA-PC	$\mathbf{1.89\pm 1.14}^{***}$	$1.70\pm 1.15$	$\mathbf{2.00\pm 1.21}^{***}$
	MPC	$1.70\pm 1.01$	$\mathbf{0.86\pm 0.61}^{\dagger\dagger\dagger}$	$2.28\pm 1.37$
	FGS	$\mathbf{1.11\pm 0.94}^{\dagger\dagger\dagger}$	$\mathbf{0.90\pm 0.81}^{\dagger\dagger\dagger}$	$\mathbf{1.30\pm 1.18}^{\dagger\dagger}$
	ASPCR-DAG	$2.34\pm 0.89^{\dagger}$	$2.34\pm 0.89^{\dagger\dagger}$	$2.34\pm 0.89$
Earthquake (5)	ABA-PC	$0.10\pm 0.34$	$0.08\pm 0.27$	$0.12\pm 0.48$

Appendix

Table 16: Average, best, and worst DAG SHD within each run of the matched primary experiments (mean

\pm

standard deviation over seeds; lower is better). ABA-PC and OptABA-PC are scored over saved compatible DAGs; MPC and FGS over consistent extensions of each valid returned CPDAG; ASPCR-DAG on its single DAG, so its average, best, and worst coincide. Subscripts mark MPC’s reduced samples: 38 on Survey and 49 on ER(8). Stars mark the significant ABA-PC–OptABA-PC winner. Daggers compare an external method with OptABA-PC; bold daggers favour that method, unbold daggers favour OptABA-PC. One, two, and three symbols denote BH-adjusted

p<0.05

0.01

, and

0.001

. Compatibility counts are in Table 18 .

Dataset	Method	F1 avg	F1 best	F1 worst
Cancer (5)	ABA-PC	$0.483\pm 0.223$	$0.688\pm 0.270$	$0.346\pm 0.264$
	OptABA-PC	$\mathbf{0.563\pm 0.262}^{*}$	$0.616\pm 0.277$	$\mathbf{0.531\pm 0.281}^{***}$
	MPC	$0.623\pm 0.222$	$\mathbf{0.856\pm 0.113}^{\dagger\dagger\dagger}$	$0.462\pm 0.326$
	FGS	$\mathbf{0.739\pm 0.215}^{\dagger\dagger}$	$\mathbf{0.794\pm 0.179}^{\dagger\dagger\dagger}$	$\mathbf{0.691\pm 0.282}^{\dagger}$
	ASPCR-DAG	$0.462\pm 0.225^{\dagger}$	$0.462\pm 0.225^{\dagger\dagger}$	$0.462\pm 0.225$
Earthquake (5)	ABA-PC	$0.989\pm 0.039$	$0.991\pm 0.030$	$0.985\pm 0.062$

Appendix

Table 17: Average, best, and worst DAG F1 within each run of the matched primary experiments (mean

\pm

standard deviation over seeds; higher is better). ABA-PC and OptABA-PC are scored over saved compatible DAGs; MPC and FGS over consistent extensions of each valid returned CPDAG; ASPCR-DAG on its single DAG, so its average, best, and worst coincide. Subscripts mark MPC’s reduced samples: 38 on Survey and 49 on ER(8). Stars mark the significant ABA-PC–OptABA-PC winner. Daggers compare an external method with OptABA-PC; bold daggers favour that method, unbold daggers favour OptABA-PC. One, two, and three symbols denote BH-adjusted

p<0.05

0.01

, and

0.001

. Compatibility counts are in Table 18 .

Dataset	Method	Time (s)	Returned CPDAGs	Returned DAGs	CI-compat. CPDAGs	CI-compat. DAGs
Cancer (5)	ABA-PC	$1.51\pm 0.48$	$15.1\pm 28.8$	$31.9\pm 57.1$	$15.1\pm 28.8$	$31.9\pm 57.1$
	OptABA-PC	$\mathbf{1.28\pm 0.38}^{***}$	$1.0\pm 0.0$	$1.5\pm 1.1$	$\mathbf{1.0\pm 0.0}^{***}$	$\mathbf{1.5\pm 1.1}^{***}$
	MPC	$\mathbf{0.01\pm 0.00}^{\dagger\dagger\dagger}$	$1.0\pm 0.0$	$3.3\pm 1.6$	$0.0\pm 0.0$	$0.0\pm 0.0$
	FGS	$\mathbf{0.08\pm 0.09}^{\dagger\dagger\dagger}$	$1.0\pm 0.0$	$1.8\pm 1.5$	–	–
	ASPCR-DAG	$12.8\pm 0.09^{\dagger\dagger\dagger}$	$1.0\pm 0.0$	$1.0\pm 0.0$	$0.0\pm 0.0$	$0.0\pm 0.0$
Earthquake (5)	ABA-PC	$1.83\pm 0.83$	$1.0\pm 0.3$	$1.1\pm 0.5$	$1.0\pm 0.3$	$1.1\pm 0.5$

Appendix

Table 18: Timing, structural output size, and CI compatibility for the matched primary experiments (mean

\pm

standard deviation over 50 seeds for each reported method–dataset pair). Returned CPDAGs and DAGs are structural output/extension counts. CI-compatible counts additionally require all retained post-repair facts for ABA-PC/OptABA-PC or every recorded native test for MPC/ASPCR-DAG; compatibility is undefined for score-based FGS (‘–’). ABA-PC and OptABA-PC times include construction, solving, and compatible-set evaluation and are directly paired. MPC/FGS time the learner call, while ASPCR-DAG reports its native end-to-end R run; their values and paired daggers describe implementations with different timing scopes and are not a controlled cross-family speed ranking. Stars identify the significant ABA-PC–OptABA-PC winner. Bold daggers favour the external method and unbold daggers favour OptABA-PC. One, two, and three symbols denote BH-adjusted

p<0.05

0.01

, and

0.001

Constraint-Based Causal Discovery

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Causal Discovery as Dialectical Aggregation: A Quantitative Argumentation Framework

Sheng Wei, Yulin Chen, Beishui Liao

The College of Computer Science and Technology, Zhejiang University · School of Philosophy, Zhejiang University · ZLAIRE, Zhejiang University+1

Causal Discovery Computational Argumentation Constraint-Based Causal Discovery

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Constraint-based causal discovery is brittle in finite-sample regimes because erroneous conditional-independence (CI) decisions can cascade into substantial structural errors. We propose Quantitative Argumentation for Causal Discovery (QACD), a semantics-driven framework that represents CI outcomes as graded, defeasible arguments rather than irreversible constraints. QACD maps statistical test outcomes to argument strengths and aggregates conflicting evidence through connectivity-mediated witness propagation, producing a fixed-point acceptability labeling over candidate adjacencies. Experiments on standard benchmark Bayesian networks suggest that QACD improves structural coherence and interventional reliability in several noisy or inconsistent CI regimes, while remaining competitive with classical constraint-based, hybrid, and prior argumentation-based baselines.

Abstract

Constraint-Based Causal Discovery

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