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Minimal Witness Reinforcement Learning
T. Y. Tsui, Zihao Ye, Pengxiang Cai, Yanchao Li, Yuqiang Li, Zhehong Ai
TL;DR
Minimal-witness tasks require recovering all irreducible sufficient sets, while standard RL rewards can return redundant supersets or only one witness. MWRL credits proposals for the certified coverage their group would lose without them. Across the reported settings, it recovers most minimal witnesses, including on unseen tasks.
Problem
Tasks that ask for sufficient factors may have multiple minimal witnesses, but standard RL methods may return only one solution or redundant supersets.
Source: §1 · pp. 1–2Method
MWRL measures the union of supersets certified by successful group proposals and credits each proposal for the coverage lost if it is removed.
Source: §3.1 · p. 3, §3.4 · p. 5Results
MWRL recovers most minimal witnesses across the reported tasks, while proposal-isolated methods return redundant supersets or a single witness, and one trained policy also recovers witnesses on unseen tasks.
Source: §5 · p. 9Takeaways & Limitations
The paper concludes that families of minimal witnesses can be learned from verifier feedback wherever a verifier certifies an outcome.
Source: §5 · p. 9The formulation assumes monotonicity; for the language-model experiment’s potentially nonmonotone raw verifier, born-minimality is measured as 1-minimality.
Source: §3.1 · p. 3
Abstract
from arXiv · showhide
``What are the irreducible conditions that are sufficient to produce an outcome?'' is one of the most common questions that recur across computation and science. Its answers, the minimal sufficient witnesses, are what we mean by explanations, mechanisms and reasons. These problems usually ask for multiple minimal witnesses, yet standard RL methods may reveal only one solution or redundant ones. We formalize this problem as minimal-witness identification and introduce Minimal-Witness Reinforcement Learning (MWRL). MWRL takes the union of the sets certified by successful proposals sampled from the policy and credits each proposal for the coverage the group union would lose without that proposal. This credit assignment, derived directly from the problem definition, unifies the demands for minimality and recovery of alternatives from a single black-box verifier bit. Under this principle, we derive a value iteration planner that recovers the entire family of witnesses and a policy gradient method that can scale to large language models. Across different experimental settings, MWRL recovers most minimal witnesses, while other methods return redundant supersets or a single witness. By making witness families learnable from verifier feedback, MWRL expands the scope of reinforcement learning beyond single-solution optimization. Our code is available at https://github.com/TSUITUENYUE/MWRL.
1 Introduction
Across many domains, the task is to identify every irreducible set sufficient for an outcome, not just one solution. MWRL uses group-level coverage credit to reward minimal witnesses while recovering alternatives from verifier feedback.
1 Introduction
- A minimal sufficient witness is an irreducible set of factors, and a task may have several incomparable witnesses.The server example includes {power}, {disk1, CPU}, and {disk1, disk2} as minimal causes; all matter for diagnosis.Source: §1 · p. 1
- Minimality is subset-based, not a preference for small sets or for varied solutions; diversity methods do not determine when the answer is complete.The required output is the full antichain, whether it contains one witness or several.Source: §1 · p. 1
- Existing enumeration methods rely on exposed formulas, constraints, or deterministic predicates, while many settings offer only a binary verifier.The paper frames the problem as recovering minimal witnesses from verifier feedback alone.Source: §1 · pp. 1–2
- MWRL recovers most of each enumerable MaxSAT instance’s antichain, while proposal-isolated methods recover at most a quarter.The figure compares finite-budget recovery; size-penalized MaxEnt RL and GFlowNet use their best penalty from a sweep.Source: Figure 1 · p. 2
- Success-only rewards treat sufficient sets and their supersets alike, whereas size penalties can collapse probability onto only the smallest witness.Because both rewards value proposals in isolation, they cannot distinguish a new minimal witness from a redundant accepted set.Source: §1 · p. 2
- MWRL measures the union of the supersets certified by successful proposals and credits each proposal for coverage lost when it is removed.Repeating or containing another accepted proposal earns zero credit; the objective rewards unnecessary-element removal and undiscovered minimal witnesses.Source: §1 · p. 2
2 Problem Formulation
The formulation seeks a policy whose sampled proposals recover the antichain of subset-minimal sufficient sets for a context. It defines recovery metrics and establishes why proposal-local objectives cannot guarantee the full antichain.
2 Problem Formulation
- A minimal witness is sufficient with no sufficient proper subset, so minimal witnesses form an antichain; under monotonicity, they are the prime implicants of the predicate.The formulation assumes that adding elements cannot turn a success into a failure.Source: §2 · pp. 2–3
- For a fixed context, minimal-witness identification seeks a policy whose proposals recover the target antichain; antichain recall measures the fraction recovered as distinct minimal witnesses.Born-minimal rate is the fraction of successful proposals that are minimal.Source: §2 · p. 3
- A group return must account for both each successful proposal’s minimality and its relation to the other proposals, including repeats or sets containing another success.The policy samples groups of K proposals from its terminal distribution.Source: §2 · p. 3
- Proposal-separable rewards evaluate proposals independently, and local rewards cannot identify minimality from a verifier bit that is identical for a set and its sufficient subset.The theorem states that, for |E| ≥ 3, each of three objective classes fails to target the minimal-witness support for some monotone predicate with multiple witnesses.Source: §2 · p. 3
- Separable objectives, local-reward proportional samplers, and success-probability objectives each have predicates for which their target law differs from the minimal-witness antichain.A size penalty leaves a reward local, so it does not escape these objective categories.Source: §2 · p. 3
3 Minimal-Witness Reinforcement Learning
MWRL values a group by the region its successful proposals certify, then credits proposals for the coverage lost when they are removed. This supports exact antichain recovery by value iteration and policy-gradient training from verifier feedback.
3.1 Monotonicity and the certified region
- Monotonicity makes each success certify its supersets; existential closure preserves minimal accepted sets when a raw predicate can reject a superset.Without closure, the paper uses 1-minimality for its non-monotone circuit verifier and reports born-minimal rates that way.Source: §3.1 · p. 3
3.2 The group return
- Under the representation theorem, group values that ignore duplicate and redundant successes can be expressed as functions of the certified region.MWRL values that region using the coverage measure of a fixed full-support measure.Source: §3.2 · p. 4
3.3 Minimal-witness value iteration
- With a full-support measure and exact maximization, value iteration selects every minimal witness and halts after recovering the complete antichain.It adds a successful proposal by its marginal increase in certified-region value.Source: §3.3 · pp. 4–5
3.4 The MWRL policy gradient
- The deletion credit is the measure of sets certified only by one proposal, making it a submodular marginal for the group’s coverage objective.The policy-gradient method uses this group return; a leave-two-out baseline can allow negative credits without changing the expected gradient.Source: §3.4 · p. 5
- If every minimal witness has sampling probability at least ϵ, group-minimal successes recover the full antichain with probability at least 1 − Le^−Kϵ.The same bound applies to deployment samples when B replaces K.Source: §3.4 · pp. 5–6
- A shared-batch Monte Carlo estimator avoids inclusion–exclusion’s exponential number of terms when computing deletion credits, without extra verifier calls.Its work per proposal scales with the Monte Carlo sample count and the number of other group-minimal successes.Source: §3.4 · p. 6
4 Experiments
Experiments test witness recovery on enumerable Boolean formulas, held-out Suzuki–Miyaura substrate pairs, and sparse language-model circuits, followed by ablations of MWRL’s credit and training choices.
4.1 Prime implicant enumeration
- MWRL covers most of the prime-implicant antichain, while success-only, size-penalized, MaxEnt, and GFlowNet methods leave redundant supersets or recover only part of it.Minimal-witness value iteration recovers the full antichain at b = |M|, whereas scalar value iteration returns one witness.Source: §4.1 · p. 6, Table 2 · p. 6
4.2 Scientific variable identification
- On held-out substrates, conditioned-policy recall is close to seen-substrate recall and reaches a large fraction of per-substrate search; removing the fingerprint cuts recall to about half.The conditioned policy also has a higher born-minimal rate than separately trained per-substrate policies; scalar reward recovers no minimal witness.Source: §4.2 · p. 7, Table 3 · p. 7
4.3 LLM sparse circuit discovery
- At matched sparsity, only verifier-guided searches find sufficient circuits on discovery prompts; MWRL returns a family with similar discovery retention to ACDC’s single circuit and better held-out behavior than ranking masks.Circuits are evaluated on questions drawn from dataset rows disjoint from discovery.Source: §4.3 · pp. 7–8
4.4 Ablations
- Raw deletion credit is non-negative and gives an unbiased update, while l2o uses signed credits without changing the expected gradient; group-mean centering is biased for finite groups.Group-mean centering’s baseline depends on the trajectory being updated.Source: §4.4 · p. 8, Figure 5 · p. 8
- Recovery rises with training group size despite fewer updates at fixed verifier calls; at deployment, small budgets favor concentrated policies and larger budgets favor broader ones.The group-size comparison holds total verifier calls fixed.Source: §4.4 · p. 8, Figure 6 · p. 9
- From N = 5,000 onward, shared-sample Monte Carlo estimates give almost the same credits and policy gradient as exact enumeration.The comparison uses a constructed K = 48 group and averages credits over twenty independent batches.Source: §4.4 · p. 9, Table 5 · p. 9
- Across the tested base-measure range, born-minimal rate changes little while antichain recall rises as larger minimal witnesses are sampled more often.The sweep varies only p in the MaxSAT protocol.Source: Base-measure choice · p. 9, Figure 7 · p. 9
5 Conclusion
The conclusion frames minimal-witness identification as recovering every minimal sufficient set from binary verifier feedback. The appendix characterizes eligible group values through certified up-sets and identifies coverage measure as the paper’s return.
5 Conclusion
- MWRL credits each proposal for the certified region a group would lose without it; its planner recovers every witness on enumerable instances, while the policy gradient scales to language models.Across evaluated tasks, it recovers most minimal witnesses, whereas methods valuing proposals in isolation return redundant supersets or a single witness.Source: §5 · p. 9
A.1 Up-set representation
- Any symmetric group value satisfying the stated success-multiset and redundancy conditions can be represented as a set function on up-sets, and monotonicity corresponds to a monotone set function.The proof uses the bijection between antichains and up-sets.Source: §A.1 · pp. 11–12
A.2 Group returns and verifier access
- The paper’s coverage measure values the certified region directly; its alternative counting measure first canonicalizes successes into minimal witnesses and requires additional verifier calls.Source: §A.2 · p. 12
A.3 Proposal-separable objectives
Separable objectives, local-reward samplers, and objectives based only on success probability can fail to target the full minimal-witness antichain. The section motivates group-dependent coverage because proposal value depends on overlap and containment.
A.3 Proposal-separable objectives
- With |E| ≥ 3, each of the three target rules can have support differing from M(c) for some monotone predicate with |M(c)| ≥ 2.Source: §A.3 · p. 12
- For separable objectives, every optimal law is supported on highest-reward sets, so a point-mass optimum misses a witness whenever |M(c)| ≥ 2.Source: §A.3 · p. 12
- A proportional sampler using a local reward can assign positive mass to a nonminimal set under one predicate because that set retains positive weight under another.In the example, X contains the witness {a} under s2.Source: §A.3 · p. 12
- When |M(c)| ≥ 2, a success-probability objective can attain its optimum with a law supported on {∅, E}, which contains no minimal witness.Source: §A.3 · p. 12
- PPO, GRPO, and RLOO are separable-objective examples, while MaxEnt RL and GFlowNet are proportional-sampler examples.Source: §A.3 · p. 12
- Coverage return is not proposal-separable: its value depends on overlap and containment, so marginal credit depends on relations among group proposals.Source: §A.3 · pp. 12–13
A.4 Nonmonotone verifiers and existential closure
With nonmonotone verifiers, accepted proposals do not certify their supersets, and finite closure provides only an approximate repair. The section also develops value-iteration and grouped objectives for recovering minimal witnesses and alternatives.
A.4 Nonmonotone verifiers and existential closure
- Without monotonicity, acceptance certifies a proposal but not its supersets, and 1-minimality can differ from minimality.In the finite-closure audit, unrepaired violations were systematic misses; canonicalizer instability did not affect coverage credit.Source: §A.4 · p. 13
B.1 Minimal-witness value iteration
- With a full-support measure, value iteration selects each minimal witness once and stops after |M(c)| steps, covering F(c).A nonminimal success has lower marginal coverage than an unselected minimal witness.Source: §B.1 · p. 14
B.2 Finite-group objective identities
- For groups larger than one, rarer certification or witness probabilities receive larger objective derivatives, so exploration determines which alternatives can be learned.A witness with zero policy probability is never sampled and receives zero score-function credit.Source: §B.2 · p. 14
B.3 The counting measure
- Coverage reaches a 0.76 born-minimal rate and two thirds of the antichain, while counting nearly saturates recall and guarantees minimal outputs at the cost of extra verifier calls.The finite-closure audit found returned circuits 1-minimal under the raw predicate; the canonicalizer returned the originating circuit in about half of repaired cases.Source: §A.4 · p. 13, §B.3 · p. 14, Table 7 · p. 14
B.4 Finite-budget optimum of the counting measure
The counting objective is optimized by distributing successful proposals uniformly across the minimal witnesses when the policy class can realize those distributions. Recovery guarantees depend on proposal budgets, positive probability for each witness, and admissible grouped policy-gradient credits.
B.4 Finite-budget optimum of the counting measure
- For K ≥ 2, the counting measure is uniquely maximized by a failure-free uniform distribution over the L minimal witnesses; for K = 1, every failure-free distribution is optimal.Under the uniform optimum, expected antichain recall is 1 − (1−1/L)^K, and K < L proposals cannot contain the full antichain.Source: §B.4 · p. 14
B.5 Deployment-budget recovery
- Deployment curves estimate distinct minimal witnesses recovered in B proposals and saturate at the witnesses with positive policy probability.A forward pass over enumerable MaxSAT subset states computes each witness probability in closed form.Source: §B.5 · p. 15
B.6 Asymptotic recovery under full support
- With full-support μ, asymptotic coverage equals μ(F(c)) exactly when every minimal witness has positive proposal probability.If each has probability at least ϵ, the probability that group-minimal successes recover the full antichain is at least 1 − Le^−Kϵ.Source: §B.6 · p. 15
C Group policy gradients and control variates
- Grouped policy-gradient derivations apply to finite-horizon episodic proposal processes with a one-bit terminal verifier and independent, identically distributed group episodes.The objective depends on the terminal proposal law rather than an episode’s internal states or path.Source: §C.1 · p. 15
C.1 Grouped objective and REINFORCE
- Recovery requires every minimal witness to be reachable and the realizable terminal laws to include every categorical distribution over them.The conditions are separate: positive probability for each witness and sufficient expressiveness of the policy class.Source: §C.1 · p. 15
C.2 Admissible baselines and leave-one-out credit
- The leave-one-out baseline withholds proposal i’s contribution, yielding its nonnegative deletion credit without changing the gradient expectation.The baseline depends only on the other trajectories, not the withheld trajectory.Source: §C.2 · p. 16
C.3 Leave-two-out baseline
- The leave-two-out estimator subtracts an average marginal contribution from deletion credit while preserving the aggregate gradient expectation.Its baseline depends only on trajectories other than the one being scored.Source: §C.3 · p. 16
C.4 Support of deletion credit
- Only successful proposals not containing another successful proposal receive positive deletion credit.This identifies unrepeated group-minimal successes as the proposals with nonzero credit.Source: §C.4 · p. 16
D.1 Witness values
The base measure determines how strongly witness value changes with set size. For large groups, shared-sample Monte Carlo estimates deletion credit with linear rather than exponential dependence on the number of other group-minimal successes.
D.1 Witness values
- Under product measures witness values decay exponentially with set size, while layer-uniform weighting assigns 1/(|S| + 1); both preserve strict-refinement ordering.Polynomial reweightings make the refinement ratio approach one as witness size grows, reducing the extra credit from removing an element.Source: §D.1 · p. 17, Figure 10 · p. 17
D.2 Exact inclusion-exclusion credit
- Exact inclusion–exclusion credit costs O(2^|Di|), and is zero when another group-minimal success is a subset of Si.The expansion uses witness values from the base measure.Source: §D.2 · p. 17
D.3 Shared-sample Monte Carlo credit
- Shared-sample Monte Carlo gives an unbiased credit estimate at cost O(N|Di|) per proposal, with no additional verifier calls.Its computation grows linearly in the number of other group-minimal successes, unlike the exponential exact sum.Source: §D.3 · p. 17
E.1 Benchmark construction and ground truth
The MaxSAT benchmark defines minimal witnesses as the ground-truth antichain of sufficient variable sets. Its subset-construction MDP gives the policy a finite-horizon sequence of add-or-stop decisions, with a single terminal verifier bit.
E.1 Benchmark construction and ground truth
- In the MaxSAT benchmark, a proposal succeeds exactly when it intersects every positive clause, making minimal witnesses the formula’s minimal hitting sets.These are equivalently the prime implicants of the induced monotone Boolean function.Source: §E.1 · p. 17
- Eight formulas use 14 variables and 10 three-variable clauses, with 16 to 19 minimal witnesses per instance and mean antichain size 18.125.The full subset space is enumerated and sufficient sets are pruned by containment to obtain scoring ground truth.Source: §E.1 · p. 17
E.2 Subset-construction MDP
- The finite-horizon MDP adds variables or stops, then returns one verifier bit; success does not end the episode automatically.Policies observe current membership bits and normalized time, and must learn when further additions are redundant.Source: §E.2 · pp. 17–18
- Each formula is fixed within a run and omitted from the observation, isolating the learning objective on that instance.A fresh policy is trained for each formula.Source: §E.2 · p. 18
E.3 Training, baselines, and evaluation
The experiments compare group-credit estimators, planning references, and learned policies for recovering minimal witnesses. They also contrast coverage valuation, which uses a base measure, with counting valuation, which requires canonicalization.
E.3 Training, baselines, and evaluation
- Extending training fourfold changes only one of twenty-four runs; size-penalized ties can leave a policy distributing probability over two equal-size witnesses.With a size penalty, the scalar planner returns the same minimum-size witness under deterministic tie-breaking; with λ = 0, it returns sufficient sets with mean size 8.0.Source: §E.3 · p. 18, §E.4 · p. 18
E.4 Executed planning references
- The planning references use backward induction: scalar value iteration operates on (S, t), while minimal-witness value iteration tracks selected witnesses and remaining proposal budget.For budget b = |M|, the latter selects the entire antichain on all eight instances; checks against enumeration have maximum discrepancy below 10^-14.Source: §E.4 · p. 18
E.5 Main learned-policy comparison
- The main learned-policy comparison makes terminal behavior observable against the complete antichain and both planning references, but these passages do not report its outcome.Figure 1 shows terminal distributions for three objective classes and recovered-family sizes on eight formulas.Source: §E.5 · p. 18
E.6 Credit estimator
- The l1o credit and l2o converge to the exact gradient, while group-mean centering retains a nonzero finite-K offset; l2o is used in the main MaxSAT comparison.The estimator test uses one enumerable instance, 8,192 groups, and sixteen independent estimates.Source: §E.6 · p. 19
E.7 Valuation and verifier access
- The coverage measure assigns group credit from terminal proposals, verifier bits, and a chosen measure; counting valuation instead requires a canonicalizer that returns minimal witnesses.The canonicalizer may add at most |S| verifier calls to a successful proposal; metrics are evaluated after canonicalization for counting valuation and on raw proposals for coverage valuation.Source: §E.7 · p. 19
E.8 Training group size and deployment budget
The experiments separate training group size from deployment proposal budget and test deletion credit across planted antichain geometries. Larger training groups yield broader terminal laws, while deletion-credit gains vary with target-family structure.
E.8 Training group size and deployment budget
- Larger training groups learn broader terminal laws and recover more of the antichain as deployment budget B grows; small-group policies saturate after fewer proposals.Training configurations use the same 36,864 verifier calls, with group sizes K ∈ {2, 8, 32, 128}.Source: §E.8 · p. 19
E.9 Antichain-geometry stress test
- The stress test varies family size, witness-size balance, overlap, and the ratio L/K while keeping the subset MDP and monotone verifier.It uses n = 20, five planted predicates per condition, three policy seeds, and 32,768 terminal verifier calls per learned run.Source: §E.9 · p. 19
- Deletion-credit gains are largest for large balanced families, where many incomparable witnesses compete for a finite proposal budget; size imbalance reduces the difference between objectives.Figure 11 reports paired antichain-recall differences against the strongest learned baseline across sixteen conditions.Source: §E.9 · p. 19
E.10 Base-measure choice for coverage credit
These sections examine how coverage-credit settings and alternative RL objectives affect witness recovery, then describe the augmented Suzuki benchmark and its verifier. The reported comparisons include trade-offs in refinement, recovery, and the use of simulated condition panels.
E.10 Base-measure choice for coverage credit
- As the per-element ratio 1/p approaches one, antichain recall rises from 0.55 to 0.87 while median proposal size rises from 3.8 to 4.2.Born-minimal rate stays between 0.71 and 0.79 across p ∈ [0.5, 0.9]; the study fixes p = 0.7 in advance, without per-domain tuning.Source: §E.10 · pp. 19–20
E.11 Credit-rescaling sensitivity
- The two credit-rescaling choices yield similar antichain recovery; mean-absolute scaling gives the higher born-minimal rate, while standard-deviation scaling slightly increases l2o recall.The comparison uses eight MaxSAT formulas, one policy seed, and the same 57,600-query training budget.Source: §E.11 · pp. 20–21, Table 10 · p. 21
E.12 Size-penalized maximum-entropy RL and GFlowNet
- At the strongest baseline setting’s born-minimal rate, it recovers about one quarter as many witnesses as MWRL; both MaxEnt RL and GFlowNet peak at intermediate penalty strengths.Weak penalties leave probability on supersets, while strong penalties concentrate on the smallest witnesses or can stop before satisfying the formula.Source: §E.12 · p. 21
F.1 Benchmark data and provenance
- The augmented condition panels are simulator outputs anchored to real reactions, not measured yields; the benchmark selects 1,024 tasks from 1,381 that pass its acceptance filters.The simulator generates condition yields over 14 discrete dimensions; the released task antichains contain 2 to 35 minimal witnesses, with mean 8.6.Source: §F.1 · p. 22
F.2 Verifier, decision process, and policy
- The reported verifier uses deterministic closed-world table lookup: a proposal succeeds if a tabulated assignment within its opened dimensions reaches the threshold, and absent assignments fail.The benchmark does not run Bayesian optimization during training or evaluation; coverage credit uses raw proposals and verifier bits.Source: §F.2 · p. 22
F.3 Protocol and evaluation
The evaluation compares conditioned and unconditioned amortized policies, scalar RL, and per-substrate search, then reports a fully synthetic amortization test. The sparse-circuit section describes component interventions and a verifier-based canonicalizer.
F.3 Protocol and evaluation
- The four Suzuki methods share the verifier and evaluation protocol; amortized policies differ by substrate fingerprint input or scalar success credit, while per-substrate search marks a recovery ceiling.Policies are evaluated within a window of 32 distinct condition sets per substrate, and recall counts only raw proposals that are themselves minimal.Source: §F.3 · p. 23
F.4 Amortization on the fully synthetic library
- On the fully synthetic Suzuki library, the conditioned policy matches per-substrate search on held-out recall; the substrate-blind ablation achieves less than half its recall, and scalar reward recovers none.The library contains 172 substrate pairs across 18 condition dimensions and is reported as a supporting instance.Source: §F.4 · p. 23, Table 13 · p. 24
G LLM sparse-circuit discovery
- Sparse-circuit discovery freezes the language model and searches attention-head and MLP components by replacing clean activations with aligned counterfactual activations.Its canonicalizer performs an EAP-guided search and iterative single-component deletion, checking each accepted deletion and returning a 1-minimal circuit.Source: G LLM sparse-circuit discovery · p. 24
G.1 Qwen3-1.7B MMLU Bench10 and matched baselines
This section examines MWRL’s search for minimal condition sets across model benchmarks and sanctioned scientific experiments, including its accuracy, distributional-faithfulness comparisons, and task-specific support recovery. It also emphasizes that sufficiency is bounded by the verifier and its probes, while recovering the full antichain makes alternative pathways explicit.
G.1 Qwen3-1.7B MMLU Bench10 and matched baselines
- For each subject, EAP, magnitude, Wanda-style, HardConcrete, and random were evaluated at MWRL’s subject-specific circuit size k, while ACDC-style greedy deletion used its own converged size; search cost differed by method.Source: §G.1 · p. 24
- Held-out four-choice accuracy ranged from 0.13 for random-k and HardConcrete to 0.21 for MWRL, versus 0.64 for the dense model; MWRL’s +0.02 difference from EAP had a 95% interval covering zero.The passage says Table 4 reports distributional faithfulness, which does separate method families, and discovery-prompt accuracy remains above chance at a retention of 0.74.Source: §G.1 · pp. 25, 27
G.2 Hierarchical circuit recovery on Qwen3-8B
- Hierarchical search recovered Qwen3-8B component supports for knowledge, arithmetic, and temporal tasks, but none for grammar; recovered supports were more accurate on their own leaves than outside their task family.Arithmetic supports transferred across most leaves, while knowledge and temporal coverage was partial; failure to find a support does not establish that none exists.Source: Figure 13 · p. 25, Figure 14 · p. 26, §G.2 · p. 27
- Qwen3-8B search uses three hierarchical passes: each child search is restricted to its parent’s support, formed from the union of accepted circuits to preserve alternatives.The passes search families, subtasks, then leaves; the policy uses add, remove, and stop actions, and each pass uses the same coverage return.Source: §G.2 · pp. 27–28
H.1 Symbolic and oracle minimal-set discovery
- Symbolic and oracle methods enumerate minimal-set families or extract individual minimal sets, while MWRL targets how a policy allocates finite rollouts across an unknown witness family; explanation and circuit work likewise motivates recovering multiple sufficient circuits.Circuit explanations need not be identifiable: distinct subnetworks can satisfy the same behavioral criterion.Source: §H.1 · p. 28, §H.2 · p. 28
H.3 Diverse and relational reinforcement learning
- MWRL derives relational credit from the verifier and subset order: it rewards refinement and recovery across incomparable witnesses, while a redundant superset receives zero credit; its coverage objective is submodular.Unlike diversity measures based on entropy or external behavior descriptors, the return uses certified up-sets and their deletion marginals.Source: §H.3 · p. 28, §H.4 · p. 29
H.5 Hypervolume and set-level objectives
- Under the geometric measure, MWRL’s coverage valuation is a hypervolume, and each proposal’s exclusive contribution corresponds to its deletion credit; acceptance into the dominated region still depends on the verifier.The antichain of minimal witnesses is the Pareto set under the stated indicator-vector mapping.Source: §H.5 · p. 29
I.1 Conditions for applying MWRL
- MWRL applies when proposals are subsets of a finite ground set, a binary verifier is available, sufficiency is monotone or canonicalized, multiple incomparable witnesses matter, and verification is costly across related contexts.Learned-policy settings include configuration reduction, perturbation biology, and data attribution; graded-utility or single-set objectives fall outside the formulation.Source: §I.1 · p. 29, §I.3 · pp. 30–31, §I.4 · p. 31
J Ethical statement
- In scientific variable identification, minimal condition sets can direct experiments, but the verifier must define the outcome, success threshold, and admissible conditions; MWRL uses only its one-bit feedback.The verifier's domain review and containment practices govern hazards; MWRL can accelerate a laboratory-sanctioned search but cannot originate one.Source: J Ethical statement · p. 31
- Recovered witnesses satisfy only their verifier’s criterion: circuits failed on held-out questions, so sufficiency depends on the verifier and probe distribution.Biological and chemical deployments should be reverified on the population of interest, and broader claims require broader probes.Source: J Ethical statement · p. 31
- The full antichain exposes alternative pathways that a single smallest witness can hide, weakening any one account’s claim to necessity.It also keeps alternatives available when a route is unsafe, unavailable, or costly to realize.Source: J Ethical statement · p. 31