Universal Catastrophic Safety Undecidability and Capability--Risk Upper Bound Frontier: Unified Theorems, Complexity Positioning, and Engineering Pathways

Universal Catastrophic Safety Undecidability and CapabilityRisk Upper Bound Frontier: Unied Theorems, Complexity Positioning, and Engineering Pathways Haobo Ma 1 Wenlin Zhang 2 1 Independent Researcher 2 National University of Singapore Abstract Establish two foundational boundaries for general learning and decision systems. First, provide catastrophic safety determination undecidability for interactive agentenvironment systems: under extremely weak modeling assumptions, for any extension-closed regular bad-prex speci- cation, whether threshold safety satised admits no global algorithm; under restricted subclass of deterministic environments and computable strategies, further position as Σ0 1 -complete/ Π0 1 - complete . Second, provide capabilityworst-risk upper bound frontier induced by joint PAC-Bayes high-probability bound, mutual information expected bound, and Wasserstein-1 distributionally robust optimization (KantorovichRubinstein duality); via point perturbation adversary establish universal geometric lower bound , together with robustaccuracy impossibility and robust generalization sample complexity lower bound, forming "upper bound lower bound" dual support. Thereby propose "scope restrictionruntime shieldingrisk budget structural prior" governance blueprint, providing minimal reproducible ImageNet-C + Shield experimental skeleton and metric conguration. Keywords : Halting; Rice theorem; Σ0 1 -complete; POMDP; PAC-Bayes; conditional mutual information; Wasserstein-DRO; KantorovichRubinstein duality; adversarial robustness; runtime shield; interruptibility 1 Introduction & Historical Context Algorithmic decidability and program semantics reveal fundamental limits of universal verication: halting problem undecidable, Rice theorem states any non-trivial semantic property undecidable. Transplanting this idea to interactive agentenvironment setting, obtain general determination unavailability for "whether triggering catastrophic specication". This direction resonates with undecidability results for innite-horizon probabilistic planning/partially observable decision at threshold and plan existence. On the other hand, modern learning theory reveals capability and robustness cannot advance without cost: PAC-Bayes and mutual information paradigm characterize complexity/information amount inuence on generalization, Wasserstein-DRO characterizes worst-risk under distribution shift; simultaneously, robustaccuracy impossibility and robust generalization sample complexity lower bound rigorously proven in natural model families. Two boundaries jointly point toward governance principles: acknowledging general static certication impossibility and capabilityrisk hard trade-o, adopt layered scheme of scope restriction, runtime shielding, and risk budget. 1 2 Model & Assumptions 2.1 Interaction Semantics and Temporal Assumptions  Action and observation alphabets A,O nite; history h1:t∈(A×O)t .  Computable policy : Agent A is function A: (A × O)⋆→ A , for any h<t exists nite time producing at=A(h<t) at step t (allowing internal randomization via sampling program implementation). This assumption satised throughout undecidability construction and completeness positioning.  Environment E specied by history conditional probability µ(ot|h<t, at) . Main results use deterministic trivial environment E0 : always returns xed observation o⊥ . 2.2 Safety Property and Regular Bad Prex  Let Σ=(A×O)⋆ . Call B⊆Σ bad prex language if for any u∈B and any extension v∈Σ , have uv ∈B ( extension-closed ). Corresponding safe prex set S= Σ \B then prex-closed .  Specication adopts regular bad prex language B (equivalent to safe prex recognized by DFA/safety automaton). Violation event Bad ={∃t:h1:t∈B}, τ(h) = inf{t:h1:t∈B}.  Threshold safety predicate : Given ε∈[0,1) , Safeε(A, E, B) := Pr µ(Bad)≤ε. 2.3 LearningEvaluation and Distributional Robustness  Data domain Z with metric d ; sample S= (Zi)n i=1 ∼Dn .  Learning algorithm outputs posterior QS∈ P(H) . Loss ℓ:H × Z → [0,1] .  Lipschitz assumption : Exists uniform constant L > 0 such that for any h∈ H , mapping z7→ ℓ(h, z) is L -Lipschitz with respect to d (0-1 loss not applicable, adopt smooth surrogates like cross-entropy/hinge; controllable via spectral norm constraint and gradient clipping).  Wasserstein-1 ball Bρ(D) = {D′:W1(D′, D)≤ρ} ; robust risk Rrob ρ(Q) := sup D′∈Bρ(D) Eh∼Q,z∼D′[ℓ(h, z)]. 3 Main Results (Theorems and Alignments) Theorem 1 (1: Universal Catastrophic Safety Determination Undecidable) . Exists regular bad prex family B such that no algorithm can determine for all computable policies A , computable environments E , B∈B , and any rational ε∈[0,1) the truth value of Safeε(A, E, B) . 2 Theorem 2 (1': Complexity Positioning of Restricted Subclass) . Under deterministic environment E0 and computable policy class, let UNSAFE ={(A, E0, B, ε) : Pr(Bad)> ε}, ε < 1. Then UNSAFE is Σ0 1 -complete , its complement SAFE is Π0 1 -complete . In this subclass Pr(Bad)∈ {0,1} , thus " Pr(Bad)> ε " equivalent to "occurrence" for any ε < 1 . Theorem 3 (2: CapabilityWorst Risk Upper Bound Frontier: PAC-Bayes + KR) . For any prior P and δ∈(0,1) , with probability at least 1−δ (over S∼Dn ) have Rrob ρ(Q)≤b RS(Q) + rKL(Q∥P) + ln(1/δ) 2n+Lρ. Right-hand three terms respectively empirical error, complexity/condence term, and distribution shift linear penalty, constituting upper bound induced capabilityrisk frontier . Theorem 4 (2': High-Probability Mutual Information Bound: Paradigmatic Statement) . Let loss ℓ∈[0,1] with sub-Gaussian constant σ for each sample point. If learning algorithm satises conditional mutual information upper bound CMI(S;QS)≤Γ or equivalent strength uniform stability, then exists constant c > 0 such that for any δ∈(0,1) , Pr RD(QS)≤b RS(QS) + r2σ2(Γ + cln(1/δ)) n!≥1−δ. Juxtaposing (2) with (1), obtain high-probability frontier expression for data-dependent posterior: take smaller of two right-hand sides as operational upper bound for capabilityrisk curve. Theorem 5 (3: Point Perturbation Geometric Lower Bound and Distribution Ball Inclusion) . For any classier f and ρ > 0 , dene Radv ρ(f) = Pr (z,y)∼D∃z′∈Bρ(z) : f(z′)=y, where Bρ(z) = {z′:d(z, z′)≤ρ} with label preserving. Then sup D′∈Bρ(D) RD′(f)≥ Radv ρ(f). (3) holds on any metric and task, providing universal lower bound "foundation" matching (1). Under Gaussian mixtures and ℓp perturbations, exist constructive lower bounds for robustaccuracy impossibility and robust generalization sample complexity lower bound. Proposition 6 (1: Tightness of KR Linear Term) . For any L, ρ > 0 and metric space, exists L -Lipschitz function f and distribution pair (D, D′) such that W1(D′, D) = ρ and sup W1(D′,D)≤ρ ED′[f]−ED[f] = Lρ. Indicates rst-order form Lρ cannot be improved under uniform Lipschitz constant condition. 3 4 Proofs 4.1 Theorem 1 (Undecidability) Take trivial environment E0 . Given Turing machineinput pair ⟨M, x⟩ , construct computable policy AM,x(h<t) = (a⋆, if M(x) halts within t steps , a0, otherwise . Let bad prex language B={h: some step action is a⋆} , regular and extension-closed. Then Pr(Bad) = 1{M(x) halts }. If universal decider exists determining Safeε(A, E, B) truth/falsity for any input ( ε < 1 ), obtain halting determination, contradiction. Proved. 4.2 Theorem 1' ( Σ0 1/Π0 1 Complete) Many-one reduction : Mapping R:⟨M, x⟩ 7→ (AM,x, E0, B, ε) polynomial-time computable, and ⟨M, x⟩ ∈ HALT ⇐⇒ (AM,x, E0, B, ε)∈ UNSAFE ( ε < 1 ). Membership : Under E0 and deterministic A , Pr(Bad)∈ {0,1} . If unsafe, exists minimal τ making h1:τ∈B , enumerate to this prex accepts, thus UNSAFE ∈Σ0 1 , complement in Π0 1 . Combining with reduction obtains completeness. Proved. 4.3 Theorem 2 (Upper Bound Frontier) PAC-Bayes (McAllester/Catoni variant) provides RD(Q)≤b RS(Q) + rKL(Q∥P) + ln(1/δ) 2n( with probability ≥1−δ). KR duality indicates for any L -Lipschitz function g , sup W1(D′,D)≤ρ ED′[g]≤ED[g] + Lρ. Applying to g(z) = Eh∼Qℓ(h, z) yields (1). Proved. 4.4 Theorem 2' (High-Probability Mutual Information) Let ℓ bounded with each point σ -sub-Gaussian. If algorithm satises CMI(S;QS)≤Γ , then via information compression and variational inequality obtain PrRD(QS)−b RS(QS)≤q2σ2(Γ+cln(1/δ)) n≥1−δ, where constant c given by tail control. Juxtaposing with (1) obtains frontier high-probability form. Proved. 4 4.5 Theorem 3 (Point Perturbation Lower Bound) For any measurable selection operator T:Z → Z with d(z, T(z)) ≤ρ almost surely, let D′= (T, y)#D . Taking coupling π(dz, dz′) = D(dz)δT(z)(dz′) , then Eπd(Z, Z′)≤ρ ; thus W1(D′, D)≤ρ . If f(T(z)) =y then errs under D′ , further sup D′∈Bρ(D) RD′(f)≥ED1{∃z′∈Bρ(z) : f(z′)=y}=Radv ρ(f). Proved. 4.6 Proposition 1 (Tightness) Take D=δ0, D′=δρu and f(z) = L|z|2 yields result. Proved. 5 Model Apply  Autonomous control and tool-using agents : Theorem 1 rules out general static certi- cation, recommend restricting policy space and interfaces to veriable sublanguages; during deployment suppress transgression via shields and interruptible protocols.  Perceptiondecision systems : According to (1)(2) establish risk budget : under given (n, ρ) enhancing capability (larger model/weaker prior) requires correspondingly increased sample size, enhanced structural prior, or compressed L .  Evaluation and calibration : Adopt corruption and perturbation benchmarks (e.g., ImageNetC) and uncertainty measures (NLL/ECE), jointly "violation ratetask accuracy" dual-axis curves exhibiting "capabilityrisk frontier" and shield interception eectiveness. 6 Engineering Proposals 1. Scope restriction : Design policy and tool invocation via veriable subsets (restricted DSL/interfaces), ensuring safety specications implemented by online discrimination via DFA/LTL synthesis safe prex recognizers. 2. Runtime shield : Synthesize pre-/post-shields via LTL → DFA → safety automaton generator; pre-shield lters unsafe action set, post-shield replaces with nearby safe action via minimal correction principle; probabilistic shield controls false rejection/false negative via condence threshold. 3. Risk budget : Treat (b RS,KL, I, ρ, L) as budget quintuple; congure "datapriorshiftLipschitz" balancing strategy respectively during development and deployment phases. 4. Structural prior and impact regularization : Adopt equivariant structures, spectral norm constraints, and reversibility penalties (AUP) reducing complexity and side-eect propensity. 5. Distributionally robust training and uncertainty governance : Combine WassersteinDRO/adversarial training with deep ensembles, temperature calibration; handle high uncertainty via rejectiondegradationhando open-loop strategy. 6. Interruptibility : Embed unbiased interruptible protocols in updating and exploration, preventing policy learning incentives to circumvent intervention. 5 7 Discussion (Risks, Boundaries, Past Work)  Boundary meaning : Undecidability negates "general, global, one-time" static proof; under restricted model families (nite horizon, fully observable, discounted MDP, etc.) strong guarantees still obtainable.  Upper boundlower bound enclosure : KR linear term with PAC-Bayes/mutual information provide operational upper bounds; point perturbation lower bound with robustaccuracy impossibility, robust generalization sample complexity lower bound indicate "zero-cost both" unattainable not artifact of loose analysis.  Relationship with existing work : Theorem 1 equivalent to Rice/halting, complements innite-horizon probabilistic planning undecidability; Theorem 2 consistent with distributionally robust optimization, PAC-Bayes, mutual information paradigm; Theorem 3 matches constructive lower bounds and sample complexity lower bounds in adversarial robustness literature; shields and interruptibility correspond to runtime enforcement systems in safe reinforcement learning and formal methods. 8 Conclusion General catastrophic safety determination unattainable in principle, capability enhancement and robustness admit hard trade-o jointly driven by complexity/information amount and distribution shift. Based on this, governance schemes should center on scope restriction, runtime shielding, and risk budget, proving within veriable subdomain, backstopping via shields and interruptibility during deployment, suppressing shift risk via structural prior and distributionally robust techniques during training. Acknowledgements, Code Availability Thank related research in decidability, distributionally robust optimization, information-theoretic generalization, and safe reinforcement learning elds. Code and data not accompanying paper; minimal reproducible experiment suggestion: based on public corruption benchmarks, LTL → DFA tools, and adversarial/robust training libraries reproduce "risk budget curveviolation rate" dualaxis diagram; repository should include data scripts, specication examples, training/inference and shield modules, hyperparameter tables, and one-click scripts. References Turing, A. M. (1936). On Computable Numbers, with an Application to the Entscheidungsproblem. Rice, H. G. (1953). Classes of Recursively Enumerable Sets and Their Decision Problems. Madani, O., Hanks, S., Condon, A. (1999). On the Undecidability of Probabilistic Planning and Innite-Horizon POMDPs. McAllester, D. (1999). Some PAC-Bayesian Theorems. Catoni, O. (2007). PAC-Bayesian Supervised Classication. Alquier, P. (2021). User-friendly Introduction to PAC-Bayes Bounds. Xu, A., Raginsky, M. (2017). Information-Theoretic Analysis of Generalization Capability of Learning Algorithms. 6 Steinke, T., Zakynthinou, L. (2020). Reasoning about Generalization via Conditional Mutual Information. Bu, Y., Zou, S., Veeravalli, V. V. (2020). Tightening Mutual Information-Based Bounds on Generalization Error. Villani, C. (2009). Optimal Transport: Old and New. Esfahani, P. M., Kuhn, D. (2018). Data-Driven Distributionally Robust Optimization Using the Wasserstein Metric. Sinha, A., Namkoong, H., Duchi, J. (2018). Certifying Some Distributional Robustness with Principled Adversarial Training. Tsipras, D., Santurkar, S., Engstrom, L., Turner, A., Madry, A. (2019). Robustness May Be at Odds with Accuracy. Schmidt, L., et al. (2018). Adversarially Robust Generalization Requires More Data. Alshiekh, M., et al. (2018). Safe Reinforcement Learning via Shielding. Koenighofer, B., et al. (2024). Shields for Safe Reinforcement Learning. Orseau, L., Armstrong, S. (2016). Safely Interruptible Agents. Turner, A. M., Hadeld-Menell, D., Tadepalli, P. (2020). Conservative Agency via Attainable Utility Preservation. Hendrycks, D., Dietterich, T. (2019). Benchmarking Neural Network Robustness to Common Corruptions and Perturbations. A Semantics and Measurability Let cylinder σ -algebra generated on Σ . Computable policy and environment jointly induce history distribution P(h1:t) = t Y s=1 A(as|h<s)·µ(os|h<s, as). Bad prex language B extension-closed and regular, event Bad ={∃t:h1:t∈B} measurable; rst violation time τ(h) is stopping time. B "First Appearance a⋆ " and Regular Bad Prex Dene B hit ={h:∃i≤ |h|, ai=a⋆}. If u∈B hit and v is any extension, then uv ∈B hit , thus extension-closed. Corresponding safe prex set S= Σ \B hit is prex-closed. C Binary Probability and Threshold Lemma Under E0 and deterministic A , Bad is event "whether appears a⋆ ", taking only values 0 or 1. For any rational ε < 1 , have Pr(Bad)> ε ⇐⇒ Pr(Bad)=1 ⇐⇒ Bad occurs . 7 D Many-One Reduction Details Mapping R sends ⟨M, x⟩ to (AM,x, E0, B hit , ε) .  Correctness : If M(x) halts, exists t0 making AM,x output a⋆ at t0 , thus Bad occurs; otherwise not.  Computability : Constructing AM,x and DFA recognition for B hit both completed in polynomial time.  Completeness : By HALT ≤m UNSAFE and membership obtain Σ0 1 -complete; complement problem obtains Π0 1 -complete. E PAC-Bayes and KR Duality Composition Let gQ(z) = Eh∼Qℓ(h, z) . If ℓ∈[0,1] and z7→ ℓ(h, z) uniformly L -Lipschitz, then gQ also L - Lipschitz. KR duality provides sup W1(D′,D)≤ρ ED′gQ≤EDgQ+Lρ. PAC-Bayes basic formula bounds EDgQ and b RS(Q) dierence with probability 1−δ , composition yields (1). F Mutual Information High-Probability Bound (CMI Paradigm) Under ℓ∈[0,1] and pointwise σ -sub-Gaussian, conditional mutual information CMI(S;QS)≤Γ induces Pr RD(QS)−b RS(QS)≤q2σ2(Γ+cln(1/δ)) n≥1−δ. Proof based on information compression inequality and PAC-Bayesian-style variational techniques; when replacing CMI with uniform stability, same-order tail bound obtainable. G Point Perturbation Lower Bound Measurable Selection and Label Preserving Let metric space separable with complete Borel σ -algebra. When selecting for each (z, y)z′∈Bρ(z) making f err, adopt Borel measurable selection lemma dening operator T(z) ; label preserving assumption ensures pushforward D′= (T, y)#D consistent with task. If error-causing perturbation non-existent, set T(z) = z . This yields (3). H Lipschitz Constant and Surrogate Loss 0-1 loss does not satisfy Lipschitz assumption; use surrogate losses like cross-entropy/hinge, control L via spectral norm constraints, gradient clipping, and Lipschitz network structures. This control enters (1) linear term, determining ρ -sensitivity. 8 I Engineering Metrics and Indicators Risk budget curve : Horizontal axis is complexity/information term (model scale or prior strength, I(S;QS) proxy), vertical axis is Rrob ρ estimate or its upper bound; overlay "violation ratetask accuracy" dual-axis curves (with/without shield two curves), exhibiting runtime shielding violation suppression eect at similar accuracy. 9