obstruction calculus

Prompt

Fully develop novel, impactful, nonstandard, technically consequential upgrades: publishable contribution must be new obstruction certificates, dual representations, or rigorous bridge continuous dynamics to finite certificates. Core question: after removing standard partial-observation safety-game machinery—belief-state DP, predecessor fixed points, viability kernels, robust controlled invariance, barrier/tangency, robust optimization duality—what theorem is new? Answer with statement, assumptions, proof, certificate semantics, computation. Must not restate belief-state DP. Must yield independently checkable certificate. Venue logic: SCL plausible only if one theorem-driven message: nonviability of partially observed safety systems certified by belief-space predecessor failure, finite obstruction witnesses, dual certificates for convex/polyhedral common-action constraints. SCL should not carry six coequal mechanisms, broad sustainability, static certification theory, certainty-equivalence examples, simultaneous continuous/discrete/finite/infinite formulations. Strong SCL: exact finite-horizon info-state recursion; common safe-action exclusion theorem; finite/sparse certificates in convex/polyhedral systems; one nontrivial example. Automatica needs substantial result beyond standard belief-state DP. Credible package: rigorous info-state viability; exact finite-horizon characterization under assumptions; continuous-time local/review-interval obstruction theorems; computable convex/polyhedral certificates; delayed-information bounds; algorithm with complexity/convergence; nontrivial numerical study and comparison. Key question: after standard machinery removed, what theorem survives? Isolate, state quantifier order/assumptions, prove, certify, compute, compare, delimit. If none, reframe as negative/counterexample/computational certification paper. Identity: framework for certifying robust nonviability under partial observation, combining exact info-state recursion with analytical and dual exclusion certificates. Central object: K_I={B0:∃π∈Π_I∀z0∈B0∀d(·),z(t)∈V∀t}, where K_I is set of information states, not physical states. Names: robust information-state viability kernel, belief-state viability kernel under partial observation. Output-feedback viability kernel acceptable if space clear. Distinguish from full-information kernel, observation-history kernel, set of beliefs from which some compatible state is viable. Define projections/quotients/equivalence maps; prove containment or strict separation. Show K_I⊆Π^{-1}(K_full) or belief-lifting relation; conditions for equality failure. Equality failure is conceptual source of partial-observation nonviability; tie to common-action exclusion, not merely uncertainty. Info-state formulation: beliefs from histories, not only current observations. Info state includes prior, control history, observation history. Discrete update: B_{k+1}=Φ(B_k,a_k,y_{k+1}), union over disturbances/latent states compatible with y_{k+1}. Do not write u(t)=π(B_t) without qualification. Record-based policy replaced by belief-based only when B_t sufficient. State: controls sampled-data or continuous; policies deterministic/randomized; record-based/belief-based/memoryless; disturbances open-loop/causal/state-feedback; whether disturbance sees current control. Quantifier order consistent: ∃π∀d differs from ∀z∃u∀d. Common-action obstruction: ∃u∀z∈B∀d generally stronger than ∀z∈B∃u∀d. State as proposition with counterexample where every compatible state individually viable under full information but no common action safe for all. Include belief, branchwise safe-action sets, empty intersection, obstruction certificate. Define observation map, partition, information filtration, policy class Π_I. Specify policies measurable/nonanticipative/causal/adapted. Specify adversary static/dynamic/nonanticipative/clairvoyant. If dynamic and observes action, use game-theoretic quantifier order; if open-loop, use signal quantifiers. Do not mix semantics silently. Theorem spine: four principal results. Theorem 1 Exact finite-horizon info-state recursion. For sampled-data/discrete-time model: Pre(C)={B:∃α∈R_I(B) s.t. all compatible trajectories safe during current stage, and Φ(B,α,y+)∈C for every possible y+}. W0={B:B⊆V_T}, W_{k+1}=Pre(W_k). In finite model, B0∈W_N iff observation-based policy keeps system safe N stages. General spaces require measurable selection, compactness, closedness. State standard assumptions. Under these, recursion exact. Infinite horizon: K_I=νZ.Pre(Z) only under complete-lattice/policy-selection framework. Finite: K_I=∩_{N≥0}W_N. General: finite-horizon viability for every N need not yield one infinite-horizon measurable policy. State counterexample or sufficient condition. Distinguish fixed-point existence from policy existence. Theorem 2 Obstruction-tree completeness. For finite spaces, B0∉W_N iff adversary has finite-depth strategy proving failure within N stages. Tree contains info-state nodes, regulator action branches, adversarial disturbance/observation branches, terminal safety violations or terminally nonviable beliefs. Tree is nonviability certificate. Finite in finite model; prune by merging observation-equivalent beliefs. Continuous action spaces may lack literal finite tree unless compactness, discretization, convex duality, or finite-witness theorem. State exact finite spaces, depth bound, branching factor, extraction algorithm. Sparse version: if belief update merges states, compress; if safe-action sets convex, replace by small active branches. Theorem 3 Nested local obstruction certificates. For held action over review interval Δ: U^B(B)=∩_{z∈B}U(z); R_V^B(B)=∩_{z∈B}{u∈U(z):Dq_i(z;f(z,u,d))≥0,∀d,i∈I(z)}; A_tube(B,Δ)={u∈U^B(B):Reach_[0,Δ](B,u)⊆V}. Under tangency assumptions, A_tube⊆R_V^B⊆U^B. Thus U^B=∅, R_V^B=∅, or A_tube=∅ is sound exclusion certificate. Do not imply instantaneous tangency and finite-interval tube safety equivalent. Common tangential action can exit later. Finite-Δ derivative theorem: uniform adverse margin and regularity. If every common action gives D^+q_i(z;f(z,u,d))≤−η for some compatible active branch and uniform η>0, then finite-time exit or tube infeasibility. Distinguish pointwise tangency, uniform tangency, tube safety. Counterexample where pointwise tangency holds but tube safety fails. Theorem 4 Blind-window/delayed-information obstruction. Controls implementable before first branch-separating observation: U_blind(B0,T_obs). σ*(B0)=sup_{u∈U_blind}inf_{z0∈B0,d(·)}τ_V(z0,u,d). Principal timing certificate: σ*(B0)<T_obs⇒B0∉K_I. Controls are strategies adapted to observations during blind interval; reduce to open-loop only if no informative observations. State observation process, blind interval, branch-separating event, policy class. Proof: if supremum survival < T_obs, every blind policy admits adversarial trajectory exiting before first informative observation. Valuable for delayed/sparse sensing. Strengthen polyhedral result: Farkas certificate beyond elementary infeasibility. Suppose R_V(z_j)={u:A_j u≤b_j}. Stack Au≤b. Common safe-action emptiness certified by λ≥0, λ^T A=0, λ^T b<0. Add: Finite-witness theorem: if safe-action sets convex in R^{m_u}, Helly-type result gives emptiness witnessed by ≤m_u+1 branchwise sets under finite-family/compactness. Conclusion: even with many compatible states, nonviability may have sparse witness. Vertex reduction: if belief polyhedral and constraints affine in state, prove conditions when checking vertices sufficient. Semi-infinite duality: for continuously parameterized beliefs, formulate robust/semi-infinite program, derive strong duality/finite certificate extraction. Certificate algorithmic: Farkas multipliers, Helly subsets, active constraints, vertex enumeration, LP/SDP hierarchy, complexity bounds. State when finite, sparse, checkable, robust to numerical error. Numerical example extracting/interpreting Farkas multiplier. Relegate weaker components: Observation-equivalence saturation K=O^{-1}(O(K)) elementary; short proposition/remark or omit from SCL. Not substantial viability theorem. If measurable certifier required, set-theoretic saturation may need O(K) measurable. Certainty equivalence: policy-class comparison Π_CE⊆Π_out, K_CE⊆K_out. Example may show strict inclusion but not main theorem. Injective sensing does not imply prescribed CE success. Sustainability: only application/interpretation. Opening language concerns safety, robust invariance, partial observation, delayed sensing, constrained control. Do not let sustainability displace kernel, theorems, certificates, computations. Be conservative about novelty: acknowledge related ideas: belief-state safety games; robust controlled invariance; viability kernels; partial-observation game recursion; predecessor fixed points; barrier/tangency; robust optimization duality; POMDP safety; info-state DP; set-valued estimation. Do not claim belief-space backward recursion new. Credible claim: connects exact info-state viability recursion with local continuous-time exclusion tests, blind-window timing bounds, finite dual certificates of common-action infeasibility. Support by literature. Introduction separates standard machinery, new theorems, new computational consequences. Related-work table columns: model, observation, certificate type, finite witness, continuous-time result, computation. State prior special/limiting cases. Add serious computational study: toy scalar insufficient for Automatica. At least one example: multidimensional constrained system; nontrivial observation partition/delayed sensor; computed belief-space viability approximation; extraction of obstruction tree/Farkas multiplier; comparison with full-information viability; comparison with barrier/reachability baseline; sensitivity to observation refinement/review interval. Report computation time; number of belief states/partitions; certificate size; max blind-window duration; reduction in viable info states from coarser sensing. Sustainability model can be benchmark, not conceptual foundation. Include parameters, discretization, solver, tolerances, reproducibility. Compare POMDP safety, barrier certificates, reachability, full-information viability. Algorithm and complexity: input dynamics, observation map, target V, horizon N, belief representation. Output W_N, obstruction tree, Farkas multipliers, blind-window bound. Steps: initialize W0; backward Pre; prune observation-equivalent beliefs; extract certificates; verify soundness; return sparse witness. Complexity O(|B||A||Y|) finite; LP per belief polyhedral; sparse witness size m_u+1; tree depth N; branching |A||Y|. Convergence for finite approximations, soundness/completeness relative to discretization, numerical conditioning. State exact, sound but incomplete, or asymptotically exact. Venue-specific packages: SCL keep finite-horizon info-state predecessor; exact nonviability/obstruction-tree for finite systems; convex/polyhedral common-action certificate; finite-witness/sparse-certificate theorem; one numerical example. Remove fibre certification; certainty-equivalence; broad institutional discussion; multiple unrelated examples; most infinite-horizon material. Strong SCL title: Finite Certificates of Nonviability for Partially Observed Safety Systems. Automatica include rigorous info-state model; finite-horizon exact characterization; infinite-horizon under explicit assumptions; continuous-time tube/tangency certificates; delayed-information survival-time theorem; polyhedral/semi-infinite dual certificates; algorithm and certificate extraction; substantial numerical evaluation. Suitable title: Robust Viability under Partial Observation: Information-State Kernels and Nonviability Certificates. Revised contribution statement: This paper studies robust safety under partial observation through info-state viability. First, info-state predecessor operator yields exact finite-horizon viability under finite-model assumptions and finite adversarial certificates when viability fails. Second, local exclusion tests derive from common admissibility, robust tangency, review-interval tube safety. Third, delayed observations treated through blind-window survival-time value function. Finally, for convex/polyhedral safe-action sets, common-action nonviability reduces to tractable feasibility with finite dual certificates. Results complete for finite-horizon finite systems; sound, generally incomplete exclusion tests for continuous-state systems. Avoid overclaiming while identifying framework. Abstract: We study robust viability of constrained systems under partial observation. State available to regulator is information set of all trajectories compatible with observation and control histories. We define info-state predecessor operator and characterize finite-horizon output-feedback viability by backward recursion. In finite systems, characterization exact and nonviability admits finite adversarial obstruction certificates. For continuous-time systems, derive sound local exclusion tests based on common admissibility, robust tangency, review-interval tube safety, survival before informative observations. For convex/polyhedral safe-action sets, common-action infeasibility admits finite dual certificates, including Farkas multipliers. Results provide checkable conditions showing no admissible output-feedback policy can maintain safety, including cases where every compatible state individually viable under full information. Highest-priority revision sequence: Fix policy/information/disturbance semantics. Define info-state predecessor precisely. Prove finite-horizon theorem with exact assumptions. Identify what is genuinely new relative to safety games. Strengthen convex/polyhedral result with finite-witness or duality. Prove delayed-information theorem using implementable blind-window policy class. Add algorithm and nontrivial numerical study. Move fibre certification and certainty equivalence to remarks/examples. Reduce sustainability rhetoric; reorganize around central kernel. Choose SCL or Automatica after determining whether new theorem supports broader claim. Final requirement: Ensure framework produces at least one result both nonstandard and technically consequential—ideally finite/sparse obstruction theorem, rigorous continuous-to-discrete certificate, or scalable dual computation method. State theorem with explicit assumptions, quantifier order, certificate semantics. Checkable on nontrivial numerical example; novelty isolated from standard partial-observation safety-game recursion. If no such theorem survives, reframe as negative result, counterexample paper, or computational certification paper. Final manuscript should make one central claim, prove it, certify it, compute it, compare it, delimit it.