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arxiv: 2603.04603 · v2 · submitted 2026-03-04 · 📡 eess.SY · cs.RO· cs.SY

Risk-Aware Rulebooks for Multi-Objective Trajectory Evaluation under Uncertainty

Tichakorn Wongpiromsarn This is my paper

Pith reviewed 2026-05-15 15:52 UTC · model grok-4.3

classification 📡 eess.SY cs.ROcs.SY
keywords risk-awarerulebookstrajectory evaluationuncertaintypreordermulti-objectiveenvironment response distributionautonomous driving
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The pith

Risk-aware rulebooks evaluate trajectories by modeling how each one shapes the uncertain environment and induce a consistent preorder without cycles.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper develops a formalism that evaluates system trajectories when interactions with the environment are uncertain rather than treating the environment as fixed noise. It explicitly models the distribution of environment responses induced by each trajectory and applies rulebooks to compare them while handling hierarchical priorities and non-comparable objectives. The central result proves that the comparison relation forms a preorder on the set of trajectories, which guarantees consistency. This matters for systems like autonomous vehicles because it produces explainable selections even when outcomes depend on unpredictable responses.

Core claim

By modeling the distribution of environment responses as a function of each system trajectory and applying risk-aware rulebooks to the resulting distributions, the formalism induces a preorder on trajectories that is transitive and reflexive while preventing cyclic preferences, as shown through an autonomous driving illustration that clarifies why one trajectory is preferred over others under uncertainty.

What carries the argument

The risk-aware rulebook, a structure that assigns and compares risk values to trajectories based on the environment response distributions each trajectory induces.

Load-bearing premise

That the distribution of environment responses can be explicitly modeled as a function of each system trajectory.

What would settle it

An example trajectory set in which the formalism produces a cycle of preferences under a concrete environment response distribution would show the preorder claim fails.

Figures

Figures reproduced from arXiv: 2603.04603 by Tichakorn Wongpiromsarn.

Figure 1
Figure 1. Figure 1: Illustrative probability density of a random cost variable [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: The interaction Eτi (ωj ) and probability measure Pr(ωj ) for each i, j ∈ {1, . . . , 4}. r1(τi, ξj ) r2(τi, ξj ) r3(τi, ξj ) r4(τi, ξj ) ξ1 ξ2 ξ1 ξ2 ξ1 ξ2 ξ1 ξ2 τ1 0 225 0 0 0 0 0 0 τ2 0 175 0 0 1.77 1.77 0 0 τ3 0 0 0 0 15 15 12.25 12.25 τ4 0 0 1 1 0 0 0 0 [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: The degree of rule violations for different AV trajectories. [Left] [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The risk of each AV trajectory with respect to rule [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
read the original abstract

We present a risk-aware formalism for evaluating system trajectories in the presence of uncertain interactions between the system and its environment. The proposed formalism supports reasoning under uncertainty and systematically handles complex relationships among requirements and objectives, including hierarchical priorities and non-comparability. Rather than treating the environment as exogenous noise, we explicitly model how each system trajectory influences the environment and evaluate trajectories under the resulting distribution of environment responses. We prove that the formalism induces a preorder on the set of system trajectories, ensuring consistency and preventing cyclic preferences. Finally, we illustrate the approach with an autonomous driving example that demonstrates how the formalism enhances explainability by clarifying the rationale behind trajectory selection.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The manuscript introduces a risk-aware formalism for evaluating system trajectories under uncertainty by explicitly modeling how each trajectory influences the environment and the resulting distribution of responses. It claims to prove that the formalism induces a preorder on the set of trajectories (ensuring reflexivity, transitivity, and absence of cyclic preferences) while handling hierarchical priorities and non-comparability among objectives, and demonstrates the approach via an autonomous driving example focused on explainability.

Significance. If the preorder proof holds under the stated modeling assumptions, the work supplies a mathematically consistent framework for multi-objective trajectory selection in uncertain environments. This could strengthen safety-critical applications such as autonomous driving by providing traceable rationale for choices among non-dominated trajectories. The explicit trajectory-dependent environment modeling distinguishes it from exogenous-noise treatments and supports the claimed consistency guarantees.

major comments (1)
  1. [Proof of preorder induction (as referenced in abstract)] The central claim rests on a proof that the formalism induces a preorder. The abstract asserts this result, but the provided manuscript text does not include the full derivation, explicit handling of edge cases (e.g., when environment-response distributions become non-identifiable or when non-comparable objectives interact with uncertainty), or verification that reflexivity and transitivity survive the distribution construction. This load-bearing step requires expansion before the consistency guarantee can be assessed.
minor comments (1)
  1. [Autonomous driving example] The autonomous driving example would benefit from explicit listing of the rulebook priorities and the precise form of the environment-response distribution used in the illustration.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive feedback and for recognizing the potential of the risk-aware rulebook formalism in providing consistent multi-objective trajectory evaluation under uncertainty. We address the major comment below and will revise the manuscript to strengthen the presentation of the core theoretical result.

read point-by-point responses

  1. Referee: [Proof of preorder induction (as referenced in abstract)] The central claim rests on a proof that the formalism induces a preorder. The abstract asserts this result, but the provided manuscript text does not include the full derivation, explicit handling of edge cases (e.g., when environment-response distributions become non-identifiable or when non-comparable objectives interact with uncertainty), or verification that reflexivity and transitivity survive the distribution construction. This load-bearing step requires expansion before the consistency guarantee can be assessed.

    Authors: We agree that the proof of preorder induction would benefit from a more complete and self-contained derivation, including explicit verification of reflexivity and transitivity as well as treatment of the identified edge cases. In the revised manuscript we will expand the relevant theoretical section to present the full step-by-step argument. The expansion will (i) detail how the preorder is constructed from the risk-aware evaluation of trajectory-environment interactions, (ii) verify that reflexivity and transitivity are preserved under the induced distribution over environment responses, and (iii) address the non-identifiability of environment-response distributions and the interaction of non-comparable objectives with uncertainty. These additions will not change the underlying formalism but will make the consistency guarantees easier to verify. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper introduces a risk-aware formalism as a new mathematical structure for evaluating trajectories under uncertainty, explicitly modeling environment responses as a function of each trajectory. It then proves that this structure induces a preorder (reflexivity and transitivity) on the set of trajectories. This proof is presented as following directly from the definitions of the formalism rather than from any fitted parameters, self-referential equations, or load-bearing self-citations. The central claim does not reduce to renaming known results or smuggling ansatzes via prior work; the modeling assumptions are stated upfront as part of the construction. Any minor self-citation on rulebooks is not load-bearing for the preorder proof itself.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the ability to define a conditional distribution over environment responses given a trajectory; no free parameters, invented entities, or additional axioms are specified in the abstract.

axioms (1)
  • domain assumption Environment responses can be represented as a probability distribution conditional on the chosen system trajectory
    This modeling choice is required to evaluate trajectories under the resulting distribution rather than treating the environment as exogenous noise.

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Reference graph

Works this paper leans on

40 extracted references · 40 canonical work pages

  1. [1]

    Assured autonomy: Path toward living with autonomous systems we can trust,

    U. Topcu, N. Bliss, N. Cooke, M. Cummings, A. Llorens, H. Shrobe, and L. Zuck, “Assured autonomy: Path toward living with autonomous systems we can trust,” 2020

    work page 2020

  2. [2]

    Baier and J.-P

    C. Baier and J.-P. Katoen,Principles of Model Checking (Represen- tation and Mind Series). The MIT Press, 2008

    work page 2008

  3. [3]

    Evaluating perception systems for autonomous vehicles using quality temporal logic,

    A. Dokhanchi, H. B. Amor, J. V . Deshmukh, and G. Fainekos, “Evaluating perception systems for autonomous vehicles using quality temporal logic,” inRuntime Verification(C. Colombo and M. Leucker, eds.), (Cham), pp. 409–416, Springer International Publishing, 2018

    work page 2018

  4. [4]

    Specifying safety of autonomous vehicles in signal temporal logic,

    N. Ar ´echiga, “Specifying safety of autonomous vehicles in signal temporal logic,” in2019 IEEE Intelligent Vehicles Symposium (IV), pp. 58–63, 2019

    work page 2019

  5. [5]

    Robustness of temporal logic spec- ifications for continuous-time signals,

    G. E. Fainekos and G. J. Pappas, “Robustness of temporal logic spec- ifications for continuous-time signals,”Theoretical Computer Science, vol. 410, no. 42, pp. 4262–4291, 2009

    work page 2009

  6. [6]

    Being correct is not enough: efficient verification using robust linear temporal logic,

    T. Anevlavis, M. Philippe, D. Neider, and P. Tabuada, “Being correct is not enough: efficient verification using robust linear temporal logic,” arXiv preprint arXiv:2102.11991, 2021

    work page arXiv 2021

  7. [7]

    Monitoring temporal properties of contin- uous signals,

    O. Maler and D. Nickovic, “Monitoring temporal properties of contin- uous signals,” inFormal Techniques, Modelling and Analysis of Timed and Fault-Tolerant Systems(Y . Lakhnech and S. Yovine, eds.), (Berlin, Heidelberg), pp. 152–166, Springer Berlin Heidelberg, 2004

    work page 2004

  8. [8]

    Robust satisfaction of temporal logic over real-valued signals,

    A. Donz ´e and O. Maler, “Robust satisfaction of temporal logic over real-valued signals,” inFormal Modeling and Analysis of Timed Sys- tems(K. Chatterjee and T. A. Henzinger, eds.), (Berlin, Heidelberg), pp. 92–106, Springer Berlin Heidelberg, 2010

    work page 2010

  9. [9]

    Model predictive control with signal temporal logic specifications,

    V . Raman, A. Donz ´e, M. Maasoumy, R. M. Murray, A. Sangiovanni- Vincentelli, and S. A. Seshia, “Model predictive control with signal temporal logic specifications,” in53rd IEEE Conference on Decision and Control, pp. 81–87, 2014

    work page 2014

  10. [10]

    Reactive synthesis from signal temporal logic specifications,

    V . Raman, A. Donz ´e, D. Sadigh, R. M. Murray, and S. A. Seshia, “Reactive synthesis from signal temporal logic specifications,” in Proceedings of the 18th International Conference on Hybrid Sys- tems: Computation and Control, HSCC ’15, (New York, NY , USA), p. 239–248, Association for Computing Machinery, 2015

    work page 2015

  11. [11]

    Time robustness in mtl and expressiv- ity in hybrid system falsification,

    T. Akazaki and I. Hasuo, “Time robustness in mtl and expressiv- ity in hybrid system falsification,” inComputer Aided Verification (D. Kroening and C. S. P ˘as˘areanu, eds.), (Cham), pp. 356–374, Springer International Publishing, 2015

    work page 2015

  12. [12]

    Robust motion planning em- ploying signal temporal logic,

    L. Lindemann and D. V . Dimarogonas, “Robust motion planning em- ploying signal temporal logic,” in2017 American Control Conference (ACC), pp. 2950–2955, 2017

    work page 2017

  13. [13]

    Control from signal temporal logic specifications with smooth cumulative quantitative semantics,

    I. Haghighi, N. Mehdipour, E. Bartocci, and C. Belta, “Control from signal temporal logic specifications with smooth cumulative quantitative semantics,” in2019 IEEE 58th Conference on Decision and Control (CDC), pp. 4361–4366, 2019

    work page 2019

  14. [14]

    Arithmetic-geometric mean robustness for control from signal temporal logic specifications,

    N. Mehdipour, C.-I. Vasile, and C. Belta, “Arithmetic-geometric mean robustness for control from signal temporal logic specifications,” in 2019 American Control Conference (ACC), pp. 1690–1695, 2019

    work page 2019

  15. [15]

    Safe control under uncertainty with prob- abilistic signal temporal logic,

    D. Sadigh and A. Kapoor, “Safe control under uncertainty with prob- abilistic signal temporal logic,” inProceedings of Robotics: Science and Systems, (AnnArbor, Michigan), June 2016

    work page 2016

  16. [16]

    Incremental reasoning in probabilistic signal temporal logic,

    M. Tiger and F. Heintz, “Incremental reasoning in probabilistic signal temporal logic,”International Journal of Approximate Reasoning, vol. 119, pp. 325–352, 2020

    work page 2020

  17. [17]

    Safe autonomy un- der perception uncertainty using chance-constrained temporal logic,

    S. Jha, V . Raman, D. Sadigh, and S. A. Seshia, “Safe autonomy un- der perception uncertainty using chance-constrained temporal logic,” Journal of Automated Reasoning, vol. 60, no. 1, pp. 43–62, 2018

    work page 2018

  18. [18]

    Stl robustness risk over discrete-time stochastic processes,

    L. Lindemann, N. Matni, and G. J. Pappas, “Stl robustness risk over discrete-time stochastic processes,” in2021 60th IEEE Conference on Decision and Control (CDC), pp. 1329–1335, 2021

    work page 2021

  19. [19]

    Reactive and risk-aware control for signal temporal logic,

    L. Lindemann, G. J. Pappas, and D. V . Dimarogonas, “Reactive and risk-aware control for signal temporal logic,”IEEE Transactions on Automatic Control, vol. 67, no. 10, pp. 5262–5277, 2022

    work page 2022

  20. [20]

    Risk-aware robotics: Tail risk measures in planning, control, and verification [focus on education],

    P. Akella, A. Dixit, M. Ahmadi, L. Lindemann, M. P. Chapman, G. J. Pappas, A. D. Ames, and J. W. Burdick, “Risk-aware robotics: Tail risk measures in planning, control, and verification [focus on education],” IEEE Control Systems, vol. 45, no. 4, pp. 46–78, 2025

    work page 2025

  21. [21]

    Risk of stochastic systems for temporal logic specifications,

    L. Lindemann, L. Jiang, N. Matni, and G. J. Pappas, “Risk of stochastic systems for temporal logic specifications,”ACM Trans. Embed. Comput. Syst., vol. 22, Apr. 2023

    work page 2023

  22. [22]

    Risk- bounded temporal logic control of continuous-time stochastic sys- tems,

    S. Safaoui, L. Lindemann, I. Shames, and T. H. Summers, “Risk- bounded temporal logic control of continuous-time stochastic sys- tems,” inAmerican Control Conference (ACC), pp. 1555–1562, 2022

    work page 2022

  23. [23]

    Minimum-violation LTL planning with conflicting specifications,

    J. Tumova, L. I. R. Castro, S. Karaman, E. Frazzoli, and D. Rus, “Minimum-violation LTL planning with conflicting specifications,” in 2013 American Control Conference, pp. 200–205, June 2013

    work page 2013

  24. [24]

    Least- violating control strategy synthesis with safety rules,

    J. Tumova, G. C. Hall, S. Karaman, E. Frazzoli, and D. Rus, “Least- violating control strategy synthesis with safety rules,” inProceedings of the 16th International Conference on Hybrid Systems: Computation and Control, pp. 1–10, 2013

    work page 2013

  25. [25]

    Incremental sampling-based algorithm for minimum- violation motion planning,

    L. I. R. Castro, P. Chaudhari, J. Tumova, S. Karaman, E. Frazzoli, and D. Rus, “Incremental sampling-based algorithm for minimum- violation motion planning,” in52nd IEEE Conference on Decision and Control, pp. 3217–3224, Dec 2013

    work page 2013

  26. [26]

    Maxi- mally satisfying LTL action planning,

    J. Tumova, A. Marzinotto, D. V . Dimarogonas, and D. Kragic, “Maxi- mally satisfying LTL action planning,” in2014 IEEE/RSJ International Conference on Intelligent Robots and Systems, pp. 1503–1510, 2014

    work page 2014

  27. [27]

    Least-violating planning in road networks from temporal logic specifications,

    J. Tumova, S. Karaman, C. Belta, and D. Rus, “Least-violating planning in road networks from temporal logic specifications,” in2016 ACM/IEEE 7th International Conference on Cyber-Physical Systems (ICCPS), pp. 1–9, 2016

    work page 2016

  28. [28]

    Minimum- violation scLTL motion planning for mobility-on-demand,

    C.-I. Vasile, J. Tumova, S. Karaman, C. Belta, and D. Rus, “Minimum- violation scLTL motion planning for mobility-on-demand,” in2017 IEEE International Conference on Robotics and Automation (ICRA), pp. 1481–1488, 2017

    work page 2017

  29. [29]

    Minimum- violation planning for autonomous systems: Theoretical and practi- cal considerations,

    T. Wongpiromsarn, K. Slutsky, E. Frazzoli, and U. Topcu, “Minimum- violation planning for autonomous systems: Theoretical and practi- cal considerations,” in2021 American Control Conference (ACC), pp. 4866–4872, 2021

    work page 2021

  30. [30]

    Liability, ethics, and culture-aware behavior specification using rulebooks,

    A. Censi, K. Slutsky, T. Wongpiromsarn, D. Yershov, S. Pendleton, J. Fu, and E. Frazzoli, “Liability, ethics, and culture-aware behavior specification using rulebooks,” in2019 International Conference on Robotics and Automation (ICRA), pp. 8536–8542, 2019

    work page 2019

  31. [31]

    Formal specification and control synthesis of autonomous robots using rulebooks,

    T. Wongpiromsarn, K. Slutsky, and E. Frazzoli, “Formal specification and control synthesis of autonomous robots using rulebooks,”IEEE Transactions on Robotics, pp. 1–20, 2026

    work page 2026

  32. [32]

    Chapter 3 - relations and orderings,

    E. Schechter, “Chapter 3 - relations and orderings,” inHandbook of Analysis and Its Foundations(E. Schechter, ed.), pp. 49–77, San Diego: Academic Press, 1997

    work page 1997

  33. [33]

    Generalized cauchy spaces,

    P. Eklund and W. G ¨ahler, “Generalized cauchy spaces,”Mathematische Nachrichten, vol. 147, no. 1, pp. 219–233, 1990

    work page 1990

  34. [34]

    A. S. Kechris,Classical Descriptive Set Theory, vol. 156 ofGraduate Texts in Mathematics. New York: Springer-Verlag, 1995

    work page 1995

  35. [35]

    Coherent measures of risk,

    P. Artzner, F. Delbaen, J.-M. Eber, and D. Heath, “Coherent measures of risk,”Mathematical Finance, vol. 9, no. 3, pp. 203–228, 1999

    work page 1999

  36. [36]

    How should a robot assess risk? towards an axiomatic theory of risk in robotics,

    A. Majumdar and M. Pavone, “How should a robot assess risk? towards an axiomatic theory of risk in robotics,” inRobotics Research (N. M. Amato, G. Hager, S. Thomas, and M. Torres-Torriti, eds.), (Cham), pp. 75–84, Springer International Publishing, 2020

    work page 2020

  37. [37]

    Some remarks on the value-at-risk and the conditional value-at-risk,

    G. C. Pflug, “Some remarks on the value-at-risk and the conditional value-at-risk,” inProbabilistic Constrained Optimization: Methodol- ogy and Applications(S. P. Uryasev, ed.), pp. 272–281, Boston, MA: Springer US, 2000

    work page 2000

  38. [38]

    The reasonable crowd: Towards evidence-based and interpretable models of driving behavior,

    B. Helou, A. Dusi, A. Collin, N. Mehdipour, Z. Chen, C. Lizarazo, C. Belta, T. Wongpiromsarn, R. D. Tebbens, and O. Beijbom, “The reasonable crowd: Towards evidence-based and interpretable models of driving behavior,” in2021 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pp. 6708–6715, 2021

    work page 2021

  39. [39]

    Convex approximations of chance constrained programs,

    A. Nemirovski and A. Shapiro, “Convex approximations of chance constrained programs,”SIAM Journal on Optimization, vol. 17, no. 4, pp. 969–996, 2007

    work page 2007

  40. [40]

    An efficient motion planning algorithm for stochastic dynamic systems with constraints on probability of failure,

    M. Ono and B. C. Williams, “An efficient motion planning algorithm for stochastic dynamic systems with constraints on probability of failure,” inProceedings of the Twenty-Third AAAI Conference on Artificial Intelligence, pp. 1376–1382, AAAI Press, 2008

    work page 2008