Qualitative State-Trajectory Constraints

• Qualitative state-trajectory constraints are defined as logic- or relation-based restrictions over sequences of states, emphasizing non-metric and symbolic representations.
• Methods such as constraint networks, factor graphs, and temporal landmark graphs enable efficient inference and search by addressing combinatorial and partial information challenges.
• Applications in qualitative SLAM, temporal planning, and energy-constrained control illustrate their practical significance in improving interpretability and computational tractability.

Qualitative state-trajectory constraints are non-metric, logic- or relation-based restrictions on the possible temporal evolution of system states, typically formulated in terms of discrete spatial, logical, or symbolic regions, and applied over sequences of states or actions. They emerge in spatial reasoning, control theory, temporal planning, and robust systems analysis, where physical, logical, or policy-driven requirements prescribe allowed (or forbidden) patterns of state evolution beyond simple pointwise constraints. Such constraints enable inference, search, and control under partial information, coarse sensing, or combinatorial objectives, with characteristic benefits in interpretability and computational tractability.

1. Formal Models and Definitions

Qualitative state-trajectory constraints can be mathematically categorized by the form of their underlying state space, transition relations, and semantic basis:

• Qualitative State Spaces: States are partitioned into equivalence classes or symbolic regions, such as the cells of a spatial map, relational tuples encoding geometric or logical relations (e.g., region adjacency, qualitative cardinal directions), or partitionings induced by landmark sets (Mor et al., 2023, Baryannis et al., 2018, Mavridis et al., 2014).
• Trajectory Constraints: Restrictions are specified not on single states but on entire sequences (trajectories) of such states, often via:
  • Allowed or forbidden subsequences (e.g., "no re-entrance," "must visit A before B")
  • Modal temporal constraints (cf. PDDL3.0: “always,” “within t,” “always-within t,” etc.) (Marzal et al., 2017)
  • Algebraic envelope bounds (e.g., all time integrals of a control variable must obey coupled inequalities for all admissible control schedules) (Rousseau et al., 22 May 2025)

Principal models include:

• Constraint Networks over base relations (such as the TC-6 and TC-10 calculi for trajectory comparison, or QTC3D for qualitative interaction semantics) (Baryannis et al., 2018, Mavridis et al., 2014)
• Factor Graphs linking qualitative pose variables over time by unary (observation) and binary (motion) factors, trimming the space of feasible qualitative state trajectories (Mor et al., 2023)
• Interval-annotated Temporal Landmark Graphs for trajectory-modal logic, associating each required state (“landmark”) with generation, validity, and necessity intervals encoded as variable domains (Marzal et al., 2017)
• Trajectory-Independent (TI) vs. Trajectory-Dependent (TD) Energy Envelopes, where TI envelopes guarantee satisfaction of state constraints for all possible trajectories with given aggregate metrics (e.g., cumulative energy), while TD only guarantees existence of at least one trajectory per bound (Rousseau et al., 22 May 2025).

2. Solution Methods and Inference Algorithms

Qualitative state-trajectory constraints necessitate reasoning, inference, and control mechanisms distinct from their metric or pointwise analogs:

• Constraint Network Satisfaction: A qualitative trajectory calculus (e.g., TC-6, TC-10) defines finite sets of jointly-exhaustive, pairwise-disjoint (JEPD) base relations for trajectory pairs and encodes system requirements as constraint satisfaction problems. Consistency is enforced by relational composition tables, with solution existence being NP-complete (Baryannis et al., 2018).
• Sampling and Pruning in Qualitative SLAM: For systems with geometric but only qualitative measurements, efficient sample-based marginalization (forward filtering) and geometric pruning are deployed in factor graphs. Binary motion factors, reflecting qualitative transitions, dramatically constrain the combinatorial hypothesis tree (Mor et al., 2023).
• Temporal Landmarks and Interval Propagation: In planning, TempLM maintains intervals for each landmark regarding generation, persistence, and necessity. Propagation of before/after and mutex relations, together with interval tightening, allows early pruning of infeasible partial trajectories, significantly reducing search (Marzal et al., 2017).
• Set-Valued and Envelope-Based Methods: In robust control and flexibility analysis, convex optimization and analytic projection are used to derive TI envelopes, ensuring that all trajectories within aggregate bounds satisfy state limits. Extremal trajectories are used for comparison functions in the construction of conservatively safe envelopes (Rousseau et al., 22 May 2025).
• Symbolic Encoding with ASP: Qualitative trajectory relations can be declaratively encoded in Answer Set Programming, with base relations and compositions mapped to logic rules. Specialized encodings scale to hundreds of trajectories (Baryannis et al., 2018).

3. Representative Calculi and Modalities

A diverse array of qualitative calculi provides the semantic backbone for expressing state-trajectory constraints:

Calculus	State Representation	Trajectory Relation Types
TC-6 / TC-10	Sequence of discrete spatial regions	Eq, Alt, Start, Finish, Intersect, Disjoint, ...
QTC3D	7-tuple over {–,0,+}^7 (A–I)	Distance, speed, angle, yaw, pitch, roll componentwise qualitative states
PDDL3.0 Trajectory Modality	States as atomic propositions	always, at-end, sometime, within, at-most-once, always-within, etc.
TI Flexibility Envelope	State, input, and integrated outputs	For-all trajectory energy bounds (TD vs. TI)

• TC-6/TC-10: Discrete sequences over map partitions, with base relations defined combinatorially between trajectory pairs, enabling reasoning about properties such as overlaps, reversals, extensions, and intersections (Baryannis et al., 2018).
• QTC3D: Encodes interaction between moving objects using signs of derivatives of distance, speed, angle, and 3D orientation (via Frenet–Serret frames and their yaw/pitch/roll decomposition). Each (A,B,C,F,G,H,I) tuple fully describes the instantaneous qualitative relation (Mavridis et al., 2014).
• PDDL3.0 Modal Operators: “always,” “within,” “at-end,” etc. correspond to sets of interval constraints (generation, validity, necessity) on specific feature landmarks within a plan, supported by the temporal landmark graph and propagation rules (Marzal et al., 2017).
• TI/TD Envelopes: Specify sets of energy or state accumulations; TI formulations enforce that all possible trajectories consistent with the envelope necessarily respect state constraints, via weighting by system loss/gain matrices (Rousseau et al., 22 May 2025).

4. Practical Applications and Performance

Qualitative state-trajectory constraints are widely relevant in robotic navigation, multi-agent interaction, planning with logical preconditions, and robust operation of constrained dynamical systems.

• Qualitative SLAM and Active Planning: Employs object-centered qualitative frames and region partitioning for mapping and localization, with motion and observation factors enforcing qualitative geometric consistency and dramatically shrinking feasible trajectory sets. Approximate inference is computationally efficient, suitable for low-quality sensor regimes (Mor et al., 2023).
• Large-Scale Spatial Reasoning in ASP: Specialized encodings for qualitative trajectory calculus handle large datasets (e.g., T-Drive taxi traces, 1,000 trajectories) with orders-of-magnitude improvement in solvability compared to baseline ASP encodings (Baryannis et al., 2018).
• Temporal Planning under Modal Trajectory Constraints: Temporal landmark methods support early detection of infeasibility, interval-based propagation, and unification of diverse modal requirements into tractable interval constraints, demonstrated in standard planning domains (Marzal et al., 2017).
• Energy-Constrained Flexibility in Power Grids: TI envelopes prevent violation of temperature or state constraints in energy-constrained loads under all possible power trajectories inside the envelope, essential for safety and comfort in applications such as building heating. The gap between TD and TI envelopes can be substantial in lossy or weakly coupled systems, with violations up to ±3.8 °C observed when TI constraints are not enforced (Rousseau et al., 22 May 2025).

5. Theoretical Guarantees and Limitations

• Completeness and Necessity/Sufficiency: TI envelope conditions are mathematically equivalent to requiring that all admissible trajectories under envelope constraints satisfy state bounds. Construction relies on extremal solutions and system-dependent weightings, formalized for linear and mildly nonlinear systems (Rousseau et al., 22 May 2025).
• Complexity: Constraint network satisfaction is NP-complete for general qualitative spatial calculi, but specialized inference (e.g., sample-based filtering with geometric pruning, interval propagation in temporal landmarks) is typically tractable in practice for structured problems (Baryannis et al., 2018, Mor et al., 2023, Marzal et al., 2017).
• Trade-Offs: Strong qualitative constraints (e.g., TI envelopes) reduce feasible flexibility (admissible region volume) but guarantee robust satisfaction of state constraints for all possible trajectory realizations. In contrast, TD envelopes yield larger regions but do not prevent boundary violations by some admissible trajectories (Rousseau et al., 22 May 2025).
• Extension to Coupled Systems: For physically coupled loads (e.g., buildings with shared walls), individual and pooled TI envelopes can be obtained, with box and simplex relaxation for distributed or centralized operation, respectively (Rousseau et al., 22 May 2025).

6. Connections to Control, Planning, and Spatial Reasoning

Qualitative state-trajectory constraints encapsulate a bridging formalism between numerical optimal control, rule-based planning, logical constraint satisfaction, and topological spatial reasoning:

• In optimal control, state constraints along a trajectory are analyzed via Hamiltonian formalism, adjoint equations, and multiplier measures; trajectory constraints are enforced on boundary subarcs and require sign-definiteness for consistency with the Pontryagin Maximum Principle (Dmitruk et al., 2017).
• In AI planning, high-level temporal modalities are mapped to sets of interval constraints and dependency graphs, unifying logical requirements with temporal and causal dependencies (Marzal et al., 2017).
• In spatial reasoning, qualitative calculi such as RCC-8, QTC, and region-based trajectory calculi provide logic- and relation-based abstraction, supporting effective combinatorial search, incremental inference, and knowledge propagation over coarsely observed or abstracted environments (Baryannis et al., 2018, Mavridis et al., 2014, Mor et al., 2023).

Qualitative state-trajectory constraints support robust, interpretable, and computationally efficient deduction and control under diverse forms of partial observability, structural uncertainty, and high-level behavioral requirements.