Computational Model for Autonomous Systems

• Computational models for autonomous systems are formal, algorithmic frameworks that represent self-governed agents using modular architectures and precise semantics.
• They integrate methods such as categorical semantics, stochastic hybrid models, and MDP controllers to ensure safety, optimality, and dynamic coordination.
• Practical implementations leverage automated planning, resource-aware adaptation, and continual learning to enhance performance across robotics, transportation, and cyber-physical systems.

A computational model for autonomous systems provides a formal, algorithmic, and mathematical representation of agents or collectives capable of self-governed behavior in dynamic or uncertain environments. Across its incarnations, such a model typically defines system architectures, agent modules, coordination protocols, planning and adaptation mechanisms, and the associated complexity classes. Modern approaches emphasize modularity, rigorous semantics, compositionality, and adaptability, all grounded in precise formal structures capable of expressing safety, optimality, and goal-directed reasoning.

1. System Architecture and Agent Models

Autonomous systems are architected as collections of interacting motifs—reconfigurable coordination environments encoding the spatial-locational relationships and permissible synchronizations among agents and objects. Each motif $M_i$ is a tuple $M_i = (A_i, O_i, {\rm Map}_i, @_i, I_i, R_i)$, where $A_i$ and $O_i$ are sets of agents and objects, ${\rm Map}_i$ defines a graph of locations, $@_i$ assigns locations, $I_i$ is a set of guarded interaction rules, and $R_i$ encodes reconfiguration (creation, deletion, migration). System configurations evolve via firing interaction or reconfiguration rules, supporting dynamic multimode coordination and mobility at scale (Sifakis, 2018).

Agent models are modular, typically comprising

• Perception: raw sensor data interpretation,
• Reflection: high-level map and environment modelling,
• Goal Management: selection and prioritization of goals,
• Planning: sequence generation of actions/interactions to satisfy goals while preserving safety,
• Self-adaptation: runtime monitoring and reconfiguration.

The knowledge repository maintains both static (design-time) and dynamic (learned or inferred) domain models, method libraries, utility functions, and goal hierarchies, enabling adaptive, goal-oriented, and trustworthy operation (Sifakis, 2018).

2. Formal Computational Frameworks

The spectrum of computational models ranges from discrete finite-state abstractions to continuous stochastic hybrid systems. A central thread is compositionality—models are assembled via structure-preserving maps (functors, natural transformations) between diverse component representations (automata, contracts, MDPs, wiring diagrams).

Categorical Semantics

AlgebraicSystems (Bakirtzis et al., 2022) formalizes model views as categorical objects and morphisms, with semantics assigned via monoidal functors $F: W \to \mathbf{Cat}$. Horizontal composition (parallel/serial wiring) and vertical composition (refinement/abstraction) are encoded categorically, enabling modular verification and rigorous handling of emergent properties via natural transformations. For example, hybrid dynamical models are horizontally composed, then refined to contracts for safety verification via a functorial pipeline.

Stochastic and Hybrid Models

Ultra-large-scale systems (ULSoS) are abstracted as Piecewise-Deterministic Markov Processes (PDMPs), with each agent's state $(q, y)$ following deterministic flows interspersed by stochastic jumps at rates $\lambda(x)$ with post-jump transitions drawn from kernels $R(x, A)$ (Bujorianu et al., 2013). Coordination among agents is separated from their local activity via additional synchronization logic (dynamic guards, neighbor communication), yielding scalable mean-field or modular descriptions for high-population scenarios.

3. Planning, Decision Making, and Goal Reasoning

Planning and goal reasoning computational models typically leverage hierarchical task networks, explicit state-transition representations, and model-checking or optimal control solvers.

Model Checking Institutionalization

GRAVITAS (Bride et al., 2019) frames planning as reachability in a Goal Task Network (GTN), with nodes embodying tasks/goals as guarded transitions and deterministic effects over integer-valued state variables. These networks are compiled into CSP$\sharp$ processes in the Process Analysis Toolkit (PAT), which synthesizes optimal and trustworthy plans by exhaustive state exploration, integrating safety invariants via linear temporal logic assertions and supporting uncertainty handling through nondeterministic program branching.

Markov Decision Process (MDP) Controllers

“Optimal by Design” (ObD) (Elrakaiby et al., 2020) defines a metamodel over system state variables, actions/capabilities, stochastic environmental events, and annotated requirements (goals). The system automatically induces a large MDP $\mathrm{MDP}_r = (S, A, P, R, \gamma)$, where the state space encodes both atomic configurations and requirement-status bits (goal automata). Solving the MDP yields a reflex controller $\pi^* : S \to A$—a lookup policy achieving timely, requirements-aware adaptation across dynamic conditions, with value- and policy-iteration or similar DP procedures.

4. Resource-Awareness, Adaptation, and Continual Learning

Resource allocation within autonomous systems is quantified via dynamic optimization of computational and energy resources, subject to context-awareness and system-wide safety/performance metrics.

Adaptive Resource Allocation

A context-driven manager coordinates multiple perception pipelines (e.g., vehicle-side cameras/detectors) according to activity patterns, distance metrics, and vehicle state (Jambotkar et al., 2021). The allocation problem is a constrained, multi-objective nonlinear program over the compute shares $C = [C_1,...,C_n]^T$: $$J(C) = \sum_{i=1}^{n} [ a\, t_i(C_i, S_i, R_i) - b\, p_i(S_i, R_i) W_i + c\, e_i(C_i, F_i) ]$$ subject to minimum safety $SF(C) \geq SF_{CCRA}$, enforced via a simplex algorithm and real-time contextual scoring. This achieves substantial improvements in both energy efficiency and operational safety.

Online Continual Learning

Online CL algorithms for real-world autonomous agents (Shaheen et al., 2021, Kim et al., 2024) incorporate replay-based resilience, regularization-based memory retention, and adaptive neural architecture growth. For instance, MAS and Latent-Replay track gradient or Fisher importance for stability, while edge-oriented continual learning leverages student–teacher–retrain loops and resource-precision partitioning for battery-limited platforms. Hybrid approaches employ experience rehearsal, task-drift detection, and local retraining, with low-latency, low-memory footprints suitable for embedded deployments and validated across automotive, UAV, and robotics domains.

5. Coordination, Collaboration, and Inter-Agent Decision Models

True autonomous multi-agent or robot systems require coordination models capable of robust communication and collective reasoning.

Minimal Mobile Robot Landscapes

The LCM model taxonomy for two-robot systems (Kitamura et al., 28 Dec 2025) demonstrates that the computational power available is a nuanced function of internal capabilities (memory, communication) and external scheduler synchrony/atomicity. Under FSYNCH, memory and communication distinctions collapse, but under various asynchrony, strict separations arise, with problems solvable in communication-only but not finite-state models, and vice versa. Complete landscapes are characterized by equivalence and separation theorems, simulation-free reductions, and minimal complexity witness tasks.

Collaborative Trustworthiness via BDD Aggregation

When autonomous agents (vehicles) share beliefs under varying sensor qualities (Saidi et al., 15 Jul 2025), collaborative aggregation is achieved by representing agents' Boolean predicates and quality attributes in ordered binary decision diagrams (OBDDs), propagating through a lattice of trust/expertise. Aggregation and propagation rules derived from social epistemology (e.g., $n$-expert, all-dissent), together with OBDD reduction steps, enable error-correcting consensus protocols scalable to large distributed collectives.

6. Computational Irreducibility, Agency, and Formal Limits

Any computational model capable of expressing genuine autonomy must confront fundamental limits—irreducibility and undecidability.

Irreducibility as Agency Foundation

A system is autonomous precisely when it can internally simulate a universal Turing machine (UTM), admitting no external shortcut for prediction (Azadi, 5 May 2025). Rigorous undecidability results demonstrate that, for goal-reachability or other nontrivial properties $P$, the prediction problem is undecidable for UTM-equivalent agents. Consequently, computational irreducibility—no algorithm can produce future states faster than stepwise simulation—becomes intrinsic to autonomy. This enables emergent information generation (linear Kolmogorov depth growth), increases mutual agent-environment information, and underpins biological notions of agency, adaptability, and even philosophical conceptions of free will.

7. Synthesis, Practical Guidance, and Open Challenges

Contemporary computational models provide scalable frameworks for trustworthy planning, multimodal perception, adaptive control, and collaborative reasoning, leveraging compositional and modular semantics, tractable optimization, and statistical learning.

Practical synthesis pipelines often integrate categorical model composition (AlgebraicSystems), automated reachability analysis (GRAVITAS), optimal policy derivation (ObD, MPC), and continual adaptation loops (online CL, DaCapo). Robustness and scalability are addressed via mean-field limits, modular process algebra, and hierarchical abstraction.

Outstanding challenges include quantifying degrees of autonomy (logical depth, empowerment), ensuring real-time guarantees under resource constraints, bridging expressiveness and tractability across formal paradigms, and rigorously accommodating nontrivial environmental stochasticity or adversarial behaviors.

These computational models underpin state-of-the-art autonomous system design and analysis, with substantial foundational and applied impact across transportation, robotics, swarm coordination, and complex cyber-physical infrastructures.