Autonomous Complexity in Modern Systems

• Autonomous complexity is the measure of inherent and emergent difficulty in systems that operate without human intervention, capturing state-space, computational, and behavioral challenges.
• It is evaluated using diverse methodologies including symbolic, probabilistic, information-theoretic, and sample complexity metrics to quantify decision processes and adaptation.
• Applications in robotics, cyber-physical systems, and DAOs demonstrate that managing this complexity enhances safety, efficiency, and system scalability.

Autonomous complexity denotes the intrinsic and emergent difficulty in engineering, quantifying, and managing the behaviors and decision processes of systems that operate with minimal or no human intervention. The concept encompasses state-space size, computational barriers, uncertainty management, and the combinatorial richness of agent-environment and agent-agent interaction, as well as the challenges posed by adaptivity and open-ended evolution. Measures of autonomous complexity span symbolic, probabilistic, information-theoretic, architectural, and sample complexity domains, applying to robotics, software agents, distributed organizations, and cyber-physical systems.

1. Formal Definitions and Foundational Metrics

Autonomous complexity is multi-dimensional, capturing:

• State-space and agent complexity: For control agents, the size of the belief base $|B|$, plan library $|\Pi|$, and per-cycle applicable plans $b_t$ drive the cardinality of the agent's reachable state set: $$C_{\mathrm{Jason}} \lesssim 2^{|B|} \times 2^{|\Pi|} \times \max_t(b_t)$$. In the limited instruction set agent (LISA) architecture, complexity collapses to a discrete-time Markov chain (DTMC) of size $|S| \leq 2^{|B|}$ by eliminating internal nondeterminism (Izzo et al., 2016).
• Emergent behavioral complexity: For autonomous robots, complexity is quantified by predictive information and excess entropy. For stationary time series $\{y_t\}$, one constructs delay vectors, calculates joint and conditional entropies, and defines

$$E(\epsilon) = \lim_{m\to\infty} \left[ H_m(\epsilon) - m\,h_{m-1}(\epsilon) \right]$$

where $E(\epsilon)$ reflects the structure and memory in temporal dynamics. For deterministic systems, $E(\epsilon)$ diverges as $\epsilon \to 0$, reflecting increased determinism and richer structure (Martius et al., 2015).

• Sample complexity in exploration: In reward-free MDPs, autonomous exploration aims to discover all incrementally $L$-controllable states and learn $\epsilon$-optimal policies efficiently. DisCo achieves sample complexity

$$\tilde{O}(L^5 S_{L+\epsilon}\, \Gamma_{L+\epsilon}\, A\, \epsilon^{-2})$$

with $S_{L+\epsilon}$ the number of reachably controlled states, $\Gamma_{L+\epsilon}$ the local branching factor, $A$ the action count, and $\epsilon$ a performance slack parameter (Tarbouriech et al., 2020).

• Architectural and module-driven complexity: In multi-agent systems, autonomic complexity is decomposed into knowledge handling (KH), decision complexity (DC), and adaptive coordination (AC), each quantified with proxy metrics and weighted to yield an overall scalar (Sifakis, 2018):

$$C_\mathrm{aut} = \alpha\,C_{KH} + \beta\,C_{DC} + \gamma\,C_{AC}$$

2. Decomposition and Measurement Methodologies

Several methodologies dissect autonomous complexity for tractability and analysis:

• Scale-dependent vs. scale-independent entropy: Behavioral complexity in robotic systems is decomposed analytically into $E_\mathrm{state}$ (information per deterministic state), $E_\epsilon$ (scaling dimension), and $E_\mathrm{mem}$ (long-range temporal correlation). The sum $E_\mathrm{core}$ is invariant under change of resolution, providing an objective, comparative measure across behaviors and learning stages (Martius et al., 2015).
• Correlation-integral estimation: Measures such as excess entropy and predictive information are computed using pairwise distances in reconstructed state-space, yielding robust estimates that avoid binning ambiguities. Automated fitting over $\log\,\epsilon$ identifies deterministic and noise-dominated scaling ranges (Martius et al., 2015).
• Design vs. run-time verification: Complexity of agent reasoning cycles can be managed via structural simplification (e.g., LISA), reducing verification cost from exponential ($O(2^{|B|})$) to polynomial in horizon and triggered-plan count ($O((b_t k)^h)$), enabling tractable run-time safety assurance (Izzo et al., 2016).
• Context enrichment and uncertainty pruning: In intralogistics, overall system state $S(t)$ quantifies complexity via context cardinality, perceptual richness, and future anomaly entropy. Three-stage mappings (context enrichment, situation analysis, strategy generation) are engineered with threshold-based pruning to contain effective computational and semantic complexity (Sigel et al., 4 Jul 2025).
• Organizational network metrics: For DAOs, organizational complexity is quantified using entropy, modularity, and centralization:
  • Shannon entropy $H = -\sum_k p_k \log\,p_k$
  • Network modularity $$Q = (1/2m)\sum_{ij}[A_{ij}-(k_ik_j/2m)]\delta(c_i, c_j)$$
  • Degree-centralization $C_D = \sum_i (C_\mathrm{max}-C_i)/[(n-1)(n-2)]$
  • Self-organization principles (collective intelligence, digital democracy, adaptation) interact to modulate emergent complexity and viability (Ballandies et al., 3 Sep 2024).

3. Computational Adaptivity and Uncertainty Management

Autonomous systems operate in dynamic, uncertain environments, requiring complexity-adaptive mechanisms:

• Motif-based architectures: The environment and agents are modeled as dynamic motifs with parametric, mobile, reconfigurable coordination and planning. Complexity arises from motif lifecycle, distributed address shifting, synchronization, and multi-agent planning under partial observability and uncertainty (Sifakis, 2018).
• Planning under sampling adaptation: For autonomous vehicles, adaptive sampling via ASAPF biases grid construction toward low-potential (safe, feasible) regions, reducing computational cost from $O(NM^2)$ to $O(NM'^2)$ without compromising solution quality or stability. Empirically, DP run times and cost values are preserved while grid sizes shrink significantly (Li, 2023).
• Uncertainty containment in strategy selection: In intralogistics, uncertainty is propagated and pruned at each stage—context enrichment leverages symbolic priors and LLMs, situation analysis filters anomalies above a threshold, and planners use digital twin simulations for cost and risk projection. Effective complexity is kept sub-linear in object count due to aggressive filtering (Sigel et al., 4 Jul 2025).

4. Case Studies and Empirical Findings

Distinct domains illustrate the practical consequences and measurement of autonomous complexity:

Domain/Setting	Key Complexity Measure	Empirical Outcome
Autonomous exploration in MDPs (Tarbouriech et al., 2020)	Sample complexity $C$ scaling in $L,\epsilon,\Gamma$	DisCo reduces sample count by up to $10 \times$ compared to prior art; discovers controllable states faster
Robotic behavior (Martius et al., 2015)	Excess entropy decomposition $E_\mathrm{core}$, attractor dimension $D$	Behavioral complexity rises with robot learning (e.g., $E_\mathrm{core}$ grows from $0.85$ to $4.8$ nats)
Motion planning (Li, 2023)	Computational grid size, DP time, optimal cost	ASAPF shrinks grid by $4.8 \times$; DP run time drops $66.7\%$; trajectory quality stable
Intralogistics automation (Sigel et al., 4 Jul 2025)	Context graph cardinality, anomaly entropy	$92\%$ of blockages resolved autonomously; latency sub-linear in anomaly count
DAO governance (Ballandies et al., 3 Sep 2024)	Entropy $H$, modularity $Q$, centralization $C_D$	Strong deliberation correlates with healthy approval rate; quadratic voting reduces proposal centralization

5. Architectural Principles and Practical Trade-offs

Engineering for manageable autonomous complexity involves explicit trade-offs:

• Expressivity vs. verifiability: Limiting agent language features (e.g., LISA) yields tractable verification at the cost of expressive richness in plan-selection and reactivity (Izzo et al., 2016).
• Combinatorial explosion containment: Through hierarchical goal management, modular knowledge handling, adaptive coordination, and environment simplification, system designers select priorities (via weights $\alpha$, $\beta$, $\gamma$) to balance performance, safety, and flexibility (Sifakis, 2018).
• Self-organization in distributed settings: In DAOs, combinatorial state-space complexity is narrowed via collective intelligence (community diversity), digital democracy (deliberative and voting protocols), and adaptation (autonomous agent action with feedback). A triangulated viability framework encapsulates these axes for health assessment and improvement (Ballandies et al., 3 Sep 2024).
• Sampling and planning adaptivity: Motion planners reduce complexity by adapting grid construction to current context, leveraging history and obstacle field information—offering faster, stable solutions, yet potentially missing global re-routing opportunities under abrupt changes (Li, 2023).

6. Future Directions and Open Problems

Research continues to address challenges and advance methodologies:

• Real-time adaptability at scale: Open problems include mechanisms for dynamic balancing of autonomy and human oversight, formal safety guarantees for mixed-initiative control, and symbolically enriched reinforcement learning for robust autonomy adjustment (Bakirtzis et al., 23 Jul 2024).
• Formal unification of metrics: There is ongoing need for standardized multi-dimensional autonomous complexity indices encompassing behavioral, computational, architectural, and adaptive facets, suited for comparative analysis across domains.
• Generalization to open, evolving environments: Techniques for modular decomposition, complexity-aware adaptation, and formal guarantees remain areas of active exploration, especially in distributed, cross-organizational systems and next-generation cyber-physical infrastructures.

Autonomous complexity thus functions as both a foundational challenge and guiding principle for the design, verification, and scaling of systems that seek to operate, adapt, and evolve with minimal human direction. Its rigorous quantification, architectural management, and empirical assessment underpin ongoing progress across robotics, multi-agent systems, automated logistics, distributed organizations, and theoretical explorations of emergent behavior.