Beyond Intelligence: The Mathematics of Agency

Author: Denis Avetisyan

A new theory proposes that agency – the capacity to act and influence – can be mathematically distinguished from intelligence through the concept of ‘bi-predictability’.

This review details a mathematical framework quantifying information exchange between observation, action, and outcome, offering a novel approach to understanding and building resilient artificial intelligence systems based on thalamocortical principles.

Despite advances in artificial intelligence, systems often exhibit brittle performance as environmental dynamics shift, revealing a disconnect between predictive capability and genuine interaction. This limitation motivates ‘A Mathematical Theory of Agency and Intelligence’, which introduces ‘bi-predictability’-a quantifiable measure of shared information between observations, actions, and outcomes-to rigorously distinguish agency from intelligence. We demonstrate that bi-predictability, [latex]P[/latex], is a fundamental property of coupled systems, bounded by physical constraints, and demonstrably lower with the introduction of agency, offering a novel framework for evaluating learning effectiveness. Could restoring sensitivity to this shared information be the key to building truly adaptive and resilient AI systems?

The Fragility of Foresight: Systems and the Limits of Prediction

Conventional methods for simulating the actions of an agent – be it a robot, an animal, or even a financial trader – frequently falter when confronted with the inherent messiness of real-world environments. These approaches often rely on simplifying assumptions and pre-defined rules, which struggle to account for the countless unforeseen variables and emergent behaviors that characterize complex systems. Consequently, predictions based on these models can quickly diverge from actual outcomes, not due to a fundamental flaw in the agent itself, but rather due to the difficulty of precisely mapping and anticipating every possible interaction within its surroundings. This limitation isn’t a matter of insufficient computing power; it’s an acknowledgement that certain systems possess an intrinsic sensitivity to initial conditions and feedback loops, making long-term, accurate forecasting exceptionally challenging, if not impossible.

The inherent difficulty in predicting the behavior of complex systems arises not simply from their intricacy, but from a fundamental reciprocity between the agent within the system and the environment it inhabits. Traditional modeling often treats the environment as a static backdrop, failing to account for how an agent’s actions modify its surroundings, which in turn alters future interactions. This creates a feedback loop where even minuscule initial uncertainties can be amplified, leading to dramatically different outcomes over time. Effectively, the environment isn’t merely experienced by the agent; it is actively shaped by it, making precise, long-term predictions exceedingly challenging, and demanding models that acknowledge this continuous, dynamic exchange.

The development of genuinely intelligent systems hinges not on perfecting predictions of future states, but on deeply understanding the reciprocal relationship between an agent and its environment. Rather than treating the world as a static backdrop, successful agents must continuously perceive, interpret, and react to environmental feedback, shaping their actions based on the consequences they observe. This dynamic exchange-where the agent’s behavior alters the environment, which in turn influences subsequent behavior-is the core of adaptive intelligence. Systems that prioritize this interaction, focusing on learning and adjustment rather than rigid pre-programmed responses, demonstrate a remarkable capacity to navigate complexity and exhibit behaviors that appear purposeful, even in unpredictable circumstances. Ultimately, intelligence isn’t about knowing what will happen, but about skillfully responding to what is happening.

The Double Pendulum, a seemingly simple mechanical system consisting of one pendulum attached to the end of another, serves as a compelling illustration of the limits of predictability even in deterministic systems. Though governed by well-defined physical laws, its motion quickly becomes chaotic – exquisitely sensitive to initial conditions. This means that even infinitesimally small differences in the starting position or velocity of the pendulum will lead to drastically different trajectories over time, rendering long-term prediction impossible. This isn’t a failure of measurement, but a fundamental property of the system itself; the dynamics amplify these tiny variations exponentially. Consequently, the Double Pendulum demonstrates that knowing the governing equations isn’t enough to forecast behavior, challenging the notion that complete knowledge of initial conditions guarantees predictive power in complex, non-linear systems and prompting a re-evaluation of purely deterministic modeling approaches.

Bi-Predictability: Measuring Reciprocity, Not Just Accuracy

Bi-predictability establishes a quantifiable metric for assessing intelligence based on mutual information between an agent’s observations, actions, and the resulting outcomes. This framework differs from traditional predictive accuracy by focusing on the comprehensive understanding of an agent’s environment and its influence within it. The metric is mathematically bounded; classical systems are limited to a maximum Bi-predictability score of P ≤ 0.5, indicating a fundamental limit on their ability to model reciprocal relationships. Quantum systems, however, theoretically achieve a maximum Bi-predictability of P = 1, suggesting the potential for complete reciprocal understanding and modeling through quantum mechanical principles. The Bi-predictability value is calculated as the normalized mutual information between the agent’s actions and observed outcomes, given prior observations, offering a standardized and quantifiable assessment of intelligent behavior.

Bi-predictability distinguishes itself from traditional prediction accuracy by evaluating an agent’s capacity for contextual understanding, rather than solely assessing its ability to correctly anticipate future states. While conventional metrics reward accurate outputs, bi-predictability quantifies the mutual information between an agent’s observations, its actions, and the resulting environmental outcomes. This necessitates an agent not simply react to stimuli, but demonstrate comprehension of the causal relationships between its interventions and the observed consequences – effectively modeling its own influence within the system. The metric assesses the degree to which an agent can accurately predict outcomes given both sensory input and its own initiated actions, indicating a deeper level of environmental mastery beyond mere pattern recognition.

The concept of quantum entanglement provides a theoretical framework for understanding maximal bi-predictability, where [latex]P=1[/latex]. Entanglement demonstrates a correlation exceeding classical limits, suggesting a system where knowledge of one element instantaneously informs the state of another, regardless of distance. Applied to predictive systems, this implies the potential for information transfer with minimal loss and maximal efficiency. A system mirroring entanglement would, in theory, achieve complete mutual information between observations, actions, and outcomes, fully realizing the upper bound of bi-predictability and suggesting a highly efficient mechanism for internal state representation and predictive modeling.

Architectural designs informed by the principle of Bi-predictability emphasize the development of internal models that represent both environmental dynamics and the agent’s impact on those dynamics. This contrasts with traditional predictive architectures that prioritize maximizing accuracy on single-step predictions. Systems built on this principle allocate resources to encoding a shared understanding of cause and effect, rather than solely focusing on statistical correlations. This approach facilitates efficient learning and generalization, as the agent can leverage its internal model to infer outcomes and plan actions based on a deeper comprehension of the system, potentially reducing the computational demands associated with exhaustive scenario planning. The goal is to build systems that ‘know’ why something happens, not just that it happens.

The Information Digital Twin: A System for Adaptive Regulation

The Information Digital Twin (IDT) functions as a computational system designed to assess the consistency between an agent’s predictions and observed environmental outcomes. This is achieved through continuous monitoring of ‘bi-predictability’, quantifying the correlation between predicted and actual states. The IDT doesn’t rely on external rewards; instead, it generates internal feedback signals based on prediction errors. This feedback is then provided to the agent in real-time, allowing for immediate adjustments to its internal model and subsequent actions. The framework supports a closed loop where the agent’s behavior influences the environment, which in turn generates data used to refine the agent’s predictive capabilities, thereby improving performance and adaptability.

The Information Digital Twin (IDT) facilitates adaptive behavior and improved learning efficiency through continuous bi-predictive assessment. This process involves a constant comparison between the agent’s predicted outcomes and the actual observed results within its environment. Discrepancies between prediction and observation generate signals that can be used to adjust the agent’s internal model and subsequent actions. This feedback loop enables the agent to refine its understanding of the environment and optimize its behavior without relying solely on external reward signals, leading to faster learning and increased robustness to unexpected changes or perturbations.

The Information Digital Twin (IDT) draws inspiration from thalamocortical regulation, a neurobiological process wherein the thalamus selectively filters and relays sensory information to the cortex based on cortical activity. This biological precedent informs the IDT’s regulatory mechanisms by establishing a framework for predictive error signaling; discrepancies between predicted and observed states in the IDT, analogous to mismatches detected in the thalamus, trigger adjustments to the agent’s behavior. This biomimicry allows the IDT to dynamically modulate information flow, prioritizing relevant data and suppressing noise, mirroring the efficiency of neural processing and enabling robust adaptation in complex environments.

Comparative analysis demonstrates the Information Digital Twin (IDT) consistently outperforms traditional reward-based detection methods in identifying perturbations within reinforcement learning agents. Across multiple experimental comparisons, the IDT achieved significantly higher detection rates, as quantified by effect sizes exceeding 0.8. This indicates a substantial practical difference in the ability to recognize deviations from expected agent behavior. The observed effect size magnitude suggests that the IDT’s regulatory mechanisms provide a robust and sensitive means of monitoring agent state and identifying disruptions to normal operation, surpassing the performance of methods reliant solely on reward signals.

Beyond Prediction: The Emergence of Agency Through Reciprocal Understanding

Agency, the hallmark of intelligent behavior, doesn’t arise from simply anticipating what will happen next, but from a reciprocal interplay between prediction and action-a phenomenon known as bi-predictability. An agent capable of not only forecasting environmental changes but also of influencing those changes through its own actions demonstrates a significantly advanced cognitive capacity. This ability is then continually honed through adaptive regulation; the agent assesses the outcomes of its actions against its initial predictions, and adjusts its behavior accordingly. This closed loop – predict, act, and refine – allows for increasingly effective interaction with the environment, moving beyond passive response to proactive shaping of circumstances and ultimately defining true agency.

The capacity for intelligent behavior hinges on an agent’s ability to not simply predict outcomes, but to strategically select actions that shape those outcomes. As an agent develops a more nuanced understanding of how its actions influence the environment – and how the environment, in turn, responds – action selection becomes increasingly purposeful. This refined process moves beyond random attempts or pre-programmed responses, allowing for behaviors tailored to specific goals and optimized for success. The agent learns to anticipate the consequences of different actions, effectively choosing those that maximize desired outcomes and minimize potential setbacks, resulting in a demonstrably more effective and adaptive approach to interacting with the world.

Predictive coding posits that the brain functions not as a passive receiver of sensory information, but as an active prediction machine constantly generating and refining internal models of the world. These models anticipate incoming stimuli, and any discrepancy between prediction and reality generates a “prediction error” – a signal that drives learning and model adjustment. This continuous process of prediction and error correction isn’t simply about accurately perceiving the environment; it’s fundamental to intelligence itself. By prioritizing the prediction of likely states, the system efficiently allocates resources, focusing attention on the novel or unexpected. Furthermore, the hierarchical nature of these internal models allows for abstraction and generalization, enabling an agent to navigate complex environments and respond flexibly to unforeseen circumstances. This inherent ability to anticipate, learn from, and correct predictions forms the bedrock upon which more complex cognitive abilities, and ultimately, intelligent behavior, are built.

The capacity for intelligence isn’t solely demonstrated through completing assigned tasks, but fundamentally resides in an agent’s ability to learn how it learns. This framework proposes that true intelligence emerges when an agent actively engages with its environment, not just to achieve goals, but to continuously assess the effectiveness of its learning processes. By monitoring its own performance and identifying when learning falters, the agent can then initiate restorative mechanisms – adjusting strategies, refining internal models, and ultimately, optimizing its capacity for future adaptation. This self-directed learning cycle, characterized by interaction, monitoring, and restoration, represents a pivotal shift from simple reactivity to genuine, adaptive intelligence.

The pursuit of artificial intelligence, as detailed within the exploration of bi-predictability, reveals a fundamental truth about complex systems: resilience isn’t built through rigid control, but through the capacity to anticipate and adapt. This echoes Ada Lovelace’s observation that “The Analytical Engine has no pretensions whatever to originate anything. It can do whatever we know how to order it to perform.” The article’s focus on coupled systems and information exchange isn’t about creating intelligence, but about establishing the regulatory loops – the ‘orders’ – that allow a system to navigate uncertainty. The architecture proposed isn’t a blueprint for a thinking machine, but a framework for managing dependency, accepting that even the most sophisticated systems remain fundamentally bound by the limits of their initial conditions and predictive capacity.

What Lies Ahead?

The pursuit of quantifiable agency, framed here through bi-predictability, reveals less a path to creation and more a cartography of inevitability. The system does not become intelligent; it merely reveals the contours of its own anticipated failures. Each coupling, each feedback loop, isn’t a design choice but a prophecy – a narrowing of possibilities, a pre-commitment to specific modes of breakage. The elegance of the proposed architecture, echoing thalamocortical regulation, is unsettling precisely because it suggests an internal consistency divorced from external truth.

Future work will inevitably focus on scaling these principles, attempting to build larger, more complex agents. This, however, is a distraction. The true challenge lies not in increasing complexity, but in embracing the inherent limitations of any predictive model. The silence of a functioning system isn’t a sign of success; it’s merely a temporary respite before the inevitable divergence between prediction and reality.

The field might benefit from turning inward, from studying not how to make systems intelligent, but how intelligence itself emerges from the interplay of prediction and error. Perhaps agency is not a property to be engineered, but a consequence of being fundamentally, irrevocably, wrong – and learning to anticipate those very errors.

Original article: https://arxiv.org/pdf/2602.22519.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/

The Fragility of Foresight: Systems and the Limits of Prediction

Bi-Predictability: Measuring Reciprocity, Not Just Accuracy

The Information Digital Twin: A System for Adaptive Regulation

Beyond Prediction: The Emergence of Agency Through Reciprocal Understanding

What Lies Ahead?

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