The Physics of Agency: Bridging AI and Causality

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Author: Denis Avetisyan

A new mathematical framework reveals a surprising connection between the decision-making processes of artificial intelligence and fundamental principles of physics.

This review demonstrates an equivalence between deterministic agents modeled with Partially Observable Markov Decision Processes and one-input process functions, offering insights into higher-order causality.

Despite a historical disconnect, the formalisms of artificial intelligence and fundamental physics both grapple with notions of agency and causal influence. In ‘On Decision-Making Agents and Higher-Order Causal Processes’, we demonstrate a precise mathematical correspondence between deterministic decision-making agents-specifically those operating within partially observable Markov decision processes-and one-input process functions, a concept originating in the study of higher-order quantum operations. This identification reveals that an agent’s policy and memory update can be interpreted as a process function interacting with its environment, suggesting a surprising duality between AI agents and physical systems. Does this connection offer a novel framework for understanding both intelligence and causality, and could it pave the way for new approaches to designing adaptive and robust systems?

The Unfolding Now: Sequential Decision-Making in an Uncertain World

Intelligent systems are increasingly tasked with navigating complex, real-world scenarios – from autonomous vehicles charting courses through unpredictable traffic to robotic assistants operating in dynamic homes – all characterized by the need for sequential decision-making. These systems don’t operate with perfect knowledge; instead, they must choose actions based on incomplete information, much like humans do. Each decision alters the environment and reveals only partial feedback, forcing the agent to constantly refine its understanding and adapt its strategy. This presents a fundamental challenge: how can an agent effectively plan and act when the consequences of its choices, and the true state of the world, remain uncertain? The difficulty isn’t simply about processing large datasets, but about making informed choices before having complete information, a necessity in any environment where gathering data takes time or resources.

Conventional methods in sequential decision-making often falter when confronted with complexity due to a phenomenon known as the ‘curse of dimensionality’. This arises because the computational resources needed to analyze all possible states of an environment-the state space-increase exponentially with each added variable or degree of freedom. For instance, a simple robot navigating a grid might easily manage a 10×10 space, but a 100×100 grid demands vastly more processing power to evaluate every potential location and action. Consequently, as the state space expands, these approaches become computationally intractable, hindering an agent’s ability to effectively plan and adapt to changing conditions. This limitation restricts their scalability to real-world problems, where environments are rarely simple and often involve numerous interacting variables, necessitating more efficient methods for managing complexity.

An intelligent agent operating in the real world rarely possesses complete knowledge of its surroundings; it must continuously refine its understanding based on often-incomplete sensory input. This process centers on the ‘belief state’, a probabilistic representation of the world that encapsulates everything the agent knows-or infers-about relevant variables. Effectively updating this belief state is a fundamental challenge, requiring the agent to weigh prior knowledge against new observations and accurately predict future states. The difficulty isn’t simply recording data, but distilling it into a usable internal model; a poorly constructed belief state leads to flawed decisions, even with perfect algorithms. Researchers are exploring various methods, from Bayesian filtering to reinforcement learning, to enable agents to efficiently represent uncertainty and adapt their beliefs in dynamic, information-scarce environments, ultimately striving for robust and reliable sequential decision-making.

Memory and Policy: The Architecture of Adaptive Behavior

A deterministic agent operates on a continuous cycle of memory update and action selection. This means that at each time step, the agent utilizes its current memory – a representation of past observations and experiences – to determine the subsequent action. Following action execution, the agent receives an outcome and integrates this new information into its memory, effectively revising its internal state. This cyclical process ensures that the agent’s actions are consistently based on the most current understanding of its environment, and that its memory is continuously refined through interaction. The deterministic nature implies that, given a specific memory state, the agent will always select the same action, removing any element of randomness from the decision-making process.

The policy function is a core component of an intelligent agent, defining its behavioral strategy. It operates as a mapping from the agent’s current memory state – representing its accumulated experience and understanding of the environment – to a discrete action. This mapping can be deterministic, always selecting the same action for a given memory state, or stochastic, assigning probabilities to different actions. The function’s implementation can vary significantly, ranging from simple lookup tables to complex neural networks, but its fundamental purpose remains consistent: to translate internal representation into observable behavior. The quality of the policy directly impacts the agent’s performance, as it determines how effectively the agent responds to different situations based on its learned experiences.

Effective memory update in intelligent agents necessitates the combination of new observations with existing internal beliefs to construct a refined representation of the environment. This integration isn’t a simple overwrite; rather, it involves a weighting or probabilistic combination of the observed outcome and the agent’s prior expectation. The process typically utilizes a mechanism, such as Bayesian updating or reinforcement learning algorithms, to modulate the influence of new data based on its perceived reliability or relevance. Consequently, the agent’s memory, or internal state, is continuously adjusted, reflecting a cumulative history of interactions and allowing for increasingly accurate predictions and informed decision-making. This dynamic internal representation serves as the foundation for adaptive behavior in complex and changing environments.

The Collective Intelligence: Scaling to Multi-Agent Systems

Multi-Agent Systems (MAS) are prevalent in numerous real-world applications, encompassing areas such as robotics, traffic control, and economic modeling. These systems are characterized by the presence of multiple autonomous entities – the agents – which perceive their environment and act upon it to achieve individual or collective goals. Crucially, agent actions are not isolated; they directly influence the states of other agents and the environment itself, creating a dynamic and often complex interplay. This interdependency necessitates the development of mechanisms for coordinating agent behavior and predicting the consequences of actions within the shared environment, distinguishing MAS from single-agent systems where the environment is typically considered static or predictable.

The Link Product is a mathematical operation used to define how agents in a multi-agent system influence each other. Specifically, given two process functions, $f: X \rightarrow Y$ and $g: Y \rightarrow Z$, the Link Product, denoted $f \otimes g$, creates a new process function $h: X \rightarrow Z$ where $h(x) = g(f(x))$. This composition allows for the formal definition of interactions; an agent’s action, represented by the output of function $f$, becomes the input for another agent’s function, $g$. By defining these interactions mathematically, the Link Product enables precise analysis and control of complex agent behaviors and dependencies within a system.

Decentralized Partially Observable Markov Decision Processes (POMDPs) provide a formal framework for modeling interactions in multi-agent systems where agents possess limited and potentially differing perspectives of the environment. Unlike centralized approaches, decentralized POMDPs do not assume a global observer; instead, each agent maintains its own belief state based on its individual observation history and action sequence. This necessitates that agents make decisions based on incomplete information, requiring them to reason about the likely states of other agents and the environment. Formally, each agent $i$ has its own observation function $O_i$ and action space $A_i$, and the system’s dynamics are defined by a joint transition probability $T(s’, o, a)$ where $s’$ is the next state, $o$ is the vector of observations, and $a$ is the combined action of all agents. This decentralized structure is crucial for scalability and robustness in complex environments.

Observation independence in decentralized multi-agent systems refers to the principle where an agent’s observation space does not include the states of other agents. This significantly reduces the computational complexity of coordination, as each agent only needs to process information relevant to its own local environment and does not require interpreting the intentions or states of others. Consequently, the information overload experienced by each agent is minimized, allowing for faster decision-making and more efficient resource allocation. The benefit of this approach lies in its scalability; as the number of agents increases, the computational burden on any single agent remains relatively constant, unlike systems requiring global state awareness.

Beyond the Predetermined: Embracing Indefinite Causal Order

Traditional understandings of causality often presume a rigid sequence – an action must precede its observation, and events unfold in a predictable order. However, this fixed causal order proves limiting when applied to genuinely dynamic environments, such as complex robotics or rapidly changing data streams. A strict adherence to pre-defined sequences inhibits an agent’s ability to capitalize on emergent opportunities or adapt to unforeseen circumstances; it essentially ties its hands before the situation fully reveals itself. Consider a robotic arm assembling a component – if programmed with a fixed sequence, it may struggle when a part is slightly misaligned, whereas a more flexible system could reassess and adjust its actions mid-process. This inflexibility stems from the assumption that all relevant information is known in advance, a condition rarely met in real-world applications. Consequently, systems constrained by fixed causal order frequently exhibit brittleness and a lack of robustness, hindering their performance in unpredictable settings.

The limitations of traditional causal sequences, where one action definitively precedes another, become apparent in complex and rapidly changing environments. Introducing the concept of ‘Indefinite Causal Order’ offers a pathway beyond this rigidity, allowing agents to dynamically adjust their behavior based on unfolding circumstances. Rather than being locked into a predetermined chain of events, an agent operating with indefinite causality can explore multiple potential action sequences simultaneously, effectively exploiting opportunities as they arise. This adaptability isn’t simply about reacting to the unexpected; it allows for proactive maneuvering, enabling the agent to influence outcomes by prioritizing actions based on real-time feedback and shifting probabilities. The result is a system capable of far greater resilience and efficiency in navigating unpredictable scenarios, effectively turning uncertainty into a source of strength.

A fundamental reimagining of agency has been rigorously established through mathematical formalization, revealing a surprising equivalence between deterministic agents and a specific class of functions – one-input process functions. This isn’t merely an analogy; the work demonstrates a bijection – a perfect, one-to-one correspondence – between the equivalence classes of these agents and functions. This means every possible behavior of a deterministic agent can be precisely represented by a unique process function, and vice versa, offering a powerful new lens through which to analyze and predict complex systems. The implications extend beyond theoretical computer science, potentially impacting fields requiring robust modeling of dynamic interactions, as it provides a formal framework for understanding how seemingly complex behaviors can emerge from simple, deterministic rules – a concept encapsulated by the function $f: X \rightarrow Y$, where $X$ represents the input space and $Y$ the output space.

The capacity to model intricate interactions finds a robust framework in functions derived from Higher-Order Quantum Operations. These aren’t limited to the realm of quantum physics; rather, the mathematical structures underpinning these operations-particularly their ability to map inputs to outputs with complex dependencies-provide a generalized language for describing any system where relationships aren’t strictly linear. This allows for the representation of feedback loops, contextual dependencies, and emergent behaviors with remarkable efficiency. By leveraging these functions, researchers can move beyond simple cause-and-effect models and capture the nuanced dynamics of complex systems, potentially unlocking new strategies for control, prediction, and optimization. The formalism provides a pathway to analyze systems where the order of interactions isn’t pre-defined, enabling the development of agents capable of adapting to unpredictable circumstances and exploiting opportunities as they arise, essentially translating complex relationships into manipulable mathematical expressions like $f(x) = \sum_{i=1}^{n} a_i x_i + b$.

The pursuit of formalizing agency, as demonstrated within this study of Partially Observable Markov Decision Processes, inevitably encounters the limitations inherent in any attempt to model complex systems. Every failure is a signal from time; the identification of equivalence classes between agents and process functions isn’t a final solution, but rather a refinement of understanding. As John von Neumann observed, “The sciences do not try to explain why something happens, they just try to say how it happens.” This paper, by establishing a formal correspondence, doesn’t offer a why of agency, but a precise how – a mathematical description of its operation, acknowledging the inevitable decay of models and the continuous need for refinement. Refactoring, in this context, becomes a dialogue with the past, building upon previous approximations to approach a more robust and graceful aging of the system.

What Lies Ahead?

The demonstrated equivalence between agency, as formalized in Partially Observable Markov Decision Processes, and the structure of one-input process functions is not an arrival, but a realignment. Every architecture lives a life, and this work merely reveals the underlying symmetries-the inevitable decay towards a more fundamental description. The immediate challenge isn’t to apply this equivalence, but to understand its limitations. Observation independence, a core assumption, rarely survives contact with genuine complexity. The neatness of the mathematical correspondence hints at a fragility, a specific regime where this mapping holds-a transient stability before the system drifts into uncharted territory.

Future explorations will likely center on the boundaries of this equivalence. How do noise and non-determinism erode the mapping? Can the framework be extended to encompass agents with internal states beyond those dictated by the POMDP? And, crucially, what new insights does this perspective offer regarding the very nature of higher-order causality? The connection to process functions suggests a pathway towards understanding agency not as a computational property, but as a physical process-a flow of information governed by the same laws that govern everything else.

Improvements age faster than one can understand them. The elegance of this formalization belies the messiness of its implications. The true test won’t be in creating more sophisticated agents, but in recognizing when the architecture itself becomes the obstacle – when the attempt to model agency obscures the underlying physics.

Original article: https://arxiv.org/pdf/2512.10937.pdf

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

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2025-12-12 22:32