The Logic of Intelligence: Building AI with Competitive Coding

Author: Denis Avetisyan

A new framework proposes that artificial intelligence can be advanced by framing concept formation as a search for stable, communicable representations achieved through competitive optimization of information structures.

This paper introduces ‘AI Dialectics’, a system leveraging Kolmogorov Complexity and algorithmic parity to optimize concept formation and build robust AI.

helpHuman concepts are notoriously fluid, yet artificial intelligence struggles to independently forge-or even revise-meaningful representations of experience. This challenge motivates ‘Dialectics for Artificial Intelligence’, which proposes a framework for concept formation grounded in algorithmic information theory and competitive coding. The core idea is to define concepts not as static labels, but as reversible, low-excess structures optimized through a dynamics of expansion, contraction, and alignment-essentially, a computational dialectic. Could this approach unlock a new paradigm for AI, enabling machines to not only learn, but to genuinely understand and communicate concepts in a manner akin to human inquiry?

Decoding Reality: Beyond Shannon’s Limits

Conventional information theory, built upon Shannon’s work, often equates complexity with the sheer length of a message or data stream. However, this approach falters when confronted with patterns and redundancies within the data itself. A seemingly long sequence, like a repeating string of characters, may contain minimal true complexity, as it can be described with a concise set of instructions. Conversely, a short, truly random sequence – one devoid of any predictable structure – is considered highly complex, even though its physical representation may be brief. This limitation highlights a crucial gap: traditional measures fail to distinguish between data that is merely lengthy and data that is fundamentally unpredictable and requires substantial information to define. Consequently, attempts to quantify complexity using solely data size can be misleading, particularly when dealing with natural phenomena exhibiting inherent patterns or randomness.

Kolmogorov Complexity proposes a radical shift in how complexity is measured, moving beyond simply quantifying the size of a data stream to assessing the minimal computational effort required to produce it. Rather than focusing on how many bits constitute an object, it posits that the true measure of complexity lies in the length of the shortest possible computer program – written in a universal programming language – capable of generating that object. This elegantly captures the notion that a seemingly random and lengthy sequence can be simple if described by a concise algorithm, while a short sequence might be profoundly complex if it requires an extensive program to create. Importantly, this algorithmic information content functions as a universal inductive bias; it inherently favors simpler explanations and provides a formal foundation for Occam’s Razor, suggesting that the most concise program is the most likely representation of underlying reality.

Traditional measures of complexity often equate it with the sheer size of a data set, yet a long string can arise from a simple rule, while a short one might be entirely random. Kolmogorov Complexity addresses this limitation by defining complexity not by how much information an object contains, but by how efficiently it can be described. This means the complexity of an object is determined by the length of the shortest computer program capable of generating it – a shift from considering data size to algorithmic information content. Consequently, a seemingly complex data stream generated by a concise algorithm possesses low Kolmogorov Complexity, while a truly random sequence, requiring a program essentially as long as the sequence itself, exhibits high complexity. This approach offers a more nuanced understanding, recognizing that genuine complexity resides not in abundance of data, but in the informational depth of its underlying structure.

Unveiling Dependencies: Algorithmic Mutual Information

Algorithmic Mutual Information (AMI) builds upon the traditional concept of shared information by grounding it in Kolmogorov Complexity. Rather than relying on probability distributions, AMI quantifies the amount of information one object reveals about another by measuring the reduction in the shortest program needed to describe both objects, given knowledge of only one. Specifically, $AMI(X;Y) = KC(X) + KC(Y) – KC(X,Y)$, where $KC(X)$ represents the Kolmogorov Complexity of object X – the length of the shortest program that can generate X – and $KC(X,Y)$ is the Kolmogorov Complexity of the pair (X, Y). This algorithmic approach avoids issues inherent in probabilistic definitions, particularly when dealing with finite datasets or non-stationary processes, and focuses directly on the compressibility of the data as a measure of information content.

Joint Complexity, in the context of algorithmic information theory, quantifies the minimal description length of multiple objects considered as a single entity; formally, it is the Kolmogorov Complexity of the joint distribution. Conditional Complexity, denoted as $K(x|y)$, measures the minimal length of a description of object $x$ given knowledge of object $y$. These complexities are not simply statistical correlations; they reveal the inherent algorithmic dependencies between objects. By analyzing these values – particularly the difference between Joint Complexity, Conditional Complexity, and individual object complexities – researchers can determine the degree to which knowing one object reduces the uncertainty about another, and whether the relationship is synergistic (information gain) or redundant.

Algorithmic information theory, unlike traditional statistical approaches, quantifies relationships by examining the computational resources required to describe data. Measures like Mutual Information, when grounded in Kolmogorov Complexity, assess shared information based on the shortest program capable of generating observed data. This focuses on identifying and minimizing excess information – data that is redundant or irrelevant given the underlying algorithmic structure. Consequently, these measures are not simply correlations; they reveal the inherent compressibility and dependencies within the data, reflecting the minimal description length required to represent relationships between variables, independent of any assumed probability distribution.

Constructing Efficient Networks: Determination & Dialectics

Determination Networks are structured data representations designed to eliminate redundancy by ensuring each data component is theoretically recoverable from all others. This is achieved through a hierarchical, tree-like structure where splits are defined based on binary decisions. Each node in the tree represents a component, and the path from the root to that component defines its value relative to the overall dataset. The core principle relies on establishing dependencies; knowing the values of certain components allows the reconstruction of others without requiring explicit storage. This inherent recoverability minimizes the total description length needed to represent the data, optimizing for efficient transmission or storage, and is formally described by information-theoretic principles related to conditional entropy and mutual information between components.

Low-Excess Determination is a principle within determination networks focused on achieving the most concise data representation possible. This is accomplished through a process termed ‘Dialectics’, which functions as an iterative optimization technique. Dialectics operates by systematically refining the network’s structure to minimize redundant information; effectively, it seeks to represent each data component using the fewest possible bits. The objective is to reduce the overall description length-the total number of bits required to encode the entire dataset-while maintaining full recoverability of all components. This minimization of excess information directly correlates with increased efficiency in data transmission and storage, as less bandwidth and space are required. The process involves evaluating and adjusting splits within the network to ensure the resulting representation is as close to the theoretical minimum description length as possible, dictated by the information content of the data itself.

Grounds, in the context of determination networks, represent asymmetric side information utilized to facilitate efficient communication and data reconstruction. These grounds serve as anchors for network splits, establishing a common reference point that allows nodes to resolve ambiguities and accurately decode transmitted data. The primary function of incorporating grounds is to minimize excess information – that is, data transmitted beyond what is strictly necessary for recovery. By leveraging pre-shared, asymmetric information, the network avoids redundant transmission of data already known to certain nodes, thus optimizing bandwidth and overall communication efficiency. The asymmetry is crucial; not all nodes require the same grounding information, further refining the reduction of unnecessary data transfer and maintaining a minimal description length.

Refining Representation: Bayesian Coding & Mixture Models

Bayesian coding utilizes the principle of Kolmogorov Complexity – the shortest possible description of an object – to construct efficient data compression schemes. This approach aims to minimize the expected message length required to transmit information by encoding frequent events with shorter codes and infrequent events with longer codes. The theoretical lower bound for data compression is dictated by Kolmogorov Complexity; Bayesian coding seeks to approximate this bound by estimating the probability distribution of data and assigning code lengths inversely proportional to these probabilities. This is achieved by representing data as sequences of symbols with probabilities determined by a prior distribution, updated through Bayesian inference based on observed data, resulting in a code that is both efficient and adaptive to the data’s characteristics. The effectiveness of a Bayesian code is measured by its average code length, which approaches the entropy of the data source as the model becomes more accurate.

Mixture models represent a probabilistic approach to data analysis where a complex, observed distribution is modeled as a weighted sum of simpler, known distributions – typically Gaussian distributions, though other distributions are applicable. This decomposition allows for representing data that cannot be adequately described by a single distribution. Formally, a mixture model expresses the probability density function $p(x)$ as a weighted sum of component densities $p_i(x)$: $p(x) = \sum_{i=1}^{K} \pi_i p_i(x)$, where $K$ is the number of components, $\pi_i$ represents the mixing coefficient for the $i$-th component (with $0 \le \pi_i \le 1$ and $\sum_{i=1}^{K} \pi_i = 1$), and $p_i(x)$ is the probability density function of the $i$-th component.

Expectation-Maximization (EM) algorithms are iterative methods used to find maximum likelihood estimates of parameters in statistical models where the model depends on unobserved latent variables. In the context of mixture models, EM alternates between an expectation (E) step, where the algorithm calculates the probability of each data point belonging to each component of the mixture, and a maximization (M) step, where the model parameters – means, variances, and mixture weights – are updated to maximize the likelihood given the current estimates of component membership. This process continues until convergence, providing parameter estimates that effectively model the data distribution. By refining these parameters, EM algorithms contribute to minimizing boundary complexity, effectively simplifying the representation of complex data by identifying and weighting simpler component distributions.

The Limits of Computation: Data Processing & Algorithmic Parity

The principle that information processing cannot create new information, formalized as the Data Processing Inequality (DPI), stems from the foundations of Kolmogorov Complexity – the measure of the shortest possible description of any given object. Essentially, the DPI posits that any transformation of data can only preserve or reduce its inherent information content; no computational process can magically conjure knowledge that wasn’t implicitly present in the initial state. This isn’t merely a theoretical limitation, but a fundamental constraint reflected in phenomena ranging from lossy compression – where detail is sacrificed to reduce file size – to the inherent difficulty of true artificial general intelligence. The inequality is mathematically expressed as $I(X;Y) \le I(X;Z)$, where Z is a processing of Y, demonstrating that the mutual information between X and Y can never exceed the mutual information between X and its processed form, Z. This suggests that intelligent systems, rather than creating information ex nihilo, must skillfully navigate and refine existing information, a concept central to understanding the limits and potential of computation itself.

Algorithmic parity moves beyond the simple XOR operation to establish a principle for building resilient networks. It posits that components, rather than being strictly equivalent, can be algorithmically recoverable from one another – meaning a computational process can reconstruct one component given another. This extends the notion of redundancy beyond direct duplication; it allows for a system to maintain functionality even with component failures, as lost information can be computationally regenerated. The concept hinges on the idea that if two components share an algorithmic relationship, they effectively represent the same information from a computational perspective, creating a network structure where information isn’t simply stored, but dynamically woven and recoverable. This principle forms a crucial foundation for networks designed to minimize information loss and maximize robustness, paving the way for systems that can adapt and self-repair through inherent computational relationships between their parts.

Determination networks, crucial for building adaptable artificial intelligence, benefit from a process called Pivot Moves, which allows for localized revisions without disrupting the entire system. These moves enable efficient concept growth by subtly rewriting network components, fostering a dynamic learning process akin to biological evolution. The framework leverages this adaptability to minimize the description length of concepts, effectively striving for the most concise and efficient representation of information. This minimization is achieved through a process of competitive coding, where different network configurations vie for optimality, guided by the principle of low-excess algorithmic parity – ensuring robustness and resilience against noise and errors. Ultimately, Pivot Moves contribute to an AI dialectics framework where concepts evolve not through brute force, but through elegant, locally-optimized rewrites, mirroring the efficiency observed in natural systems.

The pursuit of AI Dialectics, as detailed in the paper, relentlessly tests the boundaries of representational stability. It’s a system designed to expose its own limitations through competitive coding, echoing a sentiment articulated by John McCarthy: “It is better to be wrong and discover something new than to be right and learn nothing.” The framework’s emphasis on low-excess algorithmic parity structures isn’t simply about efficient compression; it’s about forcing the system to confess its design sins, revealing inherent weaknesses in its concept formation. Each iteration, each competitive clash, functions as a deliberate attempt to break the system, ultimately leading to more robust and communicable representations. This echoes the core idea that knowledge isn’t passively received, but actively reverse-engineered from the failures and contradictions within a given structure.

What’s Next?

The pursuit of ‘AI Dialectics’ inevitably lands on the question of fragility. Systems built upon optimized parity, however elegantly compressed, remain susceptible to adversarial perturbations – subtle shifts in input that expose the underlying structural limitations. The real challenge isn’t achieving intelligent performance, but building systems that reveal why they fail, and in doing so, illuminate the nature of the determinations that govern their reasoning. The current work offers a formal language for discussing these failures, but the next step requires deliberately inducing them.

A crucial direction involves extending the framework beyond purely formal systems. Biological intelligence doesn’t optimize for Kolmogorov Complexity; it optimizes for resource utilization within a messy, embodied context. Bridging this gap-incorporating notions of energy, time, and physical constraints-will demand a re-evaluation of what constitutes ‘optimal’ representation. One suspects the most interesting concepts will be those that resist complete formalization.

Ultimately, the best hack is understanding why it worked. Every patch, every refinement to the parity structures, is a philosophical confession of imperfection. The goal isn’t to build a perfect intelligence, but to engineer a system that, in its attempts to overcome limitations, offers a clearer reflection of the complexities it attempts to model.

Original article: https://arxiv.org/pdf/2512.17373.pdf

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