Grounding Occam's Razor in a Formal Theory of Simplicity (2004.05269v2)

Abstract: A formal theory of simplicity is introduced, in the context of a "combinational" computation model that views computation as comprising the iterated transformational and compositional activity of a population of agents upon each other. Conventional measures of simplicity in terms of algorithmic information etc. are shown to be special cases of a broader understanding of the core "symmetry" properties constituting what is defined here as a Compositional Simplicity Measure (CoSM). This theory of CoSMs is extended to a theory of CoSMOS (Combinational Simplicity Measure Operating Sets) which involve multiple simplicity measures utilized together. Given a vector of simplicity measures, an entity is associated not with an individual simplicity value but with a "simplicity bundles" of Pareto-optimal simplicity-value vectors. CoSMs and CoSMOS are then used as a foundation for a theory of pattern and multipattern, and a theory of hierarchy and heterarchy in systems of patterns. A formalization of the cognitive-systems notion of a "coherent dual network" interweaving hierarchy and heterarchy in a consistent way is presented. The high level end result of this investigation is to re-envision Occam's Razor as something like: When in doubt, prefer hypotheses whose simplicity bundles are Pareto optimal, partly because doing so both permits and benefits from the construction of coherent dual networks comprising coordinated and consistent multipattern hierarchies and heterarchies.

• The paper introduces a formal framework that refines Occam’s Razor by employing Compositional Simplicity Measures (CoSMs) and Operating Sets (CoSMOS).
• It utilizes combinational computational models and Pareto optimality to generate hierarchical and heterarchical patterns.
• The work offers significant implications for AI and cognitive systems by enhancing pattern recognition and adaptive processing capabilities.

An Examination of "Grounding Occam's Razor in a Formal Theory of Simplicity"

The paper "Grounding Occam's Razor in a Formal Theory of Simplicity" by Ben Goertzel introduces a formal framework for understanding and applying the heuristic of Occam's Razor within computational systems. The paper critiques the traditional understanding of simplicity, proposes a new formalization of simplicity through the idea of Compositional Simplicity Measures (CoSMs) and Compositional Simplicity Measure Operating Sets (CoSMOS), and explores its implications on the hierarchical and heterarchical organization of patterns within cognitive systems.

Formal Theory of Simplicity

Goertzel challenges existing interpretations of simplicity, which often rely on specific computational models, like algorithmic information theory, or use simplicity as an informal concept. Instead, the paper proposes a formal theory using combinational models of computation, where elements interact with binary operations to produce new elements. This approach is more general and adaptable than previous models, allowing for multiple measures of simplicity to coexist. These measures, or vectors, form CoSMs and CoSMOS, where a set of simplicity measures is employed together to assess the complexity of entities.

Key Concepts and Framework

A significant contribution of the paper is the introduction of CoSM and CoSMOS. In essence, the CoSM framework extends conventional measures of simplicity by encompassing new operations termed "auto-ops," that facilitate the measurement of complexity across a broader scope. This notion crucially generalizes the idea of simplicity to form a multifaceted evaluation of complexity, abandoning the search for a universal simplicity measure in favor of multiple criteria evaluated collectively through Pareto optimality. This concept of CoSMOS allows for an exploration of "simplicity bundles" -- sets of Pareto-optimal simplicity values.

Moreover, this theory supports the creation of pattern and multipattern definitions stemming from these new measures. It views patterns as entities that offer reduced complexity through additional combinatory operations; this simplicity optimization leads to hierarchical pattern formations.

Hierarchical and Heterarchical Structures

Another pivotal aspect explored is the emergence of hierarchical structures from simplicity measures. Goertzel demonstrates that, under specific conditions, combinational operations can lead to a natural hierarchy amongst patterns, characterizing a near partial order when operators exhibit properties like associativity and cost-associativity. Alternately, hierarchy can arise from abstract combinatory logic where associativity is not present. These structural insights hold significant implications for cognitive architectures, suggesting perception and cognitive processes might align along these differing routes to hierarchy.

In contrast, heterarchical structures are derived from extending hierarchical patterns into metric spaces, with these interconnected structures formulated as dual networks. This conceptualization illustrates the alignment between hierarchical and heterarchical formations, offering a metric-based framework to support cognitive models that embody these dual network principles.

Practical and Theoretical Implications

By framing computational simplicity through CoSMs and CoSMOS, Goertzel's work offers pathways for reconsidering foundational principles underlying cognitive patterns and structures. The implications of this research are manifold. It prompts reconsideration of existing AI models by underpinning them with multipattern and simplicity vector evaluations rather than singular, unidimensional measures. The multi-measure approach may enhance systems' ability to adapt and interpret complex data streams effectively, reflecting a more nuanced view of Occam's Razor that supports both the practical application and theoretical grounding in cognitive and computing contexts.

Future research endeavors may explore the exploration of cognitive phenomena through this new lens, potentially refining artificial intelligence architectures to emulate the cognitive traits observed in biological systems more effectively. Additionally, the theory offers fertile ground for developing more robust cognitive frameworks embracing the dual network approach, fostering coordinated pattern recognition and abstraction capabilities within AI and cognitive computational models.