Computationally-Embedded Perspective

• Computationally-embedded perspective is a framework that defines natural and informational systems as hierarchies of computational layers embedded in physical substrates.
• It integrates classical Turing models with oracle machines and type theory to capture emergent phenomena and address the limits of bit-level control.
• The approach informs practical modeling by emphasizing material embodiment and stratified information-types to simulate complex, real-world systems.

A computationally-embedded perspective is an ontological and methodological framework for understanding natural and informational systems as intrinsically composed of—and constrained by—layers of computation, logically structured and materially embodied. Rather than viewing computation as an external tool applied to phenomena, this perspective posits that informational structure and causal patterns emerge, persist, and become humanly actionable via computational processes that are directly grounded in physical systems. This approach draws on extended Turing frameworks, type theories, and the rejection of “flat digital ontology,” favoring stratified layers of information-types and computation that accommodate discontinuity, emergence, and higher-order phenomena (Cooper, 2015).

1. Foundational Commitments and Ontology

The computationally-embedded perspective, as developed by Cooper (Cooper, 2015), is anchored in three key commitments:

• Stratification of information-types: The world is not exclusively made up of Type-0 (bit-level) computable objects; rather, it comprises hierarchies of information—from bit sequences to higher-type functionals and relations. Information-types are often modeled via extended Turing or Russell-style type theory.
• Material embodiment: Every computational process is instantiated in some physical substrate—whether in natural systems (e.g., brains, biological networks) or informational systems (e.g., language, legal precedent, social rituals). Abstract information is always grounded in material representation.
• Scientific realism and logical structure: Physical, mental, and social systems are real informational objects, whose interrelations exhibit logical structure and are susceptible to mathematical modeling. This often necessitates frameworks that go beyond classical Turing machines to include oracle Turing machines, partial computable functionals, and continuous or approximate computation.

2. Formal Models: Turing Machines, Oracles, and Type Theory

A computationally-embedded system is best described by a hierarchy of computational models:

• Classical Turing Machine: Encodes bit-level computable functions $f: \mathbb{N} \to \mathbb{N}$ via finite programs acting on infinite tapes.
• Oracle Turing Machine: $M^A$ augments the machine with oracle access, allowing $M$ to query bits of a real $A \in 2^{\mathbb{N}}$, capturing computations relative to higher-type data.
• Partial Computable Functionals: Type-2 mappings $$\Phi: 2^{\mathbb{N}} \times \mathbb{N} \rightharpoonup \{0,1\}$$, representing computations of functionals on reals via oracle queries. The effective enumeration $\{\Phi_e\}_{e \in \mathbb{N}}$ defines the “Turing Universe,” which models all relative computations of reals from reals.
• Turing Degrees & Reducibility: Information-objects $A, B \in 2^{\mathbb{N}}$ are compared by Turing reducibility:

$$A \leq_T B \iff \exists e \forall n: \Phi_e(B, n) = A(n)$$

establishing a hierarchy of computational “power.”

This stratification allows modeling not only discrete but also continuous, chaotic, and emergent informational processes, preserving a coherent framework for relating simulation, human intuition, semantic constructs, and physical causality.

3. Emergence, Definability, and Loss of Computational Control

A central insight is that higher-order information and emergent phenomena often exceed Type-0 computability, leading to loss of computational control: situations where bit-level computation cannot capture or constrain the behavior of complex systems. Examples span chaos, randomness, and “big data”; re-establishing computational control sometimes requires reductions between types or restricting attention to semantically tractable substructures.

Key problems addressed within this perspective include:

• Characterizing information complexity: Assessing the balance between the intricacy of informational structure and the computational mechanisms available to “control” or simulate it.
• Modularity and type reduction: Demonstrating that, in certain cases, loss of control at higher types can be restored by suitable reductions, often guided by formal type theory as in Turing’s 1936 and Russell’s foundational work.
• Limits of simulation and computation: Addressing scenarios where chaos, big data, and emergent phenomena force researchers to rely on approximations, statistical sampling, or semantic constructs—moving beyond algorithmic computability.

4. Implications for Information Theory and Natural Systems

This framework extends standard information-theoretical approaches by identifying the computational structure of information in real-world contexts. It supports:

• Modeling of layered causality: Informational structure is not “flat”; causality and emergence operate at and across multiple computational types.
• Integration of approximation and continuity: Accommodating systems where information is encoded not just in discrete bits, but in continuous variables and functional relations; this is essential for modeling physical, social, and mental phenomena.
• Oracle-style access to higher types: Allowing simulations and mathematical models to access data at higher types when needed, supporting scientific realism and effective modeling of interfaces between systems.

5. Relation to Broader Philosophical and Scientific Traditions

The computationally-embedded perspective rejects pure digital reductionism and instead adopts a nuanced stratified ontology, informed by both classical computability and modern information theory. Its commitments resonate with scientific realism, viewing the interface between nature, cognition, and social order as grounded in real informational and computational structure.

This approach enables:

• Bridging digital ontology and higher-order reasoning: Uniting bit-level computational frameworks with structures capable of representing human reasoning, language, and legal precedent.
• Extension and generalization of classical computability theory: Providing methods for modeling and analyzing systems whose complexity defies classical algorithmic approaches, especially in the presence of emergence, nonlinearity, and semantic ambiguity.

6. Practical Methodologies and Limitations

From the computationally-embedded perspective, practical modeling involves aligning simulation, coding, statistical analysis, and semantic interpretation with the inherent computational structure of the system under study. Limitations—imposed by emergence, chaos, and “big data”—are not viewed as failures of computation, but as consequences of the layered, embedded nature of reality. Restoration of computational control, when possible, is achieved through type reductions and the strategic restriction of semantic domains.

A plausible implication is that future methodologies for information-theoretical modeling, simulation, and analysis must internalize the constraints and stratifications identified by this perspective. Rather than artificially flattening systems to bit-level abstraction, researchers are encouraged to recognize and leverage the layered computational landscape inherent in natural and informational systems (Cooper, 2015).