‍—The Boundary Between Linear Computability and Nonlinear Relative Computability

I. Introduction: Two Paradigms of Computation

At the origin of computing history, Alan Turing defined the “Turing Machine” in 1936, establishing the foundational framework of modern computation theory and fundamentally answering the question: “What is computable?” Two years later, in his 1938 doctoral dissertation, Turing introduced the “Oracle Machine”—a computational system that, compared to the Turing Machine, can reason with the help of external information sources, aiming to overcome the incompleteness of formal logical systems.

Today, when we re-examine these two computational models, we find that they are not merely theoretical distinctions, but rather mark the watershed between two paradigms of system design in technological practice:

• Turing Machine: deterministic, linear, top-down, reductionist system
• Oracle Machine: non-deterministic, nonlinear, bottom-up, evolutionary system
  

The most representative manifestation of this paradigm divide in the modern era is the emergence of the Bitcoin system.

II. Turing 1936: Turing Machine and Linear Computability

In his landmark 1936 work On Computable Numbers, Turing proposed the concept of the Turing Machine. This is a top-down, deterministic computational model, characterized by:

• Computability: emphasizes whether a problem can be solved by a strictly defined algorithm within a finite number of steps
• Linear structure: state transitions are sequential and static
• Closed logic: computation occurs entirely within the machine, with no access to the external environment
• Reductionist thinking: complex problems can be broken down into combinations of basic components
  

Traditional programmatic computing, formal logic proofs, and automatic control systems all fall under the extension of this paradigm.

III. Turing 1938: Oracle Machine and Relative Computability

However, Turing himself soon realized that not all mathematical truths could be exhausted by Turing Machines. In 1938, he proposed the “Oracle Machine” in his doctoral thesis Systems of Logic Based on Ordinals:

• The Oracle Machine does not break the boundaries of incomputability, but introduces external information sources (oracles) as the basis for relative judgment
• This model describes a kind of relative computability or decidability
• It is characterized by openness (relying on external context), nonlinearity (not exhaustible by sequential procedures), and adaptability (structure changes with input)
• The corresponding philosophical perspective shifts from reductionism to evolutionary and complex systems thinking
  

In short:

The Turing Machine answers “what we can compute”;

The Oracle Machine explores “how to find locally decidable structures within incomputability”

IV. Bitcoin: The Real-World Emergence of a Distributed Oracle Machine

Bitcoin is not built upon the linear computational logic of the traditional Turing Machine. Its system behavior is more akin to a distributed network of Oracle Machines, with the following characteristics:

• Bottom-up consensus formation: each miner node acts as a local judgment unit; in the absence of central coordination, global state is formed through the Proof-of-Work mechanism
• Nonlinear evolutionary system: the system’s state is constantly reconstructed by changing participants, hash rate, and time—resulting in a stable yet non-static overall order
• Openness and adaptability: the Bitcoin protocol itself is merely a minimal set of rules; its ecology and behavior patterns emerge over time within an open environment
• Decidability takes precedence over computability: the essence of the system is not about solving a specific algorithmic problem, but about achieving consensus on whether a set of historical records is accepted by all nodes
  

Therefore, Bitcoin becomes a real-world “Oracle system”: It does not rely on single-point computing power or logical inference, but on a dynamic sense of order formed by the system’s game structure and evolutionary process.

V. Systematic Comparison: Two Computational Worldviews

VI. Conclusion: Bitcoin Is Not a Program, But a Shift in Computational Perspective

We can regard Bitcoin as a profound computational paradigm revolution: It is not an “algorithmic tool” in the Turing sense, but an “evolutionary mechanism” in the oracle sense.

It demonstrates a new computational logic that does not rely on central authority, does not pursue deterministic solutions, but achieves decidability through game theory, emergence, and consensus.

This computational model not only transcends traditional program structures but also foretells the foundational paradigm for building complex systems in the decentralized era.

In this sense, Bitcoin is not only a revolution in the form of money—It is an evolution in computational philosophy.