‍Introduction

The traditional Turing machine model precisely delineates the boundary of what is “computable” within formal systems. However, in constructing a real-world system that truly possesses intelligence and the ability to determine trust, mere Turing-style computability is far from sufficient. Intelligent behavior relies not only on pure deductive reasoning but must also include effective evaluation and decision-making regarding the consequences of signals. This profound insight can be traced back to the foundational proposition made by Wiener & Bigelow (1943) during the early development of cybernetics:

“The ability to evaluate and record the effects of specific signals is a necessary condition for the evolution of intelligent behavior.”

This viewpoint offers us a new lens through which to re-examine Turing’s two foundational papers—On Computable Numbers (1936) and Systems of Logic Based on Ordinals (1938), and inspires us to draw deep connections between these theories and the operational logic of real-world distributed systems such as Bitcoin.

I. Turing’s Logical Structure: From Computability to Relative Decidability

In his groundbreaking 1936 paper, Turing defined the domain of “computability”: processes that can be mechanically executed and yield results via algorithms (Turing machines). This forms a model of deductive reasoning and deterministic computation.

However, in his 1938 doctoral dissertation, Turing introduced a more complex logical structure: (∀x)(∃y)R(x,y)

This is a typical Q₂-class (second-order double quantifier) problem structure. Turing aimed to construct a system in which such structures could achieve “partial completeness” in a certain sense. Its internal structure can be deeply interpreted as follows:

• x: Represents logical propositions that can be formalized and computed by a Turing machine—i.e., the domain that the 1936 model could handle.
• y: Symbolizes an Oracle Machine, envisioned to resolve “decision problems” that standard formal systems cannot solve. It does not perform computation but instead provides instantaneous, external “judgments.”
• R(x,y): Expresses a kind of transfinite inductive selection algorithm. It is no longer a simple recursive relation but a structure that handles issues of trust or systemic consistency. It lies outside the computable realm of 1936 and is part of the enhanced mechanism introduced in 1938, transcending conventional computational capabilities.
  

Turing, by externalizing the “decision problem” into the concept of “relative computability,” proposed an enlightening framework: A truly intelligent system must internally embed a judgment mechanism that transcends purely formal (deductive) systems.

II. Bitcoin: An Engineering Interpretation of Turing’s Logic

Bitcoin, as the world’s first decentralized digital currency system, is precisely a large-scale integration and engineering interpretation of the two Turing models in the real world:

1. Turing Machine and Computability: The UTXO Script System

Bitcoin’s UTXO (Unspent Transaction Output) model uses its built-in scripting language (Script) to express transaction validation rules. These rules fully conform to the formal system of “computability” described in Turing’s 1936 model—they are behaviorally closed, deterministic, and verifiable by any standard computational device.

Script logic ≈ Turing 1936 computability framework

2. Oracle Machine and Decidability: Double-Spending Problem and Miner Behavior

However, the logic for solving the double-spending problem is fundamentally different:

• It is not something that can be resolved within a local script, but rather a global consistency issue at the system level.
• Judging whether a transaction is valid requires consensus on the global state by all network nodes—this essentially involves a structural trust judgment about the system.
• Bitcoin resolves this through the miner network and the Proof-of-Work (PoW) consensus mechanism, thereby creating a “decentralized judgment structure.”
  

Hence, we can view the miners and consensus mechanism within the Bitcoin network as a kind of relative Oracle Machine:

• There is no pre-set central authority; no single entity holds final judgment power.
• Through the probabilistic competition process of PoW, among multiple candidate ledger states, the system recursively selects and finalizes a universally accepted true state.
• This competition-and-selection mechanism is essentially a real-world manifestation of Turing’s 1938 “transfinite inductive logic” in a distributed environment.
  

PoW process ≈ Decentralized Oracle Machine’s Decider

III. Logic Structure Comparison (Revised)

IV. Philosophical Summary: Two Pillars of Intelligence Design – Computability ≠ Decidability

A truly intelligent system must not only execute predefined rules, but also determine whether the execution of these rules suffices to produce the expected behavior—this is the core of “evaluating signal effects” as stated by Wiener and Bigelow.

• ✅ Turing Machine (1936): Excels at handling deductive rules (answers: What can be done).
• ✅ Oracle Machine (1938): Focuses on judgment and trust (answers: What should be believed).
  

Bitcoin’s system architecture exemplifies this beautifully:

• The formal system (UTXO scripts) is responsible for constructing and executing precise transaction rules.
• The oracle structure (PoW consensus) is responsible for achieving global consensus and trust in a distributed environment, thereby resolving judgment problems that cannot be handled by local script logic.

Conclusion

From Turing’s concept of “computability” to “relative computability” and the profound theory of the Oracle Turing Machine, to its actual engineering implementation in Bitcoin—we are witnessing a critical cognitive shift: A truly intelligent system must incorporate a mechanism for judgment about the meaning of signals and system states, beyond strict formal logic.

Turing’s Oracle Machine model provides the earliest and most forward-looking theoretical language for building such systems capable of handling complex decision and trust problems.

Bitcoin represents the first large-scale engineering realization of this deep logical framework in the real world.