The Logical Core of Bitcoin (Part I): From Turing Machines to Oracle Turing Machines — Deconstructing Decentralized Transactions and Consensus

Introduction

Bitcoin is not just a digital asset or a payment tool. The real problem it solves is: How can trust be established in the absence of a central arbitrator?

Traditional financial systems rely on central institutions such as banks and governments to guarantee the legality of transactions and the authority of ledgers. Bitcoin, by contrast, leverages blockchain technology and distributed systems to enable value transfer without trusting a third party. This process involves not only cryptography and game theory but also conceals a deep logical structure that can be precisely analogized to Turing Machines and Oracle Turing Machines in computation theory.

This article approaches Bitcoin’s architecture from the perspective of formal systems and computational models, revealing its two-layer structure in transactions and consensus: deterministic computation and undecidable judgment.

I. Turing Machine: The “Computable” Foundation of Bitcoin

Bitcoin uses the UTXO (Unspent Transaction Output) model as its account structure. This model resembles a “digital check” in real life: every transaction input must correspond to a legitimate, unspent UTXO.

To verify a transaction, the following operations are required:

• Verify whether the digital signature matches the address;
• Check if the input UTXO exists and is unspent;
• Ensure value consistency (input ≥ output).
  

All of this is mechanically executed by Bitcoin nodes. This rule-based automation mirrors the formal reasoning of a Turing Machine:

Turing Machine = A set of deterministic rules + initial input + finite control + infinite tape.

In this sense, Bitcoin’s transaction validation process resembles parallel-running Turing Machines, each processing computable problems within a formal system.

II. Oracle Turing Machine: Bitcoin’s “Decisional” Mechanism

But the transaction layer is only part of Bitcoin. More crucial is the consensus layer.

In a decentralized network, multiple miners may mine blocks simultaneously, creating forks. At that point, the system must choose: which chain is the “true” one?

This is not a question that can be answered by a deterministic algorithm alone. It depends on:

• Which branch is extended next (i.e., chosen by more miners);
• Network synchronization, propagation delay, and hash power distribution.
  

These are not immediately knowable by a single node’s algorithm. Hence, Bitcoin adopts the longest chain rule:

“Whichever blockchain grows longer wins.”

This mechanism performs one key function: it delegates a formally undecidable question (“Which chain is true?”) to be resolved by the network’s computational behavior and synchronization.

This is precisely like the Oracle Turing Machine proposed by Turing in his 1938 dissertation:

“An extension of the standard Turing Machine with a black-box oracle that can make direct judgments on certain questions.”

Analogously, Bitcoin’s consensus mechanism acts as the system’s “oracle”: the answer is not logically derived but rather “selected” from all possible histories through competition, probability, and feedback.

III. Division of Labor: Collaboration Between Turing and Oracle Machines

• Transaction validation is the Turing Machine’s task: computable, deductive, unambiguous;
• Block consensus is the Oracle Machine’s task: undecidable, path-dependent, reliant on network behavior;
• The two are fused through the block structure, where each block contains a computable transaction set and marks a point of history added through judgment.
  

Thus, Bitcoin can be modeled as a logical system where Turing and Oracle layers collaborate.

Conclusion

Bitcoin’s system is neither chaotic nor mysterious. It exhibits high abstract elegance by integrating computability of formal systems and decidability of network behavior:

• Computable problems → Turing Machine → Transaction verification
• Undecidable but resolvable problems → Oracle Machine → Consensus
• Blockchain structure → A dual-layered, evolving machine
  

In the next section, we will explore how this system continuously evolves through the mechanism of “decision-confirmation-expansion”, entering the deeper realms of logic such as transfinite iteration and ordinal logic.

The Logical Core of Bitcoin (Part II): Transfinite Iteration and the Evolution of Decentralized Trust

Introduction

Previously, we revealed the “two-layer logic” of the Bitcoin system:

• The transaction validation layer handles computable problems, like a Turing Machine;
• The consensus layer handles judgment problems, like an Oracle Turing Machine.
  

Now we explore a deeper question: How does this system evolve stably over time? Why does it keep moving forward instead of falling into paradox or deadlock?

The answer lies in a concept from formal logic called transfinite iteration.

I. Gödel’s Incompleteness and the Limits of Formal Systems

Gödel’s incompleteness theorems tell us:

“In any sufficiently powerful and consistent formal system, there exist statements that cannot be proven true or false within the system.”

In other words, a formal system always has blind spots. The solution? Extend the system.

This was exactly the research direction of Turing’s 1938 dissertation:

• If a proposition cannot be decided within system S, can we accept it as a new axiom to generate S′?
• By repeating this process, can we construct a super-system?
  

Turing’s answer was yes—and he proposed a model of transfinite iteration, using ordinals to label the layers of system evolution.

II. Ordinal Logic and the Path of Transfinite Evolution

Core idea of ordinal logic:

• We label system extensions using natural numbers: S, S₁, S₂, …
• Upon reaching a limit (like ω), define Sω as the union of all previous systems;
• Continue building: Sω+1, Sω+2, …, Sω·2, …, Sα (where α is any recursive ordinal).
  

This process can, in theory, continue indefinitely—modeling the expanding power of human logical systems.

III. How Bitcoin’s Blockchain Structure Reflects This Logic

Returning to Bitcoin, we see that its blockchain growth structure is exactly an infinite loop of: judgment → new state → next deduction:

1. Initial system: Includes the initial UTXO set and Genesis Block;
2. Judgment event: A new block is chosen via PoW;
3. System extension: Transactions in the new block become irreversible chain history;
4. Next round: Mining and transaction collection begin again from the new chain tip.
   

This is a concrete engineering realization of a transfinite logical structure.

IV. The “Q2 Structure”: A Unified Framework of Computation and Judgment

We use the formula:

(\forall x)(\exists y)R(x, y)

This elegantly expresses the dynamic between computation and judgment, forming Bitcoin’s logical backbone:

• x: Computable elements within the system. In Bitcoin, this refers to individual transactions, which are validated via deterministic rules.
• y: Judged elements requiring an oracle. In Bitcoin, this is the block—a package of transactions discovered through probabilistic mining.
• R(x, y): A recursive, decidable relationship verifying the validity of y in relation to x. In Bitcoin, this is the longest-chain consensus rule.
  

This gives us a structural model for Bitcoin:

• Q₁: Turing Machine layer — for deterministic logic (transaction processing);
• Qω: Oracle Turing Machine layer — for undecidable consensus;
• Qα (α > ω): Chain evolution structure — every new block is a system “rebirth,” sealing past state as the new base for next iteration.
  

We call this model the Q2 Structure: a logical system formed by combining deductive computation (Turing) and decisional judgment (Oracle) through infinite iteration.

This model not only explains how Bitcoin builds trust from computation but also why it can self-evolve over time without any external arbitrator.

Conclusion

Using tools from formal logic, we’ve uncovered the deep architecture underlying Bitcoin:

• Computable problems are handled by the Turing Machine (transaction validation);
• Judgment problems are resolved by the Oracle Turing Machine (consensus);
• The entire system evolves via transfinite ordinal structure (blockchain growth).
  

This is a self-adaptive logic system without central judges. It reveals the root of decentralized trust in Bitcoin and offers a logical paradigm for understanding other future adaptive complex systems. Bitcoin is not just a financial innovation—it is a grand experiment in computational philosophy.

Note: This series is based on Turing’s doctoral dissertation “Systems of Logic Based on Ordinals” and draws from formal system evolution research by Solomon Feferman and others. It attempts to build a computability–judgment–evolution trinity framework for decentralized logic.