Turing completeness: Good or bad for blockchains? A ChatGPT perspective...
Who was Kurt Goedel?
Kurt G?del was an Austrian-born mathematician and logician who lived from 1906 to 1978. He made groundbreaking contributions to the fields of mathematical logic and the philosophy of mathematics. G?del is best known for his incompleteness theorems, which demonstrated the inherent limitations of formal axiomatic systems. His theorems showed that there will always be true statements that cannot be proven within any consistent formal system of arithmetic. This insight had profound implications for the foundations of mathematics and had a lasting impact on various areas of logic, computer science, and philosophy of mind.
Geodel's incompleteness theorem
Imagine you're in a math class, and your teacher gives you a set of rules and axioms to follow. You start with some basic facts and use those rules to prove new statements. The idea is to build a system that can prove all true mathematical statements. Now, G?del's incompleteness theorem tells us something interesting. It says that in any such system, there will always be true statements that can't be proven using the rules of that system. In other words, there will always be things that are true but can't be shown to be true within the system itself.
To understand this better, let's use a simple example. Imagine we have a system of math where we can only add numbers. We start with some basic facts, like 1+1=2. Using the addition rules, we can prove things like 2+2=4 or 3+5=8. But now, what if we have a statement like "There is no solution to the equation x+3=5." That statement is true. But we can't prove it using only addition because our system doesn't allow subtraction.
G?del's theorem shows that in more complex systems, there will always be statements like this, which are true but can't be proven within the system. It tells us that no matter how comprehensive or powerful our rules are, we can't create a system that can prove everything.
This theorem has had a significant impact on mathematics and computer science. It showed that there are inherent limits to what can be proven, and it raised important questions about the nature of truth, logic, and the foundations of mathematics itself.
Who was Alan Turing?
Alan Turing was a British mathematician, logician, and computer scientist who lived from 1912 to 1954. He is considered one of the most influential figures in modern computing and artificial intelligence development. Turing made significant contributions to mathematics, particularly in computability and the theory of algorithms. Turing's work laid the foundation for the creation of the modern computer, and his legacy continues to shape the field of computing today.
Turing completeness
Imagine you have a bunch of building blocks and want to build something cool with them. Now, Turing completeness is like having a set of building blocks that can be used to create any structure you can imagine.
In the world of computers, we have different programming languages that are like those building blocks. Some languages are simple and can only do basic things like addition or subtraction. Still, they can't do loops or make decisions. It's like having building blocks that can only stack on top of each other, but you can't use them to make a bridge.
However, other languages, like Python or Java, are like super-powered building blocks. They have all kinds of special features like loops and conditionals, allowing you to build complex structures. These languages are called Turing complete because they can do anything a computer can.
Being Turing complete means that a language is powerful because it can solve any problem a computer can solve. It's like having a toolbox filled with all the tools you need to build anything you can imagine.
So, when people talk about Turing completeness, they're saying that a language or system can do any computation or solve any problem that's possible with a computer. It's like having the ultimate set of building blocks that can create anything you can dream up in the world of computers.
Why do Turing complete machines face the Goedel's halting problem?
Turing complete machines, which include general-purpose computers, face G?del's halting problem because of the inherent limits of computation. The halting problem refers to the challenge of determining whether the program will eventually halt (finish) or run indefinitely, given a program and its input.
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G?del's halting problem arises because Turing complete machines can run programs that can analyze and manipulate other programs. This means they can potentially analyze themselves. Now, imagine we have a program that can predict whether another program will halt or run forever. We can then create a paradoxical situation.
If we run this predictive program on itself, what happens if it predicts that it will not halt? If it's correct, it should halt, contradicting its prediction. On the other hand, if it predicts that it will halt, and it's correct, then it should continue running indefinitely, contradicting the prediction again. This creates a logical dilemma; no program can consistently solve the halting problem for all possible programs.
G?del's halting problem shows the fundamental limitation of any general method for determining whether an arbitrary program will halt or run forever.?
This has significant implications in computer science, as it means that even the most powerful computers and programming languages cannot predict the outcome of all programs. It highlights the inherent complexity and unpredictability in some aspects of computation, leading to challenges in designing reliable and error-free software systems.
The risks of using a Turing complete blockchain?
Using a Turing complete blockchain platform carries risks that need to be considered. Here are some critical risks associated with Turing complete blockchains:
Ethereum and the halting problem
Ethereum, being a Turing complete blockchain platform, does not directly address the halting problem. The halting problem is a fundamental limitation of computation and cannot be fully solved by any Turing complete system. However, Ethereum introduces measures and mechanisms to mitigate the potential risks associated with the halting problem.
In Ethereum, smart contracts are the building blocks of decentralized applications (dApps) running on the blockchain. These smart contracts are written in a programming language called Solidity, which is Turing complete. While this allows for expressive and flexible programming, it also introduces challenges related to the halting problem.
To handle this, Ethereum includes a mechanism called gas. Gas is a computational resource that introduces a cost for executing smart contracts. Each computational step consumes a certain amount of gas, and users must pay for the gas consumed by their transactions.
The gas mechanism serves two purposes. Firstly, it prevents infinite loops or overly complex computations by limiting the amount of gas a transaction can consume. If a smart contract exhausts the allocated gas, the execution halts, preventing excessive resource usage.
Secondly, the gas mechanism incentivizes efficient programming. Since users pay for gas, they have an economic incentive to write optimized smart contracts that consume fewer resources. This encourages developers to avoid excessive computational demands and promotes the creation of more secure and cost-effective applications.
While Ethereum's gas mechanism helps address potential issues arising from the halting problem, it does not eliminate the halting problem itself. The responsibility of ensuring correct and secure smart contracts ultimately lies with developers and careful code auditing to identify and address potential risks or vulnerabilities.
Some well-known blockchains
Disclaimer: Please note that the Turing completeness status of a blockchain may vary depending on the specific components or features within the blockchain ecosystem, such as the scripting language, smart contract capabilities, or virtual machine architecture. Additionally, advancements and updates in blockchain platforms can introduce changes to their Turing completeness status. Therefore, it is vital to refer to each blockchain project's latest information and documentation to confirm their current Turing completeness status.
Distributed Software Architect
1 年There is no such thing as "G?del's halting problem."
A proven expert in strategic planning and the delivery of high-value, complex portfolios and projects | Director | Expert Witness l Adjudicator
1 年We like to think that Math provides us with an absolute framework to describe, predict and control nature. Whereas we now know that fortunately the laws of Nature are much more "incomplete", that is dynamic and subject to observation. Math can help us describe nature, it does not define it. Blockchains are not a panacea - they are a tool and like all tools they are constrained to a particular set of use cases. Understand their constraints and they are useful, fail to understand and they are dangerous.
Senior Managing Director
1 年Daniel LIEBAU Very informative.?Thanks for sharing.
| Veteran Central Banker | Occasional Podcaster | Mature Student | Judicious Contrarian | Personal Account | ????????????
1 年Very cool Daniel ?? I learned a lot reading this, but, some of the concepts being new to me, I will probably have to read it again …