FLP Impossibility Theorem Simplified

I am having struggle following the proof of FLP impossibility theorem. Can you please help me out?

Of course! The Fischer, Lynch, and Paterson (FLP) Impossibility Theorem is a fundamental concept in distributed computing, particularly in the context of blockchain and consensus mechanisms. I'll simplify it for you.

The FLP Impossibility Theorem states that in a distributed computer system, it's impossible to achieve consensus if just one process (or node) can fail. This is particularly relevant in blockchain networks, where achieving consensus reliably is crucial.

Here's a simplified way to understand the proof:

1. Assumptions: The theorem assumes an asynchronous system where there is no upper bound on how long it takes for a message to be delivered. Processes can fail by stopping entirely, and there's no way to differentiate between a slow process and a failed one.

2. Basic Scenario: Imagine a network of processes that need to agree on a decision (like whether to add a transaction to a blockchain). They communicate by sending messages to each other.

3. Proof by Contradiction: The proof uses a technique called 'proof by contradiction.' It starts by assuming that there is an algorithm that always reaches consensus even if one process fails.

4. Indistinguishable States: The proof then considers two different scenarios that are indistinguishable to a particular process. In one scenario, the process should decide '0', and in the other, it should decide '1'. But since the process can't distinguish between these two scenarios, it can't decide correctly in both.

5. Inevitable Delay: Due to the asynchronous nature of the system, there's always a possibility of message delay. So, a process can't be sure if it's not receiving a message because another process has failed, or because the message is just delayed. This uncertainty prevents the system from reaching a unanimous decision in all cases.

6. Conclusion: Therefore, the assumption that a consensus algorithm can always work in such a system is false. Hence, it's impossible to guarantee consensus if one process may fail.

In simpler terms, the FLP Impossibility Theorem tells us that in a network where communication delays are unpredictable and nodes can fail, it's impossible to guarantee that all nodes will reach the same decision every time. This has important implications for designing blockchain systems, as it highlights the challenges in achieving consensus in an unreliable environment.

If you need more details or have specific questions about parts of the proof, feel free to ask!

Can you expand on the Proof by contradiction, especially the part where Contradiction is proved?

Absolutely, let's delve deeper into the proof by contradiction aspect of the FLP Impossibility Theorem, focusing on how the contradiction is actually proven.

The proof by contradiction in the FLP theorem involves a few key steps:

1. Initial Assumption: The proof begins by assuming the opposite of what we want to prove. Here, the assumption is that there exists a consensus algorithm that always makes a decision (reaches consensus) correctly in an asynchronous system, even if one process (or node) can fail.

2. Considering Two Executions: The proof then considers two different executions or histories of the system. These are sequences of steps (like message deliveries and internal decisions by processes) in the system.

   • In one execution, the processes decide on the value 0.
   • In the other execution, they decide on the value 1.

3. Constructing a Critical State: The proof constructs a state of the system called a 'critical state'. This state is reached just before the system makes a decision. In this state, the next step by any process could lead to a decision being made.

4. Indistinguishable States: Here's the key part: The proof demonstrates that there is a process that can't distinguish between these two executions when it is in the critical state. This means the process doesn't have enough information to decide if the consensus should be 0 or 1.

5. The Contradiction: The contradiction arises here. Since the system is asynchronous, we cannot assume that a message from another process will arrive in time to help make the decision. If the process decides on 0, there could be a scenario where it should have decided on 1, and vice versa. The inability to distinguish between these scenarios means that the process might make the wrong decision.

6. Final Conclusion: Therefore, the initial assumption that a consensus algorithm can always reach a correct decision in such a system is contradicted. Hence, the theorem concludes that no such algorithm exists.

In simpler terms, the proof shows that due to the uncertainty and lack of timely information in an asynchronous system, processes can reach a point where they have to make a decision without being able to know for sure what the correct decision is. This inherent uncertainty contradicts the possibility of a guaranteed consensus algorithm under these conditions.

Understanding this aspect of the FLP theorem helps in appreciating the challenges in designing robust consensus mechanisms, especially in decentralized systems like blockchains.